Pinned base plates

AUSTRALIAN STEEL INSTITUTE (ABN)/ACN (94) 000 973 839

STEEL CONSTRUCTION JOURNAL OF THE AUSTRALIAN STEEL INSTITUTE VOLUME 36 NUMBER 2 SEPTEMBER 2002

Design of Pinned Column Base Plates

ISBN 0049-2205 Print Post Approved pp 255003/01614

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STEEL CONSTRUCTION - EDITORIAL This paper is one of a planned series which deals with the design and use of rationalized structural connections. It draws heavily on the excellent work done in the publication “Design of Structural Connections” by Tim Hogan and Ian Thomas. Since that time, there has been new research, some variations to the design models, new steel grades introduced and some minor changes in section properties. We have also seen the adoption of sophisticated 3D modeling software which has the capability to generate many different connection types. The ASI, through this project is endeavouring to provide an industry wide rationalized set of dimensions, models and design capacities.

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STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

Design of Pinned Column Base Plates Contents

This paper deals with the design of pinned base plates. The design actions considered are axial compression, axial tension, shear force and their combinations. The base plate is assumed to be essentially statically loaded, and additional considerations may be required in the case of dynamic loads or in fatigue applications.

1.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Design actions in accordance with AS 4100 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. BASE PLATE COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. AXIAL COMPRESSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. BASE PLATE DESIGN - LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . 4.3. RECOMMENDED MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. AXIAL TENSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. BASE PLATE DESIGN - LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . 5.3. DESIGN OF ANCHOR BOLTS - LITERATURE REVIEW . . . . . . . . . . . . . 5.4. RECOMMENDED MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. SHEAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. TRANSFER OF SHEAR BY FRICTION OR BY RECESSING THE BASE PLATE INTO THE CONCRETE LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. TRANSFER OF SHEAR BY A SHEAR KEY-- LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. TRANSFER OF SHEAR BY THE ANCHOR BOLTS LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. RECOMMENDED MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. BASE PLATE AND ANCHOR BOLTS DETAILING . . . . . . . . . . . . . . . . . . . . . . 8. ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. APPENDIX A - Derivation of Design and Check Expressions for Steel Base Plates Subject to Axial Compression . . . . . . . . . . . . . . . . . . . . . . . . 11. APPENDIX B-- Derivation of Design and Check Expressions for Steel Base Plates Subject to Axial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. APPENDIX C - Determination of Embedment Lengths and Edge Distances . . . . 13. APPENDIX D - Design Capacities of Equal Leg Fillet Welds . . . . . . . . . . . . . . . . 14. APPENDIX E - Design of Bolts under Tension and Shear . . . . . . . . . . . . . . . . . . .

1 1 1 3 3 3 4 10 12 12 12 17 21 30 30 30 30 31 34 36 38 38 40 46 49 53 53

Design of Pinned Column Base Plates Gianluca Ranzi School of Civil and Environmental Engineering The University of New South Wales Peter Kneen National Manager Technology Australian Steel Institute 1. INTRODUCTION This paper deals with the design of pinned base plates. The design actions considered are axial compression, axial tension, shear force and their combinations as shown in Fig. 1. The base plate is assumed to be essentially statically loaded, and additional considerations may be required in the case of dynamic loads or in fatigue applications. N *t

N *t

N *c V *x

Figure 1

N *c V *y

Column Design Actions: Axial and Shear Loads along minor and major axes (Ref. [26])

Firstly the requirements of AS 4100 ”Steel Structures” [11] in the calculation of the design actions for connections are outlined. Then for each design action available design guidelines and/or models are briefly presented in a chronological manner to provide an overview on how these have improved/changed over time. Attention has been given to try to ensure that the assumptions and/or limitations of each model presented are always clearly stated. Among these models, the most representative ones in the opinion of the authors are then recommended for design purposes. It is not intended to suggest that models, other than those recommended, may not give adequate capacities. The design of concrete elements is outside the scope of the present paper. Nevertheless some design considerations regarding the concrete elements still need to be addressed, i.e. bolts’ edge distances, bolts’ embedment lengths, concrete strength etc., and therefore it is necessary to ensure that such design assumptions/considerations are included in the final design of the concrete elements/structure.

1.1.

Design actions in accordance with AS 4100

Pinned type column base plates may be subject to the following design actions, as shown in Fig. 1: an axial force, N*, either tension or compression;

1

a shear force, V* (usually acting in the direction of either principal axis or both). Clause 9.1.4 of AS 4100 [11], which considers minimum design actions, does not specifically mention minimum design actions for column base plates but does require that: connections at the ends of tension or compression members be designed for a minimum force of 0.3 times the member design capacity; connections to beams in simple construction be designed for a minimum shear force equal to the lesser of 0.15 times the member design shear capacity and 40 kN. It is considered inappropriate for these provisions to be applied to column base plates, since the design of columns is usually governed by a combinations of axial loads and bending moments at other locations.

2. NOTATION The following notation is used in this work. Other symbols which are defined within diagrams may not be listed below. Generally speaking, the symbols will be defined when first used. a b = distance from centre of bolt hole to inside face of flange a e = minimum concrete edge distance (side cover) A 1 = bearing area which varies depending upon the assumed pressure distribution between the base plate and the grout/concrete (i) A 1 = bearing area at the i--th iteration in Murray--Stockwell Model A 2 = supplementary area which is the largest area of the supporting concrete surface that is geometrically similar to and concentric to A 1 A H = assumed bearing area (in the case of H--shaped sections it is a H--shaped area) in Murray-Stockwell Model A (i) = assumed bearing A H at the i--th iteration in H Murray--Stockwell Model A i = base plate area A psk = projected area over the concrete edge ignoring the shear key area A ps = effective projected area of concrete under uplift


A ps.1 = effective projected area of isolated anchor bolt (no overlapping of failure cones) A ps.2 = effective projected area of 2 anchor bolts with overlapping of their failure cones A ps.4 = effective projected area of 4 anchor bolts with overlapping of their failure cones. In this case each failure cone overlaps with all other 3 failure cones A s = tensile stress area in accordance with AS1275 [9] A sk = area of the shear key b c = width of the column section (RHS and SHS) b fc = width of the column section (H--shaped sections and channels) b fc1 = width of the column flange ignoring web thickness b i = width of base plate b s = depth of shear key b t = distance from face of web to anchor bolt location d c = column depth d c1 = clear depth between flanges (column depth ignoring thicknesses of flanges) d f = nominal anchor bolt diameter d h = diameter of bolt hole d i = length of base plate d 0 = outside diameter of CHS f′ c = characteristic compressive cylinder strength of concrete at 28 days f *p = uniform design pressure at the interface of the base plate and grout/concrete f uf = minimum tensile strength of bolt f uw = nominal tensile strength of weld metal f yi = yield stress of the base plate used in design f ys = yield stress of shear key used in design k r = reduction factor to account for length of welded lap connection L d = minimum embedment length of anchor bolt L h = hook length of anchor bolt L s = length of shear key Lw = total length of fillet weld m p = plastic moment capacity of the base plate per unit width m s = nominal section moment capacity of the base plate per unit width m sk = nominal section moment capacity per unit width of shear key m *c = design moment per unit width due to N *c m *sk = design moment to be carried by the shear key per unit width * m t = design moment per unit width due to N *t

2

n b = number of anchor bolts part of the base plate connection N *c = column design axial compression load N *b = N *t ∕n b = design axial tension load carried by one bolt N des.c = design capacity of the base plate connection subject to axial compression N des.t = design capacity of the base plate connection subject to axial tension * N p = prying action N *t = design axial tension load of the column N tf = nominal tensile capacity of a bolt in tension N *0 = portion of N *c acting over the column footprint s p = bolt pitch S i = plastic section modulus per unit width of plate t c = thickness of column section t i = base plate thickness t g = grout thickness t s = thickness of shear key t t = design throat thickness of fillet weld t w = thickness of column web v des = Ôv w = design capacity of the weld connecting the base plate to the column per unit length * v h and v *v = components of the loading carried by the weld between column and base plate in one horizontal direction in the plane of the base plate and in the vertical direction respectively per unit length * v w = design action on fillet weld per unit length V des = design shear capacity of the base plate connection * V s = design shear force to be transferred by means of the shear key W i and W e = internal and external work Ô = capacity factor = maximum bearing strength of the concrete at Ôf (i) b the i--th iteration in Murray--Stockwell Model Ôf b = maximum bearing capacity of the concrete based on a certain bearing area A 1 ÔN c = design axial capacity of the concrete foundation ÔN c.lat = lateral bursting capacity of the concrete ÔN cc = design pull--out capacity of the concrete foundation ÔN s = design axial capacity of the steel base plate ÔN t = axial tension capacity of the base plate ÔN tb = design capacity of the anchor bolt group under tension ÔN th = tensile capacity of a hooked bar ÔN w = design axial capacity of the weld connecting the base plate to the column section


Ôv w = design capacity of the fillet weld per unit length ÔV f = design shear capacity of the base plate transferred by means of friction ÔV s = design shear capacity of the shear key ÔV s.c = concrete bearing capacity of the shear key ÔV s.cc = pull--out capacity of the concrete ÔV s.b = shear capacity of the shear key based on its section moment capacity ÔV s.w = shear capacity of the weld between the shear key and the base plate ÔV w = design shear capacity of the weld connecting the base plate to the column η = ratio depth and width of column μ = coefficient of friction

There is a large variety of drilled--in anchors available, many of which are proprietary bolts whose installation and design is governed by manufacturers’ specifications. References [2], [15], [17], [31] and [33] contain information on these types of anchors. This paper deals only with cast--in--place anchors, and specifically hooked bars, anchor bolts with a head and threaded rods with a nut/washer/plate. Grade 4.6 anchor bolts are recommended to be utilised in base plate applications.

3. BASE PLATE COMPONENTS Typical base plates considered in this paper are formed by one unstiffened plate only as shown in Fig. 3. For highly loaded columns or larger structures other base plate solutions or more elaborate anchor bolt systems might be required. Guidelines for the design and detailing of more complex base plates can be found in [4], [13], [14], [16] and [34]. Two types of anchor bolts are usually used, which are cast--in--place or drilled--in bolts. The former are placed before the placing of the concrete or while the concrete is still fresh while the latter are inserted after the concrete has fully hardened. Different types of cast--in--place anchors are shown in in Fig. 2. These include anchor bolts with a head, threaded rods with nut, threaded rods with a plate washer, hooked bars or U--bolts. These are suitable for small to medium size structures considering anchor bolts up to 30 mm in diameter.

sp

sg

Figure 3

Typical unstiffened base plate (Ref. [26])

4. AXIAL COMPRESSION 4.1.

INTRODUCTION

The literature review presented covers only models regarding the design of the actual steel plate as the anchor bolts do not contribute to the strength of the connection under this loading condition. Unless special confinement reinforcement is provided the maximum bearing strength of the concrete Ôf b is calculated in accordance with Clause 12.3 of AS 3600 [10] as follows:



Ôf b = min Ô0.85f′ c (a) Hooked Bar

(b) Bolt with head

(c) Threaded Rod with Nut

(d) Threaded rod with plate washer Fillet welds Square plate

Figure 2

3

(e) U--Bolt

Common Forms of Holding Down Bolts (Ref. [26])



A2 , Ô2f′ c A1



(1)

where: Ô = 0.6 Ôf b = maximum bearing capacity of the concrete based on a certain bearing area A 1 f′ c = characteristic compressive cylinder strength of concrete at 28 days A 1 = bearing area which varies depending upon the assumed pressure distribution between the base plate and the grout/concrete A 2 = supplementary area which is the largest area of the supporting concrete surface that is geometrically similar to and concentric to A 1


4.2.

BASE PLATE DESIGN -- LITERATURE REVIEW

The main design models available in literature differ for their assumptions adopted regarding the pressure distribution at the interface between the base plate and the grout/concrete and for the relative sizes of the base plate and the connected column. For example, the first model presented, here referred to as the Cantilever Model, is adequate for base plates whose dimensions (d i × b i ) are much greater than those of the column (d c × b fc ), while other models, such as Fling and Murray--Stockwell Models, deal with base plates with similar dimensions to the ones of the connected column.

4.2.1.

This model assumes that, in the case of a H--shaped column, the axial load applied by the column is concentrated over an area of 0.95d c × 0.80b fc which corresponds to the shaded area of Fig. 4(a). This causes the base plate to bend as a cantilevered plate about the edges of such area as shown in Fig. 4(b). The pressure at the underside of the base plate is assumed to be uniformly distributed, as shown in Fig. 4(c), therefore leading to a conservative design for large base plates. a1

Cantilever Model

Historically the cantilever model was the first available approach for the design of base plates. It is well suited for the design of large base plates with the dimensions of the base plate (d i × b i) much greater than those of the column (d c × b fc). It has been present in the AISC(US) Manuals over several editions. Its formulation is suitable for the base plate design of only H--shaped columns. [5] bi b fc a1 dc

0.95d c

di

a1

a1 Dashed lines indicate yield lines

Figure 5

0.8b fc

Each of the two collapse mechanisms considered by this model assumes two yield lines to form at a distance a 1 and a 2 from the edge of the plate respectively as shown in Fig. 5. Comparing the two collapse mechanisms and according to the rules of yield line theory the governing design capacity is based on the longest cantilever length a m, being the maximum of the two cantilevered lengths a 1 and a 2 shown in Fig. 4(a). The design moment m *c and the design capacity of the plate Ôm s are calculated per unit width in accordance with AS 4100 [11] as:

a2

(a) Critical sections and assumed loaded area Critical section in bending

ti

N *c b id i (b) Deflection of the cantilevered plate N *c

ti (c) Assumed bearing pressure

Figure 4

4

Cantilever Model (Ref. [26])

N *c a 2m b id i 2

Ôm s = Ôf yiS i =

(2) 0.9f yi t 2i 4

(3)

where: N *c = column design axial compression load

am

N *c b id i

a2

Cantilever Model -- Collapse mechanisms

m *c = a2

a2

m *c = design moment per unit width due to N *c m s = plate nominal section moment capacity per unit width f yi = yield stress of the base plate used in design S i = plastic section modulus per unit width of plate a m = max(a 1, a 2) a 1 and a 2 = cantilevered plate lengths t i, d i and b i = thickness, length and width of base plate and ensuring that the plastic section modulus of the cantilevered plate S i is able to transfer the axial compression load N *c to the supporting material (verified per unit width of plate): 0.9f yi t i N *c a 2m ≤ = Ôm s 2 4 b id i 2

m *c =

(4)

yields a maximum design axial force of:


N *c ≤

0.9f yi t 2i b id i 2 a 2m

bi

(5)

bc

or equivalently requires a minimum plate thickness of: ti ≥ am



2N *c 0.9f yi b id i

a1

(6)

Provisions on how to extend this approach for channels and hollow sections columns have been provided in [21], [26] and [36]. The dimensions of the loaded areas and of the cantilevered lengths a 1 and a 2 for channels and hollow sections are shown in Figs. 6, 7 and 8 and their values are summarised in Table 1 based on the recommendations in [21], [26] and [36]. The values in Table 1 assume that the column is welded concentrically to the base plate.

di

a1 a2

Figure 7

SHS and RHS [36] SHS and RHS [21] CHS [21]

a1 d i − 0.95d c 2 d i − 0.95d c 2 di − dc + ti 2 d i − 0.95d c 2 d i − 0.80d o 2

b i − 0.80b fc 2 b i − 0.80b fc 2 bi − bc + ti 2 b i − 0.95b c 2 b i − 0.80d o 2

a1

dc

0.95d c

a1 a2

a2 0.8b fc

Figure 6

a2

Cantilevered plate lengths -- RHS and SHS (Ref. [26])

a1 di

do

0.8d o

a2

bi b fc

di

0.95b fc

bi

Table 1 Cantilever Model -- Cantilevered plate lengths a1 and a2 (refer to Figs. 4, 6, 7 and 8 for the definition of the notation) SECTION H--shaped section [21] Channel [26]

0.95d c

dc

Cantilevered plate lengths -- Channels (Ref. [26])

a1 a2

Figure 8

0.8d o

a2

Cantilevered plate lengths -- CHS (Ref. [26])

Parker in [37] notes how other possible yield line patterns could be investigated for hollow sections such as the ones shown in Fig. 9. Nevertheless in [36] he recommends to investigate collapse mechanisms similar to the ones considered by the Cantilever Model with values of a 1 and a 2 as shown in Table 1. In [36] he also recommends to specify plate thicknesses not less than 0.2 times the maximum cantilever length in order to limit the deflection of the plate. Applying this model to base plates with similar dimensions to the ones of connected column would lead to inadequate design as the capacity of the base plate would be overestimated. Utilizing equations (5) and (6) the capacity of the base plate would increase and the plate thickness t i would decrease while decreasing the cantilevered plate length a m. Other design models need to be adopted in these instances. bi Dashed lines bc indicate yield lines a1 di dc

0.95d c

a1 0.95b c

a2

5

a2


bi a1 do di

0.8d o

d1

b es

a1 a2

Figure 9 4.2.2.

0.8d o

Dashed lines indicate yield lines

a2

θ

Possible yield line pattern (Ref. [37])

β = tan θ

Fling Model

Fling, in [25], presents a design model applicable to base plates with similar dimensions to the ones of the connected column and reviews the design philosophy of the Cantilever Model. Only H--shaped columns are considered in this model. He recommends to apply both a strength and a serviceability criteria to the design of base plates. Regarding the Cantilever Method, which is based on a strength criteria, he recommends to apply also a serviceability check by limiting the deflection of the cantilevered plate. He argues that, while increasing the size of the plate, deflections of the cantilevered plate would increase reducing the ability of the most deflected parts of the plate to transfer the assumed uniform loading to the supporting material. Thus the load would re--distribute to the least deflected portions of the plate which may overstress the underlying support. His proposed deflection limit intends to prevent such overstressing. He also notes that such limit should vary depending upon the deformability of the supporting material. Fling suggests 0.01 in. (0.254 mm) to be a reasonable deflection limit to be imposed for most bearing plates, even if he clearly states that it is beyond the scope of his paper to specify deflection limits applicable to various supporting materials. [25] Regarding the design model for base plates with similar dimensions to the ones of the connected column he recommends to apply the following strength and serviceability checks. The strength check is based on the yield line theory and the assumed yield line pattern is shown in Fig. 10. The procedure is derived for a base plate with width and length equal to the column’s width and depth (therefore b i and d i equal b fc and d c respectively). The support conditions assumed for the plate are fixed along the web, simply supported along the flanges and free on the edge opposite to the web.

Figure 10 Fling Model -- Yield Line Pattern (Ref. [25]) The internal and external work produced under loading are calculated as follows: W i = 1 (2d 1 + 4βb es)Ôm p + 1 4b esÔm p (7) b es βb es W e = 2f *p(d 1 − 2βb es)b es 1 + 4 f *pβb 2es 2 3

(8)

where: m p = plastic moment capacity of the baseplate per unit width * f p = uniform design pressure at the interface of the base plate and grout/concrete which is assumed to be equal to the maximum bearing strength of the concrete Ôf b W i and W e = internal and external work d 1, β and b es = as defined in Fig. 10 Fling introduces the following parameter λ to simplify the notation: λ=

d1 b es

(9)

Equating the internal and external work yields: Ôm p(2λ + 4β + 4) = f *pb 2es( λ − 2 β) 3 β

(10)

The value of β which maximises the required moment capacity of the base plate is as follows: β=

34 + 4λ1 − 2λ1 2

(11)

which is obtained by differentiating for β the expression of the plastic moment derived from equation (10). The required base plate thickness t i is then calculated as: [25]

 

t i ≥ 0.43b fcβ

= 0.43b fcβ

6

βb es

f *p 0.9f yi (1 − β 2)

Ôf b 0.9f yi(1 − β 2)

(12)


where: b fc = column flange width Equation (12) includes a safety factor of 1 and the plastic moment capacity is increased by 10% to allow for lack of full plastic moment at the corners (as recommended in [25]). This method assumes simultaneous crushing of the concrete foundation and yielding of the steel base plate as the pressure at the interface of the base plate and grout/concrete is assumed to be equal to the maximum bearing strength of the concrete Ôf b. The serviceability check verifies the adequacy of the maximum deflection of the base plate calculated from elastic theory and assumes the same support conditions as adopted in the strength check. The maximum deflection occurs at the middle of the free edge of the plate (opposite to the web).

4.2.3.

Murray--Stockwell Model

In 1975 Stockwell presents a design model for lightly loaded base plates with base plate dimensions similar to the column’s width and depth. His formulation is suitable to only H--shaped columns. He defines a lightly loaded base plate as one wherein the required base plate area is approximately equal to the column flange width times its depth. [40] The novelty of this model is to assume that the pressure distribution under the base plate is not uniform but is confined to an area in the immediate vicinity of the column profile and is approximated by a H--shaped area characterised by the dimension a 3 as shown in Fig. 11. This pressure distribution implies that in relatively thin base plates uplift might occur at the free edge. A few years later Murray carried out a finite element study to verify the possibility introduced by Stockwell of uplift at the free edge. He established, from both modelling and testing, that thin base plates lift off the subgrade during loading and therefore the assumption of uniform stress distribution at the interface is not valid. He also concludes that experimental evidence does not support the need for the serviceability check introduced by Fling. [32] Murray further expanded Stockwell’s model to obtain the model which is known today as the Murray--Stockwell Model [41] and refines the definition of lightly loaded base plates to be relatively flexible plate approximately the same size as the outside dimensions of the connected column. [32] According to Stockwell there is only a little difference between the procedures specified in Fling and Murray--Stockwell Models as he considers both to be valid and logically derived. [41]

7

bi a3 AH

di

dc a3

a3

a3 b fc

Figure 11 Murray--Stockwell Model -- Assumed shape of pressure distribution. The Murray--Stockwell Model assumes that the pressure acting over the H--shaped bearing area is uniform and equal to the maximum bearing capacity of the concrete Ôf b. The values of A H and Ôf b are not known a priori and therefore an iterative procedure can be implemented to evaluate their values. The value of Ôf b is not known a priori as it depends upon the value of the bearing area A 1 which in this case is equal to A H. The area contained inside the column profile d c × b fc is used as a first approximation for the bearing area A H in the calculation of Ôf b as shown in equation (13).



= min Ô0.85f c′ Ôf (1) b

AA , Ô2f ′ 2 (1) 1

c

(13)

where: Ôf (1) = maximum bearing strength of the concrete at b the first iteration = bearing area at the first iteration equal to A (1) 1 d c × b fc The H--shaped bearing area A H is then calculated as the area required to spread the applied load with a uniform pressure equal to Ôf (1) . b = A (1) H

N *c Ôf (1)

(14)

b

where: A (1) = assumed H--shaped bearing area A H at the first H iteration is equal to the maximum possible concrete If Ôf (1) b bearing strength Ô2f′ c no further iterations are required and the value of the H--shaped bearing area has converged to A (1) calculated with equation (14). In the H case Ôf (1) is less than Ô2f′ c, or equivalently if the ratio b of A 2∕A 1 is smaller than (2∕0.85) 2 = 5.53, the value of the H--shaped bearing area can be further refined. Successive values of Ôf (i) and A (i) at the i--th iteration H b can be calculated as follows:




Ôf (i) = min Ô0.85f′ c b = A (i) H

AA

2 (i−1) 1



, Ô2f′ c

N *c Ôf (i)

(15) (16)

b

where: Ôf (i) = maximum bearing strength of the concrete at b the i--th iteration A (i) = bearing area at the i--th iteration equal to A (i−1) 1 H A (i) = assumed H--shaped bearing A H at the i--th H iteration The value of A H can be further refined until the difference between the values obtained from two subsequent iterations can be considered to be negligible. The use of the iterative process allows to obtain the smallest possible value of A H which yields thinner base plate thicknesses. Ignoring to refine the value of A H would simply lead to a more conservative plate design. The value of a 3 is then obtained from equation (14) observing that A H can be expressed as (refer to Fig. 11):

The Stockwell--Murray Method is recommended by DeWolf in Refs [21] and [22] and introduced in the AISC(US) Manuals in 1986. [7] [1] notes that there are cases where the value under the square root of equation (18) becomes negative. In such cases other design models should be adopted. Ref. [21] extends the application of Murray--Stockwell Model to channels and hollow section members as shown in Figs. 12, 13 and 14. For these sections the value of the bearing area A (1) (to be utilised in the first 1 and A (1) ) and the iteration while calculating Ôf (1) H b expressions of the cantilevered length a 3 and of the H--shaped area A H are summarised in Table 2. [21][26] The same iterative procedure, as outlined for H--shaped sections, can be adopted to refine the value of A H if the calculated Ôf b is less than Ô2f′ c. a3

a3

A H = 2b fca 3 + 2a 3(d c − 2a 3) = 2b fca 3 + 2d ca 3 − 4a 23

(17)

where: a 3 = cantilevered langth A H = assumed H--shaped bearing area d c and b fc = depth and width of column and solving for a 3 yields: a 3 = 1 (d c + b fc) − (d c + b fc) 2 − 4A H 4

Figure 12 Murray--Stockwell Model: Assumed pressure distribution -Channels (Ref. [26]) a3

(18)

The plate is now designed in accordance with AS4100 [11] as a cantilevered plate of length a 3 supporting a uniform pressure equal to the converged value of the maximum bearing strength of the concrete previously calculated: 2 0.9 f yi t 2i a2 N* a m *c = Ôf b 3 = c 3 ≤ = Ôm s 2 AH 2 4 The maximum axial load is then calculated as: 0.9f yi t 2i A H N *c ≤ 2a 23

a3

(19)

a3 a3

a3

Figure 13 Murray--Stockwell Model: Assumed pressure distribution -- RHS and SHS (Ref. [26]) d3

a3

or equivalently the minimum required plate thickness t i is determined as: ti ≥ a3



2N *c 0.9f yi A H

(20)

The value of the cantilevered plate length a 3 should be measured from the centre--line of the column’s plate elements as shown in Fig. 11.[21]. Nevertheless in the formulation presented here, as also carried out in [32] and [21], the full flange thickness is included in the calculation of the cantilevered plate length a 3. This only leads to a slightly more conservative design.

8

do

Figure 14 Murray--Stockwell Model: Assumed pressure distribution -- CHS (Ref. [26])


4.2.4.

Thornton’s Model

In [42] and [43] Thornton recommends that a satisfactory design of a base plate should be carried out complying with the requirements of the Cantilever, Fling (ignoring the serviceability check) and Murray--Stockwell Models. He derived a compact formulation for the design procedure which includes all three models. His formulation is suitable for the design of only H--shaped columns. In [42] he also re--derives the collapse load based on the same yield line pattern assumed by Fling in [25]. It is interesting to note that while Fling applied the principle of virtual work Thornton based his results on the equilibrium equations [35]. Obviously the results are identical. Note that Fling increased the required plate plastic moment by 10% to allow for lack of plastic moment at the corners. The design expression proposed by Thornton in [43] and currently recommended in the AISC(US) Manual [5] is as follows: ti = am

0.9f2Nb d * c

where: a m = max(a 1, a 2, λa 4)



λ = min 1,

(21)

yi i i



2 X 1 + 1 − X



Ôf b = min Ô0.85f′ c

dAb , Ô2f′  2

i i

c

a 5 = b fc + d c The concatenation of the three design models (Cantilever, Fling and Murray--Stockwell Models) is achieved in the calculation of a m. The Cantilever Model is the governing criteria in the case a m equals either a 1 or a 2. In the case a m is equal to λa 4 the Fling Model would be governing if λ equals 1 or Murray--Stockwell Model would be governing if λ is less than 1. The use of λ leads to the selection of the thinner plate obtained by using the Fling Model and Murray--Stockwell Model in order not to loose the economy in design of the latter model in the case of lightly loaded columns. Recalling the description of Murray--Stockwell Model no refinement in the calculation of A H is implemented in equation (21). It is interesting to note how this approach provides a more mathematical definition of lightly loaded column where a column is said to be lightly loaded if its λ is less than 1, or equivalently if its X is less than (4∕5) 2 = 0.64. The expression of the plate thickness of Fling Model, re--derived in [42], is simplified by Thornton in [43] in order to reduce the complexity of the yield line solution. His simplification introduces an approximation in the value of a 4 with an error of 0% (unconservative) and 17.7% (conservative) for values of d c∕b fc ranging from 3/4 to 3. The value of N *0 represents the portion of the total axial load N *c acting over the column footprint (d cb fc) under the assumption of uniform bearing pressure under the base plate. Murray--Stockwell Model is concatenated in equation (21) to carry a design axial load equal to N *0 (not on N *c) over the assumed H--shaped bearing area inside the column footprint.

a 4 = 1 d cb fc 4 N *0 = portion of N *c acting over the column footprint N* = c b fcd c b id i 4b fcd c N *c X= (d c + b fc) 2 Ôf bd ib i d cb fc = 24 N *0 = 24 N *c d ib i a 5Ôf b a 5Ôf b Table 2 Murray--Stockwell Model (refer to Figs. 4, 6, 7, 8, 11, 12, 13 and 14 for the definition of the notation) SECTION

A (1) 1

H--shaped section [21]

b fcd c

Channel [26]

b fcd c

RHS SHS [21][26] CHS [21][26]

b cd c

4.2.5.

π

d 20 4

a3 (d c + b fc) − (d c + b fc) 2 − 4A H 4  (2b fc + d c) − (2b fc + d c) 2 − 8A H 4  (d c + b c) − (d c + b c) 2 − 4A H 4  d o − d 2o − 4A H∕π 2

Eurocode 3 Model

Clause 6.11 and Annex L of Eurocode 3 deal with the design of base plates. [23]

9

AH 2b fca 3 + 2a 3(d c − 2a 3) 2b fca 3 + (d c − 2a 3)a 3 d cb c − (d c − 2a 3)(b c − 2a 3) = 2(d c + b c)a 3 − 4a 23 π(d 2o − d 23)∕4

= π(d oa 3 − a 23 )

where : d 3 = d o − 2a 3

Requirement of the EC3 is to provide a base plate adequate to distribute the compression column load over an assumed bearing area. The EC3 Model assumes an H--shaped bearing area as shown in Fig. 15(a). It requires that the pressure


assumed to be transferred at the interface base plate/foundation should not exceed the bearing strength of the joint f j.EC3 and the width of the bearing area should not exceed c calculated as follows: c = ti



c

c

f yi

(22)

3f j.EC3γ MO

c

(b) Short Projection

where: f j.EC3 = bearing strength of the joint

(c) Large Projection

Figure 15 Assumed bearing pressure distributions specified in EC3 [23]

=β jk jf cd β j = 2/3 provided that the characteristic strength of the grout is not less than 0.2 times the characteristic strength of the concrete foundation and the thickness of the grout is not greater than 0.2 times the smallest width of the steel base plate k j = concentration factor and may be taken as 1 or otherwise as

c

N *c

h

Baseplate

Concrete foundation

aabb

1 1

Elevation

a1

a 1 and b 1 = dimensions of the effective area as shown in Fig. 16 a 1 = mina + 2a r, 5a, a + h, 5b 1 ≥ a f cd = design value of the concrete cylinder compressive strength = f ck∕γ c f ck = characteristic concrete cylinder compressive strength (in accordance with Eurocode 2) γ c = partial safety factor for concrete material properties (in accordance with Eurocode 2) γ MO = 1.1 (boxed value from Table 1 of [23]) In the case of large or short projections the bearing area should be calculated as shown in Figs. 15(b) and (c). [23] [23] requires that the resistance moment m Rd per unit length of a yield line in the base plate should be taken as: m Rd =

t 2i f yi 6γ MO

(23)

No specific expression for the sizing of the steel base plate are provided. N *c

≤c

c c c

c

≤c Bearing area

≤c (a) General Case

10

b1

b

b 1 = minb + 2b r, 5b, b + h, 5a 1 ≥ b

This area not included in bearing area

br ar

a

Plan

Figure 16 Column base layout [23] 4.3. 4.3.1.

RECOMMENDED MODEL Design considerations

The recommended design model is a modified version of the one proposed by Thornton in [43] and also adjusted to suit Australian Codes AS 3600 [10] and AS 4100 [11]. The Thornton Model is currently recommended by the AISC(US) Manual [5]. Unfortunately the Thornton Model presented in [5], [42] and [43] is suitable for the design of H--shaped columns only. His formulation has been here modified for H--shaped sections and extended for channels and hollows sections adopting a similar approach as in [43] which is outlined in Section 10. The modification to the Thornton Model introduced here regards the manner in which Murray--Stockwell Model is implemented. It is in the authors’ opinion that the calculation of A H and consequently of λ (refer to the literature review for further details regarding the notation) should be calculated based on N *c (total axial compression load) and not N *0 (portion of the total load N *c acting over the column footprint under the assumption of uniform bearing pressure). This intends to ensure that Murray--Stockwell Model would govern the design only for base plates of similar dimensions to the ones of the connected columns and for lightly loaded columns, which represents the actual base plate layout for which the model has been developed. The design would then be based on only one assumed pressure STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

distribution. Calculating A H based on N *0 could lead to the design situation for lightly loaded columns where the plate thickness is governed by Murray--Stockwell Model even for plate dimensions larger than those of the connected columns as the model would select the thinner plate between the ones calculated with Fling Model and with Murray--Stockwell Model. It is interesting to note how the assumed bearing area (H--shaped in the case of H--shaped column sections) could extend also beyond the footprint of the column section as shown in Fig. 17 in the case of H--shaped sections and hollow sections. [34] No specific design guidelines are provided in [34]. A similar pressure ditribution is considered in the Eurocode 3 Model. [23] Nevertheless in the recommended model the application of Murray--Stockwell Model is always carried out based on assumed bearing areas inside the column footprint even for base plates with dimensions greater than the column’s depth and width as other bearing distributions need to be validated by testings.

a

a

a a

a

b

b

b

Figure 17 Possible assumed bearing areas (Ref. [34]) Design criteria

There are two different design scenarios which are considered here: the column is prepared for full contact in accordance with Clause 14.4.4.2 of AS 4100 [11] and the axial compression may be assumed to be transferred by bearing. Design requirements are as follows: the end of the column is not prepared for full contact and the welds shall have sufficient strength to carry the axial load. The design requirements are as follows:

11

4.3.3.

Design Concrete Bearing Strength

The maximum bearing strength of the concrete Ôf b is determined in accordance with Clause 12.3 of AS 3600 [10].




AA , Ô2f′  2 1

c

(26)

It is interesting to note from equation (26) that increasing the supplementary area A2 increases the concrete confinement which yields larger design capacities ÔN c. The loss of bearing area due to the presence of the anchor bolt holes is normally ignored. [21]

b

N des.c = [ÔN c ; ÔN s] min ≥ N *c

where: N des.c = design capacity of the base plate connection subject to axial compression ÔN c = design axial capacity of the concrete foundation ÔN s = design axial capacity of the steel base plate ÔN w = design axial capacity of the weld connecting the base plate to the column section * N c = design axial compression load

ÔN c = Ôf bA i

b b

4.3.2.

(25)

where: Ô = 0.6 A 1 = b id i The axial capacity of the concrete foundation ÔN c is then obtained multiplying the maximum concrete bearing strength Ôf b by the base plate area A i as follows:

a

Ineffective areas

b b

N des.c = [ÔN c ; ÔN s ; ÔN w] min ≥ N *c

(24)

4.3.4.

Steel Base Plate Design

The base plate thickness required to resist a certain design axial compression N *c is calculated as follow: ti = am



2N *c 0.9f yi d i b i

where: a m = max(a 1, a 2, λa 4)



λ = min 1, k

X  1+ 1−X

(27)



X = YN *c a 1, a 2, a 4, k and Y are tabulated in Table 3. When X is greater than 1, λ should be taken as 1.


Table 3 Values for the design and check specified by the recommended model for axial compression. Section

a1

a2 b i − 0.80b fc 2

a4

H--shaped sections

d i − 0.95d c 2

Channels

d i − 0.95d c b i − 0.80b fc 2 2

2d cb fc 3

RHS

d i − 0.95d c b i − 0.95b c 2 2

2d23b

SHS

d i − 0.95b c b i − 0.95b c 2 2

bc 3

CHS

d i − 0.80d 0 b i − 0.80d o 2 2

d0 2 3

d cb fc 4

ÔN s =

0.9f yi d ib i t 2i 2a′ m

(28)

2

 λ′ = max1, 





1 2 k a4 k 2 t i Y

0.9f2 d b − 1 yi i i



  

a4 λ a 1, a 2, a 4, k and Y are tabulated in Table 3. This model is applicable to column sections as outlined in Table 3 with the exception of H--shaped sections for which b fc∕2 is greater than d c as a different yield line pattern from those considered would occur. a′ m = max a 1, a 2,

4.3.5.

Weld design at the column base

The design of the weld at the base of the column is carried out in accordance with Clause 9.7.3.10 of AS 4100. [11] The weld is designed as a fillet weld and its design capacity ÔN w is calculated as follows: ÔN w = Ôv wL w = Ô0.6f uwt tk rL w

(29)

where: Ôv w = design capacity of the fillet weld per unit length Ô = 0.8 for all SP welds except longitudinal fillet welds on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100) 0.7 for all longitudinal SP fillet on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100)

12

4N *c Ôf ba 25

b fc + d c

8N *c Ôf ba 25

2b fc + d c

4N *c Ôf ba 25

bc + dc

3 d ib i 2 bc 2 d ib i d0

4N *c Ôf ba 25

2b c

4N *c Ôf bπd 20

−

i i

c fc

i i

c fc

i i

c fc

0.6 for all GP welds (Table 3.4 of AS 4100) f uw = nominal tensile strength of weld metal (Table 9.7.3.10(1) of AS 4100) t t = design throat thickness k r = 1 (reduction factor to account for length of welded lap connection) Lw = total length of fillet weld Refer to Section 13. for tabulated values of the design capacity of fillet welds Ôv w.

5. AXIAL TENSION 5.1.

where:

a5

dd bb 3 d b 2 db db 1.7  db 2

i i

Thicknesses of base plates with dimensions similar to those of the connected column section calculated with equation (27) might be quite thin, especially in the case of lighlty loaded columns (where Murray--Stockwell Model applies). It is therefore recommended to specify plate thicknesses not less than 6mm thick for general purposes and not less than 10mm for industrial purposes. Similarly a procedure to evaluate/check the capacity of an existing plate is carried out as follows:

Y

k

INTRODUCTION

There is not much guidance available in literature for the design of unstiffened base plates subject to uplift. The literature presented here outlines the available guidelines for the design of base plates and of anchor bolts. Two models presented here for the design of base plates for hollow sections, which are the IWIMM Model (named here after its authors) and Packer--Birkemoe Model, were firstly derived for bolted connections between hollow sections. [37] and [36] suggest their suitability also for the design of base plates. These models include also guidelines for determining the required number of anchor bolts. Such guidelines are incorporated in the literature review for the design of the steel base plates as their application is only suitable for the particular base plate model they refer to and as they do not account for the interaction between the anchor bolts and the concrete foundation, which is dealt with in the literature review on anchor bolts.

5.2.

BASE PLATE DESIGN -- LITERATURE REVIEW

The models presented here differ for their assumptions regarding the failure modes investigated. It is interesting to note that the design guidelines currently available deal with a limited number of base plate layouts. For each model outlined here, the column sections and the number of bolts considered by the model are specified after the model name.


5.2.1.

Murray Model (H--shaped sections with 2 bolts)

In [32] Murray presents a design procedure for base plates of lightly loaded H--shaped columns with only two anchor bolts subject to uplift. He also notes that to his knowledge no studies have been published on the design of lightly loaded column base plate subjected to uplift loading prior to his [32]. His design model is based on yield line analysis and the yield line pattern assumed is shown in Fig. 18. The expressions of the internal and external work can be written as follows:



W i = Ôm p 2 2b′ + 1 4 2 b fc b′ b fc

N *ts g N* sg 2 We = t = 2 2 b fc 2b fc

(31)

b′b fc N sg Ôm p = 2 2 b fc 4b′ + 2b 2fc * t

(32)

The presence of the flanges requires b′ to remain always less or equal to d c∕2 and therefore the value of b′ which maximises the plate plastic capacity varies depending upon the column cross--sectional geometry as follows: b b d b′ = fc for fc ≤ c (34) 2 2 2 (35)

The minimum plate thicknesses required under a certain axial load N *t are obtained substituting equations (34) and (35) into equation (32) as shown below:



N *t s g 2 b d for fc ≤ c 2 0.9f yib fc4 2

(36)

N *t s gd c b d for fc ≥ c 2 2 2 0.9f yi(d c + 2b fc) 2

(37)



Murray carried out a finite element study to investigate the adequacy of the proposed model. He also validated the reliability of equations (36) and (37) using limited 13

b fc∕2 b′ Figure 18 Murray Model Assumed Yield Line Patterns (Ref. [32])

The value of b′ which maximises the required plate plastic capacity is obtained differentiating equation (32) for b′ and is equal to: b b′ = fc (33) 2

b dc d for fc ≥ c 2 2 2

d c∕2 b′ = 2 (b fc∕2) ≤ d c∕2

b′

Equating the external and internal work the expression of Ôm p can be written as follows:

ti ≥

b′ b′

1 unit

where: N *t = design tension axial load s g and b′ = as defined in Fig. 18

ti ≥

b fc∕2 d c∕2

(30)

b′ =

b fc∕2

s g∕2 s g∕2



4b′ 2 + 2b 2fc b′b fc

= Ôm p

experimental results, which consisted of 4 base plate specimens with dimensions ranging from 8” x 6” (203.2 x 152.4 mm) to 12” x 8” (304.8 x 203.2 mm) and thicknesses varying from 0.364 in. (9.246 mm) to 0.377 in. (9.576 mm). This method is included in the design model recommended by the current AISC(US) Manual [5].

5.2.2.

Tensile Cantilever Model (Generic Model)

Tensile Cantilever Method, as it is referred here, assumes that the tension in the anchor bolts spreads out to act over an effective width of plate (b e ) which is assumed to act as a cantilever in bending ignoring any stiffening action of the column flanges.

1

bt 1 dh

bt

bt

be

Figure 19 Tensile Cantilever Model (Ref. [26]) It can be applied to generic base plate layouts. Nevertheless it provides conservative designs as it ignores the two way action of the base plates. Reference [47] suggests a 45 degree angle of dispersion as shown in Fig. 19. This is based on considerations of elastic plate theory as described in reference [13]. The design moment and the design moment capacity are then calculated as: N* m *t = n t b t b Ôm s =

0.9b e t 2i f yi 4

(38) (39)

where: m *t = design moment per unit width due to N *t


n b = number of anchor bolts b t = distance from face of web to anchor bolt location d h = diameter of the bolt hole b e = 2b t + d h The axial capacity of the base plate can then be determined equating the design moment and the section moment capacity as follows:

do N

N

* t

ti

a2 a1

* t

ti

(40)

Figure 20 Bolted CHS Flange--plate Connection (Ref. [36])

or equivalently the minimum base plate thickness t i under a certain loading condition is calculated as:

[27] also recommends to determine the number of required anchor bolts as follows:

N *t ≤

ti =

5.2.3.

0.9f yib et 2i n b 4 bt



4N *t b t 0.9f yi b e n b

(41)

IWIMM Model (CHS with varying number of bolts)

The IWIMM Model has been named here after the initials of the authors of the model. [27] The model was firstly derived for the design of CHS bolted connections. [37] and [36] suggest its use also for the design of base plates of CHS columns. The base plate layout considered by this model is shown in Fig. 20. The plate thickness is calculated based on the design axial tension load N *t as follows: ti ≥



2N *t Ôf yi π f 3

(42)

where: Ô = 0.9 d 0 = outside diameter of a CHS t c = thickness of column section f 3 = 1 k 3 + k 23 − 4k 1 2k 1 r k 1 = ln r 2 3



k3 = k1 + 2 d r2 = 0 + a1 2 d0 − tc r3 = 2 a 1 and a 2 as defined in Fig. 20 [27] recommends to keep the value of a 1 as small as possible, i.e. between 1.5d f and 2d f (where d f is the nominal diameter of the bolts), while ensuring a minimum of 5 mm clearance between the nut face and the weld around the CHS.

nb ≥

  

  

N *t 1 1− 1 + f 3 f lnr 1 ÔN tf 3 r 2

(43)

where: Ô = 0.9 N tf = nominal tensile capacity of the bolt d r 1 = 0 + 2a 1 2 d0 r2 = + a1 2 a1 = a2 This procedure does not verify the capacity of the concrete foundation and its interaction with the anchor bolts needs to be checked. Assumptions adopted by this model are an allowance for prying action equal to 1/3 of the ultimate capacity of the anchor bolt (at ultimate state), a continuous base plate, a symmetric arrangement of the bolts around the column profile and a weld capacity able to develop the full yield strength of the CHS. [28] notes that adopting the above prying coefficient for the bolted CHS connection in the base plate design is conservative due to the greater flexibility of the concrete foundation when compared to the steel to steel connection. [36]

5.2.4.

Packer--Birkemoe Model (RHS with varying number of bolts)

The Packer--Birkemoe Model is here named after the authors of the model. [36] This model deals with base plate for RHS as shown in Fig. 21 and it has been validated only for base plates with thickness varying between 12mm and 26mm. The model includes prying effects in the design procedure. The prying action decreases while increasing a 2 as shown in Fig. 21. The value of a 2 should be kept less or equal to 1.25 a 1, as no benefit in the base plate performance would be provided beyond such value. a 1 is defined as the distance between the bolt line and the face of the hollow section. Generally 4--5 bolt diameters are used as spacing of the bolts s p but shorter spacing are also possible. Based on the design loads the required number of anchor bolts should be calculated assuming that the

14


prying action absorbs about 20--40% of the anchor bolt capacity. The coefficient δ is then calculated as follows: d δ = 1 − sh

(44)

p

where: s p = bolt pitch as defined in Fig. 21 The designer should then select a preliminary plate thickness in the following range:



KN *b ≤ t i ≤ KN *b 1+δ

(45)

where: K=

N *t

a3 a4

(where f yi is in MPa)

N *b = design axial tension load carried by one bolt N* = nt b d f = nominal anchor bolt diameter The value of α represents the ratio of the bending moment per unit width of plate at the bolt line to the bending moment per unit width at the inner hogging plastic hinge. In the case of a rigid base plate α is equal to 0 while for a flexible base plate with plastic hinges forming at both the bolt line and at the inner face of the column (see Fig. 21) α is equal to 1. From equilibrium, the value α for preliminary base plate layout is calculated as follows: KÔN tf −1 t 2i

= = = = = sp

sp

Figure 21 Packer--Birkemoe Model (Ref. [36])

a 3 = a 1 − d f∕2 + t c





a 2 + d f∕2 δ(a 2 + a 1 + t c)



(46)

5.2.5.

Eurocode 3 Model (H--shaped sections with varying number of bolts)

The Eurocode 3 does not provide a specific design procedure for the design of base plates subject to tension. Nevertheless it provides very useful guidelines for the design of bolted beam--to--column connections (Appendix J.3 of [23]) which can be adapted for the design of base plates considering all anchor bolts as bolts on the tension side of the beam--to--column connection. The design of the end plate or of the column flange of the beam--to--column connection is carried out in terms of equivalent T--stubs as shown in Fig. 22.  e m 0.8a 2 a e m

α should be taken as 0 if its value calculated with equation (46) is negative. The capacity of the steel base plate is then calculated as follows: ÔN t =

t 2i (1 + δα)n b K





a N* δα N *b ≈ n t 1 + a 3 4 1 + δα b where: α=







15

l

(48)

e min e m

0.8r r

tf e min

Figure 22 T--stub connection in EC3 (Ref. [23])

KN *t −1 1 t 2i n b δ

a 4 = min 1.25a 1, a 2 +

tf

(47)

where: ÔN t = axial tension capacity of the base plate ÔN t calculated with equation (47) must be greater than N *t . The actual tension in one bolt, including prying effects, is determined as follows:



N *t

tc

=

4a 310 3 Ôf yis p

α=

The value of α previously calculated in equation (46) does not have to equal the value of α calculated from equation (48) as the former assumes the bolts to be loaded to their full tensile capacity. It interesting to note how equation (48) provides an estimate of the prying action present in the base plate. a2 a1



df 2

EC3 considers that the capacity of a T--stub may be governed by the resistence of either the flange, or the bolts, or the web or the weld between flange and web of T--stub. The failure modes considered are three as shown in Fig. 23. The axial capacity is calculated as follows:


F t.Rd = minF t.Rd1, F t.Rd2, F t.Rd3

(49)

where: 4M pl.Rd m 2M pl.Rd + nΣB t.Rd F t.Rd2 = m+n F t.Rd3 = ΣB t.Rd 0.25lt 2f y M pl.Rd = γ f MO

Mode 2

Mode 1

Ft Mode 1: Complete flange yielding Ft +Q 2

Q

Mode 2: Bolt failure with flange yielding

 B ∕2  B ∕2 t

t

β=

4M plRd m

B

t.Rd

=

β

l t 2ff y∕γ MO m

B

t.Rd

Figure 24 Prying action in T--stub for the three failure modes considered in (Ref. [23]) The tension zone of the end plate should be considered to act as a series of equivalent T--stubs with a total length equal to the total effective length of the bolt pattern in the tension zone, as shown in Fig. 26.[23] The length to be utilised in the design of the equivalent T--stub is calculated as follows: for bolts outside the tension flange of the beam

for other inner bolts l eff.c = minp, 4m + 1.25e, 2πm

(52)

for other end bolts Q

l eff.d=min(0.5p+2m+0.625e, 4m+1.25e, 2πm) (53)

Ft Mode 3: Bolt failure

 B ∕2  B ∕2 t

λ = n∕m

2

1

2λ 1 + 2λ

l eff.a = min0.5b p, 0.5w+2m x+0.625e x, (50) 4m x+1.25e x, 2πm x) for first row of bolts below the tension flange of the beam (51) l eff.b = min(αm, 2πm)

Ft

Q

Mode 3

2λ 1 + 2λ

n = e min ≤ 1.25m l = equivalent effective length calculated in equations (50), (51), (52) and (53) ΣB t.Rd = tensile capacity of bolt group γ MO = partial safety factor = 1.10 (boxed value from Table 1 of [23]) F t.Rd1, F t.Rd2 and F t.Rd3 = tensile capacities of the T--stub based on failure modes 1, 2 and 3 respectively

Ft +Q 2

t.Rd

1

F t.Rd1 =

Q

 BF

t

where: α = as defined in Fig. 27 It is interesting to note that the failure modes considered for example by equations (52) and (53) are the same as those considered to evaluate the capacity of an unstiffened flange. The yield line patterns of such failure modes are shown in Fig. 25.

Figure 23 Failure modes of a T--stub flange (Ref. [23]) It is interesting to note that the amount of prying action for a certain base plate layout can be obtained as the ratio F t.Rd∕ΣB t.Rd as shown in Fig. 24.

16


p

p

2π 65.5

1.4

e m Centreline of web (a) Combined bolt group action

5

4.5 4.75 4.45

α

1.3 1.2

λ 2 1.1 1.0 0.9

Centreline of web (b) Separate bolt patterns

0.8 0.7 0.6

Centreline of web

0.5

(c) Circles around each bolt

Figure 25 Yield line patterns for unstiffened flange (Ref. [23]) bp w

Equivalent T--stub for extension

ex mx

0.4 0.3 0.2 0.1 0

l eff.a

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ex mx l eff.b

p

l eff.c

p

bp

e m Portion between flanges b p∕2

l eff.a

l eff.a

Transformation of extension to equivalent T--stub

Figure 26 Effective lengths of equivalent T--stub flanges representing an end plate (Ref. [23])

17

m1 m1 + e

λ2 =

m2 m1 + e

m2

e

l eff.d

e m

λ1 =

0.9 λ1

m1

Figure 27 Value of Effective lengths of α to calculate equivalent T--stub flanges (Ref. [23]) 5.3.

DESIGN OF ANCHOR BOLTS -LITERATURE REVIEW

Available design guidelines regarding the behaviour of anchor bolts in tension distinguish between the behaviour of anchor bolts with an anchor head and of hooked anchor bolts and therefore these will be discussed here separately. For the purpose of this paper an anchor head is defined as a nut, flat washer, plate, or bolt head or other steel component used to transmit anchor loads from the tensile stress component to the concrete by bearing. [2]

5.3.1.

Anchor bolts with anchor head

The first detailed guidance on the design of anchor bolts is provided by the American Concrete Institute Committee 349 in 1976 in [3]. These recommendations are produced for the design of nuclear safety related structures. Some of the ACI Committee 349 members, very active in the preparation of [3], publish an article [17] where the guidelines provided in [3] are modified to suit concrete structures in general.


The design criteria at the base of [2] and of [17] is that anchor bolts should be designed to fail in a ductile manner, therefore the anchor bolt should reach yielding prior to the concrete brittle failure. This is achieved by ensuring that the calculated concrete strength exceeds the minimum specified tensile strength of the steel. [2][17] Typical brittle failure of an isolated anchor bolt is by pulling out of a concrete cone radiating out at 45 degrees from the bottom of the anchor as shown in Fig. 28. [2] and [17] recommend to calculate its nominal concrete pull--out capacity based on the tensile strength Ô4 f′ c (where f′ c is in psi) or Ô0.33 f′ c (where f′ c is in MPa) acting over an effective area which is the projected area of the concrete failure cone. In both [3] and [17] it is recommended to use a capacity reduction factor of 0.65 in the calculation of the concrete cone capacity, which can be increased to 0.85 in the case the anchor head is beyond the far face reinforcement. The value of 0.65 applies to the case of an anchor bolt in plain concrete. This intends to be a simplification of a very complex problem. [3][17] In the current version of ACI349 [2] the capacity reduction factor is equal to 0.65 unless the embedment is anchored either beyond the far face reinforcement, or in a compression zone or in a tension zone where the concrete tension stress (based on an uncracked section) at the concrete surface is less than the tensile strength of the concrete 0.4 f′ c subjected to strength load combinations calculated in accordance with current loading codes (i.e. AS1170.0 [8]) in which cases a capacity reduction factor of 0.85 can be used. [2] An embedment is defined in [2] as that steel component embedded in the concrete used to transmit applied loads to the concrete structure. The ACI Committee 349 recognises that there is not sufficient data to define more accurate values for the strength reduction factor. [2] Experimental results have generally verified the results of this approach. [31]

that the thickness of the anchor head is at least 1.0 times the greatest dimension from the outermost bearing edge of the anchor head to the face of the tensile stress component and that the bearing area of the anchor head is approximately evenly distributed around the perimeter of the tensile stress component. [2] The placing of washers or plates above the bolt head to increase the concrete pull--out capacity should be avoided as it only spreads the failure cone away from the bolt--line which may cause overlapping of cones with adjacent anchors or edge distance problems. [31] Ld 45 o

Ld Failure plane

Projected surface

Figure 28 Concrete failure cone (Ref. [26]) If reinforcement in the foundation is extended into the area of the failure cone additional strength would be present in practice since the nominal capacity of the failure cone is based on the strength of unreinforced concrete. The concrete pull--out capacity of a bolt group is calculated as the average concrete tensile strength Ô0.33 f′ c times the effective tensile area of the bolt group. This effective area is calculated as the sum of the projected areas of each anchor part of the bolt group if these projected areas do not overlap; when overlapping occurs overlapped areas should be considered only once in the calculation of the effective tensile area, thus leading to a smaller concrete pull--out capacity if compared to the sum of the concrete pull--out capacities of each anchor in the bolt group considered in isolation. [2][17]

The value of Ô0.33 f′ c represents an average value of the concrete stress on the projected area accounting for the stress distribution which occurs along the failure cone surface varying from zero at the concrete surface to a maximum at the bolt end. [31] In calculating the projected area of the failure cone the area of the anchor head should be disregarded as the failure cone initiates at the outside periphery of the anchor head. [2] Experimental results have shown that the head of a standard bolt, without a plate or washer, is able to develop the full tensile strength of the bolt provided, as specified in [2], that there is a minimum gross bearing area of at least 2.5 times the tensile stress area of the anchor bolt and provided there is sufficient side cover,

18


Tension Force s s

Ld

Transverse splitting

Ld



2 cos −1

2Ls πL

2 d

d Shaded = πL 2 − +s d 0 2 360 Area (a) Two Intersecting Failure Cones

L − s4 2 d

2

− Ld Ld

s 2

=

Ld



2 cos −1

Area = πL 2d −

2Ls πL d

360 0

2 d

+ Ld +s 2

L − s4 2 d

2

Circle -- Sector + Triangle (b) Failure Cone Near an Edge (Note: the inverse cosine term listed in the equations is in degrees)

Figure 29 Calculation of the projected area of two intersecting failure cones or one failure cone near an edge (Ref. [30]) Simple procedures to calculate the effective tensile areas of bolt groups are provided in [30], i.e. the procedure to calculate two intersecting cones is shown in Fig. 29. [30] Depending upon the bolt group layout other possible failure modes could take place such as the one shown in Fig. 30 where an entire part of the concrete foundation would pull--out. In such cases the effective tensile area should be calculated selecting the smallest projected area due to the possible concrete failure surfaces as shown in Fig. 30. A similar average tensile strength as in the case of the pull--out cones can be adopted. [2][17]

Figure 31 Transverse splitting failure mode (Ref. [2]) It is interesting to note that in the case of shallow anchor bolts the angle at the bolt head formed by the failure cone tends to increase from 90 degrees to 120 degrees. An anchor bolt is classified as shallow when its length is less than 5in. (127 mm). Nevertheless for design purposes caution should be applied is using angles greater than 90 degrees as cracks might be present at the concrete surface. It is recommended not use angles other than 90 degrees. [2][17] The previous considerations assume the concrete element to be stress--free and only subjected to the anchor bolts loading. [2] and [17] consider the case when there is a state of biaxial compression and tension in the plane of the concrete. The former loading condition would be beneficial to the anchor bolt’s strength while the latter loading state would lead to a significantly decrease in strength. Nevertheless, it is in the opinion of the ACI 349 Committee that a failure cone angle of 90 degrees can still be utilised as it is assumed that any cracking would be controlled by the main reinforcement designed in accordance with current concrete codes, i.e. AS 3600 [10]. The design procedure proposed by ACI 349 and [17] is also recommended by DeWolf in [21]. [21] notes that the use of cored holes, such as shown in Fig. 32, should not reduce the anchorage capacity based on the failure cone, provided that the cored hole does not extend near the bottom of the bolt. This situation should be avoided if the dimensions shown in Fig. 32 are followed. [26]

Tension Force

Figure 30 Potential Failure Mode with limited depth (Ref. [2]) Transverse splitting is another failure mode which can occur between anchor heads of an anchor bolt group when their centre--to--centre spacing is less than the anchor bolt depth and is shown in Fig. 31. This failure mode occurs at a load similar to the one required to cause a pull--out cone failure in uncracked concrete and therefore no additional design checks need to be considered. [2][17]

19

Projection

3d f but ≥ 75mm

Ld

df

Figure 32 Suggested layout for Cored Holes to Permit Minor Adjustments in Position on Site (Ref. [26])


45 o 45

Failure surface

Ô = 0.65 in Ref. [3], = 0.85 in Refs. [2] and [17] Adopting the capacity reduction factor Ô equal to 0.85 the minimum side cover to avoid lateral bursting of the concrete can be calculated as follows:

Blow out cone

ae = df

o

Figure 33 Failure Surface of Blow--out Cone due to Lateral Bursting of the Concrete (Ref. [31]) Lateral bursting of the concrete can occur when an anchor bolt is located close to the concrete edge as shown in Fig. 33, which is caused by a lateral force present at the bolt head location. This lateral force may be conservatively assumed to be one--fourth of the nominal tensile capacity of the anchor bolt for conventional anchor heads which can be calculated in accordance with Clause 9.3.2.2 of AS 4100 [11] as follows: N tf = A sf uf = 0.75A 0f uf = 0.75

d 2f π f 4 uf

(55)

where: Ô = 0.65 in Ref. [3], 0.85 in Refs. [2] and [17] ÔN c.lat = lateral bursting capacity of the concrete a e = side cover Equating the assumed lateral force (equal to 0.25 N tf) to the concrete lateral bursting capacity allows to express the minimum required side cover as a function of both the concrete and anchor bolt strengths as shown below: 0.25N tf = ÔN c.lat = Ô0.33 f ′c π a 2e

(56)

and solving equation (56) for a e yields: ae = df where:

20



f uf

Ô7 f′ c

f uf

(58)

6 f′ c

Equation (58) has also been recommended in [26] and [47]. Tension Force

Potential Failure Zone

Spiral reinforcement

(54)

where: A s = tensile stress area in accordance with AS1275 [9] and conservatively approximated with 0.75 A0 d2 π A 0 = f = shank area 4 f uf = minimum tensile strength of a bolt The failure surface has the shape of a cone which radiates at 45 degrees from the anchor head towards the concrete edge. The concrete capacity is calculated as the average concrete tensile strength Ô0.33 f′ c applied over the projected cone area as follows: [2][3][17] ÔN c.lat = Ô0.33 f′ c π a 2e



(57)

Figure 34 Reinforcement Against Lateral Bursting of Concrete Foundation (Ref. [2]) Based on the guidelines provided in reference [3], simplified design guidelines regarding minimum embedment lengths and minimum edge distances are presented in reference [39]. These minimum embedment lengths are calculated with an additional safety factor of 1.33 when compared to the guidelines presented in reference [3]. These simplified guidelines are as follows: for Grade 250 bars and Grade 4.6 bolts: L d ≥ 12d f a e = min(100, 5d f) for Grade 8.8 bolts: L d ≥ 17d f a e = min(100, 7d f) where: L d = minimum embedment length These minimum embedment lengths and edge distances have also been recommended in references [18], [21] and [26]. Reinforcement needs to be specified in the case anchor bolts are located too close to a concrete edge (the edge distance a e is less than the one required by equation (58)) or their embedment length is less than the one required to develop the bolt’s full tensile strength. Such reinforcement should be designed and located to intersect potential cracks ensuring full development length of the reinforcement on both sides of such cracks. The placement of the reinforcement should be concentric with the tensile stress field. [2] STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

In the specific case of insufficient embedment length a possible reinforcement layout to enhance the concrete pull--out capacity is detailed in Fig. 35 using hairpin reinforcement. The hairpins need to be placed as specified in Fig. 35 in order to effectively intercept potential failure planes. Other reinforcement configurations can be specified in accordance with AS 3600 while still complying with the specifications previously outlined for hairpin reinforcement to consider the reinforcement to be effective. These specifications are the maximum distance from the anchor head and the minimum embedment length equal to 8 reinforcement diameters. Tension Force 8x diameter of the hairpin reinforcement

Ld

Development length from AS3600

Ld 3

Maximum distance from anchor head for reinforcement to be considered effective

Ld 3

Locate legs of hairpin reinforcement in this region

In the case of insufficient side cover a e there are no experimental results to validate a design procedure to include reinforcement to avoid lateral bursting of the concrete. The ACI 349 Committee recommends the use of spiral reinforcement as shown in Fig. 34 while also suggesting to refer to accepted practices for prestressing anchorages to resist the lateral bursting force. [2] [2] and [17] recommend that if proper anchorage of the reinforcement cannot be accomplished in the available dimensions, the anchorage configuration should be changed.

Hooked bars

There are different opinions regarding the ability of hooked anchor bolts to carry tensile loading. Some authors do not recommend to use them to resist uplift loads, while others have provided some design guidelines. The major concern regarding the use of hooked bars in tension is that they tend to fail by straightening and pulling out of the concrete as shown by research carried out by the PCI.[24] [24] and [31] discuss the behaviour of smooth anchor bolts and recommend to use hooked anchor bolts with a bearing head as smooth bars are less able to develop their strength along their length than deformed bars. [24] recommends to use the following formula to determine the pull--out capacity of a hooked anchor bolt:

21

(59)

where: Ô = 0.80 (as recommended in [26]) ÔN th = tensile capacity of a hooked bar d f = nominal diameter of the hooked bar L h = length of the hook DeWolf in [22] recommends to use hooked anchor bolts only under compressive axial loading, and where no fixity is needed at the base except during erection. Even for this case he recommends to design the hook to resist half the design tensile capacity of the bolt using equation (59). He also recommends to use anchor bolts with a more positive anchorage which is formed when bolts or rods with threads and nut are used. [22] Similar design considerations are presented in reference [47]. The recommendations of the AISC(US) Manuals have changed over time. In reference [6] the design of hooked anchor rods under tension is recommended to be carried out based on the design procedure presented in [24] as outlined in equation (59) while in reference [5] the use of hooked anchor rods is recommended only for axially loaded members subject to compression only.

5.4. 5.4.1.

Figure 35 Possible Placement of Reinforcement for Direct Tension (Ref. [2])

5.3.2.

ÔN th = 0.7f′ cd f L h

RECOMMENDED MODEL Introduction

Available design guidelines have been included in the recommended design models where possible. Additional design models/provisions are here provided for those instances, to the knowledge of the authors, not covered by available design guidelines. Their use has been clearly stated and their derivations are illustrated in Section 11. It is interesting to note that depending upon the magnitude of the plate flexural deformation and the bolt elongation which occur in the loaded base plate connection, a prying action might be present. The possible collapse mechanisms which can occur are similar to those which can occur in bolted connections. These are shown in Fig. 36. N *b

N *t N *p

N *b

N *t

N *b

N *t

N *p

Schematic failure modes

Bending moment diagrams showing plastic hinges

Figure 36 Possible plate deformations and anchor bolt elongations (modified from Ref.[13]) In the case the plate flexural deformation is smaller than the bolt elongation no prying action would take place as shown in Fig. 36(a). In the case the plate flexural STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

deformation is of similar or of greater magnitude as the bolt elongation, as shown in Fig. 36(b) and (c), prying actions N *p should be accounted for in the design. Possible bending moment diagram occurring in the plate in all three collapse mechanisms are also shown in Fig. 36. [13] For design purposes the use of a prying factor of 1.4 is conservatively recommended as suggested in [37] and [36].

5.4.2.

N des.t = [ÔN t ; ÔN w ; Ô Ô pN tb] min ≥ N *t

(60)

with the following constraint to ensure a ductile failure of the anchorage system (connection of anchor bolt to concrete): ÔN cc > ÔN tb

(61)

and complying with the anchor bolts’ embedment lengths and concrete edge distances specified in Sections 5.4.5. and 5.4.6. and where: N des.t = design capacity of the base plate connection subject to axial tension ÔN t = design tensile axial capacity of the steel base plate ÔN w = design axial capacity of the weld connecting the base plate to the column ÔN tb = design capacity of the anchor bolt group under tension Ô p = 1/1.4 = 0.72 prying reduction factor as recommended in references [36] and [37] unless noted otherwise in 5.4.3. ÔN cc = design pull--out capacity of the concrete foundation N *t = design axial tension load

Anchor bolt design

The tensile design capacity of the anchor bolt group ÔN tb is calculated in accordance with Clause 9.3.2.2 of AS4100 [11] as the sum of the design capacities of each single bolt ÔN tf. ÔN tb = n bÔN tf = n bÔA sf uf

(62)

where: Ô = 0.8 Refer to Section 14. for tabulated values of the tensile capacities of anchor bolts. In the case the base plate is designed based on Packer--Birkemoe Model the preliminary number of bolts required is obtained from equation (62) which is then refined in the section describing the steel plate

22

ÔN tb =

Design Criteria

The recommended model for axial tension is based on the following design criteria:

5.4.3.

design. Once the steel plate design is complete the capacity of the anchor bolt groups needs to be re--checked. The value of Ô p to be adopted in the Packer -- Birkemoe model is specified in equation (95). In the case the design of the base plate is carried out base on IWIMM Model (refer to Section 5.4.7.) the tensile design capacity of the anchor group should be calculated as follows: n bÔN tf 1 1 1− + f 3 f lnr 1 3 r2

(63)

where: Ô = 0.9 Ô p = 1 to be used in equation (60) as prying effects are already included in equation (63) d r 1 = 0 + 2a 1 2 d0 r2 = + a1 2 a 1 = a 2 (condition to apply equation (63)) f 3 = 1 k 3 + k 23 − 4k 1 2k 1 r k 1 = ln r 2 3



k3 = k1 + 2 d r2 = 0 + a1 2 d0 − tc r3 = 2 a 1, a 2 and d 0 are defined in Fig. 20

5.4.4.

Design of concrete pull--out capacity

The pull--out capacity of the concrete ÔN cc varies depending upon the anchor bolts layout and it can be calculated in accordance with AS 3600 as follows: ÔN cc = Ô0.33 f′ c A ps

(64)

where: Ô = 0.7 (based on Ô required for Clause 9.2.3 of AS 3600) A ps = effective projected area Equation (64) is similar to the expression provided in Clause 9.2.3 of AS 3600 to calculate the concrete capacity of a slab against punching shear, which involves a similar failure mechanism as the one of the pull--out cone. The value of β h to be calculated in Clause 9.2.3 of AS 3600 would be equal to 1 as the shape of the effective loaded area is a circle. AS 3600 recommends a strength reduction factor under shear of 0.7 (Table 2.3 of AS 3600). The capacities of a few common bolt layouts as shown in Fig. 37 are here outlined. [47]


L1

  

L2

a e = max 100, d f

s

Projected area

L1

45 o

L2

(a) Single Cone

(b) Two Intersecting Cones

L4

(c) Four Intersecting Cones

s

Figure 37 Common bolt layouts (Ref. [47]) The effective projected areas of each anchor bolt layout shown in Fig. 37 is calculated as follows: A ps.1 = effective projected area of isolated anchor bolt (no overlapping of failure cones) as shown in Fig. 37(a) = πL 21 A ps.2 = effective projected area of 2 anchor bolts with overlapping of their failure cones as shown in Fig. 37(b);



= πd 22 × 1 −



2 cos −1(s∕2L 2) + s L 22 − s 2∕4 2 360

A ps.4 = effective projected area of 4 anchor bolts with overlapping of their failure cones. In this case each failure cone overlaps with all other 3 failure cones as shown in Fig. 37(c).



= πd 24 0.75 −



2 cos −1(s∕2L 4) 360

+ s L 24 − s 2∕4 + s 2∕4 2 where the inverse cosine term is in degrees.

5.4.5.

Concrete cover requirements

The cover requirements for an anchor bolt are determined in accordance with [2] and [17] in order to prevent lateral bursting of the concrete which can occur when a bolt is located close to a concrete edge as shown in Fig. 33. The minimum cover to be provided is calculated as follows: [17][2]

23

f uf



 6 f′ c

(65)

Tabulated values of equation (65) are presented in Section 12. The following simplified expressions, which have been derived in Section 12., can be used in place of equation (65) leading to slightly more conservative side covers than those calculated with equation (65). for Grade 4.6 bolts and Grade 250 rods a e = 4 d f when f′ c = 20, 25 and 32 MPa ≥ 100 when f′ c = 20, 25 and 32 MPa for Grade 8.8 bolts a e = 6 d f when f′ c = 20 and 25 MPa = 5 d f when f′ c = 32 MPa ≥ 100 when f′ c = 20, 25 and 32 MPa The requirement of a minimum side cover of 100 mm is based on recommendations of [21], [26] and [39].

5.4.6. L4



Minimum embedment lengths

The recommended minimum embedment length Ld of an anchor bolt is determined in accordance with the design guidelines specified in [2] adjusted to suit AS 3600. ae

Ld

Lh

Edge of Concrete Foundation

Figure 38 Hook, embedment lengths and edge distances for anchor bolts (Ref. [26]) The minimum embedment length Ld for an isolated anchor bolt should be calculated as follows: (refer to Fig. 38) − d 2f + d 2f + 4γ

Ld = ≥ 100 (66) 2 where: Ô = 0.7 (based on Ô in Clause 9.2.3 of AS 3600) f ufA s γ= Ô0.33 f′ c π Even if it has been observed that for shallow anchors the angle at the bolt head formed by the concrete failure cone tends to increase from 90 degrees to 120 degrees (therefore increasing the concrete pull--out capacity) a minimum limit of 100mm is here introduced in equation (66) as cracks might be present at the concrete surface. Refer to Section 12. for the derivation of equation (66) and of the simplified expressions shown below which can be used in place of equation (66). for Grade 4.6 bolts and Grade 250 rods


L d = 9 d f when f′ c = 20, 25 and 32MPa for Grade 8.8 bolts L d = 13 d f when f′ c = 20 MPa = 12 d f when f′ c = 25 MPa = 11 d f when f′ c = 32 MPa Hooked anchor bolts, as shown in Fig. 38, need to be detailed with a minimum embedment length as specified for bolts with an anchor head of same nominal diameter (specified by equation (66) or by its alternative simplified expressions) and with a minimum hook length calculated as follows:[24][26] A sf uf Lh ≥ (67) 0.7f′ cd f where: L h = hook length of anchor bolt The anchorage length (embedment length and hook length) should be such as to prevent bond failure between the anchor bolt and concrete prior to yielding of the bolt. When possible, a more positive anchorage should be adopted at the end of the hook, for example by means of a nut.

5.4.7.

y y

H--SHAPED COLUMN -- 2 anchor bolts The yield line pattern considered by the recommended model is shown in Fig. 39 and is similar to the one considered in Murray Model modified to account for the reduction in plate capacity due to the anchor bolt holes.

d c1 2 d c1 2

b fc

Figure 39 Yield line pattern -- H--shaped column section with 2 anchor bolts The plate thickness required to resist a design axial force ÔN *t is calculated as follows: ÔN t = 0.9f yit 2i α ti ≥

(68)



N *t 0.9f yiα



y = min

Design of the Steel Base Plate

The recommended procedure to design or check the steel base plate varies depending upon the column section and number of bolts considered. Recommended models are illustrated below for the following combinations of column section and number of bolts: H--shaped column section -- 2 anchor bolts (*) H--shaped column section -- 4 anchor bolts (*) Channel -- 1 anchor bolt (*) Channel -- 2 anchor bolts (*) Hollow section (RHS, SHS, CHS) -- 2 anchor bolts (*) Hollow section (RHS, SHS) -- 4 anchor bolts (*) Hollow section (CHS) -- varying no. of anchor bolts (IWIMM Model described in the literature review) Hollow section (RHS) -- varying no. of anchor bolts (Packer--Birkemoe Model described in the literature review) The derivation of the models marked with (*) is illustrated in Section 11. It is important to note that, similarly to Murray Model, in the case of open sections the derived models to determine the capacity of the steel base plate capacity account only for the strength of plate present inside the column footprint. The reduction in plate capacity due to the bolt hole has been included in the model. The yield line patterns considered for open sections are assumed to develop inside the internal faces of the column profile.

24

s

b

d c1 , 2

(69) fc1 − d h

2

b fc1



(70)

where: ÔN t = axial tension capacity of the base plate b fc1 = width of the column flange ignoring web thickness = b fc − t w d c1 = clear depth between flanges (column depth ignoring thicknesses of flanges) t w = thickness of web d h = diameter of bolt hole 2b 2 − 2b fc1d h + 4y 2 α = fc1 4sy y and s = as defined in Fig. 39 In this model the reduction in plate capacity due to the presence of a bolt hole along the yield line perpendicular to the web has been included. Further reductions due to other yield lines intersecting bolt holes have not been considered as they are very unlikely to occur and a more detailed analysis should be carried out in such situation. The critical yield line pattern is a function of the value of y calculated from equation (70). To ensure that none of the oblique yield lines intersects the bolt hole, as assumed in the model derived, the following condition needs to be satisfied: (71)

y > l2 where: l1 =

dh 2

1 − 4sd

2

h 2

l 1l 3

l2 = s−



d2 h 4

− l 21


and the notation is defined in Fig. 40. l3

y c = mina b, y



y d = min a b,

b

fc1 − d h

2

a b = distance from bolt hole to inside face of flange s

s Web

diameter of hole = d h

l2

d 2h∕4 − l21

y

Edge of plate

The yield line patterns considered by the recommended model are shown in Figs. 41, 42, 43, 44 and 45. In the case of yield line patterns (a), (b) and (c) the derived model does not assume that the oblique lines intersect the bolt hole. This should be verified and considered in a similar manner as previously outlined in the case of H--shaped column with 2 anchor bolts (refer to equation (71) and Fig. 40). The recommended design procedure is as follows: (72)



2

y sp y y

ab

b fc

Figure 42 Yield line pattern (b) H sections

(73)

ab y

b fc1

(74)

and the value of α is calculated as follows: sp α = max(α a, α b) when y < 2 sp = α b when y < and y > a b 2 sp = max(α c, α d, α e) when y ≥ 2 where: 2b 2 − 2b fc1d h + 4y 2 α a = fc1 2sy b fc1(b fc1 − d h)(a b + y) + 2(y + a b)a by αb = 2sa by b 2fc1 − d hb fc1 + 2y 2c + s py c αc = 2sy c b fc1s − d hs + 2y 2d + s py d − d hy d αd = sy d b fc1s − 2d hs + 4a 2b + 2a bs p − 2a bd h αe = 2a bs

25

ab

y

s

N *t 0.9f yiα fc1 − d h

Figure 41 Yield line pattern (a) H sections s

H--SHAPED COLUMN -- 4 anchor bolts

b y=

d c1 2

b fc

Figure 40 Yield line layout near the bolt hole

ÔN t = 0.9f yit 2i α

d c1 2

y

l1

ti ≥



s

sp y ab b fc

Figure 43 Yield line pattern (c) H sections s y

ab sp

y

ab

b fc

Figure 44 Yield line pattern (d) H sections


s y

ti ≥

ab

(76)

d2 , (2b c1

fc1 − d h)b fc1



2b 2fc1 − b fc1d h + y 2 2sy y and s = as defined in Fig. 47 α=

CHANNEL -- 2 anchor bolts

Figure 45 Yield line pattern (e) H sections

The yield line patterns considered by the recommended model are shown in Figs. 48, 49, 50, 51 and 52. In the case of yield line patterns (a), (b) and (c) the derived model does not assume that the oblique lines intersect the bolt hole. This should be verified and considered in a similar manner as previously outlined in the case of H--shaped column with 2 anchor bolts (refer to equation (71) and Fig. 40). The recommended design procedure is as follows:

s ab sp ab

ÔN t = 0.9f yit 2i α

b fc

Figure 46 Yield line pattern (f) H sections

The yield line pattern considered by the recommended model is shown in Fig. 47 and is similar to the one considered in the case of H--shaped sections with 2 anchor bolts. The derived model does not assume that the oblique lines intersect the bolt hole. This should be verified and considered in a similar manner as previously outlined in the case of H--shaped column with 2 anchor bolts (refer to equation (71) and Fig. 40). s

y y

d c1 2 d c1 2

(78)



N *t 0.9f yiα

(79)

y = (2b fc1 − d h)b fc1

(80)

ti ≥

CHANNEL -- 1 anchor bolt

and the value of α is calculated as follows: sp α = max(α a, α b) when y < 2 sp = α b when y < and y > a b 2 sp = max(α c, α d, α e) when y ≥ 2 where: 2b 2 − b fc1d h + y 2 α a = fc1 sy b (2b − d h)(a b + y) + (y + a b)a by α b = fc1 fc1 2sa by 4b 2fc1 − 2d hb fc1 + 2y 2c + s py c 4sy c 2b fc1s − d hs + 2y 2d + s py d − d hy d αd = 2sy d αc =

b fc

Figure 47 Yield line pattern -- Channel with 1 anchor bolt The plate thickness required to resist a design axial force ÔN *t is calculated as follows: ÔN t = 0.9f yit 2i α

(77)

where: ab

b fc

26

N *t 0.9f yiα

y = min

sp y



(75)

αe =

b fc1s − d hs + 2a 2b + a bs p − a bd h 2a bs

y c = mina b, y



y d = min a b,

2b

fc1 − d h

2



s


s

s y

ab

ab

sp

sp

ab

ab

y y y

b fc

b fc

Figure 48 Yield lines (a) Channels, 2 bolts s

HOLLOW SECTION (RHS, SHS, CHS) -2 anchor bolts

ab y sp y ab b fc

Figure 49 Yield lines (b) Channels, 2 bolts s y

ab sp

y

ab

b fc

Figure 50 Yield lines (c) Channels, 2 bolts s y

ab

sp y

Figure 52 Yield lines (e) Channels, 2 bolts

The yield line patterns considered by the recommended model are shown in Figs. 53 and 54. In the case of yield line pattern (a) the derived model does not assume that the oblique lines intersect the bolt hole. This should be verified and considered in a similar manner as previously outlined in the case of H--shaped column with 2 anchor bolts (refer to equation (71) and Fig. 40). The recommended design procedure is as follows: ÔN t = 0.9f yit 2i α ti ≥

(81)



N *t 0.9f yiα

(82)

y = (2s 2 − d h)s 2

(83)

and the value of α is calculated as follows: l l α = max(α a, α b) when y ≤ i = α b when y > i 2 2 where: 2s 2 − d hs 2 + y 2 α a = 2 ys 1 li αb = 2s 3 s 3 = distance from centerline of bolt hole to yield line location specified by s 4 s 4 = cantilevered lengths a 1 or a 2 of Cantilever Model depending upon orientation of the column section s2 s1

ab y

b fc

Figure 51 Yield lines (d) Channels, 2 bolts

27

li

y


s1

s2

y

li

y

In the case of yield line pattern (a) the derived model does not assume that the oblique lines intersect the bolt hole. This should be verified and considered in a similar manner as previously outlined in the case of H--shaped column with 2 anchor bolts (refer to equation (71) and Fig. 40). The recommended design procedure is as follows: ÔN t = 0.9f yit 2i α

s2 ti ≥

s1 y

(84)



N *t 0.9f yiα

(85)

y = (2s 2 − d h)s 2

(86)

and the value of α is calculated as follows: l − sp α = max(α a, α b) when y ≤ i 2 li − sp = α b when y > 2 where: 4s 2 − 2d hs 2 + 2y 2 + s py αa = 2 2ys 1 li αb = 2s 3 s2 s1

li

y

Figure 53 Yield lines (a) Hollows, 2 bolts s4 s3 li

y s4

sp

s3

li y

li

s2 s1

s3

s4

y li

sp li

y

Figure 55 Yield lines (a) Hollows, 4 bolts Figure 54 Yield lines (b) Hollows, 2 bolts HOLLOW SECTION (RHS and SHS) -4 anchor bolts The yield line patterns considered by the recommended model are shown in Figs. 55 and 56.

28


s4

HOLLOW SECTION (RHS) -varying no. of anchor bolts (Packer--Birkemoe Model)

s3

li

s4 s3

li

RHS COLUMNS -- varying no. of bolts The model recommended here is Packer--Birkemoe Model. This model is applicable only to base plates between 12mm and 26mm. The design procedure is as follows (refer to the literature review for further details regarding the model and to Fig. 21 regarding the notation): a preliminary number of bolts required is determined from equation (62) a bolt spacing s p equal to 4--5 d f should be used (even if smaller spacing are possible) and that: (89) a 2 ≤ 1.25a 1 Calculate δ: d δ = 1 − sh

(90)

p

The designer should then select a preliminary plate thickness in the following range:

Figure 56 Yield lines (b) Hollows, 4 bolts

1KN+ δ ≤ t ≤ KN

HOLLOW SECTION (CHS) -varying no. of anchor bolts (IWIMM Model)

*

b

The recommended model for the design of base plates of CHS with a symmetric arrangement of bolts around the column profile as shown in Fig. 20 is based on IWIMM Model previously outlined in the literature review. The recommended design procedure is as follows: Ôf yi π f 3t 2i ÔN t = 2 ti ≥



2N *t Ôf yi π f 3

(87) (88)

where: Ô = 0.9 f 3 = 1 k 3 + k 23 − 4k 1 2k 1 r k 1 = ln r 2 3



k3 = k1 + 2 d r2 = 0 + a1 2 d0 − tc r3 = 2 a 1, a 2 and d 0 are defined in Fig. 20 [27] recommends to keep the value of a 1 as small as possible, i.e. between 1.5d f and 2d f (where d f is the nominal diameter of the bolts), while ensuring a minimum of 5 mm clearance between the nut face and the weld around the CHS. Assumptions adopted by this model are a continuous base plate and a weld capacity able to develop the full yield strength of the CHS.

29

i

(91)

*

b

where: K=

4a 310 3 Ôf yis p

(where f yi is in MPa)

a 3 = a 1 − d f∕2 + t c calculate α: α=



KÔN tf −1 t 2i



a 2 + d f∕2 δ(a 2 + a 1 + t c)



(92)

with the constraint of α ≥ 0 The capacity of the steel base plate is then calculated as follows: t 2i (1 + δα)n b (93) K And ÔN t calculated with equation (93) must be greater than N *t . The actual tension in the anchor bolt group, including prying effects, is determined as follows: ÔN t =



a N *tb ≈ N *t 1 + a 3 4

1 +δαδα

(94)

where: N *tb = design tension in anchor bolt group including prying effects α=





KN *t −1 1 t 2i n b δ



a 4 = min 1.25a 1, a 2 +



df 2

The anchor bolt group capacity calculated with equation (62) needs to be greater than the axial loads applied to the bolt group calculated with equation (94). This is


achieved adopting a value Ô p to be used in equation (60) equal to:





a δα Ôb = 1 + a3 4 1 + δα



−1

≤1

Design of weld at column base

The design of the weld at the base of the column is carried out in accordance with Clause 9.7.3.10 of AS 4100. The weld is designed as a fillet weld and its design capacity ÔN w is calculated as follows: ÔN w = Ôv wL w = Ô0.6f uw t t k rL w

(96)

where: Ô = 0.8 for all SP welds except longitudinal fillet welds on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100) 0.7 for all longitudinal SP fillet on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100) 0.6 for all GP welds (Table 3.4 of AS 4100) k r = 1 (reduction factor to account for length of welded lap connection) Refer to Section 13. for tabulated values of Ôv w. The fillet weld is recommended to be placed all around the column section profile.

where: Ô = 0.8 μ = coefficient of friction ÔV f = shear capacity of the base plate transferred by friction Coefficients of friction μ available in literature are shown in Fig. 57 and are specified as follows: [2][21][22] 0.9 -- concrete or grout against as--rolled steel when the contact plane is the full base plate thickness below the concrete surface (i.e. recessed); 0.7 -- for concrete or grout placed against the as--rolled steel surface with the contact plane coincidental with the concrete surface; 0.55 -- for grouted conditions with the contact plane between the grout and the as--rolled steel exterior to the concrete surface (normal condition). μ = 0.9

6. SHEAR 6.1.

6.2.

TRANSFER OF SHEAR BY FRICTION OR BY RECESSING THE BASE PLATE INTO THE CONCRETE -LITERATURE REVIEW

There is general agreement regarding the determination of the shear capacity of a base plate which can be

30

μ = 0.7

INTRODUCTION

The shear action may be assumed to be transferred from the column to the concrete base either: 1. by friction between between base plate and concrete/grout base or by recessing the base plate into the concrete footing; 2. by a shear key (or shear lug); 3. by the anchor bolts; 4. by a combination of two or more of the above. Available design information regarding the transfer of shear by each of these means with and without axial loading is now outlined. It is interesting to note how there are still very different opinions regarding the ability of anchor bolts to transfer shear actions. For clarity, the literature review regarding the behaviour of anchor bolts is further divided into the case of anchor bolts subject to shear only or to shear and axial compression and the case of anchor bolts subject to shear and axial tension.

(97)

ÔV f = ÔμN *c

(95)

The evaluation of the capacity of an existing base plate is carried out following the design procedure previously outlined. Instead of the preliminary values the actual number of bolts and plate thickness are utilised.

5.4.8.

transferred by means of friction when the column is subject to axial compression loading. The shear capacity is calculated as follows:

μ = 0.55

Figure 57 Coefficients of Friction (Ref. [26]) 6.3.

TRANSFER OF SHEAR BY A SHEAR KEY-- LITERATURE REVIEW

Available design guidelines agree that in the presence of a shear key, the shear force is transferred through the shear key acting as a cantilever and bearing against the concrete surface as shown in Fig. 58 while no bearing is assumed to occur against the grout. The bearing capacity of the concrete is calculated in accordance with AS 3600 [10]. Uniform bearing pressure is assumed to occur at the interface between the shear key and the concrete equal to the maximum bearing capacity of the concrete. The shear key is designed as a cantilever to carry the assumed bearing pressure. [26] The required area of the shear key is determined based on the bearing concrete strength 0.85Ôf′ c as shown in Fig. 58: A sk =

V *s 0.85Ô cf′ c

(98)

where: Ô = 0.8 A sk = area of the shear key STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

V *s = design shear force to be transferred by means of the shear key The actual length of the shear key L s is then determined based on the available plate depth in contact with the concrete, which, referring to Fig. 58, is equal to (b s − t g). The design moment per unit width of plate m *sk carried by the shear key can then be calculated as follows: m *sk =

V *s b s + t g Ls 2

(99)

where: m *sk = design moment to be carried to the shear key L s = length of shear key b s = depth of shear key t g = grout thickness Equating the design moment to the plastic nominal section moment capacity of the shear key the following is obtained (per unit width of plate): m *sk =

0.9f ys t 2s V *s b s + t g = = Ôm sk Ls 2 4

(100)

where: m sk = nominal section moment capacity per unit width of shear key f ys = yield stress of shear key used in design t s = thickness of shear key from which the minimum thickness for the shear key t sk can be calculated in accordance with AS4100 as follows: ts =



4m *sk = 0.9f ys



V *s b s + t g 2 L s 0.9f ys

(101)

or equivalently the shear capacity of a shear key is calculated as: ÔV s =

0.9f ys t 2sL s bs + tg 2

(102)

where: ÔV s = design shear capacity of the shear key

to resist the part of the design shear force that cannot be resisted by friction. For shear keys located near a free concrete edge it should be verified that the concrete is able to carry the applied shear action. The possible failure surface is the one which radiates at 45 degrees from the shear key’s edges towards the concrete edge. The concrete capacity should be determined by multiplying the effective concrete stress area, determined as the projected area of the failure surface on the concrete edge ignoring the shear key area, by the average concrete tensile stress of Ô0.33 f′ c (where f′ c is in MPa) with Ô is equal to 0.85. [2] The weld of the shear key shall be designed to carry both design shear and moment actions acting on the shear key. It is interesting to note that the shear key can be welded to the underside of the base plate at any angle even if it is common to choose directions parallel to one or both of the principal axes of the column as these are usually the axes along which the shear needs to be transferred. Reference [26] extends this design procedure for shear keys in two orthogonal directions applying the same design procedure in both orthogonal directions.

6.4.

TRANSFER OF SHEAR BY THE ANCHOR BOLTS -- LITERATURE REVIEW

6.4.1.

Shear only or Shear and Axial Compression

An anchor bolt located away from a concrete edge and with sufficient embedment length would typically transfer the shear through bearing at the surface of the concrete and testing has shown that this transfer mode could cause a concrete wedge to form as shown in Fig. 59. It has been observed that the depth of the concrete wedge can be approximated to be one quarter of the anchor bolt diameter. In the presence of a base plate the translation of the concrete wedge is prevented by a clamping force provided by the base plate and anchor bolts. While the anchor’s behaviour remains in the elastic range the clamping force applied by the anchor bolt and base plate is proportional to the shear force. Applied Shear Concrete Wedge

d f∕4

ts V *c tg

Shear Key

bs

0.85f′ c

Figure 58 Forces acting on Shear Keys (Ref. [26]) In the presence of combined shear and axial compression actions, the shear key is normally assumed

31

df

Figure 59 Concrete wedge failure mode under anchor bolt shear force (Ref. [31]) Locating an anchor bolt near the concrete free edge could lead to another failure mode to occur as shown in Fig. 60. The concrete failure surface is determined by radiating at 45 degrees from the anchor bolt at the STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

concrete surface towards the free edge. The concrete capacity is calculated by multiplying the projected area of the failure surface at the concrete edge by the concrete average tensile strength of Ô0.33 f′ c. Applied Shear

Failure Surface

Side

Front

Figure 60 Concrete failure surface under bolt shear force near a concrete edge (Ref. [31]) The minimum side cover required to ensure a ductile failure requires the concrete wedge capacity to carry a shear load equal to the nominal shear capacity of the anchor bolt. The concrete capacity of the wedge cone can be calculated as follows: ÔV u.c = Ô0.33 f′ c

πa 2e 2

(103)

where: Ô = 0.65 in [3] and 0.85 in [17] ÔV u.c = concrete capacity against wedge cone failure Experimental results have shown that equation (103) provides a good estimate of the concrete wedge capacity using Ô equal to 0.65. [44][45] Based on [2], [3] and [17] the nominal shear capacity of the anchor bolt is calculated assuming that the shear is transferred by friction between the steel and the concrete with a friction coefficient of 0.7: V u.b = 0.7

πd 2f f 4 uf

(104)

where: V u.b = nominal shear capacity of an anchor bolt assumed to be transferred by friction between anchor and concrete with a friction coefficient of 0.7 The minimum side cover a e to be adopted for the anchor bolt to avoid the concrete wedge failure can be determined ensuring that the concrete capacity against wedge failure ÔV u.c is able to carry the shear capacity of the bolt transferred by friction V u.b and equating equation (103) to equation (104): [2] πa 2e ÔV u.c = Ô0.33 f′ c 2 = 0.7

πd 2f f = V u.b 4 uf

and solving equation (105) for a e:

32

(105)

ae ≥ df



f uf

(106)

Ô0.94 f′ c

where: Ô = 0.65 in [3] and 0.85 in [17] Based on the guidelines provided in reference [3], simplified design guidelines of the minimum edge distances calculated with equation (106) using Ô equal to 0.65 are presented in reference [39] which are as follows: for Grade 250 bars and Grade 4.6 bolts: a e ≥ 12d f minimum bolt spacing ≥ 16d f for Grade 8.8 bolts: a e ≥ 17d f minimum bolt spacing ≥ 24d f These minimum bolt spacings intend to avoid overlapping of anchors’ concrete failure cones. These have also been recommended in reference [26]. For completeness minimum edge distances have been derived in Section 12. based on equation (106) with Ô equal to 0.65 and 0.85. Also simplified expressions have been derived as shown in Tables 4 and 5. Table 4 Grade 4.6 bolts and 250 Grade rods Ô 0.65

f′ c

ae

20

13 df

0.65

25

12 df

0.65

32

11 df

0.85

20

11 df

0.85

25

10 df

0.85

32

10 df

Table 5 Grade 8.8 bolts Ô 0.65

f′ c

ae

20

18 df

0.65

25

17 df

0.65

32

16 df

0.85

20

16 df

0.85

25

15 df

0.85

32

14 df

References [26] and [47] recommend edge distances based on Ô values equal to 0.85. In the case the side cover is less than a e (calculated with equation (106)) caution should be placed in the design and positioning of the reinforcement. The shear capacity of an anchor bolt located at a distance less than a e∕3 from a concrete edge should be ignored. Adopting a similar reinforcement layout as suggested in Fig. 35 to resist direct tensile loading it has been observed by limited testing that concrete failure would occur when STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

anchor bolts are located with a side cover less than 2a e∕3. A possible reinforcement layout to be utilised in the case the side cover is in between a e∕3 and 2a e∕3 is shown in Fig. 61. Allowance for the full development of the reinforcement should be allowed for in accordance with AS 3600 regardless of the reinforcement layout adopted and in the case such allowance is not feasible the shear capacity of the anchor bolt with edge distance problems should be disregarded. [2][17] Experimental studies have shown that possible failure modes which can occur by transferring shear actions by means of anchor bolts are concrete failure with and without wedge cone, concrete failure with pull--out cone and shear failure of the anchor bolt. [45] Shear force

* Potential failure zone

*

* -- Development length from AS3600

Figure 61 Reinforcement for Shear Near an Edge of Concrete Foundation (Ref. [2]) [45] notes that by ensuring sufficient embedment length of the anchor bolt no concrete pull--out can occur. The concrete edge cone failure can be prevented if either an edge distance a e as determined in equation (106) or adequate reinforcement are provided. From test data, [45] concludes that among available guidelines the one of [3], outlined in equation (106), is the most appropriate. [45] shows that equation (106) is not applicable to anchor bolt groups as it can lead to unsafe design particularly for large edge distances and that the nominal concrete capacity is related to both edge distance and bolt spacing. [45] provides no alternative design guidelines but notes that from experimental results the nominal capacity of a two bolt group may only be 60% more than that of a single bolt for the same edge distance.[45] No guidance is currently available for calculating the nominal shear capacity of anchor bolt groups. It is interesting to note that for the case where a grout pad exists between the base plate and the concrete, the grout pad allows bending deformation of the anchor bolt to occur under an applied shear force. The lateral

33

deformation of the bolt leads to tensile stress in the bolt but this is generally insufficient to cause pullout. [38] Some authors do not recommend that shear be resisted by the anchor bolts. Ricker in [38] specifically notes that anchor bolts should not be used to resist shear forces in a column base. In his opinion bolts have a low bending resistance and that if a plate eases sideways to bear against a bolt, bending is induced in the bolt which acts as a cantilever with a lever arm equal to the grout thickness plus an additional distance should the concrete foundation crush locally. Fischer in [24] notes that in his opinion no more than two anchor bolts for each anchor group would transfer shear. He explains that under normal loading condition only one bolt would be carrying shear in bearing as shown in Fig. 62. The column would then rotate subject to a shear action till a second anchor would go into bearing. Due to the oversize holes specified in base plates it is not possible to ensure that the bolts of the bolt group would deform sufficiently to allow all bolts to go into bearing. [24] Ref. [31] considers that, in the case of base plates, there is not enough data available to precisely quantify the shear strength of an individual anchor bolt, much less a group of anchor bolts.

Figure 62 Transfer of shear by bearing of anchor bolts DeWolf in [22] recommends to avoid the use of anchor bolts to resist shear and suggests that the transfer of shear through anchor bolts takes place by either shear friction or bearing. In the former instance the transfer of shear occurs once a clamping force is developed to the base plate. [22] Even if the anchor bolts are not tightened properly the clamping force can still develop as a consequence of a wedge concrete failure which would tend to lift the base plate up and therefore tensioning the anchor bolts. [31] No specific guidelines are available to evaluate the contribution of the clamping force to the shear resistance of the bolt and in practice this clamping force may not necessary be available. The other transfer mode of anchor bolts described by DeWolf is by bearing between the anchor and the bolt hole, but he regards this very unlikely to occur in practice in more than one or two anchors as the bolt holes of base plates are usually oversized holes. [22] He also notes that a more reliable method of shear transfer through the anchor bolts can be achieved by welding the nuts to the base plate or by providing special washers


with normal size holes (bolt dia + 2 mm) which fit over the oversize holes and are welded to the base plate. [21] Projected area of wedge cone

Anchor bolt Top of concrete block

ae

ae

α = 45 o Anchor bolt Overlapped area

ae

α

45 o

ae

Anchor bolts

Figure 63 Concrete edge failure cones (Ref. [45]) Ref. [34] notes that it is common and successful industrial practice to use anchor bolts of pinned--base portals to resist the shear forces while recommending the following design guidelines: if shear force is less than 20% of the axial load, then no special provisions are required; for higher levels of shear force, it suggests that great attention be paid to ensuring good grouting under the base plate and around the anchor bolts using a mix of minimum shrinkage; excessive clearance between the anchor bolts and the holes in the base plate should be avoided; to avoid possible horizontal deformation of the column the shear actions should be transferred either by recessing the base plate into concrete, or by means of a shear key or by tying the steel columns to share the load among adjacent columns. 6.4.2. Shear and Axial Tension The ability of anchor bolts to transfer shear actions was considered in the previous paragraph. Here only available models to describe the interaction of shear and tension are considered. [39] notes that most references suggest the use of a parabolic interaction equation, similar to the one adopted for conventional bolts as also specified in AS

34

4100 [11], for the design of the anchor bolts. Shipp and Haninger suggest in [39] that the total area of anchor bolt required should be the sum of that required to resist tension and that required to resist shear. They argue that the shear force causes a bearing failure near the concrete surface and translates the shear load on the anchor bolt into an effective tension load by friction, so that the bolt must have enough tension capacity to resist both effects. [30] notes that for an anchor bolt subject to both shear force and axial tension, design difficulties exist because the interaction of shear and tension is not understood and generally a straight line interaction relationship is assumed, which requires the total steel bolt area be obtained by adding the area required for shear force and the area required for tension. [30] notes that this approach is conservative but is warranted since test data concerning combined shear and tension are lacking for most anchors. Reference [20] suggests an elliptical interaction relationship between tension and shear for the design of anchor bolts while considering the linear interaction relationship to be conservative. References [2] and [17] recommend, in the case of anchor bolts subject to combined shear and tension, to adopt the design recommendations regarding minimum embedment length and edge distances provided in the case of anchor bolts subject to tension and shear separately.

6.5. 6.5.1.

RECOMMENDED MODEL Introduction

The recommended design model allows shear action to be transferred by friction between the base plate and the concrete/grout base, by recessing the base plate into the concrete footing, by a shear key or by a combination of the above. It is in the authors’ opinion that due to the uncertainty regarding the ability of anchor bolts to transfer shear it is left up to designer to decide whether or not to design the anchor bolts to carry shear actions.

6.5.2.

Design criteria

The recommended model for the design of base plate subject to shear or combined shear and axial actions is base on the following design criteria: V des = ÔV f + ÔV s, ÔV w min ≥ V *

(107)

N des.c ≥ N *c N des.t ≥ N *t v des = Ôv w ≥ v *w where: V des = design shear capacity of the base plate connection ÔV f = design shear capacity of the base plate transferred by means of friction ÔV s = design shear capacity of the shear key


ÔV w = design shear capacity of the weld connecting the base plate to the column N des.t = design capacity of the base plate connection subject to axial tension as determined in Section 5.4. N des.c = design capacity of the base plate connection subject to axial compression as determined in Section 4.3. N *t = design axial tension load N *c = design axial compression load v des = Ôv w = design capacity of the weld connecting the base plate to the column per unit length of weld v *w = design load per unit length acting on the weld connecting the base plate to the column. Its direction depends upon the combined shear and axial loading The additional check on the weld capacity is required as the critical action acting on the weld (between column and base plate) is caused by a combination of shear and axial loading.

6.5.3.

Design of shear transfer by friction and by recessing the base plate in the concrete

The design shear capacity of the base plate transferred by means of friction and by recessing the base plate into the concrete footing is calculated as follows: (108)

ÔV f = ÔμN *c

where: Ô = 0.8 μ = coefficient of friction = 0.9 -- concrete or grout against as--rolled steel when the contact plane is the full base plate thickness below the concrete surface (i.e. recessed) = 0.7 -- for concrete or grout placed against the as--rolled steel surface with the contact plane coincidental with the concrete surface = 0.55 -- for grouted conditions with the contact plane between the grout and the as--rolled steel exterior to the concrete surface (normal condition)

6.5.4.

Design of the column weld

The design action applied to the weld between the column and the base plate is calculated as follows: v *w = v *h + v *v 2

2

(109)

where: v *h and v *v = components of the loading carried by the weld between column and base plate in one horizontal direction in the plane of the base plate and in the vertical direction respectively per unit length * v *h = V Lw 35

N *c if the column end is not prepared for full Lw contact =0 if the column end is prepared for full contact (under axial compression only) The fillet weld capacity between the column and the base plate Ôv w is designed in accordance with Clause 9.7.3.10 of AS 4100 [11] as follows: Ôv w = Ô0.6f uwt tk r (110) v *v =

where: Ô = 0.8 for all SP welds except longitudinal fillet welds on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100) 0.7 for all longitudinal SP fillet on RHS/SHS with t < 3 mm (Table 3.4 of AS 4100) 0.6 for all GP welds (Table 3.4 of AS 4100) Refer to Section 13. for tabulated values of the fillet weld capacity Ôv w.

6.5.5.

Design of shear transfer by a shear key

The shear capacity of a shear key can be calculated once the bearing and pull--out capacity of the concrete, the shear capacity of the shear key due to its nominal section moment capacity and the weld capacity between the shear key and the base plate are determined as shown below. ÔV s = ÔV s.c; ÔV s.cc; ÔV s.b; ÔV s.w min ≥ V * (111) where: ÔV s = design shear capacity of the shear key ÔV s.c = concrete bearing capacity of the shear key ÔV s.cc = pull--out capacity of the concrete ÔV s.b = shear capacity of the shear key based on its section moment capacity ÔV s.w = shear capacity of the weld between the shear key and the base plate The concrete bearing capacity of the shear key ÔV s.c is calculated as follows: ÔV s.c = Ô0.85f c′L s(b s − t g)

(112)

where: Ô = 0.6 L s and b s = length and depth of the shear key as shown in Fig. 64 tg bs

Shear Key ts Ls

Figure 64 Shear Key Details (Ref. [26]) In the case the shear key is located near a concrete edge the capacity of the concrete could be reduced by the


sp

formation of a failure surface radiating at 45 degrees from the shear key’s edges towards the concrete edge. The concrete capacity calculated over the projected area of such failure surface ignoring the shear key area is determined as follows: ÔV s.cc = Ô0.33 f c′ A psk ≤ ÔV s.c

(113)

where: Ô = 0.7 (based on as Ô required for Clause 9.2.3 of AS3600) A psk = projected area over the concrete edge ignoring the shear key area The shear capacity of the shear key based on its nominal section moment capacity ÔV s.b is calculated as follows: ÔV s.b =

0.9f ys t 2sL s bs + tg 2



Ôv w2L s

bs+ts 1+ t s

7. BASE PLATE AND ANCHOR BOLTS DETAILING Typical base plate layouts considered in this paper are shown in Figs. 65, 66, 67 and 68. Typical anchor bolts used in base plate applications are cast--in anchors of category 4.6/S and of diameter either M16, M20, M24 or M30. Masonry anchors of diameter M16, M20, M24 may also be used.

Component to suit Grout pad Typical Typical

Figure 65 2--bolt base plate to UB /UC column (Ref. [26])

36

sp

Figure 67 2--bolt base plate to channel column (Ref. [26])

(115)

2

where: Ôv w = design capacity of the fillet weld per unit length (as calculated in equation (110) or as tabulated in Section 13.)

sg

Figure 66 4--bolt base plate to UB/UC column (Ref. [26])

(114)

The capacity of the fillet weld connecting the shear key to the base plate ÔV s.w calculated in the direction perpendicular to the shear key is determined as follows (assuming the shear key is welded all around): ÔV s.w =

sg

Legend: Anchor Bolt Location Hole to allow grout egress

Figure 68 2--bolt base plate to hollow columns (Ref. [26]) Preferred anchor bolt gauge (sg) and pitch (sp) are given in Reference [12]. The ”weld all round” philosophy sometimes adopted in the weld design of base plates can lead to over--welding and can become very expensive. The details shown in Figs. 65, 66, 67 and 68 can, if designed for light loadings, tend to the other extreme and some fabricators may prefer to increase the amount of welding above that shown on the design drawings in order to prevent damage during handling and shipping. There is usually a compromise possible between these two extremes. Another design consideration is the likelihood of a nominally pinned base being subjected to some bending moment in a real situation. [26] Prior to erecting the column/base plate assembly, the level of the base plate area should be surveyed and shims placed to indicate the correct level of the underside of the base plate as shown in Fig. 69. For heavier column / base plate assemblies, levelling--nut arrangements may be used in order to allow accurate levelling of the base plate as outlined in [7] and [38]. Hole sizes in base plates


may be up to 6mm larger than the anchor bolt diameter in accordance with Clause 14.3.5.2 of AS 4100 [11].

Shims

Level of U/S Baseplate Concrete surface

Tack weld 10mm reinforcing bars to form cage -- no tacks on HS bolts.

Figure 69 Use of shims for levelling purposes (Ref. [26]) Holes require a special plate washer of 4 mm minimum thickness under the nut if the bolt hole is more than 3 mm larger than the anchor bolt diameter. Base plates should be provided with at least one grout inspection hole through which the grout will rise indicating a satisfactory grouting operation. Anchor bolts are usually galvanized, even for an interior application, in order to avoid corrosion during the construction period where the steel columns may stand for some time in the open air. The size and location of any permanent steel shims under the base plate should be shown on the drawings. Temporary packers which are used for erection purposes until the underside of the base plate is grouted or concreted should be left to the erector to detail. The minimum space between the underside of the base plate and the concrete foundation should be: 25 mm for grouting; 50 mm for mortar bedding; 75 mm for concrete bedding. Tolerances on anchor bolt positions and level of base plate should conform to the provisions of Clause 5.12 of AS 4100.[11] [24] notes that possible design and detailing problems for base plates include: inadequate development of the anchor bolts for tension and of concrete reinforcing steel; improper selection of anchor bolt material; inadequate base plate thickness; poor placement of anchor bolts; shear and fatigue loading on anchor bolts. Based on a survey carried out in the UK [29] notes that poor fit of base plates onto holding down bolts is among one of the four most commonly reported problems of lack of fit on site. To ensure that the bolt centres match the nominated centres and the hole centres drilled in the base plate, the bolts are often caged into a group as shown in Fig. 70. Also useful is the provision of cored holes usually formed by using polystyrene which allow the adjustment of anchor bolt positions once the concrete is cast in order to exactly match the hole centres in the base plate as already shown in Fig. 32. Anchor bolt centres must comply with the tolerances set out in Clause 15.3.1 of AS 4100 as shown in see Fig. 71.

37

Figure 70 Locating Holding Down Bolts with a Cage (Ref. [26]) 1

2

3

Specified dimension (+/-- 6 in every 30m but not greater than +/-- 25 overall) Max deviation +/-- 6

C/L Anchor bolts Max deviation +/-- 6 Max deviation +/-- 6 C/L Anchor bolts +/-- 3

C/L Grid

Detail of off--centre location of anchor bolts

Unless otherwise specified, dimensions are in millimetres

+/-- 3 C/L Grid

4

Main column C/L grid

Max deviation +/-- 6 if column offset from main column line.

Figure 71 Tolerances in Anchor Bolt Location after AS 4100 (Ref. [26]) [19] and [38] present a discussion of a number of practical aspects of the use of anchor bolts and should be referred to if problems arise on site. [19] deals with general aspects regarding design, installation, anchorage, corrosion of anchor bolts, bedding and grouting as well as the responsibilities of all parties in


the construction process but no firm recommendations are made on design however.

8. ACKNOWLEDGEMENTS This paper started from the very significant work carried out by Tim Hogan and Ian Thomas who collated the majority of the research results on steel connections from around the world in Ref [26]. Valuable input and support for this current work has come from OneSteel -- in particular Anthony Ng, Gary Yum and Nick van der Kreek. The ASI State Managers -- Leigh Wilson, Rupert Grayston, John Gardner and Scott Munter have all contributed industry insights. Several overseas researchers, notably Jeffery Packer and John DeWolf, have contributed significantly in this area and their work and comments are acknowledged.

[14] [15] [16]

[17]

[18] [19]

9. REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8] [9] [10] [11] [12]

[13]

38

Ahmed, S. and Kreps, R.R., “Inconsistencies in Column base Plate design in the New AISC ASD Manual”, Engineering Journal, American Institute of Steel Construction, Vol. 27, No. 3, 1990, pp 106 -- 107. American Concrete Institute, ”Code Requirements for Nuclear Safety Related Structures”, ACI 349 -- 90, Manual of Concrete Practice (1994). American Concrete Institute, ”Code Requirements for Nuclear Safety Related Structures”, ACI 349 -- 1976, Manual of Concrete Practice. American Institute of Steel Construction, “Detailing for Steel Construction”, Second Edition, 2002. American Institute of Steel Construction, “Manual of Steel Construction -- Load and Resistance Factor Design”, Third Edition, 2001. American Institute of Steel Construction, “Manual of Steel Construction -- Volume II Connections”, Ninth Ed./First Edition, 1992. American Institute of Steel Construction, “Manual of Steel Construction -- Load and Resistance Factor Design”, First Edition, 1986. AS/NZ 1170.0:2002 -- “Structural design actions -- Part 0: General principles”, 2002 AS 1275 -- ”Metric Screw Threads for Fasteners”, 1985. AS 3600 -- ”Concrete Structures”, 2001. AS 4100 -- ”Steel Structures ”, 1998. Australian Institute of Steel Construction, ”Standardized Structural Connections”, Third Edition, 1985. Ballio, G. and Mazzolani, F.M., “Theory and Design of Steel Structures”, Chapman and Hall, 1983.

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

Bangash, M.Y.H., “Structural detailing in Steel”, Thomas Telford, 2000 Bickford, J.H. and Nassar, S., Handbook of Bolts and Bolted joints”, Marcel Dekker, 1998 Blodgett, O., Design of Welded Structures”, The James F Lincoln Arc Welding Foundation, Fifth Printing, 1972, Section 3.3. Cannon, R.W., Godfrey, D.A. and Moreadith, F.L., ”Guide to the Design of Anchor Bolts and Other Steel Embedments”, Concrete International, July 1981, pp 28 -- 41. Chen, W.F., “Handbook of Structural Engineering”, CRC Press, 1997 Concrete Society/British Constructional Steelwork Association/Constructional Steel Research and Development Organisation, ”Holding Down Systems for Steel Stanchions”, 1980. Cook, R. and Klingner, R., “Behaviour of Ductile Multiple--Anchor Steel--to Concrete Connections with Surface--Mounted Baseplates”, from “Anchors in Concrete -Design and Behavior” edited by Senkiw, G.A. and Lancelot III, H.B., American Concrete Institute, 1991 DeWolf, J.T, ”Column Base Plates”, American Institute of Steel Construction, Design Guide Series No. 1, 1990. (Publication also contains Refs. [38] and [42]) DeWolf, J.T, ”Column Anchorage Design”, American Institute of Steel Construction, National Eng Conf., New Orleans, Proceedings, Paper 15, April/May 1987. Eurocode 3: Design of steel structures DD ENV 1993--1--1 Part 1.1 General rules and rules for buildings, 1992 Fischer, J.M., “Structural details in Industrial buildings”, Engineering Journal, American Institute of Steel Construction, Vol. 18, No. 3, 1981, pp 83--89. Fling, R.S., ”Design of Steel Bearing Plates”, Engineering Journal, American Institute of Steel Construction, Vol. 7 No. 2, April 1970, pp 37 -- 40. Hogan, T.J. and Thomas, I.R., “Design of structural connections”, Fourth Edition, Australian Institute of Steel Construction, 1994. Igarashi, S., Wakiyama, K., Inove, R., Matsumoto, T. and Murase, Y., “Limit Design of high strength Bolted Tube Flange joint -Parts 1 -- 2”, Journal of Structural and Construction Engineering Transactions of AIJ, Department of Architecture reports, Osaka University, Japan, 1985. Jaspart, J.P. and Vandegans, D., “Application of the component method to column bases”, Proceedings of the International Conference on STEEL CONSTRUCTION VOLUME 36 NUMBER 2 SEPT 2002

[29] [30]

[31]

[32]

[33]

[34]

[35] [36]

[37]

[38]

[39]

[40]

[41]

[42]

39

Advances in Steel Structures, Hong Kong, Vol.1, 1996, pp 139--144. Mann, A.P. and Morris, L.J., “Lack of fit in steel structures”, CIRIA Report 87, 1981 Marsh M.L. and Burdette, E.G., ”Multiple Bolt Anchorages: Method for Determining the Effective Projected Area of Overlapping Stress Cones”, Engineering Journal, American Institute of Steel Construction, Vol. 22 No. 1, 1985, pp 29 -- 32. Marsh, M.L. and Burdette, E.G., ”Anchorage of Steel Building Components to Concrete”, Engineering Journal, American Institute of Steel Construction, Vol. 22 No. 1, 1985, pp 33 -- 39. Murray, TM., Design of Lightly Loaded Steel Column Base Plates”, Engineering Journal, American Institute of Steel Construction, Vol. 20 No. 4, 1983, pp 143 -- 152. National Institute of Standards and Technology, “Post--Installed Anchors -- A Literature Review”, NISTIR 6096, 1998. Owens, G.W. and Cheal, B.D., ”Structural Steelwork Connections”, Butterworths, London, 1989. Park, R. and Gamble, W.L., “Reinforced Concrete Slabs”, Wiley, 1980. Parker, J.A. and Henderson, J.E., “Hollow structural section connections and trusses -- A design guide”, Second Edition, Canadian Institute of Steel Construction, 1997. Parker, J.A., “Design with structural steel hollow sections -- Australian Institute of Steel Construction Seminar”, Australian Institute of Steel Construction, March 1996. Ricker, D.T, ”Some Practical Aspects of Column Base Selection”, Engineering Journal, American Institute of Steel Construction, Vol. 26 No. 3, 1989, pp 81 -- 89. Shipp, J.G. and Haninger, E.R., ”Design of Headed Anchor Bolts”, Engineering Journal, American Institute of Steel Construction, Vol. 20 No. 2, 1983, pp 58 -- 69. Stockwell, F.W., ”Preliminary Base Plate Selection”, Engineering Journal, American Institute of Steel Construction, Vol. 12 No. 3, 1975, pp 92 -- 93. Stockwell, F.W., ”Base Plate Design”, American Institute of Steel Construction, National Eng Conf, Proceedings, Paper 49, April/May 1987. Thornton W.A., ”Design of Small base Plates for Wide Flange Columns”, Engineering Journal, American Institute of Steel Construction, Vol. 27, No. 3, 1990, pp 108--110.

[43]

[44]

[45]

[46]

[47]

Thornton W.A., ”Design of Base Plates for Wide Flange Columns -- A Concatenation Method”, Engineering Journal, American Institute of Steel Construction, Vol. 27, No. 4, 1990, pp 173--174. Ueda, T, Kitipornchai, S. and Ling, K., ”Experimental Investigation of Anchor Bolts Under Shear”, Journal of Structural Engineering, 1990 Ueda, T, Kitipornchai, S. and Ling, K., ”An Experimental Investigation of Anchor Bolts Under Shear”, University of Queensland, Dept of Civil Eng., Research Report No. CE93, Oct. 1988. Wood, R.H. and Jones, L.L., “Yield--line analysis of slabs”, Thames and hudson, Chatto & Windus, London, 1967. Woolcock, S.T, Kitipornchai, S. and Bradford, M.A., ”Limit State Design of Portal Frame Buildings”, Second Edition, Australian Institute of Steel Construction, 1993.


10. APPENDIX A -- Derivation of Design and Check Expressions for Steel Base Plates Subject to Axial Compression The design model for base plates subject to axial compression recommended in this paper is a modified version of Thornton Model presented in [43] which is suitable for H--shaped columns only. Its derivation has also been extended here for channels and hollow sections. The recommended model concatenates the Cantilever, Fling and Murray--Stockwell Models as follows: ti ≥ am



2N *c 0.9f yi d i b i

a m = max(a 1, a 2, λa 4) For clarity the model which describes the design of base plates subject to uniform pressure using yield line theory is referred to throughout this section as Yield Line Model. In the case of H--shaped sections Fling Model and the Yield Line Model coincide. The assumed yield line patterns are based on the external dimensions of the column profile. Values of a 1 and a 2 are available in [21], [26] and [36] for H--shaped columns, channels and hollow sections while values of λ and a 4 are available in [5] and [43] for only H--shaped sections. In the recommended model presented here the values of λ and a 4 have been re--derived and modified for H--shaped sections and have been derived for channels and hollow sections. The derivation of such values is outlined below based on a procedure similar to the one utilised by Thornton in [43]. The values of λ and a 4 allow the inclusion in the recommended model of the results obtained with Murray--Stockwell Model and with the Yield Line Model respectively. It is important to note that, similarly to Thornton Model, the recommended model always adopts the thinnest plate determined using Murray--Stockwell Model and the Yield Line Model. In the following derivation the values of a 4 are firstly determined to include the Yield Line Model and then the value of λ to include Murray--Stockwell Model is determined. A.1

DERIVATION FOR DESIGN PURPOSES -- H--SHAPED SECTIONS

A.1.1

DETERMINATION OF a 4 (Yield Line Model -- Fling Model)

The base plate is designed assuming a yield line pattern as shown in Fig. 72. The present derivation is suitable for H--shaped sections for which b fc∕2 is less than d c as a different yield line pattern would otherwise occur.

d1


b es

θ

Figure 72 Yield line pattern for H--shaped sections The base plate is considered to be simply supported along the flanges, fixed along the web and free along the edge opposite to the web. Solutions from yield line theory are available for this kind of support conditions carrying a uniformly distributed load f *p and based on the results presented in [35]:



24Ôm p 1 + f *p =



b 2fc

4+48η 2−2 4η 2

4+48η 2−2 3− 4η 2





(116)

where: η = d c∕b fc In this case the uniform load f *p is calculated as follows: f *p =

N *c d ib i

The required design plastic moment Ôm p to support a uniform pressure of f *p is obtained by re--arranging equation (116) as follows: Ôm p = f *p

b 2fc 6η 2 − 1 + 12η 2 + 1 24 2η 2 + 1 + 12η 2 − 1

= 1 f *pb 2fcα 2 8

(117)

where: 6η 2 − 1 + 12η 2 + 1 α2 = 1 3 2η 2 + 1 + 12η 2 − 1 The value of α 2 introduced in equation (117) is approximated by the following expression with an error of --0% (unconservative) and +17.7% (conservative) for values of η (which is equal to d c∕b fc ) between 3/4 and 3: α = 1 η 2

(118)

The required plate thickness to support f *p can be determined by equating the nominal section moment capacity of the plate Ôm s (per unit width) to the required design plastic capacity (per unit width) as follows:

40


Ôm s =

0.9f yit 2i ≥ 1 f *pb 2fcα 2 = Ôm p 8 4

(119)

and re--arranging equation (119) in terms of the required plate thickness yields: t i = 1 d cb fc 4 = a4



Substituting equations (122) and (123) into equation (124) and solving for a 3 the following expression for a 3 is obtained: a3 =



2f *p 0.9f yi

=

2N *c 0.9f yid ib i

(120)

A.1.2

X=

DETERMINATION OF λ (Murray--Stockwell Model)

ti = a3



2N *c 0.9f yiA H

(121)

It is interesting to note how, in the formulation presented in [5], [42] and [43], the load adopted in equation (121) would have been equal to N *0 instead of N *c, where N *0 is the portion of full column load N *c acting over the column footprint under the assumption of uniform bearing pressure, while in the derivation presented the full column load N *c is assumed to be applied on the H--shaped area A H. Referring to Fig. 11 the H--shaped bearing area A H can be expressed as follows: A H = 2a 3a 5 − 4a 23



AA , Ô2f′  2 1

c

(123)

where: Ô = 0.6 A 1 = bearing area equal to the base plate area A i The H--shaped area A H is defined as the area able to support the applied axial compression load N *c at a uniform pressure of Ôf b. AH =

N *c Ôf b

4N *c Ôf ba 25

AH =

a 25X 4

(126)

The required plate thickness can now be calculated substituting the values of A H and a 3 calculated from equations (125) and (126) into equation (121). ti =

a5 1 − 1 − X 4

= λa 4



λ=2





8N *c 0.9f yi a 25X

2N *c 0.9f yid ib i

(127)

where:

(122)

where: a 5 = b fc + d c In this derivation, similarly to Thornton Model, the iterative procedure for the calculation of A H and Ôf b described in the literature review is not implemented and is terminated at the first iteration. The value of the maximum bearing strength of the concrete Ôf b is calculated as follows: Ôf b = min Ô0.85f′ c

(125)

Substituting the value of a 3 calculated in equation (125) into equation (122) yields, after simplifying, the following expression for the H--shaped bearing area A H:

The thickness of the base plate calculated according to Murray--Stockwell Model is determined as follows:

41

a5 1 − 1 − X 4

where:

where: a 4 = 1 d cb fc 4

Ôf ba 5 − (Ôf ba 5) 2 − 4Ôf bN *c 4Ôf b

X d ib i d cb fc 1 + 1 − X

A.2

DERIVATION FOR DESIGN PURPOSES -- CHANNELS

A.2.1

DETERMINATION OF a 4 (Yield Line Model)

The yield line pattern assumed in the case of channels is similar to the one assumed in the case of H--shaped column sections as shown in Fig. 73 and it is suitable for channels with b fc less than d c , as a different yield line pattern would otherwise occur.


(124)

Figure 73 Yield line pattern for Channels


The base plate is considered to be simply supported along the flanges and the web and free at the edge opposite to the web. Available solutions as proposed in [35] for a uniformly distributed load f *p are utilised. 8Ôm p4+9η 2−2 fp = *

b 2fc

η2



(128)



4+9η 2−2 3−4 3 η2

N *c d ib i

The required design plastic moment Ôm p to support a uniform pressure of f *p is obtained by re--arranging equation (128) as follows: Ôm p = f *pb 2fc

3


The thickness of the base plate calculated according to Murray--Stockwell Model is determined as follows:

(129)

where: 9η 2 − 4 4 + 9η 2 + 8




AH =

Ôm s =

0.9f yit 2i ≥ f *pb 2fcα 2 = Ôm p 4

(131)

and re--arranging equation (131) in terms of the required plate thickness yields: ti =

2d cb fc

= a4 where:

42

3





2f *p 0.9f yi

2N *c 0.9f yid ib i

a3 =

2 1

c

(135)

N *c Ôf b

(136)

=

Ôf ba 5 − (Ôf ba 5) 2 − 8Ôf bN *c 4Ôf b

a5 1 − 1 − X 4

(137)

where: X=

8N *c Ôf ba 25

Substituting the value of a 3 calculated in equation (137) into equation (134) yields, after simplifying, the following expression for the assumed bearing area A H: AH =

(132)

AA , Ô2f′ 

Substituting equations (134) and (135) into equation (136) and solving for a 3 the following expression for a 3 is obtained:

(130)


(134)

where: Ô = 0.6 A 1 = bearing area equal to the base plate area A i The assumed area A H is defined as the area able to support the applied axial compression load N *c at a uniform pressure of Ôf b.

2

α = 1 η 3

(133)

where: a 5 = 2b fc + d c The value of the maximum bearing strength of the concrete Ôf b is calculated as follows:

24 4 + 9η 2 − 48

The value of α introduced in equation (129) can be approximated by the following expression with an error of --0% (unconservative) and +6.7% (conservative) for values of η (which is equal to d c∕b fc ) between 1.25 and 4 (which include the channel sections available in Australia):

2N *c 0.9f yiA H

A H = a 3a 5 − 2a 23

2

244 + 9η 2 − 2



Referring to Fig. 12 the assumed bearing area AH can be expressed as follows:

9η − 4 4 + 9η + 8 2

= f *pb 2fcα 2

α2 =

A.2.2

2d cb fc

ti = a3

where: η = d c∕b fc Similarly to the case of H--shaped column sections the uniform load f *p is calculated as follows: f *p =

a4 =

a 25X 8

(138)

The required plate thickness can now be calculated substituting the values of A H and a 3 calculated from equations (137) and (138) into (133). ti =

a5 1 − 1 − X 4



16N *c 0.9f yi a 25X


= λa 4



2N *c 0.9f yid ib i

(139)

A.3



X d ib i d cb fc 1 + 1 − X

ti =

DERIVATION FOR DESIGN PURPOSES -- RECTANGULAR HOLLOW SECTION

A similar procedure to the ones adopted in the case of H--shaped sections and channels is adopted for rectangular hollow sections. A.3.1

DETERMINATION OF a 4 (Yield Line Model)

The yield line pattern considered in the case of rectangular hollow sections is shown in Fig. 74 and the required design plastic moment Ôm p under a uniform pressure f *p can be expressed as follows (based on [35]): Ôm p = f *pb 2c

1 + 3η 2 − 1 24η 2

1 + 3η 2 − 1

A.3.2

2f 2d23b 0.9f c c

= a4



a4 =

2d23b

= f *pb 2cα 2

(140)

2

(143)

c c


(144)

where: a5 = bc + dc The value of the maximum bearing strength of the concrete Ôf b is calculated as follows:



AA , Ô2f′  2 1

c

(145)

where: Ô = 0.6 A 1 = bearing area equal to the base plate area A i The assumed area A H is defined as the area able to support the applied axial compression load N *c at a uniform pressure of Ôf b.

dc

bc

AH =

Figure 74 Yield line pattern for Rectangular Hollow Sections The plate is assumed to be simply supported along all the edges. The value of α 2 introduced in equation (140) can be approximated by the following expression with an error of --0% (unconservative) and +11.1% (conservative) for values of η (which is equal to d c∕b c) between 3/4 and 4: (141)


43

yi

2N *c 0.9f yid ib i


η = d c∕b c



p

A H = 2a 3a 5 − 4a 23

N* f *p = c d ib i

α=

*


24η 2

η 23

(142)

where:

2

where: α2 =

0.9f yit 2i ≥ f *pb 2cα 2 = Ôm p 4

and re--arranging equation (142) in terms of the required plate thickness yields:

where: λ=3 2

Ôm s =

N *c Ôf b

(146)

Substituting equations (144) and (145) into equation (146) and solving for a 3 the following expression for a 3 is obtained: a3 = =

2Ôf ba 5 − 4(Ôf ba 5) 2 − 16Ôf bN *c 8Ôf b

a5 1 − 1 − X 4

(147)

where: X=

4N *c Ôf ba 25

Substituting the value of a 3 calculated in equation (147) into equation (144) yields, after simplifying, the following expression for the assumed bearing area A H:


AH =

a 25X 4

(148)

The required plate thickness can now be calculated utilising the values of A H and a 3 calculated from equations (147) and (148) as previously carried out for H--shaped sections and channels. ti =

a5 1 − 1 − X 4

= λa 4

Ôm s =

ti =

8N *c 0.9f yi a 25X

2N *c 0.9f yid ib i

(149)

dd bb 238 1 + 1X− X

≈ 1.7 A.4

 

i i

c c

A.4.2

X d ib i  d cb c 1 + 1 − X

A similar procedure to the one previously adopted is carried out for square hollow sections. A.4.1 DETERMINATION OF a 4 (Yield Line Model) The yield line pattern considered in the case of rectangular hollow sections is shown in Fig. 75 and the required design plastic moment Ôm p under a uniform pressure f *p can be expressed as follows (based on [35] and [46]): f *pb 2c 21.4

(150)

where: N *c d ib i

= a4



a4 =

1 b 10.7

yi

2N *c 0.9f yid ib i

(152)

c

≈ 1 bc 3


(153)

where: a 5 = 2b c The value of the maximum bearing strength of the concrete Ôf b is calculated as follows:




AA , Ô2f′  2 1

c

(154)

where: Ô = 0.6 A 1 = bearing area equal to the base plate area A i The assumed area A H is defined as the area able to support the applied axial compression load N *c at a uniform pressure of Ôf b. AH =

bc

N *c Ôf b

(155)

In a similar manner as previously carried out the value of a 3 can be determined as follows:

bc

Figure 75 Yield line pattern for Square Hollow Sections The plate is assumed to be simply supported along all the edges. The required plate thickness to support f *p can be determined by equating the nominal section moment capacity of the plate Ôm s (per unit width) to the required design plastic capacity (per unit width) as follows: 44

p

A H = 2a 3a 5 − 4a 23

DERIVATION FOR DESIGN PURPOSES -- SQUARE HOLLOW SECTION

f *p =

*

c




Ôm p =

1 b 10.7 0.9f2f

where:

where: λ=

(151)

and re--arranging equation (151) in terms of the required plate thickness yields:





0.9f yit 2i f *pb 2c ≥ = Ôm p 4 21.4

a3 =

bc  1 − 1 − X 2

X=

4N *c Ôf ba 25

(156)

where:

and the value of the assumed bearing area A H can be expressed as follows: AH =

a 25X = b 2cX 4

(157)

The required plate thickness can now be calculated. ti =

bc  1 − 1 − X 2



2N *c 0.9f yi b 2cX




2N *c 0.9f yid ib i

= λa 4

(158)

A H = π [d 20 − (d 0 − 2a 3) 2]= π(a 3d 0 − a 23) 4

d ib i X λ=3 2 b c 1 + 1 − X DERIVATION FOR DESIGN PURPOSES -- CIRCULAR HOLLOW SECTION

DETERMINATION OF a 4 (Yield line theory)

The yield line pattern considered in the case of circular hollow sections is shown in Fig. 76 and the required design plastic moment Ôm p under a uniform pressure f *p can be expressed as follows (based on [35]): Ôm p =

f *pd 20 24




f *p =

AA , Ô2f′  2 1

c

(163)

where: Ô = 0.6 A 1 = bearing area equal to the base plate area A i The assumed area A H is defined as the area able to support the applied axial compression load N *c at a uniform pressure of Ôf b. AH =

(159)

where:

(162)

The value of the maximum bearing strength of the concrete Ôf b is calculated as follows:

A similar procedure to the one previously adopted is carried out for circular hollow sections. A.5.1


Referring to figure 14 the assumed bearing area A H can be expressed as follows:

where:

A.5

A.5.2

N *c Ôf b

(164)

In a similar manner as previously carried out the value of a 3 can be determined as follows:

Nc d ib i *

a3 =

d0 1 − 1 − X 2

X=

4N *c d 20πÔf b

(165)

where: do

and the value of the assumed bearing area A H can be expressed as follows: AH = π

Figure 76 Yield line pattern for Circular Hollow Sections

0.9f yit 2i f *pd 20 ≥ = Ôm p 4 24

(160)

ti =

d0 2 3



2f *p = a4 0.9f yi

where:

2N *c 0.9f yid ib i

d0 2 3



8N *c 0.9f yi πd 20X



2N *c 0.9f yid ib i

(167)

where: λ=

(161)

≈2 A.6

a4 =

45



d0 1 − 1 − X 2

= λa 4

and re--arranging equation (160) in terms of the required plate thickness yields: ti =

(166)

The required plate thickness can now be calculated.

The plate is assumed to be simply supported along all the edges. The required plate thickness to support f *p can be determined by equating the nominal section moment capacity of the plate Ôm s (per unit width) to the required design plastic capacity (per unit width) as follows: Ôm s =

d 20X 4

12π d ib i d0

d ib i d0

X  1+ 1−X

X 1 + 1 − X

DERIVATION FOR CHECK PURPOSES -- ALL SECTIONS

The base plate capacity for a given base plate according to each Model considered is first determined and then a unique expression which concatenates them is derived.


The following notation is used in the derivation: ÔN c.1 = design capacity based on a 1 of the Cantilever Model ÔN c.2 = design capacity based on a 2 of the Cantilever Model ÔN c.3 = design capacity based on the Yield Line Model ÔN c.4 = design capacity based on Murray -Stockwell Model ÔN c.1 =

0.9f yid ib it 2i 2a 21

(168)

ÔN c.2 =


(169)

ÔN c.3 =


(170)

The calculation of the design capacity ÔN c.4 based on Murray--Stockwell model requires the following derivation: t i = λa 4

=



2ÔN c.4 0.9f yid ib i

ÔN c.4 Y

1 + 1 − ÔN c.4Y

ka 4

0.9f2 d b

yi i i

(171)

where: λ=k

X 1 + 1 − X

X = ÔN c.4Y and re--arranging equation (171) yields: ÔN c.4 =

0.9f yib id i t 2i λ′ 2a 24

(172)

where: λ′ = 12 k



2ka 4 t Y i

0.9f2 b d − 1 yi i i

The design capacity of the base plate is then calculated as follows: ÔN c = min(ÔN c.1, ÔN c.2, ÔN c.5)

11. APPENDIX B-- Derivation of Design and Check Expressions for Steel Base Plates Subject to Axial Tension The derivation of the expressions for the design and check of base plate subject to axial tensile loading has been here carried out for common base plate layouts when no design guidelines were found in literature. Yield line theory, based on conservative yield line patterns (in the authors’ opinion), has been utilised in the derivation. The plate moment capacity per unit length of yield line has been calculated here based on the plastic section modulus of the plate as also carried out in Australian and American guidelines [5], [21] and [26]. It is interesting to note that [23] recommends to use the elastic section modulus. The reduction of plate capacity due to the anchor bolt holes has been accounted for. Ignoring the effects of bolt holes is a substantial simplification as also noted in [37]. Murray Model, which considers the design of base plates for lightly loaded H--shaped columns with two anchor bolts, has been here re--derived and modified to include the plate reduction capacity due to bolt holes. Here the yield lines are conservatively assumed to remain inside the internal faces of the column profile, while in Murray Model they extend to the centerline of the web and to the outside faces of the flanges. The derivations of the capacity or required thickness for the yield line patterns considered have been carried out for various combinations of column sections and number of anchor bolts as listed in Section 5.4.7. The derivation for the case of a H--shaped column with anchor bolts, as shown in Fig. 77, is outlined below. All other cases are considered in a similar manner and the relevant expressions of their derivation are summarised in Table 6. Similar considerations outlined for the validity of the Yield Line Model for the case of a H--shaped column section with 2 bolts can be applied to the other base plate configurations considered. B.1

H--SHAPED COLUMN WITH 2 ANCHOR BOLTS

In the case of H--shaped column sections with two anchor bolts the yield line pattern assumed is shown in Fig. 77. It is the same as the one considered in Murray Model. The base plate dimensions are conservatively assumed to be equal to the outside column dimensions unless noted otherwise.

(173)

where: ÔN c.5 = max(ÔN c.3, ÔN c.4) and ÔN c.1, ÔN c.2, ÔN c.3 and ÔN c.4 area calculated as shown in equations (168), (169), (170) and (172).

46


s tw

where: y = as calculated from equation (179) or equivalently the minimum plate thickness required for a certain design tension load N *t :

d c1 2

y y

d c1 2

ti ≥

b fc

Figure 77 Yield line pattern: H--shaped column with 2 bolts Considering the symmetry about the column web the derivation of the internal work and external work is carried out only considering half the plate area:





b W i = Ôm p 1y 4 fc1 − 2d h + 2 2y 2 b fc1 = Ôm p W e = 2N

* b

2b

fc1 − 2d h

y

+

4y b fc1



s

 (174) (175)

b fc1

where: b fc1 = b fc − t w y and s are defined in Fig.77 Equating the internal and external work the expression of the design axial tension load per bolt N *b is obtained as follows:





2b fc1 − 2d h b 4y N *b = fc1 + Ôm p y 2s b fc1

(176)



0.9f yi2b 2fc1 − 2b fc1d h + 4y 2

(177)

(182)

y > l2 where: l1 =

dh 2

1 − 4sd

2

h 2

l 1l 3

l2 = s−



d2 h 4

− l 21

and the notation is defined in Fig. 78. l3

s Web

diameter of hole = d h l1

Solving equation (177) for y yields: y=

l2



b fc1 − d h b fc1 2

(178)

The presence of the flanges requires the value of y to be always less or equal to d c∕2 and therefore y is re--defined as follows:



y = min

d c1 , 2

b

fc1 − d h

2

b fc1



(179)

The design axial tension capacity of the base plate ÔN t is then obtained re--arranging equation (176) as follows: ÔN t = 2b 2fc1 − 2b fc1d h + 4y 2

47

(181)

In this model the reduction in plate capacity due to the presence of a bolt hole along the yield line perpendicular to the web has been included. Further reductions due to other yield lines intersecting bolt holes have not been considered as they are very unlikely to occur and a more detailed analysis should be carried out in such situation. The critical yield line pattern is a function of the value of y calculated from equation (179). To ensure that none of the oblique yield lines intersects the bolt hole, as assumed in the model derived, the following simplified condition needs to be satisfied:

The value of y which minimises N *b (or equivalently that maximises the required Ôm p) is determined differentiating equation (176) for y. dN *b 2b − 2d h = − fc1 2 + 4 =0 b fc1 dy y

4syN *t

0.9f yit 2i 4sy

d2h∕4 − l 21

Edge of plate

Figure 78 Yield line layout near the bolt hole Substituting a nil value for the diameter of the bolt hole d h in equations (179) and (181) would lead to the determination of plate thicknesses t i similar to those obtained with Murray Model.

(180)


Table 6 Summary of Internal and External Work for the Various Base Plate Configurations (refer to figures of Section 5.4.7. to view the yield line patterns considered) Section / No. Bolts H--shaped section 2--bolts H--shaped section 4--bolts (a) H--shaped section 4--bolts (b) H--shaped section 4--bolts (c) H--shaped section 4--bolts (d) H--shaped section 4--bolts (e) Channel 2--bolts

Wi Ôm p

2Ôm p Ôm p

Channel 4--bolts (c) Channel 4--bolts (d) Channel 4--bolts (e) Hollow 2--bolts (a) Hollow 2--bolts (b) Hollow 4--bolts (a) Hollow 4--bolts (b)

48

fc1 − 2d h

2b

fc1 b

Ôm p

fc1 − 2d h

Ôm p



Ôm p

y



4N *b s b fc1

4y+4a b b fc1

4y + 2s p b fc1

+



4b

fc1 − 2d h

y

p

+

fc1 − 2d h

y

2y b fc1



2N *b N *b s b fc1



2N *b s b fc1

fc1

Ôm p

4b

 2b Ôm  p



2N *b

fc1 − 2d h

y

+

2y + s p b fc1





 − 2d 4a + 2s − 2d  + a s 4s − 2d 2y Ôm  +s  y

4b fc1 − 2d h 4y + 2s p − 2d h + y s fc1

p

b

h

h

b

fc1 − d h



min a b,



2

2

2

fc1 --d h

2

b

b fc1



fc1 --d h

2

s

d c1 , (2b fc1 --d h)b fc1 2

 

y ≤ a b,

(2bfc1 − d h)b fc1

2N *b s b fc1

mina b, (2b fc1 --d h)b fc1

2b

sp 2

 

2N *b s b fc1





b fc1



min a b,

sp 2

y≤

b fc1

(2bfc1 − d h)b fc1

2N *b



y ≤ a b,

b fc1

fc1 − d h

b

min a b,

fc1 --d h

fc1 − d h

2

sp 2

y≤



sp 2



s

2N *b

b

2

u

h

2

l Ôm p s i

4s −y 2d + 2y +s s  2

s N *b s 1 2

(2s2 − dh)s2

2y ≤ l i

s N *b s 3 4

4

Ôm u

b

b

min

Restraints

d c1 , 2

4N *b s b fc1



2y + b fc1



min

4N *b s b fc1

4b fc1 --2d h 4b fc1 --2d h 2y+2a b + + y ab b

Ôm p





b fc1 − 2d h 4a b + 2s p − 2d h + ab s

2b 4Ôm 



y

2N *b s b fc1

4y b fc1

+

y



2b fc1 − 2d h 4y + 2s p − 2d h + y s

Ôm p

Ôm p

fc1 --2d h

fc1 − 2d h

4y b fc1

+

y

h

2b

+

y

2b a --2d + 2b

Channel 4--bolts (a) Channel 4--bolts (b)

2b

We

p

h

2

l Ôm p s i 2

s 2N *b s 1 2

(2s2 − dh)s2

2y + s p ≤ l i

s 2N *b s 3 4


12. APPENDIX C -- Determination of Embedment Lengths and Edge Distances

The tensile capacity of the anchor bolt is determined in accordance with Clause 9.3.2.2. of AS 4100 as follows:

The recommended guidelines regarding the minimum embedment lengths and concrete edge distances are here derived in a similar manner as carried out in references [39] and [47]. The guidelines derived in [39] are also recommended in [21] and [26]. Differences between the derivations carried out here and those presented in references [39] and [47] are noted. C.1

MINIMUM EMBEDMENT LENGTH OF ANCHOR BOLTS

The recommended model requires the anchorage system (anchor to concrete connection) to fail in a ductile manner. This is achieved by ensuring that the concrete capacity is greater than the tensile capacity of the anchor bolt. [2] Minimum embedment lengths are here derived, similarly to [39], for isolated anchor bolts. Anchor bolts in bolt groups might require longer embedment lengths due to overlapping of the concrete failure envelopes. The calculation of the concrete capacity is based on the procedure described in the recommended model. The concrete cone projected area is calculated ignoring the area of the bolt calculated using the nominal bolt diameter d f. In [39] the projected area is calculated ignoring the area of a circle equivalent to the projected area of a heavy hexagonal head. Comparing the ratios L d∕d f (where L d is the minimum embedment length required and d f is the nominal bolt diameter) regarding the same types of bolts, the results obtained here appear to be of the order of 1% more conservative than the ones obtained in [39]. The further simplification of simply considering the cone as starting at the embedded end of the anchor bolt has been adopted in reference [47]. The concrete capacity is calculated as follows: ÔN cc = Ô0.33 f′ c A ps

(183)

where: Ô = 0.7 (based Ô required for Clause 9.2.3 of AS 3600) instead of 0.65 as adopted in references [39] and [47]





d A ps = π L d + f 2

= π(L d + d fL d) 2

49

2

 =

d −π f 2

N tf = A sf uf

(184)

where: A s = tensile stress area in accordance with AS 1275 [9] The minimum embedment length is calculated equating equations (183) and (184) as follows: Ô0.33 f′ c πL 2d + d fL d = A sf uf

(185)

and solving for L d: Ld =

− d f + d 2f + 4γ 2

≥ 100

(186)

where: γ=

f ufA s

Ô0.33 f′ c π The minimum embedment lengths derived and recommended in [39] have been calculated adding an additional safety factor of 1.33. The recommended embedment lengths recommended here do not include the additional safety factor of 1.33 (similarly to reference [47]). For completeness the embedment lengths have been here calculated with and without the safety factor of 1.33. The calculation of the minimum embedment lengths for anchors with different bolts tensile strengths and for different concrete strengths is carried out in Tables 7 and 8 in order to explicitly show how this additional safety factor of 1.33 introduced in references [39] is incorporated in the results. The tabulated results are smaller than those presented in reference [47] due to the different procedure utilised to determine the projected area even if here a Ô equal to 0.7 has been adopted. Including the additional factor of safety Ô sf = 1.33 recommended in reference [39] equation (186) can be re--written as : L d = Ô sf

2

where:

− d 2f + d 2f + 4γ 2

≥ 100

(187)

Ô sf = 1.33 f ufA s γ= Ô0.33 f′ c π


Table 7 Minimum embedment lengths for Grade 4.6 bolts and Grade 250 rods (fuf = 400 MPa) Bolt Type

df mm

As mm2

f’c MPa

Ld mm

Min ratio Ld/df

1.33 Ld

1.33 Ld/df

12

84.3

20

100.0

8.4

127.8

10.7

M16

16

157

20

131.3

8.2

174.7

10.9

M20

20

225

20

164.1

8.2

218.2

10.9

M24

24

324

20

196.9

8.2

261.9

10.9

M30

30

519

20

248.4

8.3

330.3

11.0

M36

36

759

20

299.8

8.3

398.8

11.1

M12

12

84.3

25

100.0

8.4

120.5

10.0

M16

16

157

25

123.8

7.7

164.7

10.3

M20

20

225

25

154.6

7.7

205.7

10.3

M24

24

324

25

185.6

7.7

246.9

10.3

M30

30

519

25

234.1

7.8

311.4

10.4

M36

36

759

25

282.6

7.9

375.9

10.4

M12

12

84.3

32

100.0

8.4

112.8

9.4

M16

16

157

32

115.9

7.2

154.2

9.6

M20

20

225

32

144.8

7.2

192.6

9.6

M24

24

324

32

173.8

7.2

231.2

9.6

M30

30

519

32

219.3

7.3

291.6

9.7

M36

36

759

32

264.7

7.4

352.1

9.8

Table 8 Minimum embedment lengths for Grade 8.8 bolts (fuf = 830 MPa except fuf = 800 MPa for M12 bolts ) As f’c mm2 MPa

Ld mm

M12 M16 M20 M24 M30 M36 M12

12 16 20 24 30 36 12

84.3 157 225 324 519 759 84.3

20 20 20 20 20 20 25

138.3 192.5 240.5 288.7 364.1 439.5 130.5

Min ratio Ld/df 11.5 12.0 12.0 12.0 12.1 12.2 10.9

M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

16 20 24 30 36 12 16 20 24 30 36

157 225 324 519 759 84.3 157 225 324 519 759

25 25 25 25 25 32 32 32 32 32 32

181.7 226.9 272.4 343.5 414.7 122.3 170.3 212.8 255.4 322.1 388.8

11.4 11.3 11.4 11.5 11.5 10.2 10.6 10.6 10.6 10.7 10.8

1.33 Ld

1.33 Ld/df

mm

183.9 256.1 319.9 384.0 484.2 584.5 173.5

15.3 16.0 16.0 16.0 16.1 16.2 14.5

241.6 301.8 362.3 456.9 551.5 162.7 226.6 283.0 339.7 428.4 517.1

15.1 15.1 15.1 15.2 15.3 13.6 14.2 14.2 14.2 14.3 14.4

f′ c (MPa) 20 25 32 20 25 32

Ô sf 1 1 1 1.33 1.33 1.33

mm

M12

Bolt df Type mm

Table 9 Grade 4.6 bolts and 250 grade rods where Ô sf is a safety factor introduced in reference [39] Ld 9 df 9 df 9 df 12 df 11 df 10 df

Table 10 Grade 8.8 bolts where Ô sf is a safety factor introduced in reference [39]

C.2

Ô sf

f′ c (MPa)

Ld

1 1 1 1.33 1.33 1.33

20 25 32 20 25 32

13 df 12 df 11 df 17 df 16 df 15 df

MINIMUM CONCRETE EDGE DISTANCES -Anchor bolt subject to tension

[2] provides a design procedure to determine the minimum concrete edge distances to avoid lateral bursting of the concrete as discussed in the literature review of anchor bolts subject to tension. This has been included in the recommended model. The minimum edge distance is calculated as follows: ae = df



f uf

6 f′ c

(188)

The required minimum edge distances a e calculated with equation (188) are tabulated in Tables 11 and 12 for different combinations of anchor bolts and concrete strengths.

Observing the results of Tables 7 and 8 the embedment lengths requirements can be simplified as shown below.

50


Table 11 Minimum concrete edge distances for anchor bolts Grade 4.6 bolts and Grade 250 rods (fuf = 400 MPa) subject to tension Bolt df (mm) f’c ae a e / df type (MPa) (mm) M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

12 16 20 24 30 36 12 16 20 24 30 36 12 16 20 24 30 36

20 20 20 20 20 20 25 25 25 25 25 25 32 32 32 32 32 32

46.3 61.8 77.2 92.7 115.8 139.0 43.8 58.4 73.0 87.6 109.5 131.5 41.2 54.9 68.7 82.4 103.0 123.6

3.9 3.9 3.9 3.9 3.9 3.9 3.7 3.7 3.7 3.7 3.7 3.7 3.4 3.4 3.4 3.4 3.4 3.4

Table 12 Minimum concrete edge distances for anchor bolts Grade 8.8 bolts (fuf = 830 MPa except fuf = 800 MPa for M12 bolts ) subject to tension Bolt df (mm) f’c ae a e / df type (MPa) (mm) M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

12 16 20 24 30 36 12 16 20 24 30 36 12 16 20 24 30 36

20 20 20 20 20 20 25 25 25 25 25 25 32 32 32 32 32 32

65.5 89.0 111.2 133.5 166.9 200.2 62.0 84.2 105.2 126.2 157.8 189.4 58.3 79.1 98.9 118.7 148.4 178.0

5.5 5.6 5.6 5.6 5.6 5.6 5.2 5.3 5.3 5.3 5.3 5.3 4.9 4.9 4.9 4.9 4.9 4.9

The values of minimum edge distances required expressed in terms of d f can be summarised as follows: for Grade 4.6 bolts and Grade 250 rods a e = 4 d f when f′ c = 20, 25 and 32 MPa for Grade 8.8 bolts

51

a e = 6 d f when f′ c = 20 and 25 MPa = 5 d f when f′ c = 32 MPa The recommended model requires the minimum edge distance a e to be always at least equal to 100mm as recommended in [21], [26] and [39]. Minimum edge distance recommended in reference [47] is 50mm. C.3

MINIMUM CONCRETE EDGE DISTANCES -Anchor bolt subject to shear

Guidelines on minimum edge distances to be adopted in the case of bolts in shear are provided in [2], [3], [17], [26], [39] and [47]. These are all based on the design procedure presented in [2], [3] and [17] which requires the minimum edge distance to be calculated as (refer equation (106)): ae ≥ df



f uf

(189)

Ô0.94 f′ c

where: Ô = 0.65 according to references [3] and [39] = 0.85 according to references [17], [26] and [47] For completeness edge distances calculated with both values of Ô have been considered and tabulated here. It is up to designer to decide whether or not to design the anchor bolts to carry shear and to select a value of Ô. These values of a e are tabulated in tables 13, 14, 15 and 16 for different combinations of anchor bolts and concrete strengths and for different values of Ô. Table 13 Minimum concrete edge distances for anchor bolts Grade 4.6 bolts and Grade 250 rods (fuf = 400 MPa) subject to shear with Ô = 0.65 Bolt type M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

df (mm)

f’c (MPa)

ae (mm)

a e / df

12 16 20 24 30 36 12 16 20 24 30 36 12 16 20 24 30 36

20 20 20 20 20 20 25 25 25 25 25 25 32 32 32 32 32 32

145.2 193.6 242.0 290.4 363.0 435.6 137.3 183.1 228.9 274.6 343.3 411.9 129.1 172.1 215.2 258.2 322.7 387.3

12.1 12.1 12.1 12.1 12.1 12.1 11.4 11.4 11.4 11.4 11.4 11.4 10.8 10.8 10.8 10.8 10.8 10.8


Table 14 Minimum concrete edge distances for anchor bolts Grade 8.8 bolts (fuf = 830 MPa except fuf = 800 MPa for M12 bolts) subject to shear with Ô = 0.65 Bolt type M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

df (mm) 12 16 20 24 30 36 12 16 20 24 30 36 12 16 20 24 30 36

f’c (MPa) 20 20 20 20 20 20 25 25 25 25 25 25 32 32 32 32 32 32

ae (mm) 205.3 278.9 348.6 418.3 522.9 627.4 194.2 263.7 329.7 395.6 494.5 593.4 182.6 247.9 309.9 371.9 464.9 557.9

a e / df 17.1 17.4 17.4 17.4 17.4 17.4 16.2 16.5 16.5 16.5 16.5 16.5 15.2 15.5 15.5 15.5 15.5 15.5

Table 15 Minimum concrete edge distances for anchor bolts Grade 4.6 bolts and Grade 250 rods (fuf = 400 MPa) subject to shear with Ô = 0.85 Bolt type M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

52

df (mm) 12 16 20 24 30 36 12 16 20 24 30 36 12 16 20 24 30 36

f’c (MPa) 20 20 20 20 20 20 25 25 25 25 25 25 32 32 32 32 32 32

ae (mm) 127.0 169.3 211.6 253.9 317.4 380.9 120.1 160.1 200.1 240.2 300.2 360.2 112.9 150.5 188.1 225.8 282.2 338.7

a e / df 10.6 10.6 10.6 10.6 10.6 10.6 10.0 10.0 10.0 10.0 10.0 10.0 9.4 9.4 9.4 9.4 9.4 9.4

Table 16 Minimum concrete edge distances for anchor bolts Grade 8.8 bolts (fuf = 830 MPa except fuf = 800 MPa for M12 bolts) subject to shear with Ô = 0.85 df (mm) f’c (MPa) 12 20 16 20 20 20 24 20 30 20 36 20 12 25 16 25 20 25 24 25 30 25 36 25 12 32 16 32 20 32 24 32 30 32 36 32

Bolt type M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36 M12 M16 M20 M24 M30 M36

ae (mm) 179.6 243.9 304.8 365.8 457.2 548.7 169.8 230.6 288.3 345.9 432.4 518.9 159.6 216.8 271.0 325.2 406.5 487.8

a e / df 15.0 15.2 15.2 15.2 15.2 15.2 14.2 14.4 14.4 14.4 14.4 14.4 13.3 13.6 13.6 13.6 13.6 13.6

Re--arranging equation (189) the ratios a e∕d f for different combinations of concrete and bolt strengths for different values of Ô are obtained as shown below. Table 17 Grade 4.6 bolts and 250 Grade rods Ô 0.65 0.65 0.65 0.85 0.85 0.85

f′ c (MPa)

ae

20 25 32 20 25 32

13 df 12 df 11 df 11 df 10 df 10 df

Table 18 Grade 8.8 bolts Ô 0.65 0.65 0.65 0.85 0.85 0.85

f′ c (MPa) 20 25 32 20 25 32

ae 18 df 17 df 16 df 16 df 15 df 14 df


13. APPENDIX D -- Design Capacities of Equal Leg Fillet Welds

14. APPENDIX E -- Design of Bolts under Tension and Shear

Table 19 Category SP, Ô =0.8, kr=1.0

Table 23 Design Capacities Commercial Bolts 4.6/S Bolting Cat. fuf=400MPa, Ô =0.8

Weld size (mm)

tw 2 3 4 5 6 8 10 12

Design Capacity per unit length of fillet weld except for RHS/ SHS with thickness less than 3 mm (kN/mm) E41XX/W40X E48XX/W50X 0.278 0.326 0.417 0.489 0.557 0.652 0.696 0.815 0.835 0.978 1.11 1.30 1.39 1.63 1.67 1.96 fuw=410 MPa fuw=480 MPa

tt 1.41 2.12 2.83 3.54 4.24 5.66 7.07 8.49

Table 20 Category SP, Ô =0.7, kr=1.0 Weld size (mm) tw 2 3 4 5

tt 1.41 2.12 2.83 3.54

Design Capacity per unit length of longitudinal fillet weld in RHS/ SHS with t < 3mm (kN/mm) E41XX/W40X E48XX/W50X 0.244 0.285 0.365 0.428 0.487 0.570 0.609 0.713 fuw=410 MPa fuw=480 MPa

Table 21 Category GP, Ô =0.6, kr=1.0 Weld size (mm) tw 2 3 4 5 6 8 10 12

tt 1.41 2.12 2.83 3.54 4.24 5.66 7.07 8.49

Design Capacity per unit length of fillet weld (kN/mm) E41XX/W40X E48XX/W50X 0.209 0.244 0.313 0.367 0.417 0.489 0.522 0.611 0.626 0.733 0.835 0.978 1.04 1.22 1.25 1.47 fuw=410 MPa fuw=480 MPa

Bolt Si Size

Axial Tension T i ÔNtf (kN)

M12

Shear (single shear) Threads included in shear plane N ÔVfn (kN)

Threads excluded from shear plane X ÔVfx (kN)

27.0

15.1

22.4

M16

50.1

28.6

39.9

M20

78.3

44.7

62.3

M24

113

64.3

89.8

M30

179

103

140

M36

261

151

202

4.6N/S

4.6X/S

Table 24 Design Capacities High Strength Structural Bolts 8.8/S, 8.8/TB, 8.8/TF Bolting Categorys, Ô =0.8 Bolt Si Size

Min. Tensile T il Strength of Bolt fuf (MPa)

Axial Tension T i ÔNtf (kN)

M12

800

M16

Shear (single shear) Threads included in shear plane N ÔVfn (kN)

Threads excluded from shear plane X ÔVfx (kN)

53.9

30.3

44.9

830

104

59.3

82.8

M20

830

163

92.7

129

M24

830

234

133

186

M30

830

372

214

291

8.8N/S

8.8X/S

Table 22 Minimum Fillet Weld Sizes Thickness of thickest part t (mm) t≤7 7 < t ≤ 10

53

Minimum size of a fillet weld tw (mm) 3 4

10 < t ≤ 15

5

15 < t

6


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54


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