Background to Eurocodes

CEN / TC250 / SC3 / N1639E - rev2 Institute for Steel Structures Univ. Prof. Dr.-Ing. Markus Feldmann Feldmann Mies-van-der-Rohe-Str. Mies-van-der-Rohe-Str. 1 D-52074 Aachen

Excerpt from the Background Document to EN 1993-1-1 Flexural buckling and lateral buckling on a common basis: Stability assessments according to Eurocode 3

G. Sedlacek, J. Naumes

Aachen, 17.03.2009

Tel.: +49-(0)241-8025177 Fax: +49-(0)241-8022140

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page II / 142

Table of content

Table of content Executive summary

1

1

General

3

2

Reference models for flexural buckling

5

3

4

5

6

2.1

Use of 2nd order theory with imperfections

5

2.2

Reference model of Maquoi-Rondal

5

2.3

European Standard flexural buckling curves

9

2.4

Use of the European buckling curves for other boundary conditions

12

2.5

Conclusions

16

Consistent determination determination of the flexural buckling resistance of columns with nonuniform cross-sections and non-uniform compression loads on elastic supports

17

3.1

Approach for solution

17

3.2

Options for assessment

20

3.3

Determination Determination of the relevant location x d (option 1)

21

3.4

Modification of the buckling curve (option 2)

22

Consistent determination of the resistance to lateral-torsional lateral-torsional buckling

27

4.1

Application of the reference model of Maquoi-Rondal Maquoi-Rondal

27

4.2

Application of the „European lateral torsional buckling curves“ for the general loading case for lateral torsional buckling

33

Conclusions Conclusions for „Recommendations „Recommendations for NDP´s“ in EN 1993-1-1

43

5.1

Procedure in EN 1993-1-1, section 6.3.1

43

5.2

Procedure according to EN 1993-1-1, section 6.3.2.1 and section 6.3.2.2

43

5.3

Procedure according to EN 1993-1-1, section 6.3.2.3

45

5.4

Procedure according to EN 1993-1-1, section 6.3.2.4

46

5.5

Procedure according to EN 1993-1-1, section 6.3.4

46

5.6

Imperfection according to EN 1993-1-1, section 5.3.4 (3)

47

Consideration Consideratio n of out of plane loading

49

6.1

Transverse loads on the standard column in compression

49

6.2

Out of plane bending and torsion for the basic situation for lateral torsional buckling

53

6.3

General case of out of plane bending and torsion

54

6.4

Proof of orthogonality orthogonality for the series-development series-development

55

6.5

Comparison with test results

56 page I

Excerpt from the Background Document to EN 1993-1-1 7

Guidance for applications

57

7.1

General

57

7.2

Design aids

60

7.3

Examples to compare the results of the general method using the European lateral buckling curve with results of the component method in Eurocode 3Part 1-1, section 6.3.2

71

7.4

Examples for sheet-piling

74

7.5

Lateral torsional buckling of beams with fin-plate connections

82

7.6

Verification of haunched beams

86

7.7

Assessment of gantry-girders

91

7.8

Channel sections

94

8

Analysis of imperfections and conclusions for tolerances for fabrication

101

8.1

General

101

8.2

Approaches to determine geometrical imperfections for tolerances

104

9

Design principles for obtaining sufficient reliability by numerical assessments in EN 1990 – Basis of structural design

111

9.1

Objective

111

9.2

First order reliability method (FORM)

113

9.3

Example for the application of FORM

117

9.4

Assumption for semi-probabilistic design

120

9.5

Determination of design values of resistances and action effect in semiprobabilistic design

126

9.6

Examples for determining the design values of combined action effects

127

9.7

Determination of γ M-values for steel structures

130

10

Literature

page II / 142

141

Executive summary

Executive summary (1)

This document is an excerpt from the background document to EN 1993-1-1, that is being prepared for publication through the Joint Research Centre (JRC) of the Commission in Ispra for the maintenance, further harmonisation, further development and promotion of Eurocode 3. It has the status of an information and technical guidance under the responsibility of the authors G. Sedlacek and J. Naumes. This document is extensively discussed and commented between the authors and Ch. Mueller, F. Bijlaard and R. Maquoi in the meeting of 22 July 2008 at the RWTHAachen. Contributions of Prof. D. Ungermann, Prof. F. Bijlaard, Dr. A. Schmitt, Prof. C. Seeßelberg and Prof. I. Baláž to the examples and design aids in section 7 have been included.

(2)

The document gives: 1. an explanation of the European flexural buckling curves and their background (Maquoi-Rondal) 2. an explanation of the European lateral torsional buckling curves and their background (Stangenberg-Naumes) consistent with the European flexural buckling curves 3. an explanation of the extension of the out-of-plane buckling verification to the beam-column with biaxial bending and torsion (Naumes) 4. the explanation of the workability of these verification methods by worked examples.

(3)

The document completes the design rules for the use of the “general method” in EN 1993-1-1 in the form of a “Non-contradicting complementary information”. G. Sedlacek, J. Naumes, F. Bijlaard, R. Maquoi, Ch. Mueller

page 1 / 142

Excerpt from the Background Document to EN 1993-1-1

page 2 / 142

General

1

General (1)

For the development of the design rules of Eurocode 3 the basic reliability requirements, laid down in EN 1990 – Eurocode – Basis of structural design – [1], have been applied, that lead to the following principles: 1. The basis of resistance rules R are the results of large scale tests. The resistance rules are presented as formulae R(X ) i deducted from mechanical models used to describe the behavior of the test specimens dependant on relevant parameters X i at the ultimate state. The resistance formulae have been calibrated to the test results. 2. This calibration has been carried out by a statistical evaluation of the test results Rexp with the resistance model Rcalc so that it gives characteristic values. Also partial factors of EN 1990.

γ Mi have been derived, that fulfill the reliability requirements

3. The models for resistances are presented in terms of a hierarchy with a reference model Rref on the top, which is used as a basis for simplifications. Any simplified model R simpl is conservative in relation to the reference model Rref . 4. All reference models are consistent, i.e. they do not give conflicting results when compared with other reference models. (2)

This also applies to the design models for flexural buckling and lateral torsional buckling, as presented in the following.

page 3 / 142


page 4 / 142


2


2.1

Use of 2nd order theory with imperfections

(1)

The highest rank in the hierarchy for stability rules for bar-like structures and structural components has the use of 2 nd order theory with imperfections.

(2)

Imperfections are composed of structural imperfections (e.g. from residual stresses from fabrication) and of geometrical imperfections.

(3)

First historical attempts to explain the results of column buckling test and lateral torsional buckling tests were based on a model with deterministic assumptions for residual stress pattern, geometrical imperfections and material properties for calculating buckling coefficients that permitted a „smaller-equal“-comparison with test results.

(4)

A breakthrough were such calculations of Beer and Schulz, that assumed standardized residual stress distributions, a geometrical imperfection of ℓ/1000 and the minimum value of the yield strength for their finite-element calculations, to produce the “European buckling coefficients”, published by the ECCS.

(5)

For the preparation of Eurocode 3 [2] these values have not been applicable because of the following reasons: 1. there was no justification by a reliability analysis with test results, 2. the numerical values produced for a set of slendernesses could not be described by a formula with a mechanical background without a certain scatter.

(6)

2.2 (1)

Therefore these „European buckling coefficients“ were not used as a Eurocodereference model.

Reference model of Maquoi-Rondal A new approach for a reference model in conformity with the Eurocode-requirements was prepared by Maquoi-Rondal [3]. These authors described the column-buckling tests with the model of a column simply supported at its ends with an equivalent geometrical imperfection in the form of a half-sinus wave, that included both structural and geometrical imperfections, see Figure 2.1.

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Figure 2.1: Simply supported column with initial imperfection

(2)

ini

The amplitude of this equivalent geometrical imperfection was defined by

=

e0

M R N R

⋅ (λ − 0,2)⋅ α

(2.1)

where M R

-

gives the influence of the cross-sectional shape and the resistance

N R

model, e.g. for I-Profiles and an elastic model M R N R

-

≈

A Fl ⋅ h

2 ⋅ A Fl

≈

h

2

⎯λ gives the influence of the slenderness, e.g. for I-Profiles 2 A Fl f y l 2 l 4 f y λ = = h π E EA Fl h 2 2 ⋅ π 2

-

α0 is the imperfection factor, that covers all parameters not included in the

simple model in Figure 2.1 (e.g. structural imperfections from residual stresses, model uncertainties, and in particular the reliability correction of the imperfection e0 on the basis of evaluations of column tests, according to EN 1990 – Annex D, to obtain characteristic values with the resistance formula. For certain I-Profiles the equivalent geometrical imperfection is e.g. with α0 = 0,34 and f y = 235 N/mm² for large slenderness values e0 l

(3)

λ :

f 1 4 1 1 ⋅ 0,34 ⋅ y = 0,108 ⋅ = 2 π 30 280 E

≈ ⋅

As the correction factor α 0 for the equivalent geometrical imperfection has been determined from a comparison of resistances R exp determined from tests and resistances R cal determined from calculations, the equivalent geometrical imperfection is

page 6 / 142

Reference models for flexural buckling only defined in association with the resistance model used. Both, the resistance model and the choice of the equivalent geometrical imperfections for the column with uniform cross-section and uniform compression load constitute the reference model for stability checks with the highest rank in the hierarchy for flexural buckling. (4)

Figure 2.2 shows the resistance model for the cross-sectional assessment which includes a linear interaction of the resistances for compression and for bending. If the action-effects from Figure 2.1 are inserted in this model, the formula for the “European column buckling curves”

χ (λ ) are obtained, that yield to the assessment for-

mula for column buckling N Ed =

Rk

γ M

=

⋅ N pl γ M

(2.2)

The old „European buckling coefficients“ of Beer and Schulz have been replaced by the new „European buckling curves“ calibrated to tests.

Figure 2.2: Derivation of -value

(5)

The comparison of the „basic equation“ and the „e0-assumption“ in Figure 2.2 makes clear that the fractures N R /M R and M R /N R compensate each other. This means, that the assumption for the equivalent geometrical imperfection (2.1) and the cross-sectional assessment in Figure 2.2 must use the same definition of the resistance M R (elastic or plastic).

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Excerpt from the Background Document to EN 1993-1-1 (6)

To illustrate this requirement, Figure 2.3 shows the determination of a value of the European buckling curve

χ (λ ) via the intersection of the load-deformation curve and

the resistance-deformation curve: 1. The curves for action effects are based on two equivalent geometrical imperfections a. for elastic resistance 1 b. for linear plastic resistance 2

NEd / Npl

χ 2 1

1

MR = Mel

3

MR = Mpl ε

Figure 2.3:

[‰]

Load deformation curves acc. to Marquoi-Rondal-model using different cross sectional resistances

2. The intersection points of the load-deformation curves with the relevant resistance-deformation curves are on the same level

χ (λ ) , only the deformations

are different. 3. FEM-calculations with a more accurate resistance model with geometrical and material non-linearities and suitably adjusted structural (residual stresses) and geometrical (measured) imperfections are given in Figure 2.4. The results confirm: 1. the levels of χ (λ ) determined with the resistance models 3

1, 2

and

are very accurate,

2. the assumption of a linear elastic cross-sectional resistance is sufficient as large plastic deformations only form in the post-critical part of the load-deformation curves. 3. the residual stress patterns for rolled sections 4 and welded sections 5

give about the same χ -values, however the „deformation capacity“

on the level of χ is different.

page 8 / 142


NEd / Npl

1

3

χ

4

Figure 2.4:

2.3 (1)

5

1

MR = Mel

3

MR = Mpl / (1 - 0,5  a) ; acc. to [2] equ. 6.36

4

FEM

rolled profile

5

FEM

welded profile

ε [‰]

comparison between load-deflection curves acc. to Marquoi-Rondal-model and FEM-calculations

European Standard flexural buckling curves Figure 2.5 shows the European flexural buckling curves together with the imperfection factors α0, and Table 2.1 gives the allocation of these imperfection factors to various shapes of cross-section and ways of fabrication. 1,2

Knickspannungslinie 1,0

a

b

c

d

Imperfektionsbeiwert α 0,13 0,21 0,34 0,49 0,76 a0 a b c

0,8

] [

a0

Euler

d 0,6

0,4

0,2

0,0 0,0

0,5

1,0

1,5

_ [-]

2,0

2,5

3,0

Figure 2.5: European column buckling curves [2]

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Excerpt from the Background Document to EN 1993-1-1 Table 2.1: Selection of buckling curve for a cross sections [2]

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Reference models for flexural buckling (2)

Figure 2.6 gives a visual impression of the test results and buckling curves, and Figure 2.7 shows the

necessary to obtain the design values of resistances. γ M-values

1,2 KSL a0 KSL a KSL b

1,0

KSL c KSL d Euler 0,8

A5.1: IPE160, S235 A5.2: IPE160, S235 A5.3: IPE160, S235

] [

A5.4: IPE160, S235

0,6

A5.5: IPE160, S235 A5.6: IPE160, S235 A5.7: IPE160, S235 0,4

A5.10: HEM340, S235 A5.11: HEM340, S235

0,2

0,0 0

0,5

Figure 2.6:

1

_ 1,5 [-]

2

2,5

3

Test results and column buckling curves for buckling about weak axis (buckling curve b) [4]

1,15 Versuchsauswertung 1,13

Normenvorschlag 1,10 1,08

1,08

M1,05

1,00

1,00

0,95 0,0

0,2

0,4

0,6

0,8

Figure 2.7: Partial factor

1,0

M1

1,2

1,4 _ 1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

[4]

page 11 / 142


2.4 2.4.1

Use of the European buckling curves for other boundary conditions General

(1)

The use of the sinus-function as shape of imperfection for columns is restricted to the simply supported column with hinged ends, uniform cross-section and constant compression force as illustrated in Figure 2.1.

(2)

For the column with uniform cross-section and constant compression force and other end conditions the imperfection depends on the buckling mode expressed by

η crit = a1 sin (κ x ) + a2 cos(κ x ) + a3κ x + a4

η crit , that can be (2.3)

where

κ 2 =

N crit

(2.4)

EI

a1, a2, a3, a4 = constants depending on the boundary conditions

(3)

The differential equation can be written in the form

′′′′ + κ 2η el ′′ = η el

qinit EI

=−

N Ed EI

′′ η init

(2.5)

where

η init ( x) = c0 c0

(4)

η crit ( x) ′′ ,max η crit

(2.6)

= e0 κ 2

(2.7)

In conclusion the equivalent geometrical imperfection is

η init ( x) =

e0 ⋅ N crit

′′ ,max EI η crit

⋅ η crit ( x)

(2.8)

The loading from the imperfection is qinit ( x) = N Ed

e0 ⋅ N crit

′′ ,max EI η crit

′′ ( x) ⋅ η crit

(2.9)

and the bending moment from the imperfection is

′′ = M II ( x) = − EI η el

2.4.2 (1)

e0 ⋅ N Ed N Ed

1−

⋅

′′ ( x) η crit ′′ , max η crit

N crit

Examples For the simply supported column, see Figure 2.1, the values are:

page 12 / 142

(2.10)


κ =

π l

π x ⎞ η crit ( x) = a1 sin⎛ ⎜ ⎟ ⎝ l ⎠ 2

π ⎞ ⎛ π x ⎞ ′′ ( x) = −a1 ⎛ η crit ⎟ ⎜ ⎟ ⋅ sin⎜ ⎝ l ⎠ ⎝ l ⎠ η init ( x) = eo

⎛ π ⎞ ⎜ ⎟ ⎝ l ⎠

2

⎛ π ⎞ ⎜ ⎟ ⎝ l ⎠

2

⎛ π x ⎞ ⎛ π x ⎞ ⎟ = eo ⋅ sin⎜ ⎟ ⎝ l ⎠ ⎝ l ⎠

⋅ sin⎜ 2

⎛ π x ⎞ ⎛ π ⎞ qinit ( x) = eo ⋅ ⎜ ⎟ N Ed ⋅ sin⎜ ⎟ ⎝ l ⎠ ⎝ l ⎠ N Ed ⎛ π x ⎞ sin⎜ M II ( x) = e0 ⎟ N Ed l ⎠ ⎝ 1−

π 2 EI l 2

(2)

For a column with clamped ends, see Figure 2.8 the values read:

κ =

2π l

⎛

2π ⎞ ⎞ x ⎟ ⎟⎟ ⎝ l ⎠ ⎠

η crit ( x) = a1 ⎜⎜1 − cos⎛ ⎜ ⎝

2

2π ⎞ ⎛ 2π ⎞ ′′ ( x) = a1 ⎛ η crit ⎜ ⎟ cos⎜ x ⎟ ⎝ l ⎠ ⎝ l ⎠ 2

η init ( x) =

⎛ 2π ⎞ ⎜ ⎟ l eo ⎝ ⎠ 2 ⎛ 2π ⎞ ⎜ ⎟ ⎝ l ⎠

⎛

2π ⎞ ⎞ x ⎟ ⎟⎟ = eo ⎝ l ⎠ ⎠

⋅ ⎜⎜1 − cos⎛ ⎜ ⎝

⎛

2π ⎞ ⎞ x ⎟ ⎟⎟ ⎝ l ⎠ ⎠

⋅ ⎜⎜1 − cos⎛ ⎜ ⎝

2

⎛ 2π ⎞ ⎛ 2π ⎞ qinit ( x) = eo ⋅ ⎜ ⎟ N Ed ⋅ cos⎜ x ⎟ ⎝ l ⎠ ⎝ l ⎠ N Ed 2π ⎞ M II ( x) = e0 ⋅ cos⎛ ⎜ x ⎟ N Ed ⎝ l ⎠ 1− 2 EI ⋅ (2π l )

Figure 2.8: Column with clamped ends under compression force NEd

(3)

For a column with a hinged end and a clamped end, see Figure 2.9 the values are

page 13 / 142


κ =

ε l

where

ε = 4,4937

⎧⎛ ε ⋅ x ⎞ ⎞ ⎛ ε ⋅ x ⎞ ε ⋅ x ⎫ η crit ( x) = a1 ⎨⎜⎜1 − cos⎛ ⎜ ⎟ ⎟⎟ ⋅ ε + sin⎜ ⎟− ⎬ ⎝ l ⎠ ⎠

⎩⎝

⎝ l ⎠

l ⎭

2 ⎧⎪ ε 3 ε ⋅ x ⎞ ⎛ ε ⎞ ⎛ ⎛ ε ⋅ x ⎞⎫⎪ ′′ ( x) = a1 ⎨ 2 ⋅ cos⎜ η crit ⎟ − ⎜ ⎟ sin ⎜ ⎟⎬ ⎝ l ⎠ ⎝ l ⎠ ⎝ l ⎠⎪⎭ ⎪⎩ l

⎛ ⎛ ε ⋅ x ⎞ ⎞ ⎛ ε ⋅ x ⎞ ε ⋅ x ⎜⎜1 − cos⎜ ⎟ ⎟⎟ ⋅ ε + sin ⎜ ⎟− l l l ⎝ ⎠ ⎝ ⎠ ⎠ η init ( x) = eo ⎝ ⎛ ε ⋅ x d ⎞ ⎛ ε ⋅ x d ⎞ ε ⋅ cos⎜ ⎟ − sin⎜ ⎟ l l ⎝ ⎠ ⎝ ⎠ with xd = xηcrit, ′′ max

≈ 0,65 ⋅ l the loading q and the bending moment M II reads ⎛ ε ⎞

qinit ( x) = eo ⋅ N Ed ⋅ ⎜ ⎟ ⎝ l ⎠

= eo ⋅ N Ed ⋅

M II ( x) = e0

= e0

1−

ε ⋅ x ⎞ ⎛ ε ⋅ x ⎞ ε ⋅ cos⎛ ⎜ ⎟ − sin⎜ ⎟

2

⋅

⎝ l ⎠ ⎝ l ⎠ ε ⋅ cos(0,65 ⋅ ε ) − sin (0,65 ⋅ ε )

− 4,3864 ⎛ ⎛ ε ⋅ x ⎞ ⎛ ε ⋅ x ⎞ ⎞ cos sin ⋅ ⎜ ⋅ − ε ⎜ ⎟ ⎜ ⎟ ⎟⎟ ⎜ 2 ⎝ l ⎠

⎝

l

⎝ l ⎠ ⎠

ε ⋅ x ⎞ ⎛ ε ⋅ x ⎞ ε ⋅ cos⎛ ⎜ ⎟ − sin⎜ ⎟

N Ed N Ed

⋅

⎝ l ⎠ ⎝ l ⎠ ε ⋅ cos(0,65 ⋅ ε ) − sin (0,65 ⋅ ε )

2

EI ⋅ (ε l )

− 0,2172 ⋅ N Ed 1−

N Ed

⎛ ε ⋅ x ⎞ ⎛ ε ⋅ x ⎞ ⎞ ⋅ ⎜⎜ ε ⋅ cos⎛ ⎜ ⎟ − sin ⎜ ⎟ ⎟⎟ ⎝

⎝ l ⎠

⎝ l ⎠ ⎠

2

EI ⋅ (ε l )

The relevant location for the cross-sectional assessment xd is at the point of maximum curvature, which compared to the previous examples (Euler-Column I and IV) no longer corresponds to the point of maximum deflection. With xd = xηcrit, ′′ max

≈ 0,65 ⋅ l

follows M II ( xd ) = e0

1−

N Ed N Ed

⋅ 1,0 2

EI ⋅ (ε l )

The bending moment at the point of maximum deflection xηcrit,max M II ( xη crit ,max ) = M II ( xd ) ⋅ 0,98

page 14 / 142

≈ 0,6 ⋅ l results to


Figure 2.9: Column with one hinged and one clamped end under compression force N Ed

(4)

For a column on elastic foundation, see Figure 2.10 the differential equation reads:

′′′′+ κ 2η el ′′ + η el

c

η el =

EI

qinit EI

=

′′ − N Ed η init EI

Figure 2.10: Elastic embedded column under compression force NEd

The shape of the eigenmode results from the assumption

⎛ π x ⎞ ⎟ ⎝ l ⎠

η crit = a1 sin⎜

where ℓ is the wave-length. This gives from the differential equation 2 ⎡ ⎛ π ⎞4 ⎤ ⎛ π ⎞ ⎛ π ⎞ ⎢ EI ⎜ ⎟ − N crit ⎜ ⎟ + c ⎥ ⋅ a1 ⋅ sin⎜ x ⎟ = 0 ⎝ l ⎠ ⎝ l ⎠ ⎢⎣ ⎝ l ⎠ ⎥⎦

and 2

⎛ π ⎞ ⎛ l ⎞ N crit = EI ⋅ ⎜ ⎟ + c ⋅ ⎜ ⎟ ⎝ l ⎠ ⎝ π ⎠

2

the minimum of which is obtained for

∂ N crit ⎡ π ⎞ ⎛ l ⎞ = ⎢− EI ⋅ ⎛ ⎜ ⎟ + c⋅⎜ ⎟ ∂l ⎝ l ⎠ ⎝ π ⎠ ⎢⎣ 2

2

⎤ 2 ⎥⋅ =0 ⎥⎦ l

and hence l

π

=4

EI c

page 15 / 142

Excerpt from the Background Document to EN 1993-1-1 so that N crit = EI ⋅

1 EI

EI

+c⋅

c

=2

EI ⋅ c

c

Therefore the values read:

κ 2 =

N crit EI

= 2⋅

c EI

⎛ EI ⎞ x ⎟ ⎟ c ⎝ ⎠ ⎛ ⎞ ′′ = − a1 EI sin⎜⎜ 4 EI x ⎟⎟ η crit c ⎝ c ⎠

η crit = a1 sin⎜⎜ 4

η imp = eo q imp

2 2

c

1−

(1)

c

= eo ⋅ N Ed ⋅ 2 ⋅

M II ( x) = e0

2.5

⎛ EI ⎞ ⎛ EI ⎞ c ⎜ 4 x ⎟ sin x ⎟ = eo ⋅ ⋅ ⎟ ⎜ c ⎟ c EI ⎝ ⎠ ⎝ ⎠

⋅ sin⎜⎜ 4 EI EI

c EI

N Ed N Ed

⎛ EI ⎞ x ⎟ ⎟ ⎝ c ⎠

⋅ sin ⎜⎜ 4

⎛ EI ⎞ x ⎟ ⎟ ⎝ c ⎠

⋅ sin⎜⎜ 4

2 EI ⋅ c

Conclusions The „reference model“ for determining the flexural buckling resistance of columns with uniform cross-section and uniform compression load according to Figure 2.1 and Figure 2.2 is not only the reference model for any simplification, but also the reference model for other design situations because of the consistency requirement: 1. flexural buckling of columns with non-uniform distribution of cross-section and compression force and also with elastic support, 2. lateral-torsional buckling of columns and beams, 3. plate buckling of unstiffened and stiffened plate fields. This is because the „reference model“ is included in these design situations for particular configurations of parameters.

(2)

In the following it is demonstrated, how flexural buckling of columns with non-uniform cross-sections and non-uniform compression forces and lateral torsion buckling of columns and beams with whatever given loads can be assessed in compliance with the „reference model“ of the simple column: The application for plate buckling is not included in this report.

page 16 / 142

Consistent determination of the flexural buckling resistance of columns with non-uniform CS

3

Consistent determination of the flexural buckling resistance of columns with non-uniform cross-sections and non-uniform compression loads on elastic supports

3.1 (1)

Approach for solution The differential equation for the column with non-uniform cross-section and nonuniform compression force on continuous elastic supports reads:

( EI ( x)η ′′)″ + α crit ( N E ( x)η ′ )′ + c( x) ⋅ η = 0

(3.1)

where αcrit = factor to the compression load NE(x) to obtain the bifurcation-value.

(2)

The solution is obtained by numerical methods and leads to the eigen-value αcrit and the first modal buckling deformation

′ and η crit ′′ , that all satη crit and its derivates η crit

isfy the boundary conditions, see equation (3.2):

″

′′ ) + c( x) ⋅η crit + q = ( EI ( x)η crit 144 4 4 244 4 4 3 innerer Widerstand

Rcrit

(3)

′ )′ = 0 α crit ⋅ (1 N E ( x)η crit 42 4 43 4

(3.2)

{

Konstante äußere Einwirkung

+ α crit ⋅

E crit

The imperfection reads according to EN 1993-1-1, 5.3.1 (11) equation (5.9) in a more generalized way:

⎡

α crit ⋅ N E ( x) ⎤ ⋅η crit ( x) ⎥ ′ ′ ( ) ( ) EI x ⋅ η x crit ⎣ ⎦ x = x

η init = ⎢e0

(3.3)

d

where x = xd is the reference point. The function (3.3) also satisfies the differential equation and the boundary conditions, see equation (3.4)

⎡ α crit N E ( x) ⎤ ′′ ( x) )″ + c( x) η crit ( x) + α crit ( N E ( x) η crit ′ ( x))′ = 0 ( EI ( x) η crit ⎢e0 ⎥ ′′ ( x) ⎦ x = x ⎣ EI ( x ) η crit d

{

}

(3.4)

144 4 4 244 4 4 3 Konstante

In the specific case: N E (x) = N E = const . EI(x) = EI = const . c(x) = 0

η crit = sin

π x l

for hinged ends of the column

the values are:

α crit =

EI ⋅ π 2 2

l N E

page 17 / 142

Excerpt from the Background Document to EN 1993-1-1 2

π ⎞ π x ′′ = ⎛ η crit ⎜ ⎟ sin l ⎝ l ⎠ and therefore at x = ℓ/2:

η init = e0 [1] sin (4)

π x l

If the loading is:

α E ⋅ N E ( x) ≤ α crit N E ( x)

(3.5)

the „resistance“ R E in equation (3.2) reads R E =

(5)

α E α ′′ )″ + c( x) ⋅ η crit = E α crit ⋅ ( N E ( x)η crit ′ )′ ( EI ( x )η crit α crit α crit

{

}

{

}

(3.6)

Hence the bending moment along the length of the member due to the imperfection

η imp is according to 1st order theory: M 0 ( x) =

α E ⎡ α crit N E ( x) ⎤ ′′ ( x) ⋅ e ⋅ EI ( x) η crit ′′ ( x) ⎥⎦ x = x α crit ⎢⎣ 0 EI ( x) η crit

(3.7)

d

This bending moment takes the following value at t he point x = xd : M 0 ( x) =

α E ⋅ e ⋅ α ⋅ N ( x) α crit 0 crit E

(3.8)

= α E ⋅ N E ( xd ) ⋅ e0 (6)

If the x = xd is defined as the location relevant for the assessment of the member (because of the most onerous conditions), than the cross-sectional assessment, taking into account 2nd order effect, reads:

⎡ α E N E ( x) ⎤ ⎢ N ( x) ⎥ ⎣ R ⎦ x= xd 144 4 244 4 3 in plane

(7)

⎡ α E e0 N E ( x) ⎤ 1 ⋅ =1 ⎥ α ( ) M x E R ⎣ ⎦ x= xd 1 −

+⎢

(3.9)

α

crit 144 4 4 4 244 4 4 4 3

out of plane

With the simplifications:

⎡ N R ( x) ⎤ ⎥ ⎣ N E ( x) ⎦ x = x

α ult , k ( xd ) = ⎢

(3.10) d

it follows from (3.9):

⎡ ⎤ ⎢ α 1 ⎥ α E N R ( x) E ⎢ ⎥ + ⋅ ⋅ e0 =1 α ( ) ( ) ( ) α x α x M x E ⎥ ⎢ ult , k ult , k R 1− ⎢⎣ α crit ⎥⎦ x = xd (8)

Using the symbols:

page 18 / 142

(3.11)


⎡

χ ( xd ) = ⎢

α E ⎤

⎥ ⎢⎣ α ult , k ( x ) ⎥⎦ x = xd

(3.12)

⎡ α ult , k ⎤ ⎥ α ⎥ x = x d crit ⎦ ⎣⎢

λ ( xd ) = ⎢

e0

(3.13)

⎡ M R ( x ) ⎤ ⋅ α ⋅ (λ − 0,2) ⎥ ( ) N x ⎣ R ⎦ x = x

=⎢

(3.14)

d

equation (3.11) may be transferred to:

χ ( xd ) + χ ( xd ) ⋅ α ⋅ (λ ( xd ) − 0,2)⋅

1 =1 1 − χ ( xd ) ⋅ λ 2 ( xd )

(3.15)

which is the same basic equation for χ (λ ) as given in Figure 2.2, that leads to the European Standard buckling curves. (9)

Thus it has been proved, that the European Standard flexural buckling curves are also applicated to columns with non-uniform distributions of stiffness and compression force, with any elastic supports and any boundary conditions without any modification, if the cross-sectional data and the force N E (x) are taken at the relevant location x = xd . According to equation (3.3) also the relevant equivalent geometrical imperfections are

′′ ( x)] x = xd at that relevant location. referred to the characteristic moment [ EI ( x ) ⋅η crit

page 19 / 142


3.2 (1)

Options for assessment The following rules apply for taking the relevant cross-section into account, see Figure 3.1: 1. If the cross-sectional properties and the compression forces are uniform and αult,k is constant, then the relevant location xd is where

value

′′ takes the maximum η crit

′′ , max . η crit

The imperfection reads:

η imp = e0 ⋅

α crit N E ⋅ η ( x) ′′ , max crit EI ⋅ η crit

(3.16)

see EN 1993-1-1, equation (5.9). 2. If α ult,k (x) varies along the member length due to variable cross-sections and/or variable compression forces N E (x), the value xd in general is located between - xult,k , where α ult,k takes the minimum value - xη ′crit ′ , where the curvature (2)

′′ takes a maximum value. η crit

There are two options for a solution by design aids: 1. For standardized cases design aids give the location xd , so that the assessment can be carried out without a modification of the χ (λ ) -formula,

α ult,k , e.g and the design aids give modifications of the χ (λ ) for-

2. for standardized cases particular locations xmin are given to determine to determine

α ult,k,min

mula, so that the right results are achieved. (3)

Normally option 1 is the most simple approach; because of its relevance for lateraltorsional buckling verifications hereafter also the option 2 is explained.

Figure 3.1: Determination of the relevant location x d

page 20 / 142


3.3 (1)

Determination of the relevant location xd (option 1) The relevant location x = xd for applying the European flexural buckling curve according to formula (3.15), is, where the utilization rate ε(x), expressed by

ε ( x) =

α E

+

α E

α ult ,k ( x) α ult ,k ( x)

⋅ α ∗ ⋅ (λ ( xd ) − 0,2)⋅

1

α 1 − E α crit

′′ ( x) EI ( x) ⋅η crit

⋅

′′ ( xd ) EI ( xd ) ⋅η crit

,

attains the maximum value, see Figure 3.2.

α α

*

(x)

α α

d

1-

ult,k

α α

1

α ⎯( λ (x ) - 0,2)

E

· E

EI(x) η’’ crit (x) EI(xd) η’’crit(xd)

crit

E

(x)

ult,k

ε(x)

xd Figure 3.2: Determination of the relevant location x d, if (x) has an extremum

(2)

This leads to

∂ε ( x) ! = 0 ∂ x (3)

Figure 3.2 shows that considering the function of

′′ ( x) would lead the “true” η crit

values:

χ true ( x) from χ true + χ true ⋅ α ∗ ⋅ (λ ( xd ) − 0,2)⋅ α E ,true ( x) = α ult , k ( x) ⋅

true

1

⋅ 2

′′ ( x) EI ( x) ⋅η crit

′′ ( xd ) 1 − χ true ⋅ (λ ( xd )) EI ( xd ) ⋅η crit

=1

( x)

whereas the use of the European buckling curve would lead to:

χ calc ( x) from χ + χ ⋅ α ∗ ⋅ (λ − 0,2)⋅

1 1 − χ ⋅ λ 2

=1

α E ,calc ( x) = α ult , k ( x) ⋅ χ calc ( x) see Figure 3.3

page 21 / 142


1.6

1.2

1.4

1.0

1.2

true

0.8

1.0

η ” fl

η''fl 0.8

0.6

χ

0.6

calc

0.4

0.4

0.2

0.2

0.0

0.0 0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

0

25

50

75

100 125 150 175 200 225 250 275 300 325 350 x [cm]

x [cm]

2.5

1.4 1.2

2.0

1.0

ε

ε calc

αEd

0.6

α E,calc

1.0

ε true

0.4

0.5

0.2 0.0

0.0 0

25

50

75

100

125

150

175 200 x [cm]

225

250

275

300

325

350

0

Figure 3.3: Functions of η crit ′′ ( x) ,

(4)

α E,true

1.5

0.8

25

50

75

100

125

150

175 200 x [cm]

225

250

275

300

325

350

( x) , α E ( x) and ε ( x)

It is evident from Figure 3.3 that at the point x = xd , where

α E,true(x) has an extremum,

both α E,true(x) and α E,calc(x) and χ true(x) and χ calc(x) are identical. (5)

α E,true(x) has no extremum along the length of the member, then the crosssectional verification with χ = 1.0 applies, see Figure 3.4 α EI(x) η’’ (x) 1 α ⎯λ ( ( x ) - 0,2) · α α (x) EI(x ) η’’ (x ) 1α α α (x) In case

E

*

crit

d

E

ult,k

d

crit

d

crit

E

ult,k

Figure 3.4: Determination of the relevant location x d, if

(6)

3.4 (1)

E (x)

has no extremum

The values xd may be determined as design aids for practical verification.

Modification of the buckling curve (option 2) A practical solution for the modification of buckling curves is, to use the values

α ult,k,min

and α crit , which are available from the modal analysis. (2)

In defining

χ =

α E α E α ult , k , min = ⋅ α ult , k α ult , k , min α ult , k

1 424 3 1 424 3 χ mod

and

page 22 / 142

f

(3.17)


α ult , k α α ult , k = ult , k , min ⋅ α crit α crit α ult , k , min

λ =

(3.18)

14243 14243 λ mod

1 f

it follows:

⎛ λ mod

χ mod ⋅ f + χ mod ⋅ f ⋅ α ∗ ⎜⎜

⎝ f

(3)

⎞

1

− 0,2 ⎟⎟ ⋅

⎠ 1 − χ ⋅ f ⋅ mod

2 λ mod

=1

(3.19)

f

The modified buckling curve therefore reads:

χ mod =

1 f

1

φ + φ 2 −

(3.20)

2 λ mod

f

and 2 ⎤ ⎡ ⎛ ⎞ λ mod ∗ ⎜ λ mod ⎟ φ = 0,5 ⋅ ⎢1 + α ⋅ ⎜ − 0,2 ⎟ + ⎥ f f ⎢⎣ ⎥⎦ ⎝ ⎠

(4)

(3.21)

Figure 3.5 shows the unmodified buckling curve and the modified buckling curve. Either of them produce with different assumptions for α ult,k the same solution:

α ult , d = χ mod ⋅ α ult , k , min = = χ ⋅ α ult , k

f

⋅ f ⋅ α ult , k

(3.22)

χ , χ mod 1.4

Euler 1.2

χ mod

1.0

0.8

1 f

χ 0.6

0.4

0.2

0.0 0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

λ , λ mod Figure 3.5: Modified buckling curve

mod and

unmodified buckling curve

page 23 / 142


One can see in Figure 3.5, that the modified buckling curve χ mod is always above the unmodified buckling curve, so that a calculation with α ult,k,min and the unmodified buckling curve is always safe-sided. This second-fence solution on the safe side is in most cases the easiest and most suitable way of verification.

(6)

Figure 3.6 gives a worked example for the application of the column buckling curve based on formula (3.15) to a column with a non-uniform cross-section and a nonuniform distribution of the compression force, that has a length of 10,00 m. The eigenmode analysis based on the distributions of N E and of the cross-sectional values gives two important results for the further “exact” verification: 1. the distribution of the curvature cross-section, where

′′ , that indicates the location of the relevant η crit

′′ , max is attained: xd = 0,855 m η crit

At this location the values for verification are N E (x) = 341 kN N R (x) = 946 kN

This gives

α ult , k =

946 = 2,774 341

2. the critical value α crit = 1,6376 This gives

λ =

α ult , k 2,774 = = 1,302 1,6376 α crit

(α = 0,34 ) = 0,426 The verification then reads 

(7)

α Ek = ⋅ α ult ,k = 0,426 ⋅ 2,774 = 1,182 > 1,0

A simplified check, that works with

α ult , k , min =

α ult , k , min and α crit would take

705 = 1,996 353

at x = 0 m . Hence it follows

λ mod =

α ult , k , min 1,996 = = 1,104 1,6376 α crit

χ = 0,533 and 

page 24 / 142

α Ek = χ ⋅ α ult ,k ,min = 0,533 ⋅1,996 = 1,064 > 1,0

Consistent determination of the flexural buckling resistance of columns with non-uniform CS In case the modified buckling curve according to Figure 3.5 would be used: f =

α ult , k , min 1,996 = = 0,720 2,774 α ult , k

λ mod =

α ult , k , min 1,996 = = 1,104 1,6376 α crit

χ mod (α = 0,34; f = 0,720) = 0,592 

(8)

α Ek = χ mod ⋅ α ult ,k ,min = 0,592 ⋅1,996 = 1,182 > 1,0

A Finite Element calculation with a geometrically and material non-linear analysis would take an effective geometrical imperfection proportional to mum value at x = xd

⎡ α ⋅ N ( x) ⎤ η imp = e0 ⎢ cr E ⎥ EI ( x) ⋅η ′′ ⎣

⎦

η crit with the maxi-

= 13,672mm

crit x = x d

It gives 

α Ek = 1,206 > 1,0

Figure 3.6: Tapered column under non-uniform axial load

page 25 / 142


Table 3.1 gives a survey on all results. Table 3.1: Summary of calculation steps and results for tapered column example Verification at x( η '' crit,max )

Verification at x( α ult,k,min )

Verification with FEM η imp = f ( η crit )

x = xd

0.855 m

0m

0m

0.855 m

NE (x)

341 kN

353 kN

353 kN

341 kN

NR (x)

946 kN

705 kN

705 kN

946 kN

αult,k

2.774

1.996

1.996

-

αcrit

1.6376

1.6376

1.6376

-

⎯λ

1.302

1.104

1.104

-

f

-

-

0.72

-

χ (α = 0.34)

0.426

0.533

0.592

-

1.182

1.064

1.182

1.206

E,k

page 26 / 142

Consistent determination of the resistance to lateral-torsional buckling

4


4.1 (1)

Application of the reference model of Maquoi-Rondal The basic model for lateral-torsional buckling that corresponds to the basic model for flexural buckling in Figure 2.1, is a beam with fork-conditions at its ends and a constant bending moment along the length, see Figure 4.1, [5] [6]. Mzy M

l Mzy M Figure 4.1: Basic model for lateral-torsional buckling of an I-girder

(2)

This case is governed by two coupled differential equations for the deflection

η and

the twist ϕ that cause displacements perpendicular to the main loading plane, see Figure 4.2 . (3)

The adoption of sinus-functions for η crit and ϕ crit leads to the eigen-value M y ,crit =

π 2 EI z l2

⋅

I w I z

⋅ 1+

GI t l 2 EI wπ 2

,

(4.1)

In this formula one can identify the moment M y,crit,Fl,o leading to lateral flexural buckling of the top flange in compression N z,crit,Fl,o M y ,crit , Fl ,o

π 2 EI z = ⋅2 2 ⋅ l2

I w I z

= N z ,crit , Fl ,o ⋅ h

(4.2)

if the St. Venant torsional stiffness is neglected and also the enhancement of this moment due to the torsional stiffness by the factor:

ε It = 1 + (4)

GI t l

EI wπ 2

The eigen-mode

ϕ crit = sin η crit =

2

≥1

(4.3)

η crit and ϕ crit is characterized by:

π x l

I w I z

⋅ ε It ⋅ sin

π x

(4.4)

l

which give the eigen-displacements of the top and bottom flanges:

page 27 / 142


I w

η crit , Fl = η crit ± = (5)

I w I z

I z

⋅ ϕ crit

⋅ (ε It ± 1) ⋅ sin

(4.5)

π x l

Using:

π ⎞ ′′ ,max, Fl = ⎛ η crit ⎜ ⎟ ⎝ l ⎠

2

I w I z

⋅ (ε It + 1)

(4.6)

the imperfections of the flanges according to equation (3.3) read:

π 2 EI Fl η init , Fl = e0

= e0 (6)

l

2

I w I z

π 2 EI Fl

I w

l2

I z

⋅ (ε It ± 1) ⋅ sin ⋅ (ε It + 1)

π x l

(4.7)

ε It ± 1 π x ⋅ sin l ε It + 1

Hence the imperfection for the top flange is:

η init , Fl ,o = e0 sin

π x

(4.8)

l

i.e. it is identical with the imperfection of the column in Figure 2.1. The imperfection for the bottom flange is

η init , Fl ,u = e0

ε It − 1 π x sin l ε It + 1

(4.9)

i.e. a value that is zero where the St. Venant-torsional stiffness is zero ( ε It  1) and that takes the same value as for the top flange, if the torsional stiffness is very large. (7)

The imperfections related to the deformations of the cross-section read:

ϕ init = e0

1 I w I z

η init = e0 (8)

sin

⋅ ε It

π x l

(4.10)

ε It π x sin l ε It + 1

When inserting these imperfections into the coupled differential equations to obtain the elastic deformations resulting from them

′′′′⎤ ⎡ 0 M y , E ⎤ ⎡η el ′′ ⎤ ⎡ 0 M y , E ⎤ ⎡η init ′′ ⎤ ⎡ EI z 0 ⎤ ⎡η el − = ⎥ ⎢ ′′ ⎥ ⎢ M ⎥ ⎢ ′′ ⎥ ⎢ 0 EI ⎥ ⎢ϕ ′′′′⎥ ⎢ M w ⎦ ⎣ el ⎦ ⎣ ⎦ ⎣ y , E 0 ⎦ ⎣ϕ init ⎦ ⎣ y , E GI t ⎦ ⎣ϕ el one obtains:

page 28 / 142

(4.11)

Consistent determination of the resistance to lateral-torsional buckling M y , E M crit M y , E

η el = e0

1−

ε It π x sin l ε It + 1

⋅

M crit

(4.12) M y , E

1

ϕ el = e0 ⋅

M crit M y , E

I w

1−

I z

⋅

1

ε It + 1

sin

π x l

M crit

which gives the elastic curvature of the top flange: M y , E I w

′′ , Fl ,o = η el ′′ + η el

(9)

I z

π ⎞ ′′ = e0 ⎛ ⋅ ϕ el ⎜ ⎟

2

M crit π x ⋅ sin M y , E l

⎝ l ⎠

1−

(4.13)

M crit

Hence the bending moment in the top flange is: 2

′′ , Fl ,o M E , Fl ,o = EI Fl ,o ⋅η el

=

EI Fl ,oπ 2

l 4 1 42 3

⋅ e0

M y , E M crit

N crit , Fl , o

= E ⋅

where EI Fl ,o

t b 3 12

1

⋅ 1−

M y , E

⋅ sin

π x l

(4.14)

M crit

.

(10) One can obtain this bending moment easier than with equation (4.11) by applying the equations (3.6) and (3.7): M E , Fl ,o

=

M y , E ′′ , Fl EI Fl ,o ⋅η init M crit 123

1

⋅ 1−

M y , E

M crit 1 424 3

α E α crit

144 4 244 4 3

1

Moment nach Theorie 1. Ordnung

1−

α E α crit

(4.15)

144 4 4 4 4 244 4 4 4 4 3 Moment nach Theorie 2. Ordnung

=

π 2 EI Fl ,o l

2

⋅ e0 ⋅

M y , E M crit

1

⋅ 1−

M y , E

⋅ sin

π x l

M crit

(11) Figure 4.2 summarizes the derivation of the equations (4.14) and (4.15). The further derivations to get the equation for the assessment of the top flange in compression is performed in the same way as that for the column in Figure 2.1 and Figure 2.2, see Figure 4.3, by using the substitution: N E , Fl N R , Fl

=

M y , E M y , R

(4.16)

page 29 / 142


Figure 4.2: Lateral torsional buckling problem and initial imperfection [5]

Figure 4.3: Derivation of

LT -value

[5]

(12) The result is the „European lateral-torsional buckling curve“

χ LT (λ ) , that differs from

the „European flexural buckling curve by the imperfection factor α ∗ , which is derived from α by taking the influence of the torsional stiffness into account by the ratio of the slenderness of the full beam 2 λ LT α α = 2 α = ε It λ Fl

∗

2 2 to the slenderness of the mere top flange λ Fl [7]. λ LT

(4.17)

(13) This modification effects a shift from the flexural buckling curve to the Euler-curve, see Figure 4.4, that is the stronger, the smaller the beam depth in relation to the

page 30 / 142

Consistent determination of the resistance to lateral-torsional buckling beam width and the greater the slenderness is (enhancement of ε It according to equation (4.3)). The use of the flexural buckling curve instead of the modified “European lateral torsional buckling curve” is however on the safe side. χ 1.2 1.1

Biegedrillknicken für einen Querscchnitt mit εIt = ∞

1.0 0.9

Biegedrillknicken für ein Profil HEB 200

0.8 0.7 KSL a

0.6

KSL b

0.5 0.4 0.3 0.2

Momentenverteilung:

0.1

Trägerprofil: HE 200 B

⎯λ

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Figure 4.4: Comparison between lateral torsional buckling curve (for a beam HEB 200 under pure bending) and column buckling curves a and b

(14) Figure 4.5 shows a comparison of test results from [8] [9] with beams with a constant buckling moment M y with the European lateral torsional buckling curve, and Table 4.1 shows the determination of the

according to EN 1990 Annex D. γ M-values

r e /r t 1.6

1.2

IPE 200

1.4 1.0 1.2 0.8

1.0

A B D F G H I J Z

A

0.8

B C D

0.6

E F

0.4

G H

0.2

I J Z

0.6

0.4

0.2

0.0

0.0 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

1 .1

1 .2

1 .3

1 .4

1 .5

1 .6

1.2

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

1.2

IPE 80

H 200 x 100 x 5,5 x 8*

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

Figure 4.5: Lateral torsional buckling of rolled beams; test-results and lateral torsional buckling curves with corresponding

* -values;

page 31 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 4.1: Statistical evaluation for lateral torsional buckling of rolled beams Eingangsdaten υrt = 0,08 (Geometrie und Streckgrenze) υfy = 0,07 (Streckgrenze)

EC3 Background Document 5.03P - Appendix I (N = 142) Standardnormalverteilung

log-Normalverteilung

2.0 g n u l i 1.5 e t r e v 1.0 l a m r o 0.5 n d r a d 0.0 n a 0.8 t S-0.5 r e d e-1.0 l i t n a u-1.5 Q

2.0

0.9

1.0

1.1

1.2

1.3

1.4

g n u l i e t r e v l a m r o N g o l r e d e l i t n a u Q

1.5 1.0 0.5 0.0 -0.1

-0.5

0.0

0.1

-1.5

ln re/rt

sδ = 0.083

b = 1.160

page 32 / 142

0.4

-1.0

re/rt

γM = 1.223

0.3

-2.0

-2.0

υδ = 0.0 71

0.2

( Mod ell) Δk = 0.897

υR = 0. 107

sδ = 0.089

b = 1.165 (gesam t) * γM = 1.096

υδ = 0. 07 6 γM = 1.167

(Mode ll) Δk = 0.898

υR = 0 .11 1

(gesa mt ) *

γM

= 1.047


4.2

4.2.1 (1)

Application of the „European lateral torsional buckling curves“ for the general loading case for lateral torsional buckling Definition of the general loading case The general loading case for lateral torsional buckling is defined by the following: 1. Loading in the main plane of the beam-column: The loading E d in the main plane of the beam-column comprises any combination of longitudinal and transverse forces applied to the structural member or the full structure. The effect of this loading is taken into account by the normal force N Ed (x) in the compression flange relevant for the lateral torsional buckling assessment. The force N Ed (x) is non uniform along the member length and has been determined taking 2 nd order effects in the main plane into account. The strength exploitation of the compression flange is defined by:

α E ⋅ N Ed ( x) Rk , Fl ( x)

=

α E ⋅ N Ed ( x) α E = α ult ,k ( x) ⋅ N Ed ( x) α ult ,k ( x)

(4.18)

2. Loading transverse to the main plane (out of plane): The loading out of main plane is effected by the equivalent geometrical imperfections η init (x) and ϕ init (x). The load-effect in the compression flange relevant for the assessment of the full beam-column is the flange-moment (see equation (4.15)): M Fl ( x) =

The value leading to

α E ′′ , Fl ,o ⋅ 1 EI Fl ,o ⋅η init α α crit 1 − E α crit

(4.19)

α crit is the eigen-value determined by numerical means, e.g. FEM,

Rcrit ( x) = α crit ⋅ N E ( x )

(4.20)

′′ , Fl ,o is the curvature of the imperfection of the compression flange deη init ′′ and ϕ crit ′′ , also calculated with numerical termined with the eigen-modes η crit and

means. These eigen-modes η crit and ϕ crit satisfy the coupled differential equations at any point x and also the boundary conditions, which may be different to the situation in Figure 4.1; e.g. they may be independent from each other or coupled as in the case of point support. Therefore the eigen-functions ϕ crit and η crit may have fully different shapes and not be proportional as given in Figure 4.2. (2)

In the following the general assessment formula for beam-columns subject to lateral torsional buckling are derived in two steps: 1. neglecting the St. Venant torsional stiffness, 2. taking the St. Venant torsional stiffness into account.

page 33 / 142

Excerpt from the Background Document to EN 1993-1-1 4.2.2 (1)

Basic equation with neglection of the torsional stiffness The differential equations for the case of general longitudinal and transverse heading in the main plane for lateral torsional buckling without the consideration of the St. Venant-torsional stiffness reads:

⎡ EI z 0 ⎤ ⎡η ∗ //// ⎤ ⎡0 ⎤ ∗ ∗ ∗ ∗/ ∗/ ⎢ 0 EI ⎥ ⎢ ∗//// ⎥ − α crit [ E (η ,ϕ ,η , ϕ )] = ⎢0⎥ w ⎦ ⎣ϕ ⎣ ⎦ ⎣ ⎦ 144 4 244 4 3

Rk ∗

(4.21)

144 4 244 4 3

∗ − α crit

⋅

E d ∗

= 0

and the solutions obtained numerically are: ∗ α crit ∗ ∗/ ∗// , η crit , η crit , ... η crit ∗ ∗/ ∗// , ϕ crit , ϕ crit , ... ϕ crit

(2)

(4.22)

The eigen-mode of the compression flange is: ∗ ∗ ∗ η crit , Fl = η crit + z M ϕ crit

(4.23)

and the related eigen-deformation reads: ∗ η crit , Fl =

∗ ∗ η crit + z M ϕ crit ∗// ∗// [η crit ] x= x + z M ϕ crit

(4.24)

d

(3)

From (4.2.4) the curvature-imperfection of the flange can be deducted: ∗ ⎡ α crit N E , Fl ( x) η init , Fl = e0 ⎢ ∗// ∗// ⎢ EI Fl ,o [η crit + z M ⋅ ϕ crit ] ⎣ ∗//

(4)

⎤ ∗// ∗// ⎥ ( x) + z M ⋅ ϕ crit ( x)] [η crit ⎥ ⎦ x = xd

(4.25)

This gives the bending moment in the flange ∗ ⎡ α crit N E , Fl ( x) α E M E , Fl = ∗ ⋅ EI Fl ,o ⋅ e0 ⎢ ∗// ∗// ⎢ EI Fl ,o [η crit α crit ] + z M ⋅ ϕ crit ⎣

⎤ ∗// ∗// ⎥ [η crit ( x) + z M ⋅ ϕ crit ( x)] ⎥ ⎦ x = xd (4.26)

= α E ⋅ e0 ⋅ N E , Fl ( x)

∗// ∗// ( x) + z M ⋅ ϕ crit ( x) η crit ∗// ∗// [η crit ( x) + z M ⋅ ϕ crit ( x)] x= x

d

(5)

In using: e0

=

M R , Fl N R, Fl

∗ (λ LT − 0,2)⋅ α

(4.27)

the flange moment reads: M E , Fl = α E ⋅

page 34 / 142

M R, Fl N R, Fl

⎡ ∗ (λ LT − 0,2 )⋅ α ⎢

⎤ ′′ + z M ⋅ ϕ crit ′′ ]⋅ 1 [η crit (4.28) ⎥ α ′ ′ ′ ′ ( ) η + z ⋅ ϕ E ⎣ crit M crit ⎦ x= xd 1− N E , Fl ( x )

∗ α crit

Consistent determination of the resistance to lateral-torsional buckling (6)

This flange-moment is inserted into the interaction formula for resistance of the flange: N E , Fl N R, Fl

+

M E , Fl M R , Fl

=1

(4.29)

which gives: N E , Fl N R , Fl

(7)

+ α E ⋅

M R , Fl N R , Fl

N ∗ (λ LT − 0,2)⋅ α ⋅ E , Fl ⋅ M R , Fl

′′ + z M ⋅ ϕ crit ′′ η crit 1 ⋅ =1 ′′ + z M ⋅ ϕ crit ′′ ] x= x 1 − α E [η crit ∗ α crit

(4.30)

d

Using equation (4.18) gives:

′′ ( x) α E α η ′′ ( x) + z M ⋅ ϕ crit 1 ∗ + E (λ LT − 0,2)⋅ α ⋅ ⋅ crit =1 α E [η crit α ult ,k , Fl α ult ,k ′ ′ ′ ′ ] ϕ + z ⋅ M crit x = x 1− ∗ α crit

(4.31)

d

i.e. if the design point x on the axis of the beam is identical with the reference point x = xd for the imperfection, then with

χ =

α E

(4.32)

α ult ,k , Fl , x

d

the final equation is:

χ + χ ⋅ (λ ∗ − 0,2 )⋅ α ⋅

1 1 − χ ⋅ λ ∗2

=1

(4.33)

Equation (4.33) demonstrates, that the standard European flexural buckling curves are applicable for solving the problem. 4.2.3 (1)

Basic equation with consideration of the torsional stiffness When taking account of the St. Venant-torsional stiffness the differential equations read:

0 ⎡ EI z ⎤ η ′′′′ 0 ⎢ ′ ′ ϕ ⎥ ⎡⎢ ⎤⎥ − α crit [ E (η , ϕ ,η ′, ϕ ′)] = ⎡⎢ ⎤⎥ ⎢ 0 EI w − GI t ⎥ ′′′′ ⎣0⎦ ϕ ′′′′ ⎦ ⎣ϕ ⎦ ⎣144 4 4 4 244 4 4 4 3 144 244 3 Rk

− α crit ⋅

E d

(4.34)

= 0

which give other numerical solutions than equation (4.21):

α crit ′ , η crit ′′ , ... η crit , η crit ′ , ϕ crit ′′ , ... ϕ crit , ϕ crit (2)

(4.35)

The further derivation follows in principle the derivation in section 4.2.2, however the imperfection for the flange reads:

page 35 / 142

Excerpt from the Background Document to EN 1993-1-1 ∗ α crit N E , Fl ( x) ′′ , Fl = e0 [η ′′ + z ⋅ ϕ ′′ ] η init ′′ + z M ⋅ ϕ crit ′′ ] crit M crit EI Fl [η crit

where

(4.36)

∗ is the eigenvalue obtained from equation (4.21) without considering the α crit

St. Venant torsional stiffness, see equation (4.25). (3)

Hence the flange-moment is different to the one in equation (4.26): ∗ ⎡ α crit ⋅ N E , Fl ( x) ⎤ α E ′′ + z M ⋅ ϕ crit ′′ ] [η crit ⋅ e0 ⋅ ⎢ M E , Fl = ⎥ ′ ′ ′ ′ α crit η + z ⋅ ϕ ⎣⎢ crit M crit ⎦⎥ x= x

(4.37)

d

that reads finally: ∗ α crit 1 ⎡ N E , Fl ( x) ⎤ [η ′′ + z ⋅ ϕ ′′ ] M E , Fl = α E ⋅ (λ LT − 0,2) ⋅ α ⋅ ′′ + z M ⋅ ϕ crit ′′ ⎥⎦ x= x crit M crit N R, Fl α crit 1 − α E ⎢⎣η crit α crit

M R, Fl

(4.38)

d

(4)

Hence the assessment formula for the design point x = xd reads [7], [10]: ∗ α crit 1 ⋅ =1 2 α 1 − χ ⋅ λ crit LT 1 424 3

χ + χ ⋅ (λ LT − 0,2 )⋅ α ⋅

(4.39)

α ∗

This equation is identical with the equation for the European lateral torsional buckling curve in Figure 4.3 and gives for the specific case in Figure 4.1 the equation (4.17). (5)

By this derivation the general applicability of the standard European flexural buckling curves and of the Standard European lateral torsional buckling curves is proved.

4.2.4

Modification of the lateral torsional buckling curves to agreed simplified assumptions

(1)

Where the real design point x = xd is not known a priori, an assumption for a substitutive design point can be made, e.g. x = xmin, where α ult,k,min is obtained. [11]

(2)

The lateral torsional buckling curve then reads following equation (3.20) and (3.21) valid for flexural buckling

χ mod =

1 f

1

φ + φ 2 −

2 λ mod

(4.40)

f

and 2 ⎤ ⎡ ⎛ ⎞ λ mod ∗ ⎜ λ mod ⎟ φ = 0,5 ⋅ ⎢1 + α ⋅ ⎜ − 0,2 ⎟ + ⎥ f f ⎢⎣ ⎥⎦ ⎝ ⎠

page 36 / 142

(4.41)

Consistent determination of the resistance to lateral-torsional buckling 4.2.5

Worked example [5]

(1)

A support frame of the „Schwebebahn“ in Wuppertal according to Figure 4.6 is taken as an example. The supports at the feet of the columns may be modeled as forks, and the beam is laterally supported by excentric point supports.

(2)

The loading is asymmetrical and effects non uniform distributions of the axial forces and bending moments in the main plane. The cross-section also varies along the length.

(3)

With FEM, see Figure 4.7 the numerical values are

α ult,k,min = 1,69 α crit = 3,41 For the verification flexural buckling curve c has been used as safe-sided approach. All relevant calculation steps are given in Figure 4.6. 2150 kN 4 5 0 / 6 0

4 5 0 / 5 0

4 5 0 / 6 0

3000

=26

R i e g e l

298 kN

Ermittlung des Abminderungsbeiwertes:

4 5 0 / 40

950

λ mod =

α ult ,k ,min 1,69 = = 0,704 α crit 3,41

β = 1 α LT = 0,49 χ LT = 0,722

s=18

s=18

Innerer Flansch: 450/60

6901 450/40

Ergebnisse der FEM-Berechnung:

α crit = 3,41 α ult , k , min = 1,69

950

450/60 s

4 5 0 / 4 0

Abstützung gegen Verformungen aus der Ebene

Alle Steifen: 450/18

Nachweis: 400 3000

2264

3136

χ LT ⋅ α ult ,k ,min ≥ γ M 1 0,722 ⋅ 1,69 = 1,22 > 1,10

3000

Figure 4.6: Example for the lateral torsional buckling verification acc. to the general method 2150 kN

298 kN

Figure 4.7: First eigenmode of the support frame from FEM-analysis (

crit =

3,41)

page 37 / 142

Excerpt from the Background Document to EN 1993-1-1 4.2.6

Application to non-symmetric cross-sections

4.2.6.1 Derivation of the assessment formula (1)

Non-symmetrical cross-sections are such sections as e.g. channels according to Figure 4.8 for which the limit state conditions for out of plane buckling depend on the direction of the deformation. Pz

E

D

D

+ +

-

η

M

y

S

y

y

-

y

-

M

ϕ +

ϕ

z

ζ

z

+

+

+

z

z

z

y

Figure 4.8:

(2)

The elastic assessment for the design point D on the cross-section, related to stresses for deformations to the left hand side in Figure 4.8 reads:

σ Eip f y

+

σ Eop f y

=1

(4.42)

where the following applies:

σ Eip f y

σ Eop f y

α E α ult ,k ∗ α E α crit y ⋅η ′′ + ω ϕ ′′ 1 ( ) 0 , 2 = λ LT − ⋅ α ⋅ ⋅ D crit D crit = 1 α ult ,k α crit 1 − α E [ y D ⋅η crit ′′ + ω Dϕ crit ′′ ] x= x α crit =

(4.43)

d

(3)

In conclusion the assessment formula (4.43) for the design point x = xd along the member length is the same as for symmetrical cross-sections given in equation (4.39).

(4)

The assumption of a deformation to the right hand side in Figure 4.8 would lead to the following equation for the point E on the cross-section:

σ Eop f y

∗ α crit y ⋅η ′′ + ω ϕ ′′ 1 α E (λ LT − 0,2)⋅ α ⋅ = ⋅ E crit E crit = 1 α ult ,k α crit 1 − α E [ y E ⋅η crit ′′ + ω E ϕ crit ′′ ] x= x α crit

(4.44)

d

ergo the same equation as equation (4.39), however with the difference, that for the design points D and E different reference values of imperfection apply. These differences of reference values may require different imperfection factors erence direction for out of plane instability may occur.

page 38 / 142

α , so that a pref-

Consistent determination of the resistance to lateral-torsional lateral-torsional buckling (5)

Hence the lateral lateral torsional torsional buckling for unsymmetrical unsymmetrical cross-sections may with with regard regard to the dependence on the direction of deformation be similar to the flexural buckling of symmetrical cross-sections, e.g. as for I-profiles. For such I –profiles –profiles flexural buckling in the main plane requires according to section 2.2 (2) of this report an imperfection ( α α = 0,34) e0 l

f 1 4 1 1 ⋅ 0,34 ⋅ y = 0,108 ⋅ = 2 π 30 280 E

= ⋅

and in the out of main plane direction with: b

2 ⋅ A Fl ⋅ f y M R 4 ≈b ≈ 2 ⋅ A Fl ⋅ f y 4 N R

λ =

( plastic) ≈

b

6

(elastic)

2 ⋅ A Fl ⋅ f y l 12 f y = EA Fl ⋅ b 2 6 ⋅ π 2 b π E

an imperfection ( α α= 0,49) e0 l

1 4

= ⋅

12

π

⋅ 0,49 ⋅

f y E

= 0,135 ⋅

1 1 ( plastic) = 30 220

= 0,090 ⋅

1 1 (elastic) = 30 333

or e0 l

1 6

= ⋅

12

π

⋅ 0,49 ⋅

f y E

4.2.6.2 Justification by tests (1)

Channels are in general loaded such by transverse loads that that the load plane plane does does not go through the shear centre M , but is in the plane of the web, so that initial eccentricities and hence additional loading by torsion has to be considered, see Figure 4.9 a).

(2)

To prove the applicability applicability of the European lateral torsional torsional buckling curve for channels without additional out-of-plane action effects an initial loading situation as given in Figure 4.9 b) is necessary for the test results, which is rather academic and can only be provided by particular test conditions in the laboratory. a)

b)

Figure 4.9: Loading conditions for channels

page 39 / 142


Tests that satisfy the conditions of Figure Figure 4.9 b) are listed in Table 4.2. 4.2.

(4)

A first comparison between calculative and test results on channel sections loaded through their shear centre are given in Table 4.3. The given tests have been performed on very compact beams with a relative slenderness of

λ ≈ 0.2 . To prove the

applicability of the European lateral torsional buckling curve, further tests (e.g. [12], [13]) on channel section loaded through its shear centre will be investigated and published in the next revised version of this report. (5)

The assessment assessment of beams made made of channels channels that that are loaded with transverse transverse loads loads and torsion caused by eccentricities of these loads and also with longitudinal compression forces is demonstrated in section 7.8. Table 4.2: Tests on channels with load application in shear centre; configurations configurations and results

Nr.

Test

Steel

Test set-up

1

2

F Load applicationexp [kN]

407.9

RWTH Aachen [9] UPE 200

S355

215.6

f y = 410 N/mm² L = 898.5 mm Ü =50 mm

3

4

page 40 / 142

114.2

…

Consistent determination of the resistance to lateral-torsional lateral-torsional buckling Table 4.3: Calculative results and comparison with tests

χ ⋅ χ ⋅ α ult ,k

βMz

ΔnE = ΔnR

r e/r t

1.000

1.000

0.000

1.000

1.015

0.487

1.000

0.331

0.663

0.994

1.406

0.483

0.994

0.142

0.852

0.995

1.004

Test

αEk *)

αult,k

αcrit

α*crit

α∗

χ

1

0.985

1.000

36.5

36.3

0.487 0 .487

2

0.711

3.025

110.3

109.7

3

0.996

7.066

156.8

154.4

4

…

1

*) load amplifier αEk = FEd / Fexp = (r e / r t)-1 which leads to an utilization level of 100%  ΔnE = ΔnR

page 41 / 142


page 42 / 142

Conclusions for „Recommendations for NDP´s“ in EN 1993-1-1

5


5.1

Procedure in EN 1993-1-1, section 6.3.1

(1)

The procedure in EN 1993-1-1, section 6.3.1 is the procedure with standardized European flexural buckling curves according to chapter 2 of this report.

(2)

The note to clause (3) refers to the application of the European standardized flexural buckling curves and lateral torsional buckling curves, that are specified in EN 1993-11, section 6.3.4. An explicit assessment of a non uniform member with the application of 2 nd order theory according to 5.3.4 (2), as mentioned in the note, is not necessary, as this application is already included in the flexural buckling curves and lateral torsional buckling curves. These buckling curves do contain the assumptions for imperfections as given in section 5.3.2 (11) equation (5.9), (5.10) and (5.11) and therefore can also be used for non-uniform members.

5.2

Procedure according to EN 1993-1-1, section 6.3.2.1 and section 6.3.2.2

(1)

The procedure in EN 1993-1-1, section 6.3.2.1 and section 6.3.2.2 is the procedure with standardized European lateral torsional buckling curves.

(2)

The note to clause (2) „The National Annex may determine the imperfection factors

α LT “ opens the door for modification of the α LT -values according to chapter 4 of this report. (3)

According to chapter 4 of this report, the values in EN 1993-1-1, table 6.3 and table 6.4 are for most cases on the safe side. An improvement by the modification ∗ α crit α LT = α LT ⋅ α crit ∗

is possible by the National Annex. (4)

The choice of the design point x = xd for different moment shapes may be taken from Table 5.1 of this report. As an alternative the given factor f can be used to modify the lateral torsional buckling curve.

page 43 / 142


Table 5.1: Bemessungsstelle x d in Abhängigkeit von der Momentenverteilung und

x d

Momentenverteilung A

mod

f

l

B

0,5

ψ = 1

1,0

0,1 ⋅ψ 2 + 0,18 ⋅ψ + 0,22

− 1 ≤ ψ ≤ 1

x

A

0,78 + 0,04 ⋅ψ

+ 0,08 ⋅ψ 2 + 0,1 ⋅ψ 3

0,5

1,0

0,5

1,0

B

λ mod ≤ ξ → λ mod > ξ →

xd l xd l

= 0 → χ LT ,mod = 1 0,5

= 0,5

0,5

λ mod ≤ ξ → a

λ mod > ξ →

b

A

xd l xd l

1,0

= 0 → χ LT ,mod = 1

2 ⋅ α

= α

B

λ mod ≤ ξ → λ mod > ξ → λ mod ≤ ξ → λ mod > ξ → λ mod ≤ ξ → a

b

λ mod > ξ →

Hinweis: Für alle Lagerungen A und B gilt:

xd l xd l xd l xd l xd l xd l

= 0 → χ LT ,mod = 1 = 0,61 = 0 → χ LT ,mod = 1

0,833

= 0,5 = 0 → χ LT ,mod = 1

3 − α 2 ⋅ α 1 − β 2

= α

η, ϕ = gehalten und η’, ϕ’ = frei

Verwendete Kürzel: α = a l ; β = b l ; l = a + b ; ξ = α 0 ⋅ f + 2 ⋅ ( f − 1)

page 44 / 142

0,562

2

⎛ α 0 ⋅ f ⎞ ⎜ ⎟ + f ⋅ (1 − 0,2 ⋅ α 0 ) − 1 ⎜ 2 ⋅ ( f − 1) ⎟ f − 1 ⎝ ⎠


5.3 (1)

Procedure according to EN 1993-1-1, section 6.3.2.3 The procedure in EN 1993-1-1, section 6.3.2.3 may be adapted to the procedure with standardized European lateral torsional buckling curves in one of the following ways: 1. The following choices are made: -

λ LT ,0 = 0,2 according to equation (6.57)

-

β = 1,0 according to equation (6.57)

-

Table 6.4 instead of Table 6.5

-

f = 1,0 according to equation (6.58)

-

λ LT at design point x = xd according to Table 5.1 of this report.

2. The function for the lateral torsional buckling curve in (6.57) and (6.58) is modified in the following way:

(2)

α ult ,k ,min α crit

-

λ LT ,mod =

-

χ LT ,mod =

-

χ LT =

-

2 ∗ ] φ = 0,5 ⋅ [1 + α LT ⋅ ( β ⋅ λ LT ,mod − λ LT ,0 )+ β ⋅ λ LT

-

β =

-

λ LT ,0 = 0,2

-

Table 6.5 and Table 6.6 are cancelled.

LT

f

however χ LT ,mod

≤ 1,0

1 2 2 φ LT + φ LT − β ⋅ λ LT

1 f

The second way is justified by the following: 1. The modified lateral torsional buckling curve in EN 1993-1-1, section 6.3.2.3 has not been derived from the standardized European flexural buckling curve based on a mechanical model. 2. The amplitudes of the imperfections used for the FEM-calculations were not consistent with the amplitudes determined for flexural buckling from tests, which would be relevant in the case

GI t ⋅ l

2

EI w ⋅ π 2

⇒0.

3. The procedure has not been verified by a reliability analysis according to Annex D of EN 1990. (3)

Some comparisons between the results of the procedure in EN 1993-1-1, section 6.3.2.3 together with the recommendations for numerical values given therein

page 45 / 142

Excerpt from the Background Document to EN 1993-1-1 ( χ LT,mod ) and the results of the standardized European lateral torsional buckling curves ( χ ∗ LT ) according to the recommendation in section 5.3 (1) 2 of this report, as well as the results of the flexural buckling curve (  χ LT ) in DIN EN 1993-1-1 are given in Figure 5.1. χ

χ

1.2

χ .LT.mod

1.1 1.0

χ .LT

0.9

1.2 1.1

χ .LT.mod

1.0

χ .LT

0.9

χ .LT*

0.8 0.7

0.7

0.6

0.6

0.5

0.5

0.4

χ .LT*

0.8

0.4

Momentenverteilung:

0.3

Momentenverteilung:

0.3

0.2

0.2

Trägerprofil: IPE 200

0.1


0.1

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

⎯λ χ

1.1

χ .LT.mod

1.0

χ .LT

⎯λ χ

1.2

0.9

3.0

1.2 1.1

χ .LT.mod

1.0

χ .LT

0.9

χ .LT*

0.8

χ .LT*

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

Momentenverteilung:

0.3

Momentenverteilung:

0.3

0.2

0.2

Trägerprofil: IPE 200

0.1


0.1

0.0

0.0 0. 0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2. 0

2. 2

2 .4

2 .6

2 .8

3 .0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

2 .2

2 .4

⎯λ

2 .6

2 .8

3 .0

⎯λ

Figure 5.1: Vergleich der diskutierten Biegedrillknickkurven für ausgewählte Beispiele

5.4 (1)

5.5

Procedure according to EN 1993-1-1, section 6.3.2.4 The approximative procedure in EN 1993-1-1, section 6.3.2.4 (1) B should be checked in view of the hierarchy of rules in relation to the standardized European flexural and lateral-torsional buckling curves within the limits of this approximation.

Procedure according to EN 1993-1-1, section 6.3.4

(1)

The procedure in EN 1993-1-1, section 6.3.4, is the procedure with standardized European flexural and lateral torsional buckling curves, which is dealt with in this report.

(2)

Using the results presented in this report clause (4) could be modified as follows: „(4) The reduction factor χ op may be determined from either of the following methods: a) From the flexural buckling curve according to 6.3.1. Then the value χ op should be calculated for the slenderness

λ op .

b) From the lateral-torsional buckling curve according to 6.3.2. Then the value χ op may be determined with the reduced imperfection factor

page 46 / 142

Conclusions for „Recommendations for NDP´s“ in EN 1993-1-1 ∗ α crit α = α ⋅ α crit where α crit is the critical amplification factor with considering the torsional ∗ stiffness and α crit is the critical amplification factor without considering the ∗

torsional stiffness. “ (3)

The equation (6.66) may be deleted, because the interaction between flexural buckling and lateral torsional buckling is included in the procedure for determining (through

5.6

LT

α ∗ ), so that no further interaction is necessary.

Imperfection according to EN 1993-1-1, section 5.3.4 (3)

(1)

Section 5.3.4 (3) of EN 1993-1-1 deals with the amplitude of the equivalent geometrical imperfection for lateral torsional buckling, for which according to section 5.3.2 (11) the eigen-mode shall be used. The note to this clause opens the door for national choices.

(2)

The wording of clause (3) is: „For a second order analysis taking into account of lateral torsional buckling of a member in bending the imperfections may be adopted as k·eo,d , where eo,d is the equivalent initial bow imperfection of the weak axis of the profile considered. In general an additional torsional imperfection need not to be allowed for.”

(3)

This wording aimed at a substitution of the imperfection defined by the eigen-mode with combined displacement and twist by a „more simple“ assumption of an „equivalent imperfection“ defined by a displacement only. To this end a single „equivalent value of k·eo,d for defining

′′ of the full profile should be chosen instead of different η init

amplitudes for the top and bottom flanges. (4)

To determine the value k is not simple, because comparative studies should be carried out with the standardized European lateral torsional buckling curve to find out what numerical value k should be taken. The value k = 0,5 given in the recommendation has been suggested by ECCS-TC8 that has taken this value form DIN 18800Part 2 as a first hint (knowing that it is evidently too small).

(5)

It would be better to define a safe-sided equivalent imperfection as a mix of displacement and twist (e.g. from GI t = 0) instead of looking for k .

page 47 / 142


page 48 / 142

Consideration of out of plane loading

6


6.1

Transverse loads on the standard column in compression

(1)

For the assessment of columns for flexural buckling with additional transverse loads Roik has developed a procedure, to come to a simple assessment formula for taking the supplementary effect into account.

(2)

A prerequisite for the accuracy of the procedure of Roik [14] is, that the shape of the in-plane bending moment M y I according to 1st order theory is equal to the shape of the eigen-mode

′′ , so that the following formula applies: η crit

M y I ( x ) = M 0 ⋅

′′ ( x) η crit ′′ ,max η crit

(6.1)

An example for the simple column with hinged ends is:

⎛ π x ⎞ ⎟ ⎝ l ⎠

M y I = M 0 ⋅ sin ⎜

(3)

This leads to the assessment formula: N E N E N R

(4)

(6.2)

+

N R

⋅

α ⋅ (λ − 0,2) 1−

N E N R

+

2

⋅ λ

M 0 M R

1

⋅ 1−

N E N R

=1

(6.3)

2

⋅ λ

In order to transfer this formula into the form of the assessment formula for columns in compression: N E

χ ⋅ N R the term

≤1

(6.4)

α ⋅ (λ − 0,2 ) is replaced by a function of χ from the basic equation for χ :

χ + χ ⋅ α ⋅ (λ − 0,2 )⋅

1 1 − χ ⋅ λ 2

=1

(6.5)

This gives:

(1 − χ ) (1 − χ ⋅ λ 2 ) α ⋅ (λ − 0,2) = χ

(6.6)

so that the formula (6.3) adopts the form: N E ⎛

N E

N R ⎝

N R

⋅ ⎜⎜1 −

⎞

λ 2 ⎟⎟ +

N E

⎠ χ N R

⋅ (1 − χ ) (1 − χ ⋅ λ 2 ) +

M 0 M R

⎛

N E

⎝

N R

= ⎜⎜1 −

⎞

λ 2 ⎟⎟

⎠

(6.7)

page 49 / 142


By rearranging the various terms in formula (6.7) one receives: N E

χ N R

+

M 0 M R

⎛

= Δn = ⎜⎜1 − ⎝

= 1−

N E ⎞ ⎛

⎞ N N ⎟⎟ ⎜⎜1 − E λ 2 ⎟⎟ + E ⋅ (1 + λ 2 − χ λ 2 ) N R ⎠ ⎝ N R ⎠ N R

N E

⎛

N E ⎞

⋅ ⎜⎜1 −

⎟⎟ ⋅ χ 2 ⋅ λ 2 χ N R ⎝ χ N R ⎠

(6.8)

1. Stufe

424 3 144 4 244 4 3 1 ≤ 0, 4

≤ 0, 25

144 4 4 4 244 4 4 4 3

2. Stufe (6.9)

≤ 0,1

144 4 4 4 4 244 4 4 4 4 3 ≥ 0,9

3. Stufe

so that an accurate solution (1 st step) and two steps of simplification (2 nd step and 3rd step) can be obtained. The maximum simplification leads to N E

χ N R (6)

+

M 0 M R

≤ Δn = 0,9

(6.10)

In order to consider also other moment shapes M 0 than those according to equation (6.1), the equation (6.10) is extended: N E

χ N R (7)

+

M 0 (1 − q ) M R

≤ Δn

(6.11)

To determine q a development of M z , p y and η in series based on of the various eigenmodes

η crit,m is performed:

∑

⎧M y I ( x) = pm ⋅η crit ′′ ,m ( x)⎫ ⎪ ⎪ m ⎪ ⎪ ⎨ ⎬ external load ⎪ p ( x) = p ⋅η ′′′′ ( x)⎪ m crit ,m ⎪⎩ z ⎪⎭ m ⎧ ⎫ ⎨ η ( x) = η m ⋅η crit ,m ( x ) ⎬ displacement m ⎩ ⎭

∑

(6.12)

∑

From the differential equation:

′′′′ + N η crit ′′ = p z ( x) EI y η crit

(6.13)

which gives the equation:

∑η ( EI ⋅η ′′′′ m

y

crit ,m

′′ ,m ) = ∑ pm η crit ′′′′ ,m + N ⋅η crit

m

(6.14)

m

the solution for η m is obtained:

η m = pm (8)

′′′′ ,m η crit ′′′′ ,m + N E ⋅η crit ′′ ,m EI y ⋅η crit

Using the orthogonality-equations (see 1.3)

page 50 / 142

(6.15)


∫ η

crit , j

′′′′ ,i ⋅η crit

dx = 0

für i ≠ j

(6.16)

′′ ,i ⋅η crit

dx = 0

für i ≠ j

(6.17)

l

and

∫ η ′′

crit , j

l

it follows:

∫ M ( x) ⋅η ′′ I y

pm

=

( ) dx

crit x

l

(6.18)

′′ ( x) ⋅η crit ′′ ( x) dx ∫ η crit l

e.g. for the simple column with hinged ends with:

η crit ,i = sin

m π x l 2

m π ⎞ m π x = −⎛ ⎜ ⎟ sin l ⎝ l ⎠

′′ ,i η crit

(6.19)

4

m π ⎞ m π x ′′′′ ,i = ⎛ η crit ⎜ ⎟ sin l ⎝ l ⎠

and for a bending moment M 0 constant along the length of the column;

pm

=

⎛ mπ ⎞ ⎜ ⎟ ⎝ l ⎠

= − M 0 ⋅ (9)

mπ x

∫

M 0 sin 2

l

mπ x

∫

sin 2

l

4l 2 m3

dx

=

M 0

dx

2l mπ

l

2

(6.20)

(m = 1, 3, 5, ...)

π 3

The bending moment M y II according to 2 nd order theory results from: M y II = EI yη ′′ = EI y

∑η

m

′′ ,m ⋅η crit

m

= ∑ EI y ⋅

′′′′ ,m p m η crit

′′ ,m η crit ′ ′ ′ ′ ′ ′ η + η N crit , m E crit ,m m ′′′′ ,m EI z η crit ′′ ,m = ∑ p m ⋅ η crit ′ ′ ′ ′ ′ ′ η + η EI N y crit , m E crit , m m = ∑ p m ⋅ m

EI y

1 N E

1−

EI y

= ∑ p m ⋅ m


′′ ,m η crit ′′′′ ,m η crit

1 1−

(6.21)

N E


N crit ,m

page 51 / 142

Excerpt from the Background Document to EN 1993-1-1 (10) With this bending moment the following equation instead of equation (6.3) is obtained:

⎧ ⎫ ⎧ ⎧ N E ⎫ ⎪⎪ N E α (λ − 0,2)⎪⎪ ⎪⎪ ⋅ ⎨ ⎬+⎨ ⎬+⎨ N N N 2 ⎩ R ⎭ ⎪ R 1 − E λ ⎪ ⎪ ⎪⎩ ⎪⎭ ⎪⎩ N R

∑ m

⎫ ⎪ pm 1 ′′ ,m ( xd )⎪⎬ = 1 ⋅ η crit N M R ⎪ 1 − E N crit ,m ⎪⎭

(6.22)

(11) Equation (6.22) can by using the series-development M y I =

∑ p

m

′′ ,m ( xd ) η crit

(6.23)

m

be brought into the form:

⎡ ⎢ I N E N E α (λ − 0,2 ) M y ⎢ + ⋅ + − N E 2 M R ⎢ N R N R 1− λ ⎢ N R ⎢ ⎣

′′ ,m ( xd ) ∑ pmη crit m

M R

⎤ ⎥ ′′ ,m ( xd ) ⎥ pm η crit =1 ⎛ N E ⎞ ⎥⎥ ⎟ M R ⎜1 − ⎜ N crit ,m ⎟ ⎥ ⎝ ⎠ ⎦

−∑ i

144 4 4 4 4 4 4 244 4 4 4 4 4 4 3 N E

+∑

′ ,m N crit , m pmη crit ⋅

M R

1−

N E

N crit , m

144 4 4 4 4 4 4 4 244 4 4 4 4 4 4 4 3 ⎧

M y I ⎪ ⎪

⎨1+ ∑

M R ⎪

⎪⎩

⎫ ⎪

N E

′′ , m N crit , m ⎪ pmη crit ⋅ ⎬ N E ⎪ M I y

1−

⎪

N

crit , m ⎭ 144 4 4 4 4 4 4 4 244 4 4 4 4 4 4 4 3

(6.24)

⎧ N E ⎞ ⎫ ⎛ ⎜ ⎟ ⎪ I ′′ , m N crit , m ⎟ ⎪⎪ M y pmη crit 1 N E 2 ⎞ ⎜ ⎪⎛ ⎜ ⎟ 1 1 − λ ⋅ + ⋅ ⎨ ⎬ ∑ M I N ⎟ M R N E 2 ⎪⎜⎝ N R ⎠⎟ ⎜ y ⎜ 1− 1− E ⎟ ⎪ λ ⎜ ⎟ ⎪ N R N crit , m ⎠ ⎪ ⎝ ⎩ ⎭

to accelerate the convergence. (12) By using the first element of the series only

M m

′′ , m ( xd ) = pm ⋅η crit

one gets a conservative solution:

⎧ N E ⎞⎫ ⎛ ⎜ ⎟⎪ ⎪⎛ N ⎞ M m N crit ⎟⎪ N E N E α (λ − 0,2 ) 1 ⎪⎜ ⎜ E ⎟ ⋅ 1 + I ⋅ + ⋅ + ⎨ 1− ⎬ =1 ⎟ N E 2 M R N E 2 ⎜⎝ N crit ⎠⎟ ⎜ N N R N R M E y 1 − 1− 1− λ λ ⎪ ⎜ ⎟⎪ N R N R ⎪⎩ N crit ⎠⎪ ⎝ ⎭ M y I

144 4 4 4 4 244 4 4 4 4 3 N E

⎛ N ⎞ M N 1− +⎜ 1− E ⎟⋅ m ⋅ crit N N crit ⎜⎝ N crit ⎠⎟ M y I 1− E N E

N

144 4 4 4 4 244 4 crit 4 4 4 3 1−

N E

+

M m N E

⋅

N crit M y I N crit

144 4 4 4 4 244 4 4 4 4 3 M m ⎞⎟ 1− I ⎜ ⎟ N crit ⎝ M y ⎠

1−

page 52 / 142

N E ⎛ ⎜

(6.25)

Consideration of out of plane loading (13) From (6.25) and using equation (6.11) one obtains q=

N E N R

⎛ ⎜ ⎝

M m ⎞⎟

⋅ λ 2 ⋅ ⎜1 −

(6.26)

M y I ⎠⎟

(14) For the example in (6.2) follows q=

N E N R

⋅ λ 2 ⋅ (1 − 1) = 0

(6.27)

and for the example in (6.20) q=

N E N R

4 ⎞ N 2 ⋅ λ 2 ⋅ ⎛ ⎜1 − ⎟ = −0,27 E ⋅ λ N R ⎝ π ⎠

(6.28)

(15) When using equation (6.11) it is presumed, that the maximum values of the effects of the out of plane imperfections and the out of plane bending are approximate at the same spot x = xd . This presumption applies in case of equation (6.3) and also in case of equation (6.11) if the maxima for in plane stressing coincide with the maxima of out of plane stressing. Therefore the results are either safe sided or the actual design point x = xd should be determined.

6.2

Out of plane bending and torsion for the basic situation for lateral torsional buckling

(1)

For the assessment of the standard beam with the standardized European lateral torsional buckling curves the method of Roik is also applicable. [15]

(2)

For the standard beam it is assumed in the first step, that the shapes of the out of I plane bending moments M z and warping bi-moments M I w follow the shape of the

eigen-mode for lateral torsional buckling: I M z = M z ,m

T w I = T w, m ⋅

⋅

′′ η crit ′′ ,max η crit

(6.29)

′′ ϕ crit ′′ , max ϕ crit

(6.30)

For the example of the simple beam with hinged ends and constant bending moment M y this means I M z = M z ,m ⋅ sin

T w I = T w, m ⋅ sin

(3)

π x

(6.31)

l

π x

(6.32)

l

According to the equation (6.8) and (6.9) the action effects using 2nd order theory are: II M z = M z ,m

1

⋅ 1−

M E , y

⋅ sin

π x l

(6.33)

M crit

page 53 / 142


1

T w II = T w, m ⋅

1− (4)

M E , y

⋅ sin

π x

(6.34)

l

M crit

The bending moments in the top flange are

σ edge f y

=

II M z b

T w II

f y I z 2

f y I w

⋅ +

M II fl , top

b

⋅ z M ⋅ = 2

f y ⋅ b

2

⋅ t

⋅6 (6.35)

= (5)

II M z

M R, z

+

T w II

=

T R , w

M II fl , top M R , fl , top

Therefore the assessment formula reads I α (λ − 0,2) ⎧⎪ M fl ⎫⎪ 1 + ⋅ +⎨ ⎬ M = 1 M E , y N R , fl N R , fl M ⎪ ⎩ R , fl ⎪⎭ 1 − E , y 1−

N E , fl N E , fl

M crit

(6)

(6.36)

M crit

Because of the analogy to equation (6.3) the conclusions in equations (6.8) and (6.9) can be transferred, so that the assessment reads M E , y

χ M R, y

+

I M E , fl , z

M R, fl , z

≤ Δn = 1 −

M E , y ⎛

M E , y ⎞ ⎜1 − ⎟ χ 2 λ 2 χ M R , y ⎜⎝ χ M R , y ⎠⎟

(6.37)

144 4 4 244 4 4 3 123 ≤ 0 , 25

≤ 0, 4

144 4 4 4 244 4 4 4 3 ≤ 0,1 144 4 4 4 4 4 244 4 4 4 4 4 3 ≤0 , 9

(7)

If the bending moments in the top flange do not follow the eigen-modes

′′ and ϕ crit ′′ η crit

II

correction factors may be applied to M E , Fl , y , so that equation (6.37) is transferred using equation (6.26) into: M E , y

χ M R, y

+

I M E , fl , z

M R , fl , z

(1 − q )+ M y

I T E , fl , w

T R , fl , w

(1 − qT ) ≤ Δn w

(6.38)

where the following applies: qM , z =

qT , w

6.3 (1)

=

M E , y M R, y M E , y M R , y

⎛

⋅ λ 2 ⋅ ⎜⎜1 − ⎝

⎛

⋅ λ 2 ⋅ ⎜⎜1 − ⎝

M z ,m ⎞

⎟

I ⎟ M z ⎠

T w, m ⎞

⎟

T w I ⎠⎟

General case of out of plane bending and torsion In the general case, see section 4.2.3, the assessment equation reads:

page 54 / 142

(6.39)

(6.40)


1

χ α ult , k

+

M E , fl , z M R , fl , z

(1 − qM ) + z

T E , fl , w T R , fl , w

(1 − qT ) ≤ Δn

(6.41)

w

=1−

⎛ ⎞ ⎜1 − 1 ⎟ χ 2 λ 2 χ α ult ,k ⎜⎝ χ α ult ,k ⎠⎟ 424 3 144 4 244 4 3 1

1

≤0 , 4

≤0, 25 144 4 4 4 244 4 4 4 3 ≤0,1 144 4 4 4 4 244 4 4 4 4 3 ≥0 , 9

6.4 (1)

q M z

=

qT w

=

⎛

1

α ult ,k

⋅ λ 2 ⋅ ⎜⎜1 −

α ult , k

⎟

⋅ λ 2 ⋅ ⎜⎜1 −

(6.42)

I ⎟ M z ⎠

⎝

⎛

1

M z ,m ⎞

T w, m ⎞

⎝

⎟

(6.43)

T w I ⎠⎟

Proof of orthogonality for the series-development The differential equation: EI z η ′′′′ + N η ′′ = 0

(6.44)

is satisfied by:

′′′′ ,i + κ i2 η crit ′′ ,i = 0 η crit 2 ′′′′ , j + κ j η crit ′′ , j = 0 η crit (2)

(6.45)

It follows:

∫ η ′′ ∫ η ′′

′′′′ ,i + κ i2 ∫ η crit ′′ , j η crit ′′ ,i = 0 η crit 2 ′′′′ ′′ ′′ crit ,i η crit , j + κ j ∫ η crit ,i η crit , j = 0 crit , j

(3)

(6.46)

By substraction and partial integration it follows

η ′′ ⋅ η ′′′′ ) − (η ′′ ⋅ η ′′′′ ) + (κ − κ )∫ (η ′′ ⋅ η ′′ ) = 0 ∫ (1 42 4 43 4 142 4 43 4 1 424 3 144 244 3 crit , j

crit ,i

crit ,i

′′ , j ⋅η crit ′′ ,i η crit R ′′ , j ⋅η crit ′′ ,i + ∫ η crit

crit , j

′′ ,i ⋅η crit ′′ , j −η crit R ′′ ,i ⋅η crit ′′ , j − ∫ η crit

144 4 4 4 244 4 4 4 3 =0

2

2

i

j

≠0 ≠ j

für i

crit ,i

crit , j

(6.47)

=0 ≠ j

für i

144 244 3 ′ , j ⋅η crit ′ ,i η crit R ′′′ ,i −η crit , j ⋅η crit R ′′′′ ,i + ∫ η crit , j ⋅η crit 144 244 3 =0 ≠ j

für i

(4)

This proves the orthogonality necessary for the serial development.

page 55 / 142


6.5 (1)

Comparison with test results The reliability of the formulae (6.41), (6.42) and (6.43) for the verification of beamcolumns with compression, biaxial bending and torsion has been determined according to the procedure given in EN 1990 – Annex. Figure 6.1 gives a comparison of test results from [9] with calculative results [15].

(2)

Table 6.1 gives the * γ M = 1,1

* * -values related to the results which are between γ M = 1,0 γ M

and

as required.

r e /r t 2.0

Lindner - IPE 200

1.8

Lindner - HEB 200

1.6

Kindmann - Vers. II

1.4

Kindmann - Vers. III

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Figure 6.1: Comparison between test results from [9] and calculative results

Table 6.1: Determination of the

-value according to EN 1990 – Annex D

M

Eingangsdaten υrt = 0,08 (Geometrie und Streckgrenze) υfy = 0,07 (Streckgrenze)

Research Project Fosta P 554 (N = 32) Standardnormalverteilung


2.0 g n 1.5 u l i e t r e 1.0 v l a m r o 0.5 n d r a d 0.0 n a 0.8 t S-0.5 r e d e l i -1.0 t n a u Q-1.5

2.0

1.0

1.2

1.4

1.6

1.8


1.5 1.0 0.5 0.0 -0.2

-0.5

0.0

0.2

-1.5 ln re/rt

sδ = 0.146

b = 1.298

page 56 / 142

0.8

-1.0

re/rt

γM = 1.333

0.6

-2.0

-2.0

υδ = 0.112

0.4

(Modell) Δk = 0.846

sδ = 0.159

b = 1.307

υR = 0.138

(gesamt) *

γM = 1.127

υδ = 0.122 γM = 1.225


υR = 0.146

(gesamt) * γM

= 1.041

Guidance for applicationsGuidance for applications

7

Guidance for applications

7.1

General

(1)

In the following design aids and worked examples for specific applications are given, that illustrate the workability of the „general method“ for flexural buckling, lateral torsional buckling and combination of both, more generally described as „out-ofplane“-buckling of members that are “in-plane” loaded in their strong plane.

(2)

A distinction between flexural buckling and lateral torsional buckling is no more necessary as the European lateral torsional buckling curve defined by

χ LT (λ ) =

1

φ + φ 2 − λ 2

φ = 0.5 ⋅ [1 + α ∗ (λ − 0.2) + λ 2 ] ∗ α crit α = ⋅ α α crit ∗

implicitly includes flexural buckling and all combinations of flexural and lateraltorsional buckling from both in-plane compression forces, eccentricities of these compression forces and any transverse loading and also can be extended to cover out-ofplane loading as well. (3)

The flow chart for the assessment of combined flexural and lateral torsional buckling is given in Table 7.1 and Table 7.2.

(4)

In order to identify

∗ without effects of St. Venant-torsional stiffness and α crit with α crit

effects of St. Venant torsional stiffness computer calculations can use assumptions as given in Figure 7.1.

Free distorsional deformation

Distortional deformation restrained

Figure 7.1: Assumptions to obtain

∗ and α crit α crit

Note: A computer program which is freely available is: LTBeam [16]

page 57 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.1: Flow chart for the lateral torsional buckling verification

Input Distribution of in-plane load effects ( N Ed , M y,Ed ) including 2

nd

order analysis

Distribution of cross- sectional data Boundary conditions for out-of-plane deflections

Analysis

α ult , k ( x) ′′ α , α ∗ , location of max η crit crit

crit

∗ ⎞ ⎛ α crit ⎟ ⋅ α α , α = ⎜⎜ ⎟ ⎝ α crit ⎠ ∗

Relevant location and assessment xd is not known xd = x( ult,k,min )

xd is known

α ult ,k ( xd ) λ ( xd ) =

α ult ,k ( xd ) α crit

χ ( xd ) = χ (α * , λ ( xd ) )

α E =

α Ed =

page 58 / 142

( xd ) ⋅ α ult ,k ( x d )

χ ( xd ) ⋅ α ult ,k ( xd ) ≥1 γ M

Guidance for applicationsGuidance for applications Table 7.2: Flow chart for the lateral torsional buckling verification with out of plane loading

β z ( x) =

α Ed ( x) =

χ ( x) ⋅ α ult ,k ( x) ≥1 γ M

Δn E =

Δn R = 1 −

β w ( x) =

1

α Ed ( xd )

M z , Ed ( x ) M z , Rd ( x ) T w, Ed ( x ) T w, Rd ( x )

(1 − q z ) (1 − qw )

+ β y ( xd ) + β w ( xd )

⎡ ⎤ 2 1 2 ⎢1 − ⎥ ⋅ χ ( x d ) λ ( xd ) α Ed ( x d ) ⎣ α Ed ( x d ) ⎦ 1

Δn E ( xd ) ≤ Δn R ( xd )

page 59 / 142


7.2 7.2.1 (1)

Design aids Hand formulae for the determination of Mcr For particular cases, e.g. those with in-plane transverse loads and in-plane moments only and without any compression forces, the values

∗ and α crit can be obtained α crit

by hand calculation from formulas, as given in Tables 7.3, 7.4, 7.5 and 7.6. 7.2.1.1 Basis (1)

The elastic critical moment for lateral-torsional buckling of a beam of uniform symmetrical cross-section with equal flanges, under standard conditions of restraint at each end and subject to uniform moment in plane going through the shear centre is given by: 2

2 π EI z =

M cr

L

L GI t

π 2 EI z

2

+

I w I z

=

π

EI z GI t L

1+

π 2 EI w L2 GI t

(7.1)

where: G=

(2)

E 2 (1 + ν )

I t

is the torsion constant

I w

is the warping constant

I z

is the second moment of area about the minor axis

L

is the length of the beam between points that have lateral restraint

ν

is the Poisson ratio

The standard conditions of restraint at each end are: -

restrained against lateral movement, free to rotate on plan (k z = 1);

-

restrained against rotation about the longitudinal axis, free to warp ( k w = 1);

-

restrained against movement in plane of loading, free to rotate in this plane (k y = 1).

7.2.1.2 General formula for beams with uniform cross-sections symmetrical about the minor or major axis (1)

In the case of a beam of uniform cross-section which is symmetrical about the minor axis, for bending about the major axis the elastic critical moment for lateral-torsional buckling is given by the general formula: M cr = μ cr

π

EI z GI t

where relative non-dimensional critical moment

page 60 / 142

(7.2)

L

μ cr is

Guidance for applicationsGuidance for applications C 1 ⎡ 1 + κ w2 t ⎢ k z ⎣

μ cr =

+ (C 2ζ g − C 3ζ j ) 2 − (C 2ζ g − C 3ζ j )⎤⎥ ,

(7.3)

⎦

non-dimensional torsion parameter is

κ wt =

π

EI w

k w L

GI t

(7.4)

relative non-dimensional coordinate of the point of load application related to shear center

ζ g =

π z g

EI z

k z L

GI t

(7.5)

relative non-dimensional cross-section mono-symmetry parameter

ζ j =

π z j

EI z

k z L

GI t

(7.6)

where: C 1, C 2 and C 3 are factors depending mainly on the loading and end restraint conditions (See Tables 7.3 and 7.4) k z and k w are buckling length factors z g

= z a − z s

z j = z s

−

(7.7)

0,5 I y

∫ ( y

2

+ z 2 ) z dA

(7.8)

A

is the coordinate of the point of load application related to centroid

z a

(see Figure 7.2) z s

is the coordinate of the shear center related to centroid

z g

is the coordinate of the point of load application related to shear centre.

NOTE 1: See section 7.2.1.2 (7) and (8) for sign conventions and section 7.2.1.4 (2) for approximations for z j . NOTE 2: z j = 0 ( y j

= 0 ) for cross sections with y-axis ( z -axis) being axis of symmetry.

NOTE 3: The following approximation for z j can be used: z j

⎛

= 0,45 ⋅ψ f h s ⎜⎜1 + ⎝

c ⎞⎟

2h f ⎠⎟

(7.9)

where: c

is the depth of a lip

h f

is the distance between centerlines of the flanges.

ψ f =

I fc − I ft I fc + I ft

(7.10)

page 61 / 142

Excerpt from the Background Document to EN 1993-1-1 I fc

is the second moment of area of the compression flange about the minor axis of the section

I ft

is the second moment of area of the tension flange about the minor axis of the section

h s

is the distance between the shear centre of the upper flange and shear centre of the bottom flange ( S u and S b in Figure 7.2).

For I -sections with unequal flanges without lips and as an approximation also with lips: I w

= (1 −ψ f 2 )⋅ I z ⋅ (h s 2 )2

(7.11)

(2)

The buckling length factors k z (for lateral bending boundary conditions) and k w (for torsion boundary condition) vary from 0,5 for both beam ends fixed to 1,0 for both ends simply supported, with 0,7 for one end fixed (left or right) and one end simply supported (right or left).

(3)

The factor k z refers to end rotation on plan. It is analogous to the ratio Lcr /L for a compression member.

(4)

The factor k w refers to end warping. Unless special provision for warping fixity of both beam ends (k w = 0,5) is made, k w should be taken as 1,0.

(C) Compression side, (T) tension side, S shear centre, G gravity centre Su, S b is shear centre of upper and bottom flange Figure 7.2: Notation and sign convention for beams under gravity loads ( F z ) or for cantilevers under uplift loads ( - F z ) (5)

Values of C 1, C 2 and C 3 are given in Tables 7.3 and 7.4 for various load cases, as indicated by the shape of the bending moment diagram over the length L between lateral restraints. Values are given in Table 7.3 corresponding to various values of k z and in Table 7.4 also corresponding to various values of k w .

(6)

For cases with k z = 1,0 the value of C 1 for any ratio of end moment loading as indicated in Table 7.3, is given approximately by: C 1

(7)

=

0.310 + 0.428 ψ + 0.262 ψ 2

The sign convention for determining z and z j, see Figure 7.2, is:

page 62 / 142

(7.12)


(8)

-

coordinate z is positive for the compression flange. When determining z j from equation (7.8), positive coordinate z goes upwards for beams under gravity loads or for cantilevers under uplift loads, and goes downwards for beams under uplift loads or cantilevers under gravity loads

-

sign of z j is the same as the sign of cross-section mono-symmetry factor ψ f from equation (7.10) . Take the cross section located at the M -side in the case of moment loading, Table 7.3, and the cross-section located in the middle of the beam span in the case of transverse loading, Table 7.4.

The sign convention for determining z g is: -

for gravity loads z g is positive for loads applied above the shear centre

-

in the general case z g is positive for loads acting towards the shear centre from their point of application.

page 63 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.3: Values of factors C 1 and C 3 corresponding to various end moment ratios

, values of

buckling length factor k z and cross-section parameters f and wt . End moment loading of the simply supported beam with buckling length factors k y = 1 for major axis bending and k w = 1 for torsion

page 64 / 142

Guidance for applicationsGuidance for applications Table 7.4: Values of factors C 1, C 2 and C 3 corresponding to various transverse loading cases, values of buckling length factors k y, k z , k w cross-section mono-symmetry factor parameter

f

and torsion

.

wt

page 65 / 142

Excerpt from the Background Document to EN 1993-1-1 7.2.1.3 Beams with uniform cross-sections symmetrical about major axis, centrally symmetric and doubly symmetric cross-sections (1)

For beams with uniform cross-sections symmetrical about major axis, centrally symmetric and doubly symmetric cross-sections loaded perpendicular to the major axis in the plane going through the shear centre, Table 7.3, z j = 0, thus

μ cr = (2)

− C 2ζ g ⎤⎥

(7.13)

⎦

For end-moment loading C 2 = 0 and for transverse loads applied at the shear centre z g = 0. For these cases:

μ cr = (3)

C 1 ⎡ 2 1 ( + κ + C 2ζ g ) 2 wt k z ⎢⎣

C 1 k z

2 1 + κ wt

(7.14)

If also κ wt = 0 :

μ cr = C 1

(7.15)

k z

Figure 7.3: Beams with uniform cross-sections symmetrical about major axis, centrally symmetric and doubly symmetric cross-sections

(4)

For beams supported on both ends ( k y

=1,

k z = 1 , 0,5 ≤ k w

≤ 1 ) or for beam segments

laterally restrained on both ends, which are under any loading (e.g. different end moments combined with any transverse loading), the following value of factor C 1 may be used in the above two formulas given in section 7.2.1.3 (2) and (3) to obtain approximate value of critical moment: C 1

=

1,7 M max M 02, 25

+ M 02,5 + M 02,75

≤ 2,5

(7.16)

where M max

is maximum design bending moment,

M 0, 25 , M 0,75

are design bending moments at the quarter points and

M 0,5

is design bending moment at the midpoint of the beam or beam segment with length equal to the distance between adjacent crosssections which are laterally restrained.

(5)

Factor C 1 defined by equation (7.16) may be used also in equation (7.13), but only in combination with relevant value of factor C 2 valid for given loading and boundary conditions. This means that for the six cases in Table 7.4 with boundary condition

page 66 / 142

Guidance for applicationsGuidance for applications k y

= 1,

k z = 1, 0,5 ≤ k w

≤ 1 , as defined above, the value

C 2

= 0,5 may be used together

with equation (7.16) in equation (7.13) as an approximation. (6)

In the case of continuous beam the following approximate method may be used. The effect of lateral continuity between adjacent segments are ignored and each segment is treated as being simply supported laterally. Thus the elastic buckling of each segment is analysed for its in-plane moment distribution (equation (7.16) for C 1 may be used) and for an buckling length equal to the segment length L. The lowest of critical moments computed for each segment is taken as the elastic critical load set of the continuous beam. This method produces a lower bound estimate.

7.2.1.4 Cantilevers with uniform cross-sections symmetrical about the minor axis (1)

In the case of a cantilever of uniform cross-section, which is symmetrical about the minor axis, for bending about the major axis the elastic critical moment for lateraltorsional buckling is given by the equation (7.2), where the values of relative nondimensional critical moment µcr are given in Tables 7.5 and 7.6. In Tables 7.5 and 7.6 non-linear interpolation should be used.

(2)

The sign convention for determining z j and z g is given in section 7.2.1.2 (7) and (8).

page 67 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.5: Relative non-dimensional critical moment µcr for cantilever (k y

= k z = k w = 2) loaded by

concentrated end load F

a) For z j = 0 , z g = 0 and κ wt0 ≤ 8 : μ cr = 1,27 + 1,14 κ wt0 + 0,017 κ w2t0 .

b) For z j = , 0− 4 ≤ ζ g ≤ 4 and κ wt ≤ 4 , μ cr may be calculated also from equation (7.13) and (7.14), where the following approximate values of the factors C 1, C 2 should be used for the cantilever under tip load F : 3 , if κ wt ≤ 2 = 2,56 + 4,675κ wt − 2,62κ wt2 + 0,5κ wt if κ wt > 2 C 1 = 5,55

C 1

C 2

= 1,255 + 1,566 κ wt − 0,931κ w2 t + 0,245κ w3 t − 0,024κ w4 t ,

if ζ g ≥ 0

C 2

= 0,192 + 0,585κ wt − 0,054 κ w2t − (0,032 + 0,102 κ wt − 0,013κ w2 t ) ζ g ,

if ζ g < 0

page 68 / 142

Guidance for applicationsGuidance for applications Table 7.6: Relative non-dimensional critical moment µcr for cantilever (k y

= k z = k w = 2) loaded by

uniformly distributed load q

2 a) For z j = 0 , z g = 0 and κ wt 0 ≤ 8 : μ cr = 2,04 + 2,68κ wt 0 + 0,021κ wt 0.

b) For z j = , 0− 4 ≤ ζ g ≤ 4 and κ ω t ≤ 4 , μ cr may be calculated also from equation (7.13) and (7.14), where the following approximate values of the factors C 1, C 2 should be used for the cantilever under uniform load q: C 1 = 4,11 + 11,2 κ wt

− 5,65 κ w2t + 0,975κ w3 t ,

if κ wt ≤ 2

C 1

= 12

if κ wt > 2

C 2

= 1,661 + 1,068 κ wt − 0,609 κ w2 t + 0,153 κ w3 t − 0,014 κ w4 t ,

if ζ g ≥ 0

C 2

= 0,535 + 0,426 κ wt − 0,029 κ w2 t − (0,061 + 0,074 κ wt − 0,0085κ w2 t ) ζ g ,

if ζ g < 0

page 69 / 142

Excerpt from the Background Document to EN 1993-1-1 7.2.2

Location of assessment xd

x d

Momentenverteilung A

B

f

l k y

=1 0,5

ψ = 1

1,0

0,1 ⋅ψ 2 + 0,18 ⋅ψ + 0,22

− 1 ≤ ψ ≤ 1

x

A

B

k y

0,78 + 0,04 ⋅ψ

+ 0,08 ⋅ψ 2 + 0,1 ⋅ψ 3

0,5

1,0

0,5

1,0

= 0,5 λ mod ≤ ξ → λ mod > ξ →

xd l xd l

= 0 → χ LT , mod = 1 0,5

= 0,5

0,5

λ mod ≤ ξ → a

λ mod > ξ →

b

A

B

k y

λ mod > ξ → λ mod ≤ ξ → λ mod > ξ → λ mod ≤ ξ → b

l xd l

= 0 → χ LT , mod = 1

λ mod > ξ →

Hinweis: Für alle Lagerungen A und B gilt: k z , k w

xd l xd l xd l xd l xd l xd l

= 0 → χ LT , mod = 1 0,562

= 0,61 = 0 → χ LT , mod = 1

0,833

= 0,5 = 0 → χ LT , mod = 1

3 − α 2 ⋅ α 1 − β 2

= α

=1

Verwendete Kürzel: α = a l ; β = b l ; l = a + b ; ξ = α 0 ⋅ f + 2 ⋅ ( f − 1)

page 70 / 142

2 ⋅ α

= α

= 0,7 λ mod ≤ ξ →

a

xd

1,0

2

⎛ α 0 ⋅ f ⎞ ⎜ ⎟ + f ⋅ (1 − 0,2 ⋅ α 0 ) − 1 ⎜ 2 ⋅ ( f − 1) ⎟ f − 1 ⎝ ⎠


7.3

(1)

Examples to compare the results of the general method using the European lateral buckling curve with results of the component method in Eurocode 3-Part 1-1, section 6.3.3 The use of the component method in Eurocode 3-Part 1-1, section 6.3.3 is illustrated in Figure 7.4.

P z,Ed M y,Ed

M y,Ed

N Ed

N Ed M z,Ed

M z,Ed

P y,Ed

compression only

inplane transverse loads and inplane moments only

out of plane transverse loads and out of plane moments only

P z,Ed M y,Ed

N Ed

M y,Ed

M z,Ed

M z,Ed

N Ed P y,Ed

N Ed

χ y N Rd

out of plane bending

lateral torsional buckling

flexural buckling

M y , Ed + ΔM y , Ed

≤1

χ LT M y , Rd

M z , Ed + ΔM z , Ed

≤1

M z , Rd

≤1

Interaction N Ed

χ y N Rd N Ed

χ z N Rd

+ k yy

+ k zy

M y , Ed + ΔM y , Ed

χ LT M y , Rd M y , Ed + ΔM y , Ed

χ LT M y , Rd

+ k zy

+ k zz

M z , Ed + ΔM z , Ed M z , Rd M z , Ed + ΔM z , Ed M z , Rd

≤1

≤1

Figure 7.4: Procedure for the “component method”

(2)

For the functions k yy, k yz , k zy and k zz there are two alternatives given in Annex A and B of Eurocode 3-Part 1-1. [2]

(3)

To compare the results of the general method with the results of the component method 5 worked examples as published in [17], are chosen, for which the various steps of calculations are given in Table 7.7.

(4)

Where the location of the design point xd /ℓ is not a priori evident, the procedure according to step 4 in Table 7.1 can be used to calculate χ (x ) i at various spots xi, from which xd is the spot where the maximum value of χ (x ) i , see distribution of χ (x) in Table 7.7, is achieved.

(5)

Where the maximum value of χ (x) is at an end of a member, see examples no. 1 and 2 in Table 7.7, lateral torsional buckling is not relevant for the design, but a crosssectional verification at the supports is necessary (with

α ult,k only).

page 71 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.7: Worked examples (from ECCS-Publication No. 119 [17])

page 72 / 142

Guidance for applicationsGuidance for applications (6)

In the calculations the reference value of the imperfection factor α is always the value associated with the flexural buckling curve for the weak axes.

(7)

Table 7.8 gives a comparison of the results of the general method (that can be considered as reliable) with the results of the component method in Eurocode 3 – Part 1-1 section 6.3.2, as published in [17]. This publication also gives results for the use of the two alternatives for interaction formulas as specified in Eurocode 3Part 1-1, Annex A and B. The choices of the reference flexural buckling curve in this publication are not always identical with the choice of α for weak axis buckling.

Table 7.8:

Utilization grades 1/ Ed and 1/ Ed,M from the general method with European lateral torsional buckling curves and from the specific method with flexural buckling curves modified with

Beispiel Nr.

(8)

Allgemeines Verfahren mit Europäischer Biegedrillknickkurve

and f and with Annex A and B of EC3 Part 1-1 Spezielles Verfahren mit der mit modifizierten BDK-Kurve

und f

Anlage A

Anlage B

1

1,603

-

-

2

0,988

0,950

0,836

3

1,111

1,131

1,112

4

0,981

1,131

0,903

5

0,950

1,045

0,946

The comparison in Table 7.8 reveals that the results of the component method, though not being fully consistent with the principles in Eurocode 3 give rather acceptable results. Criticism on the component method may be placed in view of their -

limited field of application (only particular end conditions and no torsion action),

-

complexity and lack of transparency,

-

disproportionality of design effort in relation to the win of safety and economy.

page 73 / 142


7.4 7.4.1 (1)

Examples for sheet-piling Design situation The design situation for a sheet piling is given in Figure 7.5, which indicates the dimensions, load application and the distribution of load effects for two alternatives of piles, see Figure 7.6: -

Alternative A has HZ-piles as single profiles.

-

Alternative B has HZ-piles as double profiles

Figure 7.5: Design situation for a sheet piling with two alternatives (A = single pile, B = double pile)

Figure 7.6: Dimension of pile A and pile B

(2)

The loading conditions and the 1st order action effects from earth pressure are given in Table 7.10.

page 74 / 142

Guidance for applicationsGuidance for applications 7.4.2

Assessments for resistance and stability

(1)

For in-plane loading a beam-column check is performed using s K = L = 20.0 m as a safe-sided assumption (free buckling length).

(2)

The verification is made using formulae (6.8) and (6.9), see Table 7.10.

(3)

For out of plane buckling of the piles the restraints due to the sheet piling and the passive earth pressure of the soil may be taken into account.

(4)

The assumptions made for lateral torsional buckling modes are given in Figure 7.7. mode 1 shear

soil

shear

c o m p r e s s i o n

c o m p r e s s i o n

shear

shear

mode 2

soil compression

Figure 7.7: Lateral torsional buckling modes

(5)

From the two modes 1 and 2 in Figure 7.7 mode 1 is selected because of the greater deformations due to shear in the sheet piling and in the soil.

(6)

For restraints that the HZ-piles will receive in the lateral torsional buckling mode 1 the following assumptions are made: 1. The transmission of bending moments through the locks of the sheet piling is neglected. 2. The sheet piling acts as a shear wall between the HZ-piles without contributing by its stiffness to direct transverse stresses, 3. Passive earth pressure acting to the webs and flanges in the soil is taken into account by a bedding stiffness resulting from the shear deformations in the soil.

(7)

As a consequence the HZ-pile is modelled as given in Figure 7.8. a. boundary condition at the ends of the pile b. elastic restraints for displacements, twist and lock-shear displacements

page 75 / 142


clamped end for flexural and torsional out-of-plane deformation

II

MEd

cϕ cη main axis bending

restraint to displacement and twist cη and cϕ due to passive earth pressure

point support

Figure 7.8: Modeling of the HZ-pile

(8)

For the bedding stiffness from the soil in terms of a spring stiffness k [kN/m³] depending on the magnitude of displacement the principle of active and passive earth pressure given in Figure 7.9 may be taken into account.

Figure 7.9: Active and passive earth-pressure depending on the pile-displacements

(9)

The values k may be taken from Figure 7.10 as related to the magnitude of the passive earth pressure.

Figure 7.10: Example for spring-stiffness of soil [Van Tol/Brassinga]

page 76 / 142

Guidance for applicationsGuidance for applications (10) An example for a particular soil with γ

= 19 kN/m³

ϕ d

= 30 °

cd

=0

tan δd = ± tan (2/3 ϕ d ) k is given as follows: k 1 k 2 k 3

= 20.000 kN/m³ = 10.000 kN/m³ = 5.000 kN/m³

(11) The equivalent spring stiffnesses cη and cφ may be taken from Figure 7.11.

R = cη ⋅ η

=

k ⋅ h ⋅ η [kN / m ⋅ m]

h cη

cη = k ⋅ h

η

cϕ

RM = cϕ ⋅ ϕ

=

k ⋅

h3

12

cϕ = k ⋅

ϕ ϕ

h 2

⋅ ϕ [ kNm / m]

h3

12

Figure 7.11: Determination of stiffness for springs in the verification model

(12) The assumptions for determining

α crit for the example

k 1 = 20 000 kN/m³ are as fol-

lows: -

II the in-plane bending moment M Ed is determined from the bending moment I according to first order analysis by M Ed

1 II M Ed = M Ed ⋅

-

1 1−

N Ed N crit

II the in-plane bending moment M Ed that together with N Ed causes lateral tor-

sional buckling is the effect of active earth pressure that through arching in the soil mainly acts on the tension flange of the HZ-pile. (13) The calculations have been carried out with the FEM-program Marc/Mentat. Table 7.9 gives the relevant buckling modes and values α crit and α *crit , that lead to the lateral torsional buckling curves as given in Figure 7.12. In general the first eigen-

page 77 / 142

Excerpt from the Background Document to EN 1993-1-1 mode is relevant. To demonstrate the effects of the assumption of the boundary conditions at the end of the pile also the second eigenmode has been calculated. (14) Details of the assessment for in-plane compression and bending and out-of-plane lateral torsional buckling are given in Table 7.10 with the relevant European lateral torsional buckling curve given in Figure 7.12. (15) The results χ · α ult,k in Table 7.10 demonstrate that for a bedding stiffness of 20000 kN/m³ for the soil the design concepts are safe. (16) A more refined analysis taking the relevant spot, where

′′ ( x) results in α ult,k (x) and ϕ crit

a maximum, would even give a greater safety. (17) Table 7.11 shows the distributions of α ult,k and

α E that indicate the position x = xd at

the points of minimum of α E . *

χ LT 1.4

1st Eigenmode

Euler

1.2

2nd Eigenmode

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

λ

Figure 7.12: Relevant lateral torsional buckling curve for out of plane buckling of piles.

page 78 / 142

Guidance for applicationsGuidance for applications Table 7.9: Determination of lateral torsional buckling modes Situation A: Lateral torsional buckling modes for k = 20 000 kN/m³ st

1 global Eigenmode

with torsional stiffness

α crit = 3.54

without torsional stiffness

α ∗ crit = 1.432

nd

2

with torsional stiffness

α crit = 4.839

without torsional stiffness

α ∗ crit = 2.580

global Eigenmode

page 79 / 142


Table 7.10: Cross-sectional data, l oading and verifications

Crosssection

Situation A

Situation B

HZ-975B-14/AZ13/S355 GP

HZ-775D-26/AZ13/S355 GP

f y = 355 N/mm²

f y = 355 N/mm²

B = 1.87 m

B = 2.35 m

I y = 717 400 cm

4

I z = 80 110 cm4

Actions

Resistances

I y = 963 740 cm

A = 397.3 cm²

I z = 677 850 cm4

M y,Ed = 4521 kNm / pile

N pl,k = 35.5 · 397.3 = 14 103 kN / pile

N pl,k = 35.5 · 798.3 = 28 340 kN / pile



λ =

Flexural buckling about strong axis sk = 20.0m

π 2 ⋅ 21000 ⋅ 717400 2000 2

= 37172kN

14103 = 0.616 37172

0.965

0.738

π 2 ⋅ 21000 ⋅ 963740 2000 2

28340 = 0.753 49936

7614 4521 + ≤ 1 − 0.328 ⋅ 0.672 ⋅ 0.82 2 ⋅ 0.752 ⋅ 0 . 82 28340 8028 3 144 244 3 1 424 3 144 4 4 4 4 244 4 4 4 4 0.328

=

0.891 ≤ 0.917

0.965 = 1.07 0.900



Lateral torsional buckling (k = 20000 kN/m³) 1 st global Eigenmode

0.917

0.563

0.9 ≤ 0.965  α ult ,k

= 49936kN

= 0.821 (curve a; α = 0.21)

2020 3598 + ≤ 1 − 0.162 ⋅ 0.738 ⋅ 0.884 2 ⋅ 0.616 2 ⋅ 0 . 884 14103 4874 4 244 4 4 4 4 4 3 144 244 3 1 424 3 144 4 4 4 4 0.162

N crit =

λ =

(curve a; α = 0.21)

= 0.884

(z-z-direction)

2nd global Eigenmode

α ult ,k =

0.917 = 1.03 0.891

Lateral torsional buckling (k = 0 kN/m³) 1 st global Eigenmode

2nd global Eigenmode

α ult,k = 1.07

α ult,k = 1.07

α ult,k = 1.03

α ult,k = 1.03

α crit = 3.54

α crit = 4.839

α crit = 10.76

α crit = 26.69

⎯λ

= 0.550

⎯λ

= 0.470

⎯λ

= 0.309

⎯λ

= 0.196

α ∗ crit = 1.432

α ∗ crit = 2.580

α ∗ crit = 3.854

α ∗ crit = 8.791

α

= 0.34

α

= 0.34

α

= 0.34

α

= 0.34

α ∗

= 0.1375

α ∗

= 0.181

α ∗

= 0.122

α ∗

= 0.112

χ

= 0.937

χ

= 0.942

χ

= 0.986

χ

= 1.0

·

page 80 / 142

798.3 cm²


N crit =

(y-y-direction)

A=

N Ed = 7614 kN / pile

sk = 20.0m

Verification for out offplane loading

W y = 22 615 cm³

N Ed = 2020 kN / pile

Flexural buckling about strong axis

Verification for in-plane loading

4

W y = 13 730 cm³

ult,k

= 1.003

·

ult,k

= 1.008

·

ult,k

= 1.016

·

ult,k

= 1.030


Table 7.11: Situation A: Additional information on the location x = x d (relevant design point) Lateral torsional buckling 6 5 4

1 st global Eigenmode

2

α ult,k = 1.07

1

α crit = 3.54

0

⎯λ

= 0.550

α ult,k α E

3

calc

0

2

4

6

8

10

12

14

16

18

20

4

6

8

10

12

14

16

18

20

4

6

8

10

12

14

16

18

20

6

8

10

12

14

16

18

20

1.2

∗

α crit = 1.432 α

= 0.34

α ∗

= 0.1375

χ

= 0.937

0.8

− c,,rit η

0.4 0

− 0.4 − 0.8

Verification for out off-plane loading

− 1.2

0

2

6

(y-y-direction)

5 4 nd

2 global Eigenmode

1

α crit = 4.839

0

= 0.470

α E

2

α ult,k = 1.07 ⎯λ

α ult,k

3

calc

0

2

1.2

∗

α crit = 2.580 α ∗

0.8

= 0.34

− c,,rit η

0.4

α

= 0.181

χ

= 0.942

0

− 0.4 − 0.8 − 1.2

0

2

4

page 81 / 142


7.5 7.5.1

Lateral torsional buckling of beams with fin-plate connections Objective

(1)

Subject of this section is to demonstrate the use of the European lateral torsional curves for the lateral torsional buckling assessment of coped beams by re-calculating tests, that have been carried out by F. Bijlaard and H. Bouras and TU Delft [18].

(2)

The tests were 3-point bending tests according to Figure 7.13 with a conservative load applied to the top flange

F

fork condition realized by cardan support

2040 mm application of conservative load IPE 120 S235

application of conservative load

fork condition realized by cardan support

Fin plate connection detail

activator

Span 2040 mm Figure 7.13: 3-point bending tests f or lateral torsional buckling

page 82 / 142


The test program and the results may be taken from Table 7.12. Table 7.12: Test program [18] connection to end plate

copes ℓ/s

fin plates hF/t

Fmax.exp [kN]

90 / 5

29.3

90 / 8

34.4

90 / 12

32.2

75 / 5

27.3

75 / 8

34.6

75 / 12

30.8

75 / 5

-

75 / 8

25.4

75 / 12

28.2

50 / 5

22.6

50 / 8

25.8

50 / 12

27.9

no

no

160/30

160/30

7.5.2 (1)

Calculative results The calculations are based on the following assumptions: 1. For “in-plane loading” the load-assumption is that no support reactions other than “in-plane” occur, see Figure 7.14. Hence the load F in the main axes of the beam causes a torsion T = F ⋅ e by the eccentricity e. 2. For “out-of-plane” loading two loads are considered: -

the effects of equivalent imperfections

-

the effects from eccentric loading T = F ⋅ e, so that the formulas (6.41) with (6.42) and (6.43) apply.

page 83 / 142


Figure 7.14: Assumption for calculating

(2)

crit

Table 7.13 gives calculative values of the maximum loads Fz,calc. for the test conditions and a comparison with the test data r e/r t = Fz,exp/Fz,calc. Table 7.13: Results of calculation and comparison with test-results *

ey

Fz.exp

Fcrit

F crit

Fz.Ed

r e /r t

[mm]

[kN]

[kN]

[kN]

[kN]

[-]

a) 90/5

4.7

29.3

30.08

13.21

24.00

1.221

a) 90/8

6.2

34.4

31.70

13.92

24.50

1.403

a) 90/12

8.2

32.2

33.24

14.60

24.75

1.302

b) 75/5

4.7

27.3

29.22

12.84

23.50

1.162

b) 75/8

6.2

34.6

30.92

13.59

24.10

1.436

b) 75/12

8.2

30.8

31.88

14.01

24.05

1.280

c) 75/5

4.7

-

21.31

9.36

18.20

-

c) 75/8

6.2

25.4

23.13

10.16

19.27

1.318

c) 75/12

8.2

28.2

24.20

10.63

19.65

1.435

d) 50/5

4.7

22.6

20.09

8.83

17.75

1.273

d) 50/8

6.2

25.6

23.79

10.45

19.72

1.298

d) 50/12

8.2

27.9

26.05

11.44

20.80

1.341

Type

page 84 / 142


The test evaluation to obtain Table 7.14. Table 7.14: Determination of plate connections

M

γ M-values according to EN 1990 – Annex D is given in -values for lateral torsional buckling of beams with fin-

beam with fin-plates 2.0

r e/r t

1.0

0.0 a) 90/5

a) 90/8

a) 90/12

b) 75/5

b) 75/8

b) 75/12

c) 75/5

c) 75/8

c) 75/12

uncoped

d) 50/5

d) 50/8

d) 50/12

coped Input values

υrt = 0,08 (geometrie and yield strength) υfy = 0,07 (yield strength)

Tests on coped beams with fin-plates (TUDelft)

(N = 11)

standard deviation

log-standard deviation

2.0

2.0

g n u 1.5 l i e t r e 1.0 v l a m r o 0.5 n d r a d 0.0 n 0.75 a t S-0.5 r e d e-1.0 l i t n a-1.5 u Q

1.00

1.25

1.50

1.75

-2.0

υδ = 0 .0 79 γM = 1.267

(4)

Δk = 0.805

1.0 0.5 0.0 -0.5

0.0

0.1

0.2

0.3

0.4

0.5

-1.0 -1.5

ln re/rt

sδ = 0.104 ( mo de l)

1.5

-2.0

re/rt

b = 1.315


υ R = 0 .1 12

s δ = 0.105

b = 1.317 ( to ta l)

υ δ = 0 .0 80

* γM = 1.020

γM = 1.185

( mo de l) Δk = 0.804

υR = 0 .1 13

( to ta l) γM

*

= 0.953

The results in Table 7.14 reveal that γ M = 1,00 could be used for this set of tests (The conservatism of the calculative values is mainly caused by the fact, that the actual imperfections of the test beams were smaller than assumed in the European lateral torsional buckling curve used).

page 85 / 142


7.6 7.6.1

Verification of haunched beams Objective

(1)

This section deals with the calculative determination of tests results obtained by D. Ungermann and I. Strohmann with haunched beams at TU Dortmund [19].

(2)

The objective of the re-calculation of this test-results is to demonstrate the reliability of the European lateral torsional buckling curve by a test evaluation according to EN 1990-Annex D.

7.6.2

Test set up and testing procedure

(1)

The purpose of the test set up was to examine lateral torsional buckling effects for the beam of portal frames with the span length L with haunches at the knee-points.

(2)

Fig. 7.5-1 gives a survey on the loading conditions and the restraint-conditions of the beam, for which the following parameters were varied:

max h min h

k V

=

k L

=

length of haunch

f 0

=

M F

total length

M S

Figure 7.15: Geometrical conditions and loading for the tests

page 86 / 142


The variations of parameters provided to check the assessment procedure are given in Table 7.15 Table 7.15: Variation of parameters

(4)

The cross sections at the ends of the haunches which were made by plates and welded to rolled beams are given in Table 7.16. Table 7.16: Cross-sections at the ends of the haunches for test beams VT1 to VT3 and VT4 to VT6

VT_1 –VT_3

(5)

VT_4 –VT_6

The full set of tests with various geometrical parameters may be taken from Table 7.17 Table 7.17: Full set of tests and parameters

page 87 / 142


The test set up for providing various end moments forces P applied with various cantilever lengths

Ms and midspan moments MF by

LLet is given in Figure 7.16.

Figure 7.16: Test set-up, load application P and boundary conditions

(7)

Figure 7.17 shows details of the application of loads at midspan with springs to provide an elastic torsional restraint cϕ at midspan.

page 88 / 142


Figure 7.17: Load application with provisions for c at mid span

7.6.3

Test results The test results for the maximum loads

P in Figure 7.16 limited by elastic lateral tor-

sional buckling are given in Table 7.18. These values have been obtained for

cϕ = 1000 kNcm/rad . Table 7.18: Tests results

7.6.4

Calculative results

(1)

The calculations were performed using the European lateral torsional buckling curves with the determination of Pcrit by a FEM-program. The yield strength of the material as tested was f y = 400 N/mm².

(2)

Table 7.19 gives the results of the calculations and the ratios between the experimental and calculative results.

page 89 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.19: Calculative results and comparison with test results Type

(3)

*

Pexp

Pcrit

P crit

PEd

r e /r t

[kN]

[kN]

[kN]

[kN]

[-]

VT1A

40.97

49.23

30.22

33.49

1.223

VT2A

49.00

59.52

36.54

39.88

1.229

VT3A

50.67

60.99

37.44

40.56

1.249

VT4A

34.40

39.99

24.55

27.86

1.235

VT5A

37.30

39.20

24.07

27.73

1.345

VT6A

41.87

38.35

23.54

27.56

1.519

VT1B

34.73

42.08

25.83

29.29

1.186

VT2B

38.87

53.10

32.60

35.44

1.097

VT3B

44.43

53.61

32.91

35.76

1.242

VT4B

30.23

33.51

20.57

23.91

1.264

VT5B

35.17

32.78

20.12

23.76

1.480

VT6B

33.97

32.09

19.70

23.51

1.445

The test evaluation according to EN 1990 – Annex D is presented presented in Table 7.20. 7.20. As usual the γ M-values obtained are in the same magnitude γ M ity phenomena.

≈ 1,00 as for other stabil-

Table 7.20: Test evaluation according to EN 1990 – Annex D and

M-values

2.0 re/rt

1.0

0.0 VT1A VT1A VT2A VT2A VT3A VT3A VT4A VT4A VT5A VT5A VT6A VT6A VT1B VT1B VT2B VT2B VT3B VT3B VT4B VT4B VT5B VT5B VT6B VT6B Input values υrt = 0,08 (geometrie and yield strength) υfy = 0,07 (yield strength)

Tests on haunched girders girders (TU Dortmund) (N = 12) standard deviation

log-standard deviation 2.0

2.0 g n u l i 1.5 e t r e v 1.0 l a m r o 0.5 n d r a d 0.0 n a 0.75 t S-0.5 r e d e-1.0 l i t n a u-1.5 Q

1.00

1 .2 5

1.5 0

1.7 5

-2.0

υδ = 0 .0 .0 80 80 γM = 1.268

page 90 / 142

Δk = 0.840

1.0 0.5 0.0 -0.5

0.0

0. 1

0.2

0.3

0 .4

0.5

-1.0 -1.5

ln re/rt

sδ = 0.101 ( mo mo de de l)l)

1.5

-2.0

re/rt

b = 1.262


sδ = 0.110

b = 1.269

υR = 0 .1 .1 13 13

( to to ta ta l)l)

υδ = 0 .0 .0 86 86

γM = 1.065

γM = 1.194

*

( mo mo de de l)l) Δk = 0.842

υR = 0 .1 .1 18 18

( to to ta ta l)l) γM

*

= 1.006

Guidance for applicationsGuidance applicationsGuidance for applications

7.7 7.7.1

Assessment of gantry-girders Structural system and loading

(1)

The structural structural system system of the the gantry girder may be taken from from Figure 7.18. It is a two span continuous girder with a span length of 6 m.

(2)

The steel profile is HEB 300 S235, with a rail 5 cm x 3 cm welded to the profile with fillet welds aw = 5 mm. The rail is not taken into account in the resistance of the girder.

(3)

Transverse stiffeners welded welded to the flanges and and the web of the profile are at the supports and the connections to the brackets of the frames of the industrial hall provide “fork”-conditions.

(4)

The loading results from a bridge crane with the maximum wheel loads R = 75 kN H = 22.2 kN

The wheel distance is c = 3.6 m. (5)

The dynamic factor is ϕ = 1.20, so that the vertical wheel loads are F 1 = F 2 = F = ϕ 1 ⋅ R = 1.2 ⋅ 75 = 90 kN

(6)

The self-weight of the gantry girder is g = 1.35 kN/m

Figure 7.18 Structural system and loading

7.7.2

Action effects

7.7.2.1 Maximum sagging moments (1)

The load load position and the design loads for the maximum maximum sagging moment may may be taken from Figure 7.19

page 91 / 142


F 1,Ed 1,Ed = 121.5 kN

F 2,Ed 2,Ed = 121.5 kN

H Ed = 30 kN T Ed = 5.4 kNm

a

l 1 = 2.1 m

g Ed = 1.82 kN/m c

b

c = 3.6 m

l 2

l = 6.0 m

l = 6.0 m

Figure 7.19: Load position for maximum sagging moment

(2)

The design design values values of action effects effects from the relevant relevant load combination are given given in Figure 7.20

M z,Ed = 37.3 kNm

M y,Ed = 157.7 kNm

T w,Ε d d = 3.86 kNm²

Figure 7.20: Action effects for maximum sagging moments

(3)

The plastic resistances of the girder are M y,Rk = 459.8 kNm M z,Rk = 209 kNm 2 T w,Rk w,Rk = 31.4 kNm

(4)

Hence the in-plane assessment follows from

α ult , k = α crit =

M y , Rk M y , Ed

=

M y , crit , LT M y , Ed

459.8 = 2.916 157.7

=

1191 = 7.552 157.7

* α crit = 4.216

λ =

page 92 / 142

α ult , k = 0.621 α crit

Guidance for applicationsGuidance for applications * 4.216 α crit α = α ⋅ = 0.34 ⋅ = 0.190 7.552 α crit *

χ = 0.891 χ ⋅ α ult , k 0.891 ⋅ 2.916 = = 2.362 1.1 γ M

α Ed = (5)

Taking into account out-of-plane loading (bending and torsion) leads to qM z

=

qT w

=

⎛

1

⋅ ⎜⎜1 −

α crit ⎝ ⎛

1

⋅ ⎜⎜1 −

β w, d = Δn E =

M y , Ed M y , Rd T w, Ed T w, Rd

1

α Ed

Δn R = 1 −

I M z

1 ⎟⎟ ≅ ⋅ (1 − 0.81) = 0.025 ⎠ 7.552

T w, m ⎞

α crit ⎝

β z , d =

M z , m ⎞

1 ⎟≅ ⎟ 7.552 ⋅ (1 − 0.648) = 0.047 ⎠

T w I

⋅ (1 − qM ) = z

⋅ (1 − qT ) = w

37.3 ⋅ (1 − 0.025) = 0.170 209

3.86 ⋅ (1 − 0.047 ) = 0.117 31.4

+ β z ,d + β w,d = ⎛

1

χ ⋅ α ult , k

1 + 0.170 + 0.117 = 0,710 2.362

⎞ 2 2 ⎟ ⋅ χ ⋅ λ = 0.913 ≥ 0,90 ⎟ χ ⋅ α ult , k ⎠ ⎝

⋅ ⎜⎜1 −

1

and hence:

Δn E < Δn R



0.710 < 0.913

A conservative assumption with qMw = 0, qMz = 0 and Δn R = 0.9 would lead to:

1

α Ed

+ β yd + β w d =

1 37.3 3.86 + + = 0,725 ≤ 0.9 2.362 209 31.4

7.7.2.2 Maximum hogging moment (1)

The load position and the design loads for the maximum hogging moment may be taken from Figure 7.21. F 1,Ed = 121.5 kN

F 2,Ed = 121.5 kN

H Ed = 30 kN T Ed = 5.4 kNm b

a

4.2 m

1.8 m

1.8 m

c

4.2 m

Figure 7.21: Load position for maximum hogging moment

page 93 / 142


The design values of action effects from the relevant load combination are given in Figure 7.22. M y,Ed = -138.8 kNm

M z,Ed = -17.35 kNm

T w,Ε d = 3.74 kNm²

Figure 7.22: Action effects for maximum hogging moments

(3)

7.8 7.8.1

Obviously the load case “maximum hogging moment” is not relevant for the lateral torsional buckling verification.

Channel sections Objective

(1)

Tests with beams made of channel sections are evaluated using the European lateral torsional buckling curve for lateral torsional buckling with transverse bending, torsion and in combination with compression forces, to verify the reliability of the assessment method.

(2)

The test data are given in Figure 7.21 and Figure 7.22.

page 94 / 142

Guidance for applicationsGuidance for applications Table 7.21: Tests TU-Berlin [9]; configurations and results

Test

Profile

Steel

Load application

Fexp [kN]

1

43.0

2

51.2

3

57.4

UPE200

S355 f y = 400 N/mm²

4

31.8

5

34.5

6

30.4

page 95 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.22: Tests Ruhr-Universität Bochum [9];configurations and results

Fexp [kN]

Nexp [kN]

7

45.91

74.88

8

36.76

59.03

29.48

278.37

10

24.16

227.93

11

22.80

37.01

21.01

33.86

Test

Profile

Steel

Test set-up

L=4m

9

Load application

Ü = 95 mm S355 UPE200

12

f y = 418 N/mm²

L=6m Ü = 95 mm

page 96 / 142

Guidance for applicationsGuidance for applications Still Table 7.22: Tests Ruhr-Universität Bochum [9];configurations and results

Test

Profile

Steel

Test set-up

Load application

Fexp [kN]

Nexp [kN]

17.93

80.83

15.95

74.45

S355 13

f y = 418 N/mm² UPE200 S355

14

f y = 364 N/mm²

L=6m Ü = 95 mm

Within this test series the axial forces N have been applied through cap and ball bearings, which were fixed on 20 mm thick steel plates at both ends of the beam-column, which impeded a free warping of the cross section. This effect has been taken into account for the re-calculations.

7.8.2 (1)

Calculative results The calculations were performed using the European lateral torsional buckling curves with α = 0.49. The critical values α crit for the M-N-interaction have been determined using the software LTBeamN [20]. The yield strengths which have been used for the calculations, were determined from material samples of the test specimens and are given in Table 7.21 and Table 7.22.

(2)

The different calculations have been performed: 1. using the elastic warping-resistance Tel,w,Rk of the channel section 2. using the plastic warping resistance T pl,w,Rk of the channel section. Results and calculative steps of each assessment are summarized in Table 7.23 and Table 7.24. Figure 7.23 shows the determined r e/r t-values for both assessment methods.

(3)

Table 7.25 and Table 7.26 give the tween

* -values related to the results which are beγ M

* * γ M = 1,0 and γ M = 1,1 as required.

page 97 / 142

Excerpt from the Background Document to EN 1993-1-1 Table 7.23: Calculative results for Tw,Rk = Tel,w,Rk and comparison with tests

Test

αEk *)

αult,k

αcrit

α*crit

α∗

χ

χ ⋅ α ult ,k

βMw

ΔnE = ΔnR

r e/r t

1

0.620

4.718

2.361

0.809

0.168

0.425

0.499

0.411

0.910

1.613

2

0.736

3.339

1.671

0.573

0.168

0.425

0.704

0.220

0.925

1.359

3

0.732

2.994

1.498

0.513

0.168

0.425

0.786

0.153

0.939

1.366

4

0.757

3.655

1.384

0.308

0.109

0.347

0.788

0.159

0.947

1.320

5

0.741

3.444

1.304

0.290

0.109

0.347

0.836

0.120

0.956

1.349

6

0.671

4.315

1.634

0.364

0.109

0.347

0.667

0.262

0.929

1.490

7

0.635

3.155

1.679

1.011

0.295

0.406

0.780

0.167

0.947

1.575

8

0.624

4.009

2.138

1.285

0.295

0.407

0.613

0.314

0.926

1.602

9

0.572

5.451

1.379

1.186

0.421

0.204

0.899

0.087

0.985

1.749

10

0.617

6.162

1.560

1.341

0.421

0.204

0.795

0.179

0.973

1.620

11

0.711

3.785

1.260

0.550

0.214

0.289

0.915

0.065

0.981

1.407

12

0.693

4.212

1.403

0.612

0.214

0.289

0.822

0.141

0.963

1.443

13

0.716

4.778

1.226

0.646

0.258

0.223

0.939

0.050

0.989

1.397

14

0.706

4.741

1.373

0.732

0.261

0.248

0.850

0.123

0.973

1.416

1


Table 7.24: Calculative results for Tw,Rk = Tpl,w,Rk and comparison with tests

Test

αEk *)

αult,k

αcrit

α*crit

α∗

χ

χ ⋅ α ult ,k

βMw

ΔnE = ΔnR

r e/r t

1

0.968

3.020

1.512

0.518

0.168

0.425

0.779

0.159

0.938

1.033

2

0.934

2.631

1.317

0.451

0.168

0.425

0.894

0.072

0.966

1.071

3

0.865

2.533

1.268

0.434

0.168

0.425

0.929

0.047

0.976

1.156

4

0.897

3.085

1.168

0.260

0.109

0.347

0.933

0.047

0.980

1.114

5

0.842

3.030

1.148

0.255

0.109

0.347

0.950

0.035

0.985

1.187

6

0.890

3.253

1.232

0.274

0.109

0.347

0.885

0.083

0.968

1.123

7

0.756

2.650

1.410

0.849

0.295

0.406

0.929

0.051

0.979

1.323

8

0.871

2.875

1.533

0.922

0.295

0.407

0.854

0.107

0.961

1.149

9

0.618

5.046

1.276

1.098

0.421

0.204

0.971

0.025

0.995

1.619

10

0.728

5.227

1.323

1.137

0.421

0.204

0.937

0.054

0.990

1.374

11

0.758

3.550

1.182

0.516

0.214

0.289

0.976

0.018

0.994

1.320

12

0.799

3.651

1.217

0.531

0.214

0.289

0.948

0.040

0.988

1.251

13

0.749

4.564

1.171

0.617

0.258

0.223

0.983

0.014

0.997

1.335

14

0.795

4.210

1.219

0.650

0.261

0.248

0.957

0.034

0.991

1.258

1


page 98 / 142


r e /r t 2.0

r e /r t 2.0

Tel,w,Rk

1.8 1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

UPE 200 - RWTH Aachen UPE 200 - TU Berlin

0.2

Tpl,w,Rk

1.8

UPE 200 - TU Berlin

0.2

UPE 200 - RuhrUni Bochum

UPE 200 - RuhrUni Bochum 0.0

0.0 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

1.0

1.1

1.2

1.3

1.4

1.5

1 .6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

Figure 7.23: Comparison between test results [9] and calculative results for Tw,Rk = Tel,w,Rk (left hand side) and Tw,Rk = Tpl,w,Rk (right hand side)

Table 7.25: Determination of the * M -value according to EN 1990 – Annex D ( Tw,Rk = Tel,w.Rk ) Eingangsdaten υrt = 0,08 (Geometrie und Streckgrenze) υfy = 0,07 (Streckgrenze)

Research Project Fosta P 554 - UPE200 (T el,w,Rk) (N = 14) Standardnormalverteilung


2.0

2.0

g 1.5 n u l i e t r e 1.0 v l a m r 0.5 o n d r a d 0.0 n a 1.2 t S r -0.5 e d e l i t -1.0 n a u Q

1.4

1.6

1.8

-1.5


1.5 1.0 0.5 0.0 -0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.0 -1.5 -2.0

-2.0

ln re/rt

re/rt

sδ = 0.158

b = 1.479 υδ = 0.107

υR = 0.133

(Modell)

(gesamt) *

Δk = 0.744

γM = 1.353

sδ = 0.154

b = 1.482

γM = 1.006

υδ = 0.104

υR = 0.131

(Modell)

* γM

Δk = 0.739

γM = 1.218

(gesamt) = 0.900

Table 7.26: Determination of the * M -value according to EN 1990 – Annex D ( Tw,R = Mpl,w,R ) Eingangsdaten υrt = 0,08 (Geometrie und Streckgrenze) υfy = 0,07 (Streckgrenze)

Research Project Fosta P 554 - UPE200 (T pl,w,Rk) (N = 14) Standardnormalverteilung


2.0 g 1.5 n u l i e t r e 1.0 v l a m r 0.5 o n d r a d 0.0 n a 1.0 t S r -0.5 e d

2.0

1.1

1.2

1.3

1.4

e l i t n -1.0 a u Q

-1.5


1.5 1.0 0.5 0.0 0.000 -0.5

0.125

0.250

-1.0 -1.5

ln re/rt

re/rt

sδ = 0.107

b = 1.193

γM = 1.296

0.500

-2.0

-2.0

υδ = 0.090

0.375


sδ = 0.116

b = 1.201

υR = 0.120

(gesamt) *

γM = 1.167

υδ = 0.097 γM = 1.208


υR = 0.126 * γM

(gesamt) = 1.090

page 99 / 142


page 100 / 142

Literature

10 Literature [1]

EN 1990 Eurocode: „Basis of structural design“, CEN, Brussels

[2]

EN 1993-1-1: Eurocode 3 – Part 1-1 “Design of steel structures – General rules and rules for buildings”, CEN, Brussels

[3]

Maquoi, R., R. Rondal, J.: Analytische Formulierung der neuen Europäischen Knickspannungskurven, Acier, Stahl, Steel 1/1978

[4]

Müller, Chr.: Zum Nachweis ebener Tragwerke aus Stahl gegen seitliches Ausweichen, Diss. RWTH Aachen 2003, Schriftenreihe Stahlbau, Heft 47, Shaker Verlag

[5]

Sedlacek, G., Müller, Chr.: The European Standard family and its basis. Journal of Constructural Steel Research 62/2006), 1047-1056

[6]

Stangenberg, H.: Zum Bauteilnachweis offener stabilitätsgefährdeter Stahlbauprofile unter Einbeziehung seitlicher Beanspruchungen und Torsion, Diss. RWTH Aachen 2007, Schriftenreihe Stahlbau, Heft 61, Shaker Verlag

[7]

Sedlacek, G., Müller, Chr., Stangenberg, H.: Lateral torsional buckling according to Eurocode 3, René Maquoi 65th birthday anniversary, 2007

[8]

Sedlacek, G., Ungermann, D., Kuck, J., Maquoi, R., Janss, J.: Eurocode 3 – Part 1,Background Documentation Chapter 5 – Document 5.03 (partim): “Evaluation of test results on beams with cross sectional classes 1-3 in order to obtain strength functions and suitable model factors” Eurocode 3 - Editorial Group (1984)

[9]

Sedlacek, G., Stangenberg, H., Lindner, J., Glitsch, T., Kindmann, R., Wolf, C.: Untersuchungen zum Einfluss der Torsionseffekte auf die plastische Querschnittstragfähigkeit und Bauteiltragfähigkeit von Stahlprofilen, Forschungsvorhaben P554; Forschungsvereinigung Stahlanwendung e.V., 2004

[10]

Stangenberg, H., Sedlacek, G., Müller, Ch.: Die neuen Biegedrillknicknachweise nach Eurocode 3 – Festschrift 60 Jahre Prof. Kindmann 2007

[11]

Braham, M., Maquoi, R.: Merchant-Rankine’s concept brought again in honour for web-tapered-I-section steel members, Festschrift Joachim Lindner, 1998

[12]

Poutré la, D. B., Snijder, H. H., Hoenderkamp, J. C. D.: Lateral torsional buckling of channel shaped sections, Experimental research report, University of Technology Eindhoven, April 1999

[13]

Poutré la, D.B.: Strength and stability of channel sections used as M.Sc.-thesis, University of Technology Eindhoven, December 1999

[14]

Roik, K., Kindmann, R.: Das Ersatzstabverfahren – Eine Nachweisform für den einfeldrigen Stab bei planmäßig einachsiger Biegung mit Druckstab, Der Stahlbau 12/1981, S. 353-358

beam,

page 141 / 142

Background to Eurocodes

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