Design Handbook for RautaRuukki Structural Hollow Sections

DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONS

Chapter 1


1

Chapter 1


New, revised edition 2000 ISBN

952-5010-47-3

Author

Rautaruukki Oyj, Hannu Vainio (M.Sc.Tech)

Typesetting

Lasjuma Oy

Translation

Trantex Oy

Printers

Otava Book Printing Ltd, Keuruu 2000

Orders

RAUTARUUKKI OYJ METFORM 13300 Hämeenlinna, Finland tel.

+358 3 528 60

fax

+358 3 528 5873

Internet: www.rautaruukki.com/metform

2


Chapter 1

FOREWORD This volume is a completely new, revised edition of the 1/98 handbook for Rautaruukki structural hollow sections, replacing all former editions. It is mainly based on the European prestandard on steel structures, Eurocode 3 (ENV 1993-1-1:1992). Other parts of the Eurocode were also used, and the design guidance for hollow section structures published by CIDECT (Comité International pour le Développement et l´Étude de la Construction Tubulaire) were consulted for additional reference. The aim of this handbook is to provide design guidance for structures manufactured of Rautaruukki structural hollow sections. It is also intended as a textbook. The primary scope of this handbook is building construction, but it can also be used in mechanical engineering where applicable. The handbook was written by Hannu Vainio (M.Sc.Tech). On Rautaruukki's behalf, the manuscript was supervised by Reijo Ilvonen, Janne Pirttijoki and Kristian Witting. Professor Erkki Niemi, Jouko Kouhi, Antti Helenius, Ilkka Hakola, Tiina Ala-Outinen, Arto Rokkanen, Hannu Reinikainen and Mikko Arponen also participated in the preparation of the manuscript and revised parts of it. The handbook was translated into English by Sirpa Meriläinen at Trantex Oy. Lauri Sannikka of Lasjuma Oy prepared the lay-out. The book was printed and bound by Otava. Warmest thanks are due to all contributors. In this document, a comma is used instead of a decimal point, and a decimal point is used instead of a multiplication sign, in accordance with the usual continental practice. Rautaruukki is happy to receive any comments and suggestions for improving the contents of this handbook.

Hämeenlinnassa 1.2.2000 RAUTARUUKKI METFORM

The accuracy of the contents of this manual has been carefully reviewed. Nevertheless, Rautaruukki is not responsible for any remaining errors or damage, whether direct or indirect, due to the incorrect application of the information presented in this book. The data in this book is for reference only, and the user is responsible for verifying the accuracy of the results by calculation. The information in this book is subject to change. 3

Chapter 1


CONTENTS 1 1.1 1.2 1.3 2. 2.1 2.2 2.2.1 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4 2.4.1 2.4.1.1 2.4.1.2 2.4.2 2.4.2.1 2.4.2.2 2.4.3 2.5 2.5.1 2.5.1.1 2.5.1.2 2.6 2.6.1 2.6.2 2.6.3 2.7 2.7.1 2.7.1.1 2.7.1.2 2.7.1.3 2.7.2 2.7.2.1 2.7.2.2 2.7.2.3 2.7.3 2.7.3.1 2.7.3.2 .7.3.3 2.8

INTRODUCTION ............................................................................................................... RAUTARUUKKI METFORM STRUCTURAL HOLLOW SECTIONS ............................... EN 10219 hollow sections ................................................................................................. Manufacture of hollow sections ......................................................................................... References ........................................................................................................................

8 9 9 9 11

RESISTANCE OF HOLLOW SECTION STRUCTURES .................................................. Limit state design and partial safety factors ...................................................................... Classification of cross-sections ......................................................................................... Calculating the effective cross-section .............................................................................. Resistance of hollow sections subjected to bending moment ........................................... Effect of holes on bending resistance ................................................................................ Bending resistance in bi-axial bending .............................................................................. Bending resistance of Class 4 circular hollow sections ..................................................... Lateral-torsional buckling of hollow sections ..................................................................... Examples for calculating bending resistance of various cross-sections ............................ Resistance of hollow sections subjected to shear force .................................................... Shear resistance of square and rectangular hollow sections ............................................ Plastic shear resistance of square and rectangular hollow sections ................................. Shear buckling resistance of square and rectangular hollow sections .............................. Shear resistance of circular hollow sections ...................................................................... Plastic shear resistance of circular hollow sections ........................................................... Shear buckling resistance of circular hollow sections ....................................................... Effect of holes on shear resistance ................................................................................... Hollow sections subjected to torsion moment ................................................................... Methods for calculating torsion resistance of hollow sections ........................................... Plastic torsion resistance of hollow sections ..................................................................... Torsional buckling resistance of hollow sections ............................................................... Hollow sections subjected to axial force ............................................................................ Tension resistance of hollow sections ............................................................................... Compression resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections (no buckling) ..................................................... Compression resistance of Class 4 cross-sections (no buckling) ..................................... Combined load resistance of hollow sections (no buckling) .............................................. Hollow sections subjected to bending moment and axial force (no buckling) ................... Class 1 or 2 hollow sections .............................................................................................. Square or rectangular Class 3 or 4 hollow sections and circular Class 3 hollow sections .................................................................................................................. Class 4 circular hollow sections ......................................................................................... Hollow sections subjected to shear force and bending moment ....................................... Square and rectangular hollow sections ............................................................................ Class 1, 2 or 3 circular hollow sections ............................................................................. Class 4 circular hollow sections ......................................................................................... Hollow sections subjected to axial force, shear force and bending moment (no buckling) ...................................................................................................................... Class 1 or 2 hollow sections .............................................................................................. Class 3 and 4 square and rectangular hollow sections and Class 3 circular hollow sections ...................................................................................................... Class 4 circular hollow sections ......................................................................................... Buckling resistance of hollow sections ..............................................................................

13 13 15 17 18 19 19 20 21 23 26 27 27 27 28 28 28 29 25 31 31 31 33 33

4

33 34 34 35 35 36 36 37 37 38 39 40 40 41 41 43


2.8.1 2.8.2 2.9 2.9.1 2.9.1.1 2.9.1.2 2.10 2.10.1 2.10.2 2.11 3 3.1 3.1.1

Chapter 1

Buckling resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections ...................................................................................................... Buckling resistance of Class 4 circular hollow sections ..................................................... Resistance of hollow sections subjected to combined loads (buckling) ............................ Hollow sections subjected to bending moment and axial force (buckling) ........................ Square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections .... Class 4 circular hollow sections ......................................................................................... Concentrated load resistance of hollow sections .............................................................. Concentrated load acting from one side only .................................................................... Concentrated load acting from both sides ......................................................................... References ........................................................................................................................

43 44 46 47 47 49 54 54 56 58

3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.3 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.2 3.5 3.5.1 3.5.2 3.6

DESIGN OF JOINTS IN HOLLOW SECTION STRUCTURES ......................................... Design of welded joints in lattice structures ....................................................................... Joints of circular, square or rectangular brace members to square or rectangular chords ............................................................................................................. Joints of circular brace members to circular chords .......................................................... Joints of circular, square and rectangular brace members to I profile chords ................... Welded frameworks ........................................................................................................... Joints of square and rectangular hollow sections subjected to bending ............................ Circular hollow section joints subjected to bending ........................................................... Welded end-to-end joints of hollow sections ..................................................................... Bolted hollow section joints ............................................................................................... End-to-end bolted hollow section joints ............................................................................. Flange plate connections ................................................................................................... In-line tension joint with splice plates ................................................................................ Bolted beam-to-column joints ............................................................................................ Hollow section-to-foundation joints .................................................................................... Joint between a column subjected to axial force and foundation ..................................... Joint between a column subjected to bending moment and axial force and the foundatio References ........................................................................................................................

62 72 77 77 78 80 82 84 84 85 90 94 104 104 105 108

4. 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.3.1 4.3.3.2 4.4 4.5 4.5.1 4.5.2 4.5.3 4.6 4.6.1 4.6.2 4.7

FATIGUE AND BRITTLE FRACTURE IN HOLLOW SECTION STRUCTURES ............. Fatigue loading .................................................................................................................. Stress calculation methods in fatigue design .................................................................... Design requirements for fatigue-loaded hollow sections (nominal stress method) ........... Conditions and necessity of fatigue design ....................................................................... Fatigue load design conditions .......................................................................................... Fatigue strength of hollow sections (nominal stress method) ............................................ Fatigue strength under normal and shear stress ............................................................... Fatigue strength of lattice structure joints (nominal stress method) .................................. Fatigue strength of lattice structure joints (hot spot method) ............................................. Design of fatigue-loaded hollow section structures ........................................................... Welded joints ..................................................................................................................... Bolted joints ....................................................................................................................... Lattice structures ............................................................................................................... Brittle fracture of structural hollow sections ....................................................................... Parameters affecting brittle fracture in structural hollow sections ..................................... Minimum service temperatures of Rautaruukki structural hollow sections ........................ References ........................................................................................................................

109 109 110 111 111 112 114 114 115 117 122 122 124 124 129 129 132 132

5

59 59

Chapter 1 5 5.1 5.1.1 5.1.2 5.2 5.2.1 5.2.2 5.3 5.4 5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 5.5.7


133 134 134 135 135 136 137 139 140 141 142 143 144 144 145 146

5.6 5.6.1 5.6.2 5.6.3 5.7 5.7.1 5.8

FIRE DESIGN OF HOLLOW SECTIONS ......................................................................... Development of temperature in fire compartments Standard time-temperature curve ...................................................................................... Development of temperature according to the parametric model ...................................... Development of steel temperature .................................................................................... Development of temperature in unprotected steel members ............................................. Development of temperature in fire protected steel members ........................................... Strength and modulus of elasticity of steel in fire situations .............................................. Critical temperature in hollow section structures ............................................................... Determining the strength of hollow section structures in fire situations ............................. Partial safety factors in fire design ..................................................................................... Determining the cross-section class in fire design ............................................................ Strength of hollow section subjected to tension in fire situations ...................................... Buckling strength of hollow sections in fire situations ........................................................ Bending strength of hollow sections in fire situations ........................................................ Shear strength of hollow sections in fire situations ............................................................ Strength of hollow sections subjected to bending moment and compressive axial force in fire situations ......................................................................................................... Fire retardant methods ...................................................................................................... Fire retardation by insulation ............................................................................................. Fire retardation by increasing the heat retention capacity of structural steel .................... Structural fire retardation ................................................................................................... Fire design of concrete-filled columns ............................................................................... Using tables in the fire design of concrete-filled columns .................................................. References ........................................................................................................................

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.1.6 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.4.6 6.4.7 6.5

DESIGN OF HOLLOW SECTION STRUCTURES ........................................................... Structural actions ............................................................................................................... Self-weight and imposed loads .......................................................................................... Snow load .......................................................................................................................... Wind load ........................................................................................................................... Additional horizontal forces ............................................................................................... Combined loads ................................................................................................................. Load determination in the model building .......................................................................... Designing columns ............................................................................................................ Column buckling length ..................................................................................................... Effect of joint stiffness on column buckling length ............................................................. Column-to-foundation connections .................................................................................... Column design in the model building ................................................................................. Designing the column-to-foundation joint in the model building ........................................ Designing the hollow section beams ................................................................................. Designing gable beam in the model building ..................................................................... Designing the door beam in the model building ................................................................ Design of trusses ............................................................................................................... Selection of truss type ....................................................................................................... Selection of the chord member .......................................................................................... Selection of bracing members ........................................................................................... Design of truss joints ......................................................................................................... Truss joints at the supports ............................................................................................... Estimation of the truss rigidity ............................................................................................ Designing the truss of the model building .......................................................................... Stiffening hollow section structures ...................................................................................

159 161 162 162 162 164 165 166 167 167 167 171 172 174 177 178 180 183 185 187 189 189 191 192 193 204

6

146 147 147 149 150 155 156 158


Chapter 1

6.5.1 6.6

Designing the stiffening elements in the model building .................................................... 204 References ........................................................................................................................ 208

7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.3 7.3.1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.4.6 7.4.7 7.5 7.6 7.7 7.8 7.9 7.10

SHOP FABRICATION AND ERECTION .......................................................................... Cutting of hollow sections .................................................................................................. Cutting of circular hollow sections ..................................................................................... Cutting methods ................................................................................................................ Notching of hollow section ends ........................................................................................ Bending of hollow sections ................................................................................................ Bending methods for hollow sections ................................................................................ Bolted joints ....................................................................................................................... Thermodrilling of hollow section walls ............................................................................... Expansion bolt joints .......................................................................................................... Pilot tap joints .................................................................................................................... Welding of hollow sections ................................................................................................ Quality levels ..................................................................................................................... Welding methods ............................................................................................................... Welding sequence ............................................................................................................. Fillet and butt welds ........................................................................................................... Preheating ......................................................................................................................... Residual stresses .............................................................................................................. Inspection of welds ............................................................................................................ Tolerances ......................................................................................................................... Assembly of trusses .......................................................................................................... Fire protection .................................................................................................................... Transport and storage ....................................................................................................... Erection ............................................................................................................................. References ........................................................................................................................

209 209 209 211 211 212 212 213 214 214 214 215 215 215 216 216 218 218 219 220 225 226 226 227 228

8 8.1 8.2 8.3 8.4 8.5

CORROSION PROTECTION ............................................................................................ Corrosivity categories ........................................................................................................ Surface preparation ........................................................................................................... Anti-corrosive painting ....................................................................................................... Hot-dip galvanizing ............................................................................................................ References ........................................................................................................................

229 229 230 230 232 234

9 APPENDIX .................................................................................................................. LIITTEET ........................................................................................................................... Appendix 9.1 Cross-sectional propertiesjaand resistance values for steel grade................................ S355J2H ............... Liite 9.1 Putkipalkkien poikkileikkauskestävyysarvot teräslajille S355J2H Appendix 9.2 Buckling tables for steel grade S355J2H .................................................................... Liite 9.2 Putkipalkkien nurjahduskestävyydet teräslajille S355J2H ................................................. Appendix 9.3 Calculation tables for truss joints ................................................................................ Appendix Calculation tables lattice of joints Appendix 9.4 Estimating thefor stiffness moment connections ......................................................... Liite 9.4 Kehäliitosten jäykkyyden arvioiminen ................................................................................ Appendix 9.5 Fatigue categories ...................................................................................................... Liite 9.5 9.6 Väsymisluokat ................................................................................................................... Appendix Cross-section factors in fire design ............................................................................. Appendix Minimum bending radii for square and rectangular hollow sections ........................... Liite 9.6 9.7 Poikkileikkaustekijät palomitoituksessa ............................................................................. Appendix 9.8 WinRAMI software ...................................................................................................... Liite 9.7 Neliön ja suorakaiteen muotoisten putkipalkkien minimitaivutussäteet ............................. Liite 9.8 WinRAMI-ohjelma ..............................................................................................................

235 235 235 257 255 289 287 327 325 333 331 339 343 337 345 341 343

7

Chapter 1


INTRODUCTION The structural hollow section is a modern, adaptable element for steel structures. It is also an environmentally friendly element, since it is easy to recycle and re-use. The simple shape of hollow sections and their excellent mechanical properties make them a light-weight and affordable solution. In a lattice structure, the high buckling resistance of hollow sections enables the use of long spans and a large spacing between diagonals. Due to the superior torsional stiffness of the closed section, lattice structures made of hollow sections as well as individual hollow sections have good resistance to lateral-torsional buckling. The fabrication of standard joint details is cost-efficient, and rounded corners and easily accesible joints facilitate pre-treatment. Hollow sections can easily be formed into light-weight and stiff frames grid structures, since their torsional stiffness and bending resistance in all directions is high. The torsional stiffness of hollow sections can be utilized also in various console structures and structures with projecting sections. In bracing members, the high stiffness of the hollow section serves to produce a low susceptibility to deflection. Another application for hollow sections is in composite structures. When using a concrete-filled composite column, the properties of steel and concrete can be efficiently utilized, under normal loads and in fire situations. New jointing systems, using direct attachment to the hollow section wall, enable the preparation of flexible and simple joints in hollow section structures. The design of a hollow section member or framework is easy and quick: the simple geometry can be expressed with few parameters, which makes computer-aided design a feasible option. The weight, resistance and stiffness of structures can be optimized by modifying the wall thickness, without needing to change the external dimensions of the hollow section or the geometry of the structure. This design handbook for hollow sections includes data on materials and dimensions of hollow sections manufactured by Rautaruukki. It also provides instructions for the dimensioning and design of cross-sections, joints and structures. Moreover, instructions are given for shop fabrication and site installation of hollow sections, as well as for their corrosion protection and fire design. The handbook is complemented by the WinRAMI software, designed by Rautaruukki especially for the dimensioning of hollow section structures. Additional information on WinRami software is available in Rautaruukki sales offices. The design guidance in this book is principally based on the European prestandard on steel structures, Eurocode 3 (ENV 1993-1-1:1992) and other parts of the Eurocode. The guidance for the design of hollow section structures published by CIDECT (Comité International pour le Développement et l´Étude de la Construction Tubulaire) were also used as a reference. The primary scope of this handbook is building construction, but it can be used also in mechanical engineering where applicable. The resistance values shown in formulae and tables are ultimate design values based on the basic partial safety factors used in Eurocode 3. However, the partial safety factors used in national application documents (NADs) may differ from those used in Eurocode 3, and this must be taken into account in structural design. The user is responsible for always verifying the information from the currently valid national application document. 8


Chapter 1

1

RAUTARUUKKI METFORM STRUCTURAL HOLLOW SECTIONS

1.1

EN 10219 hollow sections

Rautaruukki Metform manufactures cold-formed welded hollow sections conforming to EN 10219, made of steel grade S355J2H. The design guidance in this manual applies to Rautaruukki Merform structural hollow sections shown in Appendix 9.1. These hollow sections are manufactured according to satisfy the requirements of Eurocode 3 [1] and the latest research [2]. According to the latest research [2], the Rautaruukki Metform structural hollow sections shown in Appendix 9.1 are suitable for welded steel structures down to the operating temperature of -40°C (section 4.6). For the longitudinally welded* structural hollow sections shown in Appendix 9.1, Rautaruukki Metform guarantees the properties required by EN 10219 standard. In addition to this, the following properties are guaranteed for steel grade S355J2H: - impact toughness value of 35 J/cm2 is guaranteed at a testing temperature of -40°C (EN 10219: requires 35 J/cm2 of a testing temperature of -20°C) - mechanical properties meet the additional requirements for the application in plasticity theory presented in Eurocode 3 (ENV 1993-1-1:1992, section 3.2.2.2) when tested in accordance with standard EN 10219 - a carbon equivalent value of 0,39 maximum is guaranteed for steel grade S355J2H (EN 10219: requires a carbon equivalent value of 0,45 maximum) - chemical composition is better than required by EN 10219 - tolerance of wall thickness is better than required by EN 10219 - products meet the wall slenderness limitations presented in the latest research [2]: (B+H)/T ≥ 25 - corners are free of cracks - the products meet the following EN 10219 options: • • • •

1.2: The maximum value of the carbon equivalent is guaranteed for constructional steels. 1.3: Alloy contents are provided on the inspection certificate 1.7: Steel grade is suitable for hot-dip galvanizing 1.8: No weld repairs are made to the base material of the hollow section

*The technical delivery conditions for spiral welded hollow sections are agreed separately for each order.

1.2

Manufacture of hollow sections

Rautaruukki Metform manufactures the structural hollow sections from hot rolled steel strip by cold forming and welding. Hollow sections, with square and rectangular cross-sections and small circular (D ≤ 323,9 mm) hollow sections, are seam-welded longitudinally using high frequency welding (HFW) (Figure 1.1). Large circular sizes (D ≥ 355,6 mm) are seam-welded spirally using submerged arc welding (SAW) (Figure 1.2). The quality of hollow sections is controlled according to the ISO 9001 quality system, certified in all hollow section factories.

9

Chapter 1


Longitudinally welded hollow sections The material used in longitudinally welded hollow sections is steel strip, cut accurately to correspond with the width of the external dimensions of the section. At the beginning of the production line, the steel strip is unrolled and the strip ends are welded together. The strip is then fed into a strip accumulator to enable a continuous manufacturing process. The steel strip is shaped with forming rolls at room temperature step by step into a circular cross secton. The edges of the strip are heated to the welding temperature with high frequency current using an induction coil and pressed together. External burrs are removed from the section. Seam quality is ensured by a continuous eddy current or ultrasonic inspection. The diameter of a circular hollow section is calibrated to the final dimensions and the crosssection is formed to square or rectangular shape with profile rollers. A continuous marking is made to the hollow section, and it is cut to dimensions according to customer orders. Samples are taken for mechanical tests and flattening/expanding tests are carried out as required by the delivery condition standard, in accordance with the factory’s quality system. After cutting, the dimensions of the hollow sections are checked and the products are packed in stacks. Each stack is marked with a tag indicating the properties of the stacked hollow sections and their identification code. Based on the identification tag, the properties of the hollow section can be traced down to the steel manufacture.

Forming rolling

Welding

Burr removal Weld inspection Shaping and calibrating rolling

Figure 1.1

Fabrication of longitudinally welded hollow sections

10


Chapter 1

Spirally welded hollow sections Rautaruukki Metform also manufactures circular hollow sections with diameters from ∅ 355,6 to ∅ 1219 mm by spirally welding them from hot-rolled steel strip. At the beginning of the production line, separate steel strips are welded into a continuous sheet, which is then straightened and formed into a spiral-welded pipe using three-roll bending at room temperature. The spiral seam is welded both inside and outside using submerged arc welding. Mechanical values are tested with test coupons cut from the section. Sections are cut to dimensions according to customer orders, then inspected and delivered to customers.

Strip coil

Forming unit Welding

Feeding roll

Roll straightening

Weld inspection

Roll cutter

Figure 1.2

Fabrication of spirally welded hollow sections

1.3

References

Welding unit for strip continuity

[1]

ENV 1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993)

[2]

CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996

11

Chapter 1


12


Chapter 2

2.

RESISTANCE OF HOLLOW SECTION STRUCTURES

2.1

Limit state design and partial safety factors

The resistance of a member refers to the structure's ability to bear the loads it is subjected to without failure or excessive deformation. Resistance and load vary according to time and location. Thus, they do not have a single absolute value, but their values are distributed according to statistic probability. In design, the dispersion of resistance and load must be taken into account by using partial safety factors. The design criterion for the ultimate limit state is:

γ f ⋅ Sk ≤

Rk γM

(2.1)

where

γf γM Sk Rk

Table 2.1

is the partial safety factor for the load is the partial safety factor for steel is the characteristic value of the force or moment induced by the load is the characteristic value of resistance

Partial safety factors for actions [1]

Permanent actions Variable actions (γQ) (γG) Leading variable Accompanying Accompanying variable action action variable action Favourable effect 1,0 * * Unfavourable effect 1,35 1,5 1,5 Fatigue-inducing action 1,0 1,0 1,0 Fire design ** ** ** Serviceability limit state 1,0 1,0 1,0 · ψ0 The partial safety factors presented in this table are the basic values of Eurocode 3. National values must be checked from the national application document (NAD) [9], [10], [11], [12]. * Usually 0 [9] ** Partial safety factors for fire design are presented in chapter 5

Also the design formulae may differ nationally. The examples in this book were calculated using basic values in Eurocode 3. When calculating the design value for load in ultimate limit state, the following formulae are applicable when the structure is subjected to several variable actions [1]:

∑ γ G.j ⋅ Gk.j + γ Q.1 ⋅ Qk.1

(2.2a )

∑ γ G. j ⋅ Gk. j + 0, 9 ∑ γ Q.i ⋅ Qk.i

(2.2b )

j

j

i≥1

13

Chapter 2


where

γ G. j Gk . j γ Q.1 Qk .1 γ Q.i Qk .i

is the partial safety factor for permanent action is the characteristic value of the permanent action [5] is the partial safety factor for leading variable action is the characteristic value of the leading variable action [5] is the partial safety factor for variable action is the characteristic value of the variable action [5]

In formula (2.2a), only the primary leading variable action is taken into account. In formula (2.2b), all variable actions are taken into account. The formula to be used in the design is the one giving the most severe effect. Alternatively, a more accurate formula (2.3) can be used for ultimate limit state calculations [1].

∑ γ G. j ⋅ Gk. j + γ Q.1 ⋅ Qk.1 + ∑ ψ 0.i ⋅ γ Q.i ⋅ Qk.i

(2.3)

i >1

j

where

ψ0

is the combination factor of the action

The partial safety factors of the material depend on the class of the cross-section, the loading and the location of the designed element in the structure (Table 2.2). Table 2.2

Partial safety factors for the material [1]

Class 1, 2 and 3 cross-sections Class 4 cross-sections Resistance to buckling Net section at bolt holes (area of holes subtracted from the gross cross-section) Resistance of bolted joints Resistance of rivetted joints Resistance of pin joints Resistance of welded joints Friction joints

Partial safety factor γM0 = 1,1 γM1 = 1,1 γM1 = 1,1 γM2 = 1,25

γMb = 1,25 γMr = 1,25 γMp = 1,25 γMw = 1,25 Ultimate limit state γMs.ult = 1,25 Ultimate limit state, oversize γMs.ult = 1,40 holes parallel to load Serviceability limit state

Resistance of joints in hollow section lattices Fatigue strength

1)

2)

γMs.ser = 1,1 γMj = 1,1

Inspection and accessibility of structure normal poor

'Fail safe' components1)

Non 'fail safe' components 2)

γMf = 1,0 γMf = 1,15

γMf = 1,25 γMf = 1,35

'Fail safe' components = failure of one structural element does not lead rapidly to the collapse of the entire structure Non 'fail safe' components = failure of one structural element leads rapidly to the collapse of the entire structure The partial safety factors presented in this table are the basic values used in Eurocode 3. National values must be checked from the appropriate national application document (NADs).

14


2.2

Chapter 2

Classification of cross-sections

Cross-sections are divided into four classes. A single structure may contain structural elements with cross-sections of different classes. Elements of a single cross-section (flanges and webs) may also belong to different classes. The class of a cross-section is determined by the slenderness and the stress state of the cross-section elements. The class of a single hollow section may be different in bending and compression. Class 1: Cross-sections which can form a plastic hinge with the rotation capacity required for plastic analysis. Class 2: Cross-sections which can develop their plastic moment resistance, but have limited rotation capacity. Class 3: Cross-sections in which the calculated compression stress in the extreme fibre the cross-section can reach yield strength, but local buckling is liable to prevent the development of the plastic moment resistance. Class 4: Cross-sections are those in which it is necessary to make explicit allowances for the effects of local buckling when determining their moment recistance or compression recistance. The classification of the cross-section is usually determined by to the highest classification of compression element. The forces and resistances of the structure can be calculated for all classes using elasticity theory, if the effect of local buckling on cross-section resistance is taken into account. Plastici theory can be applied when calculating forces for class 1 cross-sections and resistances for class 1 and 2 cross-sections. In practice for simplicity forces can be calculated using the method determined by the highest class. For Class 4 square and rectangular hollow sections, the calculation of bending and compression resistance is based on the effective cross-section. Resistances of the crosssection are thus calculated using only the effective area of elements. When calculating resistance for class 4 circular hollow sections, the effective cross-section cannot be used, and Eurocode 3 does not give instructions for calculating their resistance. In this case, resistance can be assessed in relation to the buckling stress of the cylindrical casing. Circular hollow sections with Class 4 cross-sections are not recommended for load carrying structures.

15


Chapter 2 Table 2.3

Design methods for various cross-section classes [1]

Cross-section class 1

Method for calculating resistance Plastic analysis

Method for calculating actions (loads) Plastic analysis

2

; ; ; ; ; ;; ; ; ; ;

Square, rectangular and circular hollow sections

fy

Plastic analysis

Elastic analysis


3

fy

Elastic analysis

Elastic analysis


4

fy

Effective cross-section

Elastic analysis


4

Distribution of stresses when the resistance is reached

fy

Buckling stress

0,5beff

Elastic analysis

Circular hollow sections

σu < fy

16


Limit values for cross-section classes [3]

;;;;;;; ;;;;;;;

Table 2.4 b

d

h

t

Stress state

Chapter 2

t

Cross-section class 1 2 235 275 355 460 235 275 355 460

3

Loading method Compression

Crossfy section N element mm2 Web and h/t 45,0 41,6 36,6 32,2 45,0 41,6 36,6 32,2 flange b/t

235

275

355 460

45,0

41,6

36,6 32,2

Bending

Flange

b/t 36,0 33,3 29,3 25,7 41,0 37,9 33,4 29,3

45,0

41,6

36,6 32,2

Bending

Web

h/t 75,0 69,3 61,1 53,6 86,0 79,5 70,0 61,5 127,0 117,3 103,3 90,8

Compres- Entire d/t 50,0 42,7 33,1 25,5 70,0 59,8 46,3 35,8 90,0 76,9 59,6 46,0 sion and/or crossbending section fy is the yield strength for steel For other steel grades, the values in column 235 N/mm2 are multiplied by correction factor ε when using square and rectangular hollow sections and by correction factor ε2 when using circular hollow sections.

235 2 235 ,ε = fy fy

ε=

The cross-section class for a compression and bending web can be determined according to the instructions in reference [1], a conservative assessment is obtained when the cross-section class of the web is determined by compression only.

2.2.1

Calculating the effective cross-section

The width reduction factor ρ for Class 4 cross-sections of square and rectangular hollow sections is calculated as follows [1]:

ρ=1

ρ=

λ p − 0, 22 2 λp

kun λ p ≤ 0,673 when

(2.4)

when kun λ p > 0,673

(2.5)

The slenderness of a class 4 web subjected to bending and its effective width is calculated according to the guidance given in reference [1]. The slenderness of the flange or web subjected to uniform compression can be determined using the following formula (2.6):

λp =

b1 fy t = σ cr 56, 8 ⋅ ε

(2.6)

17


Chapter 2 where

t σ cr b1

is the thickness of the hollow section wall is the buckling stress = b- 3t; flange design width or h- 3t; web design heigth

ε

=

fy

is the nominal yield strength of the material [8]

Table 2.5

235 fy

Effective width beff of the flat compression elements with Class 4 cross-sections in square and rectangular hollow sections [1]

Stress distribution (compression positive) +σ

Effective width befff +σ

beff

= ρb1

be1 = 0,5 beff be1

be2 = 0,5 beff

be2 b1

2.3

Resistance of hollow sections subjected to bending moment

Hollow sections are efficient when subjected to bending about one or both principle axis. In addition, the buckling resistance about the minor axis is superior to an equivalent weight I or H profile section and therefore lateral restraints can be placed at greater spacings. The design criterrion for a member subject to bending about one axis is: MSd ≤ Mc. Rd

(2.7)

where

M Sd is the design value for bending moment Mc. Rd is the design value for bending resistance The bending resistance for hollow sections of different classes of cross-section is calculated as follows [1]: Class 1 and 2 cross-sections Mc. Rd = M pl. Rd = Wpl ⋅ f y / γ M 0 Class 3 cross-sections Mc. Rd = Mel. Rd = Wel ⋅ f y / γ M 0 Class 4 cross-sections Mc. Rd = Meff . Rd = Weff ⋅ f y / γ M 1 (square and rectangular) Class 4 cross-section (circular hollow sections)(section 2.3.3) where: Wpl is the plastic modulus of the cross-section Wel is the elastic modulus of the cross-section Weff is the effective modulus of the cross-section fy is the design strength 18


2.3.1

Chapter 2

Effect of holes on bending resistance

The effect of holes need not be taken into account in a flange subjected to tension when the following criterion is satisfied [1]:

0, 9 ⋅

A f.net f y γ M2 ≥ Af fu γ M0

(2.8)

where

Af A f.net fu γ M0 γ M2

is the cross-sectional area of the tension flange is the net cross-sectional area of the tension flange is the ultimate strength of the material is the partial safety factor for the material (Table 2.2) is the partial safety factor for the net effective cross-section (Table 2.2)

If the criterion is not satisfied, the cross-sectional area of the tension flange assumed in design, must be reduced to such an extent that the criterion is satisfied. This reduced cross-sectional area of the tension flange is then used to calculate the bending resistance. The effect of holes in the tension area of the web need not be considered, if the criterion (2.8) is met in the entire tension area. The tension area consists of the tension flange and the tension element of the web. In the compression area, the effect of holes need not be considered, unless the bolt holes are oversize or slotted [1].

2.3.2

Bending resistance in bi-axial bending

The following design criteria are applied, if the hollow section is subjected to bi-axial bending [1]: α

α

Class 1 and 2 cross-sections

 M y.Sd   M z.Sd  + M  ≤ 1, 0 M   pl.z.Rd   pl.y.Rd 

(2.9)

Class 3 cross-section

 M y.Sd   M z.Sd   ≤ 1, 0 M  +  el.y.Rd   Mel.z.Rd 

(2.10)

Class 4 cross-section (square and rectangular hollow sections)

 M y.Sd   M z.Sd  M  +  ≤ 1, 0  eff.y.Rd   Meff.z.Rd 

(2.11)

where

α α

=2 =1,66

(circular hollow sections) (square and rectangular hollow sections)

The effective cross-section modulus Weff.y is calculated assuming only My.Sd is present and Weff.z is calculated assuming only Mz.Sd is present. The bending resistances are thus calculated separately. 19


Chapter 2

2.3.3

Bending resistance of Class 4 circular hollow sections

The buckling stress must be calculated for circular hollow sections with Class 4 cross-sections. The design criterion is that the bending moment due to loading is smaller than the bending resistance of the hollow section:

M Sd ≤ Mc. Rd

(2.12)

where

M Sd is the design value for the bending moment Mc. Rd = σu ·Wel/γM1 (design value for bending resistance) σu is the buckling stress of the hollow section The buckling stress of a circular hollow section is calculated as follows, when λ ≤

(

σ u = 1 − 0, 4123(λ )

1, 2

)f

2 [2]: (2.13)

y

where

fy α b ⋅ σ cr

λ

=

σ cr

= 0,605E

r

=

t r

d −t 2

(the radius of the wall central axis)

αb is the reduction factor for buckling stress, which is calculated as follows, when r/t ≤ 212 [2]:

α b = 0, 1887 +

0, 6734

(2.14)

r 1 + 0, 01 t

r > 212 and/or λ > t calculated according to guidance given in reference [2]. For the hollow sections for which

20

2 , the buckling stress should be


2.3.4

Chapter 2

Lateral-torsional buckling of hollow sections

With hollow sections, lateral-torsional buckling does not normally govern. However, lateraltorsional buckling may govern in long hollow sections with a small b/h ratio. Figure 2.1 and Table 2.6 give the maximum lengths with which lateral-torsional buckling need not be taken into account when using square and rectangular hollow sections with Class 1-3 cross-sections. Circular hollow sections are not liable to lateral-torsional buckling. The values in Table 2.6 are determined using the following formula [3]:

Mc.Rd ⋅ γ M1 ≤ 0, 4 Mcr

(2.15)

where

Mc.Rd is the design value for bending resistance is the elastic critical lateral-torsional buckling moment Mcr If condition (2.15) is not met, the lateral-torsional buckling resistance can be calculated according to Eurocode 3 [1]. Table 2.6

Rectangular hollow sections: length ratios below which lateral-torsional buckling need not be taken into account [3]

;;

b−t h−t

M

M

L

L h−t

fy = 235 N/mm2 fy = 275 N/mm2 fy= 355 N/mm2 fy = 460 N/mm2

h

0,25 27,8 23,8 0,33 41,8 35,8 0,5 73,7 63,0 t 0,6 93,1 79,5 0,7 112,5 96,2 b 0,8 132,0 112,8 0,9 151,3 129,3 1,0 170,6 145,8 These values are determined for uniform moment which is the most severe case.

21

18,4 27,7 48,8 61,6 74,5 87,4 100,2 112,9

14,2 21,4 37,7 47,5 57,5 67,4 77,3 87,2


Chapter 2

Length L (m)

b−t h−t 45

1,0

40

0,9

35

0,8

30

0,7

25

0,6

20

0,5

15

0,33 10 0,25 5

0 50

75

100

125

150

175

200

225

250

275

300

325

350

Height (h-t) (mm)

Figure 2.1 Limit value curves for lateral-torsional buckling (fy = 355 N/mm2)

22

375

400


2.3.5

Chapter 2

Examples for calculating bending resistance of various cross-sections

Example 1a: Class 1 cross-section Consider a structural hollow section with dimensions 140 x 140 x 5. The steel grade used is S355J2H, for which the Class 1 criteria are (Table 2.4): Flange: b / t = 28 < 29,3 (compression) Web: h / t = 28 < 61,1 (bending) The plastic section modulus for the cross-section is determined by the following equation [6]: Wpl =

b ⋅ h 2 ( b − 2t )( h − 2t ) 2 − − 4 Az ⋅ hz + 4 Aξ ⋅ hξ 4 4

(2.16)

where the terms induced by corner rounding are: Az Aξ hz hξ ri r0

π =  1 −  ⋅ r02  4 π =  1 −  ⋅ r12  4 h 10 − 3π  = − ⋅ r0 2  12 − 3π  h − 2t  10 − 3π  = − ⋅r  12 − 3π  i 2 = 5 mm (internal nominal corner radius) = 10 mm (external nominal corner radius)

By inserting the hollow section dimensions 140 x 140 x 5 in the formula 2.16, we obtain the following value for section modulus: Wpl

140 ⋅ 140 2 (140 − 2 ⋅ 5)(140 − 2 ⋅ 5) 2 = − − 4 ⋅ 21, 46 ⋅ 67,77 + 4 ⋅ 5, 37 ⋅ 63, 88 4 4 = 132 ⋅ 10 3 mm 3

The bending resistance of the hollow section with dimensions 140 x 140 x 5 is thus: M pl. Rd = Wpl

fy

γ M0

= 132 ⋅ 10 3

355 = 42, 6 kNm 1, 1

Bending resistance values are presented also in tables in Appendix 9.1. 23


Chapter 2

Example 1b: Class 2 cross-sections Consider a hollow section with dimensions 160 x 160 x 5. The steel grade used is S355J2H, for which the Class 2 criteria are (Table 2.4): Flange: b / t = 32 < 33,4 (compression) Web: h / t = 32 < 70 (bending) For Class 2 cross-sections, the plastic section modulus is determined by the following formula (2.16): ri = 5 mm r0 = 10 mm Wpl = 175·103 mm3 Thus, the following bending resistance is obtained for a hollow section with dimensions 160 x 160 x 5: M pl. Rd = Wpl

fy

γ M0

= 175 ⋅ 10 3

355 = 56, 5 kNm 1, 1

Example 1c: Class 3 cross-sections Consider a hollow section with dimensions 180 x 180 x 5. The steel grade used is S355J2H, for which the Class 3 criteria are (Table 2.4): Flange: b/t = 36 < 36,6 (compression) Web: h/t = 36 < 103,3 (bending) For a Class 3 cross-section, the elastic section modulus is calculated as follows [6]:  b ⋅ h 3 ( b − 2t )( h − 2t ) 3  2 Wel =  − − 4( I zz + Az ⋅ hz 2 ) + 4( Iξξ + Aξ ⋅ hξ 2 ) ⋅ 12  12  h

(2.17)

where the terms, taking account of corner rounding, are: I zz

1 1 π  4 = − −  r0  3 16 3(12 − 3π ) 

1  4 1 π − Iξξ =  −  ri  3 16 3(12 − 3π )  ri r0

= 5 mm = 10 mm

By inserting the dimensions of the hollow section 180 x 180 x 5 in the formula (2.17), the following value for elastic section modulus is obtained:  180 ⋅ 180 3 (180 − 2 ⋅ 5)(180 − 2 ⋅ 5) 3  2 Wel =  − − 4(75, 5 + 21, 5 ⋅ 87, 8 2 ) + 4( 4, 7 + 5, 3 ⋅ 83, 9 2 ) ⋅ 12 12   180 = 193 ⋅ 10 3 mm 3 24


Chapter 2

The following bending resistance is obtained for a hollow section with dimensions 180 x 180 x 5: Mel. Rd =

Wel ⋅ f y 193 ⋅ 10 3 ⋅ 355 = = 62, 3 kNm 1, 1 γ M0

Example 1d: Square hollow section with Class 4 cross-section Consider a hollow section with dimensions 200 x 200 x 5. The steel grade used is S355J2H, for which the Class 3 criteria are (Table 2.4): Flange: b/t = 40 > 36,6 (compression) ⇒ Class 4 cross-sections Web: h/t = 40 < 103,3 (bending) ⇒ Class 1 cross-sections

1,5t

0,5beff bnon.eff 0,5beff 1,5t

δ

200 − 3 ⋅ 5 5 = 0, 801 > 0, 673 λp = 235 56, 8 ⋅ 355

; ; ;;;;; ;;;;;

As the compression flange belongs to Class 4, its effective width must be determined. The slenderness of the flange is obtained from the formula (2.6):

Now, the effective width of the flange can be calculated from the formula (2.5): beff =

0, 801 − 0, 22 ⋅ ( 200 − 15) = 167 , 5 mm 0, 8012

The neutral axis of the effective cross-section is transferred downwards. The effective section modulus of the cross-section is calculated by subtracting the section modulus of the non-effective element from the section modulus of the entire cross-section. The effective section modulus for a hollow section with dimensions 200 x 200 x 5 is obtained as follows: Weff

I + Aδ 2 − bnon.eff ⋅ t ⋅ ( 0, 5h - 0, 5t + δ ) 2 = 0, 5h + δ

( 2.18 )

2410 ⋅ 10 4 + 3840 ⋅ 2, 27 2 − 17 , 5 ⋅ 5( 0, 5 ⋅ 200 - 0, 5 ⋅ 5 + 2, 27) 2 = = 227 , 2 ⋅ 10 3 mm 3 0, 5 ⋅ 200 + 2, 27 where

δ =

bnon.eff ⋅ t ⋅ ( 0, 5 ⋅ h − 0, 5 ⋅ t ) 17 , 5 ⋅ 5 ⋅ ( 0, 5 ⋅ 200 − 0, 5 ⋅ 5) = = 2, 27 mm 3840 − 17 , 5 ⋅ 5 A − bnon.eff ⋅ t

(transfer of the neutral axis of the cross-section) 25


Chapter 2

bnon.eff = b – 3t – beff = 200 – 3· 5 – 167,5 = 17,5 mm (non-effective element of the compression web) A is the area of the entire cross-section (Appendix 9.1) I is the second moment of area of the entire cross-section (Appendix 9.1) To obtain the bending resistance value, the effective section modulus is multiplied by the yield strength: Meff.Rd = 227,2·103·355 / 1,1= 73,3 kNm Example 1e: Circular hollow section with Class 4 cross-section Consider a hollow section with dimensions 323,9 x 5. The steel grade used is S355J2H, for which the Class 3 criteria are (Table 2.4): d / t = 64,78 > 59,6 ⇒ Class 4 cross-section First, determine the slenderness of the cross-section: t 5 σ cr = 0, 605 E = 0, 605 ⋅ 210000 = 3984, 0 N/mm 2 r 159, 45 0, 6734 0, 6734 = 0, 1887 + = 0, 775 α b = 0, 1887 + r 159, 45 1 + 0, 01 1 + 0, 01 t 5 fy = α b ⋅ σ cr

λ=

355 = 0, 339 0, 775 ⋅ 3984, 0

Buckling stress is determined by the following formula (2.13):

(

)

(

)

σ u = 1 − 0, 4123 ⋅ λ 1, 2 f y = 1 − 0, 4123 ⋅ 0, 339 1, 2 355 = 315, 0 N/mm 2 To obtain the bending resistance of the cross-section, the buckling stress is multiplied by the elastic section modulus: Mc. Rd

[

]

[

]

4 4 315, 0 ⋅ π 323, 9 4 − ( 323, 9 − 2 ⋅ 5) 4 σ u ⋅ Wel σ u ⋅ π d − (d − 2t ) = = = = 112, 6 kNm γ M1 32d ⋅ γ M 1 32 ⋅ 323, 9 ⋅ 1, 1

2.4

Resistance of hollow sections subjected to shear force

The design criterion for a hollow section subject to shear force is [1]:

VSd ≤ VRd

( 2.19)

where

VSd VRd

is the design value for the shear force is the design value for the shear resistance 26


2.4.1

Chapter 2

Shear resistance of square and rectangular hollow sections

The method for calculating the shear resistance depends on the slenderness of the web of the cross-section as follows [1]:

h 235 ≤ 69 ⋅ + 3 ⇒ calculate the plastic shear resistance (section 2.4.1.1) t fy h 235 > 69 ⋅ + 3 ⇒ calculate the resistance to shear buckling (section 2.4.1.2) t fy Shear buckling need not be considered for square and rectangular hollow sections for which h/t < 59,1 and the yield strength of the material fy ≤ 355 N/mm2. In practice, shear buckling governs only for a very few hollow sections.

2.4.1.1

Plastic shear resistance of square and rectangular hollow sections

Plastic shear resistance is calculated using the following formula [1]:

Av ⋅ f y 3 ⋅ γ M0

( 2.20)

; ;;;

Vpl. Rd =

Vsd

=A

h b+h

Vsd

h

Av

h

where

(h in this case is the dimension parallel to shear force)

b

2.4.1.2

b

Shear buckling resistance of square and rectangular hollow sections

The shear buckling resistance of hollow sections is calculated using the following formula [1]:

Vba. Rd = 2(h - 3t ) ⋅ t

τ ba γ M1

( 2.21)

The web shear buckling stress τba depends on the slenderness of the web λw as follows [1]:

(

)

τ ba = 1 − 0, 625(λ w − 0, 8)

fy 3

for 0, 8 < λ w < 1, 2

( 2.22)

where

λw

h - 3t t = 235 86, 4 fy

The resistance to shear buckling is calculated according to the instructions in reference [1], ifλw ≥ 1,2.

27


Chapter 2

2.4.2

Shear resistance of circular hollow sections

The shear resistance of circular hollow sections can be assessed by the following methods Class 1, 2 and 3 cross-sections ⇒calculate plastic shear resistance (section 2.4.2.1) Class 4 cross-sections ⇒calculate shear buckling resistance (section 2.4.2.2) 2.4.2.1

Plastic shear resistance of circular hollow sections

The plastic shear resistance is calculated by the following formula [1]:

Vpl. Rd =

Av ⋅ f y 3 ⋅ γ M0

( 2.23)

where

Av

2.4.2.2

=A

2 π

Shear buckling resistance of circular hollow sections

The design value of shear buckling resistance for circular hollow sections is obtained using the following formula [2]:

Vba. Rd = π ⋅ r ⋅ t

τ ba γ M1

( 2.24)

where

τ ba r

is the shear buckling stress the central axis radius of the hollow section wall

The calculation of the theoretical shear buckling stress is a complex task. A conservative assessment is obtained by the following simplified formula [2]:

τ cr

t  t  0,75 = 0, 747 E ⋅ L  r

( 2.25)

where L

is the length of the hollow section element which is subjected to the shear Vsd

A more accurate formula for calculating the theoretical shear buckling stress is given in reference [2]. The difference to the results obtained by the formula (2.25) is, however, rather small when using normal hollow section lengths (> 1000 mm). The shear buckling stress is determined by the following formulae [2]:

τ ba = 0, 65τ cr τ ba =

fy  fy   1 − 0, 222  τ cr  3

for τ cr ≤ 0, 444f y

( 2.26 )

for τ cr > 0, 444f y

( 2.27 )

28


2.4.3

Chapter 2

Effect of holes on shear resistance

The effect of holes located in webs need not be considered when calculating the design value for plastic shear resistance, if the following criterion is satisfied [1]:

Av.net ≥

fy Av fu

( 2.28 )

where is the cross-sectional area of the web Av Av.net is the net cross-sectional area of the web The cross-sectional area Av used in design is reduced to the value (fu / fy) · Av.net, if the criterion (2.28) is not satisfied.

y;y; 200

h 400 235 = = 66, 67 > 69 + 3 = 59, 1 t 6 fy

6

400

Example 2a: Calculate the shear resistance of a hollow section with dimensions 400 x 200 x 6. The steel grade used is S355J2H. First determine whether the buckling of the web needs to be considered. Using the dimensions given in the example, the following value is obtained:

VSd

⇒ buckling of the web must be taken into account Calculate the slenderness of the web and the shear buckling stress τba using the formula (2.22). h - 3t 400 - 18 t 6 = = 0, 906 λw = 235 235 86, 4 86, 4 355 fy

(

)

τ ba = 1 − 0, 625 ⋅ (λ w − 0, 8)

fy 355 = (1 − 0, 625 ⋅ (0, 906 − 0, 8)) = 191, 4 N / mm 2 3 3 The shear resistance is obtained by inserting the shear buckling stress in the formula (2.21): Vba. Rd = 2(h - 3t )t

τ ba 191, 4 = 2( 400 - 3 ⋅ 6 ) ⋅ 6 ⋅ = 797 , 6 kN γ M1 1, 1

Example 2b: Calculate the shear resistance of a hollow section with dimensions 400 x 200 x 8. The steel grade used is S355J2H. h 400 235 = = 50 < 69 + 3 = 59, 1 ⇒ buckling of the web need not be taken into t 8 fy account 29


Chapter 2

The shear resistance is obtained directly by inserting values in the formula (2.20): Av ⋅ f y 400 355 = 9124 ⋅ ⋅ = 1133 kN 3 ⋅ γ M0 200 + 400 3 ⋅ 1, 1

;yy; y;

Vpl. Rd =

Example 3a: Calculate the shear resistance of the circular hollow section in example 1e with dimensions 323,9 X 5 and a Class 4 crosssection. The length of the hollow section is 6 m, and the steel grade used is S355J2H. Shear force is assumed constant along the entire length of the hollow section. First, determine the theoretical shear buckling stress from the formula (2.25):

τ cr

t  t  0,75 = 0, 747 E ⋅ L  r = 0, 747 ⋅ 2,1 ⋅ 10 5

5  5  ⋅  6000  159, 45 

00

60

VSd

0,75

= 337 , 4 N/mm 2

The shear buckling stress is obtained from the formula (2.27), since τcr > 0,444 fy:

τ ba =

fy  fy  355  355  2 − 1 0 , 222  1 − 0, 222  = 157,1 N/mm  = τ cr  3 3 337 , 4 

The resistance to shear buckling is calculated using the shear buckling stress: Vba. Rd = τ ba ⋅ π ⋅ r

t

γ M1

= 157 , 1 ⋅ π ⋅ 159, 45

5 = 357 , 7 kN 1, 1

Example 3b: Calculate the shear resistance for a hollow section with dimensions 323,9 x 8. The steel grade used is S355J2H. For a Class 2 cross-section, the plastic shear resistance must be calculated. (d/t= 40,5 < 46,3): Vpl. Rd =

2.5

Av ⋅ f y 2 355 = 7939 ⋅ = 941, 7 kN 3 ⋅ γ M0 π 3 ⋅ 1, 1

Hollow sections subjected to torsion moment

Hollow sections are efficient when subjected to a torsional moment. Their torsional resistance is superior to that of open sections. The design criterion for a member subjected to torsion moment is the following:

Mt .Sd ≤ Mt . Rd

( 2.29)

where

Mt .Sd is the design value for torsion moment Mt . Rd is the design value for torsion resistance 30


2.5.1

Chapter 2

Methods for calculating torsion resistance of hollow sections

As Eurocode 3 Appendix G dealing with torsion is not yet available, torsion resistance is calculated using the same method as for shear resistance. Torsional resistance for square and rectangular hollow sections can be calculated as follows:

235 h ≤ 69 ⋅ + 3 ⇒ calculate plastic torsional resistance (section 2.5.1.1) t fy h 235 > 69 ⋅ + 3 ⇒ calculate resistance to torsional buckling (section 2.5.1.2) t fy The torsional resistance of a circular hollow section depends on the slenderness and the length of the hollow section. Normally, it can be determined using the following methods: Class 1, 2 and 3 cross-sections ⇒ (calculate the plastic torsion resistance (section 2.5.1.1) Class 4 cross-section ⇒ (calculate the resistance to torsional buckling (section 2.5.1.2) 2.5.1.1

Plastic torsion resistance of hollow sections

The plastic torsional resistance of hollow sections can be expressed by the following formula [4]:

Mt . pl. Rd =

fy Wt ≈ 3 ⋅γ M0

fy 2 At ⋅ t 3 ⋅γ M0

( 2.30)

where

Wt At

2.5.1.2

is the torsional resistance of the cross-section is the area bounded by the central axis of the hollow section wall

Torsional buckling resistance of hollow sections

The resistance to torsional buckling can be calculated by the equation [2]:

Mt .b. Rd = τ ba

Wt γ M1

( 2.31)

where

τ ba

is the shear buckling stress (sections 2.4.1.2 and 2.4.2.2)

31


Chapter 2

y;y;

Example 4: Consider the hollow section in example 2a with dimensions 400 x 200 x 6 subjected to torsion moment Mt.Sd = 150 kNm. The steel grade used is S355J2H.

Wt 877 , 1 ⋅ 10 3 = 191, 4 ⋅ = 152, 6 kNm γ M1 1, 1

Mt .b. Rd > Mt .Sd

OK!

Mt.Sd

Example 5: Consider the circular hollow section in example 3a with dimensions 323,9 x 5 subjected to torsion moment Mt.Sd = 100 kNm. (τba obtained from example 3a). The torsional buckling resistance is given by the following formula: Mt .b. Rd

00

60

Wt 786, 6 ⋅ 10 3 = τ ba = 157 , 1 ⋅ = 112, 3 kNm γ M1 1, 1

Mt .b. Rd > Mt .Sd

6

y;y; y;

Mt .b. Rd = τ ba

400

The shear buckling stress τba, for the adjacent cross-section was calculated in the example 2a. Determine the torsional buckling resistance of the cross-section as follows:

200

OK!

Mt.Sd

32


2.6

Hollow sections subjected to axial force

2.6.1

Tension resistance of hollow sections

Chapter 2

Regardless of slenderness, the cross-section is fully effective when subjected to tension. Thus, the form of the cross-section does not affect the tension resistance. Hollow sections are efficient when used as tension members, as their joints can be made stronger and less complex than equivalent open sections. The design criterion for hollow section in tension is [1]:

N Sd ≤ Nt . Rd

( 2.32)

where is the design value of tensile force N Sd Nt . Rd is the design value of tension resistance The tension resistance of the cross-section is the smallest of the following [1]:

Nt . Rd = A

fy

( 2.33)

γ M0

Nt . Rd = 0, 9 Anet

fu

( 2.34)

γ M2

where

Anet fu

is the net area (the area of holes subtracted from the gross area) is the ultimate strength of the material

If ductility is required of the structure, the criterion (2.35) must be satisfied [1]:

0, 9

fy γ M 2 Anet ≥ ⋅ A fu γ M 0

( 2.35)

However, it is recommended that the criterion (2.35) should always be satisfied.

2.6.2

Compression resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections (no buckling)

The design criterion for hollow sections loaded in compression is [1]:

N Sd ≤ Nc.Rd

( 2.36 )

where

N Sd

is the design value for compressive force

Nc.Rd

= N pl. Rd = A

Nc.Rd

= Aeff

Aeff

fy

γ M1

fy

γ M0

for Class 1, 2 and 3 cross-sections for Class 4 cross-sections

is the effective cross-section area in axial compression 33


Chapter 2

2.6.3

Compression resistance of Class 4 cross-sections (no buckling)

The design criterion for circular hollow sections with Class 4 cross-sections can be expressed as [2]:

N Sd ≤ Nc.Rd

( 2.37 )

where

σu γ M1

Nc. Rd

=A

σu

= 1 − 0, 4123 λ

λ

=

(

σ cr r

( )1,2 ) fy

when λ ≤

2

fy α 0 ⋅ σ cr t = 0, 605 E r d −t (the radius of the wall central axis) = 2

Factor α0 is determined by the equation:

α0 =

2.7

0, 83

when

r 1 + 0, 01 t

r ≤ 212 t

(2.38 )

Combined load resistance of hollow sections (no buckling)

Table 2.7 presents criteria for the effect of different load combinations. Table 2.7 Combined load criteria to be checked when no risk of buckling is present Load combination

Crosssection

Class 1 and 2 cross-sections Section Formula Square and rectangular 2.7.1.1 (2.39) Circular 2.7.1.1 (2.39) Square and rectangular 2.7.2.1 (2.46) Circular 2.7.2.2 (2.47) Square and rectangular 2.7.3.1 (2.49) Circular 2.7.3.1 (2.49)

Class 3 cross-section Section Formula 2.7.1.2 (2.43) 2.7.1.2 (2.43) 2.7.2.1 (2.46) 2.7.2.2 (2.47) 2.7.3.2 (2.50) 2.7.3.2 (2.50)

Bending, compression or tension Bending Shear* Bending Compression or tension Shear* *The shear force needs to be taken into account only if VSd > 0,5 VRd The effect of torsion is allowed for by adding the following term in the interaction expression:

Mt .Sd Mt . Rd 34

Class 4 cross-section Section Formula 2.7.1.2 (2.43) 2.7.1.3 (2.44) 2.7.2.1 (2.46) 2.7.2.3 (2.48) 2.7.3.2 (2.50) 2.7.3.3 (2.51)


2.7.1

Chapter 2

Hollow sections subjected to bending moment and axial force (no buckling)

The cross-section class of a web loaded by bending moment and axial force depends on the stress distribution. In practice, the cross-section class is more easily determined by the compression element (web or flange). 2.7.1.1

Class 1 or 2 hollow sections

The interaction expression (2.39) can be applied for Class 1 and 2 cross-sections [1]: α

α

 M y.Sd   M z.Sd  +   ≤ 1, 0 M   M Nz.Rd   Ny.Rd 

(2.39)

The parameter α used in the calculation of bending resistance depends on the form of the hollow section: Circular hollow sections α = 2

; ;; ;

b

Square and rectangular hollow sections

1, 66  N  1 − 1, 13 Sd   N pl.Rd 

2

≤ 6

y

My

h

α=

z

where

N pl. Rd = A

fy

γ M0

Mz

The bending resistance is reduced by axial force and depends on the shape of the hollow section:

 N  M N . Rd = 1, 26 M pl. Rd  1 − Sd  , for square hollow sections N pl.Rd   however

M N . Rd ≤ M pl. Rd

( M pl. Rd

is the plastic bending resistance, section 2.3)

 N  M Ny. Rd = 1, 33 M pl. y. Rd  1 − Sd  , for rectangular hollow sections N pl.Rd   however

M Nz. Rd

M Ny. Rd ≤ M pl. y. Rd

M Nz. Rd ≤ M pl.z. Rd

( 2.41a )

( M pl. y. Rd is the plastic bending resistance about the y axis, section 2.3)

 1 − N Sd   N pl.Rd  = M pl.z. Rd  , for rectangular hollow sections ht   0, 5 +  A  

however

( 2.40)

( 2.41b )

( M pl.z. Rd is the plastic bending resistance about the z axis, section 2.3) 35


Chapter 2

M N . Rd

  N  1,7  = 1, 04 ⋅ M pl. Rd  1 −  Sd   circular hollow sections   N pl.Rd     M N . Rd ≤ M pl. Rd

however

2.7.1.2

( 2.42)

( M pl. Rd is the plastic bending resistance, section 2.3)

Square or rectangular Class 3 or 4 hollow sections and circular Class 3 hollow sections

The interaction expression (2.43) is derived according to the elasticity theory [1]:

M y.Sd N Sd M z.Sd + + ≤ 1, 0 Nc. Rd Mel. y. Rd Mel.z. Rd

Class 3 cross-section

( 2.43a )

M y.Sd N Sd M z.Sd + + ≤ 1, 0 Nc. Rd Meff . y. Rd Meff .z. Rd

Class 4 cross-sections

( 2.43b )

where

=A

Nc. Rd Nc. Rd Mel. y. Rd Meff . y. Rd

fy

for Class 3 cross-section

γ M0 fy for Class 4 cross-sections = Aeff γ M1 fy fy ja Mel.z. Rd = Wel.z = Wel. y and γ M0 γ M0 fy fy and ja Meff .z. Rd = Weff .z = Weff . y γ M1 γ M1

Aeff is calculated for axial compression. The effective cross-section modulus Weff.y is calculated assuming only My.Sd is present and Weff.z assuming only when Mz.Sd is present. 2.7.1.3

Class 4 circular hollow sections

The combined load criteria for Class 4 circular hollow sections can be expressed as follows:

N Sd M Sd + ≤ 1, 0 Nc. Rd Mc. Rd

( 2.44)

where

M Sd Nc. Rd

= M y2.Sd + M z2.Sd σ =A u γ M1

36


Mc. Rd = Wel

(

σu γ M1

( )1,2 ) fy

σu

= 1 − 0, 4123 λ

λ

=

σ cr

= 0, 605 ⋅ E

when kun λ ≤

Chapter 2

2

fy α ⋅ σ cr t r

The combined effect of bending moment and axial force is allowed for by the parameter α [2]:

α=

α0 ⋅σ 0 + αb ⋅σ b σ0 +σb

( 2.45)

where

N Sd A M = Sd Wel

σ0 =

the design stress due to the axial force

σb

the design stress due to the bending moment

α0 =

0, 83 1 + 0, 01

α b = 0, 1887 +

r t 0, 6734 1 + 0, 01

r t

when

r ≤ 212 t

when

r ≤ 212 t

2.7.2

Hollow sections subjected to shear force and bending moment

2.7.2.1

Square and rectangular hollow sections

If the shear force is more than half of the shear resistance of the cross-section (VSd > 0,5VRd), the effect of shear force must be accounted for when calculating bending resistance. The shear resistance value VRd is either Vpl.Rd (section 2.4.1.1) or Vba.Rd (section 2.4.1.2). The bending resistance of the cross-section is then [1]:

MV . Rd

 ρAv 2  − W  pl  fy 8t   = ≤ Mc. Rd γ M0

( 2.46 )

where

ρ

 2V  =  Sd − 1  VRd 

2

37


Chapter 2

Mc. Rd = Wpl Mc. Rd Mc. Rd t Av

fy

γ M0 fy = Wel γ M0 fy = Weff γ M1

for Class 1 and 2 cross-sections (section 2.3)

for Class 3 cross-sections (section 2.3)


is the wall thickness of the hollow section is the the area of the shear element formula [(2.20) or (2.21)]

Figure 2.2 depicts the resistance area of the combined effect of bending moment and shear force.

V

Combined effect of shear force and bending moment

VRd

0,5VRd

Mf.pl

Mpl

M

Figure 2.2 The effect of shear force on plastic bending resistance. The bending resistance of the section due to the flanges is Mf.pl.

2.7.2.2

Class 1, 2 or 3 circular hollow sections

For a circular hollow section with Class 1, 2 or 3 cross-section, the bending resistance can be expressed as follows when the shear force is more than half of the shear resistance (VSd > 0,5 Vpl.Rd) [3]:

MV.Rd = M pl.Rd

 V  1 −  Sd   Vpl.Rd 

2

( 2.47 )

however MV.Rd ≤ Mc.Rd 38


Chapter 2

where

Mc. Rd = Wpl Mc. Rd VSd

fy

γ M0 fy = Wel γ M0 =

for Class 1 and 2 cross-sections (section 2.3)


(Vy.Sd ) 2 + (Vz.Sd ) 2

Vpl. Rd is the plastic shear resistance (section 2.4.2.1) 2.7.2.3


For circular hollow sections with Class 4 cross-sections, the effect of shear force is accounted for in the combined load criterion [2]:

M Sd V + Sd ≤ 1 Mc.Rd Vba.Rd

( 2.48 )

where

Mc.Rd Vba.Rd

is the design value for bending resistance (section 2.3.3) is the design value for shear buckling resistance (section 2.4.2.2)

Example 6 Calculate the bending resistance for a hollow section with dimensions 400 x 200 x 6 when subjected to a shear force equal to 600 KN. The steel grade used is S355J2H. The resistance to shear buckling for a similar hollow section was calculated in example 2a. This value is now used. Force quantities are:

MSd

;y

VSd = 600 kN > 0,5 Vba.Rd = 0,5 · 797,6 = 398,8 kN MSd = 290 kNm Determine the classification of the cross-section: Flange: Web:

29,3 < b/ t = 200/ 6 = 33,3 < 33,4 ⇒ Class 2 61,1 < h/ t = 400/ 6 = 66,7 < 70 ⇒ Class 2

VSd

Since the cross-section of the hollow section is Class 2, the plastic bending resistance can be used. The shear force is more than half of the resistance to shear buckling. Thus, it decreases the moment resistance. Calculate the value for the parameter ρ : 2

2

 2V   2 ⋅ 600  ρ =  Sd − 1 =  − 1 = 0, 255  797,6   VRd  39


Chapter 2

Inserting the value for ρ in the expression (2.46):

MV . Rd

 ρ ⋅ AV 2  − W  pl  fy 8t   = = γ M0

 0, 255 ⋅ 4584 2   906000 −  ⋅ 355 8⋅6   1, 1

= 256, 4 kNm ≤ M pl.Rd = 292, 3 kNm The bending resistance of hollow section with dimensions 400 x 200 x 6 is not sufficient (MSd > MV.Rd). The wall thickness must be increased or a larger hollow section must be selected. 2.7.3

Hollow sections subjected to axial force, shear force and bending moment (no buckling)

2.7.3.1

Class 1 or 2 hollow sections

The interaction expression (2.49) can be applied to Class 1 and 2 cross-sections [1]: α

α

 M y.Sd   M z.Sd  +   ≤ 1, 0 M   M Nz.Rd   Ny.Rd 

( 2.49)

where

1, 66

α

=

α

=2

circular hollow sections

M N.Rd

 N  = 1, 26 MV . Rd  1 − Sd   NV.Rd 

square hollow sections

however

M Ny.Rd however

M Nz.Rd

however

 N  1 − 1, 13 Sd   NV.Rd 

2

≤ 6

square and rectangular hollow sections

M N.Rd ≤ MV . Rd  N  = 1, 33 MV . y. Rd  1 − Sd  rectangular hollow sections  NV.Rd  M Ny.Rd ≤ MV . y. Rd  1 − N Sd   NV.Rd  = MV .z. Rd  h⋅t  0 , 5 +    A 

rectangular hollow sections

M Nz.Rd ≤ MV .z. Rd 40


Chapter 2

  N  1,7  = 1, 04 MV . Rd  1 −  Sd   circular hollow sections   N pl.Rd    

M N.Rd

however M N.Rd ≤ MV . Rd

fy

N pl. Rd

=A

NV . Rd

= ( A − ρ ⋅ Av )

γ M0 fy

γ M0

MV . y. Rd

is the bending resistance with the effect of the shear force taken into account (by y axis) (section 2.7.2.1)

MV .z. Rd

is the bending resistance with the effect of the shear force taken into account (by z axis) (section 2.7.2.1)

MV . Rd

is the bending resistance with the effect of the shear force taken into account for circular hollow sections (section 2.7.2.2) or for square hollow sections (section 2.7.2.1)

2.7.3.2

Class 3 and 4 square and rectangular hollow sections and Class 3 circular hollow sections

The interaction expression for a member subjected to compression and bending in the presence of shear force can be expressed in a similar way to that in section 2.7.1.2. The effect of shear force is accounted for in bending resistance values [1]:

M y.Sd N Sd M z.Sd + + ≤ 1, 0 Nc. Rd MV . y. Rd MV .z. Rd

2.7.3.3

( 2.50)


When using circular hollow sections with Class 4 cross-sections, the resistance of the crosssection must be checked using the combined load criterion [2]:

N Sd M Sd V + + Sd ≤ 1 Nc.Rd Mc. Rd Vba.Rd

( 2.51)

where

Nc.Rd is the design value for bending resistance (section 2.3.3) Mc. Rd is the design value for compression resistance (section 2.6.3) Vba.Rd is the design value for shear buckling resistance (section 2.4.2.2)

41


Chapter 2

;y

Example 7 Determine whether a hollow section with dimensions 200 x 200 x 8 can carry the load shown in the adjacent figure. The steel grade used is S355J2H, and the load values are: = 1400 kN = 27 kNm = 24 kNm = 400 kN > 0,5 Vpl.Rd = 275,75 kN = 150 kN < 0,5 Vpl.Rd

NSd My.Sd Mz.Sd Vz.Sd Vy.Sd

My

Vy

N

Mz

Vz

Determine the classification of the cross-section (Table 2.4): h/t = b/t = 200/8= 25 < 29,3 ⇒ Class 1

The method for calculating the shear resistance also depends on the slenderness of the cross-section: h/t = 200/8 = 25 < 59,1 ⇒ calculate the plastic shear resistance The plastic shear resistance of the cross-section is the same about both axes: Vpl. Rd

fy h⋅ A = 3 ⋅ γ M0 h + b

=

355 200 ⋅ 5924 = 551, 9 kN 3 ⋅ 1, 1 200 + 200

The reduction in the bending resistance due to shear force depends on the parameter ρ: 2

ρz

2  2Vz.Sd   2 ⋅ 400  − 1 =  − 1 = 0, 203 =  551, 5   Vpl.Rd 

ρy

=0

for (Vz.Sd < 0, 5Vpl.Rd )

Next, the resistance to axial force and bending is determined, taking into account the effect of shear force:

MV . y. Rd

 ρ z ⋅ AV2  W −  pl  fy 8t   = = γ M0

 0, 203 ⋅ 2960 2  420860 −   355 8⋅8   = 1, 1

= 126, 9 kNm < M pl.y.Rd = 135, 8 kNm MV .z. Rd NV.Rd

355 ⋅ 420, 86 = 135, 8 kNm 1, 1 fy 355 = ( 5924 − 0, 203 ⋅ 2960) ⋅ = 1718 kN = ( A − ρ ⋅ Av ) γ M0 1, 1 = M pl.z. Rd =

42


Chapter 2

The effect of axial force on the bending resistance is accounted for by the following formulae:  N  1400  M Ny. Rd = 1, 26 ⋅ MV . y. Rd  1 − Sd  = 1, 26 ⋅ 126, 9 ⋅  1 − = 29, 60 kNm   NV.Rd  1718   N  1400  M Nz. Rd = 1, 26 ⋅ MV .z. Rd  1 − Sd  = 1, 26 ⋅ 135, 8 ⋅  1 − = 31, 67 kNm   NV.Rd  1718  The parameter α for a rectangular hollow section is (section 2.7.3.1):

α

=

1, 66  N  1 − 1, 13 Sd   NV.Rd 

2

=

1, 66 2 = 6, 65 > 6 ⇒ α = 6 1400  1 − 1, 13  1718 

In this case, the interaction expression (2.49) is as follows: α

α

6 6  M y.Sd   M z.Sd   27   24   +  = 0, 765 < 1, 0  =  M  +  M     29 , 60 31 , 67  Ny. Rd  Nz.Rd

OK!

2.8

Buckling resistance of hollow sections

2.8.1

Buckling resistance of square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections

Stuctural hollow sections are particularly efficient as compression members, as the hollow section material is located equally about and at a distance from the cross-section’s mid-point. Due to high torsional stiffness, torsional buckling need not be taken into account. The design criterion for the flexural buckling resistance of the compression member can be expressed as follows [1]:

N Sd ≤ N b.Rd

( 2.52)

where

N b.Rd = χ ⋅ β A ⋅ A

fy

γ M1

(the design value for flexural buckling resistance)

χ

is the reduction factor for flexural buckling

βA

=

βA Aeff A

is the effective cross-sectional area is the total cross-sectional area

Aeff for square and rectangular Class 4 hollow sections A for Class 1, 2 and 3 cross-sections =1

43


Chapter 2

The reduction factor χ for buckling is given by the equation [1]:

χ=

1

φ + φ − (λ ) 2

2

≤ 1, 0

[

φ = 0, 5 ⋅ 1 + α (λ − 0, 2) + (λ ) λ=

Lc ⋅ π ⋅i

( 2.53) 2

]

( 2.54)

fy βA E

( 2.55)

where

Lc i α

is the buckling length is the radius of gyration is the factor allowing for initial deflection and residual stresses

Buckling length depends on the type of connection at the ends of the member. Usually, it is conservative to use the actual length as the buckling length for lattice structures and the theoretical buckling length for rigid structures without allowing for joint rigidity. The determination of buckling length is presented in more detail in Chapter 6. According to Eurocode 3, there are several methods for calculating the buckling resistance of the cross-sections of cold formed hollow sections. A simple conservative method is to use the nominal yield strength fy and buckling curve c [7] for the hollow sections. For buckling curve c, the value of the imperfection factor α is 0,49 [1].

2.8.2

Buckling resistance of Class 4 circular hollow sections

The design criterion for the buckling resistance of Class 4 cross-sections is:

N Sd ≤ N b.Rd

( 2.56 )

where

N b. Rd = χ ⋅ A

(

σu γ M1

(design value for buckling resistance)

( )1,2 ) fy

σu

= 1 − 0, 4123 λ

λ

=

α0

=

σ cr

= 0, 605 E

(buckling stress [2])

fy α 0 ⋅ σ cr 0, 83 1 + 0, 01

r t

t r 44


Chapter 2

The reduction factor χ for buckling is calculated as in section 2.8.1, except that the buckling stress (fy = σu). is used instead of yield strength. The cross-sections of Class 4 circular hollow sections are always considered to be totally effective; thus βA= 1.

NSd

; y y;

Example 8a Calculate the compression resistance of a hollow section with dimensions 200 x 200 x 5. The steel grade used is S355J2H, and the buckling length is 4 m. The member is nominally pinned at both ends. Determine the classification of the cross-section of the hollow section (Table 2.4):

200

4000

200

5

b / t = 200/ 5 = 40 > 36,6 ⇒ Class 4 As the cross-section of the hollow section is Class 4, the effective cross-section must be determined. The slenderness of the compression elements is calculated using the formula (2.6): 200 - 15 b1 5 t = = 0, 801 > 0,673 λp = 235 56, 8 ⋅ ε 56, 8 355

The dimensions of the effective and non-effective elements of the cross-section are as follows [formula (2.5) and Table 2.5]: beff = heff =

λ p − 0, 22 2 λp

(b − 3t ) =

0, 801 − 0, 22 ( 200 − 3 ⋅ 5) = 167, 5 mm 0, 8012

bnon.eff = b − 3t − beff = 200 − 3 ⋅ 5 − 167 , 5 = 18 mm Using the effective cross-section, let us determine the effective area and parameter βA: Aeff

= A − ( 4 ⋅ bnon.eff ⋅ t ) = 3840 − ( 4 ⋅ 18 ⋅ 5) = 3480 mm

βA

=

Aeff 3480 = = 0, 906 A 3840

The local buckling of the cross-section has now been taken into account. Next, consider the buckling resistance of the hollow section. The cross-section slenderness is determined using the formula (2.55):

λ=

Lc i ⋅π

fy 4000 βA = E 79, 3 ⋅ π

355 0, 906 = 0, 628 210000

The reduction factor for buckling is calculated from the formulae (2.53) and (2.54) by inserting α = 0,49 ( for buckling curve c):

χ = 0,769 45


Chapter 2

The buckling resistance of the hollow section is calculated by multiplying the plastic compression resistance of the effective cross-section by the reduction factor (2.52): N b. Rd = χ ⋅ Aeff

fy

γ M1

= 0, 769 ⋅ 3480 ⋅

355 = 863, 7 kN 1, 1

Example 8b Calculate the compression resistance of a hollow section with dimensions 200 x 200 x 8. The steel grade used is S355J2H, and the buckling length is 4 m. The member is nominally pinned at both ends. Determine the classification of the cross-section (Table 2.4): b/ t= 200/ 8 = 25 < 36,6 ⇒ Class 1 cross-section The effective cross-section need not be calculated for Class 1, 2 and 3 cross-sections. The cross-section slenderness is calculated using the formula (2.55):

λ=

Lc i ⋅π

fy 4000 = E 77 , 6 ⋅ π

355 = 0, 675 210000

The reduction factor for buckling is calculated from formulae (2.53) and (2.54) by inserting α = 0,49 (for buckling curve c):

χ = 0,740 The buckling resistance of the hollow section is calculated by multiplying the plastic compression resistance by the reduction factor χ (2.52): N b. Rd = χ ⋅ A

2.9

fy

γ M0

= 0, 740 ⋅ 5920 ⋅

355 = 1414, 8 kN 1, 1

Resistance of hollow sections subjected to combined loads (buckling)

Criteria for the effect of different load combinations when buckling is taken into account are shown in Table 2.8. Table 2.8 Loading combination

Combined load criteria to be checked when buckling may be present Cross-section

Class 1 and 2 cross- Class 3 crossClass 4 crosssections sections sections Section Formula Section Formula Section Formula Bending and Square 2.9.1.1 (2.57) 2.9.1.1 (2.57) 2.9.1.1 (2.57) compression Rectangular 2.9.1.1 (2.57) 2.9.1.1 (2.57) 2.9.1.2 (2.60) The effect of torsion is accounted for by adding the following term in the interaction expression:

Mt .Sd Mt . Rd

46


2.9.1

Chapter 2

Hollow sections subjected to bending moment and axial force (buckling)

The classification of webs subjected to bending and compression depends on the stress distribution. In practice, the cross-section class is more easily determined by the compression element (web or flange). 2.9.1.1

Square and rectangular hollow sections and Class 1, 2 and 3 circular hollow sections

The interaction expression for a structure subjected to compression and bending is as follows [1]:

k y ⋅ M y.Sd k z ⋅ M z.Sd N Sd + + ≤1 N b.Rd M y.Rd M z.Rd

( 2.57 )

where

N b. Rd = χ min ⋅ β A ⋅ A

fy

γ M1

χ min is the minimum value of the reduction factor for buckling (about the y or z axis) fy fy for Class 1 and 2 cross-sections M y. Rd = Wpl. y M z. Rd = Wpl.z γ M1 γ M1 fy fy for Class 3 cross-sections M y. Rd = Wel. y M z. Rd = Wel.z γ M1 γ M1 fy f y for Class 4 square and rectangular M y. Rd = Weff . y M z. Rd = Weff .z γ M1 γ M 1 hollow sections The parameters ky and kz in expression (2.57) are determined as follows [1]:

ky = 1 −

µ y ⋅ N Sd ≤ 1, 5 χ y ⋅ A ⋅ fy

( 2.58 )

kz = 1 −

µ z ⋅ N Sd ≤ 1, 5 χ z ⋅ A ⋅ fy

( 2.59)

where

χy

is the reduction factor for buckling determined about the y axis

χz

is the reduction factor for buckling determined about the z axis

µy

= λ y ( 2β My − 4) +

Wpl. y − Wel. y ≤ 0, 9 Class 1 and 2 cross-sections Wel. y

µz

= λ z ( 2β Mz − 4) +

Wpl.z − Wel.z ≤ 0, 9 Class 1 and 2 cross-sections Wel.z 47


Chapter 2

µy

= λ y ( 2β My − 4) ≤ 0, 9 Class 3 and 4 cross-sections

µz

= λ z ( 2β Mz − 4) ≤ 0, 9 Class 3 and 4 cross-sections

β My and β Mz

are equivalent uniform moment factors allowing for the shape of the moment diagram (Table 2.9)

λy

is = the slenderness determined by the y axis

λz

= the slenderness determined by the z axis is

Table 2.9

The form factor of the moment plane βM [1]

Method of loading End moments

Moment diagram

Equivalent uniform moment factor

M1 ψM1

βM.ψ = 1,8 - 0,7ψ -1 ≤ ψ ≤ 1

In plane lateral uniform load

βMQ = 1,3 MQ

;y;y In plane lateral concentrated load

βMQ = 1,4

MQ

End moments and in plane lateral loads

βM is derived from the formulae

MQ

∆M

MQ

MQ

∆M

∆M

48

βM = βM.ψ +

(

MQ βMQ − βM.ψ ∆M

)

MQ = the moment with the greatest absolute value due to lateral loading ∆M = the moment with the greatest absolute value when the sign of the moment does not change ∆M = the sum of the absolute values of the greatest and smallest moments, when the sign of the moment changes


2.9.1.2

Chapter 2


Buckling is taken into account in the interaction expression which can be expressed as follows:

k y ⋅ M y.Sd k z ⋅ M z.Sd N Sd + + ≤ 1, 0 N b.Rd Mc.Rd Mc.Rd

( 2.60)

where

N b.Rd is the design value for buckling resistance (section 2.8.2) σ Mc. Rd = Wel u γ M1

( )1,2 ) fy

(

σu

= 1 − 0, 4123 λ

λ

=

αb

= 0, 1887 +

σ cr

= 0, 605 E

(buckling stress [2])

fy α b ⋅ σ cr 0, 6734 1 + 0, 01

r t

t r

The parameters ky and kz in expression (2.60) are determined as follows:

ky = 1 −

µ y ⋅ N Sd ≤ 1, 5 (σu is determined with parameter α0, section 2.8.2) χ y ⋅ A ⋅σu

kz = 1 −

µ z ⋅ N Sd ≤ 1, 5 (σu is determined with parameter α0, section 2.8.2) χz ⋅ A ⋅σu

where

χy

is the buckling reduction factor determined about the y axis

χz

is the buckling reduction factor determined about the z axis

µy

= λ y ( 2β My − 4) ≤ 0, 9

µz

= λ z ( 2β Mz − 4) ≤ 0, 9

β My ja β Mz are the equivalent uniform moment factors (Table 2.9) λy

=

Lc. y π ⋅ iy

σu (σu is determined with parameter α0, Table 2.8.2) E

λz

=

Lc.z π ⋅ iz

σ u (σu is determined with parameter α0, Table 2.8.2) E

Y and z axis are chosen in such a manner that the primary governing combination is obtained for bending moments. 49


NSd =500 kN 200

The slenderness and compression resistance of the effective cross-section is obtained from example 8a:

200

Example 9 Calculate the resistance for the hollow section from example 8a with dimensions 200 x 200 x 5, when it is subjected to axial force and bending moment due to a uniform transverse load. Only one of the axes is subjected to bending. The member is supported by nominally pinned connections at both ends.

4000

Chapter 2

5

MSd =10 kNm

;y

λ = 0,628

Nb.Rd = 863,7 kN (Nb.Rd = Nb.y.Rd = Nb.z.Rd, symmetrical cross-section)

The bending resistance value for a hollow section with dimensions 200 x 200 x 5 was calculated in example 1d i.e. My.Rd = Meff.Rd = 73,3 kNm

The equivalent uniform moment factor is given in Table 2.9:

βMQ = 1,3 Using factor βMQ the parameters µ and k in the interaction expression can be derived from the formula (2.58):

µy

= λ ( 2β MQ − 4) = 0, 628( 2 ⋅ 1, 3 − 4) = −0, 879

ky

= 1−

µ y ⋅ N Sd 500000 = 1 + 0, 879 = 1, 419 χ y ⋅ A ⋅ fy 0, 769 ⋅ 3840 ⋅ 355

Check the resistance of hollow section for the combined effect of axial force and moment (2.57): k y ⋅ M y.Sd N Sd 500 1, 419 ⋅ 10 + = + = 0,772 < 1, 0 OK! N b.Rd M y.Rd 863,7 73, 3

50


NSd My.Sd Mz.Sd Mt.Sd

= 400 kN = 9 kNm = 9 kNm = 3 kNm (assumed constant along the entire hollow section)

My

Mz

z-z

y-y

;y

The hollow section in classified as Class 3, since 33,4 < b/t= 36 < 36,6 (Table 2.4). Thus, the bending resistance must be determined using elasticity theory. The effect of torsion must be accounted for in the interaction expression. The reduction factor χ for the buckling of a hollow section subjected to compression is calculated using buckling curve c (section 2.8.1):

N

N

4000

Example 10 Calculate the resistance of a hollow section with dimensions 180 x 180 x 5 to the loading shown in the adjacent figure. The buckling length of the structure is 4 m, and the member is supported by nominally pinned connections at both ends. The steel grade used is S355J2H, and the loading values are:

Chapter 2

λ

χ

Lc f y 4000 = i ⋅π E 71,1 ⋅ π = 0,702 =

My

Mt

N

Mz

355 = 0,736 210000

Determine the compression and bending resistance of the hollow section: N b. Rd = χ ⋅ A

fy

γ M1

= 0, 702 ⋅ 3436 ⋅

M y. Rd = M z. Rd = Wel

fy

γ M1

= 193 ⋅

355 = 778, 4 kN 1, 1

355 = 62, 28 kNm 1, 1

The parameters µ and k depending on the shape of the moment diagram are as follows (section 2.9.1.1)

µy

= λ ( 2β MQ − 4) = 0, 736 ⋅ ( 2 ⋅ 1, 3 − 4) = −1, 03

ky

= 1-

µz

= λ ( 2β MQ − 4) = 0, 736 ⋅ ( 2 ⋅ 1, 4 − 4) = −0, 883

kz

= 1-

µ y ⋅ N Sd 400000 = 1 + 1, 03 = 1, 481 < 1, 5 0, 702 ⋅ 3436 ⋅ 355 χ y ⋅ A ⋅ fy µ z ⋅ N Sd 400000 = 1 + 0, 883 = 1, 412 < 1, 5 χ z ⋅ A ⋅ fy 0, 702 ⋅ 3436 ⋅ 355

The calculation method for torsional resistance is determined by the web slenderness (section 2.5.1): h/t = 180/ 5 = 36 < 59,1 ⇒ calculate the plastic torsion resistance 51


Chapter 2

Plastic torsional resistance is calculated using the formula (2.30): Mt . pl. Rd

=

f y Wt 355 289, 8 ⋅ = ⋅ = 54, 0 kNm 3 γ M0 3 1, 1

By adding the effect of torsion in the interaction expression (2.57), the following result is obtained: k y ⋅ M y.Sd k z ⋅ M z.Sd N Sd Mt .Sd + + + N b.Rd M y.Rd M z.Rd Mt . pl. Rd =

400 1, 481 ⋅ 9 1, 412 ⋅ 9 3 + + + = 0, 987 < 1, 0 OK! 778, 4 54, 0 62, 28 62, 28

NSd My.Sd Mz.Sd Mt.Sd

;yy; y;

Example 11 Calculate the resistance of a circular hollow section with dimensions 323,9 x 5 to the combined loading shown in the adjacent figure. The steel grade used is S355J2H, and the hollow section length is 6 m. The moment is assumed constant along the hollow section length. The hollow section is supported by hinges at both ends. The loading values are: = 500 kN (compression) = 18 kNm = 18 kNm = 6 kNm

00

60

Mt

My

N

Mz

First, the buckling resistance of the hollow section is calculated. Obtain the following value for the parameter α0 in compression only: 0, 83

α0 =

r 1 + 0, 01 t

= 0, 723

The buckling stress σu, in compression only, is calculated as follows: t 5 σ cr = 0, 605 E = 0, 605 ⋅ 2, 1 ⋅ 10 5 = 3984 N/mm 2 r 159, 45

λ=

fy = α 0 ⋅ σ cr

(

355 = 0, 351 0, 723 ⋅ 3984, 0

σ u = 1 − 0, 4123(λ )

1, 2

)f

y

(

)

= 1 − 0, 4123 ⋅ 0, 3511, 2 355 = 313, 3 N/mm 2

The slenderness is obtained for buckling by inserting the buckling stress value into the formula (2.55):

52


λ=

Chapter 2

6000 313, 3 Lc σ u = = 0, 654 π ⋅ i E π ⋅ 112, 8 210000

The buckling resistance of the compression member is derived from formulae (2.52 - 2.54): N b. Rd = χ ⋅ A

σu 313, 3 = 0, 753 ⋅ 5009 ⋅ = 1074, 2 kN γ M1 1, 1

The equivalent uniform moment factor for constant moment is (Table 2.9):

βMψ = 1,1 Parametes µ and k are derived from the formula (2.58):

µ y = µ z = λ ⋅ ( 2β Mψ − 4) = 0, 654 ⋅ ( 2 ⋅ 1, 1 − 4) = −1, 177 k y = kz

= 1-

500000 µ ⋅ N Sd = 1 + 1, 177 = 1, 498 < 1, 5 χ y ⋅ A ⋅σu 0, 753 ⋅ 5009 ⋅ 313, 3

The bending resistance for a hollow section with dimensions 323,9 x 5 was calculated in example 1e: Mc. Rd = Wel

σu = 112, 6 kNm γ M1

The torsional buckling resistance is taken from example 5: Mt .b. Rd = τ ba. Rd

Wt = 112, 3 kNm γ M1

The resistance values calculated can be inserted in the interaction expression (2.60). Allowing for the effect of torsion, the following result is obtained: k y ⋅ M y.Sd k z ⋅ M z.Sd N Sd Mt .Sd + + + N b.Rd Mc.Rd Mc.Rd Mt .b. Rd 1, 498 ⋅ 18 1, 498 ⋅ 18 500 6 = + + + = 0, 998 < 1, 0 OK! 1074, 2 112,6 112,6 112, 3

53


Chapter 2

2.10

Concentrated load resistance of hollow sections

When the hollow section is subjected to concentrated loads, the local resistance of the web must be checked, as the walls of the hollow section are relatively thin. The larger the area into which the concentrated load can be distributed, the greater the resistance. The width ss of the effective supporting surface of the area subjected to the load is directly proportional to the thickness of the material between the concentrated load and the web. Regarding the effect of the concentrated load, two cases can be distinguished [1]: a) Concentrated load acting from one side only b) Concentrated load acting from both sides

a) FSd FSd

b) FSd

2.10.1

Concentrated load acting from one side only

With a concentrated load acting from one side only, the resistance of the one web of hollow section is the smallest of the following [1]:

Ry. Rd = (ss + s y )t

Ra. Rd = 0, 5t 2

fy

( 2.61)

γ M1

 1 + 3s s   h - 3t  E ⋅ fy γ M1

( 2.62)

where

ss

sy b

the width of the effective supporting surface determined by assuming that the concentrated load is distributed in an 45° angle along continuous metal planes

 σ f . Ed  = 2 b⋅t 1−    fy 

2

is the lesser of the flange width and 25 t

σ f . Ed is the bending stress in the flange h

is the web height

ss ≤ 0, 2 (h − 3t )

54


load from hollow section

y;y;

load from I profile

Chapter 2

Detail 1 tw

Detail 2

t

FSd

FSd

FSd

ri

FSd

FSd

1 1:

1 1:

1: 1

tf

t

ri

Fillet weld

ss

ss

;y Figure 2.3 Width of effective supporting surface

The width of the effective supporting surface is calculated by the following formula, when the corner rounding of the uppermost hollow section is filled by the weld (Figure 2.3, Detail 2):

ss = 2t + ( 2 − 2 ) ⋅ ri

( 2.63)

where

t ri

is the wall thickness of the uppermost hollow section (Figure 2.3, Detail 2) is the internal corner radius of the uppermost hollow section (Figure 2.3, Detail 2)

Alternatively, when load is transmitted from the I section (Figure 2.3, Detail 1):

ss = t w + 2t f + 2( 2 − 2 ) ⋅ ri

(hot rolled I section)

( 2.64)

ss = t w + 2t f + 2 2 ⋅ ab

(welded I section)

( 2.65)

where

tw tf ri ab

is the web thickness of the I section is the flange thickness of the I section is the internal corner radius of the hot rolled I section is the throat thickness of a welded I-section

Additionally, when bending moment is present the interaction expression must be checked [1]:

FSd M + Sd ≤ 1, 5 Ra.Rd Mc.Rd FSd ≤ 1, 0 Ra.Rd M Sd ≤ 1, 0 Mc.Rd

( 2.66 ) ( 2.67 ) ( 2.68 )

55


Chapter 2

2.10.2

Concentrated load acting from both sides

With a concentrated load acting from both sides, the resistance of one web of the hollow section for one web is the smallest of the following [1]:

Ry.Rd = (ss + s y )t Rb.Rd = χ ⋅ beff

fy

( 2.69)

γ M1 fy ⋅t γ M1

( 2.70)

where

Rb.Rd χ beff

is the buckling resistance of the compression member formed by the web is the buckling reduction factor in buckling class c is the effective width of the web (Figure 2.4)

Formula (2.70) determines the compression resistance of the hollow section web by treating the web as a compression member whose width is beff and height is h - 3t. The buckling length is the web height h - 3t. The buckling load is calculated using the formulae (2.52) - (2.56) presented in section 2.8.1. Depending on the location of the concentrated load, the effective width of the web is calculated as follows [1]: a) concentrated load on the hollow section span

beff =

h 2 + ss2

b) concentrated load close to the end of the hollow section

beff = 0, 5 ⋅ h 2 + ss2 + a + 0, 5ss ≤ h 2 + ss2 Ss

a)

a

h

beff

Ss

b) h

beff beff

Figure 2.4 Effective web width 56


Chapter 2

yy;;y;

Example 12 Calculate the concentrated load resistance of the member shown in the adjacent figure using hollow sections with dimensions 200 x 200 x 10 and 100 x 100 x 5. The steel grade used is S355J2H. The resistance is derived from the formulae (2.61) and (2.62), as the concentrated load in the joint acts from one side only. Flange plates (t = 10 mm) increase the width of the effective supporting surface in proportion to their thickness. FSd

Ss

Ss

0,5FSd

0,5FSd

100x100x5

y;y;

FSd

Ss

Ss

200x200x10

For a hollow section with dimensions 200 x 200 x 10, the following result is obtained (concentrated load acting on both webs): Ry. Rd = 2(ss + s y )t

fy

γ M1

(

)

= 2 ⋅ 2 ⋅ 5 + ( 2 − 2 ) 5 + 2 ⋅ 10 + 2 ⋅ 10 + 2 200 ⋅ 10 ⋅ 10 ⋅

Ra. Rd = 2 ⋅ 0, 5t 2

355 = 918, 9 kN 1, 1

 1 + 3s s   h − 3t  E ⋅ fy ⋅ γ M1

= 1 ⋅ 10 2 210000 ⋅ 355 ⋅

(1 + 3 ⋅ 0, 2) 1, 1

 ss  ≤ 0, 2   (h - 3t ) 

= 1255, 9 kN

The resistance of the hollow section with dimensions 200 x 200 x 10 is the smallest of the values calculated above, that is, RRd = 918,9 kN. For the hollow section with dimensions 100 x 100 x 5, the following result is obtained (concentrated load is distributed on webs): Ry. Rd = 2 ⋅ (ss + s y )t

fy

γ M1

(

)

= 2 ⋅ 2 ⋅ 10 + ( 2 − 2 )15 + 2 ⋅ 10 + 2 ⋅ 10 + 2 100 ⋅ 5 ⋅ 5 ⋅

Ra. Rd = 2 ⋅ 0, 5t 2

355 = 366, 3 kN 1, 1

 1 + 3s s   h − 3t  = 5 2 ⋅ 210000 ⋅ 355 ⋅ (1 + 3 ⋅ 0, 2) = 314, 0 kN E ⋅ fy ⋅ 1, 1 γ M1 57

Chapter 2


The resistance of the hollow section with dimensions 100 x 100 x 5 is the smallest of the values calculated above, that is, RRd = 314 kN. The concentrated load resistance of the entire joint is determined by the buckling of the web of the hollow section with dimensions 100 x 100 x 5. Thus, the greatest allowed concentrated load value affecting the joint is FSd = 314,0 kN.

2.11

References

[1]

ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993)

[2]

ECCS: Technical Committee 8- Structural stability- Technical working group 8.4- Stability of shells: Buckling of steel shells, European recommendations, 4th Edition, 1988

[3]

CIDECT: Structural stability of hollow sections, Verlag TÜV Rheinland GmbH, Köln 1992

[4]

CIDECT: Design guide for structural hollow sections in mechanical applications, Verlag TÜV Rheinland GmbH, Köln 1995

[5]

ENV 1991-2-1:Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat. Osa 2-1: Rakenteiden kuormat: Tiheydet, oma paino ja hyötykuormat, 1995 (ENV 1991-2-1:Eurocode 1: Basis of design and actions on structures. Part 2-1: Actions on structures. Densities, self-weight and imposed loads, 1995)

[6]

EN 10219-2: Kylmämuovatut hitsatut seostamattomat rakenne- ja hienoraerakenneteräsputkipalkit. Osa 2: Toleranssit, mitat ja poikkileikkaussuureet, 1997 (EN 10219-2: Cold formed welded structural hollow sections of non-alloy and fine grain steels. Part 2: Tolerances, dimensions and sectional properties, 1997)

[7]

CIDECT: Research project No 2R-2-16: Buckling behaviour of a new generation of cold formed hollow sections, Draft final report-2R-2-16 final, Aachen 1996

[8]

EN 10219-1: Kylmämuovatut hitsatut seostamattomat rakenne- ja hienoraerakenneteräsputkipalkit. Osa 1: Tekniset toimitusehdot, 1997 (EN 10219-2: Cold formed welded structural hollow sections of non-alloy and fine grain steels. Part 1: Technical delivery requirements, 1997)

58


Chapter 3

3

DESIGN OF JOINTS IN HOLLOW SECTION STRUCTURES

3.1

Design of welded joints in lattice structures

Joints in lattice structure are usually assumed to be nominally pinned, and brace members are designed for axial load only. Depending on the dimensions of the chord and brace members, the effect of joint rigidity can be accounted for, by reducing the buckling length of the brace member. The transverse loads on the chord span between the braces introduce bending moments, and the chord must therefore be designed for compression and bending. In terms of compression resistance, a hollow section with thin walls is the most practical solution. However, when considering the resistance of the joint , a thin-walled, wide chord is not as good as a thick-walled, narrow chord. The design formulae for lattice structure joints are partially based on test results. When using the formulae, it must be ascertained that the hollow sections meet the validity conditions given in the tables. Appendix 9.3 includes formulae for assessing lattice structure joints for different hollow section types. Table 3.1 shows different types of uniplaner lattice structure joints. Joint in multiplaner frames are dealt with in references [2] and [3]. The figures also feature the following parameters essential for the joint design: is eccentricity is gap is overlap

g g

-e

e

e g q

q

Eccentricity value is taken as positive when the neutral axes of the brace members intersect the chord below the centre of gravity (Table 3.1). Eccentricity is negative when the intersection is located above the chord’s centre of gravity (Table 3.1). The joint gap refers to the space between the brace members. The joint is overlapped when the brace members are partially or completely overlaid by each other. The overlap can also be expressed as a negative-value. Eccentricity and gap are interrelated in the following manner [2]:

h sin(θ 1 + θ 2 ) h1 h2 g = e + 0  − −  2  sin θ 1 sin θ 2 2 sin θ 1 2 sin θ 2

(3.1)

h2  h1  sin θ 1 sin θ 2 h0 e= + + g −  2 sin θ 1 2 sin θ 2  sin(θ 1 + θ 2 ) 2

( 3.2)

where

θi hi h0

is the smaller of the angles between the brace member and the chord is the height of the brace member is the height of the chord

59


Chapter 3 Table 3.1

Joint types in lattice structures

Joint type

Gap joint

Overlap joint

N

g θ1

θ2

θ2

e

θ1 -e

q

g

K θ1

θ1

g1

θ3

θ3

θ1

θ2

e

θ1

q

g2 e=0

KT

θ2 -e

e

θ2

T

q1

θ2 q2

θ1

θ1

X

Y θ1

Table 3.2 presents different failure modes of hollow section lattice structures. The governing failure mode depends on the dimensions of the chord and brace members, and on the joint geometry.

60


Table 3.2

Chapter 3

Failure modes of lattice structure joints

Failure mode

Structure in which the failure mode is possible Thin-walled chord, brace member narrower than chord

Flexural failure of the chord face

Punching shear failure of the chord face

Thin-walled chord with great b0, brace member slightly narrower than chord

Tension failure of the bracing member or weld failure

Thick-walled brace member and thin-walled chord

Local buckling of the bracing member

Thin-walled brace member with a great bi or hi

Overall shear failure of the chord

Thin-walled chord with small h0

Local buckling of the chord walls

Thin-walled high chord of equal width as the brace member

Local buckling of the chord face

Thin-walled chord with great b0

61

Chapter 3

3.1.1


Joints of circular, square or rectangular brace members to square or rectangular chords

Before calculating the resistance of the joint, members must be designed according to their loads (chapter 2). Joints are usually assumed pinned, so the brace members are designed for axial force only. When calculating the resistance of the joint, the moments due to the eccentricity of the joint need not be taken into account, if the eccentricity is between

−0, 55h0 ≤ e ≤ 0, 25h0

(3.3)

where

h0

is the height of the chord

However, when designing the chord, the moments due to the joint eccentricity must be taken into account. The chord face resistance is affected by the axial force and the bending moment. This function is determined by using the parameter n [1]:

n=

γ M 0 ⋅ γ Mj σ 0. Ed γ M 0 ⋅ γ Mj  N 0.Sd M0.Sd  = +  fy0 1, 1 1, 1  A0 ⋅ f y 0 Wel ⋅ f y 0 

(3.4)

where

σ 0. Ed N 0.Sd M0.Sd fy0

is the greatest compression stress in the flange on the side of the joint is the axial force of the chord is the bending moment of the chord is the yield strength of the chord

In examples 13-17, the resistance of the joint is determined by using Tables 9.3.1, 9.3.2 and 9.3.3 in Appendix 9.3. When using the tables, it must be ascertained that the lattice members and the joint geometry meet the validity conditions presented in the tables. The principle is to calculate the resistance of the joint for different failure modes and select the smallest value as the final resistance of the joint. The steel designation used in all examples is S355J2H. When calculating the resistance of the hollow section lattice structure joints, the partial safety factor γMj, is used with the value of 1.1. The validity condition for the joint are met in examples 13-27, but their checking is not presented in the examples. In the tables in Appendix 9.3 and examples 13-27, it is assumed that the value of the partial safety factor γM0 of the material is 1.1. The tables in Appendix 9.3 apply for steel grades with the yield strength value of 355 N/mm2 or smaller.

62


Chapter 3

Example 13

θ

h0

Brace member: 100 x 100 x 5, A1 = 1836 mm2 ⇒ NRd = 1836 · 355/ 1,1 = 592,5 kN Chord: 200 x 200 x 8, A0 = 5924 mm2 N0.Sd = 936,4 kN (compression) N1.Sd = 590 kN (tension) θ = 45° β = b1/ b0 = 100/ 200 = 0,5 η = h1/ b0 = 100/ 200 = 0,5

h1

;; ; ;

The joint geometry and loading are as follows:

b1

N1.Sd

A Y joint with a tension brace member (Table 9.3.1).

N0.Sd

t1

t0

b0

The chord axial force N0.Sd influences the resistance of the joint in the form of the term kn: n=

γ M 0 ⋅ γ Mj N 0.Sd 1, 1 936400 = = 0, 490 A0 ⋅ f y 1, 1 1 5924 ⋅ 355

kn = 1, 3 −

0, 4n 0, 4 ⋅ 0, 490 = 1, 3 − = 0, 908 β 0, 5

Chord face yield Since β = 0,5 < 0,85 the chord face resistance must be checked: N1. Rd

f y ⋅ t 0 2  2η 1, 1 = + 4 1 − β  kn  γ Mj ⋅ γ M 0 (1 − β ) sin θ  sin θ 355 ⋅ 8 2  2 ⋅ 0, 5 + 4 1 − 0, 5  0, 908 1 = 225, 1 kN < N = Sd  (1 − 0, 5) sin 45  sin 45 1, 1

Resistance of the joint The resistance of the joint is thus N1.Rd = 225,1 kN, which is remarkably less than the brace member axial force N1.Sd = 590 kN. A larger hollow section must be selected as the brace member or the chord face must be reinforced to obtain a sufficient resistance of the joint.

63


n=

b1

N1.Sd

;;; ;

Example 14 A T joint with a compression brace member (Table 9.3.1). The joint geometry and loading are the following: Brace member: 100 x 100 x 5, A1 = 1836 mm2 ⇒ NRd = 1836 · 355/1,1 = 592,4 kN Chord: 100 x 100 x 6, A0 = 2163 mm2 N0.Sd = 400 kN (compression) N1.Sd = 350 kN (compression) θ = 90° β = b1/ b0 = 100/ 100= 1,0

h1

h0

Chapter 3

θ

t1

N0.Sd

t0

b0

γ M 0 ⋅ γ Mj N 0.Sd 1, 1 400000 = = 0, 573 A0 ⋅ f y 1, 1 1 2163 ⋅ 355

kn = 1, 3 −

0, 4n 0, 4 ⋅ 0, 573 = 1, 3 − = 1, 07 > 1, 0 ⇒ kn = 1 β 1, 0

Note that the term kn is not relevant in this example, since the chord face yield is not the governing failure mode (β > 0,85). Chord web buckling The chord web buckling may be a relevant failure mode for the design, since β is 1,0. First, determine the buckling stress using buckling curve c: fy h  1 100 355 1 λ = 3, 46  0 − 2 = 3, 46  − 2 = 0, 664  6  210000(sin 90) π  t0  E(sinθ ) π

χ = 0, 747 N mm 2 Now we can calculate the chord web resistance: fb = 0, 747 ⋅ 355 = 265, 2

N1. Rd =

1 265, 2 ⋅ 6  2 ⋅ 100 1 fb ⋅ t 0  2h1 + 10t 0  = + 10 ⋅ 6 = 376, 1 kN  γ Mj ⋅ γ M 0  1, 1 sin θ  sin θ sin 90  sin 90

Brace member failure Next, determine the effective width of the brace member: 10b1 ⋅ t 02 10 ⋅ 100 ⋅ 6 2 beff = = = 72 ≤ b1 b0 ⋅ t1 100 ⋅ 5 1 1 N1. Rd = f y ⋅ t1 ( 2h1 − 4t1 + 2beff ) = 522, 8 kN = 355 ⋅ 5( 2 ⋅ 100 − 4 ⋅ 5 + 2 ⋅ 72) 1, 1 γ Mj ⋅ γ M 0 Resistance of the joint The chord web resistance determines the resistance of the entire joint, which then is: N1.Rd = 376,1 kN > N1.Sd OK ! 64


Example 15 An X joint with compression brace members (Table 9.3.1). The joint geometry and loading are as follows:

θ

h0

Brace members: 180 x 180 x 6, A1 = 4083 mm2 ⇒ NRd = 4083 · 355/ 1,1 = 1318 kN Chord: 200 x 200 x 8, A0 = 5924 mm2 N0.Sd = 620 kN (tension) ⇒ kn =1 N1.Sd = 1000 kN (compression) θ = 30° β = b1 / b0 = 180/ 200 = 0,90 η = h1 / b0 = 180/ 200 = 0,90 γ = 0,5b0 / t0 = 0,5 · 200/ 8 = 12,5

b1

N1.Sd

;; ;

h1

Chapter 3

t1

N0.Sd

t0

b0

h1

Chord face punching shear The chord face punching shear must be checked, since 0,85 < β < 1-(1/γ)= 0,92: bep =

10t 0 ⋅ b1 10 ⋅ 8 ⋅ 180 ≤ b1 = = 72 ≤ 180 b0 200

N1. Rd =

f y ⋅ t 0  2h1 1 355 ⋅ 8  2 ⋅ 180 1 + b1 + bep  = + 180 + 72 = 2897 , 8 kN  γ Mj ⋅ γ M 0  1, 1 3 sin θ  sin θ 3 sin 30  sin 30

Chord face yield and chord web buckling The chord resistance must be determined for both chord face and web, since 0,85 < β < 1,0. The resistance of the joint is calculated for the chord face when β = 0,85 and for the chord web when β = 1,0. Then, the resistance is determined by linear interpolation when β = 0,9: a) β = 0,85 N1. Rd

f y ⋅ t 02  2η 1, 1 = + 4 1 − β  kn  γ Mj ⋅ γ M 0 (1 − β ) sin θ  sin θ

355 ⋅ 8 2  2 ⋅ 0, 9 + 4 1 − 0, 85  1 = 1418, 1 kN > N ⇒ N = 1. Rd = 1318 kN Rd  1, 1 (1 − 0, 85) sin 30  sin 30

65


Chapter 3

b) β = 1,0 fy h  200 355 1 1 λ = 3, 46  0 − 2 = 3, 46  − 2 = 1, 473  8  210000(sin 30) π  t0  E(sinθ ) π

χ = 0, 323 N mm 2 f ⋅ t 2h 1 1 45, 87 ⋅ 8  2 ⋅ 180 + 10 ⋅ 8 = 533, 8 kN = b 0  1 + 10t 0  =  1, 1  γ Mj ⋅ γ M 0 sin θ  sin θ sin 30  sin 30

fb = 0, 323 ⋅ 0, 8 ⋅ sin 30 ⋅ 355 = 45, 87 N1. Rd

Now, determine the chord resistance by interpolation using the values calculated above: N1.Rd = 533,8+ (1318 – 533,8)(1– 0,9)/ 0,15 = 1056,6 kN Chord shear Next, check the chord shear resistance: N1. Rd =

f y ⋅ Av 1 355 ⋅ 3200 1 = = 1192, 5 kN 3 sin θ γ Mj ⋅ γ M 0 3 sin 30 1, 1

Brace member failure Also the effective width of the brace member must be checked: 10b1 ⋅ t 02 10 ⋅ 180 ⋅ 8 2 beff = = = 96 mm < b1 200 ⋅ 6 b0 ⋅ t1 1 1 = 1022, 4 kN N1. Rd = f y ⋅ t1 ( 2h1 − 4t1 + 2beff ) = 355 ⋅ 6( 2 ⋅ 180 − 4 ⋅ 6 + 2 ⋅ 96 ) 1, 1 γ Mj ⋅ γ M 0 Resistance of joint The resistance of the joint is the smallest of the above values, that is, N1.Rd = 1022,4 kN > N1.Sd OK !

66


Example 16 A gapped K joint (Table 9.3.2). The joint geometry and loading are as follows:

n=

γ = 0, 5 ⋅

h2

N2.Sd

g

b1,2

θ2

N0.Sd

t1,2

t0

e

h0

θ1

b0

γ M 0 ⋅ γ Mj N 0.Sd 1, 1 1363600 = = 0, 713 A0 ⋅ f y 1, 1 1 5924 ⋅ 355

kn = 1, 3 −

h1

;; ;

Brace members:150 x 150 x 6, A1 = 3363 mm2 ⇒ NRd = 3363 · 355/ 1,1 = 1085 kN Chord: 200 x 200 x 8, A0 = 5924 mm2 θ1 =θ2 = 45° N0.Sd = 1363,6 kN (compression) N1.Sd = 600 kN (compression) N2.Rd = 600 kN (tension)

N1.Sd

Chapter 3

0, 4n 0, 4 ⋅ 0, 713 = 1, 3 − = 0, 920 β 0, 75

200 = 12, 5 8

β = 150 / 200= 0,75 < 1 – (1/γ)= 0,92, so the chord punching shear must be checked. The joint gap presented by the joint geometry is [2]: h sin(θ 1 + θ 2 ) h1 h2 g = e + 0  − − = 27 , 9 mm  2  sin θ 1 sin θ 2 2 sin θ 1 2 sin θ 2 e = 20 mm, e < 0,25h0 = 50 mm (eccentricity is within the limits allowed in Appendix 9.3) Determine the resistance of the joint by brace member 1 only, since the brace members are of equal size and carrying equal loads. Chord face yield First, calculate the resistance by chord face yield:

N1. Rd

m   m 2  ∑ bi + ∑ hi  f y ⋅ t0 i =1 1, 1 i =1 = 8, 9  kn γ  sin θ  2m ⋅ b0  γ Mj ⋅ γ M 0    

355 ⋅ 8 2  150 + 150  1 = 8, 9 = 634, 2 kN 0, 920 12, 5 sin 45  2 ⋅ 200  1, 1

67

Chapter 3


Chord shear Obtain the following value for the shear resistance of the entire chord: Av = (2h0 + α · b0)t0 = (2 · 200+ 0,241 · 200)8 = 3586 mm2

α=

1 1 = 0, 241 = 4 ⋅ 27 , 9 2 4g 2 1+ 1+ 2 3 ⋅ 82 3t 0 f y ⋅ Av 1, 1 355 ⋅ 3586 1 = = 944, 9 kN 3 sin θ γ Mj ⋅ γ M 0 3 sin 45 1, 1

N1. Rd =

Brace member failure The effective width of the brace member is: beff N1. Rd

10b1 ⋅ t 02 10 ⋅ 150 ⋅ 8 2 = = = 80 mm < b1 b0 ⋅ t1 200 ⋅ 6 1, 1 = f y ⋅ t1 ( 2h1 − 4t1 + b1 + beff ) γ Mj ⋅ γ M 0 = 355 ⋅ 6( 2 ⋅ 150 − 4 ⋅ 6 + 150 + 80)

1 = 979, 8 kN 1, 1

Chord punching shear In this case, the chord punching shear resistance must also be taken into account: bep = 10t0 · b1 / b0 = 10 · 8 · 150/ 200 = 60 ≤ 150 N1. Rd = =

f y ⋅ t 0  2h1 1, 1 + b1 + bep   γ Mj ⋅ γ M 0 3 sin θ  sin θ 355 ⋅ 8  2 ⋅ 150 1 + 150 + 60 = 1337 , 1 kN  1, 1 3 sin 45  sin 45

Resistance of the joint The chord face yield determines the resistance of the joint, which then is: N1.Rd = 634,2 kN > N1.Sd OK!

68


In a gapped KT joint, it must also be checked that the sum of the brace member vertical force components is less than the resistance of the joint. The vertical rigidity in the chord face is poor, so this condition is relevant. Four different load modes can be suggested. These are presented in Figure 3.1. In 3.1c, the lattice joint is subjected to a down-pulling point load. In 3.1d, an intermediate support is placed at the corner. In the cases presented in the figure, the conditions for the resistance of the joint are the following [2]:

a) 1

3

2

θ3

c)

3

1

b)

3

1

θ3

θ2

θ1

θ1

θ1

2

θ3

θ2

Chapter 3

d)

3

1

θ3

2

θ2

2

θ2

θ1

θ4 4

θ4 4

Figure 3.1 A gapped KT joint

a)

N 2. Rd sin θ 2 ≥ N1.Sd sin θ 1 + N 3.Sd sin θ 3

( 3.5)

b)

N1. Rd sin θ 1 ≥ N 2.Sd sin θ 2 + N 3.Sd sin θ 3

( 3.6 )

c)

N 2. Rd sin θ 2 ≥ N1.Sd sin θ 1 + N 3.Sd sin θ 3 + N 4.Sd sin θ 4

( 3.7 )

d)

N1. Rd sin θ 1 ≥ N 2.Sd sin θ 2 + N 3.Sd sin θ 3 + N 4.Sd sin θ 4

( 3.8 )

69


Chapter 3

N2.Sd

N1.Sd

b1,2

θ 2 t1,2

N0.Sd

-e

h0

Brace members: 140 x 140 x 6, A1 = 3123 mm2 ⇒ NRd = 3123 · 355/1,1 = 1008 kN Chord: 180 x 180 x 8, A0 = 5284 mm2 θ1 =θ2 = 60° N0.Sd = 1500 kN (tension) N1.Sd = 800 kN (compression) N2.Sd = 800 kN (tension)

h2

θ1

h1

;; ;

Example 17a A gapped K joint (Table 9.3.3). The joint geometry and loading are as follows:

t0

q

b0

The joint overlap expressed by the joint geometry is as follows [2]:

h sin(θ 1 + θ 2 ) h1 h2   q = −  e + 0  − − = 92, 4 mm (overlap scale) 2  sin θ 1 ⋅ sin θ 2 2 sin θ 1 2 sin θ 2  

e = –30 mm > –0,55h0 = –99 mm (joint eccentricity is within the limits allowed in Appendix 9.3) The relative value of the overlap λov is: λov = q· sin(θ1)/ h1 = 92,4 sin(60)/ 140 = 0,57 Brace member failure Now 0,5 ≤ λov < 0,8, so the following value is obtained for the effective width: beff

10b1 ⋅ t 02 10 ⋅ 140 ⋅ 8 2 = = = 83 mm ≤ b1 b0 ⋅ t1 180 ⋅ 6

10b1 ⋅ t 22 10 ⋅ 140 ⋅ 6 2 be( ov ) = = = 60 mm ≤ b1 b2 ⋅ t1 140 ⋅ 6 1, 1 N1. Rd = f y ⋅ t1 2h1 − 4t1 + be + be( ov ) γ Mj ⋅ γ M 0

(

)

= 355 ⋅ 6( 2 ⋅ 140 − 4 ⋅ 6 + 83 + 60)

1 = 772, 5 kN 1, 1

Resistance of the joint The resistance of the joint is not sufficient, since N1.Rd = 772,5 kN is less than N1.Sd. With an overlap (q) of 130 mm, a sufficient strength of the joint is obtained.

70


Example 17b A lower corner joint in a lattice structure (Table 9.3.3). The resistance of the lattice’s lower corner can be determined using the formulae for overlapped joints which consider the lower chord as continuous (Figure 3.2) [2]. The resistance of the joint can be determined for the lower corner with the following joint members:

Chapter 3

N1.Sd

θ2

θ1

Brace member: 100 x 100 x 5, A1 = 1836 mm2 Figure 3.2a A lower corner ⇒ NRd = 1836 · 355/ 1,1 = 592,4 kN joint in a lattice structure 2 Chord: 150 x 150 x 6, A0 = 3360 mm θ1 = 90° θ2 = 45° N1.Sd = 500 kN (compression) The joint must be designed in such a manner that eccentricity e = 0. N1.Sd The following overlap value is obtained: h sin(θ 1 + θ 2 ) h1 h2   q = −  e + 0  − − = 81, 1 mm 2  sin θ 1 ⋅ sin θ 2 2 sin θ 1 2 sin θ 2   sin(θ 1 ) sin( 90) λ ov = q = 81, 1 = 0, 81 θ1 h1 100 Brace member failure Now λov > 0,8, so the following value for the effective width is obtained: be( ov ) N1. Rd

)

= 355 ⋅ 5( 2 ⋅ 100 − 4 ⋅ 5 + 100 + 48)

q

Figure 3.2b The calculation model

10b1 ⋅ t 22 10 ⋅ 100 ⋅ 6 2 = = = 48 mm ≤ b1 150 ⋅ 5 b2 ⋅ t1 1, 1 = f y ⋅ t1 2h1 − 4t1 + be + be( ov ) γ Mj ⋅ γ M 0

(

θ2

1 = 529, 3 kN 1, 1

Resistance of the joint Thus, the resistance of the joint is N1.Rd = 529,3 > N1.Sd OK !

71

Chapter 3

3.1.2


Joints of circular brace members to circular chords

The joint of a circular brace member to a circular chord is designed according to Tables 9.3.4, 9.3.5 and 9.3.6 presented in Appendix 9.3. Otherwise, lattice structures constructed of circular hollow sections are designed by the same principles as square and rectangular hollow sections, presented in section 3.1.1. When determining the resistance of the joint, the moments due to joint eccentricity need not be taken into account, if the eccentricity is between:

−0, 55d0 ≤ e ≤ 0, 25d0

(3.9)

where

d0

is the diameter of the chord

However, when designing the chord, the moments due to joint eccentricity need to be taken into account. The chord axial force and bending moment influence the chord face resistance. This function is determined by parameter np.

np =

γ M 0 ⋅ γ Mj σ p. Ed γ M 0 ⋅ γ Mj  N p.Sd M0.Sd  = + A ⋅f  fy0 1, 1 1, 1  0 y 0 Wel ⋅ f y 0 

(3.10)

where

σ p. Ed

is the chord compression stress due to force Np.Sd and bending moment M0.Sd

N p.Sd = N 0.Sd − ΣNi.Sd cos(θ i ) N 0.Sd is the axial force of the chord Ni.Sd is the axial force of the brace member is the angle between the brace member and the chord θi M0.Sd is the bending moment of the chord fy0 is the yield resistance of the chord

72


Example 18 A T or a Y joint (Table 9.3.4). The joint geometry and loading are as follows:

N1.Sd

;;

; ;;;; ;;;

Brace member: 168,3 x 5, A1 = 2565 mm2 ⇒ NRd = 2565 · 355/ 1,1 = 827,8 kN Chord: 219,1 x 10, A0 = 6569 mm2 θ = 90° N1.Sd = 450 kN (compression) Np.Sd = 1018,2 kN (compression)

Chapter 3

d1

θ

d0

Np.Sd

t1 t0

The axial force of the chord Np.Sd influences the resistance of the joint in the form of the term kp. np =

γ M 0 ⋅ γ Mj N p.Sd 1, 1 1018200 = = 0, 48 f y ⋅ A0 1, 1 1 355 ⋅ 6569

(

)

(

)

k p = 1, 0 − 0, 3 n p + n p2 = 1, 0 − 0, 3 0, 48 + 0, 48 2 = 0, 79 168, 3 = 0, 77 219, 1 219, 1 = 11, 0 γ = 20

β=

Chord face yield The resistance of the joint determined by the chord face yield is: N1. Rd

f y ⋅ t 02 1, 1 2, 8 + 14, 2β 2 γ 0.2 k p = sin θ γ Mj ⋅ γ M 0

(

)

355 ⋅ 10 2 1 = 462, 1 kN 2, 8 + 14, 2 ⋅ 0, 77 2 11, 0 0,2 ⋅ 0, 79 = 1, 1 sin 90

(

)

Chord punching shear The chord punching shear resistance is given by: N1. Rd =

f y ⋅ t 0 ⋅ π ⋅ d1  1 + sin θ  1, 1 355 ⋅ 10 ⋅ π ⋅ 168, 3  1 + sin 90  1 = = 985, 2 kN 2  2 sin θ  γ Mj ⋅ γ M 0  2 sin 2 90  1, 1 3 3

Resistance of the joint The resistance of the joint is the smallest of the above values N1.Rd = 462,1 kN > N1.Sd OK !

73

Chapter 3


Example 19 An X joint (Table 9.3.4). The joint geometry and loading are as follows:

N1.Sd

;;

; ; ; ; ; ;;;; Np.Sd

d0

Chord: 219,1 x 10, A0 = 6569 mm2 θ = 90° N1.Sd = 450 kN Np.Sd = 1018,2 kN (compression) 1, 1 1018200 = 0, 48 1 355 ⋅ 6569 kp = 1,0 – 0,3(np+np2) = 1,0– 0,3(0,48+ 0,482) = 0,79 β = 193,7/ 219,1 = 0,884 ⇒ np =

θ

d1

Brace members: 193,7 x 6, A1 = 3538 mm2 ⇒ NRd = 3538 · 355/ 1,1 = 1142 kN

t1 t0

d1

N1.Sd

Chord face yield The resistance of the joint determined by the chord face yield is: N1. Rd

f y ⋅ t 02  5, 2  1, 1 5, 2 1 355 ⋅ 10 2   = = = 466, 9 kN   ⋅ 0, 79   kp 1, 1 sin θ  1 − 0, 81β  γ Mj ⋅ γ M 0 sin 90  1 − 0, 81 ⋅ 0, 884 

Chord punching shear The chord punching shear, resistance is given by: N1. Rd =

f y ⋅ t 0 ⋅ π ⋅ d1  1 + sin θ  1, 1 355 ⋅ 10 ⋅ π ⋅ 193, 7  1 + sin 90  1 = = 1133, 8 kN 2  2 sin θ  γ Mj ⋅ γ M 0  2 sin 2 90  1, 1 3 3

Resistance of the joint The resistance of the joint is the smallest of the above values N1.Rd = 466,9 kN > N1.Sd OK !

74


Example 20 An overlapped K joint (Table 9.3.5). The joint geometry and loading are as follows:

d1

g

d2

θ2 t1,2 t0

;;

θ1

d0

Chord: 219,1 x 10, A0 = 6569 Brace members: 168,3 x 5 (compression member) ⇒ N1.Rd = 2565 · 355/ 1,1 = 827,8 kN 139,7 x 5 (tension member) ⇒ N2.Rd = 2116 · 355/ 1,1 = 682,9 kN θ1 =50° θ2 = 60° N1.Sd = 600kN (compression) N2.Sd = 530,7 kN (tension) Np.Sd = 636,4 kN (compression) ⇒ np =

N2.Sd

N1.Sd

;;;; ; ;; ;

mm2

Chapter 3

Np.Sd

1, 1 636400 = 0, 30 1 355 ⋅ 6569

kp = 1,0 – 0,3(np + np 2) = 1,0 – 0,3(0,30+ 0,302) = 0,88 β = (168,3+139,7)/ (2 · 219,1) = 0,703 γ = 219,1/ 20 = 11.0 g = 25 mm e = 42,6 mm [formula (3.2)] e < 0,25d0 = 54,8 mm OK ! Chord face yield The chord face yield resistance for the compression member is given by:     1, 2 0 , 024 ⋅ γ 0, 024 ⋅ 111, 2   0, 2  0, 2  = 1, 974 = 11 kg = γ  1 + 1+   25 − 1, 33   g  1 , 33 −         1 + e 2⋅10   1 + e 2t0 N1. Rd

f y ⋅ t0 2 1, 1 = (1, 8 + 10, 2β )kg ⋅ k p γ Mj ⋅ γ M 0 sin θ =

355 ⋅ 10 2 1 (1, 8 + 10, 2 ⋅ 0, 703) ⋅ 1, 974 ⋅ 0, 88 = 656, 5 kN sin 50 1, 1

For the tension member, the corresponding resistance is:  sin(θ 1 )   sin( 50)  N 2. Rd = N1. Rd  = 656, 5    = 580, 7 kN  sin(60)   sin(θ 2 ) 

75

Chapter 3


Chord punching shear The chord punching shear resistance for the compression member is given by: N1. Rd =

f y ⋅ t 0 ⋅ π ⋅ d1  1 + sin θ 1  1, 1 355 ⋅ 10 ⋅ π ⋅ 168, 3  1 + sin 50  1 = = 1482, 4 kN   2  2 sin 2 50  1, 1 3 3  2 sin θ 1  γ Mj ⋅ γ M 0

For the tension member, the corresponding resistance is: N 2. Rd =

f y ⋅ t 0 ⋅ π ⋅ d 2  1 + sin θ 2  1, 1 355 ⋅ 10 ⋅ π ⋅ 139, 7  1 + sin 60  1 = = 1017 , 3 kN   2  2 sin 2 60  1, 1 3 3  2 sin θ 2  γ Mj ⋅ γ M 0

Resistance of the joint The resistance of the joint is the smallest of the above values: Compression member: N1.Rd = 656,7 kN > N1.Sd OK ! Tension member: N2.Rd = 580,7 kN > N2.Sd OK !

Example 21 An overlapped K joint (Table 9.3.6). The joint geometry and loading are as follows:

1, 1 636400 = 0, 30 1 355 ⋅ 6569

d1

d2

t1,2

d0

;;

⇒ np =

d1,2

; ;;;; ; ;;

Brace members: 139,7 x 5, A1 = 2116 mm2 ⇒ NRd = 2116 · 355 / 1,1 = 682,9 kN Chord: 219,1 x 10, A0 = 6569 mm2 θ1 = 40° θ2 = 50° N1.Sd = 600 kN (compression) N2.Sd = 503,5 kN (tension) Np.Sd = 636,4 kN (compression)

N2.Sd

N1.Sd

θ1 Np.Sd q

kp = 1,3 – 0,3(np+np2) = 1,0 – 0,3(0,30+ 0,302) = 0,88 β = 139,7/ 219,1 = 0,64 γ = 219,1/ 20 = 11,0 q = 85 mm (overlap) e = –53,0 mm [formula (3.2)] e > – 0,55d0 = –120,5 mm OK !

76

θ2

t0


Chapter 3

Chord face yield The chord face yield for the compression member is given by:     1, 2 1, 2 ⋅ 0 024 ⋅ γ 0 024 11 , ,    = 2, 30 = 110, 2  1 + k g = γ 0, 2  1 +  −85 − 1, 33    −q  − 1, 33        2t 0  1 + e 2⋅10  1+ e N1. Rd

f y ⋅ t0 2 1, 1 = (1, 8 + 10, 2β )kg ⋅ k p γ Mj ⋅ γ M 0 sin θ 355 ⋅ 10 2 1 = (1, 8 + 10, 2 ⋅ 0, 64) ⋅ 2, 30 ⋅ 0, 88 = 846, 3 kN sin 40 1, 1

For the tension member, the corresponding resistance is: N 2. Rd = N1. Rd

sin θ 1 sin 40 = 846, 3 = 710, 1 kN sin θ 2 sin 50

Resistance of the joint The resistance of the joint expressed by the brace members is: Compression member: N1.Rd = 846,3 kN > N1.Sd OK ! Tension member: N2.Rd = 710,1 kN > N2.Sd OK ! 3.1.3

Joints of circular, square and rectangular brace members to I profile chords

The joint is designed according to the principles presented in section 3.1.1, except that now Tables 9.3.7 and 9.3.8 in Appendix 9.3 are used.

3.2

Welded frameworks

Members in welded frameworks are subjected to both bending and axial loads. Since the joint rigidity influences the scale of the joint moment and the joint moment influences the joint rigidity, the final distribution of forces and moments should be determined by iteration. A conservative method is to determine the joint moments assuming complete rigidity of the joints, and the span moments by assuming the joints θ are pinned. Appendix 9.4 contains instructions for M b1/b0= 1 estimating joint rigidity when using square and M rectangular hollow sections. b1

;

Figure 3.3 presents the moment-rotation curve for welded joints. The curve slope represents the joint rigidity. This depends on the relation between the hollow section width and the column width, and on the wall thickness of joint elements. The greater the b1/b0 relation and the wall thickness, the stiffer the joint. 77

b1/b0< 0,85

b0

θ Figure 3.3 Rigidity of frame joints

Chapter 3

3.2.1


Joints of square and rectangular hollow sections subjected to bending

Table 9.3.9 in Appendix 9.3 includes the formulae for determining the bending resistance of square and rectangular hollow sections for loads both parallel and perpendicular to the chord axis. The combined effect of the axial force and bending moment are accounted for by the interaction expression [1]:

N1.Sd Mip.1.Sd Mop.1.Sd + + ≤1 N1. Rd Mip.1. Rd Mop.1. Rd

(3.11)

where

N1.Sd is the axial force of the brace member Mip.1.Sd is the bending moment parallel to the plane of the frame Mop.1.Sd is the bending moment perpendicular to the plane of the frame Moreover, the resistance of the joint to axial force (section 3.1) and to bending moment (section 3.2) must be checked separately.

Mip.1.Sd b1

N1.Sd

;;; ;

Example 22 A compression and bending T joint (Table 9.3.9). The joint members and loading are the following: Chord: 150 x 150 x 6 NRd = 3363 · 355/ 1,1 = 1085 kN MRd = 180 · 355/ 1,1 = 58,1 kNm Brace member: 150 x 150 x 6 NRd = 3363 · 355/ 1,1 = 1085 kN MRd = 180 · 355/ 1,1 = 58,1 kNm Loads: Mip.1.Sd = 15 kNm N1.Sd = 150 kN β = b1/ b0 = 150/ 150 = 1,0

h1

θ t1

h0

t0

b0

Brace member failure Since β =1, the effective width of the brace member is calculated first (Table 9.3.9): beff

10b1 ⋅ t 02 10 ⋅ 150 ⋅ 6 2 = = = 60 mm ≤ b1 b0 ⋅ t1 150 ⋅ 6

beff    1  Mip.1. Rd = f y Wpl.1 −  1 −  b1 ⋅ h1 ⋅ t1   b1    γ Mj 60    1 = 355 180000 −  1 − 150 ⋅ 150 ⋅ 6  = 32 kNm  150    1, 1 78


Chapter 3

Chord web yield The bending resistance determined by the chord web yield is (Table 9.3.9): Mip.1. Rd = 0, 5 f yk ⋅ t 0 (h1 + 5t 0 )

2

1, 1 1 = 31, 4 kNm = 0, 5 ⋅ 355 ⋅ 6 ⋅ (150 + 5 ⋅ 6 ) 2 1, 1 γ Mj ⋅ γ M 0

T joint, fyk = fy Bending resistance of the joint The bending resistance of the joint is the smallest of the above values. Mip.1.Rd = 31,4 kNm > MSd The brace member resistance to compression is determined according to the guidance in section 3.1.1. Since the brace member and chord are of equal width, the brace member failure and the chord face buckling due to axial force need to be checked: Brace member failure (Table 9.3.1) beff N1. Rd

10b1 ⋅ t 02 10 ⋅ 150 ⋅ 6 2 = = = 60 mm ≤ b1 150 ⋅ 6 b0 ⋅ t1 1, 1 1 = 766, 8 kN = f y ⋅ t1 ( 2h1 − 4t1 + 2beff ) = 355 ⋅ 6 ⋅ ( 2 ⋅ 150 − 4 ⋅ 6 + 2 ⋅ 60) 1, 1 γ Mj ⋅ γ M 0

Chord face buckling (Table 9.3.1) fy h  1 150 355 1 λ = 3, 46  0 − 2 = 3, 46  − 2 = 1, 041  6  210000(sin 90) π  t0  E(sin θ ) π

χ = 0, 516 fb = 0, 516 ⋅ 355 = 183, 3 N1. Rd

N mm 2

= fb ⋅ t 0 ( 2h1 + 10t 0 )

1 1, 1 = 359, 7 kN = 183, 3 ⋅ 6 ⋅ ( 2 ⋅ 150 + 10 ⋅ 6 ) 1, 1 γ Mj ⋅ γ M 0

Compression resistance of the joint The compression resistance of the joint is the smallest of the above values. N1.Rd = 359,7 kN Combined load condition of the joint Now, the calculated resistance values are substituted in the interaction expression (3.11). N1.Sd Mip.1.Sd 150 15 + = + = 0, 895 < 1, 0 OK! N1. Rd Mip.1. Rd 359, 7 31, 4

79

Chapter 3

3.2.2


Circular hollow section joints subjected to bending

Table 9.3.10 in Appendix 9.3 presents the formulae for determining the moment resistance of circular tubes for loads both parallel and perpendicular to the plane of the frame. The combined effect of axial force and bending moment is allowed for by using the following interaction expression [1]: 2

Mop.1.Sd N1.Sd  Mip.1.Sd  + + ≤1  N1. Rd  Mip.1. Rd  Mop.1. Rd

(3.12)

Moreover, the resistance of the joint for the axial force (section 3.1) and the bending moment must be checked separately (section 3.2).

Mip.1.Sd

d0

; ;;

;; ; ; ; ; ; ; ;

Example 23 A compression and tension T joint. N1.Sd The joint members and loading are as follows: Chord: 219,1 x 5 d1 NRd = 3363 · 355/ 1,1 = 1085 kN MRd = 229 · 355/ 1,1 = 73,9 kNm Brace member: 219,1 x 5 NRd = 3363 · 355/ 1,1 = 1085 kN MRd = 229 · 355/ 1,1 = 73,9 kNm Loads: Mip.1.Sd = 30 kNm Np.Sd = 272,7 kN (chord) N1.Sd = 70 kN (brace member) β = d1 / d0 = 219,1/ 219,1 = 1,0 γ = 219,1/ (2 · 5)= 21,91 First, we calculate the effect of the chord axial force: (compression) N p.Sd = 272, 7 kN (puristusta)

⇒ np =

t1

Np.Sd

t0

1, 1 272700 = 0, 251 1 355 ⋅ 3363

kp = 1,0– 0,3(np + np 2) = 1,0– 0,3(0,251+ 0,2512) = 0,906 Chord face yield The bending resistance calculated by the chord face yield is (Table 9.3.10): Mip.1. Rd = 4, 85 f y ⋅ t 02 ⋅ γ 0, 5 ⋅ β ⋅ d1

kp 1, 1 sin(θ ) γ Mj ⋅ γ M 0

= 4, 85 ⋅ 355 ⋅ 5 2 ⋅ 21, 910, 5 ⋅ 1, 0 ⋅ 219, 1 ⋅

0, 906 1 = 36, 4 kNm sin( 90) 1, 1

Bending resistance of the joint Since d1 = 219,1 mm > d0 – 2t0 = 219,1– 2 · 5 = 209,1 mm, the punching shear resistance of the chord need not be taken into account in calculating the bending resistance. The bending resistance of the joint is thus Mip.1.Rd= 36,4 kNm. 80


Chapter 3

Chord face yield The compression resistance for the brace member is calculated according to the guidance presented in section 3.1.1. By the chord face yield, we obtain (Table 9.3.4): N1. Rd

f y ⋅ t0 2 1, 1 2, 8 + 14, 2β 2 γ 2 ⋅ k p = sin θ γ Mj ⋅ γ M 0

(

=

)

355 ⋅ 5 2 1 = 230, 4 kN 2, 8 + 14, 2 ⋅ 1, 0 2 ⋅ 21, 910, 2 ⋅ 0, 906 1, 1 sin 90

(

)

Chord punching shear The chord punching shear need not be calculated, since d1 = 219,1 mm > d0 – 2t0 = 209,1 mm. Compression resistance of the joint The compression resistance of the joint calculated by the chord face is N1.Rd = 230,4 kN. Now, the calculated resistance values are substituted in the interaction expression (3.12): 2

2 N1.Sd  Mip.1.Sd  70  30  + = +  = 0, 983 < 1, 0 OK! N1. Rd  Mip.1. Rd  230, 4  36, 4 

81


Chapter 3

3.3

Welded end-to-end joints of hollow sections

The resistance of hollow section end-to-end joints should be designed to be at least equal to the hollow section plastic resistance in order to utilize the entire plastic resistance of the cross section. The resistance of the welded joint is at least equal to the hollow section resistance, if the weld is a full penetration weld and the conect weld material for the parent metal is selected. The weldability of structural steels is good with all welding procedures. Preheating is necessary only if the external temperature is below 5 °C or if the hollow sections are damp. Instructions for selecting the form of the backing and grooves are given in Tables 3.3 and 3.4.

Wall thickness

α

b

c

Backing thickness

t ≤ 3 mm

0°

t

-

-

3 ≤ t ≤ 20 mm 60°

0 ≤ b ≤ 3 mm

-

-

t ≤ 20 mm

60°

0 ≤ b ≤ 4 mm

1,5 ≤ c ≤ 4 mm -

t0 = 3 mm

0°

3 ≤ b ≤ 5 mm

-

t1 =3 mm

t0

Welded end-to-end joints with equal wall thicknesses [4]

t0 = 5 mm

0°

5 ≤ b ≤ 6 mm

-

3 < t1 ≤ 5 mm

t1

Table 3.3

t0 = 6 mm

0°

6 ≤ b ≤ 8 mm

-

3 < t1 ≤ 6 mm

t0 < 20 mm

> 60°

5 ≤ b ≤ 8 mm

1 ≤ c ≤ 2,5 mm 3 < t1 ≤ 6 mm

Groove type

t

Without backing plate

b

t

α

b

b

c

t

α

With backing plate b

t1

b

c

t0

α

82


Table 3.4

Chapter 3

Welded end-to-end joints with different wall thicknesses [4]

Groove type

β -

Unlimited

60°- 80°

≤ 30°

t2 - t1 ≤ 1,5 mm

60°- 80°

-

1,5 < t2 - t1 < 3 mm

60°- 80°

-

t2 - t1 ≥ 3 mm

60°- 80°

≤ 14,036°

t1

t2

α

Difference in thicknesses α 60°- 80° t2 - t1 ≤ 0,5 t2 ≤ 3 mm

b

β

t1

t2

α

b

t2

t1

α

b

t2

t1

α

b

t2

t1

α

b

β

Root gap b is selected as in Table 3.3

83

Chapter 3


3.4

Bolted hollow section joints

3.4.1

End-to-end bolted hollow section joints

In structures utilizig hollow section structures, it is normal practice to assemble structure elements in the workshop by welding and then connect the elements on site with bolted joints. Bolted joints are quicker and easier to prepare on site than welded joints. Alternatives for endto-end bolted joints are shown in Figure 3.4.

a)

b)

c)

d)

e)

f)

g)

Figure 3.4 Alternatives for end-to-end connections

When designing a joint, it is essential to ensure that the load is as concentric as possible in relation to the cross section and that the rigidity of the joint components is uniform. In this respect, a tension joint is best constructed using the alternatives b, c, f or g. In these joint types, the tension load is transmitted more directly to the hollow section than in joints a, d or e, which also include the risk of lamellar tearing. In the flange joints d and e, a sufficient flange thickness must be selected to minimize the bolt prying forces due to flange elasticity.

84


3.4.1.1

Chapter 3

Flange plate connections

p/2 p

p

p/2

;;; ;

A flange plate connection can be designed assuming it a bi-dimensional joint of T elements, in which the bolts are placed at opposite sides of the hollow section (Figure 3.5). Due to the tension load, a plastic hinge is formed in the flanges at the bolt rows and the hollow section walls. The tension resistance of the flange can be determined using the plastic flange moment. When determining the loading on bolts, the prying force due to flange bending must be accounted for.

a

Flange plate connections for square and rectangular hollow sections

d  δ = 1−  0   p

b

NSd

t0

bred

NSd

ared

When calculating the flange plate joint resistance, the first thing to determine is factor δ for the relative net area of the bolt row [2]:

d0

Figure 3.5 Flange plate connection

(3.13)

where

d0 p

is the diameter of bolt holes is the spacing between the bolt centres

Parameter αh accounts for the effect of holes on the flange plastic moment values at the bolt location, when the tensile force of the bolts is assumed equal to their tension resistance [2]:

 K ⋅ Bt . Rd   a + 0, 5d  αh =  − 1 2   δ ( a + b + t )   tp  0

(3.14)

where

K bred b a d t0 Bt . Rd fy

4bred ⋅ γ M 0 0, 9 f y ⋅ p b − 0, 5d + t 0 (the bolt row lever arm by the plastic hinge) =

is the distance of the bolts from the hollow section edge is the distance of bolts from the flange edge is the diameter of the bolt is the thickness of the hollow section wall is the tensile resistance of bolts or the punching resistance of the flange [Eurocode 3:6.5.5.4] [1] (select the smaller value) is the yield strength of the flange 85

Chapter 3


For the flange thickness, the following minimum and maximum values are obtained [2]:

K ⋅ Nt .Sd ≤ tp ≤ 1+δ

K ⋅ Nt .Sd

(3.15)

where

Nt .Sd tp

is the axial force in one bolt is the thickness of the flange

The resistance of the joint can be determined by expressing the work done in the plastic hinges equal to the work done by the external load [2]:

N Rd

t p2 (1 + δ ⋅ α h )n = K

(3.16 )

where n

is the number of bolts

Due to the prying effect, the axial force introduced to the bolt is greater than Nt.Sd. This axial force in the bolt is expressed with the symbol Np.Sd [2]:

  b  δ ⋅α p  N p.Sd = Nt .Sd  1 +  red     ared  1 + δ ⋅ α p 

(3.17 )

where

ared

= a + 0, 5d ≤ 1, 25b + 0, 5d

αp

 K ⋅ Nt .Sd   1 = − 1 2 δ   tp 

The axial force Np.Sd must be smaller than the tension resistance of the bolt.

86


a b

bred ared

d0 = 30 mm p = 90 mm ⇒ δ = 1– (d0/ p) = 1– (30/ 90)= 0,67 b = 45 mm d = 27 mm t0 = 8 mm ⇒ bred = b– 0,5d + t0 = 45– 0,5 · 27+ 8 = 39,5 mm K = 4bred ·γM0 /(0,9fy · p) = 4· 39,5· 1,1/ (0,9 · 345 · 90) = 6,22 mm2/ kN (tp > 16 mm ⇒ fy = 345 N/mm2) a = 45 mm ⇒ ared = a+ 0,5d = 45+ 0,5 · 27 = 58,5 mm

d0

p

t0

p/2 p

The strength grade of the M 27 bolts is 10.9. The flange hole position parameters are:

p/2

;;; ;

Example 24 Calculate the tension resistance of the adjacent flange plate joint which is subjected to axial force NSd = 1086 kN. The hollow section dimensions are 120 x 120 x 8, and the steel grade used is S355J2H. The steel grade used in the flange is S355J2.

Chapter 3

NSd

NSd

The bolt force in one bolt is: Nt.Sd = NSd / 6 = 1086/ 6 = 181 kN The tension resistance of the bolt is [1]: Bt.Rd = 0,9fub · As /γMb = 0,9 · 1000 · 459/ 1,25 = 330 kN where As fub γMb

is the tension cross section of the bolt is the ultimate strength of the bolt is the partial safety factor for bolt joints (chapter 2)

Flange resistance The minimum and maximum values for the flange thickness are obtained by the formula (3.15): K ⋅ Nt .Sd ≤ t p ≤ K ⋅ Nt .Sd 1+δ 6, 22 ⋅ 181 ≤ t p ≤ 6, 22 ⋅ 181 1 + 0, 67 26 mm ≤ t p ≤ 33,6 mm Select a flange thickness of tp = 28 mm. 87

Chapter 3


The joint resistance is determined using the formula (3.16): N Rd

t p2 (1 + δ ⋅ α h )n 28 2 (1 + 0, 67 ⋅ 1, 442)6 = = = 1487 kN > N Sd OK! K 6, 22

 K ⋅ Bt . Rd   a + 0, 5d   6, 22 ⋅ 330   45 + 0, 5 ⋅ 27  − αh =  1 −1   = 1, 442 2   δ ( a + b + t )  =  28 2    , ( ) 0 67 45 45 8 + + t   0 p

Resistance of bolts The tensile resistance of the bolts is checked, with the prying effect taken into account by the formula (3.17):  K ⋅ Nt .Sd   1   6, 22 ⋅ 181   1  αp =  1 − 2   δ  =  28 2 − 1  0, 67  = 0, 651  tp    b  δ ⋅α p    39, 5  0, 67 ⋅ 0, 651  N p.Sd = Nt .Sd  1 +  red  = 181 1 +       58, 5  1 + 0, 67 ⋅ 0, 651   ared  1 + δ ⋅ α p  = 218, 1 kN < 330 kN

OK!

Resistance of welds The resistance of a fillet weld is calculated as shown in Eurocode 3, subsection 6.6.5.3 Fw. Rd =

fu ⋅ a ⋅ Lw 3 ⋅ β w ⋅ γ Mw

(3.18 )

where fu a Lw βw γ Mw

is the ultimate strength of the weaker joint component is the thickness of the weld throat is the length of the weld is the strength factor (S355 ⇒ βw = 0,9) [1] is the partial safety factor of the welded joints (chapter 2)

In the example, the hollow section is welded to the flange from all edges. In such cases, the required throat thickness is: a=

3 ⋅ β w ⋅ γ Mw ⋅ N Sd = fu ⋅ Lw

3 ⋅ 0, 9 ⋅ 1, 25 ⋅ 1086 = 9, 0 mm 490 ⋅ 480

The joint resistance is sufficient for an axial force of 1086 kN, which is also the plastic tension resistance of the 120 x 120 x 8 hollow section.

88


Chapter 3

Flange plate joint in circular hollow sections

f3 10

; ; ;

d0

e2

tp

8

e1

6

t0

4 2 0 0

0,2

0,4

0,6

0,8

1,0

d0 − t 0 d0 − t 0 + 2e1

Figure 3.6 Shape coefficient of the flange f3 [3] The required flange thickness for a circular hollow section is calculated from the following formula [3]:

2 ⋅ N Sd ⋅ γ M 0 f y ⋅ π ⋅ f3

tp ≥

(3.19)

where

f3 N Sd fy

is the shape coefficient of the flange (Figure 3.5) is the design value for the tensile force of the joint is the yield strength of the flange

For the number of bolts, we obtain the following equation [3]:

    1 1 N Sd  1 − +  f3  r1    f 3 ⋅ ln     r2    n≥ 0, 67 ⋅ Bt . Rd

(3.20)

where

r1 r2

= 0, 5d0 + 2e1 = 0, 5d0 + e1 89


Chapter 3

In-line tension joint with splice plates

An in-line tension joint with splice plates is suitable as a splice joint for the lower chord in a lattice structure. Since the load is parallel to the plates, there is no risk of lamellar tearing of the splices (Figure 3.7).

NSd

a1

a1

NSd

NSd t

The resistance of the joint is determined separately for bolts and splices. The bolts transfer the force affecting the joint by their shear resistance. The bolt’s shear resistance per shear plane is determined from the following formula, assuming the shear plane does not pass through the threaded portion of the bolt [1]:

Fv. Rd =

NSd

a2

3.4.1.2

Lv

Lv

Figure 3.7 In-line joint with splice plates

0, 6 fub ⋅ A γ Mb

(3.21)

where

fub A γ Mb

is the ultimate strength of the bolt is the cross section of the bolt is the partial safety factor of the bolt joints (chapter 2)

The tension resistance of the splice plates is calculated by taking into account both the net cross section and the bearing resistance. The resistance of the net cross section can be calculated by the same principle as that of a hollow section in tension (chapter 2). The bearing resistance of a splice plate depends on the positioning of the holes and the strength of the bolts. This relationship is illustrated by the parameter α, obtained as the minimum value from the following equation [1]:

 e1  3d  0 1  p α = min  1 −  3d 0 4  fub   fu

( 3.22)

however, α ≤ 1,0 where

e1 d0 p1 fub fu

is the distance of the bolt from the edge parallel to force is the diameter of the bolt hole is the distance between bolts parallel to force is the ultimate strength of the bolt is the ultimate strength of the splice 90


Chapter 3

The bearing resistance of a double shear joint per one bolt is determined from the equation [1]:

Fb. Rd =

2, 5 ⋅ α ⋅ fu ⋅ d ⋅ t γ Mb

(3.23)

where

d t γ Mb

is the diameter of the bolt is the thickness of the splice is the partial safety factor of the bolt joints

Addihonally, the resistance for block shear failure of the splice in the middle must be checked. The design value for block shear failure is determined from the formula [1]:

Veff . Rd =

f y ⋅ Av.eff 3γ M 0

(3.24)

In this case, the effective shear area Av.eff can be calculated from the following formula [1]:

  f  Av.eff = t ⋅ Lv.eff = t  2( Lv + a1 ) + ( a2 − k ⋅ d0 ) u    f y    f  however Lv.eff ≤ 2( Lv + 5d ) + ( a 2 − k ⋅ d0 ) u   fy 

where is the ultimate strength of the hollow section fu is the yield strength of the hollow section fy k = 2, 5 Lv , a1 and a2 are defined in Figure 3.7

91

(3.25)

hp

NSd

p1

NSd

d0 Lw

t2 t1

The steel grade used in splices is S355J2. The strength grade of the M24 bolts is 8.8. The parameters of the joint geometry are:

e1

p2

Example 25 Calculate the tension resistance of the adjacent joint. The dimensions of the hollow section are 150 x 150 x 6,3, and the steel designation used is S355J2H.

e2


a2

Chapter 3

NSd

t1 = 20 mm t2 = 10 mm d = 24 mm d0 = 26 mm a1 = e1= e2 = 40 mm Lv = a2 = p1 = p2 = 80 mm hp = 160 mm

NSd Lv

a1

Resistance of the splice plate net cross-section The splice plates can be taken as tension cross sections. Thus, the resistance of a cross section containing holes can be obtained from formulae (2.33) and (2.34) [1]: Anet

= 20(160 − 2 ⋅ 26 ) = 2160 mm 2

Av

= 20 ⋅ 160 = 3200 mm 2 Av ⋅ f y 3200 ⋅ 345 N  = = = 1003, 6 kN  t p > 16 mm ⇒ f y = 345  1, 1 γ M1 mm 2  0, 9 Anet ⋅ fu 0, 9 ⋅ 2160 ⋅ 490 = = = 762, 0 kN 1, 25 γ M2

Ft . Rd Ft . Rd

Thus, the tension resistance of the net cross section is Ft.Rd = 762,0 kN. Bearing resistance of splice plates When the holes are situated as in the example, the bearing resistance of splice plates is as follows [1]:

α

 e p 1 f  = min  1 ; 1 − ; ub   3d0 3d0 4 fu  = min[0, 513; 0,776; 1,632] = 0, 513 ≤ 1, 0

Fb. Rd

=

2, 5 ⋅ α ⋅ fu ⋅ d ⋅ t 2, 5 ⋅ 0, 513 ⋅ 490 ⋅ 24 ⋅ 20 = = 241, 3 kN 1, 25 γ Mb

Now there are 4 bolts per plate, so the bearing resistance is: Fb.Rd = 4 · 241,3 = 965 kN 92


Chapter 3

Block shear failure resistance of splice plates First, calculate the effective shear area from the formula (3.25): Av.eff

  f  = t1 ⋅ Lv.eff = t1  2( Lv + a1 ) + ( a2 − k ⋅ d0 ) u    f y    490    = 5226 mm 2 = 20  2( 80 + 40) + ( 80 − 2, 5 ⋅ 26 )    345  

Lv.eff

f  = 261, 3 mm < 2( Lv + 5d ) + ( a2 − k ⋅ d0 ) u  = 421, 3 mm  fy 

Obtain the block shear failure resistance by substituting in formula (3.24): Veff . Rd =

f y ⋅ Av.eff 345 ⋅ 5226 =2 = 946, 3 kN 3 ⋅γ M0 3 ⋅ 1, 1

Shear resistance of bolts The shear resistance of bolts is determined by assuming that the shear plane does not pass through the threaded portion of the bolt [1]: Fv. Rd =

0, 6 ⋅ fub ⋅ A 0, 6 ⋅ 800 ⋅ 452 = = 173, 6 kN 1, 25 γ Mb

There are four bolts and the joint has two shear planes, so the bolt resistance is: Fv.Rd = 8 · 173,6 = 1389 kN Resistance of welds Design the fillet welds with a throat thickness of 5 mm applying formula (3.18). The axial force introduced into the weld is assumed equal to the hollow section plastic tension resistance. The plastic tension resistance of a hollow section with dimensions 150 x 150 x 6,3 is [1]: Npl.Rd = NSd = 3485 · 355/ 1,1 = 1125 kN The following value for the required weld length is obtained: Lw ≥

3 ⋅ β w ⋅ γ Mw ⋅ N Sd = 4 fu ⋅ a

3 ⋅ 0, 9 ⋅ 1, 25 ⋅ 1125 = 224 mm 4 ⋅ 490 ⋅ 5

Resistance of joints The entire resistance of the joint is then determined by the resistance of the net cross section: Ft.Rd = 762 kN 93

Chapter 3

3.4.2


Bolted beam-to-column joints

;

A hollow section or an I profile can be joined to a hollow section column by several different methods, as shown in Figures 3.8-3.16. Introducing rigidity to the joint requires the use of end plates, which means that the tolerance on length must be more rigorous. In structures with multiple bays, the variation of length may accumulate, and the length deviation must be evened out with intermediate plates. More flexible joints in which bolts transmit the shear forces allow for greater adjustment. However, even in flexible joints, it is important to take into account the moments due to the eccentricity of shear force in the design of the column.

Figure 3.8 A beam-to-column connection between I section and hollow section subjected to shear, bending and normal loads. The joint resistance is usually limited by the buckling or plastification of the column web.

Figure 3.8

; ;;

Figure 3.9 A typical beam-to-column connection between I section and hollow section. The end of the beam must be stiffened with a plate, which then transfers the shear force, through contact in bearing, to the stiff portion of the support component. The joint is suitable mainly for beams with minor shear force.

Figure 3.10 Semi-rigidity of the joint is obtained by using very stiff end plates. However, the most practical way is to assume the joint pinned and take the bending moment of the extension into account in the column design. Due to its simplicity, this type of joint is frequently used in lattice structures. Complex joint details are made by welding, and simple straight members are connected to the outstands, starting from the corner point, by flange plate joints. The rigidity between the column and the outstand can be estimated as shown in Appendix 9.4.

Figure 3.11 The joints shown in Figures 11a and 11b behave in a similar way. When the bolted joint is made as an ordinary joint, carrying the load by bolt shear, the clearances between the bolts and the holes makes the joint indeterminate with regard to transmission of bending moment. With a friction grip bolted joint, the moment can be transferred from the beam to the support plate, preserving the rigidity. If the joint is subjected to shear force only, the support plate can be connected directly to the column flange. In a joint carrying the bending moment or axial force, the column flange usually must be stiffened with a reinforcing plate. 94

Figure 3.9

Figure 3.10

Figure 3.11a

Figure 3.11b


; ;

Figure 3.12 In the adjacent figure, a support plate with threaded holes is welded to the column flange. Since the bolts are short and the threaded part is subjected to shear load, the plastic deformation capacity of the joint is low. The best way to design the bolted joint is to ensure that it's resistance is equal to that of the incoming hollow section. When the flange is made from ordinary structural steel, it is advisable to select such plate thicknesses and bolt dimensions that the thread of the bolt reaches up the threaded hole at least by the length of the bolt’s nominal diameter, but the end of the bolt does not touch the column wall. The column wall thickness should not be included in the effective thread length, if the holes are drilled and threaded after the plate is welded.

Chapter 3

Figure 3.12

b)

;;

; ;; ;; ;;;;

a)

Figure 3.13 The joint shown in figure 3.13 can be made by threading the holes that are made in the column wall by thermodrilling (Figure 3.13 a). In thermodrilling, the hollow section wall is thickened next to the hole, so a sufficient length for the thread is obtained. An alternative is to use expansion bolts with the bolt holes drilled as usual (Figure 3.13 b). The expansion bolts press against the tube wall when the bolt is tightened.

Figure 3.13

b)

a)

Figure 3.14

Detail 1

Figure 3.14 Figures a and b show examples of connecting a bracing to the column.

; ;; ; Detail 1

; ;;

Figure 3.15 This figure illustrates an end-to-end joint of a chord of a lattice structure. Diagonals should not be welded to the end flanges, but to the chord. The actual gap (ga) of the joint is the distance between diagonal weld and the chord flange weld. The gap must meet the requirements given in the tables in Appendix 9.3.

Figure 3.16 Figures a and b show beam-to-column joints in which the beam is continuous. For the proper functioning of this kind of joint, it is essential that the loads during erection and use of the structure are close to symmetrical. When bending moments and shear forces are unequal, the column must be sufficiently strong to resist bending. The column flange plate joint is taken as a hinge in relation to buckling, unless the rigidity of the joint is increased with specific methods.

95

ga

Figure 3.15

Figure 3.16a

Figure 3.16b


Chapter 3

Hollow section flange plate joint subjected to bending moment

The lowest of the component resistances determines the bending resistance of the entire joint. Bending resistance of the column web

MSd

;;; ;

The flange plate joint is capable of transferring both the bending moment and the shear force. When calculating the bending resistance, the joint is divided into components. - bending resistance of the column web - shear resistance of the column web - tension resistance of flanges and bolts

The column web bending resistance can be estimated using the formulae for welded lattice joints (Table 9.3.9). When the column and the beam are of equal width, the following bending resistance is obtained [1]:

Mip.1. Rd = 0, 5 f y ⋅ t 0 (h1 + 5t 0 )

2

1, 1 γ Mj ⋅ γ M 0

(3.26 )

where

fy

is the yield strength of the column

Shear resistance of the column web The shear resistance of the column must also be checked, since the moment load is transferred from beam flanges to the column as a shear force. It is assumed that the column has no external shear load, and the column shear force then consists of the joint load only. The bending resistance of the joint when governed by the shear is obtained by multiplying the shear resistance of the column web by the column height [1]:

h1

t0 Vpl.Rd Vpl.Rd

MRd t1

M Rd = Vpl. Rd (h1 − t1 )

(3.27 )

where Vpl. Rd = h1 t1

f y ⋅ Av (design value of the column shear resistance, section 2.4) 3 ⋅γ M0

is height of the column is the thickness of the beam flange

96


Chapter 3

Tension resistance of flanges and bolts

;; ; ;; ; ;;; ;

The resistance of flanges and bolts can be estimated by calculating the resistance of the joint between the flange and hollow section using equivalent T models. Equivalent T models consist of a column and a flange, and a hollow section and a flange. There are three potential failure modes for a T model. According to these modes, the tension resistance values of the bolt row are as follows. (Figure 3.17) [1]:

Ft.Rd

a)

e

m

b)

m e

Ft.Rd

Ft.Rd

c)

Ft.Rd

Ft.Rd

Ft.Rd

Failure modes for an equivalent T model. Ft.Rd is the force of the bolt row in the tension area of the joint.

Figure 3.17

a) flange yield at the location of the bolt row and at the hollow section webs

Ft . Rd =

4 M pl. Rd m

(3.28 )

b) bolt failure as the flanges yield at the hollow section webs

Ft . Rd =

2M pl. Rd + e ⋅ 2 ⋅ Bt . Rd m+e

(3.29)

c) bolt or flange failure

Ft . Rd = 2 Bt . Rd

(3.30)

where

Ft . Rd M pl. Rd Leff tp m e Bt . Rd

is the tension resistance of the bolt row

0, 25 Leff ⋅ f y ⋅ t p2 = γ M0 is the effective length of the bolt row is the thickness of the flange is the bolt’s distance from the outer edge of the hollow section is the bolt’s distance from the edge of the flange, e ≤ 1,25 m is the tension resistance of the bolt or the punching resistance of the flange [Eurocode 3: 6.5.5 (4)] [1] (select the smaller value)

97


Chapter 3

The effective length of the bolt row depends on the shape of the flange’s yield line pattern. From the following equations, select the one giving the smallest result [1]:

Leff = 2π ⋅ m

(3.31)

(all bolts)

Leff = 4m + 1, 25e

(all bolts) (the uppermost and lowermost bolt row, if there are Leff = 0, 5 p + 2m + 0, 625e several rows) (other bolt rows, if there are several rows) Leff = p

(3.32) (3.33a ) (3.33b )

where

p

is the vertical distance between the horizontal bolt rows

The bending resistance of the joint is obtained by multiplying the tension resistance values of the horizontal bolt rows by the distance of the bolt rows from the centre of compression. Only the bolt rows in the tension zone are considered. The tension zone of the joint is located above the neutral axis of the hollow section. The following value for bending resistance is therefore obtained [1]:

M Rd = ∑ ( Ft . Rd )i (hr )i

(3.34)

i

where

Ft . Rd hr

is the design value for the bolt row’s tension resistance is the distance of the bolt row distance from the compression centre

98


50

MSd

hr

300 x 200 x 8

50 200

200 x 200 x 6,3

300

50

;;; ;

Example 26 Calculate the bending resistance of the flange plate joint. The column dimensions are 200 x 200 x 6,3 and those of the hollow section are 300 x 200 x 8. The steel grade used is S355J2H. The flange thickness is 20 mm. The steel grade used in the flanges is S355J2. The strength grade of the M22 bolts is 8.8. The bending resistance values of the joint components are: 300 x 200 x 8: M1.Rd = 244 kNm 200 x 200 x 6,3: M0.Rd = 110 kNm Usually, the joint also includes shear force, which must be taken into account in the joint design.

Chapter 3

Bending resistance of the column web The resistance of the column web is determined from the formula (3.26): Mip.1. Rd = 0, 5 f yk ⋅ t 0 (h1 + 5t 0 )

2

1, 1 1 = 111, 7 kNm = 0, 5 ⋅ 355 ⋅ 6, 3( 300 + 5 ⋅ 6, 3) 2 1, 1 γ Mj ⋅ γ M 0

Shear resistance of the column web Shear resistance of the column web is determined as shown in chapter 2: Vpl , Rd =

f y ⋅ Av 355 ⋅ 2372 = = 442 kN 3 ⋅γ M0 3 ⋅ 1, 1

By multiplying the shear resistance by the height of the hollow section, the moment resistance for the column web is obtained [formula(3.27)]: MRd = Vpl.Rd(h1– t1) = 442(0,3 – 0,008) = 129,1 kNm Resistance of flanges and bolts First, calculate the effective length of the bolt row [formulae (3.31)- (3.33)]: Leff = 2π ⋅ m = 2π ⋅ 50 = 314 mm Leff = 4m + 1, 25e = 4 ⋅ 50 + 1, 25 ⋅ 50 = 262, 5 mm Since the latter formula gave the smallest value, it is used as the effective length of the bolt row Leff = 262,5 mm. Then, substitute the effective length of the bolt row in the failure mode equations for the T stub [formulae (3.28) - (3.30)]: M pl. Rd

0, 25 Leff ⋅ f y ⋅ t p2 0, 25 ⋅ 262, 5 ⋅ 345 ⋅ 20 2 = = = 8, 233 kNm 1, 1 γ M0 99

Chapter 3


a) flange yield at the location of the bolt row and at the hollow section web Ft . Rd =

4 M pl. Rd 4 ⋅ 8, 233 = = 658, 6 kN 50 m

b) bolt failure as the flanges yield at the hollow section webs Ft . Rd =

2M pl. Rd + e ⋅ 2 ⋅ Bt . Rd 2 ⋅ 8, 233 + 50 ⋅ 2 ⋅ 174, 5 = = 339, 2 kN 50 + 50 m+e

c) bolt or flange failure Ft . Rd = 2 ⋅ Bt . Rd = 2 ⋅ 174, 5 = 349 kN We see that the failure mode to be used in the design is therefore bolt failure with flange yielding Ft.Rd = 339,2 kN. It is normally recommended to design the flange joint so that the flanges yield before the bolts fail. The failure mode is then ductile. For flanges and bolts, the joint bending resistance is [formula (3.34)]: MRd = Ft.Rd · hr = 339,2 · (0,25– 0,004) = 83,44 kNm Design of welds The welds must transfer the tensile force due to bending moment into the hollow section’s upper flange NSd = MRd / 0,3 = 278,1 kN. The fillet weld is provided across the width of the entire column (200 mm), which gives a required throat thickness of: a≥

3 ⋅ β w ⋅ γ Mw ⋅ N Sd = fu ⋅ Lw

3 ⋅ 0, 9 ⋅ 1, 25 ⋅ 278, 1 = 5, 5 mm 490 ⋅ 200

Bending resistance of the joint The smallest bending resistance value is that involving failure of the bolts and flange yielding, so it is selected as the bending resistance of the entire joint: MRd = 83,44 kNm

100


40

40

200 x 200 x 6,3

40

40 120

200 x 200 x 6,3

FSd

NSd

;

Example 27 Calculate the resistance of the adjacent shear joint. The dimensions of the hollow sections are 200 x 200 x 6,3, and the steel grade used S355J2H. At the end of the hollow section, the joint is subjected to a force FSd = 150 kN. The thickness of the splice plate is 15 mm, and the strength grade of the M20 bolts is 8.8.

Chapter 3

The vertical load introduces a bending moment into the column: MSd = 150 · 0,04 = 6 kNm The axial force of the column is: NSd = 300 kN (compression)

Bearing resistance of the splice plates The bearing resistance of the splices is calculated as shown in example 25 [1]:  e p 1 f  α = min  1 ; 1 − ; ub  ≤ 1, 0  3d0 3d0 4 fu 

= min[0, 606; 1, 568; 1,632 ] = 0, 606 ≤ 1, 0 Fb. Rd =

2, 5 ⋅ α ⋅ fu ⋅ d ⋅ t 2, 5 ⋅ 0, 606 ⋅ 490 ⋅ 20 ⋅ 15 = = 178, 2 kN 1, 25 γ Mb

2 bolts ⇒ Fb.Rd = 2 · 133,7= 267,4 kN > Fsd OK ! Block shear failure resistance of splice plates First, calculate the effective cross-section [1]:   f  Av.eff = t ⋅ Lv.eff = t  Lv + a1 + ( a2 − k ⋅ d0 ) u    f y    490    = 3000 mm 2 = 15 120 + 40 + ( 40 − 0, 5 ⋅ 22)    355   f  Lv.eff = 200 mm ≤ ( Lv + a1 + a3 − n ⋅ d0 ) u  = 215, 3 mm  fy  Lv.eff = 200 mm ≤ ( Lv + a1 + a3 ) = 200 mm a1 = 40 mm ≤ 5d = 100 mm The resistance to block shear failure is obtained by substituting in the formula (3.24): Veff . Rd =

f y ⋅ Av.eff 355 ⋅ 3000 = = 559, 0 kN > FSd OK! 3 ⋅γ M0 3 ⋅ 1, 1 101


Chapter 3

Shear resistance of bolts In the equation for determining the shear resistance of bolts, it is assumed that the shear plane passes through the threaded portion of the bolt [1]: Fv. Rd =

0, 6 ⋅ fub ⋅ As 0, 6 ⋅ 800 ⋅ 245 = = 94, 1 kN 1, 25 γ Mb

Two bolts and a single-lap joint ⇒ Fv.Rd = 2 · 94,1= 188,2 kN > Fsd OK ! Resistance of the column wall The resistance of the column wall is calculated as shown in Table 9.3.11 in Appendix 9.3: n=

γ M 0 ⋅ γ Mj  N Sd M Sd  1, 1 300000 6000 + = + = 0, 260   1, 1 1 4745 ⋅ 355 292, 2 ⋅ 355  A ⋅ f y Wel ⋅ f y 

km = 1, 3(1 − n) = 1, 3(1 − 0, 260) = 0, 962 M1. Rd = 0, 5km

f y ⋅ t 02 ⋅ h1  2h1 tp  1, 1 + − 4 1   t p  b0 b0  γ Mj ⋅ γ M 0 1− b0

= 0, 5 ⋅ 0, 962

355 ⋅ 6, 3 2 ⋅ 200  2 ⋅ 200 15  1 + 4 1− = 7 , 79 kNm > M Sd OK!   15   , 200 200 1 1 1− 200

Resistance of welds The weld between the column and the splice must transfer the force FSd vertically and the tensile force due to moment MSd horizontally. Therefore determine the stress components of the weld:

τ ll =

FSd a⋅L

τ⊥ = σ⊥ =

(3.35) M Sd t 1 ⋅ ⋅ Wel a 2

(3.36 )

where

τ ll τ⊥ σ⊥

is the shear stress parallel to the weld axis is the shear stress perpendicular to the weld axis is the axial force perpendicular to the weld design surface

102


Chapter 3

The weld strength is checked with formulae in Eurocode 3 Appendix M [1]:

(

)

σ ⊥2 + 3 τ ⊥2 + τ ll2 ≤ σ⊥ ≤

fu β w ⋅ γ Mw

(3.37 )

fu

(3.38 )

γ Mw

Try a throat thickness of 3 mm, which gives the following weld stresses (fillet welds on both faces of the plate): FSd 150000 N = = 125 2a ⋅ L 2 ⋅ 3 ⋅ 200 mm 2 M t 1 N 6000 t 1 τ ⊥ = σ ⊥ = Sd ⋅ ⋅ = ⋅ ⋅ = 106, 1 Wel 2a 2 100 2 ⋅ 3 2 mm 2

τ ll =

Checking the conditions (3.37) and (3.38):

(

)

σ ⊥2 + 3 τ ⊥2 + τ ll2 = 303, 2 σ ⊥ = 106, 1

N fu 2 ≤ γ Mw mm

N fu 490 N ≤ = = 435 , 6 mm 2 β w ⋅ γ Mw 0, 9 ⋅ 1, 25 mm 2 490 N OK! = = 392 1, 25 mm 2

OK!

The same throat thickness can be used in the weld between the hollow section end plate and the splice. Resistance of the joint Compare the calculated resistance values to the force quantities: VRd = 188 kN > VSd = 150 kN OK ! MRd = 7,79 kNm > MSd = 6 kNm OK !

103


Chapter 3

3.5

Hollow section-to-foundation joints

The joint between the column and the foundation is generally assumed to be either rigid or pinned. In practice, all joints are semi-rigid. However, in practical design, it is seldom necessary to take the semi-rigidity into account. It is usual to provide a nominally pinned joint. It is seldom necessary to design a freely rotating joint (Figure 3.18a) however as the base plate (Figure 3.18b) can be made sufficiently flexible. The moment at the base of the column need not be taken into account in the design when the joint is sufficiently flexible.

; ;; ;;

Reinforcing a rigid joint with stiffeners is seldom an economical alternative. It is usually more efficient to increase the thickness of the base plate.

a)

Figure 3.18

3.5.1

b)

Joint of a hollow section to foundation

Joint between a column subjected to axial force and foundation

The joint is subjected to axial force only, so the concrete surface pressure is obtained directly by dividing the load by the base plate area:

p=

N Sd A

(3.39)

The area is calcutated for axial force concentric with base plate. When the axial force is eccentric, the base plate area must be reduced. Only the portion of the base plate concentric to the axial force is included in the reduced area. The base pressure must be less than the concrete compression resistance.

104


Chapter 3

The pressure under the baseplate induces a bending moment into the base plate. The base plate can be treated as a cantilever whose length is the distance between the hollow section wall and the plate edge. When calculating the elastic plate resistance, we obtain the following condition for the base plate thickness [1]:

3p ⋅γ M0 6M Sd ⋅ γ M 0 = a1 b ⋅ fy fy

(3.40)

;;

tp ≥ where

b a1 tp fy γ M0

is the width of the base plate is the length of the cantilever is the thickness of the base plate is the yield strength of the base plate is the safety factor of the material

b

M Sd

a1

b ⋅ p ⋅ a12 = 2

a

In addition, the load during erection should be checked, which is when the holding down bolts transfer the forces to the foundation.

3.5.2

Joint between a column subjected to bending moment and axial force and the foundation M Sd

NSd

fcd

In a rigid joint, the bending moment in the column must be taken into account. At the ultimate limit state, the compression stress is limited by the concrete compression resistance design value of fcd. The tensile force is transferred into the foundation through the holding down bolts on the tension face. The following conditions for the balance of forces are then obtained:

Ns d

y Nc

N Sd = Nc − N s = beff ⋅ y ⋅ fcd − As ⋅ f yb

(3.41)

M Sd + N Sd [0, 5a − ( a − d )] = Nc ( d − 0, 5 y)

(3.42)

where

As f yb fcd beff

is the stress cross-section of the foundation bolts (on the tension face) is the dimensioning valve for the foundation bolts is the design value for the compression strength of concrete is the effective width of the bottom plate on the compression side

105

Chapter 3


Using this pair of equations, we can determine the length of the concrete compression area y and the required stress cross-section of the holding down bolts As:

y=

beff ⋅ fcd ⋅ d ±

(−beff ⋅ fcd ⋅ d ) 2 − 2beff ⋅ fcd [ MSd + NSd (d − 0, 5a)]

(3.43)

beff ⋅ fcd

The holding down bolts have no tensile force if the formula (3.41) yields a negative value for Ns. The bending load of the base plate is created from the distribution of stresses in the compression area. The base plate thickness is determined similarly as in section 3.5.1, except the tension load for the holding down bolts most also be checked. The holding down bolts may be subjected to some tension, if the moment is high and the axial force is low. In addition, the load during erection must also be checked.

MSd

Example 28 Calculate the joint resistance of a hollow section with dimensions 200 x 200 x 8. The steel grade used is S355J2H, and the design value for the compression resistance of concrete is 14 N/mm2. The base plate dimensions are a x b = 400 x 400. The steel grade used in the holding down bolts is S355. The column is subjected to the following loads:

p1

p2

a

; ;

NSd = 1500 kN MSd = 35 kNm VSd = 100 kN

NSd VSd

b

First, determine whether the holding down bolts are subjected to tension at the ultimate limit state:

a1

y=

y=

beff ⋅ fcd ⋅ d ±

(−beff ⋅ fcd ⋅ d ) 2 − 2beff ⋅ fcd [ MSd + NSd (d − 0, 5a)] beff ⋅ fcd

[

]

400 ⋅ 14 ⋅ 350 ± ( −400 ⋅ 14 ⋅ 350) 2 − 2 ⋅ 400 ⋅ 14 35 ⋅ 10 6 + 1500 ⋅ 10 3 ( 350 − 0, 5 ⋅ 400) 400 ⋅ 14

y = 178 mm (or 522 mm) Nc = beff · y · fcd = 400·178·14 = 996,8 kN Ns = Nc – NSd = 996,8 kN – 1500 kN = –503,2 kN ⇒ holding down bolts are not subjected to tension 106


Chapter 3

Resistance of the holding down bolts Since the holding down bolts are not subjected to tension, the holding down bolts need be designed for shear only: As >

VSd 3 ⋅ γ M 0 100 3 ⋅ 1, 1 = = 552 mm 2 fy 345

4 bolts Ø 24 ⇒ AS = 1412 mm2 > 552 mm2 OK ! Base plate resistance The value of the bending moment in the base plate at the column edge is as follows: M Sd

beff ⋅ a12 ⋅ fcd 400 ⋅ 100 2 ⋅ 14 = = = 28 kNm 2 2

The thickness of the base plate is obtained by substituting the bending moment MSd into the formula (3.40): tp ≥

6 M Sd ⋅ γ M 0 6 ⋅ 28 ⋅ 1, 1 = = 36, 6 mm b ⋅ fy 400 ⋅ 345

107

⇒ select tp = 35 mm

Chapter 3

3.6


References

[1] ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu, Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (Sisältää myös liitteen K: ENV 1993-1-1:1992/ A1:1994) (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993) (Include also annex K: ENV 1993-1-1:1992/ A1:1994) [2] CIDECT: Design guide for rectangular hollow section joints under predominantly static loading, Verlag TÜV Rheinland GmbH, Köln 1992 [3] CIDECT: Design guide for circular hollow section joints under predominantly static loading, Verlag TÜV Rheinland GmbH, Köln 1991 [4] CIDECT: Design guide for structural hollow sections in mechanical applications, Verlag TÜV Rheinland GmbH, Köln 1995

108


4.

FATIGUE AND BRITTLE FRACTURE IN HOLLOW SECTION STRUCTURES

4.1

Fatigue loading

Chapter 4

; ;

Fatigue refers to failure of an element due to repeated loading. Normally, failure results from stress concentrations created at structural discontinuities. Local stress concentrations cause the propagation of minor initial defects in the weld and at the weld toe. Fatigue loading can cause failure at lower stress values than static loading. Fatigue loading varies in magnitude, direction and position. This type of loading occurs, for instance, in crane gantries, bridges and machinery foundations.

Figure 4.1 Structural discontinuities

Fatigue loading generates cracks in the element and propagates existing initial defects. In welded structures, the most susceptible position for crack propagation is normally the line between weld and parent material. Crack propagation is initiated in the weld or in its proximity, since the weld always contains minor defects. Careful welding is thus especially important for structures subjected to fatigue loading. Welded joints have a decisive role in the fatigue design of Figure 4.2 Stress concentration at the weld the entire member, since the fatigue strength undercut of a structural hollow section is rarely lower than that of the joint. The fatigue strength of a welded detail depends on the following factors: - stress range (load amplitude) - stress frequency (number of stress cycles) - shaping of structural discontinuity locations - weld geometry - size of the initial crack - residual stress state - toughness of the material In dynamically loaded members, the effect of vibration on stress values must be accounted for. The increase of stress is significant if the natural frequency of the members is close to the vibration frequency of the load. In practice, members are usually designed in such a manner that the lowest natural frequency is higher than the frequency of the dynamic load. In this way, the stress concentrations due to resonance can be prevented. The frequency of the dynamic load can also be higher than the natural frequency, if the resonance frequency is passed through rapidly (e.g. in machinery foundations). 109

Chapter 4

4.2


Stress calculation methods in fatigue design

;;; ;;; ;;

Stress values can be calculated using four methods that represent different levels: - nominal stress method - hot spot stress method - notch stress method - fracture mechanics In the nominal stress method, structural stress values are calculated, usually σm according to the elasticity theory, without accounting for the effect of structural discontinuities. This method is simple and well-suited for manual calculations. The a) Nominal stress method fatigue design values presented in Eurocode 3 are based mainly on the nominal stress σm method.

σb

+

=

b) Hot spot stress method

σm

σb

+

σnlp

+

σln

=

c) Notch stress method

σm σb σhot spot σnlp σln

is the nominal stress is the bending stress depending on the joint geometry is the hot spot stress is the non-linear stress concentration due to the notch is the notch stress

Figure 4.3 Stress calculation methods

0,4t (at least 4 mm) 0,6t σhot spot σb

σa

t

Hot spot stress is the stress present at the critical point of a structural discontinuity (Figure 4.4). It is at this location that the fatigue crack growth is assumed to start. Structural discontinuities, which appear for instance where the cross-section changes or the attachment ends, are taken into account in the hot spot method. However, this method does not account for the effect of weld geometry. Due to non-uniform stress distribution, hot spot stress values are usually higher than nominal stress values. The stress values(σa, σb) are calculated for at least two points at the proximity of the weld, and these values are used to extrapolate the hot spot stress in the edge of the weld (σhot spot). The structure must then be modeled using a suitable calculation program to determine the stresses. The stress calculation points can be selected at a distance of 0,4t and 1,0t (see Figure 4.4) from the edge of the weld [3]. Linear extrapolation is used if two extrapolation points are selected. With more extrapolation points, parabolic extrapolation is used. Hot spot stress values can also be measured from the prototype or calculated using the concentration factors (Ks) taken from reference manuals.

σhot spot

Figure 4.4 Determining hot spot stress values 110


Chapter 4

Notch stress refers to the actual stress at the bottom of the notch. The notch is usually generated next to the weld or at some other structural discontinuity. The area affected by the notch stress concentration is approximately 0,3 times the plate thickness [3]. To determine the notch stress, the structural calculations must be performed using the FE method which accurately takes into account the geometry of the structure, including the actual weld geometry and corner radii. The design must not include non-rounded corners, since FE calculation generates an infinite stress in them as the element size is decreased using shell elements (not applicable to solid-elements). The size and shape of the initial crack is also taken into account in the model based on fracture mechanics. With this model, the rate of crack propagation can be calculated using the geometry and the properties of the material.

4.3

Design requirements for fatigue-loaded hollow sections (nominal stress method)

4.3.1

Conditions and necessity of fatigue design

The guidance in Eurocode 3 can be used in fatigue design, provided that the following conditions apply [1]: - the normal stress range is lower than 1,5 fy - the shear stress range is lower than 0,866 fy - the structure is corrosion-protected so that pit depth is less than 1,0 mm - the temperature of the structure is below 150 °C Fatigue strength need not be calculated if the stress range ∆σ fulfills the following condition [1]:

γ Ff ⋅ ∆σ ≤

26 γ Mf

(4.1)

where

γ Ff γ Mf ∆σ

is the partial safety factor for fatigue loading (Table 2.1) is the partial safety factor for fatigue strength (Table 2.2) is the greatest nominal stress range [N/mm2]

Fatigue need not be taken into account, either, if the number of stress cycles fulfills the following condition [1]: 6

36

N ≤ 2 ⋅ 10   γ Mf ⋅ γ Ff

 ⋅ ∆σ E 2 

3

(4.2)

where

∆σ E 2 is equivalent constant-amplitude fatigue stress range at 2 ·106 stress cycles N

[N/ mm2] (Figure 4.5) N is the number of stress cycles 111


Chapter 4

4.3.2

Fatigue load design conditions

Constant-amplitude load For constant amplitude loading, the fatigue load design condition is presented in the following form [1]:

γ Ff ⋅ ∆σ ≤

∆σ R γ Mf

(4.3)

where

∆σ R

is the fatigue strength which is dependent on the fatigue category and the number of stress cycles

Variable-amplitude load In variable-amplitude loading, the principle is to calculate the cumulative effect of different stress ranges on the structure. The number of stress cycles featuring different stress ranges are thus compared with the numbers of corresponding fatigue strength stress cycles and the obtained quotients are added. The sum must meet the following condition (Palmgren-Miner rule) [1]:

Σ i

ni ≤1 Ni

(4.4)

where

ni Ni

is the number of stress ranges with the magnitude of ∆σi (load) is the number of the failure-inducing stress ranges with the magnitude of ∆σi for the relevant detail category

Ni can be determined for normal stresses as follows [1]: 6

 ∆σ D ⋅ γ Ff ⋅ ∆σ i 

3

6

 ∆σ D ⋅ γ Ff ⋅ ∆σ i 

5

Ni = 5 ⋅ 10   γ Mf Ni = 5 ⋅ 10   γ Mf Ni = ∞

when γ Ff ⋅ ∆σ i ≥

when

∆σ D γ Mf

∆σ D ∆σ L > γ Ff ⋅ ∆σ i ≥ γ Mf γ Mf

when γ Ff ⋅ ∆σ i <

112

∆σ L γ Mf

(4.5)

(4.6) (4.7)


Chapter 4

where

∆σ i ∆σ D ∆σ L

is the normal stress range caused by the load is the normal stress range of the constant-amplitude fatigue limit (= 5 ·106 stress cycles) for the relevant fatigue detail (Figure 4.5) is the cut off limit (= 1 ·108 stress cycles) for the relevant fatigue detail (Figure 4.5)

In the case of shear stress, the corresponding value for Ni is [1]: 6

Ni = 2 ⋅ 10   γ Mf

 ∆τ c ⋅ γ Ff ⋅ ∆τ i 

Ni = ∞

5

when γ Ff ⋅ ∆τ i ≥

∆τ L γ Mf

(4.8)

when γ Ff ⋅ ∆τ i <

∆τ L γ Mf

(4.9)

where

∆τ i ∆τ C ∆τ L

is the shear stress range caused by the load is the shear stress range at 2⋅106 stress cycles for the relevant fatigue detail (Figure 4.6) is the shear stress cut off limit (= 1 ·108 stress cycles) for the relevant fatigue detail (Figure 4.6)

The combined load condition for normal stress and shear stress is [1]:

Σ i

ni ( ∆σ i ) n ( ∆τ i ) +Σ i ≤1 Ni ( ∆σ i ) i Ni ( ∆τ i )

(4.10)

113


Chapter 4

4.3.3

Fatigue strength of hollow sections (nominal stress method)

When calculating the fatigue strength with the nominal stress method, the fatigue category of the member or element must first be determined. The fatigue category number indicates the fatigue strength of the detail [N/ mm2] at 2 ⋅106 stress cycles. 4.3.3.1

Fatigue strength under normal and shear stress

The fatigue strength curves for hollow section details (4.5) and (4.6) represent the stress ranges for details in different fatigue categories. The determination of a detail's fatigue category is explained in Appendix 9.5. Constant-amplitude fatigue limit (∆σD)

1000

Cut-off limit (∆σL)

500 400

Normal stress range (N/mm2 ) ∆σ

300

200

100

Detail category 140 125 112 100 90 80 71 63 56 50 45 40 36

50 40 30

20

10 104

105

106 2·106 5·106 107 Number of stress cycles N

Figure 4.5 Fatigue strength curves for various normal stress ranges [1]

114

108

109


(∆τC)

Chapter 4

Cut-off limit (∆τL)

1000

Shear stress range (N/mm2 ) ∆τ

500 400 300 200

100 Detail category 50 40

100 80

30 20

10 104

105

106 2·106

107

108

109

Number of stress cycles N

Figure 4.6 Fatigue strength curves for various shear stress ranges [1]

4.3.3.2

Fatigue strength of lattice structure joints (nominal stress method)

For the design of hollow section lattice structures, the fatigue curves presented in Figure (4.7) are used. Due to secondary bending moments, lattice structure members feature local stress concentrations. To account for them, the forces in the brace members and chords are multiplied by the factors presented in Table 4.1. The stresses obtained are compared with the fatigue strength ∆σR. The values in Table 4.1 are approximate empirical values or values based on testing. The nominal stress method yields only a rough estimate of the fatigue strength of the structure. For instance, the combined effect of chord and brace member stresses on the fatigue strength is difficult to account for in the nominal stress method. Chapter 4.4 presents the more accurate hot spot calculation method for the hollow section lattice joints.

115

Chapter 4


Cut-off limit (∆σL) 1000

Stress range (N/mm2 ) ∆σ

500 400 300

200

100

Detail category 50 90

40

71

30

56 50 45

20

36

10 104

105

106 2·106

107

108

109


Figure 4.7 Fatigue strength curves for lattice structure joints determined by nominal stress method [1]

116


Table 4.1

Chapter 4

Stress correction factors for lattice structure joints [1]

Joint type

Chords

Verticals

Diagonals

K joint

1,5

-

1,3

N joint

1,5

1,8

1,4

K joint

1,5

-

1,2

N joint

1,5

1,65

1,25

K joint

1,5

-

1,5

N joint

1,5

2,2

1,6

K joint

1,5

-

1,3

N joint

1,5

2,0

1,4

Circular hollow sections Gap Overlap

Square and rectangular hollow sections Gap Overlap

4.4

Fatigue strength of lattice structure joints (hot spot method)

When using the hot spot method for fatigue design, the nominal stress values of lattice members are multiplied by the concentration factors Ks. The hot spot stress range obtained is used as in the nominal stress method, except that the fatigue stress curves are selected from Figure 4.8 according to the wall thickness. The design condition for the hot spot method is:

γ Ff ⋅ K s ⋅ ∆σ ≤

∆σ R γ Mf

( 4.11)

For variable-amplitude loading, the number of stress cycles causing failure Ni is determined from the following formulae:

Ni = 10 Ni = 10

12, 476 − 3 log( ∆σ ) 16 1− 0,18 log    t  16 16 , 327 − 5 log( ∆σ ) + 2,01 log    t 

117

ni ≤ 5 ⋅ 10 6

( 4.12)

5 ⋅ 10 6 < ni ≤ 1 ⋅ 10 8

( 4.13)

Chapter 4


Constant-amplitude fatigue limit (∆σD)

Cut-off limit (∆σL)

1000

Hot spot stress range (N/mm2 ) ∆σ

500 400 300 200

100

Wall thickness mm 4 5 6 8 10 12,5

50 40 30 20

10 104

105

106

5·106 107

108

109


Figure 4.9 Hot spot fatigue strength curves for lattice structure joints of square and rectangular hollow sections [8]. With wall thicknesses 2-4 mm, the 4 mm curve can be used.

The calculation of stress concentration factors for hollow section lattice structure joints is dealt with in source [7]. Concentration factors are expressed as functions of chord and brace member dimensions and joint dimensions (gap or overlap and joint angle). Formulae for determining stress concentration factors (Ks) for T, X and K jointed square hollow sections are presented in Tables 4.2 and 4.3.

118


Chapter 4

In the case of T or X joints, the concentration factors must be calculated for several points. The governing point is the one with the greatest stress range. The stress range of the members in T and X joints can be expressed as follows: Chord:

∆σ 0 =

K s 0. N ⋅ ∆N 0. max K s 0. M (ip) ⋅ ∆Mip.0. max K s 0. M ( op ) ⋅ ∆Mop.0. max + + + A0 Wip.0 Wop.0 K si. N ⋅ ∆Ni K si. M (ip) ⋅ ∆Mip.i K si. M ( op ) ⋅ ∆Mop.i + + Ai Wip Wop

(4.14)

Brace member:

∆σ i =

K s 0. N ⋅ ∆N 0 K s 0. M (ip) ⋅ ∆Mip.0 K s 0. M ( op ) ⋅ ∆Mop.0 + + + A0 Wip.0 Wop.0 K si. N ⋅ ∆Ni. max K si. M (ip) ⋅ ∆Mip.i. max K si. M ( op ) ⋅ ∆Mop.i. max + + Ai Wip Wop

(4.15)

where is the stress range of the chord ∆σ 0 is the concentration factor for the stress due to the axial force of the chord K s 0. N K s 0. M (ip) is the concentration factor for the stress due to the chord bending moment parallel to the lattice plane

K s 0. M ( op ) is the concentration factor for the stress due to the chord bending moment

∆N 0 ∆Mip.0 ∆Mop.0 A0 Wip.0 Wop.0

perpendicular to the lattice plane is the normal stress range of the chord

is the stress range of the chord bending moment parallel to the lattice plane is the stress range of the chord bending moment perpendicular to the lattice plane is the area of the chord is the chord section modulus parallel to the lattice plane is the chord section modulus perpendicular to the lattice plane

The quantities marked with the subscript i are for the brace member, respectively. The stresses in chord and brace members do not necessarily vary at the same phase. The stress range of the chord is determined by using the maximum values for the chord’s stress range. The brace member stress ranges are calculated from the forces acting at the same time as the chord maximum forces and are added to the chord stress ranges. This is done similarly for the brace members, except that the maximum values are attributed to the brace member stress ranges. With K joints, the maximum concentration factor values need not be calculated at several points, since Table 4.3 gives the formulae needed for calculating the maximum concentration factor for chords and brace members. An axially loaded K joint also generates secondary bending moments, the effect of which is accounted for by multiplying the nominal stresses by the correction factors given in Table 4.4 [7]. In other cases, the formulae (4.14) and (4.15) are applied. 119


Chapter 4 Table 4.2

Concentration factors for T and X joints with square hollow sections [7] M1.ip

M1.ip N1 b1

;;; ; ;; ; ;

b1

N1 t1

t1

E

A

E

A

B

D

B

D

C

t0

M0.ip

b0 T joint Load In-plane bending moment in bracing member

C

t0

N0

N0

M0.ip b0

X joint

Concentration factor SCF Chord:

( ) (1,722+1,151β −0,697 β ) τ 0,75 ( −0,690 + 5, 817 β − 4,685 β ) 0,75 = (0, 952 − 3, 062β + 2, 382β 2 + 0, 0228 ⋅ 2γ )( 2γ ) τ ( 2,084 − 1,602β + 0, 527 β ) 0,75 = ( −0, 054 + 0, 332β − 0, 258 β 2 )( 2γ ) τ

K s 0. M (ip ). B = −0, 011 + 0, 085β − 0, 073β 2 ( 2γ )

2

2

K s 0. M (ip ).C

2

K s 0. M (ip ). D

Brace member:

(

)

K si. M (ip ). A, E = 0, 390 − 1, 054β + 1, 115β 2 ( 2γ ) Axial force in brace member

Chord:

( −0,154 + 4, 555 β − 3, 809 β 2 )

( ) (1,377 +1,715β −1,103β ) τ 0,75 ( 1, 565 + 1, 874 β − 1,082β ) 0,75 = (0, 077 − 0, 129 β + 0, 061β 2 − 0, 0003 ⋅ 2γ )( 2γ ) τ ( 0, 925 + 2, 389 β − 1, 881β ) 0,75 = (0, 208 − 0, 387 β + 0, 209 β 2 )( 2γ ) τ

K s 0. N . B = 0, 143 − 0, 204β + 0, 064β 2 ( 2γ )

2

2

K s 0. N .C

2

K s 0. N . D

Brace member:

(

)

K si. N . A, E = 0, 013 + 0, 693β − 0, 278 β 2 ( 2γ ) Normal stress caused Chord: by axial force and K s 0. M (ip).C bending moment in chord K

s 0. M ( ip ). D

( 0,790 + 1, 898 β − 2,109 β 2 )

= K s 0. N .C = 0, 725( 2γ ) ( 0, 248 β ) τ 0,19 = K s 0. N . D = 1, 373( 2γ ) ( 0, 205 β ) τ 0, 24

Minimum concentration factor value is 2 X joints for which β = 1,0 ⇒ Ks0.N.C and Ks0.M(ip).C multiply by 0,65 X joints for which β = 1,0 ⇒ Ks0.N.D and Ks0.M(ip).D multiply by 0,50 Fillet welds: ⇒ Ksi.M(ip)A.E and Ksi.N.A.E multiply by 1,40 Parameters: Conditions: τ = t1 / t0 0,25 ≤ τ ≤ 1,0 γ = b0 / (2t0) 12,5 ≤ 2γ ≤ 25,0 β = b1 / b0 0,35 ≤ β ≤ 1,0; θ = 90°

120


Table 4.3 N1

Concentration factors for K joints with square hollow sections [7] h1 g

N2

h2

N1

b1,2

N2

b1,2

1

t1,2

θ1

h

θ2

h2

;; ; ;; ; θ2

t1,2

t0

N0

t0

h0

θ1 h0

Chapter 4

q

b0

Gap joint Joint type Gap

b0

Overlap joint

Member Brace member

Concentration factor SCF

3,62τ ( 2 - τ ) + 0, 336ξ ⋅ γ 2 (0, 3 − 0, 01ξ ⋅ γ ) +

(

0, 044γ ⋅ β 6, 38 − γ ⋅ β

Gap

Chord

2

)

2

 γ ⋅g  − 4, 18  − 2, 2  100t0 

  g g 1, 1τ  0, 00288γ 3 +  + 5, 73ξ  1 − 0, 178ξ 2  −   t0  t0  2

 g 0, 166 β 3   − 1, 73  t0  Overlap

Brace member

(

)

0, 144β ⋅ γ 2 1 − 0, 813β 2 + 1, 84β ⋅ τ + 2

γ  3  3, 23ξ  1, 94τ − 1, 9τ − − 0, 26  10    Overlap

Chord

2

(

3

)

−40, 22ξ ⋅ β 2 1 − 0, 59ξ 2 + 0, 028γ 2 ( 8, 9 β + τ ) − 5, 41γ ⋅ β 3 − 0, 008ξ 2 ⋅ γ 3 + 2, 109ξ 6 − 4, 24

The minimum concentration factor value is 2 Parameters: τ = ti / t0 ξ = g / bi gapped ξ = -q / bi overlapped γ = b0 / (2t0) β = bi / b0

Conditions for gapped joints: 0,25 ≤ τ ≤ 1,0 0,25 ≤ ξ ≤ 0,75 12,5 ≤ 2γ ≤ 25,0 0,35 ≤ β ≤ 1,0

Conditions for overlapped joints: 0,4 ≤ τ ≤ 1,0 -1,0 ≤ ξ ≤ -0,4 12,5 ≤ 2γ ≤ 25,0 0,35 ≤ β ≤ 0,7

g ≤ 7,0 t0 35° ≤ θ ≤ 55°

2,5 ≤

1,5 ≤

q ≤ 17,0 t0

35° ≤ θ ≤ 55°

Table 4.4

Nominal stress correction factor for K joints (hot spot stress method) [7]

Joint type

Chord

Brace member

Gapped

1,5

1,5

Overlapped

1,3

1,3

Source [7] includes parametric concentration factor formulae also for circular hollow section joints. 121

Chapter 4


4.5

Design of fatigue-loaded hollow section structures

4.5.1

Welded joints

Structural welded joints should be placed in locations subjected to the lowest load, whenever this is feasible (Table 4.5). In welded joints, the fatigue strength does not depend significantly on the yield strength of steel, which means that the use of high strength steel does not necessarily bring the same advantages as it does in static design. The level of stress can be decreased by expanding the cross-sectional area, but this increases the weight of the structure. The optimal dimensions of a structure must therefore be determined on case by case basis, taking into account the effects of weight and service life of the structure. From the fatigue point of view, it is more advantageous to construct the welded hollow section end-to-end joints without using intermediate plates, since a better fatigue category can be obtained. It is essential for the weld strength that the weld is free of defects especially at the root side, which is the side more difficult to inspect. If there is a possibility of defects remaining at the root side (e.g. when using a high material thicknesses), it is advisable to use backing plates. However, it should be kept in mind that the use of backing plates generates a stress concentration which may decrease the fatigue strength of the structure. A high weld convexity increases the stress concentration at the weld toe, so a concave weld provides a better fatigue strength. Other factors improving the fatigue strength of the weld include a low-gradient weld joint angle and a great corner radius of the weld toe. Post-weld heat treatment Post weld heat treatment of the weld toe can be used to improve the fatigue strength of the joint. In the post treatment, the weld toe is re-melted with, for instance, a TIG torch or plasma torch. Another possibility is to remove minor initial cracks by grinding the weld toe. At the same time, the corner radius of the weld’s joining location is increased (Table 4.5). Grinding depth is typically at the most 10% of the plate thickness. A recommended grinding corner radius for plates with thickness less than 20 mm is 10 mm [5].

Grinding

It is also possible to generate, in the welded detail, a compression stress that counteracts the crack initiation and decelerates the propagation of the crack. A compression stress can be generated, for instance, by shot blasting. Stress relieving or annealing is a method intended for reducing the internal residual tension stresses due to the weld. In annealing, the yield strength of the material is temporarily lowered by increasing the temperature, which relieves the internal stresses of the element. To prevent the generation of new stresses, heating and cooling should be performed as slowly as possible. Post weld treatment can be profitable when the treated area is small or when the treatment can be automated in the shop. This treatment can also be used for improving the fatigue resistance of old structures. The effect of post weld treatment has not been accounted for in the design guidance of Eurocode 3 [1]. An empirical observation is that grinding increases fatigue strength by 30-100% (2⋅106 stress cycles) and re-melting by 10-170% (2⋅106 stress cycles). Shot blasting can yield a fatigue strength improvement of 30-170% (2⋅106 stress cycles) [5]. 122


Methods for improving fatigue strength of details

Method

Improved solution

σ2 < σ1

σ2 < σ1

σ1

σ1

Reduction of bending moment in flange plates

Connection placement in a location subjected to lower load

h1

t2

t1

Stress reduction

Original detail

> t1

Table 4.5

Chapter 4

h2 < h1

connection

connection

Smoother attachment plate geometry

grinding

Smoother splice plate geometry

Overlapping of a truss connection

L joint reinforcement with an intermediate plate M

;; ;; ;

Welding method selection Weld end rounding by grinding

M

Manual welding

Machine welding

Det 1

Det 1

Det 1

123

grinding Det 1


Chapter 4

4.5.2

Bolted joints

In bolted joints subject to tension, the use of prestressed bolts increases the fatigue strength of the joint significantly. The fatigue-inducing stress decreases, if the prestressing force is greater than the tension load on the joint. Long and elastic bolts increase the fatigue strength, since the elasticity of flange plates does not significantly reduce the prestressing force of long bolts. In flange joints subject to tension, it is advisable to place the bolts as close to the weld as possible in order to reduce additional stress caused by eccentricity. The prestressing of bolts is also a useful practice in bolted joints that are subjected to shear force. Prestressing reduces the stress in the material at the edge of the hole, since part of the force is transferred through in friction between the splices. However, it is important to ensure that the friction between the splices is sufficient to prevent the bolt from slipping towards the edge of the hole, which would result in the loss of the prestressing benefits.

4.5.3

Lattice structures

In lattice structure joints, increasing the chord wall thickness and decreasing the brace member width improves the fatigue strength of the structure, as the bending stresses on the chord wall are then reduced. The proportion of brace member width to chord width is expressed by the parameter β. For fatigue strength, the optimal solution would be to have chords and brace members of equal width, which would yield a β value of 1.0. In that case, the load is transmitted from the brace member directly to the chord web. With a thick-walled chord, however, the great corner radius may make it difficult to weld a brace member with equal width to the chord. The increase in the β value improves fatigue strength in cases in which β is greater than 0.5-0.7 [6].

; ;;

Welds in lattice structures must be made so that the initiation and termination points do not coincide with the brace member corners. To obtain the required weld throat thickness when the joint angle is smaller than 60°, the brace member end must be tapered (chapter 7). The throat thickness of lattice structure joints must be sufficient to prevent the weld root side from governing the fatigue strength. The effect of joint eccentricities is more significant in fatigue design than in static design, since the fatigue design of lattice structure joints must also account for the secondary bending moments due to eccentricities. The need for reinforcing plates in lattice structure joints must be judged on a case by case basis. Reinforcing plates increase the static strength of the joint; on the other hand, they also generate discontinuities at which stress concentrations are generated.

1 2 3

∆NSd(kN) 220 350 50

Number of stress cycles(ni) 1,5 ·107 1 ·106 1 ·107

124

NSd

75

Example 29 Calculate the fatigue strength of a hollow section with dimensions 200 x 200 x 8, subjected to a fatigue load ∆NSd. A non-loaded plate is welded to the side of the hollow section. The load on the member fluctuates as follows:


Chapter 4

When determining the fatigue strength, stresses are calculated using elastic theory. Calculate the nominal stress range caused by the first load fluctuation for the hollow section:

∆σ 1 =

∆N1.Sd 220000 N = = 37 , 13 A 5924 mm 2

The fatigue strength of the structure is calculated using the Palmgren-Miner rule, [formula (4.4)], since the load is not of constant-amplitude. The fatigue category for the non-load bearing joined element is 71, when the plate length is 75 mm (Appendix 9.5, Table 9.5.2). In fatigue category 71, the normal stress ranges of fatigue limits are the following (Figure 4.5):

∆σD = 52 N/ mm2 ∆σL = 29 N/ mm2

(stress range of the constant-amplitude fatigue limit) (stress range of the cut off limit)

Let us assume a fail-safe structure and normal accessibility. A value of 1,0 (Table 2.2) is obtained for the safety factor of the material in fatigue design γMf The number of stress cycles resulting in failure is obtained by substituting in the formula (4.6):   ∆σ D N1 = 5 ⋅ 10 6    γ Mf ⋅ γ Ff ⋅ ∆σ i  6

N1 = 5 ⋅ 10   γ Mf

5

, when

∆σ D ∆σ L > γ Ff ⋅ ∆σ i ≥ γ Mf γ Mf

5

5  ∆σ D 52  6 7 = 5 ⋅ 10   = 2, 69 ⋅ 10   1, 0 ⋅ 1, 0 ⋅ 37 , 13  ⋅ γ Ff ⋅ ∆σ 1 

Load fluctuations 2 and 3 are calculated in a similar manner:

∆σ 2 =

N 350000 = 59, 08 5924 mm 2 6

N 2 = 5 ⋅ 10   γ Mf

3

3  ∆σ D 52  6 6 = 5 ⋅ 10   = 3, 41 ⋅ 10   1, 0 ⋅ 1, 0 ⋅ 59, 08  ⋅ γ Ff ⋅ ∆σ 2 

N 50000 = 8, 44 5924 mm 2 N3 = ∞

( ∆σ 2 > ∆σ D )

∆σ 3 =

( ∆σ 3 < ∆σ L )

The strength of the hollow section is calculated from the summation equation (4.4): 1, 5 ⋅ 10 7 1 ⋅ 10 6 1 ⋅ 10 7 ni Σ + = 0, 85 < 1, 0 = + i Ni ∞ 2, 69 ⋅ 10 7 3, 41 ⋅ 10 6

125

OK !


N1.Sd

h1

θ1

b1

N1.Sd

g

θ1

t0

e

N0.Sd

t1

b0

N (kN) 590 190

The dimensions of the joint are as follows:

h1

;; ;

Example 30 Check the fatigue strength of a gapped K joint using the nominal stress method. Dimensions of the chord are 200 x 200 x 10 and those of the brace members are 140 x 140 x 5. The axial force range for brace members ∆N1.Sd is 190 kN and for the chord ∆N0.Sd is 590 kN. The axial forces in the chord and the brace member fluctuate, the chord axial force being 100 kN when the axial force in the brace members is 190 kN. Similarly, the axial force in the brace members is 49 kN when the axial force in the chord members is 590 kN.

h0

Chapter 4

100 49

θ1 = 37° g = 35 mm e = 0,8 mm (≈ 0 mm)

Chord

Brace member Time

When determining the stresses on lattice structure joints, the effect of secondary bending moments must be taken into account. The corrected stress range value is obtained by multiplying the uniform stress range by the correction factor presented in Table 4.1: Brace member: Chord:

1, 5 ∆N1.Sd 1, 5 ⋅ 190000 N = = 108 A1 2636 mm 2 1, 5 ∆N 0.Sd 1, 5 ⋅ 590000 N ∆σ 0 = = = 122 A0 7260 mm 2

∆σ 1 =

The fatigue category of the joint is determined by the ratio of the thickness of the chord walls to brace member walls (Table 9.6.3): t0 / t1 = 2,0 ⇒ fatigue category 71 Assume a fail-safe structure and a normal accessibility, which yields a material factor for fatigue designl (γMf) of 1,0 (Table 2.2) The fatigue strength at 5 ·104 stress cycles, for fatigue category 71, is as follows (Figure 4.7):

∆σR = 148 N/ mm2

126


Chapter 4

Compare the fatigue strength with chord and brace member stress ranges (4.3):

γ Ff ∆σ 0 = 1, 0 ⋅ 122, 0 = 122 γ Ff ∆σ 1 = 1, 0 ⋅ 108 = 108

N ∆σ R 148 N = = 148 2 < γ Mf 1, 0 mm mm 2

N ∆σ R 148 N < = = 148 1, 0 mm 2 γ Mf mm 2

OK ! OK !

Example 31 Calculate the fatigue strength of the joint shown in example 30 using the hot spot stress method. First determine the values of the parameters required in Table 4.3: ti 5 = = 0, 5 t 0 10 g 35 ξ= = = 0, 25 bi 140 b 200 γ = 0 = = 10 2t 0 20 b 140 β= i = = 0, 7 b0 200

τ=

The stress concentration factor (Ks) values for the brace member and the chord are obtained separately by substituting in the formulae given in Table 4.3: Brace member: K si. N = 3, 62τ ( 2 − τ ) + 0, 336ξ ⋅ γ 2 (0, 3 − 0, 01ξ ⋅ γ ) +

(

0, 044γ ⋅ β 6, 38 − γ ⋅ β

2

)

2

 γ ⋅g  − 4, 18  − 2, 2  100t 0 

= 3, 62 ⋅ 0, 5( 2 − 0, 5) + 0, 336 ⋅ 0, 25 ⋅ 10 2 (0, 3 − 0, 01 ⋅ 0, 25 ⋅ 10) +

(

0, 044 ⋅ 10 ⋅ 0, 7 6, 38 − 10 ⋅ 0, 7

2

)

10 ⋅ 35  2  − 4, 18 − 2, 2 = 2, 77  100 ⋅ 10 

Chord: 2

K s 0. N

   g g g = 1, 1τ  0, 00288γ 3 +  + 5, 73ξ  1 − 0, 178ξ 2  − 0, 166 β 3   − 1, 73    t0  t0  t0  35 35 = 1, 1 ⋅ 0, 5 0, 00288 ⋅ 10 3 +  + 5, 73 ⋅ 0, 25 1 − 0, 178 ⋅ 0, 25 2  −   10  10  0, 166 β

3  35 

 10 

2

− 1, 73 = 2, 46

127

Chapter 4


Now determine the hot spot stresses, multiplying nominal stress values by the concentration factor and by the correction factors given in Table 4.4 (in this example, 1.5): Brace member:

γ Ff

Ni. max N K si. N + γ Ff 0 K s 0. N Ai A0

190000  100000  N = 1, 0 ⋅ 1, 5 ⋅ 2, 77 + 1, 0 ⋅ 1, 5 ⋅ 2, 46 = 350  2636   7260  mm 2 Chord:

γ Ff

Ni N K si. N + γ Ff 0. max K s 0. N Ai A0

49000  590000  N = 1, 0 ⋅ 1, 5 ⋅ 2, 77 + 1, 0 ⋅ 1, 5 ⋅ 2, 46 = 377  2636   7260  mm 2 Considering the wall thickness, obtain the following fatigue strength values at 5 ·104 stress cycles (Figure 4.9): Brace member: ∆σR.1= 543 N/ mm2 Chord: ∆σR.0= 447 N/ mm2 Now compare the fatigue strength with the chord and brace member stress ranges. Brace member:

γ Ff ⋅ ∆σ 1 = 350

N ∆σ R.1 543 N = = 543 2 < γ Mf 1, 0 mm mm 2

OK !

N ∆σ R.0 447 N = = 447 2 < γ Mf 1, 0 mm mm 2

OK !

Chord:

γ Ff ⋅ ∆σ 0 = 377

128


4.6

Chapter 4

Brittle fracture of structural hollow sections

Brittle fracture is the rapid failure of an element with no clearly distinguishable plastic deformation. Brittle fracture initiates from a small crack initiated by fatigue or a weld defect and propagates rapidly even in a defect-free structure. The probability of brittle fracture depends on the following factors: - steel strength grade - thickness of the material - loading rate - service temperature - steel toughness A hollow section with high strength and thick walls is more sensitive to brittle fracture than one with low strength and thin walls. A high loading rate increases the risk of brittle fracture, and so does a low service temperature. A tougher steel grade is, however, better in low temperatures. Vulnerability to brittle fracture is indicated by the parameters of transition temperature and impact toughness. The methods for calculating the minimum service temperature presented in different versions of Eurocode 3 are contradictory and thus not recommended. This manual presents a simplified method for calculating the minimum service temperature, which is based on tests made to cold formed hollow sections at low temperatures [2].

4.6.1

Parameters affecting brittle fracture in structural hollow sections

The probability of brittle fracture in cold-formed hollow sections depends on both the material and the dimensions of cross-sections and joints. This section presents conditions that must be met to prevent brittle fracture. The conditions are valid for the Eurocode 3 service condition category C, defined as follows: C2: Fracture of critical members or joints, where local failure would cause complete structural collapse with serious consequences to life or very high costs. Mechanical properties of structural hollow sections The ultimate strength and yield strength of structural steel must meet the following condition [2]:

fu ≥ 1, 2 fy

(4.16 )

where

fu fy

is the nominal ultimate strength of steel is the nominal yield strength of steel

129


Chapter 4

The elongations, measured on coupons cut longitudinally from the hollow section flange, must fulfill the following conditions [2]:

A5 ≥ 15 %

(4.17 )

Ag ≥ 20ε y

(4.18 )

where is the ultimate elongation when the measured length of the test piece is 5,65 is the cross-sectional area of the test piece is the uniform elongation corresponding to the ultimate tensile strength (%) is the elongation at yield strain (%)

A5 S0 Ag εy

S0

For the steel grade S355J2H, the required uniform elongation is Ag ≥ 3,38%. The impact toughness KV measured on coupons cut longitudinally from the hollow section flange must fulfill the following condition [2]: The test is carried out at the minimum service temperature of the structure:

KV ≥ 35

J cm 2

(4.19)

Dimensions of the cross-section Table 4.6 Wall thickness (mm) t ≤ 6 mm 6 < t ≤ 10 mm t > 10 mm

Minimum values for the corner radius of hollow sections Minimum internal corner radius

Minimum external corner radius

0,6 t 1,0 t 1,4 t

1,6 t 2,0 t 2,4 t

Furthermore, the manufacturer must show that the manufacturing method used is feasible for the constant production of hollow sections whose internal corners do not have cracks exceeding the allowed values. The maximum depth allowed for a flaw with a blunt notch is 0.2 mm. For flaws with a sharp crack-like tip, the maximum depth allowed is 0,05 mm [2]. The slenderness of the walls of the hollow section must fulfill the following condition [2]:

b+h ≥ 25 t

(4.20)

where

b h t

is the width of the hollow section is the height of the hollow section is the wall thickness of the hollow section

130


Chapter 4

Minimum gap value In gapped N, K and KT joints, the actual gap ga (Figure 4.9) must fulfill the following condition [2]:

ga ≥ 1, 5t 0

(4.21)

where is the wall thickness of the chord

t0

The gap g between brace members (Figure 4.9) must fulfill the following conditions [4]:

g ≥ t1 + t 2

(4.22)

 b  g ≥ 0, 5 1- i   b0  b0

(4.23)

 b  g ≤ 1, 5 1- i   b0  b0

(4.24)

where

t1 and t 2 are the wall thicknesses of the brace members bi is the width of the brace member b0 is the width of the chord

Detail 1

;; ; g

θ

θ

t0

Detail 1 (θ ≤ 60°)

Detail 1 (θ > 60°)

t0

ga

t0

ga

g

Figure 4.9

g

Actual gap 131

Chapter 4

4.6.2


Minimum service temperatures of Rautaruukki structural hollow sections

The Rautaruukki Metform hollow sections shown in Appendix 9.1 (steel grade S355J2H) fulfill all the requirements for hollow section properties (section 1.1) stated in section 4.6.1. According to the latest research [2], they can be used in welded structures down to a temperature of - 40 °C.

4.7

References

[1] ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993) [2] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996 [3] Niemi, E.:Stress determination for fatigue analysis of welded components, IIW/ 115-199193, 1995 [4] CIDECT: Design guide for rectangular hollow section joints under predominantly static loading, Verlag TÜV Rheinland GmbH, Köln 1992 [5] Tarjavuori, P.: Hitsin väsymislujuuden parantaminen jälkikäsittelyllä, Lappeenrannan teknillinen korkeakoulu, Konetekniikan osasto, 1995 [6] CIDECT: Design guide for structural hollow sections in mechanical applications, Verlag TÜV Rheinland GmbH, Köln 1995 [7] CIDECT: Research Project 7M: Working draft: Design guide for hollow section structures under fatigue loading, Aachen 1996 [8] IIW International conference on Performance of dynamically loaded welded structures: Scale effects on the fatigue behaviour of tubular structures, San Francisco, 1997

132


5

Chapter 5

FIRE DESIGN OF STRUCTURAL HOLLOW SECTIONS

In fire situations, the temperature of the steel increases together with the temperature of the gases in fire compartment. As the temperature of the steel increases, its strength and deformation properties are transformed. According to their use, structures have different fire resistance requirements (e.g. requirements for bearing capacity and compartmentation). Often it is necessary to protect steel components in order to slow down the increase in temperature during fire. Several fire retardant methods are applicable for use with hollow sections, for instance, the use of protective materials such as mineral wool or fire-retardant paint. The heat retention capacity of hollow sections can be improved for instance with a concrete infill. Hollow sections are efficient in fire design, since their section factor (the ratio of fire-exposed area to unit mass) is smaller than that of open sections. In addition, hollow sections with their rounded corners are well-suited for fire-retardant painting. Fire retardant methods are described in more detail in Section 5.6. The strength of a hollow section in a fire situation can be calculated by two different methods: either by the properties of the material (yield strength and modulus of elasticity) in a fire situation or by determining the critical temperature of structural steel as a function of degree of utilization. The non-uniform temperature distribution of the steel component can be taken into account when carrying out fire design by the properties of the material. When using the critical temperature method, the temperature distribution of structural steel is assumed uniform. The methods are illustrated in Figures 5.1 and 5.2.

Improve structural fire retardation or select larger hollow section size

Required fire resistance period

Kuva 5.1

Calculate max steel temperature during fire resistance period θ a.max.

Determine properties of material fy.θ and Eθ

Calculate strength in fire situation Rfi.d

Figure 5.2

Efi.d ≤ Rfi.d

yes

OK

Fire design based on properties of material

Improve structural fire retardation or select larger hollow section size

Required fire resistance period

no

Calculate max steel temperature during fire resistance period θ a.max.

Determine utilization ratio µ0

Calculate critical structural temperature θa.cr

Fire design based on critical temperature 133

no

θa.cr ≤ θa.max

yes

OK

DESIGN HANDBOOK FOR RAUTARUUKKI HOLLOW SECTIONS

Chapter 5

5.1

Development of temperature in fire compartments

Factors affecting the development of a real fire include the mode of combustion, the shape of fire compartment, the magnitude and type of fire load, the supply of air needed for combustion and the fire extinguishing system. However, the models used in practical design are simpler. Fire design can be based on either the standard time-temperature curve [1] common to all fire situations or on parametric temperature-time curves.

5.1.1

Standard time-temperature curve

The temperature of fire compartment varies with time. In ISO-834, this is expressed with the following formula [1]:

θ g = 20 + 345 log ( 8t + 1)

(5.1)

where

θg t

is the temperature of the gases in the fire compartment (°C) is the time (min)

1100 1000 900

Temperature (°C)

800 700 600 500 400 300 200 100 0 0

10

20

30

40

50

60

70

80

Time (min)

Figure 5.3

Standard time-temperature curve according to ISO-834

134

90

100

110

120


5.1.2

Chapter 5

Development of temperature according to the parametric model

An alternative method for calculating the evolution of temperature in fire compartment is the parametric model presented in Eurocode 1, Section 2.2 [1]. This model also accounts for openings in the fire compartment, the thermal properties of the wall materials and the magnitude of the fire load in determining the development of the temperature. The parametric model can be used if the fire compartment area is less than 100 m2, there are no openings in the fire compartment ceiling, and the height of the fire compartment does not exceed 4 m [1]. The temperature of the fire compartment increases as long as there is flammable material. Finally, the temperature reaches a maximum value of Θmax (Figure 5.4), after which the fire compartment temperature starts decreasing. The parametric model is not explained more extensively in this manual, since many buildings do not meet the conditions for using it.

Θg

Standard time-temperature curve

Θmax

Parametric fire curve

Heating phase

Cooling phase

t

Figure 5.4

Temperature-time curves in standard and parametric fire models (sketch)

5.2

Development of steel temperature

In fire situations, the temperature of steel members increases slower than that of the fire compartment. The development and distribution of steel temperature depends on the shape of the steel member and its thermal properties. It is always necessary to calculate the steel temperature up to the required fire resistance period, since a steel member may reach its maximum temperature during fire at a point where the fire compartment temperature starts decreasing according to the parametric fire model. By using a fire retardant material, the evolution of steel temperature can be slowed down, which lengthens the fire resistance period.

135

Chapter 5

5.2.1


Development of temperature in unprotected steel members

The increase of temperature in unprotected steel members can be determined from the formula (5.2), when the temperature distribution in the cross-section is assumed uniform [2]:

∆θ a.t

Am = V h˙net .d ⋅ ∆t ca ⋅ ρ a

(5.2)

where

Am V Am V ca h˙net .d ∆t ρa

is the section factor of an unprotected steel member (m-1), at least 10 m-1 (Appendices 9.1 and 9.6) is the exposed surface area of the member per unit length (m2) is the volume of the member per unit length (m3) is the specific heat of steel (J/ kgK) is the net heat flux per unit area (W/ m2) is the time interval (s), maximum 5 s is the unit mass of steel (7850 kg/m3)

The net heat flux consisting of radiation and convection can be expressed as follows [1]:

W h˙net .d = γ n.c ⋅ h˙net .c + γ n.r ⋅ h˙net .r  2  m 

(5.3)

where

γ n.c

is a factor allowing for the differences in national testing (Eurocode 1 default value γn.c = 1,0)

γ n. r

is a factor allowing for the differences in national testing (Eurocode 1 default value γn.r = 1,0)

h˙net .c h˙net .r

is the convective net heat flux (W/ m2) is the radiative net heat flux (W/ m2)

The convective net heat flux is obtained from the following formula [1]:

W h˙net .c = α c (θ g − θ m ) 2  m 

(5.4)

where

αc θg θm

is the convective heat transfer coefficient (Eurocode 1 default value αc= 25 W/m2K is the ambient gas temperature (°C) is the steel surface temperature (°C)

136


Chapter 5

Respectively, the radiative net heat flux is determined from the following equation [1]:

[

]

W 4 4 h˙net .r = Φ ⋅ ε res 5, 67 ⋅ 10 −8 (θ r + 273) − (θ m + 273)  2  m 

(5.5)

where is the configuration factor (Eurocode 1 default value Φ = 1,0) is the resultant emission factor (Eurocode 1 default value εres = 0,50) is the ambient radiation temperature

Φ ε res θr

The constant ca= 600 J/ kgK can be used for the specific heat of steel. Alternatively, specific heat can be determined from the following formulae [2]:

ca = 425 + 0, 773θ a − 1, 69 ⋅ 10 −3 θ a2 + 2, 22 ⋅ 10 −6 θ a3 13002 ca = 666 + 738 − θ a 17820 ca = 545 + θ a − 731

when 20 ≤ θ a < 600 °C

(5.6)

when 600 ≤ θ a < 735 °C

(5.7)

when 735 ≤ θ a < 900 °C

(5.8)

ca = 650

when 900 ≤ θ a < 1200 °C (5.9)

where

θa

is the steel temperature

5.2.2

Development of temperature in fire protected steel members

The increase of temperature in fire protected steel members is calculated from the formula, when the temperature distribution in the cross section is considered uniform [2]:

∆θ a.t

Ap (θ g.t − θ a.t )  10φ  V = ∆t −  e − 1 ∆θ g.t ≥ 0 φ    d p ⋅ ca ⋅ ρ a 1 +  3

λp

(5.10)

where

Ap V Ap V ca dp

is the section factor of the fire protected steel member (m-1), (Appendices 9.1 and 9.6) is the interior area of the fire retardant material in the member per unit lengt (m2) is the specific heat of steel (J/kgK) is the volume of the member per unit length (m3) is the thickness of the fire retardant material (m) 137


Chapter 5

∆t θ a.t θ g.t ∆θ g.t λp

is the time interval (s), maximum 30 s is the temperature of the steel member (°C) is the temperature of the fire compartment (°C) is the increase in the fire compartment temperature within interval ∆t (°C) is the thermal conductivity of the fire retardant material (W/mK) depending on the temperature of the fire retardant material

ρa

is the unit mass of steel (kg/ m3)

The parameter φ is determined as follows [2]:

φ=

cp ⋅ ρ p Ap dp ca ⋅ ρ p V

(5.11)

where

cp ρp

is the specific heat of the fire retardant material (J/kgK) is the unit mass of the fire retardant material (kg/m3)

The delay in the increase in temperature of the steel member due to moisture evaporation can be taken into account if the moisture content of the fire retardant material is great. During moisture evaporation, the steel member temperature is constant (= 100°C). The delay time can be expressed as follows [3]:

p p ⋅ ρ p ⋅ d p2 tv = ( min) 5λ p

(5.12)

where is the moisture content of the fire retardant material (%) is the unit mass of the fire retardant material (kg/m3) is the thickness of the fire retardant material (m) is the thermal conductivity of the fire retardant material (W/mK)

Temperature (°C)

pp ρp dp λp

Fire compartment temperature

Steel member temperature

100 tv t (min)

Figure 5.5

Effect of the moisture content of the fire retardant material to increase in temperature 138


5.3

Chapter 5

Strength and modulus of elasticity of steel in fire situations

As the temperature increases, the strength and the modulus of elasticity of the steel changes. However, the room temperature values for yield strength can be used up to 400 °C. The yield strength correponds to a total elongation of 2%. The modulus of elasticity is constant up to 100 °C. The dependence of strength and modulus of elasticity on temperature is given in Table 5.1 and Figure 5.6.

Table 5.1 Effect of temperature on the strength and modulus of elasticity of steel [2] Temperature θa (°C)

Reduction factor for yield strength ky.θ (fy.θ/ fy)

Reduction factor for modulus of elasticity kE.θ (Ea.θ/ Ea)

20 1,0 100 1,0 200 1,0 300 1,0 400 1,0 500 0,78 600 0,47 700 0,23 800 0,11 900 0,06 1000 0,04 1100 0,02 1200 0,0 Intermediate values can be determined by linear interpolation

1,0 1,0 0,9 0,8 0,7 0,6 0,31 0,13 0,09 0,0675 0,045 0,0225 0,0

1 0,9

k y.θ

k E.θ

0,8

k y.θ , k E.θ

0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0

100

200

300

400

500

600

700

800

900

1000

1100

1200

Temperature (°C)

Figure 5.6

Relative strength of steel and modulus of elasticity as a function of temperature 139


Chapter 5

5.4

Critical temperature in hollow section structures

The critical temperature of hollow sections can be determined as a function of the degree of utilization for Class 1, 2 and 3 cross-sections and for hollow sections with Class 4 crosssections loaded in tension [2]:

  1 θ a.cr = 39, 19 ln − 1  + 482 3,833  0, 9674 µ 0 

(5.13)

In critical temperature calculations, it is assumed that the temperature of the steel member is distributed uniformly through the entire cross-section. However, the method gives a conservative value even if the steel temperature distribution is non-uniform. For Class 4 crosssections of other than hollow sections in tension, the strength in fire situations is sufficient if the steel temperature during fire is lower than 350 °C. The degree of utilization µ0 is determined from the following formula:

µ0 =

E fi.d R fi.d .0

(5.14)

where

E fi.d is the design value for the loads in fire situations (Section 5.5.1) R fi.d .0 is the design value of the strength at room temperature The critical temperature for different degrees of utilization is presented in Table 5.2 and Figure 5.7.

Table 5.2 µ0 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32

Critical temperature of steel θa.cr as a function of the degree of utilization µ0 [2] θa.cr 829 802 779 759 741 725 711 698 685 674 664 654

µ0 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48 0,50 0,52 0,54 0,56

θa.cr 645 636 628 620 612 605 598 591 585 578 572 566

µ0 0,58 0,60 0,62 0,64 0,66 0,68 0,70 0,72 0,74 0,76 0,78 0,80

140

θa.cr 560 554 549 543 537 531 526 520 514 508 502 496

µ0 0,82 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00

θa.cr 490 483 475 467 458 448 436 421 398 349


Chapter 5

900 800

Critical temperature (°C)

700 600 500 400 300 200 100 0 0,1

0,2

0,3

0,4

0,5

0,6

Degree of utilization

0,7

0,8

0,9

1 1,0

µ0

Figure 5.7

Critical temperature of steel

5.5

Determining the strength of hollow section structures in fire situations

The fire design criterion is expressed as follows [2]:

E fi.d ≤ R fi.d

(5.15)

where

E fi.d R fi.d .

is the design value for the effect of loads in a fire situation is the design value for hollow section strength in a fire situation (varies according to time and temperature)

141


Chapter 5

5.5.1

Partial safety factors in fire design

The design value of loads in a fire situation is affected by the expansion and deformation of the material due to temperature. The simpler method is to calculate the fire situation loads by multiplying the design value of loads in normal temperature by the fire situation reduction factor ηfi, so the effect of the structural heat expansion need not be taken into account. The design value for the fire situation load is as follows [2]:

E fi.d = η fi ⋅ Ed

(5.16 )

where

η fi Ed

is the design load reduction factor in a fire situation is the design value for force or moment at room temperature

The fire design reduction factor ηfi is determined from the following formula [2]:

η fi =

γ GA ⋅ Gk + ψ 1.1 ⋅ Qk .1 γ G ⋅ Gk + γ Q ⋅ Qk .1

(5.17 )

where

γ GA γG γQ ψ 1.1 Gk Qk .1

is the partial safety factor for permanent loads in an accident situation [7] is the partial safety factor for permanent load [7] is the partial safety factor for variable load [7] is the combination factor for variable loads [7] is the permanent load is the principal variable load

Figure 5.8 shows various curves of the fire design reduction factor ηfi with different values of combination factor ψ1.1 for γGA = 1,0, γG = 1,35 and γQ= 1,5 (Eurocode 3 basic values).

The values of partial safety factors may vary country by country. The partial safety factors must be checked from national application documents (NADs).

142


Chapter 5

0,8 0,7 0,6

ψ1,1 =

0,9

ψ1,1 =

0,7

ψ1,1 =

0,5

ψ1,1 =

0,2

η fi

0,5 0,4 0,3 0,2 0,1 0 0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

Qk Gk Figure 5.8

Various reduction factor ηfi curves in fire design

5.5.2

Determining the cross-section class in fire design

For compression members, the cross-section classification is calculated as in section 2.2 (Table 2.4). For other structural members, the correction factor ε is used when determining the cross-section limit values. The correction factor is calculated from the following formula [2]:

ε=

235 k E.θ f y k y.θ

(5.18 )

The formulae given in Sections 5.5.3 - 5.5.7 for the determination of strength are valid only for hollow sections with Class 1, 2 and 3 cross-sections, and for hollow sections with Class 4 cross-sections in tension. Class 4 cross-sections of other than hollow sections in tension must be fire protected so that the hollow section temperature does not exceed 350 °C during the fire.

143


Chapter 5

5.5.3

Strength of hollow section subjected to tension in fire situations

The strength of a hollow section in tension is given by the following formula [2]:

N fi.θ . Rd = k y.θ ⋅ N Rd

γ M1 γ M . fi

(5.19)

where

k y.θ N Rd γ M1 γ M . fi

is the reduction factor for the yield strength of steel in temperature θa is the tension strength in normal temperature (Appendix 9.1) is the partial safety factor of the material is the partial safety factor of the material in a fire situation (Eurocode 3 default value γM.fi= 1,0)

5.5.4

Buckling strength of hollow sections in fire situations

The buckling strength for a hollow section in compression is given by the following formula [2]:

fy  χ fi  N fi.θ . Rd =   ⋅ A ⋅ k y.θ  1, 2  γ M . fi

(5.20)

where

k y.θ χ fi

γ M . fi

is the reduction factor for the yield strength of steel in temperature θa is the reduction factor for buckling in a fire situation (always calculated by the buckling curve c) is the partial safety factor of the material in a fire situation (Eurocode 3 default value γM.fi = 1,0)

The modified strength properties of steel are taken into account when calculating the slenderness of a hollow section in a fire situation [2]:

λθ =

L fi k y.θ ⋅ f y π ⋅ i k E.θ ⋅ E

(5.21)

where

L fi i k y.θ k E.θ

is the buckling length in a fire situation is the moment of inertia is the reduction factor for the yield strength of steel in temperature θa is the reduction factor for the modulus of elasticity of steel in temperature θa

144


Chapter 5

The buckling length in a fire situation is calculated as at room temperatures, excluding the following cases [2]: - The buckling length of a column in a non-sway frame can be determined by assuming a semi-rigid column having rigid supports at both ends above or below the fire compartment. A pre-requisite for this is that the fire resistance of the structural members susceptible to buckling, which limit the fire compartment, is at least the same as the fire resistance of the columns. - In a non-sway building in which each floor constitutes a separate fire compartment, the buckling length of the column can be determined as follows: a) columns in intermediate floors Lfi = 0,5 L b) columns in the top floor Lfi = 0,7 L The value of these buckling length may vary country by country. The values should be checked from the national application documents (NAD's)

5.5.5

Bending strength of hollow sections in fire situations

The bending strength of a hollow section is determined by the following formula [2]:

M fi.θ . Rd = k y.θ ⋅ M Rd

γ M1 γ M . fi ⋅ κ 1 ⋅ κ 2

(5.22)

where

k y.θ M Rd γ M1 γ M . fi

is the reduction factor of the yield strength of steel in temperature θa is the bending strength at room temperature (Appendix 9.1) is the partial safety factor of the material is the partial safety factor of the material in a fire situation (Eurocode 3 default valueM.fi = 1,0)

With a non-uniform temperature distribution of the hollow section cross-section, the values of the adaptation factor κ1 are the following: - hollow section exposed to fire on all sides - hollow section exposed to fire on three sides with a concrete or a composite plate on the fourth side

κ1 = 1,0 κ1 = 0,7

With a non-uniform longitudinal temperature distribution of the hollow section, the values of the adaptation factor κ2 are as follows: - supports of a statically undetermined hollow section - other cases

145

κ2 = 0,85 κ2 = 1,0


Chapter 5

5.5.6

Shear strength of hollow sections in fire situations

The shear strength of a hollow section is given by the following formula [2]:

M fi.θ . Rd = k y.θ ⋅ VRd

γ M1 γ M . fi ⋅ κ 1 ⋅ κ 2

(5.23)

where

k y.θ VRd γ M1 γ M . fi

is the reduction factor for the yield strength of steel in temperature θa is the shear strength in normal temperature is the partial safety factor of the material is the partial safety factor of the material in a fire situation (Eurocode 3 default value γM.fi = 1,0)

The adaptation factors κ1 and κ2 are given as in the case of bending strength.

5.5.7

Strength of hollow sections subjected to bending moment and compressive axial force in fire situations

The interaction expression for hollow sections subjected to bending moment and compressive axial force is as follows [2]:

N fi. Ed k y ⋅ M y. fi. Ed k z ⋅ M z. fi. Ed + + ≤1 N fi.θ . Rd M y. fi.θ . Rd M z. fi.θ . Rd

(5.24)

where

N fi.θ . Rd

is the compression strength of a hollow section in a fire situation in temperature θa (5.18)

M y. fi.θ . Rd is the bending strength of a hollow section in a fire situation in temperature θa (by y axis)

M z. fi.θ . Rd is the bending strength of a hollow section in a fire situation in temperature θa (by z axis)

N fi. Ed M y. fi. Ed M z. fi. Ed k y and k z

λ y ⇒ λ y.θ

is the compressive axial load in a fire situation is the bending load in a fire situation (by y axis) is the bending load in a fire situation (by z axis) are calculated as in Section 2.9.1.1, with the following modifications:

λ z ⇒ λ z.θ

χy ⇒

146

χ y. fi 1, 2

χz ⇒

χ z. fi 1, 2


5.6

Chapter 5

Fire retardant methods

Steel structures can be protected against fire by insulating them or increasing their heat retention capacity. Structural solutions can also be used to increase the fire resistance period. In addition to the cost of materials, installation and maintenance costs should be considered when selecting the fire retardant method. Table 5.3

Fire retardant methods

Principle Heat insulation

Improvement of heat retention capacity

Structural fire retardation

5.6.1

Methods - insulation boards - fire retardant paints - sprayed insulation materials - concrete infill - water infill - sprinkler systems - ceiling screens - placing the columns outside the fire compartment - placing the columns inside the wall

Fire retardation by insulation

As compared to unprotected structures, insulated structures are slower to heat and slower to reach the critical temperature. Structures can be insulated with boards or sprayed materials. The thermal conductivity λp of the fire retardant material depends on the temperature of the material, which must be taken into account when calculating the temperature of the steel member. The thermal conductivity characteristics of the fire retardant material are usually shown in manufacturers' brochures. The following is a description of the properties and use of the most common fire retardant materials. Mineral wool boards The fire retardant properties of mineral wool are based on its good thermal conductivity. The sintering temperature of fire protective mineral wool, that is, the temperature in which the fibres melt, must be sufficiently high. Depending on the fire resistance period, a sintering temperature of 800-1000 °C is required. The density of the boards varies between 100-400 kg/ m3, and their thickness varies between 10-100 mm. Mineral wool boards can be fixed mechanically or with glue. In mechanical fixing, nails and bolts are used. Steel spikes and lock plates are also an alternative. Nails are fixed in place by shooting or by resistance butt welding. Steel spikes must be fixed before installing the insulation material. The insulation material is attached to the steel member by lock plates. When using glue, the steel surface must be dry and clean from any dust or oil.

Vermiculite boards The basic material in vermiculite boards is exfoliated mica. The moisture content of the boards is high, as the binding agent is a mineral containing silicate. Vermiculite boards have a good thermal insulation capacity, and the evaporation of moisture in fire situations increases the fire resistance period. The density of boards varies between 350-500 kg/m3 and their thickness between 16-80 mm. 147

Chapter 5


Vermiculite boards can be fixed with glue or nails. Normally, boards are fixed to form a casing around the tube. When using nails, an air slot of approximately 3 mm must be left between the casing and the tube. When using glue, the work temperature must be over 0 °C. The surface of a vermiculite board is smooth and fit for painting. Calcium silicate boards The fire retardant properties of calcium silicate boards are almost as good as those of vermiculite boards. The density of calcium silicate boards varies between 430 - 950 kg/ m3 and their thickness between 6 - 65 mm. Calcium silicate boards are fixed to form a casing around the tube, normally with bolts. Plaster board and profiled elements The use of plaster board as a fire retardant is based on the high content of absorbed water in plaster. After the evaporation of absorbed water, non-reinforced board fails and loses its thermal conductivity. The strength of the board can be improved by reinforcing it with glass fibre, which secures the insulation capacity of plaster board even after the evaporation of absorbed water. The density of boards varies between 770 - 980 kg/m3. The profiled thickness is normally 13 mm, and it can be installed in several layers. Plaster board is fixed to the hollow sections with bolts. One to four board layers can be installed. A mixture of plaster, perlite and glass fibre can be used to pre-form profiled elements, to the shape of the hollow section, which are then fixed around the tube on-site. The profiled elements are fixed with glue or with separate cover plates. Wood fibre plaster boards Wood fibre plaster boards are made by pressing a mixture of wood fibres and plaster to form a 3 hard-surface board. The density of wood fibre boards is approximately 1200 kg/m . Boards are fixed in place with bolts or nails. Cement cellulose boards The material of cellulose cement boards is cellulose, and binding agents include various materials which contain silicate. The density of the board is approximately 1100 kg/m3. Boards are used primarily in light-weight fire-resistant partition walls. Cellulose cement boards are usually fixed directly to the frame with bolts. Sprayed mineral fibre In sprayed mineral fibre, mineral wool fibres and cement are sprayed together with water on the hollow section surface. The density of the sprayed layer is 220 - 500 kg/ m3, and the thickness of the layer is 10 - 60 mm. If the thickness of the layer is more than 35 mm, a reinforcement is installed around the hollow section. The finished surface of the sprayed layer is porous. The sprayed surface can be painted, plastered or clad with boards. The surface of the mineral fibre layer must be protected from oil and, when used outdoors, from moisture. Sprayed vermiculite The sprayed mass consists of vermiculite aggregate and cement, lime or plaster binding agent. The mass is sprayed in a similar manner to mineral fibres. The thickness of the insulation layer is normally 10 - 60 mm, and it can be sprayed in one or more layers of 10 - 15 mm. The density of the sprayed layer is 300 - 800 kg/m3. Sprayed masses with greater density form a stronger surface. 148


Chapter 5

Fire retardant paints The protective effect of fire retardant paints is based on the heat-insulating foam generated as the temperature increases to 250 - 300 °C. However, the foam layer cannot resist long periods of fire, but is peeled off as the fire advances. Fire retardation by paint is thus suited for maximum fire resistance periods of one hour. The thickness of a paint layer is 0,2 - 3 mm, and several layers can be applied. Fire retardant paint can be applied to the tube surface in the same way as normal anti-corrosive paint. Normally, an anti-corrosive undercoat must be applied to the hollow section before the application of fire retardant paint. During on-site work and transport, it is important to keep in mind that the fire retardant coat must not be subjected to mechanical stress or moisture. The compatibility of undercoat and top coats must be checked with the paint manufacturer. The advantage of a fire retardant paint is the thin coat and a finished surface which retains the original shape.

5.6.2

Fire retardation by increasing the heat retention capacity of structural steel

Concrete infill of hollow sections A concrete-filled column is a simple and effective fire retardant method which retains the appearance and the dimensions of the hollow section. The use of reinforcement significantly improves the fire resistance period of the hollow section. The amount of reinforcement can be adjusted to regulate the strength of the column at normal temperature and in fire situations. This way, the same column size can be used in multistory buildings from the ground floor to the top. Since the concrete infill is usually carried out on-site, the light weight of hollow sections can be fully utilized during erection. In normal temperature, a concrete-filled hollow section functions as a composite structure, and in a fire situation, the majority of loads is transferred by the concrete filling and the reinforcement. For fire situations, the tube must be provided with steam exhaust openings. During the fire, the steam pressure is then dissapated through the openings without damaging the section. When placing the concrete infill, sufficiently thin layers must be used and consolidation performed with great care. Fire design tables for concrete-filled columns are shown in reference [4]. Water infill of hollow sections Water infill in a hollow section functions as a cooling agent. The thermal energy generated by fire is consumed by heating and vaporizing the water contained in the hollow section. The effect of water cooling can be enhanced by connecting the hollow sections with an overhead water tank. In a fire situation, the vaporized water ascends to the tank and returns to the hollow sections cooled. To prevent the water from freezing, an agent such as calcium carbonate or calcium nitrate must be added. Water cooling is an effective fire retardant method. By arranging a water circulation, the temperature of hollow sections normally stays at 200 - 250 °C for the entire duration of the fire. Water cooling can be applied only when protecting columns. To prevent leakage, special attention must be paid to sealing the tube joints of the tubes trougt which water circulates.

149


Chapter 5 Sprinkler systems

A sprinkler is an automatic fire extinguishing system which starts operating as the temperature increases in a fire situation. The fire compartment temperature does not increase after the sprinkler system has started operating. National regulations include instructions on allowing for sprinkler systems in the fire design. The profitability of installing a sprinkler system depends on the ratio of its installation costs to the cost of other fire retardation methods.

5.6.3

Structural fire retardation

With appropriate structural solutions, separate fire retardation of hollow section structures can be reduced or completely omitted. The use of structural solutions to improve the fire resistance of structural elements and joints reduces the need for fire retardant materials which increase material and installation costs. Structural fire retardation must be applied individually for each case, and it should be taken into account at the planning stage.

Fire proof ceilings To obtain the space required for Heating, Ventilating and Air Conditioning installations, the room height can be reduced with a false ceiling. False ceilings are also used to cover pipe installations and other services, and a fire proof ceiling can be utilized for the fire protection of structural components (e.g. floor joists) in the intermediate space. In such a case, the fire proof ceiling must be designed and dimensioned appropriately. Also the fixtures that connect the the fire proof ceiling to the floor above must be sufficiently strong to bear the loads during the fire. In practice, the fixtures often constitute governing elements in a fire situation.

Placing columns outside the fire compartment When placing columns outside the external walls, the increase in fire compartment temperature need not be taken into account in column design. A prerequisite for this is that the column is placed sufficiently far from window openings. In a fire situation, the hot gases and flames exiting through window openings increasing the temperature of steel columns that are close to the openings. Window openings are usually placed so close to one another that a flame retardant must be used in the columns. Sheet steel is an example of a flame retardant material.

Placing columns inside the wall The size of the column cross-section exposed to fire is reduced if the column can be placed partially or completely inside the fire retardant material used in the wall structure. The materials used in the wall structure at column locations must be fire-resistant in order to be able to take their protective effect into account in the fire design. A problem may be the connection of the bracing members to the column placed inside the wall.

150


5000

15000

600

The fire resistance period required for the building is 15 min. The steel grade used is S355J2H and the buckling length of the column is Lfi 4,0 m. The temperature evolution in the fire compartment is determined with the standard timetemperature curve [Formula (5.1)].

4000

Example 32 Calculate the resistance of columns (180 x 180 x 5) to axial loads in the building shown in the adjacent figure. In a fire situation, the compressive load on a hollow section is Efi.d = Nfi.Ed = 550 kN.

Chapter 5

Development of temperature in an unprotected hollow section The temperature increase of unprotected steel is obtained from the formula (5.2):

∆θ a.t


The net heat flux per area consists of convection and radiation: h˙net .d = γ n.c ⋅ h˙net .c + γ n.r ⋅ h˙net .r h˙net .c = α c (θ g − θ m ) = 25(θ g − θ m )

[

4 4 h˙net .r = Φ ⋅ ε res 5, 67 ⋅ 10 −8 (θ g + 273) − (θ m + 273)

[

]

= 1, 0 ⋅ 0, 5 ⋅ 5, 67 ⋅ 10 −8 (θ g + 273) − (θ m + 273) 4

4

]

By replacing the material constants and the section factor for steel in the formula (5.2), we obtain: Am V

= 205m −1

ca

= 600

ρa

= 7850

∆t

= 5s

(Appendix 9.6)

J kgK kg m3

151


Chapter 5


∆θ a.t

[ [(θ

] + 273) ]

= 1, 088 ⋅ 10 −3 (θ g − θ m )∆t + 1, 2339 ⋅ 10 −12 (θ g + 273) − (θ m + 273) ∆t = 5, 4406 ⋅ 10 −3 (θ g − θ m ) + 6, 1696 ⋅ 10 −12

∆θ a.t

4

+ 273) − (θ m 4

g

4

4

Figure 5.9 presents the evolution of an unprotected 180 x 180 x 5 hollow section in a standard fire. The curve is calculated using the formula above with time steps of 5 seconds. The maximum temperature conforming to the required fire resistance period (15 min) is:

θ a. max = 673 °C Development of temperature in a protected hollow section A column of 180 x 180 x 5 is protected with 15 mm mineral wool boards. The temperature increase of the fire protected steel structure conforms with the formula (5.10): Ap θ g.t − θ a.t ) (  φ  V ∆θ a.t = ∆t −  e 10 − 1 ∆θ g.t ≥ 0 φ   d p ⋅ ca ⋅ ρ a  1 +   3 Calculate the properties of the fire retardant material and steel using the formula above.

λp

Ap = 210 m -1 V J ca = 600 kgK dp

= 15 mm

∆t

= 5s

λp

= 0, 25

W mK

(appendix 9.6)

(for simplicity, heat conductivity is assumed constant)

kg m3 The parameter φ is determined with the formula (5.11):

ρa

= 7850

φ

=

cp

= 1000 J/kgK

ρ

= 150 kg/m 3

Ap 1000 ⋅ 150 cp ⋅ ρ p dp = 0, 015 ⋅ 210 = 0, 1003 ca ⋅ ρ a V 600 ⋅ 7850

152


Chapter 5

By replacing the values of material properties and the parameter φ in the formula (5.10), the following expression is obtained:

∆θ a.t = 0, 003595(θ g.t − θ a.t ) − 0, 01008 ∆θ g.t Figure 5.9 shows the increase in temperature of a 180 x 180 x 5 hollow section protected with 15-mm mineral wool boards during a standard fire. The curve is calculated using the formula above with time steps of 5 seconds. The maximum temperature conforming to the required fire resistance period (15 min) is:

θa.max = 301 °C

800 Fire compartment temperature 700

Temperature (°C)

600 500

Unprotected hollow section temperature

400

Fire protected hollow section temperature

300 200 100 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

t (min)

Figure 5.9

Increase in temperature of unprotected hollow sections and hollow sections protected with mineral wool boards (t=15 mm) of dimensions 180 x 180 x 5

Calculating structural strength When calculating the compression resistance, the changes in the steel strength and modulus of elasticity caused by temperature must be taken into account. The strength is determined by the maximum temperature during fire resistance period. The adaptation factors for strength and the modulus of elasticity ky.θ and kE.θ are determined by linear interpolation from Table 5.1. The results are listed in Table 5.4.

153


Chapter 5 Table 5.4

Compression strength of columns in a fire situation

Fire situation Unprotected hollow section Fire protected hollow section

0,1786

χfi λfi Formula(5.21) 0,946 0,572

Nfi.θ.Rd (kN) Formula(5.20) 171,4

0,7990

0,824

657,7

θa.max °C 673

ky.θ

kE.θ

0,2948

301

1,0

0,647

The column can resist a 15 minute fire if a 15 mm layer of mineral wool is used, because the compression resistance at 301°C is: fy 355  χ fi   0, 647  N fi.θ . Rd =  = = 657 , 7 kN > N fi. Ed = 550 kN  3436 ⋅ 1, 0  A ⋅ k y.θ  1, 2  γ M . fi  1, 2  1, 0 An unprotected column does not meet the fire resistance requirements, because, because the compression resistance at 673°C is: N fi.θ . Rd = 171, 4 kN < N fi. Ed = 550 kN

Critical temperature by the degree of utilization The fire resistance period for the above structure can also be determined by the degree of utilization. Substituting values in formula (5.14), we obtain: N fi.0. Rd = χ fi.0 ⋅ A

µ0 =

fy 355 = 0, 702 ⋅ 3436 = 713, 6 kN 1, 2 1, 2

N fi. Ed 550 = = 0, 771 N fi.0. Rd 713, 6

The critical temperature is obtained from formula (5.13)   1 1   ln θ a.cr = 39, 19 ln − 1 482 39 , 19 1 + = −   + 482  3, 833  0, 9674 ⋅ 0,7713,833   0, 9674 µ 0  = 505 °C Using the critical temperature, we obtain the same result as above: Unprotected hollow section: θa.max = 673 °C > θa.cr = 505 °C Fire protected hollow section: θa.max = 301 °C < θa.cr = 505 °C

154

OK !


5.7

Chapter 5

Fire design of concrete-filled columns

Relative yield strength

The strength of a concrete-filled column in a fire situation is based on the slower heating of the concrete filling and the reinforcement. The hollow sections walls and the external casing of concrete filling heats rapidly as the fire compartment temperature increases, but the reinforcement and the inner part of the concrete filling retain the normal temperature for a longer period (Figure 5.10).

reinforcement

concrete steel time

Figure 5.10 Decrease in yield strength of a concrete-filled column in a fire situation When designing concrete-filled columns, it is recommendable that hollow sections with large external dimensions and thin walls are used. In this way, the portion of the column's concrete-filled interior area is large. In a fire situation, the concrete and the reinforcement bear more load in a larger column, since the interior temperature is lower and the strength is higher (Figure 5.11).

;; ;

temperature

R = 100 mm

y

R = 150 mm 90 min 60 min

R

30 min

y/R 0

0,25

0,50

0,75

1,0

Figure 5.11 Temperature distribution in cross-sections of concrete-filled hollow sections of various sizes 155


Chapter 5

5.7.1

Using tables in the fire design of concrete-filled columns

A simple conservative design method is presented in Table 5.5 which lists the minimum dimensions for concrete-filled columns. Minimum dimensions depend on the degree of utilization ηfi.t, calculated from the formula [5]:

η fi.t =

N fi. Ed ≤1 N b. Rd

(5.25)

where

N fi. Ed is the design value for axial force in a fire situation N b. Rd is the buckling strength of a concrete-filled column in normal temperature [6]

In addition, reference [5] presents theoretical formulae for the fire design of concrete-filled columns. An alternative design method for concrete-filled columns is to use pre-calculated design tables. Reference [4] lists the buckling strength values in a fire situation for hollow sections with various reinforcement ratios and strength values of concrete. The tables include buckling strength values for fire resistance periods of 60, 90 and 120 minutes.

156


Table 5.5

Chapter 5

Minimum dimensions for columns in a fire situation [5] Fire resistance period

; ;;

As

t

h

us

us

Ac

b

d

As is the area of reinforcement Ac is the area of concrete filling Degree of utilization ηfi.t 0,3

0,5

0,7

Minimum dimension

R30

R60

R90

R120

R180

b or d (mm)

160

200

220

260

400

3

6

6

Reinforcement ratio As / (Ac+ As) (%)

0

Reinforcement position us (mm)

-

30

40

50

60

260

260

400

450

500

Reinforcement ratio As / (Ac+ As) (%)

0

3

6

6

6

Reinforcement position us (mm)

-

30

40

50

60

260

450

500

-

-

3

6

6

-

-

25

30

40

-

-

b or d

b or d Reinforcement ratio As / (Ac+ As) (%) Reinforcement position us (mm)

1,5

N/mm2

- when calculating the degree of utilization, the assumed yield strength value of steel is 235 - the steel grade used as reinforcement is S500 - when calculating the buckling resistance Nb.Rd in normal temperature, greater reinforcement ratios than 3% are not taken into account - the following conditions must be met: b/ t ≥ 25 and d/ t ≥ 25

157

Chapter 5

5.8


References

[1] ENV 1991-2-2: Eurocode 1. Suunnitteluperusteet ja rakenteiden kuormat: Osa 2-2 Palolle altistettujen rakenteiden kuormat, 1995 (ENV 1991-2-2: Eurocode 1. Basis of design and actions on structures. Part 2-2: Actions on structures. Actions on structures exposed to fire, 1995) [2] ENV 1993-1-2: Eurocode 3: Teräsrakenteiden suunnittelu: Osa 1-2: Rakenteellinen palomitoitus, 1996 (ENV 1993-1-2: Eurocode 3: Design of steel structures. Part 1-2: General rules. Structural fire design, 1996) [3] ECCS: Tecnical Committee 3- Fire safety of steel stability: Design manual on the European recommendations for the fire safety of steel structures, First edition, 1985 [4] CIDECT: Design guide for structural hollow section columns exposed to fire, Verlag TÜV Rheinland GmbH, Köln 1994 [5] ENV 1994-1-2: Design of composite steel and concrete structures: Part 1.2: Structural fire design, 1994 [6] ENV 1994-1-1: Design of composite steel and concrete structures: Part 1-1: General rules and rules for buildings, 1994 [7] ENV 1991-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat: Osa 1: Suunnitteluperusteet, 1995 (ENV 1991-1: Eurocode 1: Basis of design and actions on structures. Part 1: Basis of design, 1995)

158


6

Chapter 6

DESIGN OF HOLLOW SECTION STRUCTURES

A structural hollow section is a versatile structural element suitable for various parts of a building. Hollow sections show their best characteristics in columns and lattice structures. The design of structures made of hollow sections is uncomplicated since, due to their good torsional stiffness, lateral-torsional buckling and torsional buckling are usually not governing factors. This also enables the efficient utilization of design software. In this chapter, the design procedure of hollow section structures is handled in its entirety. First, the central issues affecting the design solutions are handled in detail, after which a design solution for a model building is presented.

10500

gable beam

14500

eaves beam door beam primary frame column

48

00

Figure 6.1

0

00

00

10

The model building

The model building is an exhibition hall shown in Figure 6.1. The building is used for arranging fairs and meetings. The frame of the building consists of hollow section columns and of primary and secondary lattices. The building is stiffened with a horizontal lattice located in the roof and by wind bracing in the walls. Hollow sections were chosen as columns, because the wall structure is constructed of lightweight wool elements which do not support the columns about the minor axis. Hollow sections have high torsional and bending stiffness about the minor axis, which makes them a good solution in this case. 159

Chapter 6


Usually, it is advantageous to select a normal K, N or KT type lattice as the roof-supporting structure. In these structures, the chords are hollow sections and the brace member joints are uncomplicated. Usually the full depth of the roof height would be used for the lattice girder. In this case the height between the eaves and the apex is 4,0 m, which would provide an efficient lattice girder or truss. Due to transport requirements however the maximum depth of prefabricated unit must not exceed 2,5 m; therefore a reduced depth parallel sides N girder has been used, which follows the roof profile. A tie is provided at eaves level across the widtt of the building. See figure 6.1 A frame spacing of 10 m is chosen, as it produces an efficient solution for the model building. Between the primary frames, there are purlin trusses with a spacing of 4 m, which enables the use of a shallow profile for the roof. In addition, the purlin trusses are an easy way to provide lateral restraint to the upper and lower chords of the primary lattice. The building is braced using horizontal lattices in the roof plane (Figure 6.2). This solution produces smaller foundations and external column dimensions than using a rigid portal solution. Hollow sections are used as bracing members due to their excellent compression resistance.

bracing lattice

Static model of the column

Figure 6.2

Horizontal stiffening in the model building

On the external walls, there are bracing elements constructed from hollow sections, functioning as tension members. Here, too, hollow sections are an efficient solution, as their stiffness facilitates installation and the elements retain their shape well. However, regarding the foundations, the best solution would be to dimension the bracing elements both as tension and compression members, as the load on the foundations can then be divided into two (Figure 6.3).

160


Chapter 6

apex rafter

compression member tension member

Figure 6.3

Stiffeners at the end wall of the building (load in the rafter is transferred to the tension diagonal)

The wind columns on the side walls of the building, carrying wind load, are supported at the upper end by roof bracing to make them non-sway. The supporting force at the upper end of the wind columns is transferred to primary frames through the roof profile.

6.1

Structural actions

The actions to which the structure is subjected are divided into permanent, variable and accidental actions. Permanent actions include: - self-weight of the structure - fixed equipment Variable actions include: - imposed loads - snow loads - wind loads Accidental actions include: - fire loads - seismic loads The design values of the loads are used in structural design. A design value is obtained by multiplying the characteristic value of the load by the partial safety factor. For load calculation, only the formulae applicable to the model building are shown. Load calculation is handled in more detail in references [1], [2], [3] and [4].

161

Chapter 6

6.1.1


Self-weight and imposed loads

Design values for weights of materials and for imposed loads are given in reference [1]. The weight of the partition walls can be distributed to generate a uniform load. When designing the structural floor elements of a single storey building, the load must be taken into account in the weakest area. The effect of concentrated load must be considered separately. In some cases, the imposed loads on the, structural floor in a single storey building, can be reduced. [1]. When designing columns, the loads on storeys are assumed uniformly distributed. Also in the case of multi-storey buildings, imposed loads can sometimes be reduced [1].

6.1.2

Snow load

The characteristic value of the snow load depends on the geographical location and shape of the building. The following equation is obtained for the characteristic value of snow load on roof structure [2]:

s = µ i ⋅ sk

(6.1)

where

µi sk

is the snow load shape coefficient 2 is the characteristic value of the snow load on the ground (kN/ m ) depending on the geographical location of the building [2].

National application documents (NADs) may present different methods for calculating the characteristic value of the snow load. Reference [2] gives characteristic values for a snow load on the ground and snow load form factor values.

6.1.3

Wind load

As in the case of snow load, the magnitude of wind load depends on the geographical location of the building and on the shape of the structure. The total force due to wind can be expressed as follows [3]:

Fw = qref ⋅ ce (ct , cr ) ⋅ cd ⋅ c p ⋅ Aref

(6.2)

where

ce ct cr cd cp Aref

is the exposure coefficient (Figure 6.4) dependent on ct and cr is the topography coefficient [3] is the roughness coefficient [3] is the dynamic pressure coefficient [3] is the pressure coefficient [3] is the area perpendicular to the wind 162


qref

Chapter 6

2 = 0, 5 ρ ⋅ vref ρ = 1,25 kg/mm3 (air density) vref is the average wind velocity in a 10-minute period measured at the distance

of 10 m above the ground in terrain category II (Table 6.1). The annual probability for exceeding the wind velocity vref is 2% [3]. National application documents (NADs) may include different methods for calculating the characteristic value of wind load.

Table 6.1

Terrain categories

Terrain category I II III IV

Description Rough open sea, lakes with at least 5 km fetch upwind and smooth flat country without obstacles Farmland with boundary hedges, occasional small farm structures, houses or trees Suburban or industrial areas and permanent forests Urban areas in which at least 15% of the surface is covered with buildings and their average height exceeds 15 m

ry

II

go

ry

te

go

ca in

ca

Te

Te

Te

rra

rra

rra

in

ca in

in rra

40

te

go te

te ca

50 Te

Height of building (m)

go

ry

ry

III

IV

60

I

70

30 20 10 0 1,4

1,6

1,8

2

2,2

2,4

2,6

2,8

3

3,2

3,4

3,6

3,8

4

ce

Figure 6.4

Exposure coefficient ce when ct = 1,0 (building stands on level ground)

The formula for calculating the exposure coefficient ce when ct is not equal to 1,0 is given in reference [3]. The values of the pressure coefficient cp for wall structures with wall area Aref greater than 10 m2 are shown in Table 6.3. The values for cp with Aref smaller than 10 m2 are given in reference [3]. 163


Chapter 6

Pressure coefficient cp in vertical walls (Aref > 10 m2)

Table 6.2

Wall A B C d/h ≤ 1 -1,0 -0,8 -0,5 d/h ≥ 4 -1,0 -0,8 -0,5 The intermediate values can be determined by linear interpolation

A

0,2e

B

0,8e

D

wind

A

A

B

d-e

d

d

D

B

wind

A

0,2e

d-0,2e

B

C E b

e h b

E -0,3 -0,3

2. d < e

1. d > e

C

D 0,8 0,6

E b

is min (b, 2h) is the height of the building is the width of the building

Figure 6.5

Pressure coefficients for the wall

6.1.4

Additional horizontal forces

In addition to wind load, horizontal forces in the structure are generated by eccentricities and installation tolerances. A further factor to be taken into account in the design are the horizontal forces transmitted from structural member in compression to the members proving restraint.

Transverse force due to a compression structural element The load caused by a compression structural member in compression (e.g. the upper chord of the lattice) on restraining members (e.g. horizontal diagonal members) is determined as follows [5]:

N 50 L N (1 + α ) q= 60 L

q=

L 2500 L when δ q > 2500

when δ q ≤

(6.3) (6.4)

164


Chapter 6

where is the the axial compression is the length of the compression member is the horizontal deflection in the stiffening system caused by the force q and

N L δq

external loads

α

=

500δ q ≥ 0, 2 L

The use of the formula (6.4) leads to iteration. Formulae (6.3) and (6.4) are applicable for stiffening system that support one member only. When there are several supported members, the horizontal load is determined as follows [5]:

kr + 0, 2 60 L k +α q = ΣN r 60 L

L 2500 L when δ q > 2500

q = ΣN

when δ q ≤

(6.5) (6.6)

where

ΣN

is the sum of the compressive forces of members

kr

= 0, 2 +

nr

is the number of members

6.1.5

1 nr

Combined loads

Chapter 2 presents combined loads in the ultimate limit state [4] [formulae (2.2a, 2.2b and 2.3)]. In the serviceability limit state, the equation for combined loads is the following [4]:

∑ Gk. j + Qk.1 + ∑ ψ 0.i ⋅ Qk.i

(rare combination)

(6.7)

∑ Gk. j + ψ 1.1 ⋅ Qk.1 + ∑ ψ 2.i ⋅ Qk. j

(normal combination)

(6.8)

∑ Gk. j + ∑ ψ 2.i ⋅Qk.i

(long-time combination)

(6.9)

i >1

j

i >1

j

j

i≥1

ψ0, ψ1 and ψ2 are load combination factors obtained from Eurocode 1 [4] or from national application documents (NADs).

Permitted deflections for various structural elements are presented in Table 6.3. 165


Chapter 6 Table 6.3

Permitted deflections [5]

Structure

Recommended limits for deflection

Horizontal deflection Frame structure without crane gantry rails

h/ 150

Other one-story building

h/ 300

One story in a multi-story building

h/ 300 (h is the height of one story)

Total height of a multi-story building

h0/ 500 (h0 is the height of the building)

Vertical deflection

δmax

δ2

Roofs in general

L/ 200

L/ 250

Roofs with person load

L/ 250

L/ 300

Intermediate floors in general

L/ 250

L/ 300

Intermediate floors supporting brittle structures

L/ 250

L/ 350

Intermediate floors supporting columns

L/ 400

L/ 500

δmax

δ2

6.1.6

δmax = end deflection δ0= pre-camber δ1 = deflection due to permanent loads δ2 = deflection due to variable loads

δ0

δ1

Deflection which can affect the appearance of the building L/ 250

Load determination in the model building

Self-weight The self-weight of the purlin trusses and roof is estimated at Gk = 0,5 kN/ m2. Snow load The characteristic value of the snow load on the ground in the building area is 1,5 kN/ m2. The roof angle is 1:6 (α = 9,5°). For a pitched roof, the shape coefficients µ1 = µ2 = 0,8, when 0° ≤ α1 = α2 ≤ 15°. Of the load combinations in Figure 6.6, the most onerous one is chosen. In this case, the snow load is determined by the combination of a) and c), since the structure is symmetrical.

a) µ2, α2 b)

0,5 µ1, α1

c)

µ1, α1

µ1, α2

µ2, α2

d)

0,5 µ1, α2 α1

α2

Figure 6.6 Snow loads on a pitched roof

s = µ · sk = 0,8 · 1,5 = 1,2 kN/m

2

166


Chapter 6

Wind load In the building area, the wind velocity vref = 23 m/s. The reference mean velocity pressure qref is determined from the formula: 2 qref = 0, 5 ρ ⋅ vref = 0, 5 ⋅ 1, 25 ⋅ 23 3 = 0, 33

kN m2

In terrain category III, the exposure coefficient ce has the following value with a building height of 10,5 m measured at the eaves level (Figure 6.4): ce = 1,816 Thus the basic value for the wind load is obtained using the formula (6.2): qwk = qref ⋅ ce ⋅ cd ⋅ c p = 0, 33 ⋅ 1, 816 ⋅ 1, 0c p = 0, 6 c p The following pressure coefficient values are obtained for the model building (Figure 6.5): Wind parallel to the side wall (d/h = 9,52 > 4) A = – 1,0, B = – 0,8, C = – 0,5, D = +0,6, E = – 0,3 Wind parallel to the end wall (d/h = 4,57 > 4) A = – 1,0, B = – 0,8, C = – 0,5, D = +0,6, E = – 0,3 Reference [3] also shows separate pressure coefficients for roof structures.

6.2

Designing columns

As was shown in Chapter 2, a hollow section is an efficient cross-section for a column. The mass of a hollow section is located far from the centre, so the radius of gyration of the hollow section is relatively large in all directions. In column design, the essential factors are the buckling length, the effect of the joint stiffness and the column-to-foundation joint.

6.2.1

Column buckling length

The buckling length of the column is influenced by the actual length, the fixing method of the ends and the lateral support to the member. Theoretical buckling lengths for columns are presented in Table 6.4.

167


Chapter 6

Theoretical buckling length Lc of compression members

Lc = 1,0 L

Lc = 2,0 L

Fixed at both ends, one sway joint

Lc = 0,5 L

Fixed at one end, pinned at the other

Lc = 1,0 L

L

L

Fixed at both ends

L

Fixed at one end

L

Pinned at both ends

L

Table 6.4

Lc = 0,7 L

In frame structures with rigid joints, the benefits of structural hollow sections can be utilized when determining the column buckling length values. Another factor influencing the buckling length in frames is the lateral support of the frame. A non-sway structure can be stiffened either with lattices or by supporting it with a rigid structural element (a lift shaft or a stair well). Generally speaking, a frame structure can be classified non-sway if the following condition is met [5]:

VSd ≤ 0, 1 Vcr

(6.10)

where

VSd Vcr

is the design value of the vertical total load is the buckling load according to the frame elasticity theory in case of sway buckling mode

The stiffening of a sway structure is based on columns functioning as cantilevers and fixed to foundations with a rigid joint, or on the rigidity of the joints. In the case of a continuous column, the buckling length can be determined using Figures 6.7 and 6.8. The distribution factors η1 and η2 in the figures are determined as follows [5]:

Kc + K1 (upper assembly point) K c + K 1 + K 11 + K 12 Kc + K 2 (lower assembly point) η2 = K c + K 2 + K 21 + K 22

η1 =

168

(6.11) (6.12)


Chapter 6

where K1

I Kc = L I K1 = 1 L1 I K2 = 2 L2 I , I1 and I 2 are the values of inertia for

K11

K12

η1

column in question

Kc

K21

η2 K22

corresponding columns parallel to frame

L, L1 and L2 are the values of height for corresponding columns K 11 , K 12 , K 12 and K 22 are the effective stiffness

K2

coefficients for corresponding hollow sections (Table 6.5)

Table 6.5

Effective stiffness coefficients K for hollow sections [5]

Conditions of rotational restraint at the far end of the hollow section Effective stiffness coefficient Fixed

1,0 Ib / Lb (1-0,4 N / Ne)

Pinned Rotation as at near end (double curvature)

0,75 Ib / Lb (1-1,0 N / Ne) 1,5 Ib / Lb (1-0,2 N / Ne)

Rotation equal and opposite to that of near end (single curvature)

0,5 Ib / Lb (1-1,0 N / Ne)

Ib is the hollow section’s moment of inertia parallel to frame Lb is the length of the hollow section N is the compressive force of the hollow section Ne = π2 · E · Ib / Lb2

In Table 6.5, the moments in the hollow section are assumed to be elastic (MSd < Wel · fy /γM0). The hollow section is assumed pinned if its moment exceeds the elastic moment [5]. The buckling length of columns in rigid jointed structures is obtained from Figure 6.7 for nonsway frames and from Figure 6.8 for sway frames. The curve values represent the relation of buckling length to the actual column length.

169


Chapter 6

Lc = 0 1,

η1

1,0 95 0,

pinned

L

9 0,

0,9 85 0,

0,8 8 0,

0,7 75 0,

0,6 7 0, 65 0,

5 67 0,

0,5

6 0,

0,3

5 62 0,

0,4

5 57 0, 55 0,

0,2

rigid

η2

5 0,

Figure 6.7

0

5 52 0,

0,1

0 0,1 rigid

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9 1,0 pinned

Column buckling length values of non-sway frames Lc /L [5]

L

Lc

∞

0,9

0 3, ,8 2 ,6 2 4 2, 2 2,

pinned

=

1,0

0 5, 0 4,

η1

0,8 0 2, 9 1, 8 1,

0,7

7 1, 6 1,

0,6

5 1, 4 1,

0,5 3 1, 25 1,

0,4 2 1,

0,3

15 1,

0

05 1,

0,1

0 0,1 rigid

Figure 6.8

η2

0 1,

rigid

1 1,

0,2

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Column buckling length values of sway frames Lc /L [5]

170

0,9 1,0 pinned


6.2.2

Chapter 6

Effect of joint stiffness on column buckling length

The joints of a frame structure can be considered rigid if the joints are stiffened as shown in Figure 6.9. A non-stiffened joint must be considered semi-rigid when determining the column buckling length. The calculation of the rigidity for non-stiffened hollow section frame structures is dealt with in Appendix 9.5. The effect of a semi-rigid joint on the column buckling length (formulae 6.11 and 6.12) is accounted for in the value of effective inertia of the hollow section. This is determined by the following formula [6]:

Ib.eff =

1 I 3 E ⋅ Ib b 1+ S j ⋅ Lb

(6.13)

;;

where

Sj Ib Lb

is the stiffness of the joint (Nm/rad) is the inertia of the hollow section parallel to frame is the length of the hollow section

≥0,75b0 h

h

M

3h

h

2,5h

;; ;

The stiffness of the joint varies according to the applied moment, as the increasing moment causes the plastification of the joint components that are subjected to the greatest loads. The total moment-rotation curve should thus be known in order to utilize the effect of semi-rigid joints on the calculation of the column buckling length. In case this is not known, the assumption that the support of the column is pinned at the location of the joint is conservative.

≥0,75b0 b0

0,75b0

M

0,7h

1,4t

h

t

M

t

M

t

Figure 6.9

0,85h

Stiffened hollow section frame structures

171


Chapter 6

6.2.3

Column-to-foundation connections

Connections between the column and the foundation were dealt with in Section 3.5. If a rigid column-to-foundation connection is assumed in the structural model the moments transferred into the foundation by the column must be accounted for when designing the holding down bolts and the base plate. If a pinned joint is assumed in the model, moments need not be taken into account. The holding down bolts must be designed such that they are able to carry the construction loads the column is subjected to. The thickness of the second stage concrete layer is taken into account when calculating the buckling length for the design of the holding down bolts.

6.2.4

Column design in the model building

Design the columns in a primary frames. The column-to-foundation connection is in this case assumed rigid. The moment transferred from the column to the foundation must thus be taken into account when designing the connection. The horizontal loads on the building are carried by the bracing lattices, so the columns are supported by hinges at the upper end. The buckling length of the columns can thus be obtained directly from Table 6.4, giving Lc.y = 0,7· 10,3 = 7,21 m. Force quantities The forces on the columns are determined simply by the area of load carried. The resistance must be checked separately for two different load combinations, since at this stage it is not known whether the dominant load is the snow load or the wind load. In case the wind load is dominant: qwd = γ Q.2 ⋅ qwk

kN ⋅ L f = 1, 5 ⋅ 0, 8 ⋅ 0, 6 ⋅ 5 = 3, 6 m

NSd qwd

10, 3 2 Msd = 3, 6 = 47 , 7 kNm (at the lower end of the column) 8 5 Vsd = ⋅ 3, 6 ⋅ 10, 3 = 23, 2 kN 8 kN sd = ψ 0.2 ⋅ γ Q.2 ⋅ Qk .2 = 0, 6 ⋅ 1, 5 ⋅ 1, 2 = 1, 08 2 m (snow load) kN Gd = γ G.1 ⋅ Gk .1 = 1, 35 ⋅ 0, 5 = 0, 68 2 (self-weight) m N Sd = 0, 5(Gd + sd ) L f ⋅ B = 0, 5(0, 68 + 1, 08)10 ⋅ 48 = 422 kN where Lf B

is the frame spacing is the width of the building 172

10,3 m

(wind load)


Chapter 6

In case the snow load is dominant: qwd = ψ 0.2 ⋅ γ Q.2 ⋅ qwk ⋅ L f = 0, 6 ⋅ 1, 5 ⋅ 0, 8 ⋅ 0, 6 ⋅ 5 = 2, 16 M Sd

kN (wind load) m

10, 3 2 = 2, 16 ⋅ = 28, 6 kNm 8

VSd =

5 ⋅ 2, 16 ⋅ 10, 3 = 13, 9 kN 8 kN m2

(snow load)

; ;

sd = γ Q.1 ⋅ Qk .1 = 1, 5 ⋅ 1, 2 = 1, 8

Gd = γ G.1 ⋅ Gk .1 = 1, 35 ⋅ 0, 5 = 0, 68

kN (self-weight) m2

N sd = 0, 5(Gd + sd ) L f ⋅ B = 0, 5(0, 68 + 1, 8) ⋅ 10 ⋅ 48 = 595 kN

For the column in this example, the equivalent uniform moment factors shown in Table 2.9 are obtained as follows: 1 MQ = q ⋅ L2 8 1 9  25 ⋅ qL2 ∆M =  + q ⋅ L2 =  128  8 128  β MQ = 1, 3 (uniform transverse load)

M0

∆M

β Mψ = 1, 8 (restraint moment at the lower end of the column)

The final value for the equivalent uniform moment factors is obtained from the following formula (Table 2.9):

z

y

 MQ  β M . y = 1, 8 +   ⋅ ( β MQ − β Mψ )  ∆M  128  = 1, 8 +  ⋅ (1, 3 − 1, 8) = 1, 48  8 ⋅ 25 

The column is designed using the formula shown in Section 2 (2.57). Try a hollow section with dimensions 300 x 200 x 6 and steel grade S355J2H. The cross-section of the hollow section is Class 4, as h/ t= 300/ 6 = 50 > 36,6. The calculation of the effective cross-section and the determination of column compression and bending resistance are detailed in Chapter 2, so only the results are presented here (Appendices 9.1 and 9.2): N b. Rd = N b.z. Rd = 815, 3 kN (about the z axis) N b. y. Rd = 1090 kN (about the y axis) M y. Rd = 189, 7 kNm 173


Chapter 6

Next, determine the value for parameter ky. The wind load and the snow load are calculated separately, since NSd affects the value of parameter ky . In case the wind load is dominant, [formula (2.58)]:

µ y = λ y ( 2β M . y − 4) = 0, 776( 2 ⋅ 1, 48 − 4) = −0, 807 ky = 1 − µ y

N Sd 422000 = 1 + 0, 807 = 1, 246 < 1, 5 χ y ⋅ A ⋅ fy 0, 677 ⋅ 5763 ⋅ 355

In case the snow load is dominant, [formula (2.58)]:

µ y = λ y ( 2β M . y − 4) = 0, 776( 2 ⋅ 1, 48 − 4) = −0, 807 ky = 1 − µ y

N Sd 595000 = 1 + 0, 807 = 1, 347 < 1, 5 χ y ⋅ A ⋅ fy 0, 677 ⋅ 5763 ⋅ 355

Now we have determined the necessary parameters. The column resistance can be checked using the following condition (2.57): k y ⋅ M y.Sd N Sd 422 1, 246 ⋅ 47 , 7 + = + = 0, 831 < 1 OK ! (if wind load is dominant) N b. Rd M y. Rd 815, 3 189, 7 k y ⋅ M y.Sd N Sd 595 1, 347 ⋅ 28, 6 + = + = 0, 933 < 1 OK ! (if snow load is dominant) N b. Rd M y. Rd 815, 3 189, 7 Regarding shear resistance, the wind load is dominant. We need to calculate the plastic shear resistance, since h / t = 47,6 < 59,1 (Section 2.4.1 and Appendix 9.1): Vpl. Rd = 644, 3 > VSd = 23, 2 kN OK! Thus, the resistance of a 300 x 200 x 6 hollow section is sufficient.

6.2.5

Designing the column-to-foundation joint in the model building

The column-to-foundation joint is made as shown in Figure 6.10. Since the joint was assumed rigid, it is designed according to the guidance in Section 3.5.2. Force quantities The dominant forces and moments are: M Sd = 47 , 7 kNm N Sd . min = 162 kN (construction loads) N Sd . max = 422 kN VSd = 23, 2 kN 174


Chapter 6

The column location is eccentric to the base plate (e1=75 mm), so the moment due to axial force must be taken into account: M Sd .tot = M Sd − N Sd ⋅ e1 = 47 , 7 − 422 ⋅ 0, 075 = 16, 05 kNm (This moment direction is dominant in the design of the base plate.) MSd

Designing the holding down bolts The design value for the compressive resistance of concrete in the foundation is fcd = 14 N/ mm2. The width of the compression area in the concrete is determined by the following formula (example 28):

NSd

;

300

225

tp

e1=75

 2 ⋅ 16, 05  = 1− 1−  350 = 8, 29 mm  400 ⋅ 14 ⋅ 0, 35 2 

450

200

The axial force in the bolts is determined from the vertical equilibrium condition (example 28):

400

300

Ns = by · fcd – NSd = 400· 8,29· 14 – 422000 = –395,6 kN ⇒ no tension in holding down bolts

During installation, however, the holding down bolts carry the total load the column is subjected to. The compressive force of a holding down bolt due to construction loads is:

300

y

150

 2M Sd .tot  = 1− 1− d b ⋅ fcd ⋅ d 2  

e2= 100

100

300

100

Figure 6.10 Base plate

N Sd . min M + N Sd . min ⋅ e2 162 47 , 7 + 162 ⋅ 0, 1 + 0, 5 Sd = + 0, 5 = 147 , 0 kN 4 0, 3 4 0, 3 (The direction of the dominant moment is now reversed as compared to the base plate design. e2 is the eccentricity of the column to the group of holding down bolts.) Fc.Sd =

Design the holding down bolts of steel grade S355J2 for shear force and axial force (example 28): Considering Ø 30 ⇒ As = 561 mm2: As ⋅ f y V 561 ⋅ 345 Fv. Rd = = = 101, 6 kN > Sd 3 ⋅γ M0 3 ⋅ 1, 1 4  Ø > 16 mm ⇒ f = 345 N  y  mm 2  175

OK !

(shear resistance per bolt)

Chapter 6


The buckling length of the holding down bolts is equal to the thickness of the second stage concrete layer, that is, 70 mm: Lc = 70 mm i = 7, 5 mm ⇒ λ = 0,12 < 0, 2 ⇒ χ = 1, 0 Fb.Rd = Ft.Rd =

As ⋅ f y 345 = 561 ⋅ = 176, 0 kN > Fc.Sd OK! γ MO 1, 1

The size of holding down bolts is usually between Ø 24-36 mm. The combined load criterion for a holding down bolt subjected to shear force and axial force is expressed as follows: Fv.Sd Fc.Sd 0, 25 ⋅ 23, 2 147 , 0 + = + = 0, 612 < 1 OK! Fv. Rd Fb. Rd 101, 6 176, 0

Designing the base plate The resistance of the base plate is determined using the value of contact pressure (example 28): N Sd 6 M Sd .tot N 422000 6 ⋅ 16050 (at the base plate edge) , + 2 = + = 3 53 a⋅b 450 ⋅ 400 450 2 ⋅ 400 a ⋅b mm 2 N 12M a 12 ⋅ 16050 150 p2 = p1 − 2 Sd 1 = 3, 53 − = 2, 74 (at the column edge) 2 a ⋅b a mm 2 450 ⋅ 400 450 p1 =

Using the previously calculated stresses, obtain the value for the base plate bending moment at the column edge as follows:  a1 2  a12  150 2 150 2  M Sd =  p2 + ( p1 − p2 )  b =  3, 53 + ( 3, 53 − 2, 74)  400 = 18, 26 kNm 2 3 2 3     The thickness of the base plate is obtained by substituting the bending moment MSd in the formula (3.40): tp ≥

6 ⋅ M Sd ⋅ γ MO 6 ⋅ 18, 26 ⋅ 1, 1 = = 29, 6 mm ⇒ b ⋅ fy 400 ⋅ 345

provide

t p = 30 mm

 t > 16 mm ⇒ f = 345 N  y  p mm 2  Thus, the resistance of the base plate tp = 30 mm and that of the holding down bolts of Ø 30 are sufficient.

176


6.3

Chapter 6

Designing the hollow section beams

When designing a beam, the use of plastic theory is recommended whenever feasible. Plasticity theory can be used when calculating force quantities for Class 1 cross-sections and when calculating resistance values for Class 1 and 2 cross-sections. Rectangular hollow sections are a more efficient alternative when the axial force is small compared to the bending moment and the bending moment is uniaxial. Even with hollow sections with a high h/b ratio, the resistance for lateral-torsional buckling is rarely governing. Allowing for the continuity of the hollow sections reduces span moments, which often makes it possible to select a smaller hollow section size. It is thus recommended that continuous hollow sections as long as possible are used. However, the effect of shop manufacture, transport and site installation on the length of structural elements must be taken into account. By placing the splices appropriately, the forces affecting the joints can be kept to a minimum, which makes it possible to also select a pinned joint. F

F

a)

; ;;;

It is recommended that the end of the hollow section is stiffened with an end plate so that cross-sectional deformation is prevented in transferring the reaction from the webs to the supporting plate (Figure 6.11a). The corner rounding in the hollow section increases the risk of buckling in bending, even when intermediate supports are used, if the notch below the corner rounding is not filled with the weld (Figure 6.11b). The cross-section of a hollow section tends to become distorted if it is subjected to torsion moment. The cross-sectional distortion can be prevented if the torsional moment is transferred to the hollow section as shown in Figure 6.11c.

The semi-rigidity of the joints can be taken into account when designing hollow sections in a frame structure. However, in such a case, the moment-rotation curve of the joint must be known, since the rigidity of the joint varies according to the joint moment. Reference [6] and Appendix 9.4 include equations for determining the moments in a hollow section with semi-rigid joints at both ends and uniform load distribution. Appendix 9.4 also deals with the estimation of rigidity in hollow section joints.

177

b)

F

c)

F

Weld

F

Figure 6.11 Preventing the distortion of a hollow section

Chapter 6

6.3.1


Designing gable beam in the model building

Now design the gable beam of the model building. The width of the building is 48 m and the column spacing at the end of the building is 6 m. Divide the gable beam into four parts which are joined to the end columns on-site using bolted joints. The gable beam is vertically loaded by the support reactions of the purlin trusses. The gable beam transfers the transverse wind load of the hall to the bracing lattices, which means it is also subjected to axial force. Consider a load combination with dominant snow load. The wind load is multiplied by the combination factor ψ0= 0,6. Loads The compressive force on the gable beam is assumed constant along the entire length of the section: qd = ψ 0.2 ⋅ γ Q.2 (c p.1 + c p.2 )qref ⋅ ce = 0, 6 ⋅ 1, 5 ⋅ (0, 6 + 0, 3) ⋅ 0, 6 = 0, 486 N Sd = qd (0, 375 H1 + H 2 )

kN (wind load) 2 m

L 100 = 0, 486 ⋅ (0, 375 ⋅ 10, 5 + 4) = 193 kN 2 2

where L

is the length of the building Cp.2

The total portion of the roof is included in the value of the horizontal load, which is conservative. A more accurate result can be obtained by taking into account the effect of the pressure coefficients of the roof’s wind load.

H1

H2

Cp.1

With dominant snow load, the following restraining force for the purlin truss is obtained: Fy.Sd = 0, 5(γ G.1 ⋅ Gk .1 + γ Q.1 ⋅ Qk .1 ) L p ⋅ L f = 0, 5(1, 35 ⋅ 0, 4 + 1, 5 ⋅ 1, 2) ⋅ 4 ⋅ 10 = 46, 8 kN (self-weight and snow load) where Lp is the purlin spacing Lf is the frame spacing The self-weight Gk.1 can be assumed smaller (0,4 kN/ m2), since the weight of the primary lattice need not be taken into account.

178


Chapter 6

Calculate the forces and moments using plastic theory. For vertical loading, the following static model is obtained: 4000

FSd

FSd

4000

L = 6000

L = 6000

FSd

FSd

θ

4000

2θ 3θ

2θ

θ

3θ

By using equalizing the internal and external work, the plastic moment can be determined: 2 L M Sd ( 3θ + 2θ ) = FSd  L ⋅ θ  ⇒ M Sd = 2FSd ⋅ = 37 , 44 kNm 3  15 2M Sd VSd = = 37 , 44 kN L 3 Resistances at the ultimate limit state The resistance of the hollow section is determined from the formula (2.39), since the hollow section is also subjected to compression load. Consider a hollow section with dimensions 150 x 100 x 8 and steel grade S355J2H. The cross-section of the hollow section is Class 1. The resistance values are as follows (Appendix 9.1): M pl. y. Rd = 54, 59 kNm Vpl.z. Rd = 394, 0 kN > VSd OK! N pl. Rd == 1137 kN The bending resistance, reduced by the axial force, is (Section 2.7.1.1):  N Sd  193  M N . y. Rd = 1, 33 M pl. y. Rd  1 − = 1, 33 ⋅ 54, 59 1 − = 60, 28 kNm > M pl.y.Rd   N pl. Rd  1137   ⇒ M N . y. Rd = M pl. y. Rd = 54, 59 kNm In the interaction expression, there is now only one bending load, so the effect of the parameter α is omitted:  M Sd  M  = 0, 686 < 1, 0 OK!  N . y. Rd  179

Chapter 6


Check that the hollow section will not buckle before the mechanism is generated. The buckling length is 6 m. The buckling resistance is as follows (Appendix 9.2): N b. y. Rd = 369, 9 kN > N Sd OK! The gable beam is restrained laterally supported by the roof profile, so lateral buckling need not be checked. In this case, the effect of bending moment on buckling need not be taken into account, since vertical buckling leads to the expected failure mechanism. Stresses and deflection at the serviceability limit state Check that the stresses do not exceed the yield strength of the material with serviceability limit state loads. The partial safety factors for loads are given by the formula (6.7): FSd = 0, 5(Gk + Qk .1 ) ⋅ L p ⋅ L f = 0, 5(0, 4 + 1, 2) ⋅ 4 ⋅ 10 = 32 kN (self-weight and snow load) 10  10    32 = 35, 6 kNm Mel. y.Sd = F =  9  Sd  9  100  = 128, 6 kN (wind load) N Sd = ψ 0.2 ⋅ Qk 2 = 0, 6 (0, 6 + 0, 3) ⋅ 0, 6(0, 375 ⋅ 10, 5 + 4)  2  Mel. y.Sd N Sd 35, 6 128, 6 N N + = + = 301, 4 σ max = 2 < f y = 355 134, 4 3524 Wel. y A mm mm 2 We determine the deflection of the gable beam using elasticity theory: 1 FSd ⋅ a( L − a) 2 a 1 32000 ⋅ 4(6 − 4) 2 4 L δ= = = 20, 2 mm = 11 −8 6 E⋅I 2 L + a 6 2, 1 ⋅ 10 ⋅ 1008 ⋅ 10 2⋅ 6 + 4 297 L OK! (Table 6.3) δ< 200 Thus, the resistance of a 150 x 100 x 8 hollow section is sufficient.

Designing the door beam in the model building

a

;

Design the side wall door beam which is joined to the intermediate column and to the column of the primary frame. The height of the door is 5 m and its self-weight is 0,75 kN/ m2. This is a sliding door, and the assembly rail is placed at a distance of 200 mm from the hollow section’s z axis and at a distance of 100 mm from its y axis. The door beam is subjected to the self-weight of the door and to the wind load. It is assumed that the lower edge of the door is supported by the floor, in which case only half of the wind load is transferred to the door beam. 180

a-a

a

s

or

Mt

do

Vy

My

100

door beam

5000

6.3.2

Mz

5000

Vz

200

qwd


Chapter 6

Loads The hollow section is subjected to biaxial bending and to torsional moment. No axial force is present. The force quantities are determined using elasticity theory. Consider a hollow section with dimensions of 180 x 100 x 6 and steel grade S355J2H. (torsional load due to the weight of the door and the wind load) qtd = γ G.1 ⋅ Gk .1 ⋅ e1 + 0, 5γ Q.1 ⋅ qwk ⋅ H d ⋅ e2 = 1, 35 ⋅ 0, 75 ⋅ 5 ⋅ 0, 2 + 0, 5 ⋅ 1, 5 ⋅ 0, 6 ⋅ 0, 6 ⋅ 5 ⋅ 0, 1 = 1, 15

kNm m

where e1 is the eccentricity of the vertical load e2 is the eccentricity of the wind load H d is the height of the door Mt .Sd qd M y.Sd Vz.Sd qwd M z.Sd Vy.Sd

= 2, 88 kNm (torsional moment at the support) = γ G.1 ⋅ Gk .1 ⋅ H d = 1, 35 ⋅ 0, 75 ⋅ 5 = 5, 06 kN/m (load causing bending due to the weight of the door) = 15, 8 kNm (vertical bending moment at the centre of the span) = 12, 7 kN (vertical shear force at the support) = 0, 5γ Q.1 ⋅ qwk ⋅ H d = 0, 5 ⋅ 1, 5 ⋅ 0, 6 ⋅ 0, 6 ⋅ 5 = 1, 35 kN/m (load causing bending due to wind) = 4, 22 kNm (horizontal bending moment at the centre of the field) = 3, 38 kN (horizontal shear force at the support)

Resistances at the ultimate limit state Since the hollow section has a Class 2 cross-section, the resistance values are determined using plasticity theory (Appendix 9.1). Mt.Rd = 33,3 kNm Mpl.y.Rd = 58,4 kNm Mpl.z.Rd = 38,8 kNm

Vpl.z.Rd Vpl.y.Rd

= 374,1 kNm > Vz.Sd OK! = 207,8 kNm > Vy.Sd OK!

The interaction expression based on plasticity theory can be used, since there is no axial force present and the hollow section has a Class 2 cross-section [formula (2.9)]. The portion of torsion is checked separately, since the maximum value of the torsion moment is at a different location than that of the bending moment:  M y.Sd  M   pl. y. Rd 

1,66

 M z.Sd  +   M pl.z. Rd 

1,66

 15, 8  =   58, 4 

1,66

 4, 22  +   38, 8 

Mt .Sd 2, 88 = = 0, 086 < 1, 0 OK! Mt . Rd 33, 3 181

1,66

= 0, 139 < 1, 0 OK!


Chapter 6

Stresses and deflection at the serviceability limit state Check that the stresses do not exceed the yield strength of the material with serviceability limit state loads. The load values are given by the formula (6.7):

Mel. y.Sd

kN (self-weight) m = 11, 7 kNm (vertical bending moment at the centre of the field)

Qk

= 0, 90

= 3, 75

Gk

Mel.z.Sd

kN (wind) m = 2, 81 kNm (horizontal bending moment at the centre of the field)

σ max

=

Mel. y.Sd Mel.z.Sd N N 11, 7 2, 81 OK! + = + = 107 , 2 < = 355 f y 145, 5 104, 8 Wel. y Wel.z mm 2 mm 2

Determine the deflection:

δy

δz

5 qk ⋅ L4 5 3750 ⋅ 5 4 = = = 11, 1 mm 384 E ⋅ I 384 2, 1 ⋅ 10 11 ⋅ 1310 ⋅ 10 −8 L L = = OK! (Table 6.3) < 424 250 L L OK! = 6, 7 mm = < 751 250

The resistance of a 180 x 100 x 6 hollow section is thus sufficient for a door beam. It is often necessary to restrict the deflection of the door beam to ensure smooth functioning of the door mechanism.

182


6.4

Chapter 6

Design of trusses

When designing trusses, the joints between brace members and chords are usually assumed pinned, so the brace members are subjected to axial force only. Bending moments need not be taken into account in the design of joints if the conditions shown in the tables of Appendix 9.3 are met. However, the chords as continuous members are also subjected to bending stresses. The chord moment can be reduced by directing the load to the joints of the lattice. Hollow sections function efficiently as compression and tension members, which makes the lattice a light-weight structure in relation to its load-bearing capacity.

Table 6.6

Steps in truss design

Task 1. Determine the loads in the structure. Determine the most severe load combination. The direction of the load is an important consideration, as the tension and buckling resistance values of a hollow section differ.

Illustration 1 2

2. Determine the height of the lattice. Often, this depends on the need for space and on the functional requirements of the building, as well as on the requirements of transport and installation. Select the type and the purlin spacing of the lattice. With these data, generate a static model of the lattic

h

3

3. After calculating the lattice moment by treating the

183

Mmax

h

N0 = Mmax /h = q · L2 /(8h)

Vmax

lattice as a section, make a preliminary selection of the members. Divide the maximum moment value by the lattice height, which gives the initial chord force value (N0 ≈ Mmax/h). Calculate the initial value for the brace member load using the shear force value of the section (V0 ≈ Vmax ). Check the resistance of the joint subjected to the largest load. Adapt the dimensions of the brace members and chords so that the relation of their widths is approximately 0,7-0,8.

V0 = 2 Vmax = 2 (q · L/2) M

V


-1431

kN

kN

0k

N

-1431

68

N

5. Calculate the local resistances of joints. Decide whether to reinforce the joints subjected to the greatest loads or to select stronger members. The reinforcement of joints is efficient if the reinforcement costs are smaller than the increased material costs of the members.

N

2k

0k -48

4. Determine the actual member forces using an appropriate design software. Calculate the member forces for all load combinations. Use a safety factor of 1,0 for the self-weight in cases where the effect of the self-weight is advantageous. In the calculation model, assume that the chord is continuous and that the brace members is pinned. Check the resistance of selected members to actual member forces. Calculate the resistance of brace members in terms of either tension or buckling resistance and check the resistance of chords from the interaction expression for moment and axial force. If the member sizes need to be adjusted, recalculate the member forces using the new dimensions. Check that the dimensions of the members meet the validity criteria shown in the tables of Appendix 9.3.

-48

Chapter 6

1166 kN

N

2k

68

detail 5

detail 5

b0 t0

e

h0

; ; ; ;

N0

θ1

t1, 2

θ2

h1

b1, 2

h2

N1

N2

δmax

6. Calculate the deflection and compare it with the permitted value.

7. Design the transverse support of the lattice and the purlin-to-lattice joints. Determine the location of assembly joints of the lattice, taking transport into account

Purlin

Primary lattice

184


Chapter 6

In lattice design, it is recommended that design software package are used which include resistance data for structural hollow sections (e.g. WinRAMI, Appendix 9.8). The costs of a lattice does not only consist of the weight of the steel, but also of shop fabrication and on-site installation. A lattice with gapped joints and few members may thus be less costly than a lighter weight lattice with several members and overlapped joints. The most advantageous type of lattice and joint shape must be decided on a case by case basis.

6.4.1

Selection of truss type

The most commonly used lattice types are K, KT and N trusses.

K truss

KT truss

N truss

Figure 6.12 Various truss types

A K truss is suitable for long-spanned structures where loads can be transferred directly to lattice joint locations. In K type trusses, the number of members is small and joints are simple. Wide spacings between members also leaves room for tube lead-throughs. However, the buckling length of the upper chord is large, which may result in a heavier chord than in the other lattice types. In general, a K truss is simple and very affordable in terms of fabrication costs. In a KT truss, the spacing of the upper chord supports is more dense, so the resistance of the chord is better than that of a K truss. The joints in a KT truss are, however, more complex to prepare. The joints in the lower chord must often be made overlapped, which increases fabrication costs. In an N truss, the number of members is larger as compared to that of a K truss. In deep and short lattices, the brace member forces are great in comparison with the chord forces. In such a case, an N truss is efficient, since the compressed brace members are shorter than the ones in KT trusses. The joints must usually be made overlapped to avoid high values of eccentricity. 185

Chapter 6


In long-spanned structures, there are big differences in the brace member loads. Close to the support, the loads are greater than in the central area of the lattice. To reduce the weight of the structure, the brace members subjected to smaller loads can be made of lighter-weight hollow sections. However, to facilitate shop fabrication, it is not advisable to use more than 2 to 4 different brace member sizes. With hollow sections having the same external diameter, only one wall thickness should be used in one lattice to avoid confusing them during construction. For simplicity's sake, the chord is usually made of one hollow section size, although axial force varies according to the length of the chord. The chord section can be constructed of hollow sections of different sizes if the lattice is divided into assembly blocks. In such a case, the size of the chord changes at the assembly joint. At the intermediate support of a continuous lattice girder, the supporting force and the moment are at their maximum values. The buckling length of the lower compression chord can be reduced by placing a vertical brace member above the support. It is an advantage if the supporting force is received by a vertical member, since the longer diagonal members are subjected to tension. In subsequent diagonal compression members, axial force is reduced from the value at the support (Figure 6.13).

Figure 6.13 Intermediate support of a continuous lattice girder

Laterally joined trusses Supports for pipes, ducts and working platforms are usually shaped as a bridge by joining the primary lattices with horizontal wind lattices (Figure 6.14). The lattices can also be joined with plates whose joints are designed according to the horizontal loads. It is advisable to brace a structure constructed of two lattices with lacing perpendicular to the lattice plane if the lattices are also subjected to torsion load. (Figure 6.14a). The distance between the lattice nodes perpendicular to the lattice plane can be used as the buckling length of a laterally nonsupported upper chord. In such a case, the transverse force generated by the upper chord must be taken into account in the design. The transverse force can be estimated using the formulae shown in Section 6.1.4. A laterally nonsupported upper chord is presented, for instance, in Figure 6.14b. 186


a)

Chapter 6

b) FN

FN

Lc

;;; ;; ; ;; FN

FN

Mt

Figure 6.14 Laterally joined trusses (Fn is the transverse force preventing buckling)

Due to services, the lattice must sometimes be perforated. To guarantee a sufficient shear resistance, the lattice must be reinforced at the openings. A lower lattice can be constructed between the upper chord and the opening if the height of the opening is smaller than that of the lattice. If the opening and the lattice are of equal height, the opening must be reinforced with a frame. The bending moment due to shear force must be taken into account in the design of the frame and chords. Usually, it is advisable to place the openings in areas with the least possible shear force.

Figure 6.15 Reinforcing openings in trusses

6.4.2

Selection of the chord member

A decisive factor in the selection of chord section are the buckling lengths about different axes. When the buckling lengths are close to equal in both directions, square hollow sections are the most advantageous. The use of rectangular hollow sections can be efficient if the buckling length values differ significantly. However, a broad and shallow chord is not a good solution because of the local strength of the chord face and the shear resistance. It is advisable to select a deep chord when the chord bears significant bending load between the lattice nodes. 187


Chapter 6

The in-plane buckling length of the chord equals the distance between nodes. When the direction of buckling is out of the plane, the buckling length equals the distance between laterally supported points. The above buckling length values can be multiplied by 0,9 if the joints are welded all round and the brace members are not flattened [5]. Lateral stiffening elements are designed for transverse loads and transverse forces due to compression chords (Section 6.1.4). The bending moments generated by joint eccentrities must be taken into account in the chord design. The joints of square and rectangular hollow sections are simpler than the joints of circular hollow sections. Exceptional cases in which the use of circular hollow sections is advisable are triangular lattice sections and space frames. A thick chord wall is efficient in terms of joint resistance, but in terms of the compression resistance of the chord the situation is quite the contrary. A feasible compromise must be reached in the design, or the chord face must be reinforced. Lattices that are subjected to heavy loads and that have a large spacing between the lateral restraints of the compression chord can be constructed using a double chord. In a double chord lattice, the chords are joined to each other directly (Figure 6.16b) or through diaphragm members (Figures 6.16a and c). The horizontal inertia of the chord increases significantly if the chords are joined by diaphragm members. Regarding the resistance of the joint, chord face yield is not possible, since the forces from brace members are transferred directly to chord webs. In the design of a joint shown in Figure 6.16c, forces due to three-dimensionality must be taken into account. When designing the joint shown in Figure 6.16b, the same formulae as with an I profile chord can be used if the space between corner roundings is welded at joint locations. The web thickness value of the I profile tw is replaced with the combined thickness value of both chord webs (2 t0), and the corner radius is the inner corner radius of the chord. The dimensions of the joint shown in Figure 6.16a must be selected so that the brace member welds are accessible. Regarding the shear resistance of the chords in Figure 16a, the following sectional areas are used [7]:

Av = 2, 6 ho ⋅ to

when ho / bo ≥ 1

(6.14)

Av = 2, 0 ho ⋅ to

when ho / bo < 1

(6.15)

where

ho bo

is the height of the chord is the width of the chord

a)

b)

hitsi

; ;;; ;; ; ; c)

Figure 6.16 Double chord

188


6.4.3

Chapter 6

Selection of bracing members

The selection of brace members is less complex, since a brace member with thin walls and large outer dimensions is better in terms of both joint strength and member resistance. However, the slenderness of a brace member should be kept within the limits specified in the tables in Appendix 9.3. Special attention must be paid to the weld between the chord and the brace member when the chord and brace member are of almost equal width (Section 7.4.4). Usually, it is advisable to select such dimensions for the brace member so that the ratio between its width and the chord width falls between 0,7 and 0,8. It is always conservative to take the actual length of the brace member as its buckling length. However, joints welded at all sides have rigidity, and thus a buckling length of 0,75 times the actual length of the member can be used. The buckling length of the brace members can also be calculated with the formulae presented in reference [5], if the b1/b0 ratio is less than 0,6. However, in joints of completely overlapped (λov = 100 %) and flattened brace members, the buckling length is always the actual length of the member. With large values of the joint angle θi , it is advisable to use rectangular brace members to keep the joint eccentrities to a minimum (Figure 6.17a). With small joint angles or near the support of a single span lattice square brace members can be used. (Figure 6.17b). Near the support of a single span lattice, the axial forces in the chord are small, so the chord can more easily carry the bending moments due to joint eccentrities. a)

b) θ

θ

θ

θ

; ;;; ; Figure 6.17 The effect of brace member shape

6.4.4

Design of truss joints

Truss joints can be divided into two main groups: gapped and overlapped. Gapped joints are easier to make, since the brace members can be cut to the conect angle in one go. There is also some tolerance when assembling the lattice. In a gapped joint made of square brace members, the eccentricity is usually large when the brace members are of the same width as the chord. The eccentricity increases the bending load on the chord. The shear resistance of the joint may also govern in a gapped joint. An overlapped joint is more complex to prepare, because the overlapping member must be cut to two different angles. The tolerances of the elements are more restricted than in gapped joints. Respectively, the resistance of the joint is greater, and eccentricity can be removed completely if an appropriate overlap is used.

189

Chapter 6


The smallest angle permitted for brace members in the tables is 30°. In practice, small joint angles should be avoided, as they make the welding of the acute angle side quite difficult. With small angles, even minor flaws in cutting the hollow section can result in large root gaps in the joints. If the joint angle θI is smaller than 60°, the ends of brace members must be chamfered. According to tables in Eurocode 3 Appendix K, the minimum gap for gap joints is (t1+ t2). It is also advisable to check that the gap meets the condition given in reference [8], that is, ga ≥ 1,5 t0, to obtain a sufficient plastic deformation capacity of the chord (ga is the distance between the weld toes, see Figure 6.18).

t1

t1

t2 ga >1,5 t0

ga >1,5 t0

θ2 > 60°

g > t1 + t2

θ2 ≤ 60°

θ1 ≤ 60°

t0

t0

θ1 > 60°

t2

g > t1 + t2

Figure 6.18 Minimum gap values

In the joints of hollow section lattices, local stress concentrations are generated at the joint. However, the stress concentrations are somewhat evened out by the yielding of the brace members and chords. Due to this, the welds at the joints must be designed to have the same strength as the members. The welds that qualify are full penetration single-V butt welds or fillet welds with a throat thickness meeting the following conditions [7]:

a ≥ 0, 95 t a ≥ 1, 00 t a ≥ 1, 07 t

( fy = 235 N mm 2 ) ( fy = 275 N mm 2 ) ( fy = 355 N mm 2 )

(6.16 ) (6.17 ) (6.18 )

In overlapped joints, the lower brace member need not usually be welded at the side which remains hidden. However, in totally overlapped joints (λov = 100 %) even the hidden side must always be welded. This is also the case when the components of the forces in the brace member, parallel to the chord, differ from each other by more than 20 % [7].

Reinforcing truss joints Reinforcing the joint is advantageous if the number of joints to be reinforced is small compared to that of members in the lattice. On one hand, the reinforcement of joints increases fabrication costs; on the other, it reduces the weight of the structure and removes the need to use too many hollow section sizes.

190


Chapter 6

The chord face can be reinforced with plates (Figure 6.19a). This is an effective method for structures with brace members distinctly narrower than the chords. When determining the resistance of joints, the thickness of the chord face t0 is replaced with the thickness of the bracing tp, and the width of the chord is replaced with the width of the bracing bp. The resistance of joints reinforced on the chord face is shown in Tables 9.3.13 and 9.3.15. The bracing is prone to lamellar tearing, which must be taken into account when selecting the bracing material. The shear resistance of the chord can be improved with plates welded to the chord side (Figure 6.19b). The height of the plates is equal to that of the chord. When calculating the sectional area, the thickness of the chord web t0 is replaced with the sum t0+ tp. The resistance of joints reinforced by the chord webs is given in Tables 9.3.14 and 9.3.16. The shear resistance of the joint often governs when the brace member and the chord are of equal width.

; ;;;

Overlapping of brace members increases both the resistance of the chord face and the shear resistance of the chord. Brace members of different widths can be overlapped with reinforcement (Figure 6.19c). The thickness of the reinforcing plate must be at least twice the thickness of the brace member wall (tp ≥ 2t1). A further advantage gained by the use of reinforcement is a symmetrical joint. Preparing the joint shown in Figure 6.19c without reinforcement is not advisable, as the resistance of the joint is smaller than the results shown in the tables in Appendix 9.3. The resistance of reinforced overlapped joints is shown in Table 9.3.17.

a)

b)

c)

Figure 6.19 Reinforcement of truss joints

6.4.5

Truss joints at the supports

The transmission of the lattice shear force to the column must be examined carefully. If the centre of gravity of diagonal member, chord and column do not intersect at the same point, the reaction is transferred as a bending moment to the column. In practice, it is often advantageous to allow a small eccentricity if this facilitates the preparation of the joint. The end of the chord is usually sealed with a plate to achieve sufficient resistance to concentrated load. The joint location of the chord must be designed taking into account the combined effect of axial force, shear force and bending moment. Especially at intermediate support location in continuous lattice sections, the effect of axial force is significant. Various methods are presented in Figure 6.20. 191

Chapter 6


Figure 6.20 Truss joints at the support

6.4.6

Estimation of the truss rigidity

Usually, the deflection of the lattice is obtained directly from the output of the lattice design software. In the software, the lattices can be modelled using continuous chords and pinned brace members. In gapped joints, due to the flexibility of the joints the actual deflection can be more than 12-15% greater than the calculated deflection [7]. In the preliminary planning stage it may be necessary to estimate the lattice deflection by manual calculations. The stiffness of the lattice can be calculated taking into account the effect of the chords only: 2 2    A2  1   A1  2 + ⋅ I = A1 ⋅ H 2  A H 2  A2  A2   1+   1+    A1  A1 

(6.19) A1

where H

A1 and A2 are the cross-sectional areas of chords H is the distance between the centre of gravity axes of chords A2

192


6.4.7

Chapter 6

Designing the truss of the model building

Loads Now design the primary trusses in the model building. Loads on the lattice consist of the self-weight of the structure and the snow load. The purlin spacing is 4 m, so it is an advantage to use the same spacing in the brace member joints. With the 10 m spacing of primary lattices, the following load on the lattice is obtained: qd = (γ G,1 ⋅ Gk ,1 + γ Q,1 ⋅ Qk ,1 ) L f = (1, 35 ⋅ 0, 5 + 1, 5 ⋅ 1, 2)10 = 24, 8 FSd = qd ⋅ L p = 24, 8 ⋅ 4 = 99, 2 kN

kN m

(load on the node)

where Lf Lp

is the frame spacing is the purlin spacing

Truss shape In a roof lattice, the lower chord can be made either straight or bent. Axial forces are greater in a bent lower chord. With a straight lower chord, the height of the lattice is greater in the centre of the span, which is also the location with the greatest moment. Thus, the axial force is smaller in a straight lower chord. With to a straight lower chord, the length of brace members is greater, which may increase the weight of the lattice as compared to a lattice with a bent lower chord. In the model building, the lower chord is bent, but the axial force of the lower chord is resisted by a tension rod. The use of the tension rod reduces the amount of steel in the brace members, but increases the number of joints in the lattice. The span is long, which makes the shear force small compared to the bending moment. An N lattice is thus too heavy to be used in the model building. For a better upper chord resistance, a KT lattice is selected, since the spacing of the purlins is large. The height of the lattice is estimated by the span. In practice, the optimal height varies between L/9L/12. In the model building, a lattice height of 6,5 m is selected. diagonal

N0.Sd = -1431 kN upper chord lower chord N0.Sd = -462 kN

α = 9,5°

H

vertical

tension rod

N0.Sd = 1166 kN

Figure 6.21 Primary truss 193

L

N1.Sd = -480 kN N1.Sd = 682 kN

Chapter 6


Determining the member forces The initial load on chords and brace members can be determined by calculating the lattice forces with the formulae of a simply supported section: N 0.Sd ≈

qd ⋅ L2 24, 8 ⋅ 48 2 = = 1099 kN (load on upper chord) (8 H ) ( 8 ⋅ 6 , 5)

N1.Sd ≈ 0, 5qd ⋅ L 2 = 0, 5 ⋅ 24, 8 ⋅ 48 2 = 841, 7 kN (load on brace member) Based on these values, select an upper chord of 200 x 200 x 8 and brace members of 120 x 120 x 6. For the buckling length of the upper chord, take 90% of the horizontal distance between purlins and, 90% of the vertical distance between the lattice nodes. Thus the following buckling length for the upper chord are obtained: 2 = 1, 83 m cos 9, 5 4 = 0, 9 = 3, 65 m cos 9, 5

Lc. y = 0, 9 Lc.z

The size of the lower chord is more complex to define manually in this particular case, so by considering the brace member joint, select a lower chord of dimensions 140 x 140 x 5. The axial force on the tension rod can be assumed equal to that on the upper chord. Select a tension rod of Ø80 mm (fy = 345 N/mm2). The output from the design software gives the following maximum values for the forces: Upper chord:

N 0.Sd M0.Sd Lower chord: N 0.Sd Tension rod : N 0.Sd Brace members: N1.Sd N1.Sd

= −1431 kN = 4, 0 kNm = −462 kN = 1166 kN = −480 kN = 682 kN

(compression) (compression) (tension) (compression) (tension)

Resistance of the upper chord First, consider the resistance of the upper chord with a 200 x 200 x 8 hollow section and the steel designation S355J2H. The hollow section has a Class 1 cross-section and its resistances are as follows (Appendix 9.2): N b. Rd = N b.z. Rd = 1484 kN (buckling resistance, Lc= 3,65 m) (buckling resistance, Lc= 1,83 m) N b. y. Rd = 1806 kN M pl. y. Rd = 135, 8 kNm Substituting the resistances in the interaction expression (2.57): k y ⋅ M y.Sd 1431 1, 182 ⋅ 4, 0 N Sd + = + = 0, 999 < 1, 0 OK ! N b. Rd M y. Rd 1484 135, 8 194


Chapter 6

Designing the lower chord and the tension rod

a a

;

The use of the tension rod generates compression in the lower chord. Compression is present only in the first diagonal spacing, after which the lower chord is subjected to tension. Support the compression element of the lower chord laterally to the lattice purlin with stays. This way, the buckling length of the lower chord of dimensions 140 x 140 x 5 is equal in both directions:

Lc

a

a

a-a

 4  = 0, 9  = 3, 65 m  cos 9, 5 

χ ⋅ A ⋅ fy 355 = 0, 617 ⋅ 2636 γ M1 1, 1 OK! = 525, 1 kN > 462 kN The resistance of the tension rod (Ø80 mm) is determined by the area and the yield strength:

N b. Rd =

Nt . Rd

=

A ⋅ fy 345 = π 40 2 ⋅ γ M0 1, 1

= 1576 kN > 1166 kN

( Ø > 16 mm ⇒ fy = 345 N/mm2 ) OK!

Resistance of the brace members The resistance of a tension brace member of dimensions 120 x 120 x 6 is determined in a similar way to that of the tension rod: Nt . Rd =

A ⋅ fy 355 = 2643 = 853 kN > 682 kN 1, 1 γ M1

The buckling length of a 120 x 120 x 6 compression brace member is 0,75L: Lc = 0, 75 ⋅ 3, 54 = 2, 66 m This buckling length is used to determine the buckling resistance: N b.Rd =

χ ⋅ A ⋅ f y 0, 690 ⋅ 2643 ⋅ 355 = = 589 kN > 480 kN 1, 1 γ M1

In brace members subjected to smaller loads, a smaller hollow section can be used.

195


Chapter 6

Resistance values of joints To simplify shop fabrication, the joints of the upper chord in the model building are designed gapped and those of the lower chord overlapped. Thus, the eccentricities can be kept small. In a simply supported lattice section, the shear force is at its greatest at the supports, which is also where the brace member forces are at their greatest. Due to this, the resistance values of joints at the two outermost purlins (corners 1 and 5 in the calculation model) must be checked. In practical design, the resistance values of all joints must be checked. detail 1

b0

Joint of corner 1

;

h0

N0

detail 1

t0

t1

θ

b1

h1

N1

Figure 6.22 Joint at corner 1

The joint at corner 1 is a Y joint with a tension brace member. The formulae for this type of joint are shown in Table 9.3.1 The geometry and forces at the joint are as follows: Chord: 200 x 200 x 8 (A0 = 5924 mm2) Brace member: 120 x 120 x 6 N 0.Sd

0, 4 479000  = −479 kN (compression) = 1, 13 ⇒ kn = 1, 0 ( puristusta) ⇒ kn = 1, 3 −    0, 6 5924 ⋅ 355 

N1.Sd

= 682 kN

θ

= 54o b = 1 = b0 h = 1 = b0

β η

(Figure 6.21)

120 = 0, 6 200 120 = 0, 6 200

196


Chapter 6

Chord face yield Since β = 0,6 < 0,85, the governing failure mode is the yielding of the chord face:

N1. Rd

f y ⋅ t 0 2  2η 1, 1 + 4 1 − β  kn =  γ Mj ⋅ γ M 0 (1 − β ) sin θ  sin θ 355 ⋅ 8 2  2 ⋅ 0, 6 + 4 1 − 0, 6  1, 0 1 = 256, 1 kN =  1, 1 (1 − 0, 6 ) sin 54  sin 54

The joint resistance is not sufficient, since N1.Sd = 682 kN > N1.Rd. When the chord face value governs, reinforcement welded to the chord face helps improve the resistance of the joint. In the model building, reinforcement is an affordable solution, since only the outermost joints must be reinforced.

b0 t0

h0

N0

bp

Lp

;; tp

θ

t1

h1

b1

N1

Figure 6.23 Reinforcing the joint at corner 1

Yield of the reinforced chord face Determine the resistance of the joint which has a 275 x 185 x 15 reinforcing plate welded to the chord face (Table 9.3.13): N1. Rd

355 ⋅ 15 2  2 ⋅ 0, 65 + 4 1 − 0, 65  1, 0 1 = 1019 kN > 682 kN =  1, 1 (1 − 0, 65) sin 54  sin 54

Check the length of the bracing: Lp

120   h  = 275mm ≥  1  + b p (b p − b1 ) =  + 185(185 − 120) = 257 mm OK!  sin 54   sin θ 1 

Lp

120  = 222 mm = 275mm ≥ 1, 5  sin 54  197

Chapter 6


The dimensions of the chord and brace members must fall within the validity area given in Table 9.3.13: b0 + h0 400 = = 50 > 25 OK! t0 8 b1 h1 120 = = = 20 < 35 OK! t1 t1 6 b1 + h1 240 = = 40 > 25 OK! t1 6 b p = 185 mm > b0 − 2t 0 = 184 mm OK!

b1 h1 120 = = = 0, 6 > 0, 25 OK! b0 b0 200 h 0, 5 < 1 = 1 < 2 OK! b1 h 0, 5 < 0 = 1 < 2 OK! b0 b0 h0 200 = = = 25 < 35 OK! t0 t0 8

Joint at corner 5 b0

det 5

; g

e

h0

det 5

θ1

ga

h1

N1

N0

t0

t1, 2

θ2

b1, 2

h2

N2

Figure 6.24 Joint at corner 5

The joint at corner 5 is a gapped K joint. The formulae for this type of joint are given in Table 9.3.2. The geometry and forces at the joint are as follows: Chord: 200 x 200 x 8, A0 = 5924 mm2 Brace members: 120 x 120 x 6 N 0.Sd = −859, 1 kN (compression) ( puristusta)

 0, 4   859100 ⋅ 1, 1 ⇒ k n = 1, 3 −  = 1, 00   0, 6   5924 ⋅ 355 

N 1.Sd = −480 kN N 2.Sd = 480 kN (θ < 60°, so the brace member must be welded to the chord with a V θ 1 = θ 2 = 54 o groove)  1 120 + 120  β = = 0, 6, β < 1 −   = 0, 92, so the punching shear of the chord must  2 ⋅ 200  γ  be checked

γ =

b0 200 = = 12, 5 2t 0 16 198


Chapter 6

The eccentricity at the joint created by the joint gap is as follows: g = 50 mm h2  h1  sin θ 1 ⋅ sin θ 2 h0 − = 36, 5 mm (eccentricity at the joint) e= + + g  2sinθ 1 2 sin θ 2  sin(θ 1 + θ 2 ) 2 Yield of the chord face First, determine the resistance by the yield of the chord face:

N1. Rd

m   m b + hi ∑ 2∑ i  f y ⋅ t0 i =1 1, 1 i =1 = 8, 9  kn γ  sin θ  2m ⋅ b0  γ Mj ⋅ γ M 0    

= 8, 9

355 ⋅ 8 2  120 + 120 + 120 + 120  1 1, 0 12, 5 = 482, 0 kN OK!  sin 54  4 ⋅ 200 1, 1

Chord shear Obtain the value for the shear resistance of the entire chord as follows: Av = ( 2h0 + α ⋅ b0 )t 0 = ( 2 ⋅ 200 + 0, 137 ⋅ 200)8 = 3419 mm 2

α=

1 1 = 0, 137 2 = 4 ⋅ 50 2 4g 1+ 1+ 2 3 ⋅ 82 3t 0

N1. Rd =

f y ⋅ Av 1, 1 355 ⋅ 3419 1 = = 787 , 4 kN 3 sin θ γ Mj ⋅ γ M 0 3 sin 54 1, 1

Vpl. Rd =

f y ⋅ Av = 637 , 1 kN 3 ⋅γ M0

VSd = 480 sin( 54) = 388, 3 kN In the determination of the compression resistance of the chord, the effect of shear force must be taken into account, since: VSd > 0, 5VRd N 0. Rd

N 0. Rd

2   2VSd   fy =  A0 − Av  − 1   Vpl. Rd   γ M 0  2   2 ⋅ 388, 3   355 =  5934 − 3419 − 1  = 1862 kN  637 , 1   1, 1  > N 0.Sd = 859, 1 kN OK!

199

Chapter 6


Failure of the brace member The effective width of the brace member is: beff N1. Rd

10b1 ⋅ t 02 10 ⋅ 120 ⋅ 8 2 = = = 64 mm < 120 mm b0 ⋅ t1 200 ⋅ 6 1, 1 = f y ⋅ t1 ( 2h1 − 4t1 + b1 + beff ) γ Mj ⋅ γ M 0 = 355 ⋅ 6( 2 ⋅ 120 − 4 ⋅ 6 + 120 + 64)

1 = 774, 5 kN 1, 1

Shear failure of the chord face In this case, the shear failure resistance of the chord must also be taken into account: bep N1. Rd

10t 0 ⋅ b1 10 ⋅ 8 ⋅ 120 = = 48 ≤ 120 mm b0 200 f y ⋅ t 0  2h1 1, 1 = + b1 + bep   γ Mj ⋅ γ M 0 3 sin θ  sin θ =

=

355 ⋅ 8  2 ⋅ 120 1 + 120 + 48 = 856, 1 kN  1, 1 3 sin 54  sin 54

Resistance of the joint Thus, the chord face yield was the most important governing failure mode N1.Rd = 482,0 kN > N1.Sd = 480 kN OK ! Check the validity criteria in the joint table: b1 120 = = 0, 6 > 0, 35 OK! b0 200 b1 0, 01b0 = 0, 6 > 0, 1 + = 0, 35 OK! b0 t0 h 0, 5 < 1 = 1 < 2 OK! b1 h 0, 5 < 0 = 1 < 2 OK! b0 b0 h0 200 = = = 25 < 35 OK! t0 t0 8 b0 + h0 400 = = 50 > 25 OK! t0 8

b1 h1 120 = = = 20 < 35 OK! t1 t1 6 b1 + h1 240 = = 40 > 25 OK! t1 6

200


Chapter 6 g

t0

b1 h1 E = = 20 < 1, 25 = 30, 4 OK! t1 t1 fy

L

θ1 e 36, 5 OK! −0, 55 < = = 0, 18 < 0, 25 200 h0 50 g = = 0, 25 > 0, 5(1 − β ) = 0, 5(1 − 0, 6 ) = 0, 2 OK! b0 200 50 g = = 0, 25 < 1, 5(1 − β ) = 1, 5(1 − 0, 6 ) = 0, 6 OK! b0 200

L

L

L θ2

ga t1

t2

ga = g − 2 L = g − 2 tan( 90 − θ )t1 = 50 − 2 tan( 90 − 54)6 OK! ga = 41, 3 mm > 1, 5t 0 = 1, 5 ⋅ 8 = 12 mm OK!

Lattice joint at the support a e

2 x 15

200 x 200 x 8

a

;

300 x 200 x 6

a-a

tp

120 x 120 x 6

Figure 6.25 Lattice joint at the support

The lattice-to-column joint is made using an end plate. To facilitate the fabrication of the joint, eccentricity is allowed about the neutral axis of the column. As a result, the upper chord must transfer the shear force to the column. The eccentricity also causes bending moment in the upper chord. When the eccentricity is 200 mm, the forces loading the end of the upper chord at the diagonal/chord intersection are as follows: VSd = 595 kN M Sd = 595 ⋅ 0, 2 = 119 kNm N Sd = 78 kN

(joint at corner 1) 201


Chapter 6

Resistance of the end of the chord to the combined load The chord needs to be reinforced, because the shear resistance of a 200 x 200 x 8 hollow section is not sufficient (Vpl.Rd = 552 kN < VSd). Reinforce the chord by plates welded to the side of the chord. The thickness of the plates is tp= 5 mm. Reinforcing plates extend over the joint of the diagonal member. The plates welded to the sides of the chord are taken into account when determining the resistance of the chord end, and the plate welded to the lower flange of the chord is taken into account when determining the resistance of the joint (joint at corner 1). The resistance values of the reinforced upper chord are: fy 200 355 h Vpl. Rd =  A + 2(t p ⋅ h) =  5924 + 2( 5 ⋅ 200) = 924, 6 kN  b + h  3 ⋅γ M 0   3 ⋅ 1, 1 200 + 200 fy 355 = 420900 + 2 0, 25 ⋅ 5 ⋅ 200 2 = 168, 1 kNm M pl. Rd = Wpl + 2 0, 25t p ⋅ h 2 1, 1 γ M0

[

)]

(

[

]γ

N pl.Rd = A + 2(t p ⋅ h)

fy M0

[

(

= [ 5924 + 2( 5 ⋅ 200)]

)]

355 = 2557 kN 1, 1

The shear force now exceeds half of the shear resistance; thus, its effect must be taken into account in the interaction expression. The effect of shear force is taken into account when calculating the resistance to bending and axial force [formula (2.46)]:

MV . Rd

 ρ ⋅ Av2  W −  pl  fy 8t   = = γ M0

NV . Rd = ( A − ρ ⋅ Av )

fy

γ M0

 0, 126 ⋅ 4962 2  − 520900   355 8 ⋅ 13   = 158, 5 kNm 1, 1

= 2356 kN

The effect of axial force is taken into account in the interaction expression (2.49). Bending moment now occurs only in the other direction, so the latter term in the condition is omitted. Buckling need not be taken into account, as it is the local resistance of the chord which is considered here. The following value for bending resistance with the presence of axial force is obtained:  N  78  M N . Rd = 1, 26 MV . Rd  1 − Sd  = 1, 26 ⋅ 158, 5 1 − = 193, 1 kNm > M pl , Rd   NV . Rd  2356  ⇒ M N . Rd = MV . Rd = 158, 5 kNm ⇒ M N . Rd > M Sd = 119 kNm

OK!

Resistance of the chord end to concentrated load Next, calculate the resistance of the upper chord webs to the reaction using the formulae shown in Section 2.10. The end of the upper chord is sealed with a plate, and the gap between the corner rounding and the splice is filled with weld. The deformation of the hollow section and the resulting secondary bending loads therefore need not be taken into account. 202


Chapter 6

The magnitude of the concentrated load to each web is:

VSd = 297 , 5 kN 2 The resistance to concentrated load is determined by formulae (2.61) and (2.62), which give the following results: fy 355 Ry. Rd = (ss + s y )(t 0 + t y ) = ( 2 ⋅ 6 + 4 ⋅ 15 + 2 200 ⋅ 13 )( 8 + 5) = 730, 0 kN > FSd OK! 1, 1 γ M1 FSd =

Ra. Rd = 0, 5(t 0 + t p )

2

E ⋅ fy

(1 + 3 ⋅ 0, 2) γ M1

= 0, 5( 5 + 8) 2 210000 ⋅ 355

(1 + 3 ⋅ 0, 2) 1, 1

= 1060, 1 kN > FSd OK!

Ra. Rd

In the previous resistance calculations, the web reinforecement was included in the thickness of the webs. The web thickness used in calculations is then t = 8 + 5 = 13 mm. Deflection of the lattice The second moment of area of the lattice is estimated with the formula (6.19): 2 2    A2   1   A1  + A2 ⋅ H 2  I = A1 ⋅ H 2   A2 A2  + + 1 1       A1  A1  2 2    5027  1  = 0, 1149 m 4 = 5924 ⋅ 6500 2  5924  + 5027 ⋅ 6500 2    5027  5027    1+   1+   5924  5924  where

A1 A2

is the area of the upper chord is the area of the tension rod

At the serviceability limit state, loads are calculated using the characteristic values (6.7), which gives the following load on the lattice: qk = (Gk + Qk ) L f = (0, 5 + 1, 2)10 = 17

kN m

where Lf

is the frame spacing

The deflection is calculated by assuming the lattice section is simply supported:

δ=

L L 5qk ⋅ L4 5 ⋅ 17000 ⋅ 48 4 mm = = 47 = < 384 E ⋅ I 384 ⋅ 2, 1 ⋅ 10 11 ⋅ 0, 1149 1032 200 203

OK ! (Table 6.3)

Chapter 6

6.5


Stiffening hollow section structures

To keep the displacements generated by the horizontal loads within permissible limits, the structure must be stiffened. The stiffening methods can be divided into five groups: - mast stiffening - frame stiffening - tower stiffening - plate stiffening - lattice stiffening In mast stiffening, the columns function as masts fixed to the foundation with a rigid joint, receiving the horizontal loads. When the height of the building increases, the foundations are subjected to larger moments, and the buckling length factor of the columns is great. This makes mast stiffening suitable only for low buildings. The advantage of this method is the simplicity of the installation, as no separate supports are needed during installation. Frame stiffening is a natural method to use when frame joints are rigid and lattice stiffening is not feasible due to, for instance, openings in the walls. In frame stiffening, horizontal loads are transferred as moments to the corners of the frame. Rigid joints are, however, more costly to make than pinned ones. In tower stiffening, horizontal loads are transferred through rigid elements such as lift shafts or stairwells to the foundation. This method is well-suited for buildings in which stiffening elements are made using slip casting. The problem is to produce a sufficiently firm joint when joining the building frame to the stiffening elements. In plate stiffening, horizontal forces are transferred to the ground through plates joined to the frame. Plates can be vertical (e.g. walls cast on-site) or horizontal (e.g. profile). Plate stiffening is an affordable method: since the plate already forms a part of the building, no separate stiffening elements are needed. Horizontal loads must also be taken into account when dimensioning the joining of the stiffening plates, as this makes the joining firmer than it would be without taking the stiffening effect into account. Lattice stiffening is an effective method in high buildings. Stiffening lattices transfer the horizontal loads to the members as axial forces. Hollow sections make excellent stiffening lattice members, since their radius of gyration is large about both axes. A disadvantage of the lattice stiffening is the space needed for lacings at doors, windows and other openings made to walls. Several stiffening methods can be used in the same building. The buckling length values of the structure can be determined using the model for non-sway frames (Figure 6.7) when using tower, plate or lattice stiffening. In mast and frame stiffening, the buckling length values must be determined using the sway-frame model (Figure 6.8).

6.5.1

Designing the stiffening elements in the model building

In the model building, the lattice stiffening method is used. This manual deals with the design of transverse stiffening elements only for the model building. The longitudinal stiffening elements are designed according to the same principles. 204


Chapter 6

Lateral stiffening The wind loads in the side wall of the building are transferred to the stiffening lattices in the end wall through the horizontal lattice in the roof parallel to the side walls. However, the horizontal force of the side wall wind columns is transferred to the primary columns through the roof profile. This way, the eaves section of the side wall can be made lighter. The joints of the stiffening profile must be checked for the loads created by the horizontal forces. End wall bracing Design the bracing elements of the end wall. The horizontal load is divided evenly in the end walls, as the building is symmetrical and the bracing in the end walls is similar. In the pressure coefficient of the wind load, the effect of negative pressure must also be taken into account, so cp= 0,6 + 0,3= 0,9. Thus the following value for the horizontal force of the end wall is obtained: FSd = γ Q.1 ⋅ c p ⋅ qref ⋅ ce (0, 375 ⋅ H1 + H 2 )

L 100 = 1, 5 ⋅ 0, 9 ⋅ 0, 6(0, 375 ⋅ 10, 5 + 4) = 321, 5 kN 2 2

where H1 is the height of the eaves H2 is the height of the roof structure L is the length of the building The bracing is made with two members, one subjected to compression and the other to tension. The total horizontal force is transferred by the tension lacing. The tensile force is as follows: NSd = 321,5 / cos 48 = 480 kN Consider a hollow section of dimensions 100 x 100 x 4: Nt.Sd = 1495⋅355 / 1,1 = 482,5 kN FSd

Det 1

;;; ; 100 x 100 x 4

250 x 150 x 12,5

Det 1

Figure 6.26 Connecting the bracing

205


Chapter 6

The bracing-to-column connection is made as a single-lap joint through a plate. The thickness of plates is 20 mm, and their width is 230 mm. The strength grade of the M30 bolts is 8.8. The shear resistance of bolts is calculated assuming that the shear plane does not pass through the threaded portion of the bolt (example 25): Fv. Rd = 2 ⋅ 707 ⋅ 0, 6 ⋅

800 = 543 kN > N Sd OK! 1, 25

The bearing resistance of the plate is limited by the edge distance. In the joint shown in the example, the edge distance is 50 mm. The following bearing resistance is obtained: Fb. Rd = 2 ⋅ 2, 5 ⋅ 0, 521 ⋅ 490 ⋅ 30

20 = 612, 7 kN > N Sd OK! 1, 25

The resistance of the column-to-end plate connection is calculated from the formulae in Table 9.3.11: Column flange shear failure: N1.Rd = 628,9 kN Splice failure: N1.Rd = 463,9 kN Column web yield: N1.Rd = 665,6 kN The resistance of the joint is sufficient, since the horizontal component of the lacing’s axial force is NSd = 321,5 kN < N1.Rd Roof Bracing

qd.2

D2

D1

qd.1 eaves section

wind column

purlin section

main column

Figure 6.27 Roof bracing 206


Chapter 6

The roof bracing is subjected to transverse wind load. The loads on the wind columns are transferred as reactions, so part of the wind load is transferred directly to the foundation. The total wind load on the roof is transferred to the roof bracing. When determining the wind load, it is important to distinguish between the effects of negative and positive pressure due to the difference between the compression and tension resistance values of hollow sections. The following wind load is obtained on the side wall: qd 1

= γ Q.1 ⋅ c p ⋅ qref ⋅ ce (0, 375 ⋅ H1 + H 2 ) = 1, 5 ⋅ 0, 6 ⋅ 0, 6(0, 375 ⋅ 10, 5 + 4) = 4, 28

qd 2

= γ Q.1 ⋅ c p ⋅ qref ⋅ ce (0, 375 ⋅ H1 + H 2 ) = 1, 5 ⋅ 0, 3 ⋅ 0, 6(0, 375 ⋅ 10, 5 + 4) = 2, 14

kN (pressure) m kN (negative pressure) m

where H1 is the height of the eaves H2 is the height of the building at the apex The axial force values in diagonal members D1 and D2 (Figure 6.27) is obtained from the reaction at end of the roof bracing: 0, 5 L 0, 5 ⋅ 100 = 4, 28 = 279, 4 kN (tension) cos 40 cos 40 0, 5 L 0, 5 ⋅ 100 = qd 2 = 2, 14 = 139, 7 kN (compression) cos 40 cos 40

D1 :

N Sd.1 = qd 1

D2 :

N Sd.2

The diagonal is supported at purlin trusses, so the buckling length is: 20 2 + 24 2 Lc. y = Lc.z = = 5, 2 m 6 Consider a hollow section of dimensions 120 x 120 x 5. The following values for tension and buckling resistance is obtained: 355 = 721, 6 kN > N Sd.1 OK! 1, 1 355 = 0, 328 ⋅ 2236 = 236, 4 kN > N Sd.2 OK! 1, 1

Nt.Rd = 2236 N b.Rd

The resistance of the upper chord of the purlin truss and that of the eaves beam, to the axial force due to wind load, must also be checked, as well as the deflection of the roof bracing. 207

Chapter 6

6.6


References

[1] ENV 1991-2-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-1: Rakenteiden kuormat, tiheydet, oma paino ja hyötykuorma, 1995 (ENV 1991-2-1:Eurocode 1: Basis of design and actions on structures. Part 2-1: Actions on structures. densities, self-weight and imposed loads, 1995) [2] ENV 1991-2-3: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-3: Rakenteiden kuormat, lumikuormat, 1995 (ENV 1991-2-3: Eurocode 1: Basis of design and actions on structures. Part 2-3: Actions on structures, Snow loads, 1995) [3] ENV 1991-2-4: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 2-4: Rakenteiden kuormat, tuulikuormat, 1995 ENV 1991-2-4: Eurocode 1: Basis of design and actions on structures. Part 2-4: Actions on structures, Wind loads, 1995) [4] SFS-ENV 1991-1: Eurocode 1: Suunnitteluperusteet ja rakenteiden kuormat, osa 1 Suunnitteluperusteet, 1995 (ENV 1991-1: Eurocode 1: Basis of design and actions on structures. Part 1: Basis of design, 1995) [5] SFS-ENV-1993-1-1: Eurocode 3 Teräsrakenteiden suunnittelu, Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (Sisältää myös liitteen K: ENV 1993-1-1:1992/ A1:1994) (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993) (Include also annex K: ENV 1993-1-1:1992/ A1:1994) [6] ECCS: Technical Committee 8- Structural stability- Technical working group 8.1/ 8.2 Skeletal structures: Analysis and design of steel frames with semi-rigid joints, First edition 1992 [7] CIDECT: Design guide for rectangular hollow section joints under predominantly static loading, Verlag TÜV Rheinland GmbH, Köln 1992 [8] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-596,1996

208


7

Chapter 7

SHOP FABRICATION AND ERECTION

In addition to structural design, the cost of a hollow section structure includes shop fabrication and erection. Design aims at minimizing the weight of the structure, since the price of a hollow section is almost directly proportional to its weight. However, the lightest structure is not necessarily the most economical solution with regard to the whole of the construction project. A maximally optimized structure may be expensive to manufacture and install, which may result in losing the savings gained in material costs to increased manufacturing and erection costs. To reach an optimal result, it is important that the designer, the shop and the site all work in close cooperation and that all parties have sufficient knowledge about on all aspects of the construction project. The general principle is to perform the most demanding and complex phases in the workshop to make the erection quick and cost-efficient. In practice this means that all welded joints are made at the shop, and erection then consists of joining of preassembled units with bolts.

7.1

Cutting of hollow sections

Hollow sections can be ordered either standard-length ( 6, 12 or 18 m) or cut to size (even 24 m). No waste pieces are produced when ordering cut-to-size sections. The cutting is often made at the fabricators if the structure requires slanted cutting surfaces (e.g. lattices). Square and rectangular hollow sections are easy to cut, since the cut can usually be made in one plane. However, various cutting planes are needed in overlapped joints. Circular hollow sections are more complex to cut particularly for joints between several circular sections. In such a case, the end of the circular hollow section must often be profiled.

7.1.1

Cutting of circular hollow sections

A circular hollow section can sometimes be joined to the edge of another circular section by cutting the hollow section in one plane. A prerequisite for this is that the external dimensions of the hollow sections differ distinctly. This way, the root gap remaining at the edges of the smaller hollow section is sufficiently small with regard to welding. A circular hollow section can be cut in one plane if the following conditions are met [1]:

g1 ≤ t 0 and g1 ≤ t1

(7.1)

g2 ≤ 3 mm

(7.2)

where

t1

g2

t0 t1

d1

; ;

g2

is the root gap at the external edge of the smaller hollow section is the root gap at the internal edge of the smaller hollow section is the wall thickness of the larger hollow section

g1

g1

is the wall thickness of the smaller hollow section

In practice, most structures made from circular hollow sections do not meet the conditions of formulae (7.1) and (7.2).

209

d0

t0


Chapter 7

The size of the root gap in circular holllow sections can also be decreased by cutting the hollow sections at different angles. Using the expressions in Figure 7.1, the following equations for cutting angles αg and αd are obtained [1]:

h ⋅ sin θ   α g = 90 o − θ + arctan   r1 + h ⋅ cos θ − L ⋅ sin θ 

(7.3)

h sin θ   α d = −90 o + θ + arctan   r1 − h ⋅ cos θ − L ⋅ sin θ 

(7.4)

where

L = r12 − (r1 − t1 )

2

d0 d02 2 − − (r1 − t1 ) h= 2 4 d − 2t1 r1 = 1 2 is the diameter of the larger hollow section d0 is the diameter of the smaller hollow section d1 is the wall thickness of the smaller hollow section t1

L

θ

d1

Figure 7.1

Cutting of circular hollow sections

210

; ;

αg

h

d0

;; ;;;;

L

αd

t1


7.1.2

Chapter 7

Cutting methods

Sawing Sawing is the most commonly used method for cutting hollow sections. Usually, a disc saw or a band saw is used. An increase in the sawing speed usually decreases the accuracy in cutting and generates burrs which then need to be removed. In addition to the sawing speed, the easy transport of sections to and from the sawing site is a factor worth considering. It is also possible to cut both ends of a hollow section simultaneously to save time. Saw blades must be changed often, as the decreased sharpness of the saw blade increases the dimensional deviation and the quality of the cut seam deteriorates. Other cutting methods a) Thermal cutting Thermal cutting, when made free-hand, is a less accurate cutting method than sawing. It is suitable, for instance, for shaping the ends of circular hollow sections in lattice joints, particularly if a profile cutting machine is used. b) Cutting by punching This method is feasible only with thin-walled hollow sections. Its advantage is the possibility of generating the most complex of cutting surfaces. In circular hollow section joints, the ends can be shaped in one go with punching. c) Laser cutting Laser cutting is an accurate method. A further advantage is the reduced area of temperature change in the area of the cut. A disadvantage of laser cutting is the expensive equipment needed.

7.1.3

Notching of hollow section ends

The use of splices in bolted joints may require notches in the hollow section wall (Figure 7.2a). In joints subjected to small loads however it is more better cut the plate rather than the hollow section (Figure 7.2b). In order to transfer forces in joints subjected to heavy loads, it is, conversly, better to cut the hollow section and keep the splice intact, since it is difficult to make a sufficiently long weld inside the hollow section. Also in joints using small hollow sections, it is necessary to cut the section, since welding the interior is practically impossible. Notches can be made using the cutting methods described above. a)

Figure 7.2

b)

Joining splice plates to hollow sections

211


Chapter 7

7.2

Bending of hollow sections

A hollow section can be bent either by cold or hot bending. Cold bending is more cost-efficient and simpler, and thus more commonly used. Circular hollow sections are easier to bend than square and rectangular hollow sections, since in the latter case the shape of the cross-section tends to get distorted. Factors affecting the success of bending include: - the ultimate strength and yield strength of the material - the chemical composition and ultimate elongation of the material - the relation of the wall thickness to the height and width of the hollow section (t/h, t/b, t/d) - the relation of the bending radius to the height and width of the hollow section (r/h, r/d, r/b) When using square and rectangular hollow sections, the effect of the distortion of the crosssection shape to structural appearance must be evaluated on a case by case basis. The more critical the appearance of the structure, the greater the bending radius must be. Hollow sections with lower ultimate strength and yield strength are easier to bend. In addition, a greater wall thickness in relation with the height and width of the hollow section facilitates bending. A small radius is always more complicated to produce than a greater one. The cross-sectional properties of square and rectangular hollow sections decrease during bending. Reduced values for second moment of area are presented in Appendix 9.7. The distortion of a hollow section’s wall during bending must also be taken into account when determining its compression resistance. More demanding bending procedures usually require practical expertise, a workshop that specializes in bending and is equipped with appropriate machinery. For successful bending, the cooperation and expertise of the designer and the manufacturer is important.

7.2.1

Bending methods for hollow sections

Roller bending In roller bending, the hollow section is directed through three or four rollers. The size of the rollers is determined by the size of hollow sections. The middle rollers determine the magnitude of the bending radius. Normally, one of the rollers is freely rotating. Minimum bending radii for square and rectangular hollow sections are shown in Appendix 9.7.

Induction bending In induction bending, the hollow section is heated during bending with an induction coil. A small portion at a time is simultaneously heated and bent. This is repeated until the entire hollow section is processed. Compared to roller bending, induction bending is a more expensive method, but it has the advantage of producing smaller minimum bending radii.

212


Chapter 7

Curved lattice structures can be made from straight elements. The curved shape is produced by joining the straight chord member elements together in an angle corresponding to the bending radius (Figure 7.3).

Figure 7.3

Curved hollow section lattice

7.3

Bolted joints

The bolts used in load-bearing structures are normally of strength grade 8.8. The recommended clearances of bolt holes are presented in Table 7.1.

Table 7.1 Bolt diameter (mm) 12 14 16

Recommended clearances of bolt holes [2] Clearance (mm) 1 1 2

Bolt diameter (mm) 18 20 22

Clearance (mm) 2 2 2

Bolt diameter (mm) 24 27 30

Clearance (mm) 2 3 3

Bolts are tightened such that the normal stress (or shank tension) Fp generated in the bolt is equal to (7.5) [2]:

Fp = 0, 7 fub ⋅ As

(7.5)

where

fub As

is the ultimate strength of the bolt is the tension cross-section area of the bolt

It is advisable to prevent the loosening of bolts by locking the nuts with an appropriate method (e.g. torque type nuts, special washers or glue). Holes can be made either by drilling or punching. However, in tension members the holes for bolts must always be made by drilling. In joints subjected to lighter loads, self tapping bolts or drill bolts can be used. With these types of bolts, the most commonly used sizes are 5,5 and 6,3 mm. The bolt is always fixed at the side of the thinner element side of the joint. The core hole for a self tapping bolt must be slightly smaller than the bolt itself. The use of drill bolts is simpler and faster, since no core hole needs to be drilled. When using self tapping bolts and drill bolts, the instructions provided by the bolt manufacturer must be followed. 213

Chapter 7


In Chapter 3, we looked at various joint details for bolted joints. The openings were usually located in the structural element, usually a plate, joined to the hollow section. The simplest way would be to drill the hole directly in the hollow section, but in practice, the slender walls of hollow sections cannot bear much load. However, by using special drilling methods or special bolts, the hollow section joints can be made directly via the hollow section wall.

7.3.1

Friction Drilling in the Wall of the Hollow Section Tube

When using friction drilling a special hard metal tool is used, which heats the wall of the hollow tube due to friction. The heated material is first pressed to the outside of the tube surface and after the penetration to the inside. Thus the basematerial itself forms a bushing, which can be threaded. The threads are made in a separate workphase, either with a traditional thread cutting tool or with a thread rolling peg. Due to the bushing, the thread length is almost doubled compared to the wall thickness. The method has been tested up to material thicknesses of 12, 5 mm. With increasing material thicknesses and bolt diameters special friction drilling settings are required from the drilling equipment. For building purposes sizes M16 - M20 are suitable.

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yyyy ;;;; ;;;; yyyy ;;;; yyyy ;;;; yyyy ;;;; yyyy ;;;; yyyy

Phase 1 Friction drilling

7.3.2

Expansion bolt joints

Phase 2

In expansion bolt joints, the opening to the section wall is drilled using normal methods. The opening is then equipped with an expansion bolt which functions in a similar way those used in connections to concrete structures. Inside the expansion bolt is a threaded hole in which the bolt is placed. When the bolt is tightened, the conical end of the expansion bolt clamps against the section wall. Simultaneously, the cone spreads the blanket of the expansion bolt, which generates the necessary tightening torque in the bolt. The method is suited for M8-M20 bolts and for all wall thicknesses [4].

7.3.3

Pilot tap joints


;;; ; ;

Kuva 7.4


Joints subjected to light loads only can also be made using tapped studs which are welded to the section wall. The opening on the joined element must be enlargened from the section wall side, so that the weld does not bear against it after the bolts are tightened. Another alternative is the use of washers between the element to be joined and the section. Special attention should be paid to the protection of tapped studs during transport and installation. 214


7.4

Chapter 7

Welding of hollow sections

The structural steel used in Rautaruukki hollow sections has good weldability with all welding methods. Weldability depends on the welding method and the chemical composition of the material. The chemical composition of steel with regard to welding is best described by the carbon equivalent value (CEV):

CEV = C +

Mn Cr + Mo + V Ni + Cu + + 6 5 15

(7.6 )

With normal wall thickness values of Rautaruukki longitudinally welded hollow sections (< 16 mm) and with steel grade S355J2H, no special welding methods are needed (CEV ≤ 0,39). Special methods are necessary only when the carbon equivalent value is higher than 0,4. If the carbon equivalent value is greater than 0,45, preheating is required. When welding two different materials together, the parameters of the welding process (temperature, welding method and filler material) must be selected according to the material which has higher strength. The welded elements must be dry and free from grease and oil.

7.4.1

Quality levels

Table 7.2

Quality levels for arc welding according to the European standard

Level symbol D C B

Quality level Moderate Intermediate Stringent

The quality level is determined by the designer responsible, who must consider structural safety requirements and the ease of inspection and manufacture. When determining the weld quality level, factors to be considered include post-weld surface finishing, type of structural loading (static or dynamic), operating conditions (temperature and environment) and the consequences of potential failure. In addition to affecting the welding costs, the choice of quality level also has an effect on the weld inspection and testing costs.

7.4.2

Welding methods

Two principal methods used in the welding of hollow sections are manual metal arc welding with covered electrode and gas shielded arc welding. Metal arc welding is used principally in on-site installations. Its advantage is the light-weight and easily transportable equipment required. Gas shielded arc welding is the most common method used in shop fabrication. Advantages associated with gas shielded arc welding are better productivity and the possibility for the automation of the welding procedure. In the welding of hollow section members, the requirements set for the welder’s performance by the standard EN-287-1 must be considered [9]. 215

Chapter 7

7.4.3


Welding sequence

Due to the stresses generated by welding and to the deformations of the joined elements, the correct welding sequence is important. The welding of hollow sections should not be started or finished in a comer. The maximum throat thickness generated in one pass is 5 mm. The welding sequence used depends on the accessability of the weld and the ability to turn the section.

;;;;

;;;; ;

In Figure 7.5a, the section can be rotated horizontally. The welding point thus remains the same and the element is rotated a full circle. In Figure 7.5b, the element is positioned horizontally and can also be rotated about its axis. The welding direction is at first from the bottom upwards. The lower side is welded after rotating the section. In Figure 7.5c, the section is fixed in a vertical position. The welding is performed continuously over the entire section. In Figure 7.5d, the section is fixed in a horizontal position. Now, the lower seam must be welded from below. a)

b)

1 3

2

4

180°

360°

;;;; ; 3

4

2

1

c)

180°

d)

1

Figure 7.5

Welding sequence

7.4.4

Fillet and butt welds

2

2

1

3

1

1

2

4

With hollow sections, the aim is to design joints in such a manner that fillet welds can be used. Fillet weld are the simplest and most cost-efficient weld type, since no weld preparation is needed. However, depending on the joint geometry, a groove must, in some cases, be made to ensure a sufficient throat thickness of the weld. It is advisable to make the weld as symmetrical as possible to minimize the consumption of the weld metal. Weld preparations for end-to-end connections are treated in section 3.3. Fillet and butt welds for lattice joints made of square and rectangular hollow sections are shown in Table 7.3.

216


Table 7.3

Chapter 7

Fillet and butt welds in joints made of square and rectangular hollow sections b1 Z

X

Y

t1

;;

r0 t0

θ

b0

X

t1 < 8 mm <2

>60°

<2

θ ≤ 60° End chamfered after cutting

Z

<2

θ > 60° t1 < 8 mm

θ > 60°

1-2,5 1-2,5

t1

<2

t1

t1 ≥ 8 mm

<2

t1

> t1

t1 ≥ 8 mm

Y <2

b0-b1 ≥ 2r0

<2

<2

0 ≤ b0- b1 < 2r0 and maximum root gap * 2 mm minimum groove angle 60°

r0

>60°

0 ≤ b0- b1 < 2r0 and root gap * more than 2 mm The brace member end must be shaped

∗

root gap = r0 −

>60°

>60°

(r0 − t1)(b0 − b1) − 0, 25(b0 − b1)2 + 2t1 ⋅ r0 − t12 , when b0 − b1 < 2(r0 − t1)

r0 is the external corner radius

217

Chapter 7

7.4.5


Preheating

The structural steels used in hollow sections do not require preheating when the maximum nominell yield strength of the steel is 355 N/ mm2 and the wall thickness is less than 13 mm. Hollow sections must, however, be preheated to the temperature of 40 °C for a minimum of 75 mm away from the joint, if the surface of the hollow section is damp or the temperature of the structure to be welded is below 5 °C [1]. In thermal cutting, preheating is usually not needed if the carbon equivalent value of steel is less than 0,45 [5].

7.4.6

Residual stresses

During welding, residual stresses due to the temperature are always generated in the steel. This may result in permanent deformations of the member. Deformations are generated both parallel and perpendicular to the weld direction. The magnitude of the deformation depends on the welded material, the number of passes and the rigidity of the structure. In addition to deformation, there is also angular distortion, in which the angle between welded elements changes due to welding stresses. Figure 7.6 shows the angular distortions due to welding in their unrestricted form.

Figure 7.6

Angular distortions due to welding

In practice, the elements are supported during welding in such a manner that the deformations are restricted, which generates stresses in the material. The deformation is significantly reduced if a fillet weld is made with a both side groove, or a double V groove is used instead of a single V groove. In this case, welding is made from both sides in turns. A single side V groove can also be made by placing the welded elements in an angle which eliminates the deformation due to welding. The estimation of the angle magnitude is based on empirical know-how. When welding an entire structural element such as a lattice, it is recommendable to first tack weld the joints, then make the fillet welds and finally the full penetration single-V butt welds [1]. Stress relief by heat-treatment is usually needed only in special cases. The normal stress relief temperature is 550-650 °C [5].

218


Chapter 7

For instance the following factors increase the residual stresses due to welding: - large throat thickness of weld - large distance of the weld from the neutral axis of the hollow section - supporting the hollow sections during welding - high rigidity of the hollow sections - wrong weld sequence - weld methods with high heat input

7.4.7

Inspection of welds

The determination of the weld quality depends on the required quality level, the structural loading and the consequences of a potential failure. Non-destructive weld inspection methods are presented in EN 10246. In fatigue loaded structures, welds are usually the weakest links, and the improvement of the weld quality therefore increases the service life of the entire structure. The weld quality can be ensured by various methods, which are selected on a case by case basis.

Visual inspection Visual inspection is the least expensive and least accurate inspection method. The inspector must have sufficient experience in welding and the necessary expertise for the task. Visual inspection can detect errors in the throat thickness and the largest flaws which reach the surface. Visual inspection is not sufficient for detecting minor defects such as hairline cracks.

Magnetic particle testing Magnetic particle testing is a method for detecting surface flaws. The inspected surface is covered with a fine-grained ferrite powder, and a magnetic field is generated in the material by an electric coil. The powder particles form a line at crack locations. Cracks as small as 0,1 µm wide can be detected in the test.

Penetrant testing Penetrant testing is another method for locating surface defects in welds. First, the inspected surfaces are carefully cleaned, after which dye penetrant is applied on the surface. The dye has low surface tension and high capillarity and can thus penetrate even the narrowest of cracks. The dye is left on the surface for 5-10 minutes, after which time excess dye is wiped off. The surface is dried and covered with developer which reacts with the dye penetrant, bringing it to the surface. The pattern generated by the developing agent indicates the defect types and their severity.

219

Chapter 7


Ultrasonic inspection Ultrasonic inspection also serves to detect internal defects. Ultrasonic inspection is a very rapid method but it requires an experienced and qualified inspector. The test consists of sending high-frequency waves to the point being inspected. The spectrum of the waves is examined with an oscilloscope which shows the weld defects as intensity peaks. The minimum material thickness feasible for ultrasonic inspection is approximately 6 mm, but with current equipment, welds connecting plate thicknesses as low as 4 mm can be inspected provided that the weld material itself is thicker than 6 mm. The disadvantage of the method is that defects parallel to the sound waves are difficult to detect. Due to this, ultrasonic inspection must always be performed in various angles.

Radiographic inspection Radiographic inspection can also detect internal defects in the weld. In this method, the weld is radiated with x-rays or gamma rays, and the passage of rays through the weld location is photographed. Radiographic inspection is not feasible for inspecting joints in structures with varying wall thickness or lattice joints. X-rays are hazardous to health, and the inspection must be conducted in a controlled environment.

7.5

Tolerances

Tolerances in the industrial manufacture of hollow sections are dealt with in Chapter 1. Tables 7.4, 7.5 and 7.6 give the tolerances for the geometry of fabricated elements. Erection tolerances are shown in Table 7.7. The tables concentrate on the most important tolerances concerning hollow sections. Quality requirements regarding fabrication procedures and erection are discussed in more detail in references [2], [6] and [7].

220


Table 7.4

Chapter 7

Tolerances for singular members in workshop fabrication [2]

Tolerance

Parameter Hollow section length measured at flange centre line

Permitted deviation Cutting length in general: ∆ = ± (2+ L / 5000) (mm) Cutting length when force is transferred by contact in bearing: ∆ = ± 1 mm

∆

L+∆ Hollow section straightness by horizontal and vertical axes

∆ = L / 1000 (mm) ∆ = 3 mm Select whichever is greater

Deviation of curvature in a bent hollow section measured at the centre point of span

∆ = L / 1000 (mm) ∆ = 6 mm Select whichever is greater

Squareness of hollow section end

Squareness in general: ∆ = ± h / 300

∆

L

L

Squareness when force is transmitted by contact in bearing: ∆ = ± h / 1000

h

∆

221


Chapter 7 Table 7.5

Tolerances for openings and cuttings in workshop fabrication [2]

Tolerance ∆

a

Parameter Deviation of a single bolt opening from the designed location

Permitted deviation ∆ = ± 2 mm

Deviation of a bolt group from the designed location

* dimension a: ∆ = +5 mm –0 mm

dimension b: ∆ = ± 2 mm b

dimension c: ∆ = ± 5 mm dimension d: ∆ = ± 2 mm, when h ≤ 1000 mm

c

dimension d: ∆ = ± 4 mm. when h > 1000 mm

h

d

L

Deviations in the dimensions of a cut element

+2 mm

dimension d: ∆ = -0 mm

+2 mm

d

dimension L: ∆ = -0 mm

Squareness of the cutting of hollow section wall

∆ = ± 0,1t

t

∆

* In reference [2] tolerance is ∆ = ± 5 mm, but a negative tolerance may cause a bearing resistance remarkably smaller than the one determined by calculations

222


Table 7.6

Chapter 7

Tolerances for lattices in shop fabrication [2], [6]

Tolerance p

Parameter Straightness of brace member

Permitted deviation ∆ = L / 500 (mm) ∆ = 6 mm L is the length of the brace member Select whichever is greater

Distance between corner points

∆p = ±5 mm

∆ ∆ L

∑∆p = ±10 mm

g ≥ t1 + t2 and ga ≥ 1,5 t0

Gap tolerance

∆g, ∆ga

∆g =

+5 mm* –0 mm

and ∆ga =

*

+5 mm* –0 mm

t1 and t2 are the thickness values of brace member walls t t0 is the wall thickness of the chord

ga g

Lattice height, width and diagonal ∆D = ±3 mm, when D ≤ 300 mm ∆D = ±5 mm, when 300 < D < 1000 mm ∆D = ±10 mm, when D ≥ 1000 mm

W + ∆W

D + ∆D

;; ; X + ∆X

∆W = ±3 mm, when W ≤ 300 mm ∆W = ±5 mm, when 300 < W < 1000 mm ∆W = ±10 mm, when W ≥ 1000 mm ∆X = ±3 mm, when X ≤ 300 mm ∆X = ±5 mm, when 300 < X < 1000 mm ∆X = ±10 mm, when X ≥ 1000 mm

Straightness parallel to the lattice ∆max = L / 500 plane ∆max = 12 mm Select whichever is greater

∆1

*

∆2 ∆3 ∆4

∆5

∆6

Deviates from tolerances in reference [6]

223

Chapter 7 Table 7.7


Installation tolerances of steel structures [2], [7]. The smallest tolerance value is selected from the ones presented in references. Structure

Permitted deviation e = ± 5 mm

;;

Tolerance Deviation of the distance between adjacent columns

L+e

e = ± 0,002h

e

h is the height of a floor

h

Slope of columns in a multi-story building between successive intermediate floors

Transverse deviation of a column in a multi-story building at all levels of intermediate floors

e

Slope of a column in a one-story building when the column is not supporting a crane gantry

h3 h2

Σh 300 n

Σh is the height in question measured from the base level n is the number of floors from the level in question to the base level

h1

Σh

e=±

e

e=±

h 300

h

h is the height of the column

Slope of a column in a portal frame when the column is not supporting a crane gantry

e1 + e 2 = ±0, 002h (average) 2 h

e = ± 0,010h (singular value) e1

e2

Slope of a column in one-story building or in portal frame when the column is supporting a crane gantry

h

e

224

h < 5 m:

e = ± 5 mm

5 ≤ h ≤ 25 m:

e = ± 0,001h

h > 25 m:

e = ± 25 mm


7.6

Chapter 7

Assembly of trusses

In lattice assembly, an important factor is the number of lattices to be fabricated. With large series, it is worthwhile speding more time in the design and preparation of the assembly frame (or jig). Also the accuracy requirements must be taken into account when designing the jig. Regarding fabrication costs, it is important to be able to store the hollow sections close to the assembly site, since this saves time during transport. The jig must be made of sufficiently strong elements so that thermal and mechanical stresses generated during the assembly do not distort it. The distances between lattice member supports must be sufficiently short to avoid deformations in the lattice due to welding. In the jig, it is advisable to join the flange plates of the lattice to the plates welded to the assembly frame with bolts (Figure 7.7).

Figure 7.7

Assembly frame

To speed up the shop fabrication phase, the members in the jig can be tack welded together, and the final welding can be done at another worksite. In such a case, attention must be paid to the firmness of tack welding to avoid the generation of high deformations during the final welding. In gapped joints, attention must be paid to retaining a sufficiently large gap. Even small gaps must meet the minimum gap tolerances shown in Table 7.6. In overlapped joints, the larger brace member that is overlapped must always be welded first.

225

Chapter 7

7.7


Fire protection

It is not always an advantage to prepare the fire protection in the workshop. Most fire retardant materials will be damaged during transport and must then be repaired on-site. The fire protection method best suited for shop fabrication is fire retardant (intumescent) painting. Even then, sufficient care must be taken during transport to keep the damage to the painted surface to a minimum. The damaged spots must be repaired on-site, and it may, in extreme cases be necessary to apply a new coat of paint to ensure the required fire resistance period. Concrete filling can be carried out in the shop, but this increases the weight of hollow sections, thus complicating their transport and installation. Pouring of concrete is usually done on site, since most intermediate floor slabs require a surface-cast layer in any case.

7.8

Transport and storage

During the structural design phase, the transport of structural elements must also be taken into account. Large pre-assembled units make the erection quicker but may increase tranport costs, especially if special transport is needed. When planning special transport, it is necessary to keep in mind the limits to the size of units set by bridges and roads at the proximity of the site. In international projects, it must be considered that regulations concerning maximum weight and dimensions of the transport vary country by country. Road transport is the simplest way of moving the material when shop and site are located relatively close to each other.

The following issues should be considered in the transport of hollow sections [3]: - the tarpaulins used to cover the load must be dry, clean, undamaged and sufficiently large - hollow sections must be placed in the platform of the transport vehicle so that they are not exposed to bumps, abrasion or any other type of damage - when loading hollow sections one over the other, the sections with the thickest walls and the greatest length and weight must be placed the lowermost - heavy products which might cause damage to hollow sections must not be piled on them - the platform must be clean, dry and even - the load must be secured to prevent shifting during transport - the load is tied with straps so that it does not touch the side or end columns of the platform - straps or chains must not be attached so tightly that they cause dents on the hollow sections; if necessary, the points of contact must be covered - suitable props must be used in the transport The following issues must be kept in mind in the storage of hollow sections [3]: - the depot must be clean, dry and properly ventilated - the entry of water from condensation into the sections must be prevented - a sufficient amount of props below and within the stack must be used

226


7.9

Chapter 7

Erection

The erection of hollow section structures is similar to the erection of other steel structures. However, the greater torsional stiffness of hollow sections and the greater bending stiffness about both axes make them easier to lift and erect and the need for temporary lateral supports is less than with conventional sections. Structures utilizing hollow sections are also less prone to the effect of wind during erection compared to open sections. A hollow section structure is erected according to an erection plan made by the designer or fabricator. The plan must take into account the routes of vehicles and the craneage. Connections on site are usually bolted to allow speedy erection. it is usual to commence erection from one of the braced bays, usually at one end. The first frame may require temporary guys, but the usual practice is to provide 4 holding down bolts in the baseplate to give temporary stability. Erection then proceeds witt the next frame are the permanent bracing is erected between the frames. Purlins are then connected to provide rafter stability. Having ensured that the end two frames are lined and levelled, erection of the other frames can proceed. The purlins and bracing being erected to provide stability to each frame in turn. The stability of the roof structure under its own dead weight should be checked to ensure that it can safetly be lifted into place. If necessary frames can be lifted in braced pairs to ensure stability of individual frames. Hollow section structures are light-weight, and even large units can be easily lifted and installed on the site. The bolts are tightened only after the positions of structural elements have been checked. After this, the bolts are tightened. (Section 7.4). The assembly of large lattice structures, especially that of space frames, is usually most easily carried out at ground level, after which the finished lattice is erected to its final position.

227

Chapter 7

7.10


References

[1] CIDECT: Design guide for fabrication, assembly and erection of hollow section structures, 1996 [2] ENV 1090-1: Teräsrakenteiden valmistus ja asennus- Osa 1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1996 (ENV 1090-1: Execution of steel structures- Part 1: General rules and rules for buildings, 1996) [3] Rautaruukinterästuotteiden käsittelyohje (The Handling Manual of Rautaruukki´s Steel Products, in finnish), 1995 [4] CIDECT: Report 6G-14(A)/96: Hollofast and hollobolt system for hollow section connections, 1996 [5] Rautaruukin teräkset: Hitsaajan opas (Rautaruukki´s Steels. The Welder Manual, in finnish), 1995 [6] prENV 1090-4 Execution of steel structures: Part 4: Supplementary rules for hollow section lattice structures, 1997 [7] ENV-1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993) [8] EN-25817: Terästen kaarihitsaus: Hitsiluokat, 1993 (EN-25817: Arc-welded joints in steel. Guidance on quality levels for imperfections, 1993) [9] EN-287-1: Hitsaajan pätevyyskoe: Sulahitsaus: Osa 1: Teräkset, 1992 (EN-287-1: Approval testing of welders. Fusion welding. Part 1: Steels, 1992)

228


8

Chapter 8

CORROSION PROTECTION

Corrosion of steel surfaces is caused by oxygen and moisture in the air. The corrosion rate of steel depends on air temperature and moisture as well as air pollution. The pollution factor having the greatest effect on corrosion is the chloride and sulphur content of the air. The decrease of temperature below zero decelerates corrosion or may prevent it completely. The shape of a hollow section is advantageous regarding corrosion protection, since the area requiring protection is small compared to the weight of the section.

8.1

Corrosivity categories

Based on the corrosivity of the climate, environments can be divided into categories. The corrosivity categories presented in reference [1] are shown in Table 8.1. The temperature and moisture content of the building also influences corrosion. The propagation of corrosion is likely when relative humidity exceeds 80 % and temperature exceeds 0 °C. However, the likelihood of corrosion is great even at lower humidity levels if the amount of air pollution is great or salinity is high. The effect of strong chemical agents (acids, alkalis, salts, organic solvents) must be taken into account when designing the corrosion protection. The corrosion rate increases substantially if the structure is exposed to simultaneous mechanical and chemical stress.

Table 8.1 Corrosivity categories [1] Corrosivity category C1

Thickness loss of low-carbon steel (µm/ year) * ≤ 1,3

Thickness loss of zinc (µm/ year) * ≤ 0,1

C2

1,3- 25

0,1- 0,7

C3

25- 50

0,7- 2,1

C4

50- 80

2,1- 4,2

C5-I

80- 200

4,2- 8,4

C5-M

80- 200

4,2- 8,4

* after first year of exposure

229

Examples

Heated buildings with clean atmospheres, e.g. offices, shops, schools, hotels. Rural areas with low level of air pollution. Unheated buildings where condensation may occur, e.g. depots, sports halls. Urban areas with moderate amount of air pollution. Coastal areas with low salinity. Production rooms with high humidity and some air pollution, e.g. foodprocessing plants, laundries, breweries, dairies. Industrial and coastal areas with moderate salinity. Chemical plants, swimming pools, coastal ship- and boatyards. Industrial and coastal areas with high humidity and aggressive climate. Buildings with almost permanent condensation and with high level of pollution. Coastal and sea areas with high salinity. Buildings with almost permanent condensation and with high level of pollution.

Chapter 8

8.2


Surface preparation

Hollow sections are normally delivered unprotected or with a light protective oil coating. Cleaning the surface before applying paint is thus essential for successful corrosion protection. The cleaning method selected depends on the amount and quality of the impurity and the shape and size of the member. Cleaning methods are summarized in Table 8.2. In workshop and paint shop conditions, the most commonly used cleaning method is shot blasting (preparation grades Sa 21/2 and Sa 3). Grease and salt must be removed from the surfaces before cleaning if acid pickling or shot blasting is used. When renovating old structures, wire brushing can be used in some special cases (St 2 and St 3).

Table 8.2

Cleaning methods for steel surfaces [3]

Preparation grade Sa 2 1/2

Cleaning method

Surface quality

Shot blasting

Sa 3

Shot blasting

St 2

Wire brushing

St 3

Wire brushing

Be

Acid pickling

Mill scale, rust, paint coatings and foreign matter are removed. Any remaining traces of contamination shall show only as slight stains in the form of spots or stripes. Mill scale, rust, paint coatings and foreign matter are removed. The surface shall be metallic clean. Poorly adhering mill scale, rust, paint coatings and foreign matter are removed. Poorly adhering mill scale, rust, paint coatings and foreign matter are removed. The surface shall have a metallic sheen. Mill scale, rust and residues from paint coatings are removed completely. Paint coatings shall be removed prior to acid pickling.

8.3

Anti-corrosive painting

Paint is applied on a dry and clean surface according to the paint manufacturer’s instructions. The best results are obtained in controlled paint shop conditions. During site painting, the relative humidity of air should be less than 80 % and the temperature should exceed +5 °C and be at least 3 °C above dew point. With some paint types (e.g. epoxy paints), a higher painting temperature is required. Recommended paint combinations for hollow sections are presented in Table 8.3. Paint selection also depends on the special characteristics required of the paint surface. Table 8.4 presents the applicability of different paints in cases where special characteristics are required of the painted surface. Prefabrication primer is a rapidly drying paint applied on the section surface in a thin layer of approximately 15 µm. The purpose of the primer is to protect hollow sections during storage and transport, and it must always be removed before applying the final paint. In normal conditions, internal corrosion of hollow sections is rare. The insides of hollow sections usually need not be protected against corrosion. However, the seepage of rain water into the sections must be prevented. Hollow section structures must be provided with openings for the removal of water from condensation, especially if there is a possibility of freezing.

230


Chapter 8

In designing to prevent corrosion, one of the central issues is to avoid recesses and pockets where water might be trapped. For instance, reinforcing the lattice corner joints with plates welded to brace member side generates a space where water and debris may accumulate. The joining of two metals with different electrochemical potential (e.g. steel and copper) must be avoided, or the metals must be insulated from each other, since corrosion will take place at the joint in the less noble of metals. The metal surfaces to be painted must be as smooth and rounded at corners as possible. Due to their rounded corners, hollow sections are also suitable for blast-cleaning. Weld splatter and other irregularities must be removed before painting. The welds must also be as smoothsurfaced as possible to avoid spots remaining inadvertently unpainted. Joints must be designed so that the structure can easily be painted from all sides. When painting with a brush, the space between splices must be at least as wide as the brush. The corrosion resistance of bolts and nuts must be at least as high as that of structural materials.

Table 8.3 Corrosivity category

Recommended paint combinations in various corrosivity categories [4] Surface preparation grade

Priming coats Paint type

C1, C2 Sa 2 1/2 C1, C2 St 2 C1, C2 Sa 2 1/2 C1, C2 Sa 2 1/2 C3 Sa 2 1/2 C3 Sa 2 1/2 C3 Sa 2 1/2 C3 Sa 2 1/2 C4 Sa 2 1/2 C4 Sa 2 1/2 C4 Sa 2 1/2 C5-I Sa 2 1/2 C5-I Sa 2 1/2 C5-I Sa 2 1/2 C5-M Sa 2 1/2 C5-M Sa 2 1/2 AK = Alkyd CR = Chlorinated rubber AY = Acrylic EP = Epoxy ESI = Ethyl silicate

Number Nominal Dry Paint type of coats FilmThickness(µm) AK 1- 2 80 AK AK 2 80 AK AY, CR, PVC 1- 2 80 AY, CR, PVC EP 1- 2 80 EP, PUR AK 1- 2 80 AK EP 1 160 AY AY, CR, PVC 1- 2 80 AY, CR, PVC EP 1- 2 80 EP, PUR EP, PUR, Zn(R) 1 40 EP, PUR ESI. Zn(R) 1 80 EP, PUR EP 1- 2 80 EP, PUR EP, PUR 4 160 AY, CR, PVC EP, PUR 1- 2 80 EP, PUR ESI, Zn(R) 1 80 EP, PUR EP, PUR, Zn(R) 1 40 EP, PUR EP, PUR 1- 2 80 EP, PUR PVC = Polyvinyl chloride PUR = Polyurethane Zn(R) = Zinc rich primer

231

Top coats

Number of Nominal Dry coats FilmThickness(µm) 1- 2 80 2- 3 120 1- 2 80 1- 2 80 2- 3 120 1 40 2- 3 120 2- 3 120 2- 3 200 2- 3 160 2- 3 200 1 40 3- 4 240 3 200 2- 3 280 3- 4 240


Chapter 8 Table 8.4

Applicability of different paint types for conditions requiring special characteristics [4]

Required characteristic Gloss retention Colour retention Resistance to: - water immersion - condensate/rain - solvents - solvents (splash) - acids - acids (splash) - alkalis - alkalis (splash) Resistance to dry heat: 60- 70 °C 70- 120 °C 120- 150 °C >150 °C Physical properties: - abrasion resistance - impact resistance - flexibility - hardness Application by: - brushing - rollercoating - spraying PVC = Polyvinyl chloride CR = Chlorinated rubber AY = Acrylic B = Bitumen AK = Alkyd

8.4

Paint type PVC CR ++ ++ ++ ++

AY +++ +++

B + +

AK ++ ++

PURp PURa EP + +++ + + +++ +

ZnS -

PURc CTV + + -

++ +++ + + + ++ + ++

++ +++ + + + ++ + ++

+ +++ + + + ++ + ++

+++ +++ + + + + + ++

+ ++ + ++ + + + +

+ +++ ++ +++ ++ ++ + +

++ +++ + +++ + ++ + +++

+++ +++ ++ +++ + ++ +++ +++

++ +++ +++ +++ + + + +

+++ +++ + + + ++ ++ +++

+++ +++ + + + + + ++

+++ + + +

+++ + + +

+++ ++ + +

++ + + +

+++ ++ + +

+++ +++ ++ +

+++ +++ ++ +

+++ +++ ++ +

+++ +++ +++ +++

+++ ++ + +

+++ + + +

+ ++ ++ ++

+ ++ ++ ++

+ ++ ++ ++

+ ++ ++ ++

++ + + +++

+++ +++ ++ +++

++ +++ +++ ++

+++ ++ +++ +++

+++ + + +++

++ +++ ++ ++

+ ++ ++ +

++ ++ ++ +++ + + + +++ +++ +++ +++ +++ PURp = Polyurethane, polyester type PURa = Polyurethane, acryl type EP = Epoxy

+++ ++ ++ +++ +++ ++ ++ ++ +++ +++ +++ +++ Zns = Zinc silicate PURc = Coal tar polyurethane CTV = Coal tar vinyl

+ ++ ++ + ++ ++ +++ +++ +++ - = not relevant + = poor ++ = good +++ = excellent

Hot-dip galvanizing

Protecting hollow section structures with galvanization is an efficient anti-corrosive method, and in many climates it can be sufficient on its own. The protective effect of zinc is based on its oxidation. Due to the cathodic protective effect of zinc, minor surface flaws in the zinc layer do not lead to the corrosion of the steel. The durability of the protective effect of zinc is directly proportional to the thickness of the zinc layer. In severe environmental conditions (e.g. corrosivity categories C5-I and C5-M), even a galvanized structure requires an anti-corrosive painting. Proper oxidation of zinc is prevented if hollow sections are placed in poorly ventilated and humid premises directly after galvanizing. This may be the case, for instance, in tightly packed stacks of sections where rain or water from condensation can accumulate. In such conditions, white rust is generated on the galvanized surface. White rust can be removed by brushing or using an appropriate detergent. To avoid white rust, hollow sections must always be stored on props. It is also important to allow enough space between hollow sections to ensure sufficient circulation of air and evaporation of moisture. If the zinc layer has already been oxidized, white rust is no longer generated. 232


Chapter 8

Hot-dip galvanized steel surfaces can be painted, for instance, using the paint combinations shown in Table 8.5. However, the galvanized surface must be treated before painting by, for instance, sand blasting (= light shot blasting). Sand blasting removes the impurities on the surface and makes the surface rougher, which improves the adhesion of the paint coat. In case shot blasting cannot be used, the galvanized surface must be cleaned with an alkaline degreasant. After the cleaning, the surface is rinsed and dried and is then ready for painting.

Table 8.5

Recommended paint combinations for hot-dip galvanized steel surfaces [4]

Corrosivity category

C1, C2 C1- C3 C1- C4 C1- C4 C1- C4, C5-I, C5-M

Priming coat(s) Paint type Number of Nominal Dry coats Film Thickness µm PVC 1 80 PVC 1 80 EP, PUR 1 80 EP, PUR 1 80

Top coat(s) Paint type Number of coats PVC PVC PVC EP, PUR EP, PUR

1 1 2 1 2

Nominal Dry Film Thickness µm 80 80 160 80 160

Hot-dip galvanized structures must be made as open as possible to produce a smooth zinc layer. Galvanized hollow section structures must be designed so that zinc can flow freely into the hollow section and out of it. If necessary, the structure must be provided with openings to ensure a sufficient flow of zinc. Site connections of hot-dip galvanized hollow sections should preferably be bolted. Welded joints made on-site must be protected by applying zinc-rich paint or by spraying them with zinc. Where galvanized sections are bolted the male thread must be made slightly smaller than the female thread of the joined element to allow sufficient space for the zinc layer. The female thread is usually made to the hot-dip galvanized element only after galvanizing, since the galvanizing of the joined element protects the inner thread from corrosion.

233

Chapter 8

8.5


References

[1]

ISO/ FDIS 12944-2: Paints and varnishes- Corrosion protection of steel structures by protective painting systems. Part 2: Classification of enviroments, 1997

[2]

ISO/ FDIS 12944-3: Paints and varnishes- Corrosion protection of steel structures by protective painting systems. Part 3: Design considerations, 1997

[3]

ISO/ FDIS 12944-4: Paints and varnishes- Corrosion protection of steel structures by protective painting systems. Part 4: Types of surfaces and surface preparation, 1997

[4]

ISO/ FDIS 12944-5: Paints and varnishes- Corrosion protection of steel structures by protective painting systems. Part 5: Protective paint systems, 1997

234


9

Appendix 9.1

APPENDIX

Appendix 9.1

Cross-sectional properties and resistance values for steel grade S355J2H

235

M A Au Am/V It

b

h

t y r0 z

series 1) h

mm

236

x x x x x x x x x x x x x x x x x x x x x x

x x x x x x

40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

b mm 40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

t mm 2 2,5 2 2,5 3 2,5 3 4 2 2,5 3 4 2 2,5 3 4 5 2 2,5 3 4 5 6 6,3 2 2,5 3 4 5 6 6,3 7,1 8

Wt I Wel Wpl i

r0 = 2,0 x t when t ≤ 6,0 mm r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm r0 = 3,0 x t when t > 10,0 mm M kg/m 2,31 2,82 2,93 3,60 4,25 4,39 5,19 6,71 4,19 5,17 6,13 7,97 4,82 5,96 7,07 9,22 11,3 5,45 6,74 8,01 10,5 12,8 15,1 15,5 6,07 7,53 8,96 11,7 14,4 17,0 17,5 19,4 21,4

A mm2 x 102 2,94 3,59 3,74 4,59 5,41 5,59 6,61 8,55 5,34 6,59 7,81 10,15 6,14 7,59 9,01 11,75 14,36 6,94 8,59 10,21 13,35 16,36 19,23 19,73 7,74 9,59 11,41 14,95 18,36 21,63 22,25 24,65 27,24

Au m2/m 0,153 0,151 0,193 0,191 0,190 0,231 0,230 0,226 0,273 0,271 0,270 0,266 0,313 0,311 0,310 0,306 0,303 0,353 0,351 0,350 0,346 0,343 0,339 0,333 0,393 0,391 0,390 0,386 0,383 0,379 0,373 0,370 0,366

= torsional section modulus = moment of inertia = elastic section modulus = plastic section modulus = radius of gyration

PL Nc.Rd Mc.Rd Vpl.Rd

= cross-section class in concentric compression = compression resistance without buckling = bending resistance = shear resistance

The calculated resistance values are design values (section 2.1) based on the default value (1.1) of the partial safety factor of material γM0 used in Eurocode 3. The partial safety factor values may differ in each country. National values must be checked from the NAD (National Application Document) of the country in question.

Am/V 1/m 521 422 517 417 351 414 348 265 512 412 345 262 510 410 344 261 211 509 409 343 259 210 176 169 508 408 342 258 209 175 168 150 134

It Wt I Wel Wpl i mm4 mm3 mm4 mm3 mm3 mm x 104 x 103 x 104 x 103 x 103 x 10 11,28 5,23 6,94 3,47 4,13 1,54 13,61 6,21 8,22 4,11 4,97 1,51 22,63 8,51 14,15 5,66 6,66 1,95 27,53 10,22 16,94 6,78 8,07 1,92 32,13 11,76 19,47 7,79 9,39 1,90 48,66 15,22 30,34 10,11 11,93 2,33 57,09 17,65 35,13 11,71 13,95 2,31 72,64 21,97 43,55 14,52 17,64 2,26 63,96 17,48 40,73 11,64 13,52 2,76 78,49 21,22 49,41 14,12 16,54 2,74 92,42 24,74 57,53 16,44 19,42 2,71 118,5 31,11 72,12 20,61 24,76 2,67 96,34 23,16 61,70 15,42 17,85 3,17 118,5 28,22 75,15 18,79 21,90 3,15 139,9 33,02 87,84 21,96 25,78 3,12 180,4 41,84 111,0 27,76 33,07 3,07 217,8 49,68 131,4 32,86 39,74 3,03 138,1 29,64 88,86 19,75 22,78 3,58 170,3 36,23 108,6 24,12 28,00 3,56 201,4 42,51 127,3 28,29 33,04 3,53 260,8 54,17 161,9 35,98 42,58 3,48 316,3 64,70 192,9 42,87 51,41 3,43 367,8 74,16 220,5 49,00 59,54 3,39 382,3 76,21 221,1 49,14 60,30 3,35 190,5 36,92 123,0 24,60 28,30 3,99 235,2 45,23 150,6 30,13 34,86 3,96 278,7 53,19 177,0 35,41 41,21 3,94 362,0 68,10 226,4 45,27 53,30 3,89 440,5 81,72 271,1 54,22 64,59 3,84 514,2 94,12 311,5 62,29 75,10 3,79 536,0 97,02 314,2 62,83 76,38 3,76 589,2 105,6 340,1 68,03 83,59 3,71 644,5 114,2 365,9 73,19 91,05 3,67

PL Nc.Rd kN 1 1 1 1 1 1 1 1 3 1 1 1 4 2 1 1 1 4 3 2 1 1 1 1 4 4 2 1 1 1 1 1 1

94,78 115,8 120,6 148,1 174,5 180,4 213,3 275,9 172,2 212,7 252,0 327,5 180,1 244,9 290,7 379,1 463,3 187,9 277,2 329,5 430,8 527,9 620,7 636,7 194,0 281,4 368,2 482,4 592,4 698,2 718,1 795,6 879,2

Mc.Rd kNm 1,33 1,60 2,15 2,61 3,03 3,85 4,50 5,69 3,76 5,34 6,27 7,99 4,70 7,07 8,32 10,67 12,82 5,73 7,79 10,66 13,74 16,59 19,22 19,46 6,84 9,17 13,30 17,20 20,85 24,24 24,65 26,98 29,38

Vpl.Rd kN 27,36 33,44 34,81 42,75 50,38 52,07 61,56 79,64 49,72 61,39 72,74 94,54 57,17 70,70 83,92 109,5 133,8 64,63 80,02 95,10 124,4 152,4 179,2 183,8 72,08 89,33 106,3 139,3 171,0 201,5 207,3 229,7 253,8


1) = recommended

= weight = cross-section area = external area = cross-section factor in fire design = torsional modulus

Appendix 9.1

Cross-sectional properties and resistance values for square hollow sections of steel grade S355J2H. (fy = 355 N/mm2).

Table 9.1.1

x x x x x x x x x x

237

x x x x x x x x x x x x x

mm x

b mm

2,5 8,31 10,59 0,431 3 9,90 12,61 0,430 4 13,0 16,55 0,426 5 16,0 20,36 0,423 6 18,9 24,03 0,419 6,3 19,4 24,77 0,413 3 10,8 13,81 0,470 4 14,3 18,15 0,466 5 17,6 22,36 0,463 5,6 19,5 24,82 0,461 6 20,8 26,43 0,459 6,3 21,4 27,29 0,453 7,1 23,8 30,33 0,449 8 26,4 33,64 0,446 8,8 28,6 36,48 0,442 4 16,8 21,35 0,546 5 20,7 26,36 0,543 5,6 23,0 29,30 0,541 6 24,5 31,23 0,539 6,3 25,4 32,33 0,533 7,1 28,3 36,01 0,529 8 31,4 40,04 0,526 8,8 34,2 43,52 0,522 10 38,1 48,57 0,517 4 18,0 22,95 0,586 5 22,3 28,36 0,583 6 26,4 33,63 0,579 6,3 27,4 34,85 0,573 7,1 30,5 38,85 0,569 8 34,0 43,24 0,566 8,8 36,9 47,04 0,562 10 41,3 52,57 0,557 4 19,3 24,55 0,626 5 23,8 30,36 0,623 6 28,3 36,03 0,619 6,3 29,3 37,37 0,613 7,1 32,7 41,69 0,609 8 36,5 46,44 0,606 8,8 39,7 50,56 0,602 10 44,4 56,57 0,597 12 50,9 64,86 0,578 12,5 52,6 67,04 0,576 t M A Au Am/V mm kg/m mm2 m2/m 1/m x 102

407 341 258 208 175 167 340 257 207 186 174 166 148 132 121 256 206 185 173 165 147 131 120 106 255 206 172 164 147 131 120 106 255 205 172 164 146 130 119 106 89 86

314,9 373,5 486,5 593,6 694,9 725,8 487,7 636,6 778,5 860,3 913,5 955,5 1056 1163 1252 1023 1256 1391 1479 1550 1719 1901 2055 2274 1265 1554 1833 1922 2134 2364 2560 2839 1541 1896 2239 2349 2611 2897 3141 3490 3997 4114 It mm4 x 104

55,23 65,07 83,63 100,7 116,5 120,4 78,15 100,8 121,8 133,6 141,2 146,2 160,1 174,6 186,5 139,8 169,8 186,9 197,9 205,4 226,0 247,7 265,8 290,9 161,7 196,8 229,8 238,8 263,1 289,0 310,7 341,0 185,3 225,8 264,2 274,7 303,2 333,6 359,2 395,1 443,1 454,6 Wt mm3 x 103

202,4 238,3 305,9 367,9 424,6 430,1 312,3 402,3 485,5 532,3 562,2 571,6 623,3 676,9 719,9 651,6 790,6 869,6 920,4 940,8 1031 1127 1205 1312 808,0 982,0 1146 1174 1289 1412 1513 1653 987,2 1202 1405 1442 1587 1741 1870 2048 2224 2275 I mm4 x 104

36,80 43,33 55,62 66,90 77,19 78,21 52,06 67,05 80,91 88,71 93,69 95,26 103,9 112,8 120,0 93,09 112,9 124,2 131,5 134,4 147,3 161,0 172,1 187,4 107,7 130,9 152,8 156,5 171,9 188,2 201,7 220,3 123,4 150,3 175,7 180,3 198,4 217,7 233,7 256,0 278,0 284,4 Wel mm3 x 103

42,47 50,27 65,21 79,27 92,46 94,36 60,24 78,33 95,45 105,3 111,6 114,2 125,7 137,8 147,9 108,2 132,3 146,3 155,3 159,6 176,3 194,2 209,2 230,4 124,9 153,0 179,9 185,2 204,8 226,0 243,9 269,2 142,8 175,2 206,2 212,6 235,4 260,1 281,1 311,0 346,1 355,7 Wpl mm3 x 103

4,37 4,35 4,30 4,25 4,20 4,17 4,76 4,71 4,66 4,63 4,61 4,58 4,53 4,49 4,44 5,52 5,48 5,45 5,43 5,39 5,35 5,30 5,26 5,20 5,93 5,89 5,84 5,80 5,76 5,71 5,67 5,61 6,34 6,29 6,25 6,21 6,17 6,12 6,08 6,02 5,86 5,83 i mm x 10

4 291,3 3 406,9 1 534,1 1 657,0 1 775,6 1 799,4 4 405,1 2 585,7 1 721,5 1 800,9 1 853,1 1 880,7 1 978,7 1 1086 1 1177 3 689,0 1 850,6 1 945,5 1 1008 1 1043 1 1162 1 1292 1 1405 1 1567 4 701,2 2 915,1 1 1085 1 1125 1 1254 1 1396 1 1518 1 1696 4 720,2 2 979,7 1 1163 1 1206 1 1345 1 1499 1 1632 1 1826 1 2093 1 2164 PL Nc.Rd kN

10,78 13,98 21,05 25,58 29,84 30,45 15,85 25,28 30,80 33,97 36,02 36,86 40,55 44,48 47,73 30,04 42,70 47,20 50,13 51,51 56,89 62,67 67,52 74,35 33,61 49,37 58,05 59,75 66,09 72,92 78,70 86,87 37,57 56,53 66,56 68,60 75,97 83,95 90,73 100,4 111,7 114,8 Mc.Rd kNm

98,65 117,5 154,2 189,6 223,9 230,8 128,6 169,1 208,3 231,2 246,3 254,2 282,5 313,4 339,9 198,9 245,5 273,0 291,0 301,2 335,5 373,1 405,5 452,5 213,8 264,2 313,3 324,7 361,9 402,9 438,3 489,7 228,7 282,8 335,7 348,2 388,4 432,7 471,1 527,0 604,3 624,6 Vpl.Rd kN

Appendix 9.1

1) h

110 110 110 110 110 110 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 150 150 150 150 150 150 150 150 160 160 160 160 160 160 160 160 160 160


110 110 110 110 110 110 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 150 150 150 150 150 150 150 150 160 160 160 160 160 160 160 160 160 160

x x x x

M A Au Am/V It

b h

t y r0 z series

1) h

mm x x x x x

238

x x x x x x x x x

180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220

b mm 180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220

t mm 5 6 6,3 7,1 8 8,8 10 12 12,5 5 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 8 8,8 10 12 12,5

Wt I Wel Wpl i

r0 = 2,0 x t when t ≤ 6,0 mm r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm r0 = 3,0 x t when t > 10,0 mm M kg/m 27,0 32,1 33,3 37,2 41,5 45,2 50,7 58,5 60,5 30,1 35,8 37,3 41,6 46,5 50,8 57,0 66,0 68,3 39,6 41,2 46,1 51,5 56,3 63,2 73,5 76,2

A Au Am/V mm2 m2/m 1/m x 102 34,36 0,703 40,83 0,699 42,41 0,693 47,37 0,689 52,84 0,686 57,60 0,682 64,57 0,677 74,46 0,658 77,04 0,656 38,36 0,783 45,63 0,779 47,45 0,773 53,05 0,769 59,24 0,766 64,64 0,762 72,57 0,757 84,06 0,738 87,04 0,736 50,43 0,859 52,49 0,853 58,73 0,849 65,64 0,846 71,68 0,842 80,57 0,837 93,66 0,818 97,04 0,816


PL Nc.Rd Mc.Rd Vpl.Rd

= cross-section class in concentric compression = compression resistance without buckling = bending resistance = shear resistance


205 171 163 146 130 118 105 88 85 204 171 163 145 129 118 104 88 85 170 163 145 129 117 104 87 84

It mm4 x 104 2724 3223 3383 3768 4189 4551 5074 5865 6050 3763 4459 4682 5223 5815 6328 7072 8230 8502 5976 6277 7010 7815 8514 9533 11149 11530

Wt mm3 x 103 289,8 340,1 354,1 391,7 432,2 466,6 515,3 583,7 600,1 361,8 425,5 443,5 491,6 543,6 588,1 651,5 743,4 765,5 520,6 543,0 602,9 667,9 723,6 803,6 922,3 950,8

I mm4 x 104 1737 2037 2096 2313 2546 2742 3017 3322 3406 2410 2833 2922 3232 3566 3850 4251 4730 4859 3813 3940 4366 4828 5221 5782 6487 6674

Wel mm3 x 103 193,0 226,3 232,8 257,0 282,9 304,6 335,2 369,1 378,5 241,0 283,3 292,2 323,2 356,6 385,0 425,1 473,0 485,9 346,7 358,2 396,9 438,9 474,7 525,7 589,7 606,7

Wpl mm3 x 103 224,0 264,4 273,1 303,1 335,7 363,6 403,5 453,6 467,1 278,9 329,7 341,2 379,3 420,9 456,6 508,1 575,6 593,5 402,2 416,8 464,0 515,6 560,2 624,7 712,0 734,9

i mm x 10

PL Nc.Rd kN 7,11 7,06 7,03 6,99 6,94 6,90 6,84 6,68 6,65 7,93 7,88 7,85 7,81 7,76 7,72 7,65 7,50 7,47 8,70 8,66 8,62 8,58 8,53 8,47 8,32 8,29

3 2 1 1 1 1 1 1 1 4 2 2 1 1 1 1 1 1 3 3 2 1 1 1 1 1

1109 1318 1369 1529 1705 1859 2084 2403 2486 1125 1473 1531 1712 1912 2086 2342 2713 2809 1628 1694 1895 2118 2313 2600 3023 3132

Mc.Rd kNm 62,28 85,31 88,13 97,81 108,3 117,3 130,2 146,4 150,7 73,38 106,4 110,1 122,4 135,8 147,4 164,0 185,8 191,5 111,9 115,6 149,7 166,4 180,8 201,6 229,8 237,2

Vpl.Rd kN 320,1 380,4 395,1 441,3 492,3 536,7 601,5 693,7 717,8 357,3 425,1 442,1 494,2 551,9 602,2 676,1 783,1 810,9 469,9 489,0 547,1 611,6 667,8 750,6 872,6 904,1


1) = recommended


Appendix 9.1

Cross-sectional properties and resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.

Table 9.1.1

x x x x

x x x

239

x 1) h

mm x

250 250 250 250 250 250 250 250 260 260 260 260 260 260 260 260 300 300 300 300 300 300 300 300 b mm

6 45,2 6,3 47,1 7,1 52,8 8 59,1 8,8 64,6 10 72,7 12 84,8 12,5 88,0 6 47,1 6,3 49,1 7,1 55,0 8 61,6 8,8 67,3 10 75,8 11 81,9 12,5 91,9 6 54,7 6,3 57,0 7,1 63,9 8 71,6 8,8 78,4 10 88,4 12 104 12,5 108 t M mm kg/m

57,63 60,05 67,25 75,24 82,24 92,57 108,1 112,0 60,03 62,57 70,09 78,44 85,76 96,57 104,4 117,0 69,63 72,65 81,45 91,24 99,84 112,6 132,1 137,0 A mm2 x 102

0,979 0,973 0,969 0,966 0,962 0,957 0,938 0,936 1,019 1,013 1,009 1,006 1,002 0,997 0,983 0,976 1,179 1,173 1,169 1,166 1,162 1,157 1,138 1,136 Au m2/m

170 162 144 128 117 103 87 84 170 162 144 128 117 103 94 83 169 161 144 128 116 103 86 83 Am/V 1/m

8843 9290 10388 11598 12653 14197 16691 17283 9970 10475 11717 13087 14283 16035 17498 19553 15434 16218 18161 20312 22195 24966 29514 30601 It mm4 x 104

681,2 711,2 791,0 878,2 953,3 1062 1226 1266 739,5 772,3 859,4 954,7 1037 1156 1247 1381 996,8 1042 1161 1293 1406 1572 1829 1892 Wt mm3 x 103

5672 5873 6522 7229 7835 8707 9859 10161 6405 6635 7373 8178 8869 9865 10479 11548 9964 10342 11514 12801 13902 15519 17767 18348 I mm4 x 104

453,8 469,8 521,7 578,3 626,8 696,5 788,8 812,9 492,7 510,4 567,1 629,1 682,2 758,8 805,8 888,3 664,2 689,5 767,6 853,4 927,4 1035 1184 1223 Wel mm3 x 103

524,5 544,4 607,0 675,8 735,3 822,0 943,6 975,2 568,8 590,8 658,9 734,0 799,0 893,8 956,5 1063 764,2 794,9 887,9 990,7 1080 1211 1402 1451 Wpl mm3 x 103

9,92 9,89 9,85 9,80 9,76 9,70 9,55 9,52 10,33 10,30 10,26 10,21 10,17 10,11 10,02 9,93 11,96 11,93 11,89 11,84 11,80 11,74 11,60 11,57 i mm x 10

4 1646 4 1770 3 2170 2 2428 1 2654 1 2987 1 3487 1 3616 4 1669 4 1797 3 2262 2 2532 1 2768 1 3116 1 3368 1 3777 4 1746 4 1887 4 2303 4 2787 3 3222 2 3633 1 4262 1 4423 PL Nc.Rd kN

136,0 143,4 168,4 218,1 237,3 265,3 304,5 314,7 145,3 153,4 183,0 236,9 257,9 288,5 308,7 343,0 184,6 195,4 228,6 266,2 299,3 390,8 452,3 468,2 Mc.Rd kNm

536,9 559,4 626,5 701,0 766,2 862,4 1007 1044 559,3 582,9 653,0 730,8 799,0 899,6 972,3 1090 648,7 676,8 758,8 850,0 930,2 1049 1230 1277 Vpl.Rd kN


250 250 250 250 250 250 250 250 260 260 260 260 260 260 260 260 300 300 300 300 300 300 300 300

Appendix 9.1

M A Au Am/V It

b

h

t y r0


r0 = 2,0 x t when t ≤ 6,0 mm

z series

r0 = 3,0 x t when t > 10,0 mm

1) h

b t mm mm mm

M kg/m

x

240

x x x x x x x x x x x

x x

40 50 60 60 70 70 70 80 80 80 80 80 80 80 80 80 90 90 90 90 90 90 90 90 90 90 90

30 30 40 40 50 50 50 40 60 60 60 60 70 70 70 70 50 50 50 60 60 60 70 70 70 70 70

2 2 2 2,5 2 2,5 3 2,5 2 2,5 3 4 2,5 3 4 5 2 2,5 3 2,5 3 4 2 2,5 3 4 5

1,99 2,31 2,93 3,60 3,56 4,39 5,19 4,39 4,19 5,17 6,13 7,97 5,56 6,60 8,59 10,5 4,19 5,17 6,13 5,56 6,60 8,59 4,82 5,96 7,07 9,22 11,3

A mm2 x 102 2,54 2,94 3,74 4,59 4,54 5,59 6,61 5,59 5,34 6,59 7,81 10,15 7,09 8,41 10,95 13,36 5,34 6,59 7,81 7,09 8,41 10,95 6,14 7,59 9,01 11,75 14,36


PL Nc.Rd Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd

= cross-section class in concentric compression = compression resistance without buckling = bending resistance by y axis = bending resistance by z axis = shear resistance in the direction of y axis = shear resistance in the direction of z axis


Au Am/V It Wt Iy Wel.y Wpl.y iy Iz Wel.z Wpl.z iz PL Nc.Rd Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd m2/m 1/m mm4 mm3 mm4 mm3 mm3 mm mm4 mm3 mm3 mm h b kN kNm kNm kN kN x 104 x 103 x 104 x 103 x 103 x 10 x 104 x 103 x 103 x 10 0,133 524 7,07 3,79 5,49 2,75 3,37 1,47 3,51 2,34 2,77 1,18 1 1 81,88 1,09 0,89 20,26 27,01 0,153 520 9,77 4,84 9,54 3,81 4,74 1,80 4,29 2,86 3,33 1,21 1 1 94,78 1,53 1,07 20,52 34,20 0,193 516 20,70 8,12 18,41 6,14 7,47 2,22 9,83 4,92 5,65 1,62 2 1 120,6 2,41 1,82 27,85 41,78 0,191 416 25,14 9,72 22,07 7,36 9,06 2,19 11,74 5,87 6,84 1,60 1 1 148,1 2,92 2,21 34,20 51,30 0,233 513 37,45 12,20 31,48 8,99 10,80 2,63 18,76 7,50 8,58 2,03 3 1 146,4 3,49 2,42 35,22 49,31 0,231 413 45,75 14,72 38,01 10,86 13,16 2,61 22,59 9,04 10,45 2,01 1 1 180,4 4,25 3,37 43,39 60,75 0,230 348 53,62 17,06 44,05 12,59 15,40 2,58 26,10 10,44 12,21 1,99 1 1 213,3 4,97 3,94 51,30 71,83 0,231 413 37,58 13,24 45,11 11,28 14,15 2,84 15,26 7,63 8,72 1,65 2 1 180,4 4,56 2,81 34,71 69,43 0,273 511 61,22 17,08 49,53 12,38 14,73 3,05 31,87 10,62 12,11 2,44 4 2 163,2 4,75 3,22 42,62 56,82 0,271 411 75,07 20,73 60,13 15,03 18,02 3,02 38,61 12,87 14,81 2,42 2 1 212,7 5,82 4,78 52,62 70,16 0,270 346 88,35 24,14 70,05 17,51 21,16 3,00 44,89 14,96 17,37 2,40 1 1 252,0 6,83 5,61 62,35 83,14 0,266 262 113,1 30,32 87,92 21,98 26,99 2,94 56,12 18,71 22,12 2,35 1 1 327,5 8,71 7,14 81,04 108,1 0,291 410 96,21 24,47 67,64 16,91 19,96 3,09 55,11 15,75 18,23 2,79 2 1 228,8 6,44 5,88 61,64 70,45 0,290 345 113,4 28,58 78,94 19,74 23,47 3,06 64,26 18,36 21,43 2,76 1 1 271,4 7,57 6,91 73,11 83,56 0,286 261 145,9 36,08 99,48 24,87 30,03 3,01 80,84 23,10 27,40 2,72 1 1 353,3 9,69 8,84 95,19 108,8 0,283 212 175,5 42,67 117,4 29,34 35,99 2,96 95,21 27,20 32,81 2,67 1 1 431,0 11,61 10,59 116,1 132,7 0,273 511 53,37 15,88 57,88 12,86 15,74 3,29 23,37 9,35 10,50 2,09 4 1 154,2 5,08 2,66 35,52 63,93 0,271 411 65,3 19,24 70,26 15,61 19,25 3,27 28,24 11,29 12,82 2,07 3 1 212,7 6,21 3,64 43,85 78,92 0,270 346 76,67 22,36 81,85 18,19 22,60 3,24 32,74 13,10 15,03 2,05 2 1 252,0 7,29 4,85 51,96 93,53 0,291 410 88,99 23,48 79,84 17,74 21,44 3,36 42,75 14,25 16,24 2,46 3 1 228,8 6,92 4,60 52,84 79,25 0,290 345 104,8 27,39 93,21 20,71 25,21 3,33 49,77 16,59 19,08 2,43 2 1 271,4 8,14 6,16 62,67 94,00 0,286 261 134,4 34,50 117,5 26,11 32,26 3,28 62,40 20,80 24,36 2,39 1 1 353,3 10,41 7,86 81,60 122,4 0,313 510 93,2 22,76 73,37 16,30 19,26 3,46 49,98 14,28 16,24 2,85 4 3 178,8 5,26 4,11 50,03 64,32 0,311 410 114,6 27,73 89,41 19,87 23,63 3,43 60,81 17,37 19,91 2,83 3 1 244,9 7,63 5,61 61,86 79,54 0,310 344 135,3 32,43 104,6 23,24 27,82 3,41 71,00 20,29 23,44 2,81 2 1 290,7 8,98 7,56 73,43 94,41 0,306 260 174,2 41,05 132,3 29,40 35,70 3,36 89,57 25,59 30,04 2,76 1 1 379,1 11,52 9,69 95,77 123,1 0,303 211 210,1 48,70 156,8 34,84 42,91 3,30 105,8 30,23 36,06 2,71 1 1 463,3 13,85 11,64 117,0 150,5


1) = recommended r0 = 2,5 x t when 6,0 mm < t ≤ 10,0 mm

Wt I Wel Wpl i

Appendix 9.1

Cross-sectional properties and resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2).

Table 9.1.2

x

241

x x x x

x

509 115,2 26,20 81,11 18,03 21,02 3,52 67,78 16,95 19,41 409 141,8 31,98 98,98 22,00 25,81 3,50 82,66 20,66 23,83 343 167,6 37,47 115,9 25,76 30,43 3,47 96,74 24,19 28,09 260 216,5 47,61 147,1 32,69 39,14 3,42 122,6 30,65 36,11 210 261,9 56,70 174,9 38,86 47,16 3,37 145,5 36,38 43,49 177 303,8 64,78 199,3 44,28 54,50 3,32 165,6 41,41 50,23 169 315,2 66,43 199,0 44,23 55,03 3,28 165,7 41,41 50,75 511 41,47 13,89 65,38 13,08 16,54 3,50 15,61 7,81 8,69 411 50,52 16,76 79,32 15,86 20,23 3,47 18,78 9,39 10,59 510 61,59 17,73 74,98 15,00 18,50 3,62 25,67 10,27 11,46 410 75,39 21,49 91,2 18,24 22,67 3,59 31,06 12,42 14,01 345 88,56 25,01 106,5 21,29 26,66 3,56 36,06 14,42 16,44 510 84,08 21,56 84,59 16,92 20,46 3,71 38,60 12,87 14,43 410 103,3 26,23 103,1 20,62 25,11 3,69 46,88 15,63 17,68 344 121,7 30,64 120,6 24,11 29,57 3,66 54,65 18,22 20,79 260 156,3 38,68 152,6 30,52 37,94 3,60 68,68 22,89 26,60 509 108,5 25,40 94,19 18,84 22,42 3,80 54,60 15,60 17,60 409 133,5 30,98 115,0 22,99 27,55 3,77 66,50 19,00 21,60 343 157,7 36,27 134,7 26,94 32,48 3,74 77,74 22,21 25,45 260 203,4 46,03 171,0 34,20 41,78 3,69 98,29 28,08 32,68 210 245,7 54,73 203,4 40,67 50,34 3,64 116,4 33,25 39,31 509 134,6 29,24 103,8 20,76 24,38 3,87 73,87 18,47 20,97 409 165,8 35,73 126,9 25,37 29,98 3,84 90,17 22,54 25,77 343 196,1 41,91 148,8 29,76 35,39 3,82 105,6 26,41 30,40 259 253,8 53,38 189,5 37,89 45,62 3,77 134,2 33,54 39,15 210 307,6 63,72 225,9 45,19 55,09 3,72 159,6 39,90 47,24 176 357,4 72,98 258,4 51,68 63,82 3,67 182,1 45,53 54,67 169 371,4 74,97 258,8 51,75 64,58 3,62 182,8 45,70 55,39 510 46,87 15,34 83,29 15,14 19,31 3,81 17,06 8,53 9,45 410 57,12 18,52 101,2 18,41 23,65 3,78 20,54 10,27 11,53 345 66,77 21,45 118,1 21,47 27,80 3,75 23,73 11,86 13,49 510 69,94 19,57 94,95 17,26 21,47 3,93 27,98 11,19 12,42 410 85,65 23,75 115,7 21,03 26,34 3,90 33,88 13,55 15,20 344 100,6 27,66 135,3 24,59 31,01 3,88 39,38 15,75 17,85 509 95,89 23,80 106,6 19,39 23,63 4,04 41,97 13,99 15,59 409 117,8 28,99 130,1 23,66 29,03 4,01 51,02 17,01 19,12 343 138,9 33,89 152,5 27,72 34,22 3,98 59,52 19,84 22,50 260 178,5 42,86 193,5 35,19 44,01 3,93 74,96 24,99 28,84 509 124,2 28,04 118,3 21,51 25,79 4,13 59,23 16,92 18,96 409 152,9 34,23 144,6 26,29 31,72 4,10 72,20 20,63 23,29 343 180,7 40,12 169,6 30,84 37,43 4,08 84,48 24,14 27,46 259 233,3 51,00 216,0 39,27 48,25 4,02 107,0 30,57 35,32 210 282,1 60,76 257,6 46,84 58,27 3,97 127,0 36,28 42,56 Am/V It Wt Iy Wel.y Wpl.y iy Iz Wel.z Wpl.z 1/m mm4 mm3 mm4 mm3 mm3 mm mm4 mm3 mm3 x 104 x 103 x 104 x 103 x 103 x 10 x 104 x 103 x 103

3,22 3,20 3,17 3,13 3,08 3,03 2,99 1,71 1,69 2,12 2,09 2,07 2,51 2,49 2,46 2,42 2,89 2,87 2,84 2,80 2,75 3,26 3,24 3,22 3,17 3,12 3,08 3,04 1,72 1,70 1,68 2,14 2,11 2,09 2,53 2,51 2,49 2,44 2,92 2,90 2,88 2,83 2,79 iz mm x 10

4 4 184,0 3 2 261,1 2 1 310,1 1 1 405,0 1 1 495,6 1 1 582,0 1 1 596,1 4 1 144,4 4 1 198,6 4 1 157,3 4 1 214,7 2 1 271,4 4 2 170,2 4 1 230,9 2 1 290,7 1 1 379,1 4 3 181,9 4 1 247,0 2 1 310,1 1 1 405,0 1 1 495,6 4 4 187,0 4 2 263,1 2 1 329,5 1 1 430,8 1 1 527,9 1 1 620,7 1 1 636,7 4 1 146,9 4 1 203,6 3 1 271,4 4 1 159,8 4 1 219,7 3 1 290,7 4 2 172,7 4 1 235,9 3 1 310,1 1 1 405,0 4 3 184,4 4 1 252,0 3 1 329,5 1 1 430,8 1 1 527,9 PL Nc.Rd h b kN

5,50 4,90 8,33 6,67 9,82 9,07 12,63 11,65 15,22 14,03 17,59 16,21 17,76 16,38 5,34 2,07 6,53 2,81 5,97 2,75 7,32 3,73 8,60 5,30 6,60 3,48 8,10 4,71 9,54 6,71 12,24 8,59 6,08 4,25 8,89 5,74 10,48 8,21 13,48 10,55 16,25 12,69 6,35 5,07 9,68 6,83 11,42 9,81 14,72 12,64 17,78 15,24 20,60 17,64 20,84 17,88 6,23 2,13 7,63 2,92 8,97 3,83 6,93 2,83 8,50 3,87 10,01 5,08 7,63 3,58 9,37 4,89 11,04 6,40 14,20 9,31 6,94 4,37 10,24 5,95 12,08 7,79 15,57 11,40 18,81 13,73 Mc.y.Rd Mc.z.Rd kNm kNm

57,32 64,48 70,93 79,79 84,25 94,78 110,0 123,8 134,7 151,5 158,1 177,9 162,0 182,2 28,41 71,03 35,08 87,69 35,63 71,26 44,03 88,06 52,22 104,4 42,88 71,47 53,03 88,38 62,94 104,9 82,09 136,8 50,15 71,65 62,06 88,66 73,72 105,3 96,27 137,5 117,8 168,3 57,45 71,81 71,13 88,91 84,54 105,7 110,5 138,2 135,5 169,3 159,3 199,1 163,4 204,2 28,51 78,39 35,22 96,86 41,78 114,9 35,73 78,61 44,19 97,22 52,45 115,4 42,99 78,81 53,20 97,52 63,19 115,8 82,52 151,3 50,27 78,99 62,24 97,8 73,97 116,2 96,72 152,0 118,5 186,2 Vpl.y.Rd Vpl.z.Rd kN kN

Appendix 9.1

1)

0,333 0,331 0,330 0,326 0,323 0,319 0,313 0,273 0,271 0,293 0,291 0,290 0,313 0,311 0,310 0,306 0,333 0,331 0,330 0,326 0,323 0,353 0,351 0,350 0,346 0,343 0,339 0,333 0,293 0,291 0,290 0,313 0,311 0,310 0,333 0,331 0,330 0,326 0,353 0,351 0,350 0,346 0,343 Au m2/m


x x x x x x

90 80 2 5,13 6,54 90 80 2,5 6,35 8,09 90 80 3 7,54 9,61 90 80 4 9,85 12,55 90 80 5 12,1 15,36 90 80 6 14,2 18,03 90 80 6,3 14,5 18,47 100 40 2 4,19 5,34 100 40 2,5 5,17 6,59 100 50 2 4,50 5,74 100 50 2,5 5,56 7,09 100 50 3 6,60 8,41 100 60 2 4,82 6,14 100 60 2,5 5,96 7,59 100 60 3 7,07 9,01 100 60 4 9,22 11,75 100 70 2 5,13 6,54 100 70 2,5 6,35 8,09 100 70 3 7,54 9,61 100 70 4 9,85 12,55 100 70 5 12,1 15,36 100 80 2 5,45 6,94 100 80 2,5 6,74 8,59 100 80 3 8,01 10,21 100 80 4 10,5 13,35 100 80 5 12,8 16,36 100 80 6 15,1 19,23 100 80 6,3 15,5 19,73 110 40 2 4,50 5,74 110 40 2,5 5,56 7,09 110 40 3 6,60 8,41 110 50 2 4,82 6,14 110 50 2,5 5,96 7,59 110 50 3 7,07 9,01 110 60 2 5,13 6,54 110 60 2,5 6,35 8,09 110 60 3 7,54 9,61 110 60 4 9,85 12,55 110 70 2 5,45 6,94 110 70 2,5 6,74 8,59 110 70 3 8,01 10,21 110 70 4 10,5 13,35 110 70 5 12,8 16,36 h b t M A mm mm mm kg/m mm2 x 102

b

h

t y r0

M A Au Am/V It


r0 = 2,0 x t when t ≤ 6,0 mm

z series

r0 = 3,0 x t when t > 10,0 mm

1) h

b t mm mm mm

M kg/m

x

242 x x

x x x x

110 110 110 110 110 110 110 110 110 110 110 110 110 110 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

90 90 90 90 90 90 90 100 100 100 100 100 100 100 40 40 40 50 50 50 60 60 60 60 80 80 80 80 80 80 80

2 2,5 3 4 5 6 6,3 2 2,5 3 4 5 6 6,3 2 2,5 3 2 2,5 3 2 2,5 3 4 2 2,5 3 4 5 6 6,3

6,07 7,53 8,96 11,7 14,4 17,0 17,5 6,39 7,92 9,43 12,4 15,2 17,9 18,5 4,82 5,96 7,07 5,13 6,35 7,54 5,45 6,74 8,01 10,5 6,07 7,53 8,96 11,7 14,4 17,0 17,5

A mm2 x 102 7,74 9,59 11,41 14,95 18,36 21,63 22,25 8,14 10,09 12,01 15,75 19,36 22,83 23,51 6,14 7,59 9,01 6,54 8,09 9,61 6,94 8,59 10,21 13,35 7,74 9,59 11,41 14,95 18,36 21,63 22,25





Au Am/V It Wt Iy m2/m 1/m mm4 mm3 mm4 x 104 x 103 x 104 0,393 508 186,6 36,52 141,6 0,391 408 230,3 44,73 173,5 0,390 342 272,8 52,59 204,0 0,386 258 354,2 67,31 261,0 0,383 209 430,8 80,74 312,8 0,379 175 502,6 92,94 359,6 0,373 168 523,8 95,78 362,3 0,413 507 220,1 40,76 153,3 0,411 407 271,8 49,98 187,9 0,410 341 322,2 58,83 221,2 0,406 258 419,1 75,47 283,5 0,403 208 510,7 90,74 340,4 0,399 175 596,9 104,7 392,1 0,393 167 622,9 108,1 396,2 0,313 510 52,32 16,78 104,1 0,311 410 63,77 20,27 126,7 0,310 344 74,56 23,51 148,0 0,333 509 78,39 21,41 118,0 0,331 409 96,03 26,01 144,0 0,330 343 112,9 30,32 168,6 0,353 509 107,9 26,05 131,9 0,351 409 132,6 31,75 161,2 0,350 343 156,3 37,14 189,1 0,346 259 201,1 47,05 240,7 0,393 508 175,0 35,32 159,8 0,391 408 215,8 43,23 195,8 0,390 342 255,5 50,80 230,2 0,386 258 331,2 64,93 294,6 0,383 209 402,3 77,77 353,1 0,379 175 468,5 89,40 406,1 0,373 168 487,8 92,07 408,5

Wel.y Wpl.y iy Iz Wel.z Wpl.z iz PL Nc.Rd mm3 mm3 mm mm4 mm3 mm3 mm h b kN x 103 x 103 x 10 x 104 x 103 x 103 x 10 25,75 30,11 4,28 104,4 23,19 26,30 3,67 4 4 193,4 31,54 37,09 4,25 127,7 28,38 32,38 3,65 4 3 280,1 37,09 43,85 4,23 150,0 33,33 38,26 3,63 3 2 368,2 47,45 56,73 4,18 191,5 42,56 49,46 3,58 1 1 482,4 56,87 68,77 4,13 229,1 50,91 59,91 3,53 1 1 592,4 65,38 79,98 4,08 262,9 58,42 69,62 3,49 1 1 698,2 65,87 81,29 4,04 265,4 58,97 70,85 3,45 1 1 718,1 27,87 32,27 4,34 132,6 26,52 30,26 4,04 4 4 196,5 34,17 39,78 4,32 162,5 32,50 37,30 4,01 4 4 286,3 40,21 47,06 4,29 191,2 38,23 44,12 3,99 3 2 387,5 51,54 60,97 4,24 244,8 48,96 57,14 3,94 1 1 508,2 61,88 74,02 4,19 293,7 58,74 69,34 3,90 1 1 624,7 71,29 86,22 4,14 338,0 67,60 80,74 3,85 1 1 736,9 72,04 87,82 4,11 341,9 68,37 82,29 3,81 1 1 758,7 17,34 22,28 4,12 18,50 9,25 10,21 1,74 4 1 148,9 21,12 27,32 4,09 22,30 11,15 12,47 1,71 4 1 207,7 24,67 32,16 4,05 25,79 12,89 14,60 1,69 4 1 270,5 19,67 24,64 4,25 30,28 12,11 13,38 2,15 4 1 161,8 23,99 30,26 4,22 36,70 14,68 16,39 2,13 4 1 223,8 28,10 35,67 4,19 42,69 17,08 19,26 2,11 4 1 289,8 21,99 27,00 4,36 45,33 15,11 16,75 2,56 4 2 174,7 26,87 33,20 4,33 55,15 18,38 20,56 2,53 4 1 240,0 31,52 39,18 4,30 64,40 21,47 24,21 2,51 4 1 309,2 40,12 50,49 4,25 81,25 27,08 31,08 2,47 2 1 430,8 26,63 31,72 4,54 86,04 21,51 24,09 3,33 4 4 191,6 32,63 39,07 4,52 105,2 26,30 29,65 3,31 4 2 272,2 38,37 46,20 4,49 123,4 30,86 35,02 3,29 4 1 347,9 49,10 59,77 4,44 157,3 39,32 45,23 3,24 2 1 482,4 58,86 72,45 4,39 187,8 46,94 54,74 3,20 1 1 592,4 67,68 84,25 4,33 215,0 53,76 63,55 3,15 1 1 698,2 68,08 85,57 4,28 217,1 54,28 64,68 3,12 1 1 718,1

Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd kNm kNm kN kN 7,54 10,18 14,15 18,31 22,19 25,81 26,23 7,79 10,42 15,19 19,68 23,89 27,82 28,34 7,19 8,82 10,38 7,95 9,77 11,51 8,71 10,71 12,64 16,29 8,18 12,61 14,91 19,29 23,38 27,19 27,61

6,10 8,26 10,76 15,96 19,34 22,47 22,86 7,03 9,49 12,34 18,44 22,38 26,06 26,56 2,19 3,01 3,85 2,90 3,99 5,12 3,67 5,04 6,45 10,03 5,34 7,29 9,32 14,60 17,66 20,51 20,87

64,87 80,40 95,65 125,3 153,9 181,4 186,6 72,20 89,52 106,6 139,7 171,7 202,6 208,6 28,59 35,35 41,96 35,82 44,33 52,65 43,08 53,35 63,40 82,90 57,66 71,47 85,03 111,4 136,8 161,2 165,8

79,29 98,27 116,9 153,2 188,1 221,7 228,0 79,42 98,47 117,2 153,7 188,9 222,9 229,5 85,76 106,1 125,9 85,98 106,4 126,4 86,17 106,7 126,8 165,8 86,50 107,2 127,5 167,1 205,2 241,9 248,7



Wt I Wel Wpl i

Appendix 9.1

Cross-sectional properties and resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.

Table 9.1.2

x

0,413 0,411 0,410 0,406 0,403 0,399 0,393 0,431 0,430 0,426 0,423 0,419 0,413 0,351 0,350 0,391 0,390 0,386 0,411 0,410 0,406 0,403 0,430 0,426 0,423 0,419 0,413 0,471 0,470 0,466 0,463 0,459 0,453 0,491 0,490 0,486 0,483 0,479 0,473 0,510 0,506 0,503 0,499 0,493 Au m2/m

507 211,8 39,96 173,7 407 261,5 48,98 213,0 341 309,9 57,64 250,7 258 402,8 73,88 321,5 208 490,4 88,76 386,2 175 572,7 102,3 445,1 167 597,3 105,6 449,3 407 309,4 54,73 230,3 341 367,0 64,47 271,3 257 477,8 82,83 348,4 208 582,9 99,75 419,3 174 682,0 115,3 484,1 167 712,3 119,1 490,00 409 77,20 23,79 189,3 343 90,30 27,62 221,8 408 162,7 37,26 236,6 342 191,9 43,64 278,1 258 247,1 55,42 355,6 407 213,1 44,00 260,2 341 252,0 51,66 306,2 258 326,0 65,94 392,6 208 395,1 78,88 471,5 341 317,1 59,69 334,4 257 411,6 76,48 429,6 208 500,5 91,83 517,1 174 583,8 105,8 597,00 167 608,5 109,2 602,7 406 387,3 64,23 331,1 340 459,6 75,76 390,7 257 599,3 97,57 503,6 207 732,1 117,8 608,2 174 858,0 136,5 704,8 166 896,9 141,2 715,4 406 451,3 70,98 354,7 340 536,1 83,80 418,9 256 700,0 108,1 540,6 207 856,4 130,8 653,8 173 1005 151,8 758,7 166 1052 157,3 771,8 340 615,4 91,84 447,00 256 804,5 118,7 577,6 206 985,5 143,8 699,4 173 1158 167,2 812,6 165 1213 173,3 828,1 Am/V It Wt Iy 1/m mm4 mm3 mm4 4 3 x 10 x 10 x 104

28,95 34,08 4,62 112,1 35,50 42,01 4,59 137,3 41,79 49,71 4,57 161,4 53,58 64,41 4,52 206,3 64,37 78,20 4,47 247,2 74,18 91,09 4,42 284,1 74,88 92,73 4,37 287,5 38,38 44,95 4,66 174,4 45,21 53,22 4,64 205,3 58,07 69,05 4,59 263,2 69,88 83,95 4,54 316,3 80,68 97,93 4,49 364,6 81,67 99,89 4,45 369,6 27,04 35,41 4,69 25,82 31,68 41,76 4,66 29,90 33,79 42,29 4,97 63,43 39,73 49,98 4,94 74,16 50,80 64,63 4,88 93,81 37,17 45,72 5,08 89,3 43,75 54,09 5,05 104,7 56,09 70,07 4,99 133,2 67,35 85,05 4,94 158,7 47,77 58,20 5,15 141,2 61,37 75,51 5,10 180,4 73,87 91,80 5,04 215,9 85,29 107,1 4,98 248,0 86,10 109,1 4,93 251,4 47,30 56,04 5,35 198,2 55,82 66,42 5,32 233,5 71,94 86,39 5,27 300,1 86,89 105,3 5,22 361,4 100,7 123,2 5,16 417,7 102,2 125,9 5,12 425,0 50,67 59,47 5,42 245,7 59,84 70,53 5,39 289,9 77,23 91,83 5,34 373,4 93,4 112,1 5,29 450,7 108,4 131,2 5,24 522,0 110,3 134,4 5,20 531,9 63,86 74,64 5,46 353,4 82,52 97,27 5,41 456,1 99,91 118,8 5,36 551,6 116,1 139,3 5,31 640,2 118,3 142,8 5,27 653,1 Wel.y Wpl.y iy Iz mm3 mm3 mm mm4 x 103 x 103 x 10 x 104

24,91 28,06 30,51 34,56 35,86 40,87 45,85 52,90 54,93 64,16 63,13 74,66 63,88 76,12 34,88 39,73 41,06 47,03 52,65 60,98 63,25 74,09 72,91 86,38 73,91 88,19 12,91 14,34 14,95 16,82 21,14 23,43 24,72 27,63 31,27 35,56 25,51 28,35 29,91 33,49 38,05 43,24 45,35 52,31 35,31 39,64 45,10 51,31 53,99 62,24 61,99 72,43 62,85 73,97 39,63 44,61 46,70 52,85 60,02 68,66 72,29 83,59 83,53 97,66 84,99 100,0 44,68 50,53 52,70 59,90 67,89 77,93 81,95 95,02 94,91 111,2 96,71 114,0 58,90 67,26 76,02 87,61 91,94 107,0 106,7 125,3 108,9 128,6 Wel.z Wpl.z mm3 mm3 3 x 10 x 103

3,71 3,69 3,67 3,62 3,57 3,53 3,50 4,06 4,04 3,99 3,94 3,89 3,86 1,73 1,71 2,57 2,55 2,51 2,98 2,95 2,91 2,86 3,35 3,30 3,26 3,21 3,19 4,14 4,11 4,07 4,02 3,97 3,95 4,51 4,49 4,44 4,39 4,35 4,32 4,85 4,81 4,76 4,71 4,68 iz mm x 10

4 4 195,5 8,51 6,24 4 3 284,2 11,46 8,50 4 2 367,3 16,04 10,85 2 1 508,2 20,79 17,07 1 1 624,7 25,24 20,71 1 1 736,9 29,40 24,10 1 1 758,7 29,93 24,57 4 4 290,5 11,73 9,77 4 2 386,7 17,17 12,45 2 1 534,1 22,28 19,68 1 1 657,00 27,09 23,91 1 1 775,6 31,61 27,88 1 1 799,4 32,24 28,46 4 1 214,1 11,43 3,16 4 1 281,7 13,48 4,09 4 1 246,3 13,65 5,28 4 1 320,5 16,13 6,84 3 1 482,4 20,86 10,09 4 1 262,5 14,76 6,43 4 1 339,8 17,46 8,32 3 1 508,2 22,62 12,28 1 1 624,7 27,45 16,88 4 1 359,2 18,78 9,86 3 1 534,1 24,37 14,56 1 1 657,0 29,63 20,09 1 1 775,6 34,56 23,37 1 1 799,4 35,20 23,87 4 4 296,8 14,51 10,23 4 2 397,9 21,44 13,16 3 1 585,7 27,88 19,37 1 1 721,5 33,98 26,98 1 1 853,1 39,75 31,52 1 1 880,7 40,64 32,27 4 4 301,8 14,98 11,61 4 3 409,1 19,31 14,92 3 1 611,5 29,64 21,91 1 1 753,8 36,16 30,67 1 1 891,8 42,34 35,88 1 1 921,4 43,36 36,78 4 4 416,4 19,51 16,73 3 2 637,3 31,39 24,53 1 1 786,00 38,34 34,51 1 1 930,5 44,94 40,44 1 1 962,00 46,08 41,49 PL Nc.Rd Mc.y.Rd Mc.z.Rd h b kN kNm kNm

64,98 86,64 80,57 107,4 95,89 127,9 125,8 167,7 154,6 206,1 182,3 243,1 187,7 250,3 89,68 107,6 106,8 128,1 140,2 168,2 172,4 206,9 203,5 244,3 209,8 251,7 35,56 124,5 42,27 147,9 53,60 125,1 63,77 148,8 83,56 195,00 62,66 125,3 74,58 149,2 97,81 195,6 120,2 240,4 85,43 149,5 112,1 196,2 137,9 241,4 162,8 285,00 167,8 293,7 89,97 126,00 107,2 150,1 140,9 197,3 173,6 243,00 205,2 287,3 211,9 296,6 99,11 126,1 118,1 150,3 155,3 197,7 191,5 243,7 226,5 288,3 234,1 297,9 129,1 150,6 169,8 198,1 209,5 244,4 248,0 289,3 256,4 299,1 Vpl.y.Rd Vpl.z.Rd kN kN

Appendix 9.1

1)

90 2 6,39 8,14 90 2,5 7,92 10,09 90 3 9,43 12,01 90 4 12,4 15,75 90 5 15,2 19,36 90 6 17,9 22,83 90 6,3 18,5 23,51 100 2,5 8,31 10,59 100 3 9,90 12,61 100 4 13,0 16,55 100 5 16,0 20,36 100 6 18,9 24,03 100 6,3 19,4 24,77 40 2,5 6,74 8,59 40 3 8,01 10,21 60 2,5 7,53 9,59 60 3 8,96 11,41 60 4 11,7 14,95 70 2,5 7,92 10,09 70 3 9,43 12,01 70 4 12,4 15,75 70 5 15,2 19,36 80 3 9,90 12,61 80 4 13,0 16,55 80 5 16,0 20,36 80 6 18,9 24,03 80 6,3 19,4 24,77 100 2,5 9,10 11,59 100 3 10,8 13,81 100 4 14,3 18,15 100 5 17,6 22,36 100 6 20,8 26,43 100 6,3 21,4 27,29 110 2,5 9,49 12,09 110 3 11,3 14,41 110 4 14,9 18,95 110 5 18,3 23,36 110 6 21,7 27,63 110 6,3 22,4 28,55 120 3 11,8 15,01 120 4 15,5 19,75 120 5 19,1 24,36 120 6 22,6 28,83 120 6,3 23,4 29,81 b t M A mm mm kg/m mm2 x 102


243

x x x x x x x

120 120 120 120 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 h mm

b

h

t y r0

M A Au Am/V It


r0 = 2,0 x t when t ≤ 6,0 mm

z series

r0 = 3,0 x t when t > 10,0 mm

1) h

b t mm mm mm

M kg/m

x

244 x x x x x x

150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 160 160 160

50 50 60 60 60 70 70 70 70 90 90 90 90 90 90 100 100 100 100 100 100 100 110 110 110 110 110 110 40 40 50

2,5 3 2,5 3 4 2,5 3 4 5 2,5 3 4 5 6 6,3 3 4 5 6 6,3 7,1 8 2,5 3 4 5 6 6,3 2,5 3 3

7,53 8,96 7,92 9,43 12,4 8,31 9,90 13,0 16,0 9,10 10,8 14,3 17,6 20,8 21,4 11,3 14,9 18,3 21,7 22,4 24,9 27,7 9,88 11,8 15,5 19,1 22,6 23,4 7,53 8,96 9,43

A mm2 x 102 9,59 11,41 10,09 12,01 15,75 10,59 12,61 16,55 20,36 11,59 13,81 18,15 22,36 26,43 27,29 14,41 18,95 23,36 27,63 28,55 31,75 35,24 12,59 15,01 19,75 24,36 28,83 29,81 9,59 11,41 12,01





Au Am/V It Wt Iy m2/m 1/m mm4 mm3 mm4 x 104 x 103 x 104 0,391 408 127,7 32,78 254,1 0,390 342 150,2 38,28 298,6 0,411 407 177,9 40,01 281,3 0,410 341 210,0 46,89 331,0 0,406 258 270,5 59,60 424,0 0,431 407 233,7 47,25 308,5 0,430 341 276,4 55,51 363,4 0,426 257 357,7 70,92 466,7 0,423 208 433,6 84,92 561,5 0,471 406 358,9 61,74 362,9 0,470 340 425,7 72,77 428,2 0,466 257 554,2 93,60 552,0 0,463 207 676,0 112,8 666,6 0,459 174 791,1 130,6 772,4 0,453 166 826,3 135,0 783,2 0,490 340 507,2 81,40 460,6 0,486 256 661,6 104,9 594,6 0,483 207 808,7 126,8 719,2 0,479 173 948,3 147,1 834,7 0,473 166 991,6 152,3 848,3 0,469 148 1096 166,7 926,6 0,466 132 1206 181,9 1008 0,511 406 498,8 76,23 417,3 0,510 340 592,6 90,04 493,1 0,506 256 774,2 116,3 637,3 0,503 206 947,7 140,8 771,8 0,499 173 1113 163,6 896,9 0,493 165 1165 169,6 913,4 0,391 408 90,78 27,31 269,0 0,390 342 106,2 31,74 315,9 0,410 341 162,8 40,93 352,9

Wel.y Wpl.y iy Iz mm3 mm3 mm mm4 x 103 x 103 x 10 x 104 33,88 43,52 5,15 45,17 39,81 51,43 5,12 52,65 37,50 47,21 5,28 67,56 44,13 55,84 5,25 79,04 56,54 72,31 5,19 100,1 41,13 50,89 5,40 94,99 48,45 60,25 5,37 111,4 62,22 78,15 5,31 141,9 74,86 94,98 5,25 169,3 48,38 58,27 5,60 166,0 57,10 69,07 5,57 195,4 73,59 89,83 5,51 250,7 88,88 109,5 5,46 301,4 103,0 128,0 5,41 347,7 104,4 130,8 5,36 353,8 61,42 73,48 5,65 247,6 79,28 95,67 5,60 318,6 95,89 116,7 5,55 384,0 111,3 136,7 5,50 444,2 113,1 139,9 5,45 452,7 123,6 154,1 5,40 493,5 134,4 169,2 5,35 535,7 55,64 65,64 5,76 260,2 65,74 77,89 5,73 307,1 84,97 101,5 5,68 395,9 102,9 124,0 5,63 478,3 119,6 145,3 5,58 554,5 121,8 148,9 5,54 565,8 33,63 44,5 5,30 29,34 39,49 52,57 5,26 34,02 44,11 57,28 5,42 55,97

Wel.z Wpl.z iz PL Nc.Rd mm3 mm3 mm h b kN x 103 x 103 x 10 18,07 19,95 2,17 4 1 232,7 21,06 23,49 2,15 4 1 305,5 22,52 24,87 2,59 4 1 248,8 26,35 29,34 2,57 4 1 324,9 33,37 37,80 2,52 4 1 488,6 27,14 30,04 3,00 4 1 265,0 31,84 35,50 2,97 4 1 344,3 40,54 45,88 2,93 4 1 514,4 48,37 55,56 2,88 2 1 657,0 36,89 41,13 3,78 4 3 293,1 43,43 48,70 3,76 4 2 383,0 55,72 63,22 3,72 4 1 566,0 66,99 76,91 3,67 2 1 721,5 77,27 89,78 3,63 1 1 853,1 78,62 91,94 3,60 1 1 880,7 49,53 55,76 4,15 4 2 402,4 63,71 72,50 4,10 4 1 591,8 76,80 88,34 4,05 2 1 753,8 88,84 103,3 4,01 1 1 891,8 90,53 105,9 3,98 1 1 921,4 98,69 116,5 3,94 1 1 1024,6 107,1 127,9 3,90 1 1 1137,4 47,30 53,22 4,55 4 4 304,3 55,83 63,11 4,52 4 3 413,5 71,98 82,17 4,48 4 1 617,6 86,96 100,3 4,43 2 1 786,0 100,8 117,4 4,39 1 1 930,5 102,9 120,5 4,36 1 1 962,0 14,67 16,22 1,75 4 1 218,7 17,01 19,04 1,73 4 1 290,0 22,39 24,90 2,16 4 1 309,4

Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd kNm kNm kN kN 14,04 16,60 15,23 18,02 23,34 16,42 19,44 25,22 30,65 15,61 22,29 28,99 35,33 41,32 42,22 23,71 30,87 37,67 44,11 45,14 49,72 54,59 16,49 21,22 32,76 40,01 46,90 48,07 14,36 16,97 18,49

4,27 5,55 5,38 6,99 10,32 6,55 8,51 12,55 17,93 9,08 11,74 17,29 24,82 28,98 29,67 13,46 19,79 28,51 33,34 34,18 37,61 41,26 11,83 15,25 22,38 32,36 37,89 38,88 3,27 4,27 5,67

44,67 53,14 53,71 63,93 83,84 62,78 74,75 98,11 120,7 80,98 96,48 126,8 156,2 184,7 190,7 107,4 141,2 174,1 206,0 212,8 236,6 262,7 99,24 118,3 155,7 192,0 227,3 235,0 35,73 42,51 53,27

134,0 159,4 134,3 159,8 209,6 134,5 160,2 210,2 258,6 135,0 160,8 211,3 260,4 307,8 317,8 161,1 211,8 261,1 308,9 319,2 354,9 394,0 135,3 161,3 212,3 261,8 309,9 320,4 142,9 170,1 170,5



Wt I Wel Wpl i

Appendix 9.1


Table 9.1.2

x x x

245 x x x x

x

3 9,90 12,61 4 13,0 16,55 3 10,4 13,21 4 13,6 17,35 5 16,8 21,36 3 10,8 13,81 4 14,3 18,15 5 17,6 22,36 6 20,8 26,43 6,3 21,4 27,29 3 11,3 14,41 4 14,9 18,95 5 18,3 23,36 6 21,7 27,63 6,3 22,4 28,55 7,1 24,9 31,75 3 11,8 15,01 4 15,5 19,75 5 19,1 24,36 6 22,6 28,83 6,3 23,4 29,81 4 16,8 21,35 5 20,7 26,36 6 24,5 31,23 6,3 25,4 32,33 7,1 28,3 36,01 8 31,4 40,04 8,8 34,2 43,52 10 38,1 48,57 4 16,8 21,35 5 20,7 26,36 5,6 23,0 29,30 6 24,5 31,23 6,3 25,4 32,33 7,1 28,3 36,01 8 31,4 40,04 4 18,0 22,95 5 22,3 28,36 6 26,4 33,63 6,3 27,4 34,85 7,1 30,5 38,85 8 34,0 43,24 8,8 36,9 47,04 10 41,3 52,57 t M A mm kg/m mm2 x 102

0,430 0,426 0,450 0,446 0,443 0,470 0,466 0,463 0,459 0,453 0,490 0,486 0,483 0,479 0,473 0,469 0,510 0,506 0,503 0,499 0,493 0,546 0,543 0,539 0,533 0,529 0,526 0,522 0,517 0,546 0,543 0,541 0,539 0,533 0,529 0,526 0,586 0,583 0,579 0,573 0,569 0,566 0,562 0,557 Au m2/m

341 228,2 257 294,0 341 301,0 257 389,7 207 472,5 340 380,3 257 494,1 207 601,3 174 702,1 166 732,3 340 465,4 256 606,2 207 739,7 173 866,0 166 904,7 148 997,9 340 555,5 256 724,9 206 886,4 173 1040 165 1088 256 979,5 206 1201 173 1414 165 1481 147 1641 131 1814 120 1960 106 2166 256 853,9 206 1045 185 1155 173 1227 165 1283 147 1420 131 1565 255 1160 206 1424 172 1677 164 1757 146 1949 131 2156 119 2332 106 2582 Am/V It 1/m mm4 x 104

50,14 389,9 63,79 500,4 59,36 426,8 75,90 549,0 90,96 661,6 68,59 463,8 88,03 597,7 105,9 721,7 122,3 836,0 126,3 846,5 77,82 500,8 100,2 646,4 120,9 781,8 140,0 907,2 144,8 920,9 158,4 1006 87,05 537,8 112,3 695,1 135,8 841,9 157,7 978,4 163,3 995,4 136,6 792,4 165,8 962,0 193,2 1121 200,4 1144 220,3 1255 241,4 1371 258,9 1467 283,0 1597 127,1 926,0 153,9 1124 169,1 1237 178,9 1310 185,5 1335 203,5 1463 222,5 1598 154,6 1050 187,8 1277 219,1 1491 227,6 1525 250,5 1676 274,8 1835 295,1 1967 323,3 2149 Wt Iy mm3 mm4 x 103 x 104

48,73 61,99 5,56 62,54 80,38 5,50 53,35 66,70 5,68 68,63 86,62 5,63 82,70 105,4 5,57 57,98 71,41 5,80 74,71 92,86 5,74 90,21 113,2 5,68 104,5 132,3 5,62 105,8 135,1 5,57 62,60 76,12 5,90 80,80 99,1 5,84 97,72 120,9 5,79 113,4 141,6 5,73 115,1 144,8 5,68 125,7 159,4 5,63 67,22 80,83 5,99 86,88 105,3 5,93 105,2 128,7 5,88 122,3 150,8 5,83 124,4 154,5 5,78 99,06 117,8 6,09 120,3 144,2 6,04 140,1 169,3 5,99 143,0 173,8 5,95 156,9 192,0 5,90 171,4 211,5 5,85 183,4 227,9 5,81 199,6 251,0 5,73 102,9 125,9 6,59 124,9 154,0 6,53 137,4 170,3 6,50 145,5 180,8 6,48 148,3 185,5 6,43 162,6 204,9 6,38 177,6 225,6 6,32 116,7 140,0 6,76 141,9 171,5 6,71 165,7 201,7 6,66 169,5 207,4 6,62 186,2 229,4 6,57 203,9 253,1 6,51 218,6 273,2 6,47 238,8 301,5 6,39 Wel.y Wpl.y iy mm3 mm3 mm x 103 x 103 x 10

83,91 106,4 118,2 150,6 179,9 159,0 203,5 244,1 280,9 285,7 206,8 265,5 319,5 368,9 375,9 409,2 261,8 337,0 406,6 470,7 480,4 510,0 617,8 718,3 734,6 804,5 877,9 938,0 1019 373,9 451,8 495,7 523,8 535,8 585,6 637,5 563,8 684,0 796,3 816,1 895,2 978,4 1047 1141 Iz mm4 x 104

27,97 31,05 35,46 40,04 33,76 37,51 43,04 48,52 51,39 58,81 39,76 44,26 50,89 57,39 61,03 69,74 70,22 81,31 71,43 83,25 45,95 51,31 59,01 66,66 71,00 81,16 81,98 94,82 83,53 97,21 90,93 106,9 52,35 58,67 67,40 76,34 81,32 93,09 94,15 108,9 96,07 111,8 84,99 96,89 103,0 118,5 119,7 139,0 122,4 142,9 134,1 157,7 146,3 173,7 156,3 187,0 169,9 205,8 74,78 84,02 90,35 102,6 99,14 113,3 104,8 120,2 107,2 123,6 117,1 136,3 127,5 149,9 93,97 106,2 114,0 130,0 132,7 152,7 136,0 157,2 149,2 173,8 163,1 191,6 174,5 206,6 190,1 227,8 Wel.z Wpl.z mm3 mm3 x 103 x 103

2,58 2,54 2,99 2,95 2,90 3,39 3,35 3,30 3,26 3,24 3,79 3,74 3,70 3,65 3,63 3,59 4,18 4,13 4,09 4,04 4,01 4,89 4,84 4,80 4,77 4,73 4,68 4,64 4,58 4,18 4,14 4,11 4,10 4,07 4,03 3,99 4,96 4,91 4,87 4,84 4,80 4,76 4,72 4,66 iz mm x 10

4 1 4 1 4 1 4 1 2 1 4 1 4 1 2 1 1 1 1 1 4 2 4 1 2 1 1 1 1 1 1 1 4 2 4 1 2 1 1 1 1 1 4 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 3 1 2 1 2 1 1 1 1 1 1 1 4 2 3 1 2 1 1 1 1 1 1 1 1 1 1 1 PL h b

328,7 498,1 348,1 523,9 689,2 367,5 549,7 721,5 853,1 880,7 386,8 575,5 753,8 891,8 921,4 1025 406,2 601,3 786,0 930,5 962,0 653,0 850,6 1008 1043 1162 1292 1405 1567 616,9 850,6 945,5 1008 1043 1162 1292 668,6 915,1 1085 1125 1254 1396 1518 1696 Nc.Rd kN

20,01 7,14 25,94 10,61 21,53 8,68 27,96 12,91 34,02 18,98 23,05 10,29 29,97 15,29 36,52 22,51 42,70 26,24 43,60 26,87 24,57 11,97 31,98 17,77 39,02 26,19 45,69 30,60 46,73 31,37 51,45 34,50 26,09 13,73 34,00 20,33 41,52 30,04 48,67 35,16 49,85 36,08 38,02 25,73 46,52 38,23 54,63 44,85 56,10 46,11 61,96 50,90 68,26 56,04 73,55 60,36 80,99 66,42 40,63 21,26 49,71 29,16 54,95 36,56 58,36 38,80 59,88 39,89 66,11 44,00 72,81 48,39 45,17 26,88 55,35 36,79 65,10 49,27 66,94 50,73 74,04 56,07 81,70 61,83 88,16 66,68 97,31 73,52 Mc.y.Rd Mc.z.Rd kNm kNm

64,07 170,9 84,09 224,2 74,9 171,2 98,38 224,9 121,1 276,8 85,76 171,5 112,7 225,4 138,9 277,7 164,2 328,3 169,5 339,0 96,65 171,8 127,1 226,0 156,7 278,5 185,4 329,5 191,5 340,5 213,0 378,6 107,6 172,1 141,5 226,4 174,6 279,3 206,6 330,6 213,6 341,8 170,5 227,3 210,5 280,6 249,4 332,5 258,2 344,2 287,5 383,4 319,8 426,3 347,6 463,4 387,8 517,1 142,1 255,7 175,4 315,7 195,0 350,9 207,8 374,1 215,1 387,3 239,6 431,3 266,5 479,6 171,0 256,6 211,3 317,0 250,7 376,0 259,7 389,6 289,5 434,3 322,3 483,4 350,6 525,9 391,8 587,7 Vpl.y.Rd Vpl.z.Rd kN kN

Appendix 9.1

1)

60 60 70 70 70 80 80 80 80 80 90 90 90 90 90 90 100 100 100 100 100 120 120 120 120 120 120 120 120 100 100 100 100 100 100 100 120 120 120 120 120 120 120 120 b mm


x

160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 h mm

b

h

t y r0

M A Au Am/V It


r0 = 2,0 x t when t ≤ 6,0 mm

z

b t mm mm mm

M kg/m

x

x x x

246


200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 250 250 250 250 250 250 250 250 250

80 80 80 80 100 100 100 100 100 120 120 120 120 120 120 120 120 120 120 120 120 120 120 150 150 150 150 150 150 150 150 150

4 5 6 6,3 5 6 6,3 7,1 8 5 6 6,3 7,1 8 8,8 10 5 6 6,3 7,1 8 8,8 10 5 6 6,3 7,1 8 8,8 10 12 12,5

16,8 20,7 24,5 25,4 22,3 26,4 27,4 30,5 34,0 23,8 28,3 29,3 32,7 36,5 39,7 44,4 25,4 30,2 31,3 35,0 39,0 42,5 47,5 30,1 35,8 37,3 41,6 46,5 50,8 57,0 66,0 68,3

A mm2 x 102 21,35 26,36 31,23 32,33 28,36 33,63 34,85 38,85 43,24 30,36 36,03 37,37 41,69 46,44 50,56 56,57 32,36 38,43 39,89 44,53 49,64 54,08 60,57 38,36 45,63 47,45 53,05 59,24 64,64 72,57 84,06 87,04

Au Am/V It m2/m 1/m mm4 x 104 0,546 256 663,6 0,543 206 808,4 0,539 173 944,8 0,533 165 986 0,583 206 1206 0,579 172 1417 0,573 164 1483 0,569 146 1641 0,566 131 1811 0,623 205 1652 0,619 172 1947 0,613 164 2040 0,609 146 2265 0,606 130 2507 0,602 119 2714 0,597 106 3007 0,663 205 1885 0,659 171 2222 0,653 164 2329 0,649 146 2586 0,646 130 2864 0,642 119 3102 0,637 105 3440 0,783 204 3285 0,779 171 3886 0,773 163 4078 0,769 145 4543 0,766 129 5050 0,762 118 5488 0,757 104 6121 0,738 88 7088 0,736 85 7315



Wt mm3 x 103 111,1 134,1 155,2 160,6 171,9 200,1 207,6 228,0 249,6 209,9 245,1 254,7 280,7 308,3 331,4 363,7 231,9 271,1 281,9 310,8 341,7 367,7 404,1 336,9 395,7 412,2 456,3 504,0 544,5 602,1 684,4 704,1

Iy mm4 x 104 1046 1269 1477 1503 1459 1703 1739 1910 2091 1649 1929 1976 2174 2386 2562 2806 2082 2439 2501 2756 3029 3258 3576 3304 3886 4001 4427 4886 5274 5825 6458 6633

Wel.y mm3 x 103 104,6 126,9 147,7 150,3 145,9 170,3 173,9 191,0 209,1 164,9 192,9 197,6 217,4 238,6 256,2 280,6 189,3 221,7 227,4 250,6 275,4 296,1 325,1 264,3 310,8 320,1 354,1 390,9 422,0 466,0 516,6 530,6

Wpl.y mm3 x 103 132,4 161,9 190,0 194,7 181,4 213,3 219,1 242,3 267,3 200,9 236,6 243,5 269,7 298,0 322,0 356,1 232,2 273,8 282,2 312,8 346,0 374,3 414,7 319,8 378,1 390,9 434,5 482,2 523,1 582,0 658,0 678,3

iy mm x 10 7,00 6,94 6,88 6,82 7,17 7,12 7,06 7,01 6,95 7,37 7,32 7,27 7,22 7,17 7,12 7,04 8,02 7,97 7,92 7,87 7,81 7,76 7,68 9,28 9,23 9,18 9,13 9,08 9,03 8,96 8,77 8,73

Iz mm4 x 104 249,8 300,4 346,7 354,3 496,9 576,9 591,2 647,0 705,4 750,1 874,4 897,7 985,8 1079 1156 1262 816,3 952,4 979,2 1076 1179 1265 1383 1508 1768 1825 2015 2219 2392 2634 2925 3002

Wel.z Wpl.z iz PL Nc.Rd mm3 mm3 mm h b kN x 103 x 103 x 10 62,45 69,55 3,42 4 1 577,6 75,11 84,74 3,38 4 1 794,4 86,69 99,07 3,33 2 1 1008 88,58 101,8 3,31 2 1 1043 99,39 112,1 4,19 4 1 858,9 115,4 131,5 4,14 2 1 1085 118,2 135,4 4,12 2 1 1125 129,4 149,5 4,08 1 1 1254 141,1 164,7 4,04 1 1 1396 125,0 141,5 4,97 4 1 923,4 145,7 166,3 4,93 2 1 1163 149,6 171,5 4,90 2 1 1206 164,3 189,8 4,86 1 1 1345 179,8 209,5 4,82 1 1 1499 192,7 226,2 4,78 1 1 1632 210,4 249,8 4,72 1 1 1826 136,1 153,0 5,02 4 1 943,4 158,7 180,0 4,98 3 1 1240 163,2 185,9 4,95 3 1 1287 179,4 205,8 4,92 2 1 1437 196,6 227,4 4,87 1 1 1602 210,9 245,8 4,84 1 1 1745 230,6 271,8 4,78 1 1 1955 201,1 225,5 6,27 4 2 1064 235,8 266,3 6,23 4 1 1366 243,3 275,7 6,20 4 1 1447 268,7 306,2 6,16 3 1 1712 295,9 339,6 6,12 2 1 1912 318,9 368,1 6,08 1 1 2086 351,2 409,2 6,02 1 1 2342 390,0 463,3 5,90 1 1 2713 400,3 477,5 5,87 1 1 2809

Mc.y.Rd Mc.z.Rd Vpl.y.Rd Vpl.z.Rd kNm kNm kN kN 42,72 52,24 61,31 62,84 58,53 68,83 70,72 78,19 86,25 64,83 76,34 78,60 87,03 96,17 103,9 114,9 74,95 88,36 91,06 101,0 111,7 120,8 133,8 103,2 122,0 126,2 140,2 155,6 168,8 187,8 212,3 218,9

16,58 22,50 31,97 32,86 29,87 42,44 43,70 48,25 53,14 37,69 53,68 55,36 61,25 67,61 72,99 80,62 39,09 51,23 52,67 66,42 73,39 79,31 87,72 54,35 69,73 73,55 86,70 109,6 118,8 132,1 149,5 154,1

113,7 140,3 166,3 172,1 176,1 208,9 216,5 241,3 268,6 212,1 251,8 261,1 291,3 324,5 353,3 395,2 212,8 252,7 262,3 292,8 326,5 355,7 398,3 268,0 318,9 331,5 370,7 413,9 451,7 507,0 587,3 608,2

284,1 350,8 415,7 430,3 352,2 417,8 432,9 482,5 537,2 353,5 419,6 435,2 485,5 540,8 588,8 658,7 390,1 463,4 480,9 536,8 598,5 652,1 730,2 446,7 531,4 552,6 617,8 689,9 752,8 845,1 978,9 1014


r0 = 3,0 x t when t > 10,0 mm

1) h




series

Wt I Wel Wpl i

Appendix 9.1


Table 9.1.2

6 6,3 7,1 8 8,8 10 6 6,3 7,1 8 8,8 10 12 5 6 6,3 7,1 8 6 6,3 7,1 8,8 10 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 6 6,3 7,1 8 8,8 6 6,3 7,1 8 8,8 10 12 12,5 t mm

35,8 37,3 41,6 46,5 50,8 57,0 39,6 41,2 46,1 51,5 56,3 63,2 73,5 30,1 35,8 37,3 41,6 46,5 40,5 42,2 47,2 57,7 64,8 45,2 47,1 52,8 59,1 64,6 72,7 84,8 88,0 45,2 47,1 52,8 47,1 49,1 55,0 61,6 67,3 54,7 57,0 63,9 71,6 78,4 88,4 104 108 M kg/m

45,63 47,45 53,05 59,24 64,64 72,57 50,43 52,49 58,73 65,64 71,68 80,57 93,66 38,36 45,63 47,45 53,05 59,24 51,63 53,74 60,15 73,44 82,57 57,63 60,05 67,25 75,24 82,24 92,57 108,06 112,04 57,63 60,05 67,25 60,03 62,57 70,09 78,44 85,76 69,63 72,65 81,45 91,24 99,84 112,6 132,1 137,0 A mm2 x 102

0,779 0,773 0,769 0,766 0,762 0,757 0,859 0,853 0,849 0,846 0,842 0,837 0,818 0,783 0,779 0,773 0,769 0,766 0,879 0,873 0,869 0,862 0,857 0,979 0,973 0,969 0,966 0,962 0,957 0,938 0,936 0,979 0,973 0,969 1,019 1,013 1,009 1,006 1,002 1,179 1,173 1,169 1,166 1,162 1,157 1,138 1,136 Au m2/m

171 163 145 129 118 104 170 163 145 129 117 104 87 204 171 163 145 129 170 162 144 117 104 170 162 144 128 117 103 87 84 170 162 144 170 162 144 128 117 169 161 144 128 116 103 86 83 Am/V 1/m

3646 3825 4259 4731 5138 5724 5566 5844 6523 7267 7912 8850 10328 2044 2403 2515 2787 3080 4988 5235 5834 7058 7879 8115 8524 9524 10627 11586 12987 15236 15768 3421 3579 3968 4831 5062 5628 6245 6774 12069 12673 14169 15820 17260 19368 22782 23594 It mm4 x 104

382,5 398,4 440,8 486,5 525,3 580,4 501,4 523,0 580,3 642,4 695,6 771,9 884,4 262,2 306,2 318,3 350,7 385,2 478,6 498,9 553,0 661,5 732,8 651,2 679,8 755,7 838,4 909,5 1012 1167 1204 412,4 429,1 473,5 505,2 526,3 582,5 643,1 694,6 877,1 916,2 1020 1133 1231 1373 1591 1644 Wt mm3 x 103

4082 4202 4647 5129 5536 6113 4856 5013 5556 6145 6647 7363 8245 4065 4777 4907 5422 5978 6074 6264 6946 8312 9209 7370 7624 8469 9389 10178 11313 12788 13179 10132 10447 11587 11063 11423 12683 14056 15231 14789 15330 17068 18974 20619 23003 26248 27101 Iy mm4 x 104

314,0 323,2 357,5 394,5 425,9 470,2 373,5 385,6 427,4 472,7 511,3 566,4 634,2 271,0 318,5 327,1 361,5 398,5 404,9 417,6 463,0 554,1 614,0 491,4 508,3 564,6 626,0 678,6 754,2 852,5 878,6 506,6 522,3 579,4 553,2 571,2 634,2 702,8 761,6 739,5 766,5 853,4 948,7 1031 1150 1312 1355 Wel.y mm3 x 103

385,9 399,0 443,4 492,0 533,7 593,8 446,9 462,9 515,3 572,7 622,2 693,8 789,9 348,2 411,4 424,9 472,0 523,5 499,6 517,3 576,0 695,7 775,9 587,8 609,9 680,0 757,1 823,8 920,9 1056 1091 669,6 693,6 772,8 716,9 743,2 828,6 922,4 1004 906,0 941,7 1052 1173 1279 1434 1656 1714 Wpl.y mm3 x 103

9,46 9,41 9,36 9,30 9,25 9,18 9,81 9,77 9,73 9,68 9,63 9,56 9,38 10,29 10,23 10,17 10,11 10,05 10,85 10,80 10,75 10,64 10,56 11,31 11,27 11,22 11,17 11,12 11,05 10,88 10,85 13,26 13,19 13,13 13,58 13,51 13,45 13,39 13,33 14,57 14,53 14,48 14,42 14,37 14,30 14,10 14,06 iy mm x 10

1567 1617 1785 1964 2115 2328 2763 2856 3162 3493 3774 4174 4679 722,8 842,4 868,1 953,9 1045 2080 2150 2378 2831 3125 3962 4104 4553 5042 5459 6058 6854 7060 1108 1145 1261 1655 1713 1892 2084 2246 5092 5286 5875 6517 7069 7864 8977 9260 Iz mm4 x 104

223,9 231,0 254,9 280,6 302,2 332,5 307,1 317,4 351,4 388,1 419,4 463,8 519,9 144,6 168,5 173,6 190,8 209,0 277,3 286,6 317,0 377,4 416,7 396,2 410,4 455,3 504,2 545,9 605,8 685,4 706,0 221,6 229,0 252,2 275,8 285,5 315,3 347,4 374,4 509,2 528,6 587,5 651,7 706,9 786,4 897,7 926,1 Wel.z mm3 x 103

251,8 260,7 289,5 320,9 347,8 386,4 347,9 360,6 401,3 445,8 484,1 539,5 614,9 159,6 187,9 194,5 215,5 238,3 309,5 320,9 357,0 430,2 479,2 446,1 463,2 516,2 574,5 624,9 698,1 801,2 827,9 244,3 253,5 281,4 303,1 314,8 350,1 388,7 421,9 562,5 585,2 653,2 728,1 793,1 888,1 1027 1062 Wpl.z mm3 x 103

5,86 5,84 5,80 5,76 5,72 5,66 7,40 7,38 7,34 7,29 7,26 7,20 7,07 4,34 4,30 4,28 4,24 4,20 6,35 6,32 6,29 6,21 6,15 8,29 8,27 8,23 8,19 8,15 8,09 7,96 7,94 4,38 4,37 4,33 5,25 5,23 5,20 5,15 5,12 8,55 8,53 8,49 8,45 8,41 8,36 8,24 8,22 iz mm x 10

4 1 4 1 3 1 2 1 1 1 1 1 4 2 4 1 3 1 2 1 1 1 1 1 1 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 3 1 2 1 4 2 4 2 4 1 4 1 3 1 2 1 1 1 1 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 2 4 2 4 1 4 1 4 1 4 1 2 1 2 1 PL h b

1339 1420 1712 1912 2086 2342 1494 1583 1895 2118 2313 2600 3023 930,8 1222 1302 1549 1833 1416 1506 1778 2370 2665 1609 1709 2007 2350 2654 2987 3487 3616 1283 1373 1652 1360 1455 1743 2086 2403 1670 1780 2110 2499 2857 3408 4262 4423 Nc.Rd kN

124,6 128,8 143,1 158,8 172,3 191,6 144,2 149,4 166,3 184,8 200,8 223,9 254,9 112,4 132,8 137,1 152,3 168,9 161,2 167,0 185,9 224,5 250,4 189,7 196,8 219,5 244,3 265,9 297,2 340,8 352,2 216,1 223,9 249,4 231,4 239,9 267,4 297,7 323,9 292,4 303,9 339,4 378,7 412,8 462,7 534,5 553,1 Mc.y.Rd kNm

64,79 68,41 82,27 103,6 112,2 124,7 89,50 94,52 113,4 143,9 156,2 174,1 198,5 34,14 44,43 47,03 55,30 64,48 74,32 78,73 92,65 121,8 154,6 107,7 114,1 134,0 156,5 176,2 225,3 258,6 267,2 48,10 51,20 60,98 60,57 64,50 76,84 91,12 103,9 116,5 124,0 147,3 174,4 199,0 236,2 331,4 342,8 Mc.z.Rd kNm

297,6 309,4 345,9 386,3 421,6 473,2 384,4 400,1 447,6 500,4 546,4 614,1 713,9 178,7 212,6 221,0 247,1 276,0 320,7 333,8 373,6 456,2 512,8 429,5 447,6 501,2 560,8 613,0 689,9 805,4 835,1 214,8 223,8 250,6 258,1 269,0 301,4 337,3 368,8 432,5 451,2 505,9 566,7 620,1 699,1 820,2 851,2 Vpl.y.Rd kN

552,7 574,7 642,5 717,5 782,9 878,9 555,3 577,9 646,6 722,7 789,3 887,1 1031 527,4 637,7 663,1 741,3 827,9 641,4 667,6 747,1 912,3 1026 644,3 671,3 751,8 841,2 919,5 1035 1208 1253 797,7 860,9 1002 797,7 860,9 1005 1124 1229 797,7 860,9 1012 1133 1240 1398 1640 1702 Vpl.z.Rd kN

Appendix 9.1

140 140 140 140 140 140 180 180 180 180 180 180 180 100 100 100 100 100 150 150 150 150 150 200 200 200 200 200 200 200 200 100 100 100 120 120 120 120 120 200 200 200 200 200 200 200 200 b mm


247

x 260 260 260 x 260 260 x 260 x 260 260 260 x 260 260 x 260 260 x 300 x 300 300 300 x 300 300 300 300 300 300 x 300 300 300 x 300 300 x 300 300 x 300 400 400 400 400 400 400 400 400 x 400 400 400 x 400 400 x 400 400 x 400 1) h mm x

d

t


Wt I Wel Wpl i


PL = cross-section class in concentric compression Nc.Rd = compression resistance without buckling Mc.Rd = bending resistance The effect of shear buckling has not been accounted for hollow sections of cross-section class 4 (section 2.4.2.2)


series 1) d

mm x x x x

248

x x x x x x x x x x x x

33,7 42,4 42,4 42,4 42,4 42,4 48,3 48,3 48,3 48,3 48,3 48,3 60,3 60,3 60,3 60,3 60,3 60,3 60,3 76,1 76,1 76,1 76,1 76,1 76,1 76,1 76,1

t mm

M kg/m 2 2 2,5 2,6 2,9 3 2 2,5 2,6 2,9 3 3,2 2 2,5 2,6 2,9 3 3,2 4 2 2,5 2,6 2,9 3 3,2 4 5

A mm2 x 102 1,56 1,99 2,46 2,55 2,82 2,91 2,28 2,82 2,93 3,25 3,35 3,56 2,88 3,56 3,70 4,11 4,24 4,51 5,55 3,65 4,54 4,71 5,24 5,41 5,75 7,11 8,77

1,99 2,54 3,13 3,25 3,60 3,71 2,91 3,60 3,73 4,14 4,27 4,53 3,66 4,54 4,71 5,23 5,40 5,74 7,07 4,66 5,78 6,00 6,67 6,89 7,33 9,06 11,17

Au m2/m 0,106 0,133 0,133 0,133 0,133 0,133 0,152 0,152 0,152 0,152 0,152 0,152 0,189 0,189 0,189 0,189 0,189 0,189 0,189 0,239 0,239 0,239 0,239 0,239 0,239 0,239 0,239

Am/V 1/m

It mm4 x 104 533 524 425 409 369 358 522 422 408 367 356 336 516 416 401 361 350 329 267 513 413 398 358 347 326 264 214

5,02 10,38 12,52 12,93 14,11 14,49 15,62 18,92 19,55 21,40 22,00 23,17 31,16 37,99 39,31 43,18 44,45 46,94 56,35 63,96 78,37 81,18 89,48 92,19 97,56 118,1 141,8

Wt mm3 x 103

I mm4 x 104 2,98 4,90 5,91 6,10 6,66 6,84 6,47 7,83 8,10 8,86 9,11 9,59 10,34 12,60 13,04 14,32 14,74 15,57 18,69 16,81 20,60 21,34 23,52 24,23 25,64 31,04 37,28

Wel mm3 x 103 2,51 5,19 6,26 6,46 7,06 7,25 7,81 9,46 9,78 10,70 11,00 11,59 15,58 18,99 19,65 21,59 22,22 23,47 28,17 31,98 39,19 40,59 44,74 46,10 48,78 59,06 70,92

Wpl mm3 x 103 1,49 2,45 2,95 3,05 3,33 3,42 3,23 3,92 4,05 4,43 4,55 4,80 5,17 6,30 6,52 7,16 7,37 7,78 9,34 8,40 10,30 10,67 11,76 12,11 12,82 15,52 18,64

i mm x 10 2,01 3,27 3,99 4,12 4,53 4,67 4,29 5,25 5,44 5,99 6,17 6,52 6,80 8,36 8,66 9,56 9,86 10,44 12,70 10,98 13,55 14,05 15,55 16,04 17,02 20,81 25,32

PL Nc.Rd kN 1,12 1,43 1,41 1,41 1,40 1,40 1,64 1,62 1,62 1,61 1,61 1,60 2,06 2,05 2,04 2,03 2,03 2,02 2,00 2,62 2,60 2,60 2,59 2,59 2,58 2,55 2,52

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1

64,28 81,92 101,1 104,9 116,1 119,8 93,89 116,1 120,5 133,5 137,8 146,3 118,2 146,5 152,1 168,8 174,3 185,3 228,3 150,3 186,6 193,8 215,2 222,3 236,5 292,4 360,4

Mc.Rd kNm

Vpl.Rd kN 0,65 1,05 1,29 1,33 1,46 1,51 1,38 1,69 1,75 1,93 1,99 2,10 2,19 2,70 2,80 3,09 3,18 3,37 4,10 3,54 4,37 4,53 5,02 5,18 5,49 6,72 8,17

23,63 30,11 37,17 38,56 42,69 44,05 34,51 42,67 44,28 49,06 50,64 53,78 43,45 53,85 55,91 62,03 64,06 68,09 83,92 55,23 68,57 71,21 79,11 81,72 86,93 107,5 132,5


1) = recommended

M A Au Am/V It

Appendix 9.1

Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2).

Table 9.1.3

x x x

x x x x

x

249

x x

x x x x 1) d

x

t mm

4,29 5,33 5,53 6,15 6,36 6,76 8,38 10,4 12,3 12,8 4,91 6,11 6,35 7,06 7,29 7,77 9,63 11,9 14,2 14,8 5,23 6,50 6,76 7,52 7,77 8,27 10,3 12,7 15,1 15,8 5,54 6,89 7,16 7,97 8,23 8,77 10,9 13,5 16,0 16,8 M kg/m

5,46 6,79 7,05 7,84 8,10 8,62 10,67 13,18 15,63 16,35 6,26 7,78 8,09 8,99 9,29 9,89 12,26 15,17 18,02 18,86 6,66 8,29 8,61 9,58 9,90 10,54 13,07 16,18 19,23 20,13 7,06 8,78 9,12 10,15 10,49 11,17 13,86 17,17 20,41 21,38 A mm2 x 102

0,279 0,279 0,279 0,279 0,279 0,279 0,279 0,279 0,279 0,279 0,319 0,319 0,319 0,319 0,319 0,319 0,319 0,319 0,319 0,319 0,339 0,339 0,339 0,339 0,339 0,339 0,339 0,339 0,339 0,339 0,359 0,359 0,359 0,359 0,359 0,359 0,359 0,359 0,359 0,359 Au m2/m

511 411 396 356 344 324 261 212 179 171 510 410 394 355 343 323 260 210 177 169 509 409 394 354 342 322 259 210 176 168 508 409 394 354 342 321 259 209 176 168 Am/V 1/m

103,1 126,8 131,4 145,0 149,5 158,4 192,7 232,8 269,9 280,5 155,3 191,2 198,3 219,2 226,1 239,7 292,6 354,9 413,4 430,1 187,2 230,7 239,3 264,6 273,0 289,6 353,9 430,1 501,8 522,5 222,5 274,5 284,8 315,1 325,1 344,9 422,1 513,8 600,4 625,4 It mm4 x 104

23,20 28,51 29,55 32,63 33,64 35,64 43,35 52,36 60,72 63,10 30,56 37,64 39,03 43,15 44,50 47,19 57,59 69,87 81,37 84,67 34,66 42,72 44,31 49,00 50,55 53,62 65,54 79,65 92,93 96,75 38,94 48,03 49,82 55,13 56,88 60,36 73,86 89,91 105,1 109,4 Wt mm3 x 103

51,57 63,37 65,68 72,52 74,76 79,21 96,34 116,4 134,9 140,2 77,63 95,61 99,14 109,6 113,0 119,9 146,3 177,5 206,7 215,1 93,58 115,4 119,6 132,3 136,5 144,8 177,0 215,1 250,9 261,2 111,3 137,3 142,4 157,6 162,6 172,5 211,1 256,9 300,2 312,7 I mm4 x 104

11,60 14,26 14,78 16,31 16,82 17,82 21,67 26,18 30,36 31,55 15,28 18,82 19,52 21,57 22,25 23,59 28,80 34,93 40,68 42,34 17,33 21,36 22,15 24,50 25,28 26,81 32,77 39,83 46,46 48,38 19,47 24,02 24,91 27,57 28,44 30,18 36,93 44,96 52,53 54,72 Wel mm3 x 103

15,11 18,67 19,37 21,46 22,15 23,51 28,85 35,24 41,31 43,07 19,84 24,56 25,49 28,26 29,17 31,00 38,12 46,70 54,91 57,30 22,47 27,83 28,89 32,04 33,08 35,16 43,29 53,09 62,50 65,24 25,23 31,25 32,45 36,00 37,17 39,51 48,69 59,77 70,45 73,57 Wpl mm3 x 103

3,07 3,06 3,05 3,04 3,04 3,03 3,00 2,97 2,94 2,93 3,52 3,50 3,50 3,49 3,49 3,48 3,45 3,42 3,39 3,38 3,75 3,73 3,73 3,72 3,71 3,71 3,68 3,65 3,61 3,60 3,97 3,95 3,95 3,94 3,94 3,93 3,90 3,87 3,83 3,82 i mm x 10

2 176,2 2 219,0 2 227,5 1 252,9 1 261,3 1 278,1 1 344,3 1 425,3 1 504,3 1 527,6 3 202,0 2 251,2 2 261,0 2 290,2 2 299,9 1 319,3 1 395,8 1 489,7 1 581,6 1 608,7 3 214,9 2 267,4 2 277,8 2 309,0 2 319,4 2 340,0 1 421,8 1 522,2 1 620,5 1 649,6 3 227,7 2 283,4 2 294,5 2 327,5 2 338,5 2 360,5 1 447,3 1 554,1 1 658,8 1 689,8 PL Nc.Rd kN

4,88 6,02 6,25 6,92 7,15 7,59 9,31 11,37 13,33 13,90 4,93 7,93 8,23 9,12 9,42 10,00 12,30 15,07 17,72 18,49 5,59 8,98 9,32 10,34 10,68 11,35 13,97 17,13 20,17 21,06 6,28 10,09 10,47 11,62 12,00 12,75 15,71 19,29 22,73 23,74 Mc.Rd kNm

64,77 80,49 83,62 92,94 96,03 102,2 126,6 156,3 185,4 193,9 74,23 92,32 95,92 106,7 110,2 117,3 145,5 180,0 213,8 223,7 79,00 98,29 102,1 113,6 117,4 125,0 155,0 191,9 228,1 238,8 83,70 104,2 108,2 120,4 124,4 132,5 164,4 203,7 242,2 253,6 Vpl.Rd kN

Appendix 9.1

mm

2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3


88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 108 108 108 108 108 108 108 108 108 108 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3

x

d

t


Wt I Wel Wpl i




series 1) d

mm

t mm

M kg/m

x x x

250

x

x

x

x x x x x x

127 127 127 127 127 127 127 127 127 127 133 133 133 133 133 133 133 133 133 133 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7

2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 8 10

6,17 7,68 7,98 8,88 9,17 9,77 12,1 15,0 17,9 18,8 6,46 8,05 8,36 9,30 9,62 10,2 12,7 15,8 18,8 19,7 9,78 10,1 10,8 13,4 16,6 19,8 20,7 26,0 32,0

A mm2 x 102 7,85 9,78 10,16 11,31 11,69 12,45 15,46 19,16 22,81 23,89 8,23 10,25 10,65 11,85 12,25 13,05 16,21 20,11 23,94 25,08 12,46 12,88 13,72 17,05 21,16 25,20 26,40 33,10 40,75

Au m2/m 0,399 0,399 0,399 0,399 0,399 0,399 0,399 0,399 0,399 0,399 0,418 0,418 0,418 0,418 0,418 0,418 0,418 0,418 0,418 0,418 0,439 0,439 0,439 0,439 0,439 0,439 0,439 0,439 0,439

Am/V 1/m 508 408 393 353 341 320 258 208 175 167 508 408 392 353 341 320 258 208 175 167 352 341 320 257 207 174 166 133 108

It mm4 x 104 306,9 379,1 393,3 435,6 449,5 477,2 585,2 714,3 836,9 872,4 353,2 436,5 453,0 501,8 517,9 550,0 675,1 824,8 967,4 1008,9 583,4 602,2 639,6 785,7 961,1 1129 1177 1441 1724

Wt mm3 x 103 48,33 59,70 61,94 68,59 70,79 75,15 92,16 112,5 131,8 137,4 53,11 65,64 68,12 75,46 77,88 82,70 101,5 124,0 145,5 151,7 83,52 86,21 91,56 112,5 137,6 161,6 168,5 206,2 246,8

I mm4 x 104 153,4 189,5 196,7 217,8 224,8 238,6 292,6 357,1 418,4 436,2 176,6 218,3 226,5 250,9 259,0 275,0 337,5 412,4 483,7 504,4 291,7 301,1 319,8 392,9 480,5 564,3 588,6 720,3 861,9

Wel mm3 x 103 24,16 29,85 30,97 34,30 35,39 37,57 46,08 56,24 65,90 68,70 26,56 32,82 34,06 37,73 38,94 41,35 50,76 62,02 72,74 75,85 41,76 43,11 45,78 56,24 68,80 80,78 84,27 103,1 123,4

Wpl mm3 x 103 31,25 38,76 40,24 44,67 46,14 49,06 60,54 74,46 87,92 91,86 34,32 42,58 44,22 49,09 50,71 53,92 66,59 81,96 96,85 101,2 54,28 56,07 59,63 73,68 90,76 107,3 112,2 138,9 168,6

i mm x 10

PL Nc.Rd kN 4,42 4,40 4,40 4,39 4,39 4,38 4,35 4,32 4,28 4,27 4,63 4,61 4,61 4,60 4,60 4,59 4,56 4,53 4,50 4,49 4,84 4,83 4,83 4,80 4,77 4,73 4,72 4,66 4,60

4 3 3 2 2 2 1 1 1 1 4 3 3 2 2 2 2 1 1 1 3 3 2 2 1 1 1 1 1

224,1 315,6 327,9 364,9 377,2 401,7 498,8 618,5 736,1 771,0 233,9 330,8 343,7 382,5 395,4 421,1 523,2 648,9 772,6 809,3 402,2 415,8 442,9 550,3 682,9 813,3 852,1 1068 1315

Mc.Rd kNm

Vpl.Rd kN 6,93 9,63 9,99 14,42 14,89 15,83 19,54 24,03 28,37 29,65 7,59 10,59 10,99 15,84 16,37 17,40 21,49 26,45 31,25 32,67 13,48 13,91 19,25 23,78 29,29 34,64 36,21 44,84 54,40

93,16 116,0 120,5 134,1 138,6 147,6 183,4 227,3 270,6 283,4 97,64 121,6 126,3 140,6 145,3 154,8 192,3 238,5 284,0 297,5 147,8 152,8 162,8 202,3 251,0 298,9 313,2 392,6 483,3


1) = recommended

M A Au Am/V It

Appendix 9.1

Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) continued.

Table 9.1.3

x

x x x x

251

x x x

x x x x 1) d

mm

t mm

10,7 11,0 11,7 14,6 18,1 21,6 22,6 11,2 11,5 12,3 15,3 19,0 22,6 23,7 11,8 12,2 13,0 16,2 20,1 24,0 25,2 31,6 39,0 18,7 23,3 27,8 29,1 36,6 45,3 53,8 55,9 21,2 26,4 31,5 33,1 41,7 51,6 61,3 63,7 M kg/m

13,58 14,04 14,96 18,60 23,09 27,52 28,84 14,22 14,70 15,66 19,48 24,19 28,84 30,22 15,07 15,58 16,60 20,65 25,65 30,59 32,06 40,29 49,73 23,84 29,64 35,38 37,09 46,67 57,71 68,50 71,16 27,03 33,63 40,17 42,12 53,06 65,69 78,07 81,13 A mm2 x 102

0,478 0,478 0,478 0,478 0,478 0,478 0,478 0,500 0,500 0,500 0,500 0,500 0,500 0,500 0,529 0,529 0,529 0,529 0,529 0,529 0,529 0,529 0,529 0,609 0,609 0,609 0,609 0,609 0,609 0,609 0,609 0,688 0,688 0,688 0,688 0,688 0,688 0,688 0,688 Au m2/m

352 340 320 257 207 174 166 352 340 319 257 207 173 165 351 340 319 256 206 173 165 131 106 255 205 172 164 130 106 89 86 255 205 171 163 130 105 88 85 Am/V 1/m

755,2 779,7 828,4 1019 1249 1469 1533 866,7 894,8 950,9 1171 1436 1690 1765 1031 1065 1131 1394 1712 2017 2107 2595 3128 2146 2640 3119 3260 4031 4883 5678 5869 3128 3856 4564 4772 5919 7197 8400 8689 It mm4 x 104

99,37 102,6 109,0 134,1 164,3 193,3 201,8 109,0 112,6 119,6 147,3 180,6 212,6 222,0 122,5 126,5 134,5 165,7 203,4 239,7 250,4 308,3 371,7 221,5 272,6 322,1 336,6 416,2 504,2 586,3 606,0 285,5 352,0 416,6 435,6 540,3 657,0 766,8 793,2 Wt mm3 x 103

377,6 389,9 414,2 509,6 624,4 734,5 766,6 433,3 447,4 475,4 585,3 717,9 845,2 882,4 515,5 532,3 565,7 697,1 855,9 1009 1053 1297 1564 1073 1320 1560 1630 2016 2442 2839 2934 1564 1928 2282 2386 2960 3598 4200 4345 I mm4 x 104

49,69 51,30 54,50 67,05 82,16 96,65 100,9 54,51 56,28 59,8 73,63 90,30 106,3 111,0 61,26 63,25 67,23 82,84 101,7 119,9 125,2 154,2 185,9 110,8 136,3 161,1 168,3 208,1 252,1 293,2 303,0 142,8 176,0 208,3 217,8 270,2 328,5 383,4 396,6 Wel mm3 x 103

64,48 66,61 70,86 87,64 108,1 128,0 133,8 70,67 73,02 77,69 96,12 118,6 140,5 147,0 79,34 81,98 87,24 108,0 133,4 158,1 165,4 205,7 250,9 144,0 178,1 211,5 221,3 276,1 337,8 396,8 411,1 185,1 229,2 272,5 285,4 356,7 437,6 515,3 534,2 Wpl mm3 x 103

5,27 5,27 5,26 5,23 5,20 5,17 5,16 5,52 5,52 5,51 5,48 5,45 5,41 5,40 5,85 5,85 5,84 5,81 5,78 5,74 5,73 5,67 5,61 6,71 6,67 6,64 6,63 6,57 6,50 6,44 6,42 7,61 7,57 7,54 7,53 7,47 7,40 7,33 7,32 i mm x 10

3 438,4 3 453,2 3 482,8 2 600,2 1 745,2 1 888,2 1 930,7 3 459,0 3 474,5 3 505,5 2 628,6 1 780,7 1 930,7 1 975,4 3 486,3 3 502,8 3 535,7 2 666,3 2 827,8 1 987,3 1 1035 1 1300 1 1605 3 769,3 2 956,6 1 1142 1 1197 1 1506 1 1862 1 2211 1 2296 3 872,3 2 1085 2 1296 2 1359 1 1712 1 2120 1 2520 1 2618 PL Nc.Rd kN

16,04 16,56 17,59 28,28 34,88 41,30 43,19 17,59 18,16 19,30 31,02 38,28 45,35 47,44 19,77 20,41 21,70 34,85 43,04 51,03 53,39 66,40 80,98 35,75 57,47 68,24 71,43 89,09 109,0 128,0 132,7 46,07 73,98 87,96 92,10 115,1 141,2 166,3 172,4 Mc.Rd kNm

161,1 166,6 177,4 220,6 273,9 326,4 342,1 168,7 174,4 185,8 231,1 286,9 342,1 358,5 178,8 184,8 196,9 244,9 304,3 362,9 380,3 477,9 589,9 282,8 351,6 419,7 440,0 553,6 684,6 812,5 844,1 320,6 398,9 476,5 499,6 629,3 779,2 926,1 962,4 Vpl.Rd kN

Appendix 9.1

x

2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 8 10 4 5 6 6,3 8 10 12 12,5 4 5 6 6,3 8 10 12 12,5


152 152 152 152 152 152 152 159 159 159 159 159 159 159 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 193,7 193,7 193,7 193,7 193,7 193,7 193,7 193,7 219,1 219,1 219,1 219,1 219,1 219,1 219,1 219,1

d

t


Wt I Wel Wpl i




series 1) d

mm

t mm

M kg/m

x x x x

252

x x x x x

273 273 273 273 273 273 273 323,9 323,9 323,9 323,9 323,9 323,9 323,9

5 6 6,3 8 10 12 12,5 5 6 6,3 8 10 12 12,5

33,1 39,5 41,4 52,3 64,9 77,2 80,3 39,3 47,0 49,3 62,3 77,4 92,3 96,0

A Au Am/V It mm2 m2/m 1/m mm4 x 102 x 104 42,10 0,858 204 7562 50,33 0,858 170 8974 52,79 0,858 163 9392 66,60 0,858 129 11703 82,62 0,858 104 14308 98,39 0,858 87 16792 102,3 0,858 84 17395 50,09 1,018 203 12739 59,92 1,018 170 15145 62,86 1,018 162 15858 79,39 1,018 128 19820 98,61 1,018 103 24317 117,6 1,018 87 28639 122,3 1,018 83 29693

Wt mm3 x 103 554,0 657,5 688,0 857,4 1048 1230 1274 786,6 935,2 979,2 1224 1501 1768 1833

I mm4 x 104 3781 4487 4696 5852 7154 8396 8697 6369 7572 7929 9910 12158 14320 14847

Wel mm3 x 103 277,0 328,7 344,0 428,7 524,1 615,1 637,2 393,3 467,6 489,6 611,9 750,8 884,2 916,7

Wpl mm3 x 103 359,2 427,8 448,2 562,0 692,0 818,0 848,9 508,5 606,4 635,6 798,5 985,7 1168 1213

i mm x 10

PL Nc.Rd kN 9,48 9,44 9,43 9,37 9,31 9,24 9,22 11,28 11,24 11,23 11,17 11,10 11,04 11,02

3 2 2 2 1 1 1 4 3 3 2 1 1 1

1359 1624 1704 2149 2667 3175 3301 1427 1934 2029 2562 3183 3795 3947

Mc.Rd kNm 89,39 138,1 144,6 181,4 223,3 264,0 274,0 112,6 150,9 158,0 257,7 318,1 376,9 391,4

Vpl.Rd kN 499,4 597,0 626,1 790,0 980,1 1167 1213 594,2 710,8 745,6 941,8 1170 1395 1451


1) = recommended

M A Au Am/V It

Appendix 9.1

Cross-sectional properties and resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2), continued.

Table 9.1.3


253

Appendix 9.1

d

t

M A Au Am/V It Wt

= weight = cross-section area = external area = cross-section factor in fire design = torsional modulus = torsional section modulus

I Wel Wpl i

= moment of inertia = elastic section modulus = plastic section modulus = radius of gyration

Appendix 9.1

Cross-sectional properties and resistance values for circular spirally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2). (Technical delivery conditions to be agreed when ordering)

Table 9.1.4

PL = cross-section class in concentric compression Nc.Rd = compression resistance without buckling Mc.Rd = bending resistance Vpl.Rd = shear resistance The effect of shear buckling has not been accounted for hollow sections of cross-section class 4 (section 2.4.2.2)

d mm

254

355,6 355,6 355,6 355,6 355,6 355,6 355,6 406,4 406,4 406,4 406,4 406,4 406,4 457 457 457 457 457 457 508 508 508 508 508 508 559 559 559 559 559 559

t mm 5,6 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5

M kg/m 48,3 51,7 54,3 68,6 85,2 102 106 59,3 62,2 78,6 97,8 117 121 66,7 70,0 88,6 110 132 137 74,3 78,0 98,7 123 147 153 81,8 85,9 109 135 162 169

A Au Am/V It mm2 m2/m 1/m mm4 x 102 x 104 61,58 1,117 181 18862 65,90 1,117 170 20141 69,13 1,117 162 21094 87,36 1,117 128 26403 108,6 1,117 103 32447 129,5 1,117 86 38279 134,7 1,117 83 39704 75,47 1,277 169 30257 79,19 1,277 161 31699 100,1 1,277 128 39748 124,5 1,277 103 48952 148,7 1,277 86 57874 154,7 1,277 83 60061 85,01 1,436 169 43236 89,20 1,436 161 45308 112,9 1,436 127 56893 140,4 1,436 102 70183 167,8 1,436 86 83113 174,6 1,436 82 86290 94,62 1,596 169 59623 99,30 1,596 161 62493 125,7 1,596 127 78560 156,5 1,596 102 97040 187,0 1,596 85 115072 194,6 1,596 82 119511 104,2 1,756 168 79702 109,4 1,756 161 83552 138,5 1,756 127 105130 172,5 1,756 102 130002 206,2 1,756 85 154327 214,6 1,756 82 160324

Wt mm3 x 103 1061 1133 1186 1485 1825 2153 2233 1489 1560 1956 2409 2848 2956 1892 1983 2490 3071 3637 3776 2347 2460 3093 3820 4530 4705 2852 2989 3761 4651 5522 5736

I mm4 x 104 9431 10071 10547 13201 16224 19139 19852 15128 15849 19874 24476 28937 30031 21618 22654 28446 35091 41556 43145 29812 31246 39280 48520 57536 59755 39851 41776 52565 65001 77164 80162

Wel mm3 x 103 530,4 566,4 593,2 742,5 912,5 1076,0 1117 744,5 780,0 978,1 1205 1424 1478 946,1 991,4 1245 1536 1819 1888 1174 1230 1546 1910 2265 2353 1426 1495 1881 2326 2761 2868

Wpl mm3 x 103 686,1 733,4 768,8 966,8 1195 1417 1472 962 1009 1270 1572 1867 1940 1220 1280 1613 1998 2377 2470 1512 1586 2000 2480 2953 3070 1835 1925 2429 3014 3591 3734

i PL Nc.Rd mm kN x 10 12,38 4 1757 12,36 3 2127 12,35 3 2231 12,29 2 2819 12,22 2 3504 12,16 1 4180 12,14 1 4348 14,16 4 2141 14,15 4 2256 14,09 3 3231 14,02 2 4019 13,95 2 4798 13,93 1 4992 15,95 4 2383 15,94 4 2513 15,88 3 3642 15,81 2 4532 15,74 2 5414 15,72 2 5633 17,75 4 2622 17,74 4 2767 17,68 4 3586 17,61 3 5049 17,54 2 6035 17,52 2 6280 19,55 4 2856 19,54 4 3015 19,48 4 3916 19,41 3 5566 19,34 3 6655 19,33 2 6926

Mc.Rd kNm 152,2 182,8 191,4 312,0 385,6 457,4 475,1 212,4 223,5 315,6 507,2 602,6 626,1 267,0 281,1 401,8 645,0 767,1 797,3 327,8 345,3 443,6 616,5 952,9 990,7 394,0 415,3 535,1 750,5 891,0 1205

Vpl.Rd kN 730,4 781,7 820,1 1036 1288 1537 1598 895,3 939,3 1188 1477 1764 1835 1008 1058 1339 1666 1990 2071 1122 1178 1491 1856 2218 2308 1236 1298 1643 2046 2446 2546



t mm

119 148 177 184 209 129 160 192 200 226 139 173 207 215 244 149 186 222 231 262 159 198 237 247 280 315 223 267 278 315 354 248 297 309 351 395 298 357 372 422 475 M kg/m

151,3 188,5 225,4 234,6 265,8 163,9 204,2 244,3 254,3 288,1 176,7 220,2 263,5 274,3 310,9 189,5 236,3 282,7 294,3 333,6 202,3 252,3 302,0 314,4 356,4 400,6 284,0 340,1 354,0 401,4 451,4 316,0 378,5 394,1 446,9 502,7 379,8 455,0 473,8 537,5 604,7 A mm2 x 102

1,916 1,916 1,916 1,916 1,916 2,073 2,073 2,073 2,073 2,073 2,234 2,234 2,234 2,234 2,234 2,394 2,394 2,394 2,394 2,394 2,554 2,554 2,554 2,554 2,554 2,554 2,871 2,871 2,871 2,871 2,871 3,192 3,192 3,192 3,192 3,192 3,830 3,830 3,830 3,830 3,830 Au m2/m

127 102 85 82 72 127 102 85 82 72 126 101 85 81 72 126 101 85 81 72 126 101 85 81 72 64 101 84 81 72 64 101 84 81 71 64 101 84 81 71 63 Am/V 1/m

137103 169693 201627 209509 236008 174176 215741 256534 266613 300526 218324 270603 321981 334686 377470 269366 334057 397710 413462 466542 327801 406728 484469 503721 568630 636443 580294 691779 719417 812689 910283 799699 953969 992246 1121524 1256959 1388029 1657432 1724362 1950668 2188182 It mm4 x 104

4495 5564 6611 6869 7738 5278 6538 7774 8079 9107 6141 7612 9057 9415 10618 7070 8768 10439 10852 12245 8064 10006 11918 12392 13988 15657 12698 15137 15742 17783 19919 15742 18779 19532 22077 24743 22773 27193 28291 32004 35901 Wt mm3 x 103

68551 84847 100814 104755 118004 87088 107870 128267 133306 150263 109162 135301 160991 167343 188735 134683 167028 198855 206731 233271 163901 203364 242235 251860 284315 318222 290147 345890 359708 406344 455142 399850 476984 496123 560762 628479 694014 828716 862181 975334 1094091 I mm4 x 104

2248 2782 3305 3435 3869 2639 3269 3887 4040 4553 3071 3806 4529 4707 5309 3535 4384 5219 5426 6123 4032 5003 5959 6196 6994 7828 6349 7569 7871 8892 9959 7871 9389 9766 11039 12372 11387 13597 14146 16002 17951 Wel mm3 x 103

2899 3600 4292 4463 5042 3401 4225 5039 5241 5923 3954 4914 5864 6099 6895 4548 5655 6751 7023 7942 5184 6448 7700 8011 9062 10165 8172 9764 10159 11498 12904 10121 12097 12588 14252 16001 14617 17483 18196 20613 23157 Wpl mm3 x 103

21,29 21,22 21,15 21,13 21,07 23,05 22,98 22,91 22,90 22,84 24,86 24,79 24,72 24,70 24,64 26,66 26,59 26,52 26,50 26,44 28,46 28,39 28,32 28,31 28,25 28,18 31,96 31,89 31,88 31,82 31,75 35,57 35,50 35,48 35,42 35,36 42,75 42,68 42,66 42,60 42,54 i mm x 10

4 4242 4 5398 3 7276 3 7572 2 8578 4 4555 4 5806 3 7884 3 8206 3 9298 4 4869 4 6218 3 8504 3 8852 3 10032 4 5178 4 6623 4 8068 4 8429 3 10766 4 5481 4 7024 4 8565 4 8951 3 11500 3 12929 4 7800 4 9534 4 9967 4 11440 3 14567 4 8562 4 10490 4 10972 4 12610 4 14343 4 10017 4 12328 4 12906 4 14872 4 16953 PL Nc.Rd kN

634,3 800,6 1067 1108 1627 738,9 934,7 1254 1304 1470 853,0 1081 1461 1519 1713 974,4 1237 1497 1562 1976 1103 1403 1700 1774 2257 2526 1758 2136 2230 2548 3214 2153 2622 2739 3134 3549 3040 3719 3888 4462 5066 Mc.Rd kNm

1795 2236 2674 2783 3153 1944 2422 2898 3016 3417 2096 2612 3126 3254 3687 2248 2802 3354 3491 3957 2400 2992 3582 3729 4227 4752 3369 4034 4199 4761 5354 3749 4490 4674 5301 5962 4505 5398 5620 6375 7173 Vpl.Rd kN

Appendix 9.1

8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16


255

610 610 610 610 610 660 660 660 660 660 711 711 711 711 711 762 762 762 762 762 813 813 813 813 813 813 914 914 914 914 914 1016 1016 1016 1016 1016 1219 1219 1219 1219 1219 d mm

Appendix 9.1


256


Appendix 9.2

Appendix 9.2

Buckling tables for steel grade S355J2H

257

1)

b t h

y r0

= recommended series = height = width = wall thickness = buckling length = buckling resistance


1) h

258

x x x x x x x x x x x x x x x x x x x x x x

x x x x x x

40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

2 2,5 2 2,5 3 2,5 3 4 2 2,5 3 4 2 2,5 3 4 5 2 2,5 3 4 5 6 6,3 2 2,5 3 4 5 6 6,3 7,1 8

0 94,78 115,8 120,6 148,1 174,5 180,4 213,3 275,9 172,2 212,7 252,0 327,5 180,1 244,9 290,7 379,1 463,3 187,9 277,2 329,5 430,8 527,9 620,7 636,7 194,0 281,4 368,2 482,4 592,4 698,2 718,1 795,6 879,2

0,5 83,79 101,9 112,3 137,5 161,7 173,0 204,2 263,3 169,0 208,4 246,7 320,0 180,1 244,0 289,3 376,6 459,5 187,9 277,2 329,5 430,8 527,9 620,7 636,7 194,0 281,4 368,2 482,4 592,4 698,2 718,1 795,6 879,2

1,0 59,80 71,86 89,54 109,0 127,7 145,7 171,7 220,1 147,7 181,9 214,8 277,9 162,2 217,8 257,9 334,9 407,9 175,0 253,4 300,7 392,0 478,8 561,5 574,5 185,2 264,9 343,4 448,9 549,9 646,4 663,8 733,6 808,9

1,5 37,88 45,03 64,65 78,05 90,92 114,6 134,6 170,8 123,6 151,9 178,8 230,2 142,4 189,0 223,3 288,8 350,6 158,1 225,6 267,2 347,4 423,2 495,1 505,3 170,6 241,4 310,7 405,4 495,7 581,5 596,4 657,6 723,8

2 24,44 28,91 44,80 53,78 62,42 85,13 99,58 125,3 98,25 120,4 141,1 180,6 120,7 157,4 185,4 238,6 288,3 139,6 194,8 230,2 298,2 361,8 421,9 429,1 154,9 215,9 274,9 357,7 436,0 510,0 522,1 573,8 629,9

2,5 16,75 19,78 31,75 38,02 44,04 62,71 73,20 91,61 75,92 92,84 108,4 138,1 98,89 126,8 148,9 190,6 229,4 120,0 163,0 192,2 247,9 299,4 347,8 352,5 137,8 188,5 236,8 307,0 372,9 434,6 443,9 486,1 532,0

Nb.Rd (kN) Lc (m) 3 3,5 12,12 9,156 14,30 10,79 23,39 17,86 27,96 21,33 32,36 24,67 47,16 36,44 54,98 42,46 68,61 52,89 58,73 46,16 71,72 56,32 83,59 65,56 106,2 83,15 79,76 64,39 101,0 80,79 118,3 94,51 150,8 120,2 181,0 144,0 100,9 84,03 133,8 109,3 157,4 128,4 202,2 164,6 243,4 197,5 281,9 228,4 284,8 230,2 120,3 103,4 161,0 135,8 199,5 166,4 257,9 214,5 312,2 259,0 362,6 300,0 369,6 305,3 403,3 332,2 440,1 361,7

h b t mm mm mm 4

14,05 16,77 19,39 28,90 33,65 41,87 36,99 45,11 52,46 66,48 52,49 65,49 76,54 97,23 116,3 69,99 89,9 105,5 134,9 161,6 186,6 187,9 88,28 114,2 138,8 178,5 215,1 248,7 252,8 274,6 298,6

4,5

11,33 13,52 15,63 23,43 27,27 33,91 30,21 36,82 42,80 54,20 43,37 53,89 62,95 79,87 95,45 58,67 74,67 87,54 111,8 133,8 154,4 155,3 75,35 96,39 116,4 149,6 180,0 207,8 211,1 229,0 248,7

5

19,35 22,53 28,00 25,08 30,57 35,52 44,96 36,31 45,00 52,54 66,62 79,56 49,63 62,77 73,54 93,88 112,3 129,4 130,1 64,57 81,94 98,55 126,5 152,1 175,4 178,1 193,0 209,5

6

18,05 21,99 25,54 32,30 26,40 32,61 38,05 48,20 57,52 36,56 45,87 53,71 68,49 81,81 94,22 94,63 48,37 60,75 72,67 93,14 111,9 128,9 130,7 141,5 153,5

7

20,00 24,65 28,76 36,41 43,43 27,92 34,86 40,80 51,99 62,06 71,44 71,72 37,29 46,55 55,51 71,10 85,33 98,23 99,61 107,8 116,8

8

21,97 27,34 31,99 40,75 48,62 55,95 56,15 29,52 36,71 43,69 55,93 67,09 77,20 78,26 84,62 91,68

9

23,91 29,66 35,24 45,10 54,08 62,21 63,05 68,15 73,82

10 40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

40 40 50 50 50 60 60 60 70 70 70 70 80 80 80 80 80 90 90 90 90 90 90 90 100 100 100 100 100 100 100 100 100

2 2,5 2 2,5 3 2,5 3 4 2 2,5 3 4 2 2,5 3 4 5 2 2,5 3 4 5 6 6,3 2 2,5 3 4 5 6 6,3 7,1 8


z b t mm mm mm

h b t Lc Nb.Rd

Appendix 9.2

Buckling resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.

Table 9.2.1

x x x x

x x x x x

259


x

291,3 406,9 534,1 657,0 775,6 799,4 405,1 585,7 721,5 800,9 853,1 880,7 978,7 1086 1177 689,0 850,6 945,5 1008 1043 1162 1292 1405 1567 701,2 915,1 1085 1125 1254 1396 1518 1696 720,2 979,7 1163 1206 1345 1499 1632 1826 2093 2164

291,3 406,9 534,1 657,0 775,6 799,4 405,1 585,7 721,5 800,9 853,1 880,7 978,7 1086 1177 689,0 850,6 945,5 1008 1043 1162 1292 1405 1567 701,2 915,1 1085 1125 1254 1396 1518 1696 720,2 979,7 1163 1206 1345 1499 1632 1826 2093 2164

280,0 386,0 505,7 620,9 731,6 753,1 392,3 562,5 691,9 767,3 816,7 842,3 934,5 1035 1121 676,0 833,8 926,2 986,9 1021 1136 1261 1370 1526 696 904,8 1072 1110 1236 1375 1494 1668 720,2 975,6 1157 1199 1337 1488 1619 1809 2068 2137

259,1 353,9 463,0 567,6 667,8 686,8 365,0 520,3 639,3 708,5 753,7 776,9 860,8 952,7 1030 634,2 781,9 868,2 924,8 955,8 1063 1180 1280 1425 657,6 853 1010 1045 1164 1293 1405 1567 685,4 923,8 1095 1135 1264 1407 1530 1708 1949 2013

236,8 319,2 416,7 509,7 598,4 614,6 336,0 475,2 582,9 645,3 686,1 706,5 781,5 863,6 932,0 590,7 727,6 807,4 859,8 887,9 986,5 1094 1186 1318 618,2 799,4 946,0 978,0 1088 1208 1312 1461 648,7 870,7 1032 1068 1190 1323 1438 1604 1826 1885

212,7 281,7 366,7 447,4 523,8 537,0 304,7 426,4 521,9 577,1 613,1 630,4 695,7 767,5 826,2 544,1 669,5 742,4 790,1 815,1 904,7 1002 1085 1204 576,5 742,6 877,8 907,0 1008 1118 1213 1349 610,3 814,7 964,9 998,4 1111 1234 1340 1494 1695 1749

187,5 243,4 316,1 384,5 448,8 459,3 271,7 375,5 458,5 506,3 537,2 551,6 607,2 668,4 717,7 494,5 607,7 673,2 716,0 737,7 817,6 903,7 977,3 1082 532,1 682,0 805,1 831,0 922,7 1022 1107 1230 569,7 755,3 893,7 923,9 1027 1140 1237 1377 1556 1603

0

0,5

1,0

1,5

2

2,5

3

162,8 140,1 207,4 175,8 268,6 227,3 326,0 275,2 379,5 319,7 387,8 326,3 238,7 207,5 325,7 280,1 396,7 340,5 437,4 375,0 463,8 397,2 475,5 406,8 522,1 445,8 573,6 489,0 614,4 522,6 443,3 393,0 544,1 481,7 602,1 532,5 639,9 565,5 658,3 580,9 728,6 642,0 803,7 707,0 867,8 762,3 958,6 840,2 485,6 438,4 618,8 555,2 729,4 653,5 752,0 672,8 833,9 745,2 922,0 822,7 998,0 889,1 1106 983,0 526,9 482,7 692,8 628,9 818,9 742,6 845,8 766,1 939,4 850,0 1041 940,5 1128 1018 1254 1129 1410 1264 1452 1301 Nb.Rd (kN) Lc (m) 3,5 4

120,4 149,3 192,6 232,9 270,2 275,5 179,7 240,4 291,8 321,1 339,9 347,7 380,5 416,9 444,9 345,9 423,5 467,7 496,4 509,3 562,2 618,1 665,7 732,3 392,5 494,2 580,8 597,3 660,7 728,4 786,2 867,9 438,4 566,0 667,6 688,0 762,5 842,5 911,1 1009 1124 1156

103,6 127,4 164,3 198,4 229,9 234,3 155,7 206,9 250,8 275,8 291,8 298,4 326,1 356,9 380,5 303,7 371,4 409,8 434,8 445,6 491,4 539,6 580,5 637,7 349,5 437,8 513,9 528,0 583,5 642,4 692,7 763,6 395,6 506,4 596,7 614,4 680,2 750,7 811,0 896,7 995,2 1022

4,5

5

78,12 94,97 122,3 147,5 170,7 173,8 118,3 155,8 188,6 207,2 219,1 223,8 244,3 267,1 284,3 234,9 286,9 316,3 335,3 343,2 377,9 414,3 445,1 487,9 276,1 343,3 402,2 412,6 455,3 500,4 538,8 592,6 319,0 403,0 474,2 487,4 538,9 593,6 640,4 706,4 779,4 799,7

6

60,45 73,01 93,91 113,2 130,9 133,2 91,99 120,5 145,8 160,1 169,2 172,7 188,4 205,8 218,9 184,6 225,3 248,2 263,0 269,0 295,9 324,1 347,9 380,9 219,8 272,0 318,3 326,3 359,7 394,9 424,8 466,7 257,5 322,4 379,0 389,2 429,9 472,9 509,7 561,5 617,2 632,8

7

47,97 57,69 74,17 89,35 103,3 105,1 73,22 95,61 115,6 126,9 134,1 136,8 149,1 162,9 173,1 148,0 180,4 198,7 210,5 215,2 236,6 259,0 277,8 304,0 177,6 219,1 256,3 262,5 289,2 317,3 341,1 374,4 210,0 261,3 307,0 315,0 347,8 382,3 411,8 453,2 497,0 509,4

8

38,91 46,66 59,97 72,22 83,44 84,88 59,52 77,54 93,71 102,8 108,7 110,9 120,8 131,9 140,1 120,8 147,3 162,2 171,8 175,5 192,9 211,0 226,3 247,5 145,9 179,6 209,9 214,9 236,7 259,6 279,0 306,0 173,5 215,0 252,5 259,0 285,8 314,0 338,1 371,9 407,1 417,1

9

32,16 38,49 49,45 59,54 68,77 69,95 49,27 64,08 77,42 84,94 89,75 91,56 99,73 108,9 115,6 100,3 122,3 134,6 142,6 145,6 160,0 175,0 187,7 205,1 121,6 149,5 174,7 178,8 196,9 215,8 231,9 254,3 145,2 179,5 210,7 216,1 238,4 261,9 281,8 309,9 338,9 347,1

10

110 110 110 110 110 110 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 150 150 150 150 150 150 150 150 160 160 160 160 160 160 160 160 160 160 h mm

110 2,5 110 3 110 4 110 5 110 6 110 6,3 120 3 120 4 120 5 120 5,6 120 6 120 6,3 120 7,1 120 8 120 8,8 140 4 140 5 140 5,6 140 6 140 6,3 140 7,1 140 8 140 8,8 140 10 150 4 150 5 150 6 150 6,3 150 7,1 150 8 150 8,8 150 10 160 4 160 5 160 6 160 6,3 160 7,1 160 8 160 8,8 160 10 160 12 160 12,5 b t mm mm

Appendix 9.2

1)

110 2,5 110 3 110 4 110 5 110 6 110 6,3 120 3 120 4 120 5 120 5,6 120 6 120 6,3 120 7,1 120 8 120 8,8 140 4 140 5 140 5,6 140 6 140 6,3 140 7,1 140 8 140 8,8 140 10 150 4 150 5 150 6 150 6,3 150 7,1 150 8 150 8,8 150 10 160 4 160 5 160 6 160 6,3 160 7,1 160 8 160 8,8 160 10 160 12 160 12,5 b t mm mm


x x x x x

110 110 110 110 110 110 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 150 150 150 150 150 150 150 150 160 160 160 160 160 160 160 160 160 160 h mm

Buckling resistance values for square hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.

b

1)

z

h b t Lc Nb.Rd

t h

y r0



1) h

x x

260

x x x x x x x x x

x x x x

180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220 250 250 250 250 250 250 250 250

180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220 250 250 250 250 250 250 250 250

5 6 6,3 7,1 8 8,8 10 12 12,5 5 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 8 8,8 10 12 12,5

0 1109 1318 1369 1529 1705 1859 2084 2403 2486 1125 1473 1531 1712 1912 2086 2342 2713 2809 1628 1694 1895 2118 2313 2600 3023 3132 1646 1770 2170 2428 2654 2987 3487 3616

0,5 1109 1318 1369 1529 1705 1859 2084 2403 2486 1125 1473 1531 1712 1912 2086 2342 2713 2809 1628 1694 1895 2118 2313 2600 3023 3132 1646 1770 2170 2428 2654 2987 3487 3616

1,0 1109 1318 1369 1529 1705 1859 2084 2403 2486 1125 1473 1531 1712 1912 2086 2342 2713 2809 1628 1694 1895 2118 2313 2600 3023 3132 1646 1770 2170 2428 2654 2987 3487 3616

1,5 1066 1266 1314 1466 1634 1779 1992 2288 2366 1105 1436 1492 1667 1860 2029 2275 2628 2719 1606 1671 1869 2087 2278 2558 2967 3073 1646 1770 2170 2428 2653 2984 3477 3605

2 1013 1203 1248 1392 1551 1688 1889 2167 2240 1060 1374 1427 1594 1778 1938 2172 2507 2594 1544 1606 1795 2005 2188 2456 2846 2947 1606 1722 2098 2345 2562 2881 3356 3478

2,5 959,1 1138 1180 1316 1465 1594 1782 2039 2107 1014 1310 1360 1519 1694 1846 2067 2382 2464 1481 1540 1721 1922 2096 2352 2723 2819 1554 1665 2024 2263 2471 2778 3234 3351

Nb.Rd (kN) Lc (m) 3 3,5 4 901,8 841,4 778,5 1069 996,4 920,9 1108 1033 953,8 1235 1150 1061 1374 1278 1178 1494 1389 1279 1668 1549 1425 1905 1763 1616 1967 1819 1666 966,0 916,3 864,1 1243 1173 1100 1291 1218 1141 1441 1358 1272 1605 1513 1415 1749 1647 1540 1957 1841 1720 2251 2113 1969 2328 2184 2034 1416 1348 1277 1472 1400 1326 1644 1564 1480 1835 1745 1651 2001 1902 1798 2244 2132 2014 2595 2461 2321 2686 2547 2401 1501 1447 1391 1607 1548 1486 1949 1871 1791 2178 2091 2000 2378 2283 2183 2673 2564 2451 3109 2979 2845 3221 3086 2946

h b t mm mm mm 4,5 714,5 844,3 873,9 971,4 1077 1169 1300 1469 1513 809,9 1024 1062 1183 1315 1430 1595 1821 1881 1203 1248 1393 1553 1690 1892 2176 2249 1333 1422 1707 1906 2079 2334 2704 2800

5 651,3 768,8 795,2 883,2 978,4 1060 1178 1326 1366 754,5 947,0 981,7 1093 1214 1319 1469 1673 1727 1127 1169 1304 1452 1580 1766 2027 2095 1273 1356 1620 1808 1972 2212 2559 2649

6 534,3 629,6 650,5 721,5 797,7 863,3 957,1 1071 1102 644,8 798,2 826,5 919,0 1019 1106 1229 1392 1435 974,0 1009 1124 1251 1359 1517 1733 1789 1147 1218 1441 1606 1750 1960 2260 2338

7 436,5 513,7 530,3 587,6 648,8 701,5 776,4 865,1 889,2 544,1 665,7 688,8 765,0 847,3 918,3 1019 1149 1184 829,8 859,0 956,0 1063 1153 1285 1462 1508 1019 1078 1262 1405 1530 1711 1967 2033

8 358,8 421,8 435,3 481,9 531,7 574,4 635,1 705,7 725,0 457,6 555,1 574,1 637,1 704,9 763,4 845,6 951,4 979,3 703,3 727,6 809,0 898,7 973,7 1084 1230 1268 895,6 943,7 1094 1217 1324 1480 1696 1751

9 298,0 350,2 361,2 399,7 440,7 475,9 525,8 583,2 598,9 386,3 465,7 481,4 534,0 590,4 639,1 707,3 794,2 817,2 597,1 617,3 686,0 761,5 824,5 917,1 1038 1070 782,3 821,9 945,7 1051 1143 1276 1458 1506

10 250,5 294,2 303,4 335,6 369,8 399,2 440,9 488,4 501,5 328,4 394,2 407,4 451,7 499,2 540,1 597,4 669,8 689,0 509,7 526,7 585,1 649,2 702,5 780,9 882,6 909,3 682,5 715,2 818,0 908,6 987,3 1101 1257 1297

180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220 250 250 250 250 250 250 250 250

180 180 180 180 180 180 180 180 180 200 200 200 200 200 200 200 200 200 220 220 220 220 220 220 220 220 250 250 250 250 250 250 250 250

5 6 6,3 7,1 8 8,8 10 12 12,5 5 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 8 8,8 10 12 12,5 6 6,3 7,1 8 8,8 10 12 12,5


b t mm mm mm

x x x

Appendix 9.2

Table 9.2.1

x

x x 1)

x

260 6 260 6,3 260 7,1 260 8 260 8,8 260 10 260 11 260 12,5 300 6 300 6,3 300 7,1 300 8 300 8,8 300 10 300 12 300 12,5 b t mm mm

1669 1797 2262 2532 2768 3116 3368 3777 1746 1887 2303 2787 3222 3633 4262 4422

1669 1797 2262 2532 2768 3116 3368 3777 1746 1887 2303 2787 3222 3633 4262 4422

1669 1797 2262 2532 2768 3116 3368 3777 1746 1887 2303 2787 3222 3633 4262 4422

1669 1797 2262 2532 2768 3116 3368 3777 1746 1887 2303 2787 3222 3633 4262 4422

1639 1761 2199 2459 2687 3023 3263 3655 1746 1887 2295 2766 3186 3590 4206 4363

1590 1706 2125 2376 2596 2920 3151 3529 1710 1843 2235 2690 3096 3488 4084 4236

1539 1651 2050 2292 2504 2816 3037 3398 1667 1795 2175 2613 3004 3384 3961 4108

0

0,5

1,0

1,5

2

2,5

3

1488 1435 1594 1536 1973 1894 2206 2116 2409 2310 2708 2596 2919 2797 3265 3126 1624 1580 1748 1699 2114 2051 2536 2456 2911 2816 3279 3171 3836 3707 3978 3844 Nb.Rd (kN) Lc (m) 3,5 4

1380 1475 1811 2023 2208 2480 2670 2982 1535 1650 1987 2375 2718 3059 3574 3706

1323 1412 1725 1926 2102 2359 2538 2832 1489 1599 1921 2290 2616 2944 3437 3563

1204 1281 1547 1725 1881 2109 2265 2523 1393 1492 1784 2113 2404 2702 3149 3262

1081 1146 1367 1523 1659 1858 1991 2214 1292 1380 1638 1928 2181 2450 2849 2951

4,5

5

6

7

960,0 1014 1195 1330 1448 1619 1732 1923 1187 1264 1490 1740 1958 2197 2550 2639

846,0 890,7 1039 1156 1257 1405 1501 1664 1082 1148 1343 1557 1743 1955 2263 2342

743,5 780,7 903,2 1004 1092 1219 1301 1440 979,0 1036 1203 1387 1545 1731 2001 2069

8

9

10

260 260 260 260 260 260 260 260 300 300 300 300 300 300 300 300 h mm

260 6 260 6,3 260 7,1 260 8 260 8,8 260 10 260 11 260 12,5 300 6 300 6,3 300 7,1 300 8 300 8,8 300 10 300 12 300 12,5 b t mm mm

261


x

260 260 260 260 260 260 260 260 300 300 300 300 300 300 300 300 h mm

Appendix 9.2

b

1)

t h

y r0 z



x

262

40

30

2

x

50

30

2

x

60

40

2

x

60

40

2,5

x

70

50

2

x

70

50

2,5

x

70

50

3

x

80

40

2,5

x

80

60

2

x

80

60

2,5

x

80

60

3

x

80

60

4

80

70

2,5

80

70

3

80

70

4

80

70

5

axis

1) h

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0 81,88 81,88 94,78 94,78 120,6 120,6 148,1 148,1 146,4 146,4 180,4 180,4 213,3 213,3 180,4 180,4 163,2 163,2 212,7 212,7 252,0 252,0 327,5 327,5 228,8 228,8 271,4 271,4 353,3 353,3 431,0 431,0

0,5 71,51 66,50 86,87 77,73 114,8 108,0 140,7 132,2 142,8 137,3 175,7 168,8 207,5 199,3 177,6 162,2 162,5 158,2 210,8 205,0 249,7 242,7 323,7 314,4 227,4 224,8 269,4 266,2 350,2 346,0 426,4 421,2

1,0 49,63 39,28 67,11 46,86 95,48 79,29 116,5 96,40 123,6 111,1 151,9 136,1 178,9 160,1 156,0 120,2 145,0 135,6 187,0 174,4 221,2 206,0 286,0 265,5 202,4 196,8 239,5 232,6 310,5 301,6 377,1 365,7

1,5 30,66 21,88 46,40 26,37 73,54 51,67 89,17 62,38 101,9 81,92 124,9 99,90 146,5 116,8 131,7 79,14 125,6 109,8 160,6 139,3 189,7 164,0 243,9 209,6 174,9 165,3 206,4 194,8 266,4 251,3 322,1 302,9

2 19,57 13,43 31,29 16,24 53,56 33,77 64,59 40,63 79,49 57,62 97,10 69,97 113,4 81,57 105,8 51,98 104,5 84,04 131,9 105,1 155,4 123,3 198,4 156,2 144,7 131,9 170,3 154,8 218,6 198,6 262,8 237,7

2,5 13,35 9,003 21,89 10,91 39,00 23,28 46,88 27,97 60,54 41,15 73,78 49,88 85,86 58,04 82,49 35,92 84,04 63,12 104,9 78,17 123,3 91,50 156,4 115,3 115,9 102,3 135,9 119,6 173,6 152,8 207,5 181,9

Nb.Rd (kN) Lc (m) 3 3,5 9,636 7,268 16,01 7,81 29,14 16,90 34,97 20,29 46,42 30,44 56,49 36,85 65,61 42,84 64,18 26,10 66,82 48,00 82,68 59,12 97,07 69,13 122,6 86,84 91,85 79,29 107,5 92,5 136,7 117,9 162,9 139,8

h b t mm mm mm 4

4,5

5

6

7

8

9

10 40

30 2

50

30 2

12,17

9,550

7,687

22,44 12,79 26,90 15,35 36,28 23,29 44,12 28,18 51,18 32,74 50,62 19,77 53,43 37,34 65,75 45,83 77,12 53,56 97,10 67,16 73,29 62,40 85,64 72,73 108,7 92,48 129,2 109,5

17,75 10,01 21,26 12,00 28,98 18,35 35,22 22,20 40,82 25,78 40,66 15,47 43,29 29,73 53,09 36,42 62,23 42,53 78,21 53,27 59,30 50,05 69,23 58,29 87,73 74,03 104,1 87,51

14,36

11,85

60

40 2

17,20

14,19

60

40 2,5

23,61 14,82 28,68 17,91 33,23 20,80 33,25

19,57 12,20 23,77 14,75 27,53

14,05

70

50 2

17,06

70

50 2,5

19,75

70

50 3

27,64

19,92

15,01

80

40 2,5

35,61 24,17 43,57 29,56 51,05 34,52 64,09 43,20 48,74 40,89 56,86 47,60 71,98 60,41 85,34 71,35

29,73 20,01 36,31 24,45 42,53 28,54 53,35 35,70 40,66 33,97 47,42 39,53 59,98 50,14 71,07 59,18

21,54 14,34 26,25 17,50 30,74

16,28

80

60 2

19,81

80

60 2,5

23,20

80

60 3

38,51

29,05

80

60 4

29,43 24,46 34,30 28,44 43,35 36,06 51,32 42,52

22,24

80

70 2,5

25,91

80

70 3

32,72

80

70 4

38,71

80

70 5


b t mm mm mm

h b t Lc Nb.Rd

Appendix 9.2

Buckling resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.

Table 9.2.2

90

50

2,5

x

90

50

3

90

60

2,5

90

60

3

90

60

4

90

70

2

90

70

2,5

90

70

3

90

70

4

90

70

5

90

80

2

90

80

2,5

90

80

3

90

80

4

90

80

5

90

80

6

90

80

6,3

100

40

2

x 100

40

2,5

1) h

b t mm mm mm

x

154,2 154,2 212,7 212,7 252,0 252,0 228,8 228,8 271,4 271,4 353,3 353,3 178,8 178,8 244,9 244,9 290,7 290,7 379,1 379,1 463,3 463,3 184,0 184,0 261,1 261,1 310,1 310,1 405,0 405,0 495,6 495,6 582,0 582,0 596,1 596,1 144,4 144,4 198,6 198,6

154,2 146,7 212,6 200,1 251,7 236,7 228,8 221,1 271,4 261,8 353,3 340,1 178,8 177,2 244,9 241,0 290,7 285,9 379,1 372,0 463,3 453,5 184,0 184,0 261,1 260,5 310,1 309,1 405,0 403,1 495,6 492,4 582,0 577,2 596,1 590,3 144,4 133,3 198,6 180,9

140,3 121,8 190,8 162,9 225,6 192,1 206,6 188,7 244,5 222,8 317,2 288,3 164,3 157,0 222,2 211,6 263,4 250,6 342,3 325,1 416,6 395,0 170,2 167,1 237,8 233,0 282,0 276,1 367,1 359,5 447,8 438,1 524,0 512,3 535,2 522,9 133,9 104,6 181,9 137,9

124,4 93,60 167,0 121,3 197,0 142,4 181,8 151,8 214,8 178,4 277,8 229,2 146,8 134,6 196,3 178,5 232,5 210,9 301,3 272,1 365,3 328,8 152,9 147,8 211,0 202,9 249,8 240,0 324,3 311,5 394,4 378,2 460,0 440,3 468,7 447,9 120,6 73,79 162,2 93,8

106,7 68,03 140,8 85,95 165,6 100,5 154,5 115,3 182,1 134,8 234,4 172,0 127,5 110,3 167,9 143,2 198,4 168,8 256,1 216,3 308,8 259,5 134,0 126,6 181,4 169,9 214,2 200,3 277,0 259,0 335,4 312,7 389,6 362,1 395,4 366,6 106,0 50,37 140,3 62,68

88,68 49,48 114,8 61,63 134,6 71,92 127,1 86,12 149,3 100,4 191,3 127,6 107,5 87,50 138,9 111,5 163,9 131,2 210,6 167,1 252,5 199,4 114,0 104,9 151,1 137,6 177,9 161,7 229,0 208,2 276,1 250,1 319,1 288,1 322,7 290,5 90,58 35,44 117,8 43,65

72,35 36,95 92,18 45,67 107,9 53,24 102,7 65,29 120,5 75,94 153,8 96,30 88,72 68,88 112,9 86,70 133,1 101,8 170,3 129,3 203,3 153,7 94,99 85,40 123,4 109,9 145,1 128,9 186,0 165,5 223,4 198,1 257,3 227,3 259,4 228,5 75,74 26,00 96,87 31,86

axis

x

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0

0,5

1,0

1,5

2

2,5

3

58,87 28,43 74,21 35,00 86,74 40,77 83,09 50,70 97,28 58,9 123,9 74,56 72,82 54,72 91,65 68,35 107,9 80,21 137,7 101,6 163,9 120,6 78,50 69,39 100,6 88,20 118,0 103,3 150,9 132,3 180,8 158,0 207,6 180,8 208,9 181,4 62,78 19,80 79,29 24,19 Nb.Rd (kN) Lc (m) 3,5

48,25 22,49 60,39 27,59 70,52 32,14 67,83 40,32 79,33 46,81 100,8 59,20 60,04 44,15 75,00 54,88 88,22 64,37 112,4 81,44 133,5 96,50 65,04 56,81 82,48 71,61 96,68 83,78 123,4 107,2 147,6 127,8 169,3 146,0 170,1 146,4 52,11 15,56 65,26 18,96

40,00 18,20 49,83 22,29 58,15 25,95 56,08 32,75 65,54 38,01 83,21 48,04 49,98 36,22 62,11 44,87 73,02 52,61 92,94 66,50 110,2 78,71 54,32 47,07 68,42 58,99 80,14 68,98 102,2 88,17 122,1 105,0 139,8 119,9 140,4 120,0 43,59 12,53 54,25 15,25

33,58 15,01 41,68 18,36 48,62 21,38 46,98 27,10 54,88 31,44 69,62 39,72 42,07 30,18 52,09 37,30 61,22 43,72 77,86 55,23 92,26 65,33 45,83 39,49 57,45 49,30 67,26 57,61 85,70 73,60 102,3 87,59 117,1 99,92 117,4 100,0 36,81

24,49

18,59

14,57

90

50 2

30,28

22,92

17,94

90

50 2,5

35,30

26,71

20,90

90

50 3

34,19 19,41 39,91 22,50 50,58

25,91

20,29

90

60 2,5

30,24

23,67

90

60 3

38,30

29,97

90

60 4

23,43

18,39

90

70 2

28,80

22,56

90

70 2,5

45,62

30,80 21,81 37,96 26,87 44,60 31,49 56,66

33,82

26,50

90

70 3

42,94

33,62

90

70 4

67,06

50,79

39,75

90

70 5

33,65 28,79 41,93 35,76 49,06 41,77 62,44 53,31 74,46 63,39 85,11 72,25 85,32 72,25 27,06

25,65 21,85 31,84 27,05 37,24 31,58 47,37 40,30 56,44 47,88 64,48 54,54 64,61 54,52 20,64

20,15 17,12 24,96

90

80 2

90

80 2,5

29,18

90

80 3

37,10

90

80 4

44,20

90

80 5

50,47

90

80 6

50,55

90

80 6,3

16,23

100

40 2

33,37

25,37

19,90

100

40 2,5

h b t mm mm mm 4

4,5

5

6

7

8

9

10

Appendix 9.2

2


50

263

90

b

1)

z

h b t Lc Nb.Rd

t h

y r0


x

264

100

50

x 100

50

x 100

50

x 100

60

x 100

60

x 100

60

x 100

60

100

70

100

70

100

70

100

70

100

70

2

axis

1) h

y-y z-z 2,5 y-y z-z 3 y-y z-z 2 y-y z-z 2,5 y-y z-z 3 y-y z-z 4 y-y z-z 2 y-y z-z 2,5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z

0 157,3 157,3 214,7 214,7 271,4 271,4 170,2 170,2 230,9 230,9 290,7 290,7 379,1 379,1 181,9 181,9 247,0 247,0 310,1 310,1 405,0 405,0 495,6 495,6

0,5 157,3 150,6 214,7 203,4 271,4 255,3 170,2 166,6 230,9 224,4 290,7 281,0 379,1 365,6 181,9 181,0 247,0 244,3 310,1 305,3 405,0 398,0 495,6 486,0

1,0 146,6 126,4 197,9 167,8 248,1 207,9 159,2 145,1 213,9 193,3 267,3 239,8 347,5 310,9 170,8 161,3 229,7 215,9 286,4 268,1 373,0 348,7 455,2 424,4

1,5 132,6 98,9 177,4 127,5 220,8 154,8 144,4 120,7 192,5 157,9 239,1 192,9 309,9 248,4 155,4 139,5 207,3 184,3 257,1 226,4 334,1 293,2 406,8 354,9

2 117,3 73,08 154,8 91,80 190,7 109,7 128,2 95,30 168,9 121,9 208,0 146,5 268,4 187,4 138,6 115,8 182,8 150,1 224,8 181,9 291,1 234,3 353,3 281,7

2,5 100,9 53,66 131,1 66,4 159,6 78,65 110,9 73,18 144,1 92,20 175,5 109,4 225,4 139,4 120,5 92,90 156,7 118,5 190,8 141,8 246,2 181,9 297,7 217,4

Nb.Rd (kN) Lc (m) 3 3,5 84,99 70,84 40,28 31,10 108,7 89,53 49,44 37,99 131,0 107,1 58,29 44,66 93,81 78,50 56,41 44,23 120,3 99,58 70,36 54,85 145,1 119,2 82,97 64,42 185,5 151,9 105,4 81,72 102,6 86,26 73,78 58,93 131,5 109,3 92,9 73,64 158,7 130,9 110,3 87,02 204,0 167,9 141,1 111,1 245,7 201,7 168,1 132,1

h b t mm mm mm 4 4,5 59,04 49,52 24,64 19,97 73,99 61,69 30,01 24,27 88,01 73,11 35,22 28,44 65,63 55,16 35,4 28,88 82,59 69,04 43,73 35,59 98,34 81,90 51,23 41,62 125,1 104,0 64,93 52,71 72,37 60,98 47,72 39,24 90,93 76,16 59,34 48,63 108,3 90,41 69,89 57,16 138,7 115,6 89,12 72,83 166,3 138,4 105,8 86,38

5 41,91 16,49 51,99 20,01 61,46 23,44 46,74 23,97 58,28 29,48 68,96 34,44 87,48 43,60 51,77 32,75 64,38 40,49 76,23 47,52 97,36 60,51 116,5 71,73

6 30,89

7 23,59

8 18,57

38,12

29,02

44,91 34,52 17,23 42,83 21,15 50,51 24,66 63,99 31,20 38,32 23,71 47,40 29,23 55,94 34,24 71,36 43,57 85,29 51,60

9 14,97

10 100

50 2

22,79

100

50 2,5

34,13

26,77

100

50 3

26,39

20,78

16,77

100

60 2

32,65

25,66

20,68

100

60 2,5

38,43

30,17

24,29

100

60 3

48,65

38,17

30,72

100

60 4

29,34 17,92 36,17 22,05 42,61 25,80 54,32 32,81 64,88

23,12

18,67

100

70 2

28,45

22,94

100

70 2,5

33,47

26,97

100

70 3

42,66

34,35

100

70 4

50,92

40,99

100

70 5


b t mm mm mm


Appendix 9.2

Buckling resistance values for rectangular hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.

Table 9.2.2

100

80

2,5

x 100

80

3

x 100

80

4

x 100

80

5

x 100

80

6

100

80

6,3

110

40

2

110

40

2,5

110

40

3

110

50

2

110

50

2,5

110

50

3

110

60

2

110

60

2,5

110

60

3

110

60

4

1) h

b t mm mm mm

x

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

187,0 187,0 263,1 263,1 329,5 329,5 430,8 430,8 527,9 527,9 620,7 620,7 636,7 636,7 146,9 146,9 203,6 203,6 271,4 271,4 159,8 159,8 219,7 219,7 290,7 290,7 172,7 172,7 235,9 235,9 310,1 310,1 405,0 405,0

187,0 187,0 263,1 263,1 329,5 328,9 430,8 429,4 527,9 525,2 620,7 616,8 636,7 631,8 146,9 136,5 203,6 186,6 271,4 245,0 159,8 153,7 219,7 209,3 290,7 274,0 172,7 169,7 235,9 230,1 310,1 300,2 405,0 390,9

176,7 171,1 245,4 237,0 305,5 294,5 398,5 383,6 487,0 468,2 571,1 548,7 584,2 561,0 139,0 108,5 190,5 144,2 250,7 183,3 151,7 130,2 206,5 174,2 270,4 223,8 164,5 148,7 222,6 199,4 289,7 256,9 377,5 333,1

161,4 152,3 222,1 208,2 275,2 256,9 358,1 333,4 436,8 405,4 511,0 473,6 521,5 482,6 127,2 77,93 172,8 99,75 225,2 121,9 139,2 103,4 188,1 134,3 244,1 167,4 151,3 125,0 203,4 164,6 262,6 207,6 341,5 267,0

144,9 131,6 196,5 176,4 241,8 215,5 313,7 278,3 381,4 336,7 444,7 391,6 452,3 397,3 114,3 53,81 153,4 67,28 197,0 80,46 125,7 77,52 167,9 98,00 215,2 119,1 137,1 99,80 182,4 128,6 232,8 158,4 302,0 202,1

127,1 110,2 169,3 144,6 206,5 174,8 267,0 224,7 323,3 270,4 375,5 313,2 380,4 316,5 100,5 38,08 132,7 47,07 167,4 55,73 111,1 57,43 146,2 71,39 184,6 85,55 121,7 77,44 159,8 98,02 201,0 118,8 259,9 150,7

109,1 90,51 142,7 116,6 172,6 139,9 222,4 179,1 268,4 214,8 310,7 248,0 313,6 249,9 86,52 28,02 112,3 34,43 139,3 40,55 96,13 43,33 124,7 53,37 155,0 63,47 105,8 60,07 137,1 75,21 169,9 90,21 218,9 114,1

axis

2

0

0,5

1,0

1,5

2

2,5

3

92,42 74,00 119,0 94,10 143,1 112,4 183,8 143,5 221,3 171,6 255,4 197,8 257,1 198,9 73,46 21,38 94,02 26,18 115,0 30,74 82,03 33,55 105,0 41,10 128,8 48,66 90,71 47,29 116,0 58,81 142,0 70,13 182,5 88,51 Nb.Rd (kN) Lc (m) 3,5

77,97 60,85 99,24 76,75 118,7 91,29 152,3 116,4 183,0 139,0 210,8 160,0 211,8 160,7 62,09 16,82 78,61 20,54 95,18 24,07 69,60 26,63 88,14 32,51 107,2 38,39 77,25 37,93 97,83 46,98 118,6 55,81 152,1 70,35

65,97 50,56 83,26 63,40 99,31 75,23 127,2 95,80 152,6 114,3 175,6 131,5 176,2 131,9 52,60

56,17 42,50 70,48 53,09 83,86 62,89 107,3 80,03 128,6 95,40 147,9 109,7 148,3 110,0 44,83

41,74 31,06 51,97 38,60 61,66 45,64 78,80 58,02 94,35 69,10 108,3 79,36 108,5 79,53 33,35

32,02 23,61 39,70 29,25 47,02 34,53 60,05 43,88 71,85 52,22 82,43 59,95 82,52 60,05 25,61

25,28 18,52 31,24 22,89 36,96 27,01 47,18

20,43

100

80 2

25,20

100

80 2,5

29,79

100

80 3

38,02

100

80 4

56,43

45,45

100

80 5

64,72

52,11

100

80 6

64,75

52,12

100

80 6,3

20,22

16,35

110

40 2

66,08

56,01

41,37

31,63

24,91

20,10

110

40 2,5

79,45

67,01

49,18

37,47

29,44

23,71

110

40 3

59,14 21,61 74,32 26,32 89,79 31,01 65,81 31,00 82,73 38,28 99,61 45,36 127,6 57,14

50,51 17,86 63,13 21,72 75,92 25,56 56,32 25,75 70,42 31,75 84,38 37,55 108,0 47,27

37,67

28,97

22,90

18,53

110

50 2

46,76

35,81

28,23

22,80

110

50 2,5

55,89

42,66

33,56

27,05

110

50 3

42,11 18,54 52,30 22,80 62,28 26,90 79,59 33,83

32,44

25,66

20,78

17,15 110

60 2

40,12

31,66

25,58

21,09 110

60 2,5

47,60

37,48

30,24

110

60 3

60,79

47,84

38,58

110

60 4

h b t mm mm mm 4

4,5

5

6

7

8

9


80

265

100

10

Appendix 9.2

b

1)

h b t Lc Nb.Rd

t h

y r0 z


x

266

110

70

2

110

70

2,5

110

70

3

110

70

4

110

70

6,3

110

90

2

110

90

2,5

110

90

3

110

90

4

110

90

5

110

90

6

110

90

6,3

axis

1) h

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0 184,4 184,4 252,0 252,0 329,5 329,5 430,8 430,8 636,7 636,7 193,4 193,4 280,1 280,1 368,2 368,2 482,4 482,4 592,4 592,4 698,2 698,2 718,1 718,1

0,5 184,4 184,1 252,0 250,0 329,5 324,9 430,8 423,9 636,7 623,3 193,4 193,4 280,1 280,1 368,2 368,2 482,4 482,4 592,4 592,4 698,2 698,2 718,1 718,1

1,0 176,2 164,8 238,6 222,1 309,2 286,0 403,2 372,1 591,6 542,8 186,6 182,2 266,9 260,0 347,7 338,0 454,7 441,6 557,2 540,7 655,2 635,6 672,7 652,1

1,5 162,4 143,7 218,5 191,0 281,2 242,5 366,0 313,9 533,7 451,8 173,2 166,2 245,5 234,3 317,7 301,9 414,8 393,4 507,5 480,5 595,8 563,7 610,9 577,0

2 147,5 120,6 196,6 157,3 250,6 195,8 325,1 251,9 470,0 356,6 158,9 148,7 222,3 206,1 285,0 262,0 371,3 340,3 453,3 414,0 530,9 484,2 543,3 494,1

2,5 131,5 97,80 173,1 125,3 217,8 153,3 281,5 196,1 402,6 274,0 143,4 130,0 197,4 176,2 249,9 220,6 324,7 285,3 395,2 345,6 461,4 402,8 471,0 409,6

Nb.Rd (kN) Lc (m) 3 3,5 114,9 98,89 78,24 62,82 149,1 126,8 98,96 78,78 185,2 155,6 119,7 94,50 238,5 199,8 152,5 120,2 337,5 280,3 211,2 165,6 127,2 111,1 111,1 93,80 171,7 147,2 147,5 122,3 214,5 181,7 181,9 149,2 277,8 234,7 234,5 191,8 337,1 284,1 283,1 231,0 392,4 329,7 328,9 267,8 399,6 335,0 333,5 270,9

h b t mm mm mm 4 84,50 51,04 107,2 63,65 130,5 76,02 167,1 96,50 233,0 132,6 96,13 78,97 125,3 101,6 153,3 123,0 197,6 157,8 238,7 189,7 276,5 219,6 280,5 221,9

4,5 72,18 42,06 90,91 52,26 109,9 62,22 140,5 78,92 195,1 108,2 82,91 66,69 106,8 85,00 129,6 102,4 166,9 131,2 201,3 157,4 232,9 182,1 236,0 183,8

5 61,89 35,16 77,52 43,57 93,29 51,75 119,1 65,60 164,9 89,76 71,61 56,71 91,38 71,81 110,4 86,14 142,0 110,3 171,0 132,3 197,6 152,9 200,1 154,2

6 46,39 25,51 57,70 31,51 69,03 37,32 88,01 47,26 121,3 64,53 54,19 42,06 68,33 52,81 82,00 63,05 105,3 80,63 126,7 96,6 146,2 111,5 147,9 112,4

7 35,78 19,30 44,32 23,79 52,84 28,13 67,31 35,60 92,56

8 28,34

9 22,96

10 18,95 110

70 2

35,00

28,30

23,34 110

70 2,5

41,64

33,62

27,69 110

70 3

53,01

42,78

35,23 110

70 4

72,79

58,68

110

42,03 32,24 52,63 40,28 62,92 47,96 80,74 61,29 97,08 73,38 112,0 84,69 113,2 85,32

33,40 25,43 41,64 31,67 49,66 37,64 63,69 48,07 76,55 57,54 88,23 66,38 89,15 66,85

27,13 20,54 33,71 25,53 40,13 30,30 51,46

22,44 110

90 2

27,82 110

90 2,5

33,08 110

90 3

42,41 110

90 4

61,82

50,93 110

90 5

71,23

58,67 110

90 6

71,96

59,26 110

90 6,3

70 6,3


b t mm mm mm


Appendix 9.2


Table 9.2.2

2,5

110 100

3

110 100

4

110 100

5

110 100

6

110 100

6,3

120

40

2

120

40

2,5

120

40

3

120

50

2

120

50

2,5

120

50

3

120

60

2

120

60

2,5

x 120

60

3

x 120

60

4

1) h

b t mm mm mm

x

196,5 196,5 286,3 286,3 387,5 387,5 508,2 508,2 624,7 624,7 736,9 736,9 758,7 758,7 148,9 148,9 207,7 207,7 270,5 270,5 161,8 161,8 223,8 223,8 289,8 289,8 174,7 174,7 240,0 240,0 309,2 309,2 430,8 430,8

196,5 196,5 286,3 286,3 387,5 387,5 508,2 508,2 624,7 624,7 736,9 736,9 758,7 758,7 148,9 139,4 207,7 191,5 270,5 246,5 161,8 156,4 223,8 214,2 289,8 275,1 174,7 172,4 240,0 235,0 309,2 300,9 430,8 416,6

190,4 188,5 274,1 270,9 366,8 362,3 480,1 474,0 589,0 581,5 693,3 684,3 713,0 703,2 143,2 112,3 197,7 149,9 255,1 188,1 156,1 133,4 213,9 179,9 274,8 227,7 169,0 152,0 230,1 204,8 294,3 259,8 407,1 355,9

177,2 174,2 253,0 247,9 335,8 328,4 438,8 428,9 537,5 525,4 631,7 617,1 649,0 633,1 132,6 82,32 181,7 105,4 232,9 128,1 144,8 107,2 197,2 140,7 251,9 174,0 157,0 129,1 212,5 170,6 270,5 213,0 372,2 286,7

163,3 158,9 230,2 222,8 302,0 291,3 393,9 379,5 481,4 463,8 564,5 543,1 579,2 555,9 121,3 57,65 164,4 71,80 208,7 85,62 132,8 81,34 179,2 104,0 227,0 125,8 144,3 104,5 193,8 134,9 244,7 165,1 334,3 218,0

148,2 142,3 205,7 195,9 265,7 251,8 345,6 326,8 421,2 398,4 492,4 464,9 504,3 474,5 109,0 41,08 145,8 50,47 182,6 59,65 119,9 60,75 159,7 76,35 200,1 91,26 130,6 81,97 173,5 103,7 216,9 125,2 293,4 163,1

132,3 125,1 180,2 168,6 228,9 212,9 296,8 275,5 360,6 334,9 420,3 389,4 429,7 396,4 96,17 30,35 126,6 37,01 156,5 43,54 106,3 46,04 139,6 57,31 172,8 68,04 116,2 64,05 152,3 80,02 188,4 95,70 252,1 123,7

axis

110 100


0

0,5

1,0

1,5

2

2,5

3

116,3 108,3 155,6 143,1 194,4 178,0 251,5 229,7 304,8 278,7 354,3 323,2 361,6 328,3 83,58 23,21 108,4 28,18 132,3 33,06 92,77 35,74 120,2 44,25 147,1 52,32 101,8 50,65 131,8 62,78 161,3 74,72 213,8 96,10 Nb.Rd (kN) Lc (m) 3,5

101,2 93,00 133,2 120,9 164,4 148,7 212,2 191,6 256,7 232,1 297,8 268,6 303,6 272,4 72,00 18,28 92,24 22,14 111,5 25,92 80,23 28,42 102,7 35,06 124,6 41,35 88,36 40,74 113,1 50,25 137,2 59,61 180,5 76,45

87,75 79,74 114,0 102,5 139,3 125,0 179,6 160,7 216,9 194,5 251,2 224,8 255,9 227,7 61,88

76,07 68,56 97,86 87,32 118,8 105,9 152,9 136,1 184,5 164,5 213,5 190,0 217,3 192,3 53,30

57,85 51,58 73,48 64,95 88,38 78,17 113,6 100,3 136,9 121,2 158,2 139,7 160,9 141,3 40,20

45,01 39,87 56,73 49,87 67,88 59,76 87,21 76,63 105,0 92,51 121,2 106,6 123,3 107,8 31,12

35,83 31,61 44,95 39,38 53,61 47,05 68,84 60,31 82,84 72,78 95,60 83,84 97,17 84,71 24,70

29,13 25,63 36,43 31,83 43,34 37,96 55,64 48,65 66,93 58,68 77,21 67,58 78,46 68,27 20,04

24,12 110 100 2 21,18 30,09 110 100 2,5 26,25 35,74 110 100 3

16,56 120

40 2

78,52

67,16

50,19

38,64

30,56

24,74

20,41 120

40 2,5

94,21

80,16

59,50

45,63

36,00

29,09

23,98 120

40 3

69,17 23,09 87,77 28,41 105,8 33,45 76,37 33,36 96,88 41,00 116,8 48,53 152,8 62,12

59,73 19,10 75,27 23,46 90,25 27,59 66,08 27,75 83,27 34,04 99,89 40,22 130,1 51,4

45,19

35,04

27,85

22,61

18,70 120

50 2

56,43

43,54

34,48

27,93

23,06 120

50 2,5

67,24

51,68

40,83

33,02

27,23 120

50 3

50,12 20,02 62,61 24,47 74,64 28,86 96,71 36,81

38,93

30,97

25,16

20,82 120

60 2

48,38

38,36

31,09

25,69 120

60 2,5

57,46

45,45

36,79

30,36 120

60 3

74,23

58,60

47,36

39,05 120

60 4

45,87 110 100 4 55,16 110 100 5 63,62 110 100 6 64,64 110 100 6,3

h b t mm mm mm 4

4,5

5

6

7

8

9


2

267

110 100

10

Appendix 9.2

1)

b

h

t y

h b t Lc Nb.Rd axis

268

0 191,6 191,6 272,2 272,2 347,9 347,9 482,4 482,4 592,4 592,4 698,2 698,2 718,1 718,1 195,5 195,5 284,2 284,2 367,3 367,3 508,2 508,2 624,7 624,7 736,9 736,9 758,7 758,7

0,5 191,6 191,6 272,2 272,2 347,9 347,9 482,4 481,9 592,4 591,0 698,2 695,4 718,1 714,5 195,5 195,5 284,2 284,2 367,3 367,3 508,2 508,2 624,7 624,7 736,9 736,9 758,7 758,7

1,0 186,5 177,5 262,3 248,5 333,2 314,6 459,2 431,9 562,9 528,8 661,8 620,7 679,4 636,9 191,1 185,1 274,6 265,2 352,6 339,7 485,1 466,3 595,2 571,5 700,9 672,5 720,3 691,2


1,5 174,0 159,7 243,2 220,8 307,6 277,4 422,0 377,2 516,6 460,5 606,5 538,6 621,7 551,4 178,8 169,5 255,1 240,1 326,0 305,3 446,7 416,3 547,5 508,9 643,8 597,7 660,8 613,3

2 161,0 140,2 222,8 190,2 280,1 236,2 381,9 317,1 466,6 385,6 546,4 448,7 559,0 457,9 166,0 152,5 234,4 212,5 297,6 267,5 405,4 361,0 496,0 439,8 582,1 515,0 596,4 527,2

2,5 146,9 119,6 200,8 158,8 250,3 194,7 338,5 257,7 412,6 312,2 481,7 361,5 491,5 367,8 152,3 134,3 211,9 183,1 266,9 227,6 360,8 303,7 440,3 368,4 515,5 429,9 526,8 439,0

3 132,0 99,9 177,8 130,0 219,5 157,7 294,1 206,6 357,4 249,5 415,9 287,8 423,1 292,1 137,6 115,7 188,4 154,4 234,8 189,6 314,7 250,3 383,1 302,6 447,3 352,0 455,8 358,7

Nb.Rd (kN) Lc (m) 3,5 116,8 82,67 155,0 106,1 189,6 127,7 251,7 166,0 305,2 200,2 354,0 230,3 359,3 233,4 122,7 98,40 164,9 128,9 203,5 156,7 270,5 205,2 328,4 247,4 382,5 287,2 388,8 292,2

h b t mm mm mm 4 102,3 68,58 133,8 87,12 162,4 104,3 214,2 135,0 259,1 162,5 299,8 186,7 303,7 189,1 108,1 83,20 142,9 107,5 174,9 129,8 230,9 169,1 279,8 203,5 325,2 235,9 329,9 239,7

4,5 89,16 57,32 115,3 72,31 139,0 86,31 182,3 111,3 220,3 133,9 254,4 153,6 257,2 155,5 94,70 70,53 123,5 90,22 150,1 108,4 197,0 140,7 238,3 169,1 276,6 195,8 280,2 198,8

5 77,61 48,39 99,45 60,75 119,4 72,33 155,9 93,07 188,2 111,9 217,0 128,3 219,3 129,8 82,78 60,14 106,8 76,38 129,1 91,50 168,8 118,4 204,0 142,1 236,5 164,5 239,3 167,0

6 59,35 35,56 75,16 44,37 89,67 52,66 116,5 67,57 140,4 81,15 161,7 92,95 163,1 93,98 63,68 44,76 80,92 56,33 97,19 67,18 126,4 86,61 152,6 103,9 176,6 120,1 178,4 121,9

7 46,32 27,11 58,25 33,70 69,25 39,93 89,70 51,14 108,0 61,39 124,2 70,27 125,2 71,03 49,87 34,39 62,82 43,04 75,16 51,19 97,46 65,87 117,5 78,96 135,9 91,25 137,2 92,52

8 36,96 21,31 46,27 26,43 54,88 31,27 70,95 40,00 85,39 48,00 98,16

9 30,09

10 24,94 120

80 2

37,56

31,06 120

80 2,5

44,48

36,74 120

80 3

57,42

47,39 120

80 4

69,09

57,00 120

80 5

79,39

65,47 120

80 6

98,90

79,96

65,93 120

80 6,3

39,88 27,16 49,96 33,88 59,61 40,23 77,15 51,69 92,98 61,94 107,5 71,55 108,5 72,53

32,52 21,96 40,58 27,32 48,34 32,41 62,48 41,61 75,27 49,84 86,98 57,56 87,74 58,34

26,98 120

90 2

33,58 120

90 2,5

39,95 120

90 3

51,58 120

90 4

62,13 120

90 5

71,77 120

90 6

72,38 120

90 6,3


r0 z 1) h b t mm mm mm x 120 80 2 y-y z-z 120 80 2,5 y-y z-z x 120 80 3 y-y z-z x 120 80 4 y-y z-z x 120 80 5 y-y z-z x 120 80 6 y-y z-z 120 80 6,3 y-y z-z 120 90 2 y-y z-z 120 90 2,5 y-y z-z 120 90 3 y-y z-z 120 90 4 y-y z-z 120 90 5 y-y z-z 120 90 6 y-y z-z 120 90 6,3 y-y z-z


Appendix 9.2


Table 9.2.2

120 100 120 100 120 100 120 100

140

40

140

60

140

60

140

60

140

70

140

70

x 140

70

x 140

70

x 140

80

x 140

80

x 140

80

x 140

80

x 140

80

1) h

b t mm mm mm

x

290,5 290,5 386,7 386,7 534,1 534,1 657,0 657,0 775,6 775,6 799,4 799,4 214,1 199,6 281,7 259,6 246,3 242,9 320,5 314,0 482,4 467,5 262,5 262,5 339,8 338,5 508,2 501,8 624,7 615,5 359,2 359,2 534,1 534,1 657,0 656,7 775,6 774,1 799,4 797,3

281,8 276,1 371,9 363,9 510,9 499,2 627,5 612,7 739,5 721,7 761,1 742,7 209,1 159,8 273,2 202,8 241,9 213,9 312,8 273,9 465,7 400,9 258,3 236,3 332,5 302,2 492,1 442,5 604,1 541,2 352,1 328,5 518,7 480,2 637,0 589,0 750,8 692,8 772,8 713,0

262,4 253,5 344,4 331,6 471,3 452,6 578,2 554,5 680,5 651,8 699,7 670,0 195,7 115,9 254,4 142,3 227,2 181,3 292,5 228,6 432,3 324,9 242,8 207,5 311,4 262,2 457,8 376,3 561,4 458,0 330,2 292,5 483,5 421,1 593,1 515,1 698,3 603,8 718,1 620,5

241,9 229,1 315,0 296,4 428,8 401,5 525,1 490,5 616,9 575,0 633,4 590,1 181,6 80,61 234,4 96,80 211,8 146,5 271,2 181,0 396,8 248,7 226,7 175,7 289,2 218,5 421,4 305,0 516,1 368,9 307,3 252,7 446,2 355,9 546,7 433,8 642,6 506,0 659,9 519

219,9 202,7 283,2 258,6 382,8 346,9 467,7 422,5 548,2 493,6 561,8 505,5 166,5 57,24 212,9 67,99 195,3 114,7 248,3 139,5 358,4 186,9 209,5 143,9 265,6 176,0 382,2 239,6 467,2 288,2 282,8 211,6 406,2 291,0 496,6 353,4 582,5 410,1 597,2 419,8

196,6 175,7 250,0 220,8 335,1 293,4 408,4 356,1 477,5 414,6 488,2 423,7 150,4 42,20 190,3 49,84 177,8 89,49 224,0 107,7 318,1 142,1 191,2 116,0 240,4 140,1 340,8 187,4 415,7 224,6 256,7 173,7 363,8 234,3 443,7 283,6 519,2 328,0 531,1 335,2

0

0,5

1,0

1,5

2

2,5

3

173,1 150,2 217,4 186,2 289,0 245,3 351,4 297,0 409,7 344,8 418,2 351,9 133,9 32,24 167,5 37,95 159,7 70,71 199,2 84,55 278,0 110,6 172,2 93,60 214,5 112,1 299,3 148,3 364,3 177,3 229,8 142,0 321,0 188,9 390,5 228,2 455,8 263,2 465,3 268,8 Nb.Rd (kN) Lc (m) 3,5

150,9 127,6 187,3 156,5 247,4 205,0 300,1 247,7 349,3 287,0 355,9 292,5 118,0 25,37 145,9 29,8 141,8 56,85 175,2 67,70 240,7 88,04 153,4 76,32 189,3 90,90 260,2 119,3 316,2 142,5 203,5 116,8 280,3 153,9 340,2 185,7 396,2 213,8 403,6 218,2

130,9 108,5 161,0 132,1 211,5 172,2 256,2 207,8 297,8 240,5 303,0 244,9 103,3

113,6 92,70 138,7 112,2 181,5 145,9 219,7 175,9 255,0 203,3 259,2 207,0 90,23

86,51 69,20 104,6 83,2 136,2 107,7 164,6 129,7 190,8 149,7 193,8 152,3 69,36

67,34 53,24 81,01 63,75 105,2 82,35 126,9 99,06 147,0 114,3 149,2 116,2 54,30

53,64 42,10 64,31 50,27 83,31 64,84 100,5 77,96 116,3 89,91 118,0 91,39 43,41

43,63 34,06 52,17 40,60 67,50 52,31 81,42 62,88 94,19 72,50 95,53 73,67 35,39

36,13 120 100 2,5 28,11 43,13 120 100 3 33,46 55,75 120 100 4

29,36 140

40 2,5

126,5

109,7

83,51

64,99

51,76

42,09

34,85 140

40 3

125,0 46,53 153,1 55,26 207,7 71,58 135,6 63,04 166,0 74,80 225,3 97,71 273,3 116,6 178,9 97,02 243,5 127,1 295,1 153,2 342,9 176,2 348,8 179,7

109,9 38,71 133,7 45,88 179,5 59,26 119,4 52,78 145,3 62,46 195,2 81,30 236,5 96,9 156,9 81,56 211,5 106,3 255,9 128,1 297,0 147,2 301,8 150,1

85,16 27,91 102,6 33,00 135,9 42,47 92,86 38,37 111,9 45,26 148,3 58,66 179,4 69,87 121,2 59,61 161,3 77,29 194,8 93,03 225,6 106,8 228,8 108,9

67,00

53,73

43,90

36,47 140

60 2,5

80,25

64,12

52,25

43,33 140

60 3

105,4

83,81

68,06

56,30 140

60 4

73,20 29,07 87,70 34,22 115,3 44,23 139,4 52,65 95,18 45,30 125,7 58,54 151,6 70,43 175,4 80,81 177,7 82,35

58,77

48,05

39,94 140

70 2,5

70,16

57,22

47,48 140

70 3

91,79

74,61

61,76 140

70 4

110,9

90,08

74,54 140

70 5

76,24 35,53 100,2 45,81 120,7 55,1 139,6 63,2 141,4

62,23

51,67 140

80 3

81,50

67,50 140

80 4

98,17

81,29 140

80 5

113,4

93,89 140

80 6

114,8

95,03 140

80 6,3

67,22 120 100 5 77,75 120 100 6 78,84 120 100 6,3

h b t mm mm mm 4

4,5

5

6

7

8

9

10

Appendix 9.2

40

269

140

290,5 290,5 386,7 386,7 534,1 534,1 657,0 657,0 775,6 775,6 799,4 799,4 214,1 214,1 281,7 281,7 246,3 246,3 320,5 320,5 482,4 482,4 262,5 262,5 339,8 339,8 508,2 508,2 624,7 624,7 359,2 359,2 534,1 534,1 657,0 657,0 775,6 775,6 799,4 799,4


120 100

2,5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z 6 y-y z-z 6,3 y-y z-z 2,5 y-y z-z 3 y-y z-z 2,5 y-y z-z 3 y-y z-z 4 y-y z-z 2,5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z 6 y-y z-z 6,3 y-y z-z axis

120 100

1)

b t h

y r0 z



x 140 100 140 100 140 100 140 100 140 100

270

140 100 140 110 140 110 140 110 140 110 140 110 140 110

axis

1) h

2,5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z 6 y-y z-z 6,3 y-y z-z 2,5 y-y z-z 3 y-y z-z 4 y-y z-z 5 y-y z-z 6 y-y z-z 6,3 y-y z-z

0 296,8 296,8 397,9 397,9 585,7 585,7 721,5 721,5 853,1 853,1 880,7 880,7 301,8 301,8 409,1 409,1 611,5 611,5 753,8 753,8 891,8 891,8 921,4 921,4

0,5 296,8 296,8 397,9 397,9 585,7 585,7 721,5 721,5 853,1 853,1 880,7 880,7 301,8 301,8 409,1 409,1 611,5 611,5 753,8 753,8 891,8 891,8 921,4 921,4

1,0 294,1 284,5 391,3 377,5 571,3 549,4 702,9 675,4 829,8 796,7 855,8 821,8 299,9 293,3 403,3 393,8 597,5 582,0 735,6 716,2 869,2 846,0 897,2 873,1

1,5 277,6 262,8 367,7 346,1 534,0 499,5 656,5 612,9 774,2 721,7 798,0 743,9 283,6 273,5 379,6 364,9 559,1 534,9 687,8 657,4 812,1 775,7 837,7 799,9

2 260,7 239,5 343,2 312,1 494,8 444,9 607,7 544,5 715,6 639,5 736,9 658,5 266,8 252,7 354,9 334,0 518,9 484,1 637,6 593,7 752,0 699,5 775,0 720,4

2,5 242,7 214,4 317,0 275,4 452,7 386,4 555,1 471,5 652,6 551,9 671,1 567,5 249,2 230,2 328,7 300,6 475,7 429,1 583,7 525,0 687,4 617,3 707,6 634,8

3 223,5 188,2 289,0 238,0 408,0 328,4 499,4 399,4 585,8 466,1 601,5 478,6 230,3 206,4 300,7 265,7 429,8 372,7 526,5 454,8 618,9 533,5 636,1 547,7

Nb.Rd (kN) Lc (m) 3,5 203,5 162,8 260,1 202,8 362,5 275,8 442,8 334,6 518,1 389,3 531,2 399,4 210,6 182,3 271,6 231,3 382,8 319,1 468,1 388,4 549,1 454,6 563,5 466,0

h b t mm mm mm 4 183,2 139,6 231,4 171,9 318,5 231,2 388,4 279,9 453,4 325,0 464,2 333,2 190,5 159,3 242,6 199,5 337,2 271,5 411,6 329,7 482,0 385,3 493,9 394,5

4,5 163,6 119,6 204,4 146,0 278,2 194,7 338,7 235,3 394,7 272,9 403,5 279,6 170,8 138,6 215,1 171,7 295,2 231,1 359,8 280,2 420,6 327,1 430,5 334,6

5 145,4 102,8 179,9 124,6 242,7 165,2 295,1 199,5 343,4 231,1 350,7 236,7 152,4 120,4 189,9 148,0 258,0 197,6 314,0 239,4 366,6 279,2 374,9 285,4

6 114,5 77,26 139,7 92,86 186,1 122,2 226,0 147,4 262,4 170,6 267,6 174,6 120,7 91,93 148,1 111,7 198,3 147,7 241,0 178,7 280,9 208,1 286,8 212,6

7 90,96 59,70 110,1 71,39 145,5 93,53 176,5 112,7 204,7 130,3 208,6 133,4 96,25 71,67 117,0 86,54 155,3 113,7 188,5 137,4 219,5 160,0 223,9 163,3

8 73,4 47,32 88,36 56,41 116,2 73,7 140,9 88,79 163,2 102,6 166,3 105,0 77,86 57,14 94,02 68,72 124,1 89,93 150,6 108,6 175,2 126,4 178,7 129,0

9 60,22 38,36 72,23 45,63 94,71 59,50 114,7 71,66 132,9 82,78 135,3 84,68 63,98 46,50 76,94 55,76 101,2 72,79 122,7 87,90 142,7 102,3 145,5 104,3

10 50,17 31,69 60,03 37,64 78,53 49,01 95,10 59,01 110,1

140 100 2,5 140 100 3 140 100 4 140 100 5 140 100 6

112,1

140 100 6,3

53,37 38,52 63,99 46,11 83,95 60,07 101,8 72,52 118,3 84,35 120,6 86,05

140 110 2,5 140 110 3 140 110 4 140 110 5 140 110 6 140 110 6,3


b t mm mm mm

h b t Lc Nb.Rd

Appendix 9.2


Table 9.2.2

4

140 120

5

140 120

6

140 120

6,3

150

50

2,5

150

50

3

150

60

2,5

150

60

3

150

60

4

150

70

2,5

150

70

3

150

70

4

150

70

5

150

90

2,5

150

90

3

150

90

4

150

90

5

150

90

6

150

90

6,3

1) h

b t mm mm mm

x

416,4 416,4 637,3 637,3 786,0 786,0 930,5 930,5 962,0 962,0 232,7 232,7 305,5 305,5 248,8 248,8 324,9 324,9 488,6 488,6 265,0 265,0 344,3 344,3 514,4 514,4 657,0 657,0 293,1 293,1 383,0 383,0 566,0 566,0 721,5 721,5 853,1 853,1 880,7 880,7

416,4 416,4 637,3 637,3 786,0 786,0 930,5 930,5 962,0 962,0 232,7 225,4 305,5 293,5 248,8 246,2 324,9 319,4 488,6 475,0 265,0 265,0 344,3 343,8 514,4 509,3 657,0 647,9 293,1 293,1 383,0 383,0 566,0 566,0 721,5 721,5 853,1 853,1 880,7 880,7

411,7 405,8 623,7 614,0 768,4 756,1 908,5 893,7 938,4 923,1 230,3 193,2 300,4 248,1 246,7 217,7 320,2 280,0 476,8 409,3 263,2 240,0 340,0 307,9 503,4 451,0 640,5 570,3 292,1 277,2 379,5 359,1 556,4 523,9 706,9 663,8 834,9 783,1 860,9 807,1

388,1 379,1 584,3 569,1 719,3 700,2 849,8 826,7 877,2 853,3 217,2 156,4 282,3 196,2 233,0 186,1 301,3 235,6 445,9 334,6 248,8 211,8 320,4 268,8 471,6 386,2 598,5 483,6 276,7 253,6 358,3 326,3 522,6 471,2 662,7 594,0 782,1 699,4 805,9 719,8

363,7 350,9 543,0 521,3 667,7 640,4 788,1 754,9 812,8 778,5 203,7 119,6 263,4 146,5 218,9 151,8 281,7 188,4 413,4 258,6 234 180,7 300,0 225,7 438,3 315,9 554,3 390,5 260,9 227,9 336,4 290,4 487,5 413,1 616,5 517,0 726,8 607,0 748,2 623,4

337,9 320,7 498,7 469,7 612,5 575,8 721,8 677,4 743,6 697,7 189,4 89,76 243,2 108,3 203,9 119,9 260,8 146,4 378,4 195,6 218,4 149,3 278,4 183,3 402,6 250,2 506,9 305,8 244,3 200,3 313,2 252,0 450,0 351,9 567,0 436,6 667,5 511,0 686,2 523,5

310,3 288,6 451,7 415,5 553,8 508,3 651,5 596,5 670,2 613,5 174,2 68,23 221,7 81,58 188,0 94,2 238,5 113,6 341,4 149,4 201,7 121,1 255,3 146,7 364,6 196,8 456,5 238,6 226,6 172,3 288,5 214,0 410,0 293,5 514,3 361,2 604,5 421,5 620,4 430,9

axis

140 120

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0

0,5

1,0

1,5

2

2,5

3

281,5 256,0 403,4 362,1 493,6 441,9 579,6 517,4 595,4 531,4 158,3 53,07 199,4 63,11 171,3 74,67 215,4 89,48 303,5 116,5 184,3 98,2 231,3 117,9 325,6 156,2 405,3 188,5 208,0 146,2 262,7 179,6 368,7 242,7 460,2 296,9 539,9 345,7 553,1 352,9 Nb.Rd (kN) Lc (m) 3,5

252,6 224,7 356,2 312,6 435,2 380,8 510,1 444,9 523,2 456,4 142,2 42,25 177,4 50,07 154,4 60,18 192,3 71,80 267,0 92,90 166,6 80,29 207,2 95,80 287,5 126,0 355,8 151,6 189,0 123,6 236,5 150,4 327,8 201,2 407,1 245,1 476,8 284,9 487,6 290,5

224,8 196,0 312,5 269,2 381,2 327,3 446,1 381,9 457,0 391,4 126,8 34,34 156,7 40,61 138,0 49,34 170,4 58,68 233,4 75,61 149,3 66,45 184,1 78,98 252,3 103,3 310,6 124,1 170,3 104,6 211,2 126,6 289,3 168,1 357,7 204,1 418,3 237,1 427,1 241,6

199,2 170,8 273,6 232,2 333,3 282,0 389,5 328,6 398,6 336,6 112,5 28,43 137,9 33,57 122,7 41,08 150,4 48,76 203,7 62,64 133,0 55,71 162,9 66,03 220,8 86,08 270,8 103,2 152,5 89,18 187,7 107,4 254,5 141,9 313,5 171,9 366,1 199,5 373,4 203,2

156,2 130,8 210,8 175,4 256,4 212,7 299,1 247,5 305,7 253,3 88,34

123,7 102,2 165,3 135,9 200,8 164,7 234,1 191,5 239,0 195,8 70,08

99,68 81,59 132,3 108,0 160,6 130,7 187,0 151,9 190,9 155,3 56,50

81,68 66,45 107,9 87,62 130,9 106,1 152,5 123,2 155,5 125,9 46,32

107,0

84,31

67,65

55,30

45,95 150

50 3

96,74 29,67 117,2 35,11 156,3 44,93 105,2 40,58 127,5 47,92 170,2 62,19 207,6 74,41 121,5 66,32 147,8 79,44 197,5 104,2 242,0 125,9 282,2 146,1 287,2 148,6

76,93

62,11

50,98

42,48 150

60 2,5

92,56

74,40

60,87

50,63 150

60 3

122,3

97,70

79,62

66,03 150

60 4

83,88 30,78 100,9 36,26 133,4 46,93 162,2 56,08 97,29 50,92 117,5 60,79 155,5 79,43 190,0 95,8 221,2 111,1 225,0 113,0

67,82

55,71

46,46 150

70 2,5

81,24

66,54

55,38 150

70 3

87,11

72,29 150

70 4

106,8

68,00 55,09 89,55 72,46 108,6 87,69 126,4 101,8 129,0 104,0 38,58

140 120 3 140 120 4 140 120 5 140 120 6 140 120 6,3 150

50 2,5

129,5

105,5

87,49 150

70 5

78,90 40,20 94,80 47,89 124,7 62,42 152,1 75,25 177,0 87,18 179,9 88,66

64,95 32,50 77,76 38,66 101,9 50,29 124,2 60,59 144,4 70,18 146,7 71,35

54,24 150

90 2,5

64,80 150

90 3

84,71 150

90 4

103,1

150

90 5

119,8

150

90 6

121,7

150

90 6,3

h b t mm mm mm 4

4,5

5

6

7

8

9

10

Appendix 9.2

3


271

140 120

1)

b

h

t y

h b t Lc Nb.Rd axis

272

0 402,4 402,4 591,8 591,8 753,8 753,8 891,8 891,8 921,4 921,4 1025 1025 1137 1137 304,3 304,3 413,5 413,5 617,6 617,6 786,0 786,0 930,5 930,5 962,0 962,0 218,7 218,7 290,0 290,0 309,4 309,4

0,5 402,4 402,4 591,8 591,8 753,8 753,8 891,8 891,8 921,4 921,4 1025 1025 1137 1137 304,3 304,3 413,5 413,5 617,6 617,6 786,0 786,0 930,5 930,5 962,0 962,0 218,7 206 290,0 270,0 309,4 298,3

1,0 399,2 383,3 582,8 557,5 740,0 706,5 874,6 834,4 902,6 860,9 1002 955,6 1112 1059 304,3 296,7 411,2 399,3 609,2 590,1 773,1 747,9 914,2 884,1 944,3 913,0 217,9 168,0 286,9 215,3 306,9 253,6


1,5 377,1 352,4 548,1 508,4 694,6 641,8 820,3 757,0 846,0 780,2 939,0 864,7 1040 957,0 289,6 277,3 389,0 370,8 573,5 543,9 726,4 687,2 858,4 811,5 886,2 837,4 206,3 125,5 270,6 155,4 289,9 202,5

2 354,4 319,2 512,0 454,9 647,3 571,1 763,7 672,1 786,8 691,7 872,4 765,0 965,6 844,4 274,2 256,9 366,2 340,4 536,5 494,1 677,9 621,6 800,3 732,9 825,7 755,4 194,5 89,09 253,8 107,7 272,4 152,7

2,5 330,4 283,2 473,6 397,4 596,7 495,4 703,1 581,6 723,4 597,3 801,1 658,9 885,4 725,3 258,0 235,0 342,2 307,6 497,0 440,3 626,1 550,8 738,3 648,1 760,9 667,0 182,1 63,95 235,9 76,32 253,9 113,6

Nb.Rd (kN) Lc (m) 3 3,5 304,8 278,0 246,3 211,1 432,7 390,1 339,9 286,9 542,8 487,2 420,5 352,8 638,6 572,1 492,4 412,2 656,0 586,7 504,7 421,9 725,3 647,5 555,3 463,1 800,3 713,1 609,7 507,4 240,8 222,8 211,7 187,9 316,6 289,7 273,2 238,9 455,1 411,4 384,7 331,2 570,9 513,8 478,2 409,1 672,2 603,9 561,4 479,3 692,0 620,8 576,8 491,8 168,8 154,9 47,42 36,35 216,9 197,0 56,21 42,92 234,2 213,6 85,87 66,57

h b t mm mm mm 4 250,8 179,7 347,9 241,4 432,3 295,5 506,8 344,7 518,9 352,3 571,6 386,2 628,4 422,4 204,1 165,1 262,4 206,9 367,7 283,0 457,2 347,9 536,5 407,0 550,8 417,0 140,7 28,67 177,0 33,76 192,7 52,89

4,5 224,4 153,1 307,8 203,8 380,9 248,7 445,9 289,7 455,8 295,9 501,3 324,0 550,3 354,0 185,4 144,1 235,6 178,6 326,1 241,7 403,8 296,1 473,1 345,9 485,1 354,1 126,6

5 199,8 131,0 271,4 173,3 334,6 210,9 391,2 245,6 399,4 250,7 438,7 274,3 481,0 299,4 167,3 125,7 210,3 154,4 288,1 207,2 355,5 253,2 415,9 295,5 426,0 302,3 113,4

6 157,7 97,90 211,3 128,5 259,2 156,0 302,4 181,4 308,2 185,0 338,0 202,2 369,9 220,6 135,0 96,4 166,8 116,9 224,9 155,4 276,1 189,1 322,4 220,5 329,8 225,4 90,25

7 125,5 75,43 166,7 98,50 203,8 119,3 237,6 138,7 241,9 141,4 265,0 154,5 289,6 168,4 109,0 75,31 133,2 90,75 177,7 119,8 217,5 145,6 253,7 169,6 259,3 173,3 72,23

8 101,4 59,68 133,9 77,72 163,4 94,02 190,3 109,2 193,6 111,3 212,0 121,6 231,6 132,5 88,85 60,14 107,8 72,14 142,9 94,90 174,6 115,1 203,5 134,1 207,9 137,0 58,56

9 83,22 48,31 109,5 62,78 133,5 75,89 155,4 88,15 158,0 89,83 172,9 98,1 188,8 106,8 73,41 49,00 88,61 58,60 117,0 76,87 142,7 93,19 166,3 108,5 169,8 110,8 48,19

10 69,38 39,87 91,04 51,74 110,9 62,50 129,0 72,58 131,2

157,7

139,8

109,8

87,06

70,17

57,52

47,90 160

40 3

172,3 42,93

153,4 35,51

121,0

96,30

77,79

63,86

53,23 160

50 3

150 100 3 150 100 4 150 100 5 150 100 6 150 100 6,3

143,5

150 100 7,1

156,6

150 100 8

61,46 40,62 73,93 48,48 97,31 63,47 118,6 76,90 138,1 89,51 141,0 91,39 40,24

150 110 2,5 150 110 3 150 110 4 150 110 5 150 110 6 150 110 6,3 160

40 2,5


r0 z 1) h b t mm mm mm x x 150 100 3 y-y z-z x 150 100 4 y-y z-z x 150 100 5 y-y z-z x 150 100 6 y-y z-z x 150 100 6,3 y-y z-z 150 100 7,1 y-y z-z x 150 100 8 y-y z-z 150 110 2,5 y-y z-z 150 110 3 y-y z-z 150 110 4 y-y z-z 150 110 5 y-y z-z 150 110 6 y-y z-z 150 110 6,3 y-y z-z 160 40 2,5 y-y z-z 160 40 3 y-y z-z 160 50 3 y-y z-z


Appendix 9.2


Table 9.2.2

160

60

4

160

70

3

160

70

4

160

70

5

160

80

3

x 160

80

4

x 160

80

5

x 160

80

6

160

80

6,3

160

90

3

160

90

4

160

90

5

160

90

6

160

90

6,3

x 160

90

7,1 3

160 100

4

160 100

5

160 100

6

160 100

6,3

1) h

b t mm mm mm

x

328,7 324,1 498,1 485,7 348,1 348,1 523,9 520,0 689,2 680,2 367,5 367,5 549,7 549,7 721,5 721,5 853,1 852,7 880,7 879,8 386,8 386,8 575,5 575,5 753,8 753,8 891,8 891,8 921,4 921,4 1025 1025 406,2 406,2 601,3 601,3 786,0 786,0 930,5 930,5 962,0 962,0

326,8 285,1 490,5 420,5 346,7 313,2 517,2 462,0 677,0 599,5 366,6 339,2 543,8 499,5 710,3 648,7 838,8 764,8 865,0 788,4 386,4 364,2 570,4 534,6 743,8 694,6 878,9 819,6 907,1 845,8 1008 938,4 406,2 388,3 596,9 568,4 777,0 738,0 918,9 871,8 949,1 900,1

309,1 241,5 461,3 346,6 328,2 274,8 487,2 397,7 635,6 509,3 347,4 304,4 512,9 442,2 667,9 568,9 788,1 668,9 812,2 688,6 366,5 332,0 538,6 482,2 700,4 622,4 826,9 732,7 852,9 755,4 946,8 836,5 385,5 358,0 564,1 519,7 732,4 671,4 865,7 791,7 893,6 816,6

290,9 194,8 431,0 270,3 309,3 232,5 456,1 327,8 592,5 412,4 327,8 266,1 481,0 378,9 623,9 480,9 735,3 563,3 757,1 578,8 346,2 296,9 505,7 424,6 655,4 542,8 773,0 636,8 796,7 655,7 883,6 724,0 364,5 325,4 530,4 466,7 686,4 598,6 810,6 704,1 836,0 725,1

271,8 152,4 398,7 205,9 289,4 190,2 423,1 261,4 546,5 323,6 307,3 225,8 447,1 314,3 576,9 393,2 678,9 458,8 698,3 470,5 324,8 259,2 471,0 363,5 607,5 459,5 715,6 536,9 736,6 551,9 816,1 607,4 342,4 290,1 494,7 409,7 637,4 520,5 752,0 610,4 774,7 627,4

251,4 118,8 364,2 157,8 268,3 153,2 387,9 206,5 497,4 252,9 285,4 187,6 411,0 256,0 526,8 316,5 618,9 368,2 635,6 377,1 302,2 221,5 434,0 304,6 556,5 380,9 654,4 443,5 672,7 455,3 744,2 499,6 319,0 253,7 456,7 352,1 585,3 443,0 689,5 517,8 709,5 531,2

0

0,5

1,0

1,5

2

2,5

3

230,0 93,84 328,5 123,4 246,1 123,6 351,2 164,5 446,7 200,0 262,4 154,8 373,3 208,1 474,9 255,2 556,8 296,3 570,8 303,1 278,4 186,9 395,2 252,9 503,3 313,6 590,8 364,2 606,4 373,4 669,7 409,0 294,3 218,5 416,9 298,5 530,9 372,4 624,5 434,1 641,6 444,7 Nb.Rd (kN) Lc (m) 3,5

208,2 75,43 292,9 98,5 223,3 100,7 314,4 132,9 396,7 160,9 238,8 128,0 335,3 170,5 423,3 207,9 495,3 241,0 506,9 246,4 253,9 157,1 356,0 210,2 450,3 259,2 527,5 300,3 540,5 307,8 595,9 336,6 268,9 186,8 376,5 252,1 476,2 312,4 559,3 363,4 573,7 371,8

186,8 61,72 259,2 80,28 201,0 83,18 279,3 109,1 349,8 131,7 215,5 106,8 298,7 141,3 374,4 171,7 437,3 198,8 446,9 203,2 229,6 132,6 318,1 176,0 399,6 216,0 467,2 250,1 478,1 256,1 526,3 279,8 243,6 159,6 337,2 213,4 423,7 263,2 496,9 305,8 509,0 312,6

166,7 51,33 228,5 66,56 179,8 69,63 247,0 90,99 307,4 109,6 193,2 90,07 264,9 118,5 330,0 143,7 384,8 166,3 392,8 169,9 206,3 112,8 282,7 148,7 353,1 182,0 412,3 210,5 421,3 215,5 463,3 235,3 219,2 136,9 300,3 181,7 375,3 223,5 439,6 259,3 449,7 264,9

132,2 37,01 177,9 47,79 143,1 50,61 193,2 65,81 238,2 79,05 154,3 66,08 208,0 86,43 256,7 104,4 298,8 120,8 304,4 123,3 165,3 83,61 222,7 109,4 275,9 133,5 321,4 154,2 327,8 157,8 359,8 172,1 176,1 102,7 237,3 135,1 294,2 165,4 343,9 191,7 351,2 195,7

105,5 140,4 114,4 38,34 152,8 49,70 187,4 59,6 123,8 50,33 164,9 65,59 202,5 79,08 235,3 91,43 239,5 93,35 132,8 64,07 177,0 83,48 218,1 101,6 253,7 117,3 258,5 120,0 283,5 130,8 141,8 79,22 189,0 103,7 233,0 126,6 272,1 146,6 277,6 149,6

85,34 112,8 92,75

70,14

58,52 160

60 3

92,23

76,68 160

60 4

76,31

63,71 160

70 3

83,81 160

70 4

123,0

100,7

150,3

122,8

100,5 39,54 132,9 51,4 162,6 61,89 188,8 71,53 192,1 73,02 107,9 50,53 142,9 65,65 175,4 79,79 203,9 92,06 207,6 94,17 227,5 102,6 115,4 62,74 152,8 81,85 187,7 99,80 219,0 115,5 223,2 117,8

82,76 109,0

102,0

160

70 5

69,15 160

80 3

90,75 160

80 4

133,0

110,6

160

80 5

154,3

128,3

160

80 6

156,9

130,3

160

80 6,3

88,99 40,82 117,2 52,92 143,6 64,26 166,9 74,11 169,8 75,81 186,0 95,20 50,83 125,5 66,16 153,8 80,58 179,4 93,24 182,7 95,10

74,41 160

90 3

97,69 160

90 4

119,5

160

90 5

138,8

160

90 6

141,2

160

90 6,3

154,6

160

90 7,1

79,64 41,97 104,6 54,54 128,0 66,37 149,3 76,78 152,0 78,31

160 100 3 160 100 4 160 100 5 160 100 6 160 100 6,3 h b t mm mm mm

4

4,5

5

6

7

8

9

10

Appendix 9.2

160 100

y-y 328,7 z-z 328,7 y-y 498,1 z-z 498,1 y-y 348,1 z-z 348,1 y-y 523,9 z-z 523,9 y-y 689,2 z-z 689,2 y-y 367,5 z-z 367,5 y-y 549,7 z-z 549,7 y-y 721,5 z-z 721,5 y-y 853,1 z-z 853,1 y-y 880,7 z-z 880,7 y-y 386,8 z-z 386,8 y-y 575,5 z-z 575,5 y-y 753,8 z-z 753,8 y-y 891,8 z-z 891,8 y-y 921,4 z-z 921,4 y-y 1025 z-z 1025 y-y 406,2 z-z 406,2 y-y 601,3 z-z 601,3 y-y 786,0 z-z 786,0 y-y 930,5 z-z 930,5 y-y 962,0 z-z 962,0 axis

3


60

273

160

b

1)

z

h b t Lc Nb.Rd

t h

y r0


x

274

160 120

4

160 120

5

160 120

6

160 120

6,3

160 120

7,1

160 120

8

160 120

8,8

160 120 10 180 100

4

x 180 100

5

180 100 x 180 100

5,6 6

180 100

6,3

x 180 100

7,1

x 180 100

8

axis

1) h

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0 653,0 653,0 850,6 850,6 1008 1008 1043 1043 1162 1162 1292 1292 1405 1405 1567 1567 616,9 616,9 850,6 850,6 945,5 945,5 1008 1008 1043 1043 1162 1162 1292 1292

0,5 653,0 653,0 850,6 850,6 1008 1008 1043 1043 1162 1162 1292 1292 1405 1405 1567 1567 616,9 616,9 850,6 850,6 945,5 945,5 1008 1008 1043 1043 1162 1162 1292 1292

1,0 649,9 632,9 843,4 820,2 998,5 970,7 1033 1004 1149 1117 1277 1240 1387 1346 1545 1499 616,9 586,7 850,4 800,3 944,9 888,5 1007 946,8 1041 978,8 1159 1088 1288 1208

1,5 615,2 589,1 796,5 760,7 942,4 899,7 974,3 929,8 1083 1033 1203 1146 1306 1243 1453 1383 591,3 538,9 807,1 729,2 896,5 808,8 955,2 861,6 987,5 889,8 1098 987,9 1220 1095

2 579,5 542,7 748,1 697,4 884,6 823,8 914,0 850,7 1016 944,4 1127 1046 1222 1133 1359 1257 561,4 487,3 762,9 651,7 847,1 721,8 902,4 768,4 932,3 792,5 1036 878,1 1150 971,4

2,5 542 492,7 697,0 629,0 823,3 741,9 850,0 765,2 943,6 848,1 1046 937,0 1134 1013 1258 1122 530,4 431,5 716,6 568,4 795,3 628,4 846,9 668,7 874,4 688,3 971,2 760,8 1077 839,4

3 502,1 439,9 642,5 557,3 758,0 656,0 781,9 675,7 866,8 747,5 959,5 824,1 1039 889,4 1150 982,0 497,7 374,3 667,4 485,2 740,4 535,5 788,1 569,4 812,9 585,0 902,0 645,0 998,6 709,8

Nb.Rd (kN) Lc (m) 3,5 460,1 386,9 585,5 486,2 689,7 571,4 710,6 587,7 786,7 649,0 869,5 713,8 940,3 768,9 1038 846,2 463,1 320,1 615,7 409,0 682,5 450,6 726,1 478,9 748,1 491,3 829,2 540,5 916,6 593,6

h b t mm mm mm 4 417,3 336,8 527,7 420,3 620,8 493,1 638,8 506,7 706,2 558,6 779,2 613,1 841,6 659,4 927,1 724,0 427,2 272,0 562,4 343,8 622,9 378,3 662,4 401,9 681,6 411,8 754,5 452,3 832,7 495,9

4,5 375,2 292,0 471,7 362,2 554,1 424,4 569,6 435,7 628,7 479,7 692,7 525,7 747,3 564,7 821,3 618,9 390,9 231,4 509,3 290,1 563,7 318,9 599,1 338,7 615,7 346,7 680,6 380,5 750,0 416,7

5 335,4 253,1 419,5 312,7 492,1 366,0 505,4 375,5 557,1 413,0 613,1 452,1 660,8 485,2 724,7 531,0 355,2 197,8 458,3 246,5 506,9 270,9 538,5 287,6 552,7 294,3 610,3 322,6 671,5 353,0

6 266,6 192,6 330,7 236,5 387,3 276,5 397,1 283,3 437,0 311,3 480,0 340,2 516,5 364,7 564,9 398,4 289,8 147,7 368,0 182,7 406,5 200,6 431,6 212,9 442,1 217,7 487,3 238,4 535,0 260,6

7 213,1 149,8 263,0 183,3 307,6 214,2 315,1 219,4 346,4 240,9 380,0 263,0 408,6 281,7 446,0 307,4 235,9 113,7 296,1 140,0 326,9 153,6 346,9 163,0 355,0 166,6 390,7 182,4 428,3 199,2

8 172,6 119,3 212,3 145,7 248,2 170,1 254,1 174,2 279,1 191,1 305,9 208,6 328,8 223,3 358,5 243,5 193,4 89,90 241,0 110,4 265,9 121,1 282,0 128,5 288,3 131,3 317,2 143,6 347,3 156,9

9 142,0 97,00 174,3 118,3 203,6 138,0 208,3 141,3 228,7 155,0 250,6 169,1 269,2 181,0 293,3 197,3 160,3 72,75 198,8 89,15 219,3 97,78 232,5 103,8 237,6 106,0 261,2 115,9 285,9 126,6

10 118,5 80,30 145,2 97,80 169,6 114,1 173,5 116,8 190,4 128,1 208,6 139,7 224,0 149,5 243,9 163,0 134,6 60,03 166,3 73,45 183,4 80,56 194,4 85,48 198,6 87,29 218,2 95,47 238,7

160 120 4 160 120 5 160 120 6 160 120 6,3 160 120 7,1 160 120 8 160 120 8,8 160 120 10 180 100 4 180 100 5 180 100 5,6 180 100 6 180 100 6,3 180 100 7,1 180 100 8


b t mm mm mm


Appendix 9.2


Table 9.2.2

180 120

5

180 120

6

180 120

6,3

180 120

7,1

180 120

8

180 120

8,8

180 120 10 80

4

200

80

5

x 200

80

6

200

80

6,3

275

200

x 200 100

5

x 200 100

6

200 100

6,3

200 100

7,1

x 200 100 1) h

8

b t mm mm mm

x


668,6 668,6 915,1 915,1 1085 1085 1125 1125 1254 1254 1396 1396 1518 1518 1696 1696 577,6 577,6 794,4 794,4 1008 1008 1043 1043 858,9 858,9 1085 1085 1125 1125 1254 1254 1396 1396

668,6 668,6 915,1 915,1 1085 1085 1125 1125 1254 1254 1396 1396 1518 1518 1696 1696 577,6 577,6 794,4 794,4 1008 1008 1043 1043 858,9 858,9 1085 1085 1125 1125 1254 1254 1396 1396

668,6 651,3 915,1 884,2 1085 1047 1125 1084 1254 1207 1395 1342 1516 1459 1692 1627 577,6 533,3 794,4 723,6 1008 908,2 1043 938,8 858,9 814,1 1085 1021 1125 1057 1254 1177 1396 1307

642,8 608,4 872,1 821,3 1033 972,2 1069 1006 1191 1119 1323 1243 1438 1349 1604 1503 560,9 478,6 764,7 641,9 964,2 797,9 996,8 823,7 830,4 745,9 1044 930,5 1080 962,8 1203 1070 1337 1187

611,2 563,1 826,0 754,4 978,0 892,0 1012 922,1 1126 1025 1251 1137 1359 1233 1514 1371 535,7 418,3 727,6 551,8 914,8 676,3 945,1 696,9 791,5 671,9 992,4 831,5 1026 859,6 1143 953,5 1269 1056

578,4 514,4 777,8 682,2 920,3 805,4 951,7 831,7 1058 922,8 1174 1022 1275 1107 1419 1227 509,7 355,0 689,1 459,3 863,4 554,5 891,3 570,4 751,4 592,2 939,3 725,3 970,9 748,9 1080 828,6 1199 915,4

543,9 462,8 726,8 606,2 859,2 714,5 887,9 736,8 986,4 816,0 1093 902,3 1186 975,0 1318 1078 482,5 295,1 648,6 375,3 809,0 447,4 834,4 459,5 709,3 511,1 883,3 619,1 912,2 638,5 1014 704,7 1125 776,5

axis

4

0

0,5

1,0

1,5

2

2,5

3

507,5 469,6 410,4 359,9 673,0 617,4 530,6 459,9 794,8 728,2 624,2 540,2 820,7 751,2 642,9 555,7 910,7 832,7 710,7 613,4 1008 920,4 784,4 675,9 1093 996,6 846,0 727,8 1212 1103 932,9 800,7 453,9 424,0 243,4 201,4 605,9 561,4 305,7 250,7 751,7 692,2 361,3 294,6 774,3 712,1 370,7 302,1 664,9 618,6 435,0 368,4 824,3 762,8 521,9 438,7 850,4 786,0 537,6 451,5 944,4 872,0 592,2 496,5 1046 965,0 651,1 545,1 Nb.Rd (kN) Lc (m) 3,5 4

431,1 313,9 561,6 397,2 661,6 466,0 681,8 478,9 754,8 528,0 833,1 581,1 901,1 624,9 995,2 686,2 393,2 168,0 516,0 208,0 632,0 243,4 649,4 249,4 571,0 312,5 700,2 370,1 720,6 380,7 798,6 418,2 882,6 458,6

393,0 273,5 507,4 343,5 597,1 402,5 614,8 413,4 679,9 455,3 749,3 500,6 809,7 537,9 892,6 589,8 362,2 141,7 471,1 174,6 573,3 203,8 588,2 208,8 523,6 266,6 638,4 314,5 656,2 323,4 726,4 355,0 801,8 388,9

322,5 209,4 410,2 260,3 481,8 304,7 495,3 312,6 546,7 343,9 601,2 377,6 648,7 405,3 712,9 443,6 302,9 104,0 387,5 127,4 466,4 148,2 477,5 151,8 434,1 198,6 523,9 233,1 537,4 239,6 593,8 262,7 653,9 287,5

263,5 163,5 331,7 202,1 389,0 236,4 399,6 242,4 440,5 266,4 483,6 292,4 521,3 313,5 571,8 342,8 251,0 79,19 317,2 96,80 378,7 112,3 387,1 115,0 357,4 152,6 428,2 178,6 438,4 183,5 483,9 201,1 532,0 219,9

216,6 130,5 270,7 160,7 317,2 187,9 325,6 192,6 358,6 211,6 393,4 232,1 423,7 248,7 464,2 271,8 208,4 62,20 261,0 75,85 310,0 87,90 316,6 89,99 295,4 120,5 352,0 140,8 360,0 144,7 397,0 158,5 436,0 173,3

179,9 106,3 223,8 130,5 262,1 152,6 268,9 156,3 296,0 171,7 324,5 188,3 349,3 201,8 382,3 220,3 174,3

151,2 88,13 187,4 108,0 219,5 126,2 225,1 129,3 247,7 142,0 271,4 155,6 292,1 166,7 319,5 182,1 147,2

217,0

180 120 4 180 120 5 180 120 6 180 120 6,3 180 120 7,1 180 120 8 180 120 8,8 180 120 10 200

80 4

182,5

200

80 5

256,8

215,4

200

80 6

262,0

219,7

200

80 6,3

246,2 97,50 292,4 113,7 298,9 116,8 329,3 127,9 361,5 139,8

207,5 80,40 245,8 93,7 251,1 96,20 276,6 105,4 303,4 115,2

200 100 5 200 100 6 200 100 6,3 200 100 7,1 200 100 8 h b t mm mm mm

4,5

5

6

7

8

9


180 120

10

Appendix 9.2

1)

b

h

t y

h b t Lc Nb.Rd

axis

276

0 923,4 923,4 1163 1163 1206 1206 1345 1345 1499 1499 1632 1632 1826 1826 943,4 943,4 1240 1240 1287 1287 1437 1437 1602 1602 1745 1745 1955 1955

0,5 923,4 923,4 1163 1163 1206 1206 1345 1345 1499 1499 1632 1632 1826 1826 943,4 943,4 1240 1240 1287 1287 1437 1437 1602 1602 1745 1745 1955 1955

1,0 923,4 897,3 1163 1124 1206 1165 1345 1298 1499 1444 1632 1571 1826 1754 943,4 920,5 1240 1201 1287 1245 1437 1389 1602 1546 1745 1683 1955 1881


1,5 895,9 836,7 1123 1045 1163 1082 1296 1204 1443 1339 1569 1455 1752 1623 927,8 860,6 1211 1117 1256 1157 1401 1290 1560 1435 1698 1561 1899 1743

2 855,3 772,6 1069 960,0 1107 993,4 1234 1105 1372 1227 1492 1332 1665 1482 890,5 797,6 1159 1028 1202 1064 1340 1185 1492 1317 1623 1431 1814 1595

2,5 813,4 703,6 1014 868,7 1050 898,0 1169 997,2 1300 1106 1412 1198 1575 1331 852,7 730,0 1106 932 1146 964 1277 1072 1421 1189 1546 1291 1727 1435

3 769,5 630,6 956,4 772,6 989,3 797,6 1101 884,2 1223 979,0 1328 1059 1479 1172 813,4 658,2 1051 830,3 1088 857,9 1212 953,0 1348 1055 1466 1143 1635 1268

Nb.Rd (kN) Lc (m) 3,5 4 723,4 675,2 556,9 486,6 895,4 831,8 676,8 587,1 925,5 859,0 697,8 604,6 1029 954,0 772,2 668,0 1142 1058 853,2 736,9 1239 1147 921,1 794,4 1378 1273 1017 875,3 772,4 729,5 585,0 514,3 992,6 931,7 728,9 633,6 1027 963,8 752,2 653,1 1144 1072 834,8 723,9 1271 1190 921,0 797,4 1381 1292 998,0 862,4 1539 1439 1103 952,0

h b t mm mm mm 4,5 625,5 423,1 766,7 507,4 790,9 522,0 877,6 576,1 972,3 634,7 1053 683,3 1167 751,5 684,9 449,4 868,8 548,4 898,0 564,7 998,1 625,4 1107 687,8 1201 743,2 1335 818,5

5 575,6 367,7 701,8 438,9 723,3 451,3 801,8 497,5 887,4 547,6 959,7 589,0 1062 646,8 639,2 392,1 804,9 475,0 831,2 488,8 923,2 540,9 1023 594,2 1108 641,6 1231 705,6

6 480,4 280,6 580,2 332,9 596,9 341,9 660,4 376,5 729,6 413,9 787,7 444,6 869,2 487,3 548,3 300,8 680,4 360,8 701,5 370,9 777,8 410,1 860,0 449,7 930,4 485,2 1030 532,6

7 397,6 218,7 476,7 258,5 489,7 265,4 541,2 292,0 597,1 320,7 643,7 344,3 708,8 377,0 464,1 235,3 568,8 280,5 585,7 288,2 648,5 318,4 715,9 348,9 773,4 376,1 854,6 412,5

8 329,7 174,3 393,3 205,6 403,7 211,0 445,7 232,0 491,3 254,8 529,2 273,4 581,9 299,1 391,3 187,9 475,1 223,2 488,8 229,2 540,7 253,2 596,2 277,3 643,5 298,8 709,9 327,5

9 275,6 141,9 327,5 167,0 336,0 171,4 370,7 188,4 408,3 206,8 439,6 221,8 482,9 242,6 330,9 153,1 399,1 181,4 410,3 186,3 453,6 205,7 499,7 225,2 539,0 242,6 594,0 265,7

10 232,6 117,5 275,8 138,2 282,8 141,8 311,9 155,8 343,4 171,0 369,5 183,4 405,6 200,5 281,7 127,0 338,1 150,2 347,4 154,2 383,9 170,2 422,7 186,3 455,7 200,7 501,8 219,7

200 120 5 200 120 6 200 120 6,3 200 120 7,1 200 120 8 200 120 8,8 200 120 10 220 120 5 220 120 6 220 120 6,3 220 120 7,1 220 120 8 220 120 8,8 220 120 10


r0 z 1) h b t mm mm mm x x 200 120 5 y-y z-z x 200 120 6 y-y z-z 200 120 6,3 y-y z-z 200 120 7,1 y-y z-z x 200 120 8 y-y z-z 200 120 8,8 y-y z-z x 200 120 10 y-y z-z 220 120 5 y-y z-z x 220 120 6 y-y z-z 220 120 6,3 y-y z-z 220 120 7,1 y-y z-z x 220 120 8 y-y z-z 220 120 8,8 y-y z-z x 220 120 10 y-y z-z


Appendix 9.2


Table 9.2.2

5

x 250 150

6

250 150

6,3

250 150

7,1

x 250 150

8,8

x 250 150 10 250 150 12 x 250 150 12,5 x 260 140

6

277

260 140

6,3

260 140

7,1

x 260 140 260 140

8 8,8

x 260 140 10 1) h

b t mm mm mm

x

1064 1064 1366 1366 1447 1447 1712 1712 1912 1912 2086 2086 2342 2342 2713 2713 2809 2809 1339 1339 1420 1420 1712 1712 1912 1912 2086 2086 2342 2342

1064 1064 1366 1366 1447 1447 1712 1712 1912 1912 2086 2086 2342 2342 2713 2713 2809 2809 1339 1339 1420 1420 1712 1712 1912 1912 2086 2086 2342 2342

1064 1064 1366 1364 1447 1444 1712 1701 1912 1898 2086 2070 2342 2321 2713 2683 2809 2776 1339 1330 1420 1409 1712 1690 1912 1885 2086 2056 2342 2305

1064 1015 1362 1294 1442 1368 1699 1609 1896 1794 2068 1956 2319 2192 2680 2529 2774 2617 1339 1257 1420 1331 1703 1591 1901 1775 2073 1934 2325 2167

1031 961,9 1315 1222 1391 1291 1637 1513 1826 1687 1991 1838 2232 2058 2577 2371 2667 2452 1295 1183 1371 1251 1643 1489 1833 1660 1998 1807 2241 2023

995,1 906,4 1267 1147 1339 1210 1574 1413 1755 1574 1913 1714 2144 1917 2473 2203 2558 2276 1250 1104 1323 1166 1581 1381 1764 1537 1923 1673 2155 1870

959 847,8 1218 1067 1287 1123 1509 1306 1683 1454 1833 1581 2054 1766 2365 2024 2446 2090 1204 1020 1273 1075 1518 1265 1693 1407 1845 1529 2067 1706

0

0,5

1,0

1,5

2

2,5

3

921,7 883,1 785,9 721,8 1168 1116 982,9 896,4 1232 1175 1032 939,4 1442 1371 1194 1080 1607 1528 1328 1200 1750 1663 1442 1302 1959 1861 1608 1449 2253 2136 1837 1649 2329 2207 1895 1699 1157 1108 932,4 843,0 1222 1169 980 884,2 1453 1385 1145 1024 1620 1543 1272 1136 1764 1680 1380 1232 1976 1880 1537 1369 Nb.Rd (kN) Lc (m) 3,5 4

842,9 657,4 1061 810,5 1116 847,4 1298 968,7 1446 1075 1573 1165 1759 1294 2014 1468 2080 1511 1056 755,7 1113 790,9 1315 909,0 1464 1008 1593 1091 1781 1210

801,2 594,7 1004 728,3 1055 759,8 1223 864,0 1361 957,6 1480 1037 1653 1150 1888 1301 1949 1338 1003 673,8 1056 703,7 1242 803,7 1381 889,8 1502 962,5 1679 1066

714,9 481,7 887,8 583,5 929,7 606,6 1069 684,2 1188 757,2 1291 818,6 1439 906,3 1635 1020 1686 1049 893,6 533,3 937,2 555,3 1092 628,1 1213 694,3 1317 749,9 1469 828,9

628,3 390,0 772,9 468,9 806,8 486,3 921,0 545,7 1022 603,3 1109 651,6 1234 720,3 1396 808,5 1438 830,5 783,9 425,2 819,4 441,8 945,6 496,7 1049 548,6 1138 592,0 1267 653,5

546,6 318,6 666,9 381,1 694,1 394,7 787,5 441,4 873,3 487,7 946,2 526,4 1052 581,4 1185 651,4 1219 668,8 680,8 343,8 709,6 356,9 812,3 399,6 899,9 441,1 975,0 475,7 1085 524,7

473,5 263,5 573,9 314,2 596,0 325,1 672,9 362,7 745,6 400,5 807,3 432,2 896,3 477,1 1007 533,9 1036 548,0 589,0 282,5 612,4 292,9 696,5 327,2 771,0 361,0 835,0 389,2 928,0 429,0

410,4 220,9 494,8 262,7 513,0 271,6 577,1 302,5 639,2 334,0 691,6 360,3 767,3 397,5 860,1 444,5 884,3 456,2 509,9 235,6 529,2 244,2 599,0 272,2 662,5 300,3 717,1 323,6 796,0 356,6

250 150 5 250 150 6 250 150 6,3 250 150 7,1 250 150 8 250 150 8,8 250 150 10 250 150 12 250 150 12,5 260 140 6 260 140 6,3 260 140 7,1 260 140 8 260 140 8,8 260 140 10 h b t mm mm mm

4,5

5

6

7

8

9

10


250 150

8

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z axis

x 250 150

Appendix 9.2

b

1)

z

h b t Lc Nb.Rd

t h

y r0


x x 260 180

6

260 180

6,3

260 180

7,1

x 260 180 260 180

8 8,8

278

x 260 180 10 260 180 12 x 300 100

5

x 300 100

6

300 100

6,3

300 100

7,1

x 300 100

8

axis

1) h

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z

0 1494 1494 1583 1583 1895 1895 2118 2118 2313 2313 2600 2600 3023 3023 930,8 930,8 1222 1222 1302 1302 1549 1549 1833 1833

0,5 1494 1494 1583 1583 1895 1895 2118 2118 2313 2313 2600 2600 3023 3023 930,8 930,8 1222 1222 1302 1302 1549 1549 1833 1833

1,0 1494 1494 1583 1583 1895 1895 2118 2118 2313 2313 2600 2600 3023 3023 930,8 901,7 1222 1174 1302 1248 1549 1475 1833 1735

1,5 1494 1452 1583 1537 1894 1830 2115 2044 2309 2231 2593 2504 3008 2903 930,8 839,0 1222 1086 1302 1153 1549 1356 1833 1588

2 1451 1388 1535 1468 1829 1744 2043 1946 2229 2124 2503 2383 2901 2760 921,0 772,6 1202 992 1278 1050 1513 1228 1782 1429

2,5 1403 1321 1483 1396 1764 1654 1969 1846 2149 2013 2412 2257 2793 2610 895,0 701,0 1165 890,8 1238 940,0 1464 1090 1722 1257

3 1353 1252 1430 1322 1697 1560 1895 1740 2066 1897 2318 2125 2682 2453 868,8 625,4 1129 784,9 1199 824,8 1415 947 1662 1082

Nb.Rd (kN) Lc (m) 3,5 4 1302 1250 1179 1103 1375 1318 1243 1161 1628 1557 1461 1358 1817 1737 1628 1511 1981 1893 1774 1646 2222 2122 1986 1841 2568 2448 2287 2114 842,1 814,8 549,7 478,2 1092 1053 681,1 585,9 1158 1116 713,1 611,6 1364 1312 811,8 691,1 1600 1535 919 777,5

h b t mm mm mm 4,5 1195 1024 1259 1077 1483 1252 1653 1392 1801 1516 2017 1692 2324 1939 786,6 414,3 1013 503,1 1073 523,8 1258 588,6 1468 658,8

5 1138 944,9 1198 991,5 1406 1147 1567 1274 1706 1386 1909 1546 2195 1766 757,5 359,0 972,3 433,0 1028 450,0 1202 503,6 1399 561,6

6 1020 792,2 1071 828,8 1247 948,7 1388 1052 1509 1143 1687 1272 1931 1447 696,5 272,9 886,2 326,3 934,1 338,3 1085 376,5 1254 417,9

7 900,8 658,0 943,3 686,7 1089 779,8 1211 863,5 1316 937,6 1468 1042 1673 1181 632,7 212,3 797,1 252,4 837,3 261,3 965,6 290,0 1108 320,9

8 787,2 547,0 822,2 569,9 942,2 643,7 1047 712,0 1136 772,7 1266 857,7 1438 969,7 568,5 169,0 708,9 200,3 742,1 207,1 849,9 229,4 968,6 253,5

9 684,4 458,0 713,3 476,5 812,7 536,2 902,1 592,8 978,3 643,0 1089 713,2 1233 805,0 506,6 137,4 625,8 162,4 653,1 167,9 743,4 185,7 842,4 204,9

10 594,8 387,1 618,9 402,4 701,9 451,5 778,7 499,0 844,0 541,2 939,0 599,9 1061 676,4 449,3 113,7 550,7 134,2 573,2 138,7 649,2 153,2 732,3 169,0

260 180 6 260 180 6,3 260 180 7,1 260 180 8 260 180 8,8 260 180 10 260 180 12 300 100 5 300 100 6 300 100 6,3 300 100 7,1 300 100 8


b t mm mm mm


Appendix 9.2


Table 9.2.2

6

300 150

6,3

300 150

7,1

300 150

8,8

300 150 10 6

300 200

6,3

300 200

7,1

x 300 200 300 200

8 8,8

x 300 200 10

279

300 200 12 x 300 200 12,5 400 100

6

400 100

6,3

400 100

7,1

1) h

b t mm mm mm

x

1416 1416 1506 1506 1778 1778 2370 2370 2665 2665 1609 1609 1709 1709 2007 2007 2350 2350 2654 2654 2987 2987 3487 3487 3616 3616 1283 1283 1373 1373 1652 1652

1416 1416 1506 1506 1778 1778 2370 2370 2665 2665 1609 1609 1709 1709 2007 2007 2350 2350 2654 2654 2987 2987 3487 3487 3616 3616 1283 1283 1373 1373 1652 1652

1416 1416 1506 1506 1778 1778 2370 2357 2665 2647 1609 1609 1709 1709 2007 2007 2350 2350 2654 2654 2987 2987 3487 3487 3616 3616 1283 1252 1373 1337 1652 1598

1416 1355 1506 1437 1778 1689 2370 2230 2665 2503 1609 1593 1709 1689 2007 1977 2350 2307 2654 2599 2987 2923 3487 3405 3616 3529 1283 1170 1373 1248 1652 1486

1400 1285 1486 1362 1748 1597 2315 2100 2600 2355 1597 1533 1693 1625 1982 1899 2313 2213 2606 2491 2931 2799 3415 3259 3540 3377 1283 1084 1373 1154 1652 1367

1360 1213 1443 1284 1696 1501 2241 1962 2516 2198 1553 1472 1646 1559 1925 1820 2244 2117 2527 2380 2841 2674 3309 3109 3429 3222 1280 992,0 1367 1054 1637 1238

1319 1137 1399 1201 1642 1399 2165 1816 2431 2032 1509 1409 1598 1492 1868 1738 2175 2018 2447 2264 2751 2543 3201 2953 3317 3059 1253 894,2 1338 947,0 1601 1103

0

0,5

1,0

1,5

2

2,5

3

1279 1237 1057 973,0 1355 1309 1114 1024 1588 1532 1292 1181 2088 2009 1662 1506 2343 2253 1857 1679 1464 1418 1344 1276 1550 1501 1421 1348 1809 1749 1652 1562 2104 2031 1914 1805 2365 2280 2144 2017 2658 2562 2406 2262 3090 2976 2789 2618 3202 3083 2889 2711 1226 1199 794,5 698,2 1309 1279 838,6 734,8 1564 1527 967,7 841 Nb.Rd (kN) Lc (m) 3,5 4

1193 888,8 1262 933,0 1475 1071 1926 1352 2159 1505 1371 1205 1450 1271 1687 1469 1956 1692 2193 1886 2463 2113 2858 2441 2961 2526 1172 609,9 1250 640,3 1490 727,2

1148 806,3 1214 844,1 1415 965,0 1840 1207 2062 1342 1322 1132 1397 1193 1622 1374 1878 1577 2103 1753 2360 1962 2736 2261 2833 2339 1144 532,1 1219 557,4 1452 629,6

1055 656,2 1112 684,3 1290 775,7 1662 958,0 1859 1063 1220 984 1286 1034 1488 1182 1715 1348 1913 1490 2145 1664 2480 1910 2567 1976 1087 408,1 1157 426,4 1372 478

956,8 533,0 1006 554,3 1160 625,0 1479 764,8 1652 847,2 1113 843,1 1171 883,5 1348 1004 1546 1137 1718 1252 1924 1396 2217 1597 2294 1650 1027 319,1 1091 332,8 1289 371,4

858,4 436,3 900,1 452,9 1032 508,9 1302 619,1 1452 685,2 1004 717,7 1054 750,5 1207 848,5 1378 956,9 1525 1049 1706 1169 1960 1334 2027 1378 965,0 254,9 1022 265,6 1201 295,5

763,9 361,5 798,9 374,7 910,8 420,0 1139 509,0 1269 562,9 898,5 611,2 941,3 638,1 1073 718,8 1219 807,8 1344 883,7 1502 983,6 1721 1120 1779 1157 899,6 207,7 950,9 216,2 1112 240,2

676,7 303,2 706,1 314,1 801,4 351,4 994,7 424,7 1107 469,5 799,8 523,0 836,3 545,3 949,0 612,7 1074 686,9 1181 750,0 1318 834,3 1508 948,9 1558 980,0 833,6 172,2 878,9 179,2 1022 198,8

300 150 6 300 150 6,3 300 150 7,1 300 150 8,8 300 150 10 300 200 6 300 200 6,3 300 200 7,1 300 200 8 300 200 8,8 300 200 10 300 200 12 300 200 12,5 400 100 6 400 100 6,3 400 100 7,1 h b t mm mm mm

4,5

5

6

7

8

9

10


x 300 200

y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z y-y z-z axis

300 150

Appendix 9.2

1)

b

h

t y

h b t Lc Nb.Rd

axis

280

0 1360 1360 1455 1455 1743 1743 2086 2086 2403 2403 1670 1670 1780 1780 2110 2110 2499 2499 2857 2857 3408 3408 4262 4262 4423 4423

0,5 1360 1360 1455 1455 1743 1743 2086 2086 2403 2403 1670 1670 1780 1780 2110 2110 2499 2499 2857 2857 3408 3408 4262 4262 4423 4423

1,0 1360 1354 1455 1445 1743 1725 2086 2054 2403 2356 1670 1670 1780 1780 2110 2110 2499 2499 2857 2857 3408 3408 4262 4262 4423 4423


1,5 1360 1282 1455 1367 1743 1627 2086 1931 2403 2210 1670 1670 1780 1779 2110 2102 2499 2481 2857 2828 3408 3360 4262 4179 4423 4335

2 1360 1208 1455 1286 1743 1525 2086 1803 2403 2057 1670 1616 1780 1719 2110 2028 2499 2391 2857 2722 3408 3229 4262 4007 4423 4156

2,5 1359 1130 1450 1201 1731 1418 2063 1667 2367 1893 1670 1560 1780 1658 2107 1954 2488 2299 2836 2614 3370 3096 4192 3831 4349 3973

3 1331 1047 1420 1110 1694 1303 2016 1522 2312 1720 1643 1502 1748 1596 2064 1877 2435 2205 2774 2503 3294 2958 4092 3648 4245 3783

Nb.Rd (kN) Lc (m) 3,5 4 1303 1275 959,5 870,4 1390 1359 1015 918,0 1656 1617 1184 1063 1969 1921 1372 1222 2256 2199 1540 1364 1610 1577 1443 1382 1712 1677 1532 1465 2021 1977 1798 1716 2381 2328 2108 2006 2711 2649 2388 2267 3217 3139 2814 2663 3991 3889 3457 3257 4139 4033 3584 3376

h b t mm mm mm 4,5 1247 782,8 1328 823,3 1579 947,0 1873 1081 2141 1200 1544 1318 1641 1396 1933 1630 2274 1900 2585 2142 3060 2507 3784 3050 3924 3160

5 1218 700,0 1297 734,4 1539 839,8 1823 952,5 2082 1053 1510 1252 1604 1324 1888 1541 2219 1790 2520 2012 2979 2346 3677 2838 3813 2939

6 1159 556,5 1232 581,6 1457 659,2 1720 740,9 1959 813,7 1441 1115 1529 1176 1795 1359 2104 1567 2385 1750 2812 2024 3455 2419 3581 2505

7 1096 445 1163 463,8 1371 522,8 1612 584,2 1829 639,2 1368 978,0 1451 1029 1698 1180 1985 1351 2243 1500 2635 1722 3220 2037 3337 2108

8 1031 360,5 1092 375,2 1281 421,4 1499 469,2 1694 512,1 1292 849,0 1368 891,3 1597 1016 1859 1156 2095 1278 2450 1457 2976 1711 3082 1770

9 963,0 296,6 1018 308,3 1189 345,4 1383 383,7 1556 418,1 1214 734,4 1283 769,4 1491 873,1 1729 988,6 1942 1088 2261 1236 2728 1443 2824 1492

10 894,3 247,6 943,0 257,2 1095 287,6 1268 319,0 1419 347,1 1133 635,7 1196 665,0 1384 751,8 1598 848,3 1789 931,5 2073 1054 2483 1226 2570 1267

400 120 6 400 120 6,3 400 120 7,1 400 120 8 400 120 8,8 400 200 6 400 200 6,3 400 200 7,1 400 200 8 400 200 8,8 400 200 10 400 200 12 400 200 12,5


r0 z 1) h b t mm mm mm x 400 120 6 y-y z-z 400 120 6,3 y-y z-z 400 120 7,1 y-y z-z 400 120 8 y-y z-z 400 120 8,8 y-y z-z x 400 200 6 y-y z-z 400 200 6,3 y-y z-z 400 200 7,1 y-y z-z x 400 200 8 y-y z-z 400 200 8,8 y-y z-z x 400 200 10 y-y z-z 400 200 12 y-y z-z x 400 200 12,5 y-y z-z


Appendix 9.2


Table 9.2.2


281

Appendix 9.2

d

1)

t

1) d

x x x x x x x

282

x x x x x x x x x

33,7 42,4 42,4 42,4 42,4 42,4 48,3 48,3 48,3 48,3 48,3 60,3 60,3 60,3 60,3 60,3 60,3 60,3 76,1 76,1 76,1 76,1 76,1 76,1 76,1 76,1

= recommended series = external diameter = wall thickness = buckling length = buckling resistance


t mm 2 2 2,5 2,6 2,9 3 2 2,5 2,6 2,9 3 2 2,5 2,6 2,9 3 3,2 4 2 2,5 2,6 2,9 3 3,2 4 5

0 64,28 81,92 101,1 104,9 116,1 119,8 93,89 116,1 120,5 133,5 137,8 118,2 146,5 152,1 168,8 174,3 185,3 228,3 150,3 186,6 193,8 215,2 222,3 236,5 292,4 360,4

0,5 51,07 70,98 87,25 90,53 100,0 103,2 84,3 103,9 107,8 119,3 123,2 111,1 137,6 142,7 158,3 163,4 173,6 213,5 146,5 181,6 188,7 209,5 216,4 230,1 284,0 349,5

1,0 28,85 48,39 58,89 61,12 67,21 69,31 62,29 76,32 79,15 87,34 90,13 90,33 111,7 115,7 128,0 132,2 140,2 171,8 126,7 156,9 162,9 180,8 186,8 198,4 244,3 299,9

1,5 15,74 29,46 35,58 36,93 40,47 41,72 40,87 49,73 51,56 56,72 58,52 67,08 82,77 85,47 94,39 97,47 103,1 125,7 104,3 128,8 133,7 148,2 153,1 162,4 199,1 243,4

2 9,597 18,69 22,50 23,36 25,56 26,35 26,80 32,50 33,69 37,01 38,18 47,43 58,42 60,21 66,39 68,55 72,36 87,90 81,25 100,0 103,8 114,9 118,7 125,7 153,4 186,7

2,5 6,413 12,72 15,29 15,87 17,36 17,89 18,50 22,41 23,22 25,50 26,30 33,97 41,81 43,04 47,41 48,96 51,63 62,59 61,82 75,89 78,79 87,11 89,98 95,22 115,8 140,4

3 9,166 11,01 11,43 12,50 12,88 13,44 16,27 16,86 18,50 19,09 25,16 30,95 31,85 35,07 36,21 38,16 46,22 47,36 58,07 60,28 66,61 68,80 72,76 88,31 106,9

Nb.Rd (kN) Lc (m) 3,5 6,907 8,293 8,611 9,413 9,701 10,18 12,31 12,76 14,00 14,44 19,27 23,70 24,38 26,83 27,71 29,20 35,34 37,01 45,33 47,06 51,98 53,69 56,76 68,80 83,17

d mm 4

7,965 9,631 9,980 10,95 11,29 15,19 18,68 19,21 21,14 21,83 23,00 27,83 29,55 36,18 37,56 41,47 42,84 45,27 54,84 66,23

4,5

12,27 15,08 15,51 17,07 17,62 18,56 22,45 24,07 29,46 30,58 33,76 34,88 36,85 44,61 53,86

5

10,11 12,43 12,78 14,06 14,51 15,29 18,49 19,95 24,41 25,34 27,98 28,90 30,53 36,95 44,60

6

14,33 17,52 18,19 20,08 20,74 21,91 26,50 31,96

7

8

9

t mm

10 33,7 42,4 42,4 42,4 42,4 42,4 48,3 48,3 48,3 48,3 48,3 60,3 60,3 60,3 60,3 60,3 60,3 60,3 76,1 76,1 76,1 76,1 76,1 76,1 76,1 76,1

2 2 2,5 2,6 2,9 3 2 2,5 2,6 2,9 3 2 2,5 2,6 2,9 3 3,2 4 2 2,5 2,6 2,9 3 3,2 4 5


mm

d t Lc Nb.Rd

Appendix 9.2

Buckling resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c.

Table 9.2.3

x x x x

x x x

x

283

x x

x x x x

176,2 219,0 227,5 252,9 261,3 278,1 344,3 425,3 504,3 527,6 202,0 251,2 261,0 290,2 299,9 319,3 395,8 489,7 581,6 608,7 214,9 267,4 277,8 309,0 319,4 340,0 421,8 522,2 620,5 649,6 227,7 283,4 294,5 327,5 338,5 360,5 447,3 554,1 658,8 689,8

175,0 217,5 225,8 250,9 259,3 275,8 341,1 420,9 498,5 521,3 202,0 251,2 261,0 290,2 299,9 319,3 395,8 489,7 581,6 608,7 214,9 267,4 277,8 309,0 319,4 340,0 421,8 522,2 620,5 649,6 227,7 283,4 294,5 327,5 338,5 360,5 447,3 554,1 658,8 689,8

155,6 193,3 200,6 222,8 230,2 244,8 302,3 372,4 440,4 460,3 184,2 228,8 237,8 264,2 273,0 290,5 359,5 443,9 526,1 550,3 198,6 246,8 256,5 285,1 294,5 313,5 388,3 479,9 569,0 595,3 212,7 264,4 274,7 305,5 315,7 336,0 416,4 515,1 611,2 639,7

134,2 166,6 172,7 191,7 198,1 210,4 259,2 318,4 375,5 392,1 163,6 203,0 211,0 234,3 242,1 257,4 318,1 392,1 463,9 485,0 178,3 221,5 230,1 255,7 264,0 281,1 347,6 429,0 507,7 530,9 192,7 239,4 248,7 276,4 285,7 304,0 376,2 464,8 550,7 576,2

110,9 137,4 142,4 157,8 163,1 173,0 212,4 260,0 305,5 318,6 140,9 174,5 181,4 201,2 208,0 221,0 272,3 334,9 395,3 412,8 156,0 193,5 201,1 223,3 230,4 245,3 302,7 372,9 440,0 459,8 170,7 211,9 220,1 244,6 252,8 268,8 332,1 409,6 484,2 506,3

88,59 109,7 113,5 125,7 129,9 137,7 168,5 205,6 240,8 250,9 117,5 145,3 151,0 167,4 173,0 183,7 225,7 276,9 325,9 340,1 132,6 164,2 170,6 189,3 195,2 207,8 255,8 314,4 369,8 386,1 147,4 182,6 189,7 210,6 217,7 231,3 285,3 351,1 413,8 432,4

70,1 86,77 89,69 99,27 102,6 108,6 132,6 161,5 188,7 196,5 96,18 118,7 123,4 136,7 141,3 149,8 183,8 224,9 264,2 275,4 110,3 136,4 141,7 157,2 161,9 172,4 211,8 259,7 304,6 317,8 124,5 154,0 160,0 177,5 183,5 194,8 239,8 294,5 346,2 361,5

t mm 0

0,5

1,0

1,5

2

2,5

3

55,88 69,13 71,43 79,02 81,64 86,44 105,4 128,1 149,5 155,6 78,43 96,72 100,5 111,3 115,0 121,9 149,3 182,4 214,0 223,0 91,06 112,5 116,9 129,5 133,4 142,0 174,2 213,2 249,6 260,3 104,0 128,5 133,5 148,1 153,0 162,4 199,5 244,7 287,0 299,6 Nb.Rd (kN) Lc (m) 3,5

45,19 55,89 57,72 63,84 65,96 69,81 85,02 103,3 120,4 125,3 64,38 79,33 82,48 91,26 94,31 100,0 122,3 149,2 174,8 182,1 75,38 93,08 96,68 107,1 110,3 117,4 143,8 175,9 205,6 214,3 86,81 107,2 111,4 123,5 127,6 135,4 166,1 203,5 238,3 248,7

37,12 45,90 47,40 52,41 54,15 57,31 69,75 84,68 98,69 102,6 53,43 65,80 68,42 75,68 78,21 82,87 101,3 123,5 144,6 150,6 62,92 77,65 80,66 89,36 91,95 97,89 119,8 146,4 171,0 178,2 72,89 89,96 93,45 103,6 107,1 113,6 139,2 170,4 199,3 207,9

30,96 38,28 39,52 43,70 45,15 47,77 58,11 70,53 82,16 85,43 44,88 55,25 57,45 63,54 65,66 69,56 84,98 103,6 121,2 126,2 53,07 65,47 68,00 75,32 77,49 82,50 100,9 123,3 143,8 149,9 61,74 76,16 79,12 87,68 90,62 96,10 117,8 144,1 168,4 175,6

22,40 27,69 28,59 31,60 32,65 34,54 41,99 50,94 59,30 61,65 32,77 40,33 41,93 46,36 47,91 50,75 61,95 75,48 88,27 91,91 38,95 48,03 49,89 55,25 56,83 60,50 73,97 90,27 105,2 109,6 45,55 56,17 58,35 64,65 66,82 70,84 86,75 106,0 123,8 129,1

16,92 20,92 21,59 23,86 24,65 26,08 31,69 38,43 44,73 46,49 24,89 30,62 31,84 35,20 36,37 38,52 47,01 57,26 66,93 69,68 29,67 36,58 38,00 42,07 43,27 46,07 56,30 68,68 80,02 83,35 34,82 42,91 44,58 49,39 51,04 54,11 66,23 80,92 94,43 98,46

19,52 24,00 24,96 27,59 28,51 30,19 36,84 44,85 52,42 54,57 23,31 28,74 29,85 33,04 33,98 36,18 44,20 53,91 62,79 65,40 27,41 33,78 35,09 38,87 40,17 42,58 52,10 63,65 74,24 77,40

88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 108 108 108 108 108 108 108 108 108 108 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3

18,78 23,15 24,04 26,62 27,37 29,14 35,59 43,40 50,54 52,63 22,11 27,25 28,30 31,35 32,40 34,34 42,01 51,31 59,84 62,38

d mm 4

4,5

5

6

7

8

9

10

2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 t mm

Appendix 9.2

x d mm

2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3


x

88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 88,9 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 101,6 108 108 108 108 108 108 108 108 108 108 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3 114,3

d

1)

t

x x x

284

x

x

x x x x x x

127 127 127 127 127 127 127 127 127 127 133 133 133 133 133 133 133 133 133 133 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7 152 152 152 152 152 152 152



t mm 0 2 224,1 2,5 315,6 2,6 327,9 2,9 364,9 3 377,2 3,2 401,7 4 498,8 5 618,5 6 736,1 6,3 771,0 2 233,9 2,5 330,8 2,6 343,7 2,9 382,5 3 395,4 3,2 421,1 4 523,2 5 648,9 6 772,6 6,3 809,3 2,9 402,2 3 415,8 3,2 442,9 4 550,3 5 682,9 6 813,3 6,3 852,1 8 1068 10 1315 2,9 438,4 3 453,2 3,2 482,8 4 600,2 5 745,2 6 888,2 6,3 930,7

0,5 224,1 315,6 327,9 364,9 377,2 401,7 498,8 618,5 736,1 771,0 233,9 330,8 343,7 382,5 395,4 421,1 523,2 648,9 772,6 809,3 402,2 415,8 442,9 550,3 682,9 813,3 852,1 1068 1315 438,4 453,2 482,8 600,2 745,2 888,2 930,7

1,0 215,2 299,9 311,7 346,7 358,4 381,5 473,3 586,1 696,5 729,2 226,1 316,7 329,1 366,1 378,4 402,9 500,0 619,6 736,9 771,7 387,8 400,8 426,9 530,0 657,0 781,6 818,6 1024 1259 427,6 442,1 470,8 584,9 725,6 864,2 905,3

1,5 199,0 275,4 286,2 318,2 328,9 350,1 433,9 536,9 637,3 667,1 210,1 292,3 303,7 337,8 349,2 371,7 460,9 570,7 678,3 710,1 359,7 371,7 395,9 491,2 608,6 723,3 757,4 946,5 1161 399,7 413,2 440,0 546,4 677,5 806,4 844,7

2 181,7 248,8 258,5 287,4 297,1 316,1 391,3 483,5 573,1 599,5 193,1 266,1 276,5 307,4 317,7 338,1 418,9 518,1 615,1 643,8 329,8 340,6 362,8 449,8 556,8 661,0 691,9 863,0 1057 370,4 382,9 407,6 505,8 626,8 745,6 780,8

2,5 162,9 220,1 228,7 254,2 262,7 279,4 345,3 426,0 503,8 526,8 174,8 237,7 247,0 274,5 283,8 301,8 373,4 461,3 546,9 572,1 297,4 307,1 327,1 405,0 500,8 593,6 621,1 772,7 943,8 338,8 350,3 372,8 462,2 572,3 680,1 712,0

3 143,4 190,8 198,2 220,2 227,6 241,9 298,5 367,6 433,7 453,3 155,5 208,3 216,5 240,4 248,5 264,2 326,4 402,6 476,5 498,2 263,5 271,9 289,6 358,2 442,3 523,2 547,2 678,8 826,7 305,4 315,7 335,9 416,0 514,5 610,7 639,1

Nb.Rd (kN) Lc (m) 3,5 124,3 163,0 169,4 188,0 194,3 206,4 254,3 312,8 368,3 384,7 136,2 179,8 186,9 207,5 214,4 227,9 281,1 346,2 409,2 427,6 229,9 237,1 252,6 312,0 384,7 454,2 474,8 587,4 713,3 271,3 280,5 298,3 369,0 455,8 540,4 565,3

d mm 4 106,9 138,5 143,8 159,6 165,0 175,2 215,6 264,8 311,3 325,0 118,2 154,0 160,1 177,6 183,6 195,0 240,3 295,6 348,9 364,5 198,7 204,9 218,3 269,2 331,6 390,9 408,5 504,2 610,7 238,4 246,4 262,0 323,8 399,5 473,1 494,7

4,5 91,70 117,7 122,3 135,7 140,2 148,9 183,0 224,6 263,7 275,3 102,2 131,8 137,0 151,9 157,1 166,8 205,3 252,3 297,6 310,7 171,2 176,5 188,0 231,7 285,1 335,7 350,7 432,0 522,4 208,2 215,2 228,8 282,5 348,2 412,0 430,7

5 78,89 100,6 104,5 115,9 119,8 127,2 156,2 191,6 224,8 234,6 88,39 113,2 117,6 130,4 134,8 143,1 176,1 216,2 254,8 266,1 147,8 152,3 162,3 199,8 245,7 289,1 301,9 371,4 448,4 181,6 187,8 199,5 246,1 303,2 358,5 374,6

6 59,40 75,08 78,00 86,49 89,40 94,85 116,4 142,7 167,2 174,4 67,04 84,98 88,30 97,90 101,2 107,4 132,0 162,0 190,8 199,1 111,8 115,2 122,7 150,9 185,4 217,8 227,4 279,3 336,6 139,3 144,0 153,0 188,5 232,0 274,0 286,3

7 45,94 57,76 60,01 66,54 68,77 72,95 89,52 109,6 128,4 133,9 52,07 65,61 68,18 75,58 78,13 82,92 101,8 124,9 147,0 153,4 86,67 89,30 95,09 116,9 143,6 168,6 176,0 215,8 259,9 108,9 112,6 119,6 147,3 181,1 213,8 223,3

8 36,44 45,67 47,45 52,60 54,37 57,67 70,74 86,60 101,4 105,7 41,42 51,99 54,02 59,88 61,90 65,69 80,66 98,90 116,4 121,4 68,86 70,92 75,54 92,86 114,0 133,8 139,6 171,1 205,9 86,97 89,92 95,5 117,6 144,5 170,5 178,1

9 29,55 36,95 38,39 42,56 43,99 46,66 57,22 70,03 81,97 85,48 33,65 42,14 43,78 48,52 50,16 53,23 65,34 80,10 94,22 98,31 55,90 57,57 61,33 75,37 92,49 108,5 113,2 138,7 166,9 70,86 73,26 77,80 95,76 117,7 138,8 145,0

10 24,42 30,49 31,67 35,11 36,29 38,49 47,19 57,75 67,58 70,48 27,84 34,80 36,16 40,08 41,43 43,96 53,95 66,13 77,77 81,15 46,23 47,61 50,72 62,32 76,46 89,67 93,58 114,6 137,8 58,76 60,75 64,50 79,38 97,53 115,0 120,1

127 127 127 127 127 127 127 127 127 127 133 133 133 133 133 133 133 133 133 133 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7 139,7 152 152 152 152 152 152 152

t mm 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2 2,5 2,6 2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 8 10 2,9 3 3,2 4 5 6 6,3


x d mm

d t Lc Nb.Rd

Appendix 9.2

Buckling resistance values for circular longitudinally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued.

Table 9.2.3

x

x x x x

x x x

285 x x x x x x x x x

x 1) d

mm x

459,0 474,5 505,5 628,6 780,7 930,7 975,4 486,3 502,8 535,7 666,3 827,8 987,3 1035 1300 1605 769,3 956,6 1142 1197 1506 1862 2211 2296 872,3 1085 1296 1359 1712 2120 2520 2618 1359 1624 1704 2149 2667 3175 3301 1427 1934 2029 2562 3183 3795 3947

459,0 474,5 505,5 628,6 780,7 930,7 975,4 486,3 502,8 535,7 666,3 827,8 987,3 1035 1300 1605 769,3 956,6 1142 1197 1506 1862 2211 2296 872,3 1085 1296 1359 1712 2120 2520 2618 1359 1624 1704 2149 2667 3175 3301 1427 1934 2029 2562 3183 3795 3947

450,3 465,5 495,8 616,2 764,7 910,9 954,3 480,4 496,7 529,1 657,8 816,7 973,2 1020 1280 1578 769,3 956,6 1142 1197 1506 1861 2207 2292 872,3 1085 1296 1359 1712 2120 2520 2618 1359 1624 1704 2149 2667 3175 3301 1427 1934 2029 2562 3183 3795 3947

422,5 436,8 465,1 577,8 716,8 853,3 893,9 452,7 468,0 498,5 619,5 768,9 915,8 959,5 1203 1482 733,1 910,7 1086 1139 1431 1766 2093 2173 846,6 1053 1257 1317 1658 2050 2433 2528 1354 1618 1696 2139 2652 3155 3280 1427 1934 2029 2562 3183 3795 3947

393,5 406,8 433,1 537,7 666,7 793,0 830,5 424,0 438,4 466,8 579,9 719,4 856,3 897,0 1124 1383 694,4 862,2 1028 1077 1353 1669 1976 2051 808,3 1005 1199 1257 1581 1954 2318 2408 1306 1560 1636 2063 2557 3041 3161 1414 1902 1994 2517 3125 3723 3871

362,4 374,7 398,8 494,8 613,0 728,4 762,6 393,6 406,9 433,2 537,7 666,7 792,8 830,3 1039 1276 653,9 811,4 967,2 1013 1271 1567 1854 1923 768,9 955,3 1140 1195 1502 1855 2198 2284 1258 1503 1576 1986 2460 2926 3040 1374 1844 1934 2441 3029 3609 3752

329,4 340,5 362,4 449,1 555,9 659,6 690,4 361,1 373,3 397,3 492,9 610,5 725,2 759,3 948,3 1163 611 757,7 902,6 945,6 1185 1458 1724 1788 727,6 903,6 1078 1129 1419 1751 2073 2153 1209 1444 1514 1907 2362 2807 2917 1335 1787 1874 2364 2933 3493 3632

t mm 0

0,5

1,0

1,5

2

2,5

3

295,3 261,8 305,3 270,6 324,7 287,8 402,1 356,0 497,1 439,6 589,1 520,2 616,3 544,1 327,2 293,3 338,3 303,2 360,0 322,5 446,1 399,3 552,0 493,7 655,0 584,9 685,5 612,0 854,5 761,4 1046 930,2 565,8 519,0 701,1 642,5 834,6 764,4 874,2 800,4 1094 1000 1344 1227 1587 1446 1645 1498 684,1 638,7 849,1 792,2 1012 944 1061 989,1 1331 1240 1641 1527 1941 1804 2016 1873 1158 1105 1383 1319 1450 1383 1825 1740 2260 2153 2684 2556 2789 2655 1294 1253 1728 1667 1812 1748 2285 2204 2834 2733 3375 3253 3508 3382 Nb.Rd (kN) Lc (m) 3,5 4

230,4 238,2 253,3 313,0 386,2 456,4 477,2 260,7 269,5 286,6 354,6 437,9 518,3 542,2 673,3 821,1 472,1 583,8 694,1 726,6 906,8 1110 1306 1353 591,9 733,7 873,8 915,3 1146 1409 1663 1726 1050 1253 1313 1651 2042 2423 2516 1210 1604 1682 2121 2628 3127 3250

202,3 209,1 222,2 274,5 338,4 399,5 417,6 230,7 238,6 253,6 313,5 386,9 457,5 478,4 593,3 722,4 426,6 527,1 626,2 655,4 816,8 998,4 1173 1214 544,9 674,9 803,4 841,4 1052 1292 1523 1580 992,8 1184 1241 1560 1928 2285 2373 1166 1539 1614 2034 2519 2997 3114

156,5 161,8 171,9 212,0 261,1 307,9 321,7 180,6 186,8 198,5 245,1 302,2 356,7 372,9 461,4 560,6 344,8 425,4 504,9 528,2 656,8 800,7 938,8 971,2 455,1 562,9 669,3 700,7 874,7 1071 1259 1306 875,5 1043 1093 1372 1693 2003 2079 1074 1403 1471 1852 2292 2723 2829

123,0 127,1 135,0 166,5 204,9 241,4 252,2 143,0 147,9 157,1 193,9 238,8 281,7 294,4 363,8 441,5 278,8 343,6 407,5 426,2 529,1 644,0 753,8 779,4 376,8 465,5 553,1 578,9 721,5 882,0 1035 1073 760,4 905,1 948,3 1189 1465 1730 1795 977,9 1262 1323 1664 2057 2442 2536

98,56 101,9 108,2 133,4 164,1 193,2 201,8 115,1 119,0 126,5 156,0 192,1 226,4 236,6 292,2 354,2 227,5 280,3 332,1 347,4 430,8 523,7 612,5 633,1 312,6 385,9 458,2 479,5 597,0 728,8 853,9 885,5 654,7 778,7 815,7 1021 1257 1483 1537 880,5 1123 1177 1479 1825 2165 2247

80,48 83,20 88,35 108,9 133,9 157,6 164,6 94,32 97,51 103,6 127,7 157,2 185,3 193,6 238,9 289,5 188,1 231,6 274,4 286,9 355,6 431,9 504,8 521,6 261,3 322,4 382,7 400,4 498,1 607,6 711,2 737,5 562,5 668,6 700,2 875,8 1077 1269 1315 786,2 991,4 1039 1304 1608 1905 1978

66,84 69,09 73,37 90,38 111,1 130,8 136,6 78,50 81,16 86,21 106,3 130,8 154,1 161,0 198,6 240,6 157,6 193,9 229,7 240,2 297,5 361,2 421,9 436,0 220,6 272,1 322,9 337,8 420,0 511,9 598,9 621,0 484,4 575,5 602,7 753,2 925,5 1090 1129 698,5 872,4 914,1 1147 1413 1673 1735

159 159 159 159 159 159 159 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 193,7 193,7 193,7 193,7 193,7 193,7 193,7 193,7 219,1 219,1 219,1 219,1 219,1 219,1 219,1 219,1 273 273 273 273 273 273 273 323,9 323,9 323,9 323,9 323,9 323,9 323,9 d mm

4,5

5

6

7

8

9

10

2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 8 10 4 5 6 6,3 8 10 12 12,5 4 5 6 6,3 8 10 12 12,5 5 6 6,3 8 10 12 12,5 5 6 6,3 8 10 12 12,5 t mm

Appendix 9.2

x x

2,9 3 3,2 4 5 6 6,3 2,9 3 3,2 4 5 6 6,3 8 10 4 5 6 6,3 8 10 12 12,5 4 5 6 6,3 8 10 12 12,5 5 6 6,3 8 10 12 12,5 5 6 6,3 8 10 12 12,5


159 159 159 159 159 159 159 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 168,3 193,7 193,7 193,7 193,7 193,7 193,7 193,7 193,7 219,1 219,1 219,1 219,1 219,1 219,1 219,1 219,1 273 273 273 273 273 273 273 323,9 323,9 323,9 323,9 323,9 323,9 323,9

d

1)

t

286

355,6 355,6 355,6 355,6 355,6 355,6 355,6 406,4 406,4 406,4 406,4 406,4 406,4 457 457 457 457 457 457 508 508 508 508 508 508 559 559 559 559 559 559



t mm 5,6 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5

0 1757 2127 2231 2819 3504 4180 4348 2141 2256 3231 4019 4798 4992 2383 2513 3642 4532 5414 5633 2622 2767 3586 5049 6035 6280 2856 3015 3916 5566 6655 6926

1 1757 2127 2231 2819 3504 4180 4348 2141 2256 3231 4019 4798 4992 2383 2513 3642 4532 5414 5633 2622 2767 3586 5049 6035 6280 2856 3015 3916 5566 6655 6926

2 1757 2114 2218 2801 3479 4148 4314 2141 2256 3231 4019 4798 4992 2383 2513 3642 4532 5414 5633 2622 2767 3586 5049 6035 6280 2856 3015 3916 5566 6655 6926

3 1669 1999 2097 2648 3288 3919 4075 2076 2187 3102 3855 4600 4784 2348 2475 3554 4421 5278 5491 2616 2759 3570 4990 5961 6203 2856 3015 3916 5560 6645 6915

4 1579 1882 1974 2491 3091 3683 3829 1981 2086 2948 3663 4368 4543 2255 2377 3401 4229 5048 5252 2525 2663 3443 4799 5732 5964 2788 2942 3814 5369 6416 6677

5 1484 1758 1843 2325 2884 3434 3570 1883 1983 2788 3462 4127 4292 2161 2277 3245 4033 4813 5006 2433 2566 3315 4606 5500 5722 2699 2847 3688 5177 6186 6437

6 1384 1625 1704 2149 2663 3168 3293 1781 1875 2619 3250 3873 4027 2063 2174 3081 3829 4567 4750 2339 2466 3184 4407 5261 5473 2608 2751 3561 4981 5951 6192

Nb.Rd (kN) Lc (m) 7 8 1278 1169 1487 1345 1559 1411 1963 1775 2431 2196 2890 2608 3003 2709 1673 1560 1761 1642 2441 2255 3027 2795 3605 3326 3748 3457 1962 1855 2066 1953 2910 2730 3614 3390 4310 4040 4482 4200 2242 2141 2364 2257 3048 2907 4199 3982 5011 4751 5213 4941 2514 2418 2652 2550 3430 3295 4779 4568 5707 5454 5939 5675

d mm 9 1060 1207 1265 1591 1966 2334 2424 1445 1519 2067 2560 3044 3163 1744 1836 2545 3157 3761 3909 2036 2145 2760 3756 4479 4658 2318 2444 3154 4349 5191 5401

10 955 1077 1129 1419 1752 2078 2157 1328 1397 1881 2329 2767 2874 1631 1716 2356 2922 3478 3615 1928 2030 2607 3523 4200 4367 2214 2334 3008 4122 4918 5117

11 857 959 1005 1262 1557 1846 1916 1215 1277 1704 2108 2503 2600 1517 1595 2170 2689 3199 3325 1816 1912 2452 3288 3917 4073 2108 2221 2858 3889 4639 4826

12 768 854 895 1123 1385 1640 1702 1107 1163 1540 1903 2259 2346 1404 1476 1990 2465 2930 3045 1704 1793 2295 3054 3637 3781 1998 2105 2705 3654 4356 4532

13 689 762 798 1001 1234 1461 1516 1007 1057 1390 1718 2037 2115 1295 1361 1820 2253 2677 2781 1592 1675 2140 2827 3365 3497 1888 1988 2550 3419 4075 4239

14 619 681 714 895 1103 1306 1355 915 961 1256 1551 1839 1909 1192 1252 1662 2057 2443 2538 1483 1560 1990 2610 3105 3227 1778 1871 2396 3190 3800 3953

15 558 612 641 804 990 1171 1215 833 874 1137 1403 1663 1726 1096 1151 1518 1878 2230 2316 1378 1449 1846 2406 2861 2973 1669 1756 2246 2969 3536 3677

16 504 551 578 724 891 1055 1094 759 796 1031 1273 1508 1565 1007 1058 1388 1716 2037 2116 1279 1345 1710 2217 2635 2738 1564 1645 2100 2759 3284 3416

355,6 355,6 355,6 355,6 355,6 355,6 355,6 406,4 406,4 406,4 406,4 406,4 406,4 457 457 457 457 457 457 508 508 508 508 508 508 559 559 559 559 559 559

t mm 5,6 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5 6 6,3 8 10 12 12,5


d mm

d t Lc Nb.Rd

Appendix 9.2

Buckling resistance values for circular spirally welded hollow sections of steel grade S355J2H (fy = 355 N/mm2) in buckling category c, continued. (Technical delivery conditions to be agreed when ordering)

Table 9.2.4

8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16

4242 5398 7276 7572 8578 4555 5806 7884 8206 9298 4869 6218 8504 8852 10032 5178 6623 8068 8429 10766 5481 7024 8565 8951 11500 12929 7800 9534 9967 11440 14567 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

4242 5398 7276 7572 8578 4555 5806 7884 8206 9298 4869 6218 8504 8852 10032 5178 6623 8068 8429 10766 5481 7024 8565 8951 11500 12929 7800 9534 9967 11440 14567 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

4242 5398 7276 7572 8578 4555 5806 7884 8206 9298 4869 6218 8504 8852 10032 5178 6623 8068 8429 10766 5481 7024 8565 8951 11500 12929 7800 9534 9967 11440 14567 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

4242 5398 7276 7572 8578 4555 5806 7884 8206 9298 4869 6218 8504 8852 10032 5178 6623 8068 8429 10766 5481 7024 8565 8951 11500 12929 7800 9534 9967 11440 14567 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

4179 5309 7100 7389 8366 4530 5765 7770 8087 9159 4869 6218 8453 8799 9968 5178 6623 8068 8429 10766 5481 7024 8565 8951 11500 12929 7800 9534 9967 11440 14567 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

4055 5149 6871 7150 8095 4408 5608 7541 7848 8889 4762 6069 8225 8560 9697 5110 6524 7936 8289 10506 5451 6973 8493 8872 11315 12717 7800 9534 9967 11440 14522 8562 10490 10972 12610 14343 10017 12328 12906 14872 16953

3931 4989 6638 6908 7820 4286 5450 7310 7608 8616 4641 5913 7995 8321 9426 4991 6370 7746 8090 10235 5334 6820 8304 8675 11045 12412 7694 9391 9815 11254 14217 8552 10463 10941 12563 14278 10017 12328 12906 14872 16953

0

1

2

3

4

5

6

3538 4481 5898 6136 6943 3905 4956 6584 6851 7756 4271 5432 7280 7576 8579 4628 5898 7164 7479 9399 4978 6357 7731 8075 10216 11478 7243 8833 9230 10577 13294 8112 9916 10367 11898 13517 9767 11997 12554 14448 16453

9

3399 4300 5633 5860 6629 3771 4781 6325 6582 7450 4141 5263 7028 7313 8280 4503 5734 6962 7267 9106 4856 6197 7534 7868 9928 11153 7090 8643 9031 10347 12977 7964 9732 10174 11674 13260 9626 11821 12369 14233 16206

10

3255 4113 5359 5575 6305 3632 4601 6058 6304 7134 4008 5089 6768 7042 7972 4374 5566 6754 7050 8804 4731 6034 7332 7656 9633 10820 6935 8450 8828 10112 12654 7814 9546 9979 11448 13000 9485 11644 12184 14018 15958

11

3106 3921 5080 5284 5974 3489 4416 5784 6019 6809 3871 4911 6500 6763 7655 4242 5394 6540 6826 8494 4603 5866 7125 7439 9329 10478 6776 8253 8622 9873 12324 7663 9357 9780 11218 12736 9343 11466 11997 13800 15708

12

2955 3726 4797 4989 5640 3343 4226 5504 5727 6478 3730 4727 6225 6476 7329 4106 5216 6321 6596 8175 4472 5695 6912 7216 9017 10125 6614 8052 8411 9628 11985 7508 9165 9579 10983 12467 9200 11287 11808 13581 15455

13

2803 3529 4516 4696 5306 3194 4033 5222 5433 6144 3585 4539 5945 6185 6998 3966 5034 6096 6360 7849 4338 5519 6694 6988 8697 9765 6448 7846 8195 9378 11638 7351 8969 9373 10745 12193 9055 11105 11618 13359 15200

14

2651 3333 4239 4408 4979 3043 3838 4940 5139 5810 3439 4348 5663 5891 6663 3824 4849 5866 6120 7517 4200 5338 6471 6754 8370 9396 6279 7635 7974 9121 11282 7191 8770 9163 10501 11913 8908 10922 11425 13133 14940

15

2501 3140 3970 4128 4662 2893 3644 4661 4850 5481 3290 4156 5380 5597 6329 3679 4660 5633 5875 7183 4060 5155 6243 6515 8039 9023 6105 7420 7748 8859 10919 7027 8566 8949 10252 11627 8759 10735 11228 12905 14677

16

610 610 610 610 610 660 660 660 660 660 711 711 711 711 711 762 762 762 762 762 813 813 813 813 813 813 914 914 914 914 914 1016 1016 1016 1016 1016 1219 1219 1219 1219 1219 d mm

8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 8 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16 10 12 12,5 14,2 16 t mm

Appendix 9.2

t mm

3804 3673 4824 4655 6400 6153 6659 6403 7538 7246 4162 4035 5289 5125 7075 6833 7363 7112 8338 8051 4520 4397 5756 5596 7763 7525 8079 7831 9150 8869 4872 4751 6215 6058 7555 7361 7889 7686 9962 9683 5217 5098 6667 6513 8115 7925 8477 8278 10773 10497 12105 11795 7544 7394 9206 9020 9621 9426 11030 10804 13912 13605 8406 8259 10281 10099 10749 10559 12342 12120 14025 13771 10017 9908 12328 12172 12906 12739 14872 14663 16947 16700 Nb.Rd (kN) Lc (m) 7 8


287

610 610 610 610 610 660 660 660 660 660 711 711 711 711 711 762 762 762 762 762 813 813 813 813 813 813 914 914 914 914 914 1016 1016 1016 1016 1016 1219 1219 1219 1219 1219 d mm

Appendix 9.2 DESIGN HANDBOOK FOR RAUTARUUKKI STRUCTURAL HOLLOW SECTIONS

288


Appendix 9.3

Appendix 9.3

Calculation tables for truss joints

This appendix includes formulae for calculating uniplanar lattice structure joints, based on Eurocode 3 (ENV-1993-1-1:1992) annex K. The tables also include values from references [2] and [3]. Table 9.3.1

Joint type T-, Y- and X-joint

Chord Square Rectangular

9.3.2

Gap N-, K and KT-joint

Square Rectangular

9.3.3

Overlap N-, K and KT-joint

Square Rectangular

9.3.4 9.3.5

T-, Y- and X-joint Gap N-, K and KT-joint Overlap N-, K and KT-joint Gap N- and K-joints and T-, Y- and X-joint Overlap N- and K-joint

Circular Circular

Brace member Rectangular Square Circular Rectangular Square Circular Rectangular Square Circular Circular Circular

Circular

Circular

I-profile

T and X-joint bending resistance T-, Y- and X-joint bending resistance Plate joint

Square Rectangular Circular

Rectangular Square Circular Rectangular Square Circular Rectangular Square Circular

9.3.6 9.3.7

9.3.8

9.3.9 9.3.10 9.3.11

I-profile

9.3.12 9.3.13

Plate joint T-, Y- and X-joint with reinforced chord face

Square Rectangular Circular Square Rectangular

9.3.14

T-, Y- and X-joint with reinforced chord web

Square Rectangular

9.3.15

Gap N-, K and KT-joint with reinforced chord face Gap N-, K and KT-joint, with reinforced chord web Overlap N- and K-joint, reinforced with intermediate plate

Square Rectangular

9.3.16

9.3.17

289

Square Rectangular Square Rectangular

Plate Plate Rectangular Square Circular Rectangular Square Circular Rectangular Square Circular Rectangular Square Circular Rectangular Square Circular

Appendix 9.3


Symbols Ai Av E Mip.i.Rd Mip.i.Sd Mop.i.Rd Mop.i.Sd Ni.Rd Ni.Sd Wel.i Wpl.i bi b0 beff be(ov) be.p bp bw di d0 dw e fb fyi fy0 g ga hi h0 i kg km kn kp M0.Sd m

is the cross-sectional area of the chord member is the shear area of the chord is the modulus of elasticity for steel is the design value of the joint bending resistance parallel to the plane of the joint (table 9.3.9) is the design value of the bending resistance perpendicular to the plane of the joint (table 9.3.9) is the design value of the bending resistance perpendicular to the plane of the joint (table 9.3.9) is the design value of the bending moment perpendicular to the plane of the joint (table 9.3.9) is the design value of the joint’s resistance to axial force is the design value of the brace member axial force is the elastic section modulus of the brace member is the plastic section modulus of the brace member is the width of the brace member is the width of the chord is the effective width of the brace member in calculating the brace member resistance is the effective width of the overlapping brace member in overlapped joint is the effective width of the brace member in calculating the shear failure resistance of the chord is the width of the bracing is the effective width of the chord web with an I profile chord is the diameter of the circular brace member is the diameter of the circular chord is the height of the chord when the chord is an I profile is the eccentricity of the joint is the buckling length of the chord web is the design value of the brace member yield strength (In Eurocode 3 Annex K, fyi is the design value of yield resistance) is the design value of the chord yield strength (In Eurocode 3 Annex K, fy0 is the design value of yield resistance) is the theoretical gap of the joint (table 9.3.2) is the actual gap of the joint (table 9.3.2) is the height of the brace member is the height of the chord is the number of the brace member (1, 2, 3) is the reduction factor for the resistance of gap and overlap joints with circular hollow sections is the reduction factor for the resistance of joints between plates and square or rectangular hollow sections is the reduction factor for the resistance of joints with square or rectangular chords is the reduction factor for the resistance of joints with circular hollow sections is the design value for the chord bending moment is the number of brace members

290


n np N0.Sd Np.Sd r tf ti t0 tp tw β

is the compression stress to yield resistance with square and rectangular chords is the compression stress due to Np.Sd and M0.Sd to the yield resistance with circular hollow sections is the design value with the greatest absolute value of the chord axial force is the normal force passing through the chord joint is the rounding radius of the I profile is the thickness of the I profile flange is the wall thickness of the brace member is the wall thickness of the chord is the thickness of the bracing is the thickness of the I profile is the brace member diameter or width to the chord diameter or width. The width of the brace member is taken as an average if the joint contains several brace members. m

β= βp

Appendix 9.3

∑ bi i =1

m ⋅ b0

m

tai β =

∑ di i =1

m ⋅ d0

is the brace member width to the bracing width βp = bi/ bp

γ

γMj η ηp λov θi

is the chord width to chord wall thickness divided by two γ = 0,5d0/ t0, = 0,5b0/ t0 or = 0,5b0/ tf is the partial safety factor for lattice structures is the brace member height to the chord width η = hi/ b0 is the brace member height to the bracing width ηp = hi/ bp is the relative magnitude of the overlap λov = q · sin(θi)/ hi is the smaller angle between brace member and chord

In all tables of this appendix, the following limitations apply: • fy ≤ 355 N/ mm2 • t0 ≥ 2,5 mm • ti ≥ 2,5 mm • θi ≥ 30° (also for the angle between brace members) • -0,55 ≤ e/h0 ≤ 0,25 or -0,55 ≤ e/d0 ≤ 0,25 • the cross-sections of compression elements of members must belong to Class 1 or 2 when subjected to bending only (section 2.2) • the brace member ends must not be flattened Depending on the joint type, the tables may include additional limitations. 291


Appendix 9.3 Table 9.3.1

Resistance of T, Y and X joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].

N1 h1

M0

h0

θ1 N0

N0

M0

h1 N1

b1

t1

;; ; t0

b0

292


Resistance β ≤ 0,85, chord face yield Ni.Rd =

Parameters

k n ⋅ fy0 ⋅ t 0 2  2 η  11 , + 4 1− β   (1 − β)sin θi  sin θi  γ Mj ⋅ γ M0

β = bi / b0 η = hi / b0 Tension chord: kn = 1 Compression chord: kn = 1,3- (0,4 / β)n ≤ 1  γ M0 ⋅ γ Mj  N0.Sd M n= + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0 

0,85 < β < 1,0 Use the values β = 0,85 and β =1,0 when calculating the resistance. Resistance is defined by linear interpolation based on the original value of β. β = 1,0, chord web buckling or yield Ni.Rd =

 fb ⋅ t 0  2hi 11 , + 10t 0   sin θi  sin θi  γ Mj ⋅ γ M0

Tension chord: fb = fy0 Compression chord: fb = χ· fyo (T and Y joints) fb = 0,8χ(sinθi)fy0 (X joints) χ = reduction factor for buckling using buckling curve when the slenderness is: fy0  h 1 λ = 3, 46 0 − 2 t E(sin ) θ π   0 i

0,85 ≤ β ≤ 1 - (1/γ), punching shear failure of the chord face Ni.Rd =

fy0 ⋅ t 0  2hi  11 , + 2b ep   sin θ γ ⋅   3 sin θi i Mj γ M0

b ep = γ=

10t 0 ⋅ b1 ≤ b1 b0

11 , γ Mj ⋅ γ M0

b0 2t 0

b eff =

10b1 ⋅ t 02 ⋅ fy0 b 0 ⋅ t1 ⋅ fy1

β = 1,0, X joints when θ < 90°, chord shear Ni.Rd =

fy0 ⋅ A v

11 , 3 sin θi γ Mj ⋅ γ M0

Av = 2h0 · t0

Validity area Square and rectangular brace members: Brace members in general:  bi + hi 1 ≥ 25 ) ti bi / b0, hi / b0 ≥ 0,25 3) 0,5 ≤ hi / bi ≤ 2 Tension brace member: bi / ti, hi / ti ≤ 35 Compression brace member: bi / ti, hi / ti ≤ 35 E bi hi , ≤ 1, 25 fyi ti ti

≤ b1

3)

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) E di ≤ 1, 5 fyi ti

β > 0,85, brace member failure Ni.Rd = fyi ⋅ ti (2hi − 4ti + 2b eff )

3)

Appendix 9.3

3)

Chords: b 0 + h0 ≥ 25 t0 h 0, 5 ≤ 0 ≤ 2 b0 b 0 h0 , ≤ 35 t0 t0

With circular brace members, the resistance values are multiplied by π / 4, and bi and hi are replaced with the diameter di. fy ≤ 355 N/ mm2, 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1 These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this parameter. 3) These limit values are defined in reference [3].

293



Resistance of gap N, K and KT lattice joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].

b1, 2

N2

;; ; ;

N1

Det 1 θ1

h

g

h1

M0

2

N0

e

h0

t1, 2

θ2

M0

t0

b0

N3

N1

g1

N2

g2

h

h3

h1

θ1

2

θ3

θ2 N0

e

M0

M0

Det 1

Det 1 θ ≤ 60°

θ > 60° ga

g

g

t0

ga

294


Resistance

Parameters

Chord face yield (in KT joints, select the distance between the brace members subjected to the greatest load as the gap)

Ni.Rd

 2  fy0 ⋅ t 0  = 8, 9 sin θi    

 hi  11 , i =1 i =1  k n γ γ Mj ⋅ γ M0 2m ⋅ b 0     m

∑

m

m

bi +

∑

β=

∑ bi i =1

m ⋅ b0

,η =

hi b0

 γ M0 ⋅ γ Mj  N0.Sd M + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0 

Chord shear fy0 ⋅ A v


When VSd > 0,5 Vpl.Rd N0.Rd

2   2VSd   fy0  = A0 − A v  − 1    Vpl.Rd   γ M0  

When VSd ≤ 0,5 Vpl.Rd N0.Rd = A 0

Av = (2h0 + α · b0)t0 (square and rectangular brace members) Av = 2h0 · t0 (circular brace members) fy0 ⋅ A v 1 α= , Vpl.Rd = 2 3 ⋅ γ M0 4g 1+ 2 3t 0 VSd is the chord shear failure at joint

fy0 γ M0

Brace member failure Ni.Rd = fyi ⋅ ti (2hi − 4ti + bi + b eff )

11 , γ Mj ⋅ γ M0

b eff =

10bi ⋅ t 02

⋅ fy0

b 0 ⋅ ti ⋅ fyi

bi hi E , ≤ 1, 25 ti ti fyi

3)

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension Brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) di E ≤ 1, 5 ti fyi

3)

b 0 + h0 1 ≥ 25 ) t0

β ≤ 1- (1/ γ), chord shear failure fy0 ⋅ t 0  2hi  11 , + bi + b ep    γ Mj ⋅ γ M0 3 sin θi  sin θi

≤ bi

Compression brace member: bi / ti, hi / ti ≤ 35

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 35

In a KT joint, also check the following conditions: N1.Rd · sinθ1 ≥ N1.Sd · sinθ1 + N3.Sd · sinθ3 N1.Rd · sinθ1 ≥ N2.Sd · sinθ2

Ni.Rd =

Validity area Square and rectangular brace members: Brace members in general: b 0 3) η, β ≥ 0,1 + 100t 0

β ≥ 0,35 m is the number of brace members 0,5 ≤ hi / bi ≤ 2 γ = b0 /(2t0) bi + hi ≥ 25 Tension chord: ti kn = 1 Tension brace member: Compression chord: bi / ti, hi / ti ≤ 35 kn = 1,3- (0,4/ β)n ≤ 1 n=

Ni.Rd =

Appendix 9.3

b ep = γ=

10t 0 ⋅ bi ≤ b1 b0

b0 2t 0

Gap: g / b0 ≥ 0,5(1-β) g / b0 ≤ 1,5(1-β) g ≥ t1+ t2 ga / t0 ≥ 1,51)

With circular brace members, the resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di. If g / b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in calculations. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value 3) These limit values are defined in reference [3].

295



Resistance of overlap N, K and KT lattice joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3]. With in the range of validity presented in this table, only brace member failure is considered a failure mode. The resistance needs to be checked for the overlapping brace member only.

b1, 2

N2

;; ; ;

N1

h

θ1

t1, 2

2

h1

θ2

M0 -e

h0

q

M0

N0

t0

b0

N3 N1

N2 h3

h1

θ1

θ3

M0

h

2

θ2

-e

N0 q1 q2

296

M0


Resistance

Parameters

Validity area Square and rectangular brace members: Brace members in general: bi hi , ≥ 0, 25 3) b0 b0

0,25 ≤ λov < 0,5, brace member failure N and K joints

[

N1.Rd = fy1 ⋅ t1 2λ ov (2h1 − 4t1) + b eff KT joints

N and K joint: 11 , q ⋅ sin(θ1) + b e(ov ) γ Mj ⋅ γ M0 λ ov = h1

]

[

Ni.Rd = fyi ⋅ ti 2λ ov (2hi − 4ti ) + b eff + b e(ov )

]γ

11 , ⋅ Mj γ M0

(i = 1,2)

0,5 ≤ λov < 0,8, brace member failure N and K joints

[

N1.Rd = fy1 ⋅ t1 2h1 − 4t1 + b eff + b e(ov ) KT joints

[

Ni.Rd = fyi ⋅ ti 2hi − 4ti + b eff + b e(ov )

]γ

]γ

b eff =

10b1 ⋅ t 02 ⋅ fy0

(

N1.Rd = fy1 ⋅ t1 2h1 − 4t1 + b1 + b e(ov ) KT joints

(

Ni.Rd = fyi ⋅ ti 2hi − 4ti + bi + b e(ov )

)γ

0,5 ≤ hi / bi ≤ 2

b 0 ⋅ t1 ⋅ fy1

E bi hi , ≤ 11 , fyi ti ti

11 , ⋅ Mj γ M0

3)

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2)

11 , ⋅ Mj γ M0

)γ

≤ b1

bi + hi 1 ≥ 25 ) t i 10b1 ⋅ t 22 ⋅ fy 2 N and K joints: ≤ b1 b e(ov ) = b 2 ⋅ t1 ⋅ fy1 t1 / t2 ≤ 1,0 KT joint: b1 / b2 ≥ 0,75 KT joints: q ⋅ sin(θi ) λ ov = i (i = 1,2) t / t ≤ 1,0 (i = 1,2) i 3 hi b i / b3 ≥ 0,75 (i = 1,2) 10bi ⋅ t 02 ⋅ fy0 ≤ bi (i = 1,2) Tension brace member: b eff = b 0 ⋅ ti ⋅ fyi bi / ti, hi / ti ≤ 35 2 Compression brace mem10bi ⋅ t 3 ⋅ fy 3 ≤ b1 (i = 1,2) ber: b e(ov ) = b 3 ⋅ ti ⋅ fyi bi / ti, hi / ti ≤ 35

(i = 1,2) λov ≥ 0,8, brace member failure N and K joints

Appendix 9.3

11 , ⋅ Mj γ M0

E di ≤ 1, 5 fyi ti

3)

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 35 b 0 + h0 1 ≥ 25 ) t0

11 , Mj ⋅ γ M0

(i = 1,2) In a KT joint, also check the following conditions: N1.Rd · sinθ1 ≥ N1.Sd · sinθ1 + N3.Sd · sinθ3 N1.Rd · sinθ1 ≥ N2.Sd · sinθ2

Overlap: 0,25 ≤ λov ≤ 1,0

Overlap: With circular brace members, the resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di. 0,25 ≤ λov ≤ 1,0 fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in reference [3].

297



Resistance of T, Y, and X joints. Chords and brace members are circular hollow sections [1].

;; ;; ;; ; ; ; ; ; ;

N1

θ1

t1

;;; ;

d1

t0

M0

Np

Np

M0

d0

298


Resistance

Parameters

Validity area T, Y and X joints:

T and Y joints, chord face yield

Ni.Rd =

fy0 ⋅ t 0 2 sin θi

(2, 8 + 14, 2β )γ 2

0,2

11 , kp γ Mj ⋅ γ M0

X joints, chord face yield Ni.Rd =

fy0 ⋅ t 0 2  5, 2  11 ,  kp  sin θi  1 − 0, 81β  γ Mj ⋅ γ M0

0,2 ≤ di / d0 ≤ 1,0

β = di / d0 γ = d0 / (2 t0) Tension chord: kp = 1 Compression chord: 2 kp = 1,0- 0,3(np + np ) ≤ 1 np =

10 ≤ di / ti ≤ 50

 γ M0 ⋅ γ Mj  Np.Sd M + 0.Sd  T and Y joints:  11 ,  A 0 ⋅ fy0 W0 ⋅ fy0  10 ≤ d0 / t0 ≤ 50 X joints:

β = di / d0 γ = d0 / (2 t0)

10 ≤ d0 / t0 ≤ 40

Tension chorde kp = 1 Compression chord:

(

)

k p = 1, 0 − 0, 3 np + np2 ≤ 1 np =

 γ M0 ⋅ γ Mj  Np.Sd M + 0.Sd   11 ,  A 0 ⋅ fy W0 ⋅ fy0 

di ≤ d0 - 2t0, T, Y and X joints, punching shear failure of the chord face Ni.Rd =

Appendix 9.3

fy0 ⋅ t 0 ⋅ π ⋅ di  1 + sin θ  , 11    2 sin2 θ  γ Mj ⋅ γ M0 3

fy ≤ 355 N/ mm , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 2

299



Resistance of gap N, K and KT joints. Chords and brace members are circular hollow sections [1]. N1

N2

d1, 2

; ;;; ; ; ;; ; ; ;;; ; ; ; ;; ; ; ; ; g

d2

; ;;;

t1, 2

d1

θ1

Np

t0

Det 1

M0

e

;; ;

M0

θ2

d0

N3

N1

g1

N2

g2

d3

d1

θ1

d

2

θ2

Np

M0

e

M0

θ3

Det 1

Det 1

θ ≤ 60°

θ > 60° ga

g

g

t0

ga

300


Resistance

Parameters

Validity area 0,2 ≤ di / d0 ≤ 1,0

γ = d0 / (2 t0)

10 ≤ di / ti ≤ 50

Tension chord: kp = 1 Compression chord:

10 ≤ d0 / t0 ≤ 50

Chord face yield

Ni.Rd =

fy0 ⋅ t 0 2 sin θ1

(1, 8 + 10, 2β)k g ⋅ k p

11 , γ Mj ⋅ γ M0

Tension member: N2.Rd = N1.Rd [sin(θ1)/ sin(θ2)]

(

k p = 1, 0 − 0, 3

np + np2

)≤1

m

β=

∑ di i =1

m ⋅ d0 m is the number of brace members di ≤ d0 - 2t0, punching shear failure of the chord face fy0 ⋅ t 0 ⋅ π ⋅ di  1 + sin θ  , 11    2 sin2 θ  γ Mj ⋅ γ M0 3


fy ≤ 355 N/ mm , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 2

1) These

Gap:

g ≥ t1+ t2 γ M0 ⋅ γ Mj  Np.Sd M0.Sd  np =  A ⋅f + W ⋅f  1) 11 ,  0 y0 0 y 0  ga / t0 ≥ 1,5   12 ,   0 , 024 ⋅ γ k g = γ 0,2 1 +   g   −133 ,      1+ e 2t 0

Ni.Rd =

Appendix 9.3

limit values are defined in reference [2].

301



Resistance of overlap N, K and KT joints. Chords and brace members are circular hollow sections [1].

N1

d1, 2

N2

; ;;;;

;; ; ; ;; d

t1, 2

2

θ2

1

d

θ1

t0

Np

M0

; ; ; ;;; ;;; ; ; ;; ; ; ; ; ;;; ; q

-e

d0

N3

N1

N2

d3

θ1

M0

Np

d2

θ3

d1

θ2

-e

M0

M0

q1

302

q2


Resistance

Parameters

Validity area 0,2 ≤ di / d0 ≤ 1,0

Chord face yield Compression member: N1.Rd =

fy0 ⋅ t 0

2

sin θ1

(1, 8 + 10, 2β)k g ⋅ k p

11 , γ Mj ⋅ γ M0

Tension member: N2.Rd = N1.Rd

sin(θ1) sin(θ 2 )

β=

Appendix 9.3

m

10 ≤ d0 / t0 ≤ 50

i =1

10 ≤ d1 / t1 ≤ 50

∑ di

m ⋅ d0 m is the number of brace members Overlap: λov ≥ 0,25 γ = d0 / (2 t0) Tension chord: kp = 1 Compression chord:

(

)

k p = 1, 0 − 0, 3 np + np2 ≤ 1 np =

 γ M0 ⋅ γ Mj  Np.Sd M + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0 

    k g = γ 0,2 1 +      λ ov = In a KT joint, also check the following conditions: N1.Rd · sinθ1 ≥ N1.Sd · sinθ1 + N3.Sd · sinθ3 N1.Rd · sinθ1 ≥ N2.Sd · sinθ2

fy ≤ 355 N/ mm , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 2

303

   ,  0, 024 ⋅ γ 12   −q    − 133 ,     2t 0    1 + e    

q ⋅ sin(θi ) di



Resistance of T, Y and X joints and gap N and K joints. Chords are I profiles. Brace members are square, rectangular or circular hollow sections [1]. N1

1

2

t1, 2

θ2

; ; ;;;;;

h

r

tw

tf

e

dw

Det 1

h

g

;; ;;;

θ1

b1, 2

N2

b0

b1, 2

N1

h

t1

1

θ1

r

tf

dw

tw

b0

Det 1 θ ≤ 60°

ga

tf

ga

θ > 60°

g

g

304


Resistance T, X and Y joints, chord web yield Ni.Rd =

fy0 ⋅ t w ⋅ b w sin θi

11 , γ Mj ⋅ γ M0

Parameters Square and rectangular brace members: bw = (hi / sinθi)+ 5(tf + r) bw ≤ 2ti + 10(tf + r) Circular brace members: bw = (di / sinθi)+ 5(tf + r) bw ≤ 2ti + 10(tf + r)

T, X and Y joints, brace member failure Ni.Rd = 2fyi ⋅ ti ⋅ b eff

11 , γ Mj ⋅ γ M0

 fyo  b eff = t w + 2r +   7t f ≤ bi 0r di  fyi 

N and K joints, chord web yield Ni.Rd =

fy0 ⋅ t w ⋅ b w sin θi

11 , γ Mj ⋅ γ M0

N and K joints, brace member failure Ni.Rd = 2fyi ⋅ ti ⋅ b eff

11 , γ Mj ⋅ γ M0

 fyo  b eff = t w + 2r +   7t f ≤ bi or di  fyi 

Appendix 9.3

Validity area All joint types in the table: Brace members in general: bi + hi 1 ≥ 25 ) ti Compression brace member: bi hi E , ≤ 11 , ti ti fyi 10 ≤

di E ≤ 1, 5 ti fyi

Tension brace member: hi / ti ≤ 35 bi / ti ≤ 35 10 ≤ di / ti ≤ 50 2) Chords: b0 E ≤ 0, 75 tf fy0 d w ≤ 400 mm

T and Y joints and N and K joints: b0 Brace members: = 1 = 1 i i β= ,γ = tai hi / bi = 1 m ⋅ b0 m ⋅ b0 2t f m is the number of brace members Chords: Checking is not necessary if the dw E ≤ 1, 5 following conditions are true: tw fy0 β ≤ 1,0 - (0,03γ) X joints: g/ tf ≥ 20 - (28β), Brace members: Square and rectangular brace 0 ,5 < hi / bi ≤ 2,0 members: Chords: 0,75 ≤ b1/ b2 ≤ 1,33 dw E Circular brace members: ≤ 1, 2 tw fy0 0,75 ≤ d1/ d2 ≤ 1,33 m

∑

N and K joints, chord shear fy0 ⋅ A v 11 , Ni.Rd = 3 sin θi γ Mj ⋅ γ M0

m

∑

bi

di

Av = A0 - (2- α)b0 · tf + (tw + 2r)tf Square and rectangular brace members: 1

α= 1+

4 g2 3t 2f

Circular brace members: α = 0 fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm 1) This limit value is defined in reference [2] 2) Eurocode 3 does not define a lower limit for this value.

305

Gap: ga ≥ 1,5 tf 1)



Resistance of overlap N, K and KT joints. Chords are I profiles. Brace members are square, rectangular or circular hollow sections [1]. The resistance needs to be checked for the overlapping brace member only.

N1

b1, 2

N2

h

2

t1, 2

; ;; ;

h

1

θ2

;; r

tw

tf

dw

q

-e

θ1

b0

306


Resistance

Parameters

0,25 ≤ λov < 0,5, brace member failure

[

N1.Rd = fy1 ⋅ t1 2λ ov (2h1 − 4t1) + b eff + b e(ov )

]γ

11 , ⋅ Mj γ M0

λ ov =

Validity area Brace members in general: 0,5 < hi / bi ≤ 2,0 b1/ b2 ≥ 0,75

q ⋅ sin θ1 h1

b eff = t w + 2r + b e(ov ) =

fy0 fyi

7t f ≤ b1

10b1 ⋅ t 22 ⋅ fy 2 b 2 ⋅ t1 ⋅ fy1

≤ b1


[

N1.Rd = fy1 ⋅ t1 2h1 − 4t1 + b eff + b e(ov )

]γ

11 , Mj ⋅ γ M0

λ ov =

q ⋅ sin θ1 h1

b eff = t w + 2r + b e(ov ) = λov ≥ 0,8, brace member failure

[

N1.Rd = fy1 ⋅ t1 2h1 − 4t1 + b1 + b e(ov )

]γ

11 , Mj ⋅ γ M0

λ ov =

bi + hi 1 ≥ 25 ) ti Compression brace member: E bi hi , ≤ 11 , fyi ti ti 10 ≤

fy0 fyi

7t f ≤ b1

10b1 ⋅ t 22 ⋅ fy 2 b 2 ⋅ t1 ⋅ fy1

≤ b1

10b1 ⋅ t 22 ⋅ fy 2 b 2 ⋅ t1 ⋅ fy1

E di ≤ 1, 5 fyi ti

Tension brace member: hi / ti ≤ 35 bi / ti ≤ 35 10 ≤ di / ti ≤ 50 2) Chords: E dw ≤ 1, 2 fy0 tw

q ⋅ sin θ1 h1

b e(ov ) =

Appendix 9.3

d w ≤ 400 mm ≤ b1

E b0 ≤ 0, 75 fy0 tf Overlap: 0,25 ≤ λov ≤ 1,0

With circular brace members, the resistance values are multiplied by π/ 4, and b1 and h1 are replaced with the section diameter d1. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm Member 1 = overlapping member Member 2 = overlapped member 1) This limit value is defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value.

307



Bending resistance of T and X joints. Chords and brace members are square or rectangular hollow sections [1], [3].

Mip.1

Mop.1

b1

h1

;;; ; t1

θ1

h0

M0

N0

N0

M0

t0

b0

308


Resistance In-plane bending moment, β ≤ 0,85, chord face yield

Parameters

Mip.1.Rd =

β = bi / b0 Tension chord: kn = 1 Compression brace member:  0, 4  k n = 1, 3 −  n ≤ 1  β 

 (1 − β)b 0 h1  2 11 , fy0 ⋅ t 0 2 ⋅ h1 + + kn  1 − β b 0 (1 − β)  γ Mj ⋅ γ M0  2h1

n=


In-plane bending moment, 0,85 < β ≤ 1,0, brace member failure    b  11 , Mip.1.Rd = fy1Wpl.1 − 1 − eff  b1 ⋅ h1 ⋅ t1 b γ ⋅   1   Mj γ M0

b eff =

10bi ⋅ t 02

⋅ fy0

b 0 ⋅ ti ⋅ fyi

≤ bi

X joints: fyk = 0,8fy0 Out-of-plane bending moment, β ≤ 0,85, chord face yield [3]  h1(1 + β) 2b 0 ⋅ b1(1 + β)  , 11 Mop.1.Rd = fy0 ⋅ t 02  +  kn β β γ ⋅ − − 2 1 1 ) Mj γ M0  (  Out-of-plane bending moment, 0,85 < β ≤ 1,0, brace member failure b eff =

10bi ⋅ t 02 ⋅ fy0 b 0 ⋅ ti ⋅ fyi

Out-of-plane bending moment, 0,85 < β ≤ 1,0, chord web yield Mop.1.Rd = 0, 5fyk ⋅ t 0 (h1 + 5t 0 )(b 0 − t 0 )

T joints: fyk = fy0 X joints: fyk = 0,8fy0

11 , γ Mj ⋅ γ M0

Out-of-plane bending moment, T joints, distortion of the chord section

[

Mop.1.Rd = 2fy0 ⋅ t 0 h1 ⋅ t 0 + b 0 ⋅ h0 ⋅ t 0 (b 0 + h0 )

]γ

11 , ⋅ Mj γ M0

fy ≤ 355 N/ mm2, θ ≈ 90°, ti ≥ 2,5 mm, t0 ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) These limit values are defined in reference [3].

309

Tension brace member: b1 / t1, h1 / t1 ≤ 35 Compression brace member: b1 / t1, h1 / t1 ≤ 35

Chords: b 0 + h0 1 ≥ 25 ) t0 h 0, 5 ≤ 0 ≤ 2 b0 b 0 h0 , ≤ 35 t0 t0

M0

2   b   11 , Mop.1.Rd = fy1Wpl.1 − 0, 5t1 ⋅ b12 1 − eff   b γ ⋅     Mj γ M0 1

Validity area Brace members in general: bi / b0 ≥ 0,25 0,5 ≤ h1 / b1 ≤ 2 b1 + h1 1 ≥ 25 ) t1

b1 h1 E , ≤ 1, 25 t1 t1 fyi

In-plane bending moment, 0,85 < β ≤ 1,0, chord web yield T joints: 11 , 2 Mip.1.Rd = 0, 5fyk ⋅ t 0 (h1 + 5t 0 ) fyk = fy0 γ ⋅γ Mj

Appendix 9.3

≤ bi

2)



Bending resistance of T, Y and X joints. Chords and brace members are circular hollow sections [1].

Mip.1

Mop.1

;; ;; ;; ; ; ; ; ; ; d1

d1

M0

;;; ;

θ1

t1

M0

Np

t0

Np

d0

310


Resistance

Parameters

In-plane bending moment, chord face yield kp 11 , Mip.1.Rd = 4, 85fy0 ⋅ t 02 ⋅ γ 0,5 ⋅ β ⋅ d1 ⋅ sin(θ1) γ Mj ⋅ γ M0

β = d1/ d0 γ = d0 / (2 t0)

Appendix 9.3

Validity area T ,Y and X joints: 0,2 ≤ d1 / d0 ≤ 1,0


(

10 ≤ d1 / t1 ≤ 50

)

k p = 1, 0 − 0, 3 np + np 2 ≤ 1 np =

T and Y joints:

Out-of-plane bending moment, chord face yield Mop.1.Rd =

fy0 ⋅ t 02

 γ M0 ⋅ γ Mj  Np.Sd M + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0  10 ≤ d0 / t0 ≤ 50

kp

2, 7 11 , ⋅ di ⋅ sin(θi ) 1 − 0, 81β γ Mj ⋅ γ M0

di ≤ d0 - 2 t0, In-plane or out-of-plane bending moment, punching shear failure of the chord Mip.1.Rd = Mop.1.Rd =

fyo 1 + 3 sin(θ1) 3 4 sin2 (θ1)

fyo 3 + sin(θ1) 3 4 sin2 (θ1)

t 0 ⋅ d12

X joints:

11 , γ Mj ⋅ γ M0

10 ≤ d0 / t0 ≤ 40

, 11 t 0 ⋅ d12 γ Mj ⋅ γ M0

fy ≤ 355 N/ mm , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm, t0 ≥ 2,5 mm 2

311



Joints between plates and square or rectangular hollow sections [1]. Mip.1

N1

;; ; ; ; ; ;

h1

t1

N0

N0

h0

M0

M0

t0

b0

N1

b1

t1

h0

t0

b0

312


Resistance

Parameters

Validity area Longitudinal plates:

η = h1 / b0

t1 ≤ 0, 2 b0

Plate longitudinal to hollow section β ≤ 0,85, chord face yield fy0 ⋅ t 02   t  11 , N1.Rd = 2η + 4 1−  1   ⋅ k m    t γ Mj ⋅ γ M0  b0   1− 1  b0 Mip.1.Rd = 0, 5N1.Rd ⋅ h1

Tension chord: km = 1 Compression chord: km = 1,3(1- n) ≤ 1 n=

 γ M0 ⋅ γ Mj  N0.Sd M + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0  Transverse plates: 0,5 ≤ β ≤ 1,0

Plate transverse to hollow section b1 ≥ b0 - 2t0, chord web yield N1.Rd = fy0 ⋅ t 0 (2t1 + 10t 0 )

11 , γ Mj ⋅ γ M0

β=

b1 b0

Plate transverse to hollow section bi ≤ b0 - 2t0, punching shear failure of the chord

N1.Rd =

fy0 ⋅ t 0 3

, (2t1 + 2bep ) γ Mj11 ⋅ γ M0

b ep = 10t 0

Chords: h 0, 5 ≤ 0 ≤ 2 b0 h0 ≤ 35 t0 b0 ≤ 30 t0 b 0 + h0 ≥ 25 t0

b1 ≤ b1 b0

Plate transverse to hollow section 0,5 ≤ β ≤ 1, failure of the plate

N1.Rd = fy1 ⋅ t1 ⋅ b eff

11 , γ Mj ⋅ γ M0

b eff =

2

fy ≤ 355 N/ mm , t0 ≥ 2,5 mm 1)

Appendix 9.3

This limit value is defined in reference [2].

313

10t 02 ⋅ b1 ⋅ fyo b 0 ⋅ t1 ⋅ fy1

≤ b1

1)



Joints between plates and circular hollow sections [1].

Mip.1

N1

; ; ;;; ;;;

h1

t1

Np

Np

M0

t0

;;; ; ;;; ;

M0

d0

Mop.1

N1

b1

t1

Np

t0

M0

;; ; ;; ; ;

M0

Np

314

d0


Resistance

Parameters

Validity area Longitudinal plates:

η = h1 / d0

t1 ≤ 0, 2 d0 η≤4

Longitudinal plate on one side of the hollow section, chord face yield N1.Rd = 5k p ⋅ fy0 ⋅ t 02 (1 + 0, 25 η)

11 , γ Mj ⋅ γ M0

Mip.1.Rd = 0, 5N1.Rd ⋅ h1


(

Mop.1.Rd = 0

np =


Longitudinal plate on both sides of the hollow section, chord face yield 11 , γ Mj ⋅ γ M0

Mip.1.Rd = 0, 5N1.Rd ⋅ h1 Mop.1.Rd = 0 Transverse plate on one side of the hollow section, chord face yield

(

N1.Rd = k p ⋅ fy0 ⋅ t 02 4 + 20β 2

)γ

11 , Mj ⋅ γ M0

β = b1/ d0

Mop.1.Rd = 0, 5N1.Rd ⋅ b1 Mip.1.Rd = 0 Transverse plate on both sides of the hollow section, chord face yield

N1.Rd =

)

k p = 1, 0 − 0, 3 np + np2 ≤ 1

N1.Rd = 5k p ⋅ fy0 ⋅ t 02 (1 + 0, 25 η)

Appendix 9.3

5k p ⋅ fy0 ⋅ t 02

, 11 1 − 0, 81β γ Mj ⋅ γ M0

Mop.1.Rd = 0, 5N1.Rd ⋅ b1 Mip.1.Rd = 0 Longitudinal or transverse plate, punching shear failure of the chord 2t 0 ⋅ fyo  N1.Sd M1.Sd  11 , +  t1 ≤  W1.el   A1 3 γ Mj ⋅ γ M0 fy ≤ 355 N/ mm2, t0 ≥ 2,5 mm

315

Transverse plates: β ≤ 0,4

Chords: d 10 ≤ 0 ≤ 50 t0



Resistance of T, Y and X joints with chord flange plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].

b1

N1 h1

t1

θ1

h0

N0

;;; ; M0

Lp

N0

bp

tp

M0

t0

b0

316


Resistance Yield in the tie beam surface 4) Ni.Rd =

fyp ⋅ t p2

(1− βp )

 2 ηp  11 , + 4 1 − βp    γ Mj ⋅ γ M0 sin θi  sin θi

Parameters βp =

bi bp

ηp =

hi bp

Validity area Square and rectangular brace members: Brace members in general: hi / b0, bi / b0 ≥ 0,25 0,5 ≤ hi / bi ≤ 2 bi + hi 1 ≥ 25 ) ti

Buckling or yield in the tie beam web 4) Ni.Rd =

 fb ⋅ t 0  2hi 11 , + 10t 0   sin θi  sin θi  γ Mj ⋅ γ M0

Tension chord: fb = fy0 compression chord: fb = χ · fy0 (T and Y joints) fb = 0,8χ(sinθi)fy0 (X joints) χ = reduction factor for buckling in buckling class C when slenderness is: fy0 h  1 λ = 3, 46 0 − 2  t0  E(sin θi ) π

Punching shear failure in the tie beam 4) Ni.Rd =

fyp ⋅ t p  2hi  11 , + 2b ep    γ Mj ⋅ γ M0 3 sin θi  sin θi

b ep = γ=

10t p ⋅ bi bp

≤ bi

11 , γ Mj ⋅ γ M0

bp

Ni.Rd =

11 , γ ⋅ 3 sin θi Mj γ M0

3)

bi hi E , ≤ 1, 25 ti ti fyi

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) 3)

2t p

b eff =

10bi ⋅ t p2 ⋅ fyp bp ⋅ ti ⋅ fyi

≤ bi

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 35 b 0 + h0 1 ≥ 25 ) t0 Plate:

X-joints when θ < 90°, tie beam shear yield 4) fy0 ⋅ A v

3)

Tension brace member: bi / ti, hi / ti ≤ 35 Compression brace member: bi / ti, hi / ti ≤ 35


Failure in the strut Ni.Rd = fyi ⋅ ti (2hi − 4ti + 2b eff )

Appendix 9.3

Av = 2h0 · t0

Lp ≥ 1,5 hi / sinθi bp ≥ b0 - 2t0 hi Lp ≥ + b p b p − bi sinθi

(

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in reference [3]. 4) The values in the table are valid, when β ≤ 0,85. The smallest value calculated from the different failure p modes is selected to be the joints strength.

317

)



Resistance of T, Y and X joints with chord side plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3].

b1

N1 h1

t1

N0

M0

t0

hp

h0

N0

;;; ;

M0

;; ;

θ1

tp

Lp

318

b0


Resistance β ≤ 0,85, chord face yield Ni.Rd =

Parameters

k n ⋅ fy0 ⋅ t 02  2 η  11 , + 4 1− β    γ Mj ⋅ γ M0 (1− β) sin θi  sin θi

Validity area Square and rectangular brace members: Brace members in general: hi / b0, bi / b0 ≥ 0,25 3) 0,5 ≤ hi / bi ≤ 2 bi + hi 1 ≥ 25 ) ti

β = bi/ b0 η = hi/ b0 Tension chord: kn = 1 Compression chord: kn = 1,3- (0,4/ β)n ≤ 1 n=

 γ M0 ⋅ γ Mj  N0.Sd M + 0.Sd   11 ,  A 0 ⋅ fy0 W0 fy0 

Tension brace member: bi / ti, hi / ti ≤ 35 Compression brace member: bi / ti, hi / ti ≤ 35

0,85 < β ≤1,0 Calculate the resistance using the values β = 0,85 and β = 1,0. Use the original value of β in linear interpolation. β = 1,0, chord web buckling Ni.Rd =

(

)

fb t 0 + t p  2hi  11 , + 10 t 0 + t p   sin θi  sin θi  γ Mj ⋅ γ M0

(

)

Tension chord: fb = fy0 compression chord: fb = χ · fy0 (T and Y joints) fb = 0,8χ(sinθi)fy0 (X joints) χ = reduction factor for buckling in buckling class C when slenderness is:  h0  fy0 1 λ = 3, 46 − 2 t + t E sin θ π ( )  0 p  i

0,85 ≤ βp ≤ 1- (1/ γ), punching shear failure of the chord Ni.Rd =

fy0 ⋅ t 0  2hi  11 , + 2b ep    γ Mj ⋅ γ M0 3 sin θi  sin θi

b ep =

10t 0 ⋅ b1 ≤ b1 b0

b γ= 0 2t 0 βp > 0,85, brace member failure Ni.Rd = fyi ⋅ ti (2hi − 4ti + 2b eff )

11 , γ Mj ⋅ γ M0

b eff =

10bi ⋅ t 02 ⋅ fy0 b 0 ⋅ ti ⋅ fyi

≤ bi

Appendix 9.3

3)

bi hi E , ≥ 1, 25 ti ti fyi

circular brace members: brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) di E ≤ 1, 5 ti fyi

3)

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 35 b 0 + h0 1 ≥ 25 ) t0 Plate: Lp ≥ 1,5 hi / sinθi

β = 1,0, X-joints, when θ < 90°, chord shear Ni.Rd =

fyo ⋅ A v 0 + fyp ⋅ A vp 3 sin θi

11 , γ Mj ⋅ γ M0

Av0 = 2h0 · t0 Avp = 2hp · tp

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in reference [3].

319

3)



Resistance of gap K, N and KT joints with chord flange plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1]. b1, 2

N2

N1

;;; ;

Det 1

h

h

2

t1, 2

θ2

M0

M0

h0

N0

bp

tp

θ1

1

g

e

t0

Lp

b0

N3

N2

N1

g1 1

h3 θ3

h

h2

θ2 tp

θ1

g2

M0

N0

M0

e

Det 1

Lp θ ≤ 60°

θ > 60° ga

tp

ga

t0

Det 1

g

g

320


Resistance

Parameters


Ni.Rd

  fyp ⋅ t p2  = 8, 9 sin θi    

 bi + hi  11 , i =1 i =1  k n γ 2m ⋅ bp  γ Mj ⋅ γ M0    m

∑

∑

Chord shear Ni.Rd = When

m

m

β=

∑ bi i =1

, η=

hi b0


VSd > 0, 5Vpl.Rd

2   2V   fy0 N0.Rd = A 0 − A v  Sd − 1    Vpl.Rd   γ M0  

When VSd ≤ 0, 5Vpl.Rd , N0.Rd = A 0

1

α= 1+ fy0 γ M0

4g

2

, Vpl.Rd =

fyp ⋅ t p  2hi  11 , + bi + b ep    γ Mj ⋅ γ M0 3 sin θi  sin θi

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 502) Compression brace member: 10 ≤ di / ti ≤ 50 2) di E ≤ 1, 5 ti fyi

3)

Chords: 0,5 ≤ h0 / b0 ≤ 2

b eff =


b0 / t0, h0 / t0 ≤ 35 b 0 + h0 1 ≥ 25 ) t0 ≤ bi

Plate: m −1  m h i L p ≥ 1, 5  + gi   i =1 sinθi i =1  bp ≥ b0- 2t0, tp ≥ 2ti

∑

βp ≤ 1- (1/ γ), punching shear failure of the chord

Ni.Rd =

3 ⋅ γ M0

VSd is the chord shear force at joint

Brace member failure 11 , γ Mj ⋅ γ M0

fyo ⋅ A v

3t 02


Ni.Rd = fyi ⋅ ti (2hi − 4ti + bi + b eff )

Validity area Square and rectangular brace members: Brace members in general: b 0 3) η, β ≥ 0,1+ 100t 0

β ≥ 0,35 0,5 ≤ hi / bi ≤ 2 m is the number of brace members bi + hi 1 ≥ 25 ) γ = b0/(2t0) ti Tension chord: Tension brace member: kn = 1 bi / ti, hi / ti ≤ 35 Compression chord: Compression brace kn = 1,3- (0,4/ β)n ≤ 1 member: γ M0 ⋅ γ Mj  N0.Sd bi / ti, hi / ti ≤ 35 M0.Sd  n=  A ⋅f + W ⋅f  3) 11 ,  0 y0 0 y0  bi hi E , ≤ 1, 25 ti ti fyi m ⋅ b0

Av = (2h0+ α · b0)t0 (square and rectangular brace members) Av = 2h0 · t0 (circular brace members)

fy0 ⋅ A v

Appendix 9.3

b ep = γ=

10t p ⋅ bi

bp 2t p

bp

≤ bi

∑

Gap: g / b0 ≥ 0,5(1-β) g / b0 ≤ 1,5(1-β) g ≥ t1 + t2 ga ≥ 1,5tp1)

With circular brace members, resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di . If g/ b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in the calculation. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in reference [3].

321



Resistance of gap K, N and KT joints with chord side plate reinforcement. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1], [3]. b1, 2

N2

N1

;;; ;

Det 1

h

2

;; ;

e

h0

N0

M0

t0

hp

M0

Lp

tp

N3

N1

N2

h3

h2

1

θ3

h

b0

g2

g1 θ1

t1, 2

θ2

h

θ1

1

g

θ2 N0

M0

M0

e

Det 1

θ ≤ 60°

θ > 60°

ga

ga

g

g

t0

Det 1

Lp

322


Resistance

Parameters


Ni.Rd

 2  fy0 ⋅ t 0  = 8, 9 sin θi    

m  bi hi  h 11 ,  i 1 = i =1 i =1 β= , η= i kn γ m ⋅ b0 b0 2m ⋅ b 0  γ Mj ⋅ γ M0   m is the number of brace  members Tension chord: kn = 1 Compression chord: kn = γ1,3- ⋅(0,4/ β)n ≤ 1  M M0 γ Mj  N0.Sd n= + 0.Sd   11 ,  A 0 ⋅ fy0 W0 ⋅ fy0  m

∑

m

bi +

∑

∑

Chord shear Ni.Rd =

fy0 ⋅ A v 0 + fyp ⋅ A vp 3 sin θi

11 , γ Mj ⋅ γ M0

When VSd > 0, 5VRd N0.Rd

2   2VSd   fy0  + = A 0 − A v0  − 1    Vpl.Rd   γ M0   2     f 2hp ⋅ t p − A vp 2VSd − 1  yp V    pl.Rd   γ M0  

When

VSd ≤ 0, 5VRd

N0.Rd = A 0

fy0 γ M0

+ 2hp ⋅ t p

Av0 = (2h0+ α · b0)t0 Avp = 2hp · tp (square and rectangular brace members) Av0 = 2h0 · t0 Avp = 2hp · tp (circular brace members) 1

α= 1+ Vpl.Rd =

fyp

4 g2 3t 02 fy0 ⋅ A vo + fyp ⋅ A vp

Brace member failure Ni.Rd = fyi ⋅ ti (2hi − 4ti + bi + b eff )

11 , γ Mj ⋅ γ M0

b eff =

10bi ⋅ t 02 ⋅ fy0 b 0 ⋅ ti ⋅ fyi

In a KT joint, also check the following conditions: N1.Rd · sinθ1 ≥ N1.Sd · sinθ1 + N3.Sd · sinθ3 N1.Rd · sinθ1 ≥ N2.Sd · sinθ2 βp ≤ 1- (1/ γ), punching shear of chord Ni.Rd =

fy0 ⋅ t 0  2hi  11 , + bi + b ep    γ Mj ⋅ γ M0 3 sin θi  sin θi

b ep = γ=

10t 0 ⋅ bi ≤ bi b0

b0 2t 0

Validity area Square and rectangular brace members: Brace members in general: b 0 3) η, β ≥ 0,1 + 100t 0 β ≥ 0,35 0,5 ≤ hi / bi ≤ 2 bi + hi 1 ≥ 25 ) ti Tension brace member: bi / ti, hi / ti ≤ 35 Compression brace member: bi / ti, hi / ti ≤ 35 bi hi E , ≤ 1, 25 ti ti fyi

≤ bi

3)

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) di E ≤ 1, 5 ti fyi

3 ⋅ γ M0 VSd is the chord shear force at joint

γ M0

Appendix 9.3

3)

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 35 b 0 + h0 1 ≥ 25 ) t0 Plate: m −1  m h i L p ≥ 1, 5  + gi   i =1 sinθi i =1 

∑

∑

Gap: g/ b0 ≥ 0,5(1-β) g/ b0 ≤ 1,5(1-β) g ≥ t1+ t2 ga ≥ 1,5t01)

With circular brace members, the resistance values are multiplied by π / 4, and bi and hi are replaced with the diameter di. If g/ b0 > 1,5(1-β), the joint is treated as two separate T or Y joints in the calculation. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, t0 ≥ 2,5 mm, ti ≥ 2,5 mm 1)

These limit values are defined in referencereference [2]. Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in referencereference [3]. 2)

323



Resistance of reinforced overlap N and K joints. Chords are square or rectangular hollow sections. Brace members are square, rectangular or circular hollow sections [1]. In the validity area presented in this table, only the brace member failure is a governing failure mode.

h1

N1

θ1

;;; ;

tp

h2

b1, 2

N2

t1, 2

θ2

q

-e

h0

M0

N0

bp

M0

t0

b0

324


Resistance

Parameters


[

Ni.Rd = fyi ⋅ ti 2λ ov (2hi − 4ti ) + b eff + b e(ov )

]γ

11 , Mj ⋅ γ M0

λ ov = b eff =

q ⋅ sin θi hi 10bi ⋅ t 02 ⋅ fy0 b 0 ⋅ ti ⋅ fyi

b e(ov ) =

≤ bi


≤ bi


[

Ni.Rd = fyi ⋅ ti 2hi − 4ti + b eff + b e(ov )

]γ

11 , Mj ⋅ γ M0

λ ov = b eff =

q ⋅ sin θi hi 10bi ⋅ t 02 ⋅ fy0

b e(ov ) =

b 0 ⋅ ti ⋅ fyi

≤ bi


≤ bi

Appendix 9.3

Validity area Square and rectangular brace members: Brace members in general: bi hi , ≥ 0, 25 3) b0 b0 0,5 < hi / bi ≤ 2,0 bi + hi 1 ≥ 25 ) ti b1 / bp ≥ 0,75 Tension brace member: bi / ti, hi / ti ≤ 35 Compression brace member: bi / ti, hi / ti ≤ 35 bi hi E , ≥ 11 , ti ti fyi

3)

Circular brace members: Brace members in general: 0,4 ≤ di / b0 ≤ 0,8 Tension brace member: 10 ≤ di / ti ≤ 50 2) Compression brace member: 10 ≤ di / ti ≤ 50 2) With circular brace members, the resistance values are multiplied by π/ 4, and bi and hi are replaced with the diameter di. fy ≤ 355 N/ mm2 , 30° ≤ θi ≤ 90°, ti ≥ 2,5 mm t0 ≥ 2,5 mm 1) These limit values are defined in reference [2]. 2) Eurocode 3 does not define a lower limit for this value. 3) These limit values are defined in reference [3].


3)

Chords: 0,5 ≤ h0 / b0 ≤ 2 b0 / t0, h0 / t0 ≤ 40 b 0 + h0 1 ≥ 25 ) t0 Plate: tp ≥ 2ti Overlap: 0,25 ≤ λov < 0,8

325

Appendix 9.3


References [1] ENV 1993-1-1: Eurocode 3: Teräsrakenteiden suunnittelu: Liite K: Putkipalkeista valmistettujen tasoristikoiden liitokset, 1994 (ENV 1993-1-1: Eurocode 3: Design of steel structures: Annex K: Hollow section lattice girder connections, 1994) [2] CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996 [3] CIDECT: CIDECT: Design guide for rectangular hollow section joints under predominantly static loading, Verlag TÜV Rheinland GmbH, Köln 1992

326


Appendix 9.4

Appendix 9.4

Estimating the stiffness of moment connections

In framed structures, the stiffness of the joints can be taken into account, and the bending moments transferred across the joint. This reduces the span moments and therefore produces an efficient design. The following formulae are obtained for the bending moments of a beam subjected to uniform load and with a semi-rigid joint at both ends, when the supports are assumed non-deflecting [1]: 2  c  q ⋅L M1 =    c + 2  12

(restraint moment)

(9.4.1)

2  c + 6  q ⋅ L (field moment) M0 =    c + 2  24

(9.4.2)

where c=

Sj ⋅ L E ⋅I

Sj

is the rotational stiffness of the joint is the length of the section is the second moment of area of the section

L I

Sj

q

Sj

L

M1

M1

M0 Figure 9.4.1

A beam with semi-rigid joints.

The rotational stiffness of a welded T joint in square and rectangular hollow sections can be determined according to the guidance in reference [2]: Sj =

1000 ⋅ C* ⋅ t 03

(9.4.3)

52

where Sj is the rotational stiffness of the joint (Nm/Rad) t0 is the wall thickness of the hollow section (mm) C* is the constant obtained from figures 9.4.3- 9.4.7 (N/ mm2) 327

Appendix 9.4


With square hollow sections, the constant C* is taken from figure 9.4.3 when b1/ b0 is less than or equal to 0.7. In other cases, it is taken from figures 9.4.4- 9.4.7. With rectangular hollow sections, the constant C* is calculated as for square hollow sections. This result is multiplied by the correction factor shown in figure 9.4.7. Formula (9.4.3) yields for rotational stiffness an approximation which best corresponds with the bending moment values of the joint, up to the yield moment (Mel.c) of the joint. Mc

Mel.c

Sj Ø

Figure 9.4.2 The moment-rotation curve of a semi-rigid joint.

328


Appendix 9.4

1000/ C*

h1

b1

25 t1

20

h0

t0

t0

/t

1=

15

b0

2, 1,7 0 5 1,5 1,2

5 1,0

10

0,7

5

0,5

5

0,4

Figure 9.4.3

0,45

0,50

0,55

0,60

0,65

Values of constant 1000/ C* for T joints in square hollow sections, when b1/ b0 ≤ 0,7 and b0/ t0 ≥ 10 [2].

329

0,70

b1/b0


Appendix 9.4

1000· C*

1000· C*

b0/t0 = 15

b0/t0 = 20

b1/b0 ≥ 0,7

b1/b0 ≥ 0,7

2,0 1,5

1,5

0,75

t0 / t 1

1,0 1,25 1,5 2,0 3,0

1,0 0,5

0,7

Figure 9.4.4

0,75

0,8

0,85

0,5

2,0

0,5

0,75

t0 / t 1

1,0

1,0

1,5 2,0 3,0

0,5

0,9 b1/b0

0,7

0,75

0,8

0,85

0,9 b1/b0

Values of constant C* for T joints in square hollow sections, when b1/ b0 ≥ 0,7 and b0/ t0 = 15 or b0/ t0 = 20 [2].

1000· C*

1000· C* b0/t0 = 25

b0/t0 = 30

b1/b0 ≥ 0,7

b1/b0 ≥ 0,7

0,5

2,5

2,5 0,5

2,0

2,0 t0 / t 1

1,5

0,75 t0 / t 1

0,75

1,0

1,5

1,0

1,5 1,0

1,5

1,0

0,5

2,0 3,0

0,5

0,7

Figure 9.4.5

0,75

0,8

0,85

0,9 b1/b0

2,0 3,0

0,7

0,75

0,8


330

0,85

0,9 b1/b0


1000· C*

b0/t 0 = 40

1000· C*

b0/t 0 = 35

Appendix 9.4

b1/b0 ≥ 0,7

b1/b0 ≥ 0,7 4,0

0,5

4,0 0,5

3,0

0,75

3,0 t0 / t 1

2,0

t0 / t 1

1,0

0,75

1,5

2,0

1,0

2,0

1,5 2,0 3,0

1,0

0,7

Figure 9.4.6

0,75

0,8

0,85

3,0

1,0

0,7

0,9 b1/b0

0,75

0,8

0,85

0,9 b1/b0


C* / C*

4

b1

b1

h1

h1

5

3

2

1

b1/h1 0

Figure 9.4.7

0,25

0,5

0,75

1,0

1,25

1,5

1,75

Values of constant C* for rectangular hollow sections to those for square, hollow sections [2].

331

Appendix 9.4


References [1] ECCS: Technical Committee 8- Structural stability- Technical working group 8.1/ 8.2 Skeletal structures: Analysis and design of steel frames with semi-rigid joints, First edition 1992 [2] Mang, F et al: The development of recommendations for the design of welded joints between steel structural hollow sections (T- and X- type joints). Final Report No. 5 AD. CIDECT Düsseldorf, 1983

332


Appendix 9.5

Appendix 9.5

Fatigue categories

Tables 9.5.1-9.5.4 present the fatigue categories for hollow sections which are valid when using the nominal stress method in fatigue design. Table 9.5.1 Fatigue categories for hollow sections and splice joints [3]. Structure

Description Cold-formed machine-welded hollow sections.

; ; ; ;; ;;;;

Fatigue category 140

Weld must be free from defects [1] with no stop/start positions.

; ;; ; ; ; ;

71

End-to-end joint with circular hollow sections. Weld must be free from defects [1] with no discontinuities. The height of the weld convexity must not be greater than 10% of the weld width, with smooth transition to section surface. A fatigue category of 90 can be selected if the wall thickness is greater than 8 mm.

56

End-to-end joint with square and rectangular hollow sections. Weld must be free from defects [1] with no discontinuities. The height of the weld convexity must not be greater than 10% of the weld width, with smooth transition to section surface.

50

; ;;

A fatigue category of 71 can be selected if the wall thickness is greater than 8 mm.

333

End-to-end joint with intermediate plate with circular hollow sections. Weld must be free from defects [1] with no discontinuities. A fatigue category of 56 can be selected if the wall thickness is greater than 8 mm.

Appendix 9.5

71

End-to-end joint with intermediate plate with square and rectangular hollow sections. Weld must be free from defects [1] with no discontinuities. A fatigue category of 50 can be selected if the wall thickness is greater than 8 mm.

End-to-end joint with intermediate plate (fillet weld) with circular hollow sections.

;;

36

; ; ; ;; ;; ;;

40

; ; ; ;; ; ;; ;

45


Wall thickness is less than 8 mm.

End-to-end joint with intermediate plate (fillet weld) with square and rectangular hollow sections. Wall thickness is less than 8 mm.

Hollow section welded directly to another cross-section. Weld is non-load-carrying.

The cross-section width parallel to stress is less than 100 mm.

≤ 100 mm

334


Appendix 9.5

Table 9.5.2 Fatigue categories for hollow section-to-plate joints [3]. Fatigue category 80 L ≤ 50 mm 71

50 < L ≤ 100mm

50

L > 100 mm

Structure

Description Longitudinal non-load-carrying plates welded to hollow section.

;;;; ; ; ; L

80

t ≤ 12 mm

71

t > 12 mm

Transverse non-load-carrying plates welded to hollow section.

t

> 10 mm

80

Welds end at a minimum distance of 10 mm from the hollow section edge.

> 10 mm

Shear connectors welded to hollow section flange.

335


Appendix 9.5

Table 9.5.3 Fatigue categories for joints of hollow section lattice structures [3]. Fatigue category 90 t0 / ti ≥ 2,0

Structure

Description Gap K or N joints. Members are circular hollow sections.

N2

; ;;;; ; ;; ;

N1

45 t0 / ti = 1

d1,2

ga

d1

θ1

1)

t1,2

; ;;

Intermediate values determined by linear interpolation

ga ≥ 1,5t0

d2

71

t0 / ti ≥ 2,0

36

t0 / ti = 1

θ2

t0

N0

e

d0

g

N1

N2

h1

;; ; ; ; ga

t1,2

θ2

θ1

g

t0

N0

e

h0


b1,2

h2

Gap K or N joints. Members are square and rectangular hollow sections. The gap must meet the following conditions: 0,5(b0- b1) ≤ g ≤ 1,1(b0- b1) g ≥ 2 t0 1) ga ≥ 1,5t0

b0

56

t0 / ti = 1

N1


N2

h1

b1,2

h2

θ1

q

Overlap K joints with overlap between 0,3 ≤ λov ≤ 1,0

Square and rectangular hollow sections:

t1,2

θ2

-e

t0 / ti ≥ 1,4

h0

71

t0

N0

q · sin(θ ) λov = q / sin(θ11)h1 h1

Circular hollow sections:

b0

N1

N2

λov =

q · sin(θ1) d1

d1,2

; ; ; ;; ; ;; d2

t1,2

;;

d1

θ2

N0

-e

d0

θ1

t0

q

336


≥ 1,4

t0 / ti

50

t0 / ti = 1

Overlap N joints N1 h1

b1,2

N2

;; ;

71

Appendix 9.5

h2


θ1

t0

N0

-e

q

h0

t1,2

θ2

b0

N1

;;;; ;;; ;

N2

d1,2

d2

d1

θ2

t1,2

;;

θ1

t0

N0

-e

d0

q

Validity area of this table: 0,4 ≤ bi / b0 ≤ 1,0, 0,25 ≤ di / d0 ≤ 1,0 b0 ≤ 200 mm, d0 ≤ 200 mm -0,5 h0 ≤ e ≤ 0,25 h0 -0,5 d0 ≤ e ≤ 0,25 d0 t0 ≤ 12,5 mm, ti ≤ 12,5 mm 35° ≤ θ ≤ 50°, b0 / t0 ≤ 25 b 0 + h0 bi + hi , ≥ 25 t0 ti

1)

Eccentricity perpendicular to lattice plane: eop < 0,02b0 and eop < 0,02d0 Fillet welds allowed when ti ≤ 8 mm 1) These limit values are determined in reference [2].

337

Appendix 9.5


Table 9.5.4 Fatigue categories for bolted joints [3]. Fatigue category 112

100

Description Bolted joint with prestressed or nonprestressed bolts transferring shear force. Resistance for prestressed bolts is determined by gross cross-section and for non-prestressed bolts by tension area.

Adjusting bolt

Bolted joint with bearing type adjusting bolts transferring shear force. Bolt resistance is determined by gross crosssection.

;; ;

36

Structure

Bolted joint transferring tensile force.

Bolt resistance is determined by tension area.

References [1]

ENV 1090-1: Teräsrakenteiden valmistus ja asennus. Osa 1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1996 (ENV 1090-1: Execution of steel structures- Part 1: General rules and rules for buildings, 1996)

[2]

CIDECT: Project 5AQ/2: Cold formed RHS in arctic steel structures, Final report 5AQ-5-96, 1996

[3]

SFS-ENV 1993-1-1:Eurocode 3: Teräsrakenteiden suunnittelu. Osa 1-1: Yleiset säännöt ja rakennuksia koskevat säännöt, 1993 (ENV 1993-1-1: Eurocode 3: Design of steel structures. Part 1.1: General rules and rules for buildings, 1993)

338


Appendix 9.6

Appendix 9.6

Cross-section factors in fire design

The calculation of cross-section factors is presented in this appendix as general formulae. The cross-section factors for hollow sections exposed to fire on all sides are given in the tables in Appendix 9.1.

Hollow section exposed to fire on all sides

; ; ; ;;;;

9.6.1

Fire protection does not follow the surface Am / V

Fire protection follows the surface Am / V

t

h

h

t

ri

ri

r0

r0

b

b

d

t

Hollow section

Square or

rectangular Circular

Am / V

Ap / V

2(b + h − 4r0 + πr0 )

(

2t(b + h − 2t) − (4 − π)

r02

− ri2

)

2(b + h)

(

2t(b + h − 2t) − (4 − π) r02 − ri2

d

d ⋅ t − t2

339

)

Appendix 9.6

Hollow section exposed to fire on three sides

; ;;;; ;;

9.6.2




t

h

h

t

ri

ri

r0

r0

b

b

ri

ri

r0

r0

h

Hollow section Square

h

Am / V (b + 2h − 6r0 + 2πr0 )

(

2t(b + h − 2t) − (4 − π)

Rectangular with non-exposed short side Rectangular with non-exposed long side

b

t

b

t

r02

Ap / V

− ri2

)

(b + 2h − 6r0 + 2πr0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 ) (2b + h − 6r0 + 2πr0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 )

340

b + 2h

(

)

(

)

(

)

2t(b + h − 2t) − (4 − π) r02 − ri2 b + 2h

2t(b + h − 2t) − (4 − π) r02 − ri2 2b + h

2t(b + h − 2t) − (4 − π) r02 − ri2


Hollow section exposed to fire on two opposite sides

; ;;;; ;;

9.6.3

Appendix 9.6



t

h

h

t

ri

ri

r0

r0

b

b

t

ri

r0

b

b

t

ri

r0

h

Hollow section Square

Rectangular with non-exposed short sides Rectangular with non-exposed long sides

h

Am / V

Ap / V

(2h − 4r0 + 2π ⋅ r0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 )

2h

(

)

(

)

(

)

2t(b + h − 2t) − (4 − π) r02 − ri2

(2h − 4r0 + 2π ⋅ r0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 )

2h

2t(b + h − 2t) − (4 − π) r02 − ri2

(2b − 4r0 + 2π ⋅ r0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 )

2b

2t(b + h − 2t) − (4 − π) r02 − ri2

341

Appendix 9.6

Hollow section exposed to fire on two adjacent sides

;; ;

9.6.4




t

h

h

t

ri

ri

r0

r0

b

Hollow section

Am / V

Square or rectangular

(b + h − 4r0 + 1, 5π ⋅ r0 ) 2t(b + h − 2t) − (4 − π)(r02 − ri2 )

b

Ap / V b+h

(

2t(b + h − 2t) − (4 − π) r02 − ri2

342

)


Appendix 9.7

Appendix 9.7

Minimum bending radii for square and rectangular hollow sections

;; ;

Tables 9.7.1 and 9.7.2 present the minimum bending radii for square and rectangular hollow sections when bending is made in room temperature with three-roller cold bending. The values in these tables are guideline minimum values that can be obtained with good equipment and careful workmanship. Table 9.7.1 Guideline values for minimum bending radii of square hollow sections in three-roller cold bending [1] b1

y

e

y

Pb is the flow of cross-section [%] b1 − b 100 Pe is the inverted deflection of the compression flange [%] b Rt is internal bending radius [m] e Pe = 100 h Ired is the reduction of the second moment of inertia Iy due to cross-sectional distortion [%]

Pb =

h2

y

y

h

t

h1

b

a) before bending b) after bending Pb = 1 %

h mm 40 50 50 60 60 70 70 80 80 80 90 90 90 100 100 100 100 120 120 120 120 150 150 150 150 180 180 180 200 200 200 250 250 250 300 300

b mm 40 50 50 60 60 70 70 80 80 80 90 90 90 100 100 100 100 120 120 120 120 150 150 150 150 180 180 180 200 200 200 250 250 250 300 300

t mm 4,0 4,0 5,0 4,0 5,0 4,0 5,0 4,0 5,0 6,3 4,0 5,0 6,3 4,0 5,0 6,3 8,0 4,0 5,0 6,3 8,0 5,0 6,3 8,0 10,0 6,3 8,0 10,0 6,3 8,0 10,0 6,3 8,0 10,0 8,0 10,0

Rt m

1,44 2,88 2,72 5,07 4,79 8,17 7,72 12,36 11,68 11,02 17,80 16,83 15,87 24,67 23,32 22,00 20,71 43,41 41,03 38,70 36,44 81,91 77,27 72,75 68,76 123,65 81,47 55,18 176,76 116,47 78,88 376,81 248,27 168,15 460,71 312,03

Pb = 2,5 %

Pb = 5 %

Pb = 7,5 %

Pe = 0,5 %

Pe = 1 %

Pe = 2,5 %

Pe = 5 %

Ired Rt

Ired Rt

Ired Rt

Ired Rt

Ired Rt

Ired Rt

Ired Rt

Ired

%

%

%

%

%

%

%

%

m

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 1 1 3 3 2 1 3 3 2 2 3 3 3 3 3 3 3 3 2 2 2

0,31 0,61 0,58 1,08 1,02 1,74 1,65 2,64 2,49 2,35 3,80 3,59 3,38 5,26 4,97 4,69 4,42 9,26 8,75 8,25 7,77 17,47 16,48 15,51 14,66 20,04 13,21 8,94 28,66 18,88 12,79 61,08 40,25 27,26 74,69 50,58

m

1 2 1 2 2 3 2 3 3 2 3 3 3 4 4 3 2 6 5 4 3 7 5 4 3 6 6 6 6 5 5 5 4 4 4 3

0,22 0,22 0,22 0,34 0,32 0,54 0,51 0,82 0,77 0,73 1,18 1,11 1,05 1,63 1,54 1,46 1,37 2,88 2,72 2,56 2,41 5,43 5,12 4,82 4,55 5,06 3,33 2,26 7,24 4,77 3,23 15,42 10,16 6,88 18,86 12,77

1 3 2 3 3 5 3 6 5 3 7 6 4 8 6 6 3 12 8 7 6 13 9 7 6 11 10 10 10 9 9 9 8 7 7 6

m

0,22 0,22 0,22 0,22 0,22 0,27 0,26 0,41 0,39 0,73 0,60 0,56 0,53 0,82 0,78 0,74 0,69 1,45 1,37 1,29 1,22 2,74 2,58 2,43 2,30 2,26 1,49 1,01 3,23 2,13 1,44 6,89 4,54 3,08 8,43 5,71

343

m

1 0,22 3 0,79 2 0,43 5 2,55 3 1,37 7 6,85 6 3,68 8 16,12 7 8,66 5 4,55 10 34,29 8 18,42 6 9,68 12 67,38 9 36,20 7 19,03 6 9,78 18 216,84 13 116,51 11 61,23 7 31,49 20 487,11 14 255,98 11 131,64 8 70,73 17 831,60 15 416,40 14 218,22 16 1046,76 14 524,13 13 274,69 14 1704,11 12 853,27 10 447,18 11 1270,17 9 665,71

m

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0,22 0,29 0,22 0,95 0,51 2,55 1,37 5,99 3,22 1,69 12,75 6,85 3,60 25,05 13,46 7,07 3,64 80,60 43,31 22,76 11,70 181,06 95,15 48,93 26,29 251,71 126,03 66,05 316,83 158,64 83,14 515,80 258,27 135,35 384,45 201,50

m 2 0,22 2 0,22 2 0,22 2 0,26 2 0,22 2 0,69 2 0,37 2 1,62 2 0,87 2 0,46 2 3,45 2 1,85 2 0,97 2 6,77 2 3,64 2 1,91 2 0,98 2 21,79 2 11,71 2 6,15 2 3,16 2 48,94 2 25,72 2 13,23 2 7,11 2 51,86 2 25,96 2 13,61 2 65,27 2 32,68 2 17,13 2 106,26 2 53,21 2 27,88 2 79,20 2 41,51

m 4 4 4 4 4 4 4 4 4 4 3 4 4 3 4 4 4 3 3 4 4 3 3 4 4 3 4 4 3 3 4 3 3 3 3 3

0,22 0,22 0,22 0,22 0,22 0,26 0,22 0,60 0,32 0,22 1,28 0,69 0,36 2,52 1,35 0,71 0,37 8,10 4,35 2,29 1,18 18,19 9,56 4,92 2,64 15,70 7,86 4,12 19,76 9,89 5,18 32,16 16,10 8,44 23,97 12,56

8 7 8 7 7 7 7 6 7 7 6 7 7 6 6 7 7 6 6 7 7 6 6 7 7 6 6 7 6 6 7 6 6 6 6 6

;; ; Appendix 9.7


Table 9.7.2 Guideline values for minimum bending radii of rectangular hollow sections in three-roller cold bending [1] b

b1

h1

e

h

h2

Pb is the flow of cross-section [%] b1 − b 100 Pe is the inverted deflection of the compression flange [%] b t Rt is internal bending radius [m] e y y y y Pe = 100 h Ired is the reduction of the second moment of inertia Iy due to cross-sectional distortion [%] a) before bending b) after bending

Pb =

Pb = 1 %

H mm 50 60 80 80 90 90 100 100 100 100 100 120 120 120 120 120 120 150 150 150 150 160 160 160 160 200 200 200 200 250 250 250 300 300 300 400 400

B mm 30 40 40 40 50 50 50 50 60 60 60 60 60 60 80 80 80 100 100 100 100 80 80 80 80 100 100 100 100 150 150 150 200 200 200 200 200

Pb = 2,5 % Pb = 5 %

Pb = 7,5 %

Pe = 0,5 %

Pe = 1 %

Pe = 2,5 %

Pe = 5 %

Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired Rt Ired T Rt mm m % m % m % m % m % m % m % m % 4,0 2,18 1 0,47 1 0,22 2 0,22 2 0,26 1 0,22 2 0,22 4 0,22 7 4,0 4,07 1 0,87 2 0,27 3 0,22 3 1,06 1 0,39 2 0,22 3 0,22 6 4,0 8,48 1 1,81 2 0,56 3 0,28 4 3,59 1 1,34 2 0,36 3 0,22 5 5,0 8,01 1 1,71 2 0,53 3 0,27 3 1,93 1 0,72 2 0,22 3 0,22 6 4,0 12,93 1 2,76 3 0,86 4 0,43 5 9,61 1 3,57 2 0,97 3 0,36 5 5,0 12,22 1 2,61 2 0,81 3 0,41 4 5,16 1 1,92 2 0,52 3 0,22 6 4,0 16,92 1 3,61 2 1,12 5 0,57 5 15,02 1 5,58 1 1,51 3 0,56 5 5,0 15,99 1 3,41 2 1,06 3 0,53 5 8,07 1 3,00 2 0,81 3 0,30 5 4,0 18,69 2 3,98 3 1,24 5 0,62 7 22,30 1 8,29 2 2,24 3 0,83 5 5,0 17,66 1 3,77 3 1,17 4 0,59 6 11,98 1 4,45 2 1,20 3 0,45 6 6,3 16,66 1 3,55 2 1,10 3 0,56 4 6,30 1 2,34 2 0,63 3 0,24 6 4,0 29,77 1 6,35 3 1,97 5 1,00 8 48,35 1 17,97 1 4,86 3 1,81 5 5,0 28,14 1 6,00 3 1,86 5 0,94 6 25,98 1 9,66 1 2,61 3 0,97 5 6,3 26,54 1 5,66 2 1,76 3 0,89 6 13,65 1 5,07 2 1,37 3 0,51 6 4,0 34,81 2 7,42 4 2,31 7 1,16 11 90,14 1 33,50 2 9,06 3 3,37 5 5,0 32,90 2 7,02 3 2,18 6 1,10 9 48,43 1 18,00 2 4,87 3 1,81 6 6,3 31,04 1 6,62 3 2,06 4 1,04 6 25,45 1 9,46 2 2,56 3 0,95 6 4,0 69,50 3 14,82 5 4,60 11 2,32 18 376,84 1 140,08 2 37,86 3 14,07 5 5,0 65,69 3 14,01 5 4,35 8 2,20 12 202,48 1 75,26 2 20,34 3 7,56 5 6,3 61,97 2 13,21 3 4,10 6 2,07 9 106,41 1 39,55 2 10,69 3 3,97 6 8,0 58,34 1 12,44 3 3,86 6 1,95 7 54,72 1 20,34 2 5,50 3 2,04 6 4,0 176,59 1 28,63 2 7,23 5 3,23 7 244,29 1 73,94 1 15,23 3 4,61 5 5,0 119,60 1 19,39 2 4,90 5 2,19 6 128,03 1 38,75 1 7,98 3 2,42 5 6,3 79,88 1 12,95 2 3,27 4 1,46 5 65,56 1 19,85 1 4,09 3 1,24 5 8,0 52,63 1 8,53 3 2,15 3 0,96 5 32,83 1 9,94 2 2,05 3 0,62 5 5,0 254,95 1 41,33 3 10,44 4 4,66 5 208,42 1 63,09 1 13,00 3 3,93 5 6,3 170,28 1 27,60 3 6,97 3 3,12 5 106,74 1 32,31 1 6,66 3 2,01 5 8,0 112,19 1 18,19 2 4,59 3 2,05 5 53,45 1 16,18 2 3,33 3 1,01 5 10,0 75,98 1 12,32 2 3,11 3 1,39 4 28,01 1 8,48 2 1,75 3 0,53 6 6,3 366,57 1 59,42 3 15,00 4 6,71 5 316,79 1 95,89 1 19,75 3 5,98 5 8,0 241,53 1 39,15 3 9,89 3 4,42 5 158,62 1 48,01 2 9,89 3 2,99 5 10,0 163,58 1 26,52 2 6,70 3 2,99 5 83,13 1 25,16 2 5,18 3 1,57 6 6,3 684,25 1 110,92 3 28,01 5 12,52 6 667,45 1 202,02 1 41,62 3 12,60 5 8,0 450,84 1 73,09 3 18,45 4 8,25 5 334,20 1 101,16 2 20,84 3 6,31 5 10,0 305,34 1 49,50 3 12,50 3 5,59 5 175,15 1 53,01 2 10,92 3 3,31 6 8,0 1177,85 1 190,94 1 48,21 2 21,55 3 242,86 1 73,51 1 15,14 3 4,58 5 10,0 797,71 1 129,32 1 32,65 3 14,60 3 127,28 1 38,52 1 7,94 3 2,40 5

References [1] Kennedy John B: Minimum bending radii for square & rectangular hollow sections (3-roller cold bending). CIDECT report 11C-88/14-E. 344


Appendix 9.8

Appendix 9.8

WinRAMI software

The WinRAMI software by Rautaruukki is an easy-to-use tool for the design of hollow section structures. The program is intended for calculating uniplanar frame structures. WinRAMI can be used for calculating the actions of the structure and for designing members and their joints with structures made of hollow sections. For other types of structures, WinRAMI can be used for calculating the force quantities only. The user interface of WinRAMI utilizes the latest Windows technology. You can create the structural model by drawing the image of the structure with the mouse. The profiles you want to use are defined after drawing the model in the following way. • Select the parts of the structural model you want to place the profile in. • Select the appropriate profile from the Hollow section module. • Move the selected tube profile on the structural model by using the mouse. • The program creates automatically a link (OLE2) to the Hollow section module. Also the loads, edge conditions and joints of the structure can be defined similarly, by simply pointing with the mouse. WinRAMI calculation methods The structural model is determined using the element method. WinRAMI uses an element with 7 degrees of freedom. With the element, the displacement of the centre of gravity axis in crosssection class 4 can be accounted for. The default axis is the total cross-section centre of gravity axis. Hardware requirements The program functions well with Pentium 100Mhz processor and 16MB RAM. Operating system can be either Windows 3.xx, 95 or NT. Structural element modules:

Static processor:

WinRAMI Structural model, loads, actions and displacements

Hollow section

E OL

ink 2l

Hollow section cross-sectional properties and resistance values

Joint modules:

Liicont lattice chord and brace member joints

Moncont moment-rigid joints

Figure 9.8.1 Structure of WinRAMI For additional information on WinRAMI, please contact Rautaruukki technical customer service (see page 351). 345

Design Handbook for RautaRuukki Structural Hollow Sections

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