partial shear connection design of composite slabs

Partial shear connection design composite slabs Mark Patrick

BHP Research - Melbourne Laboratories

Russell Q. Bridge

School of Civil and Mining Engineering, University of Sydney, Sydney, Australia

of

A new partial shear connection strength model has been developed which considers the equilibrium of a segment of a simply-supported composite slab under any loading condition. The model incorporates shear connection performance derived from a new test called the slip block test. This small-scale test shows that shear connection performance is affected by factors such as profile geometry, base metal thickness and concrete compressive strength. Full-scale slab tests have proved that it is a reliable physical model which can readily account for the effects of changes in shear connection performance, loading pattern, sheeting end support conditions, end anchorage devices, conventional reinforcement etc., and is being developed for possible use in an Australian Standard on the design of composite slabs. The ease with which the model can be used in design is demonstrated in a worked example. Another new ultimate strength method of design, published by the Steel Deck Institute in America, is examined in the worked example, and shown to give erratic, sometimes overly conservative but often very unsafe predictions for the particular profiled sheet and design situations studied.

Keywords:composite slabs, testing, partial shear connection strength model

A new partial shear connection strength model has been developed which considers the equilibrium of a segment of a simply-supported composite slab under any loading condition. The model incorporates shear connection performance derived from a new test called the slip block test. During this test the conditions during longitudinal slip failure in a full-size slab are simulated using a small slab element. The effects of end anchorage devices and conventional reinforcement which act in conjunction with the sheeting can be readily accounted for using the model. Good agreement between the predictions of the model and test results has been obtained. As it is a reliable physical model, it is being developed for possible use in an Aus- tralian Standard on the design of composite slabs.

No ta t ion

al,a2,a3 coefficients used to calculate CEB curves for concrete compressive strength

A~I cross-sectional area of profiled steel sheeting bottom flanges within width of slab b

0141-0296/94/050348-15 © 1994 Butterworth-Heinemann Ltd

348 Engng Struct. 1994, Volume 16, Number 5

Ap

Aw

b b'

bbj

br

b,y

d <,,

Des

&

L

cross-sectional area of profiled steel sheeting within width of slab b cross-sectional area of profiled steel sheeting webs within width of slab b width of slab average width of concrete between steel sheeting ribs within width of slab b width of bottom flange of compact monosymmetric I-section representing profiled sheet average steel rib spacing of a profile for all rib types width of top flange of compact monosymmetric 1-section representing profiled sheet effective depth of profiled steel sheeting depth of rectangular compressive stress zone in concrete measured below top of slab overall depth of composite slab inclusive of profiled steel sheeting modulus of elasticity of profiled steel sheeting strength parameter used to calculate CEB curves for concrete compressive strength

f~ f"

Lm

fst fsy.sh

h~

h. hw

nrib

H.b/b, k.

kub (kud)lim

L Lc

L.

M*

M.

M.p

Mup.O.5

R* R.

S

tb/

tbm t~

tw

T

T.

T.

Partial shear connection design of composite slabs: M. Patrick and R. O. Bridge

concrete compressive stress characteristic compressive cylinder strength of concrete at 28 days; or specified strength grade mean value or best estimate of site-cured compressive cylinder strength of concrete at relevant age steel tensile stress yield strength of profiled steel sheeting measured in direction of rolling depth of centroid of profiled steel sheeting measured below top of steel rib overall height of rib of profiled steel sheeting web height of compact monosymmetric I-section representing profiled sheet average rib resistance per unit length for all rib types of a profile at a particular value of slip, in absence of clamping force as determined from slip block test average rib resistance per unit width of slab neutral axis parameter, being ratio of depth to neutral axis from extreme compressive fibre to d, at ultimate strength neutral axis parameter for balanced conditions limiting value of k.d corresponding to yield in top fibre of sheeting flange for under-reinforced cross-sections slab span length of cantilever portion of slab measured between end face of concrete and centre of adjacent support shear span measured between centre of loading point and centre of nearer roller support applied bending moment plastic moment capacity of profiled steel sheeting flexural capacity, or nominal flexural capacity in design, of a slab cross-section with complete shear connection moment capacity, or nominal moment capacity in design, of a slab cross-section with partial shear connection value of M.p corresponding to T/T. = 0.5, i.e. 50% shear connection design vertical reaction at an end support vertical reaction at an end support when slab ultimate strength is reached longitudinal slip measured in direction of sheeting ribs bottom flange thickness of compact monosymmetric I-section representing profiled sheet base metal thickness of profiled steel sheeting top flange thickness of compact monosymmetric 1-section representing profiled sheet web thickness of compact monosymmetric I- section representing profiled sheet resultant tensile force in sheeting at a slab cross- section anchorage force per unit width of slab as determined by SDI method resultant tensile force corresponding to M., or 4,M. in design calculation term for over-reinforced cross- sections

W

w, w, w. X

7

~c I?,sl '~st

I z

Pc 4, X

module width of a profiled steel sheeting measured between lap joint centres dead load live load design ultimate load distance to slab cross-section measured from centre of closer end support coefficients ratio of depth of assumed rectangular compressive stress block to k.d; or distance from start of a diagonal crack to critical cross-section divided by overall depth of slab Des slip factor concrete compressive strain interface slip strain steel tensile strain maximum strain in extreme compressive fibre of concrete at ultimate strength proportion of end support reaction R. transmitted directly through sheeting pans into concrete coefficient of friction developed between sliding surfaces density of concrete capacity factor cross-section curvature

New partial shear connection strength model

The new partial shear connection strength model 1-4 will be discussed here in relation to the design of the simply-supported slab shown in Figure 1, which is broadly described as follows.

• The slab comprises a span L measured between support centres, and two short cantilever portions each of length Lc which extend past the centre of the supports and which may be locally reinforced in the top face to resist the negative support moments

• The torsional rigidity of the supporting beams is assumed to be sufficiently small that its effect on the negative moments which develop in the slab at the supports can be ignored.

• The sheeting is continuous over the supports whereby the entire reaction at each support is transmitted directly through the underside of the sheeting

• There is no connection between the sheeting and the supports, and there are no mechanical devices such as end anchors either fastened through or attached to the sheets

• The slab has a uniform cross-section • The vertical loads are applied through the top face of the

slab, uniformly across its width so that one-way action O c c u r s

The theory has been developed to calculate the ultimate strength of such a slab provided it fails by either flexure or longitudinal slip, this latter failure mode being also

/ Coaczem

- . -----~-D¢ s

- L _L Figure 1 Simply-supported composite slab

Engng Struct. 1994, Volume 16, Number 5 349


referred to in the technical literature as longitudinal shear. The presence of end anchors, and bottom-face deformed bars or welded-wire fabric acting as longitudinal tensile reinforcement in conjunction with the steel sheeting, can be directly taken into account with the theory, although this latter situation is not considered herein. The slab must be reinforced transversely to control shrinkage and tempera- ture effects, as well as to control longitudi~aal splitting which can occur directly above the sheeting ribs due to transverse tensile forces which develop when slip occurs.

A slab can be considered to fail by flexure when the flexural capacity M, is reached at the cross-section under peak positive bending moment, even though slip may occur at this or other cross-sections including at the end of the slab. In such situations, the strength of the shear connection between the end of the slab and the peak moment cross- section does not limit the moment capacity of that cross- section, and the shear connection can then be termed 'full ' or 'complete' .

A suitable definition for a longitudinal slip failure is when as a consequence of slip the moment capacity Mup of the governing or critical cross-section (i.e. the cross-section where the applied bending moment M* equals M,p) is less than the flexural capacity M,. In such situations, the strength of the shear connection between at least one end of the slab and the critical cross-section limits the moment capacity of that cross-section, and the shear connection may then be termed 'partial' or ' incomplete' .

The theory cannot predict the occurrence of vertical shear failures. The transverse shear capacity of composite slabs incorporating a particular Australian profiled steel sheeting Bondek 1I 5 has recently been investigated 6'7. It has been recommended that at least for this product, used in situations like that shown in Figure 1, the occurrence of vertical shear failures (viz. by diagonal splitting or flexure shear) can normally be ignored in design. Although this recommendation does not necessarily apply to the design of composite slabs incorporating other types of profiled sheets, it will be assumed herein that design for vertical shear does not need to be considered.

Shear connection performance from the slip block test The data which defines the shear connection performance of a profile for use with the theory is obtained directly from the slip block test 1'3"x. Adhesion bond is ignored and the test yields estimates for the mechanical interlock (defined by the average rib resistance per unit width of slab H~ib/br) and the coefficient of friction /z which develop between the sliding surfaces. Each feature of a profile which can affect the longitudinal slip resistance needs to be included in the slip blocks being tested. Corresponding values of H,b/br and/x are determined at regular intervals of slip s so that H, J b , versus s and /x versus s curves can be produced (see Figure 2). The variable H,b is the average rib resistance per unit length of slab (all ribs considered equal) with units of force per unit length, and b, is the average steel rib spacing.

When using the model, the shear connection performance of the profile is assumed to be perfectly ductile, i.e. H~b/br and/x are both assumed to be constant in magnitude inde- pendent of the amount of slip. The accuracy of this assumption depends on the shape of the Hr~b/br versus s and /x versus s curves, and the relative importance of each parameter for the situation being designed.

In normal design situations involving uniformly-distributed loads, the average rib resistance per unit width of slab

+ 4 0 [ I I I I - ~ ~M}

(I 2 4 6 8 I(~ 12

S l lp s I m p )

Figure 2 Typica l H.Jbr versus s and/~ versus s curves for Bon- dek II prof i led sheet

H,ib/br is the more important parameter and accordingly will be discussed further here. For some profiles it has been determined that Hriblb ~ is not only a function of slip s but also base metal thickness tbm, concrete compressive strength f , , and concrete type (e.g. normal density or lightweight). In such cases the approach taken has been to systematically test sets of slip blocks incorporating sheeting with the same base metal thickness tbm and concrete type, varying the concrete compressive strength by pouring concretes of different grades and testing the blocks at different ages.

Some typical test results from such an investigation performed on Bondek II sheeting with a nominal base metal thickness tbm = 1.00 mm and using normal-density concrete are shown in Figure 38. The results all correspond to a slip s = 5 mm, and it can be seen that assuming a relationship of the form H,b/br = a'JFm gives c~= 116.4 (while

= 118.8 and 127.2 for s = 1 mm and 3 mm, respectively, indicating reasonable ductility of the shear connection). For the purpose of this paper it will be assumed that mean values of Hrib/b~ and /x for 1.00 mm Bondek II sheeting with normal-density concrete can be estimated as Hr~b/br = 115 X/~,'. and /x- - 0.8.

Assumed equilibrium state of strength model A free-body diagram of an end segment of the simply-supported composite slab illustrated in Figure 1 is shown in Figure 4(a). It is assumed that the slab is subjected to a load consisting of a uniformly-distributed component and concentrated line loads, and has reached its ultimate strength and failed by longitudinal slip. The end segment is bounded by the critical cross-section where the applied bending moment M* equals the moment capacity mup of the cross-section (see Figure 4(b)).

The equilibrium of horizontal forces per unit width of slab acting on the steel profile over the critical end region

8OO

~ o

~oo

o

116.4 f ~

lO 20 30 40

C o m ~ v e Cylinder Strength tom ( M P a )

Figure 3 Typical H.dbr versus fc,. curve for Bondek II prof i led sheet



a

Lc 1 M*(Applied Bending Moment)

b (Moment Capacity in Posidve Bending)

(x + Le) Hrib/br

>T

C ~ laRu

Figure4 New partial shear connection strength model. "(a) free-body diagram of slab end segment bounded by critical cross-section; (b) intersection of bonding moment diagram with moment capacity line at critical cross-section; (c) horizontal force equilibrium for steel profile over critical end region.

is shown in Figure 4(c), whereby the resultant tensile force T in the sheeting is balanced by the rib resistance force HrJb~ (x + Lc) which is assumed to develop at a uniform rate along the sheeting between the critical cross-section and the end face of the concrete, and the frictional resistance IX/Y, developed at the support where the sheeting is clamped between the support and the concrete (ignoring any shear transmitted through the webs of the sheeting ribs) i.e.:

T= nrib]b r (x + Lc) + tx R, (la)

fvy.sh Ap/b (lb)

where T resultant tensile force in the sheeting per unit width of

slab which cannot exceed the tensile capacity of the sheeting f~y.sh AJb

Ap cross-sectional area of profiled steel sheeting within width of slab b

x distance to slab cross-section measured from centre of closer end support

R, ultimate vertical reaction at end support per unit width of slab

It follows that, at any cross-section distance x from the closer end support, which can potentially fail by longitudinal slip:

• The resultant tensile strength force T cannot exceed the lesser value given by equation (1)

• Its moment capacity is determined from the M,p-T relationship for the cross-section (see section on cross- sectional analysis to derive the M,p-T relationship)

• Since the limiting value of T is affected by the magnitude of the support reaction R,, the moment capacity of the cross-section is affected by the distribution and magnitude of the applied loads (see section on limiting the moment capacity curve).

Magnitude of vertical force transmitted across interface According to equation (la), the vertical force transmitted across the interface between the sheeting and the concrete equals the support reaction Ru. The distribution of vertical interfacial pressure is assumed to be such that the pressures are concentrated at the supports and equal zero everywhere else.

The finite element method has been used to investigate the accuracy of these assumptions 4. If the vertical force transmitted directly through the sheeting pans into the concrete is represented by KR,, it can be shown that for Bondek II slabs the value of K is typically somewhere between 0.9 and 1.0 (noting that the remainder of R, is transmitted as shear through the webs of the sheeting ribs). However, because there is greater uncertainty in the value of IX than there is in K for Bondek II slabs (e.g. see Figure 2), in design is it quite acceptable to assume that K = 1. The vertical interracial pressures are found to be relatively small in magnitude away from the supports and can be ignored.

Diagonal shear cracking The free-body diagram in Fig- ure 4(a) can be an over-simplification of the equilibrium state at ultimate load. The edge of this free body may be bounded by a diagonal or inclined crack when there is a biaxial state of stress due to combined flexure and shear. The diagonal crack is an extension of a flexural crack which forms in the shear span of the slab before reaching ultimate load. If the flexural crack forms a distance 3"Dcs on the low- moment side of the critical cross-section, and the contri- bution dowel action makes to moment capacity is ignored (a justifiable assumption for profiled steel sheeting), then it can be shown that the resultant tension in the sheeting at a distance (x - 3"Dcs) from the support is governed by the bending moment M* at distance x. Therefore, and also including the factor K introduced in the preceding subsection, equation (la) has been modified to

T = Hrib/br (x + Lc - 3"Dcs) + IX (mY,)

<-- fsy,h Ae/b

(2a)

(2b)

where: mY, = vertical force at the closer end support transmitted directly through the sheeting pans into the concrete such that 0 --< K --< 1 (i.e. K = 0 if the sheeting stops short of the supports); and 3'Des = the distance from the start of the diagonal crack to the critical cross-section distance x from the centre of the support, whereby normally it can be assumed that 0-< 3' -< 2 4. It should be noted that 7Dc~ should not exceed x in equation (2a)).

For the sake of simplicity, in this paper it will be assumed that 3' = 0, i.e. diagonal shear cracking will be ignored, and since it will also be assumed that K = 1, equation (2a) gives the same results as equation (la) and is only shown here for generality.



Cross-sectional analysis to derive M,p-T relationship

Analysis of a slab cross-section must be performed in order to derive the unique relationship between moment capacity M,p and resultant tensile force T, i.e. the M,p-T relationship. In the case of a prismatic slab with constant geometric and material properties along its length, which is the situation being considered here, the relationship applies equally to all cross-sections of the slab irrespective of their distance from the supports.

Moment-curvature analysis The influence that longitudinal slip has on the development of the resultant tensile force T at a slab cross-section can be studied using moment-curvature analysis. For a particular profile (defined by its geometry and base metal thickness tbm), overall slab depth De,, and set of material properties (i.e. consisting of constitutive relationships for the steel and the concrete), curves for different slip strains es~ can be computed from compatibility and equilibrium considerations.

Profile geometry and concrete section The profile geometry can normally be reasonably accurately represented by a compact, monosymmetric I-section shown in Figure 5. For simplicity the concrete section can normally be assumed to consist of two rectangles comprising a cover slab of depth (Dcs-hr) and width b, and the concrete between the sheeting ribs of depth (hr-tbs) and average width b', where hr is the overall height of the steel sheeting ribs.

The dimensions of the 1-section can be determined approximately from the nominal profile geometry defined by the manufacturer. (For the sake of this discussion it will be assumed that the profile is compact under positive bending, so that compressive parts of the steel section can fully yield.) However, because the sheeting is stretched transversely in the region of the bends during the roll-forming operation, it is thinner in these regions and therefore does not have a uniform thickness. Accordingly, it is inaccurate to estimate the cross-sectional area A e by calculating the theoretical perimeter length and assuming an average base

metal thickness equal to that of the unstretched material away from the bends. For design purposes, a better estimate for Ap can be obtained by assuming that the cross-sectional area of the profile with a module width w equals the cross- sectional area of the steel strip prior to roll-forming. This requires one to know the nominal feed width of the steel strip from which the profile is roll-formed, which can be obtained from the manufacturer.

For example, consider the nominal Bondek 1I profile geometry shown in Figure 6 with a module width w = 590 mm. The nominal width of the steel strip prior to roll-forming the product is 990 mm, and therefore the nominal cross-sectional area Ap of the profile can be assumed to be 1.678 thmb mm 2 (i.e. 1.678 = 990/590) per width of slab b. Using the nominal overall dimensions given in Fig- ure 6, initial estimates for the I-section dimensions (per module width w = 590 mm) are obtained as follows

try = tbm

b r y = 3 x 3 2 + 2 9 = 125mm

tw = 6tbm/sin 78 ° = 6.134 tb,.

hw = 52 mm

bhl = 5 9 0 - 2 × 1 3 = 5 6 4 m m

tb~-= tb,.

while

b' = 590 - 2 x 22.5 = 545 mm

It follows that

A n = bo~ t,. + hw tw + bbi tbl

= [(125 + 52 × 6.134 + 564)/590]th,, b mm 2

= 1.708 tb,, b mm 2

> 1.678 tb,, b mm 2

c s

4 b P

:s-hr

I hr-tbf

Figure 5

b' (ignoring steel sheet)

Model for composite slab cross-section



Longi~th~

Joint 200 1 ~ 200 d

t b m = 0.75 or 1.00

Average rib spacing b r = (200 + 200 + 190) / 3

= 197 torn

Figure 6 Nominal Bondek II profile geometry

F~male Lapmb

Table 1 Bondek II nominal profile dimensions for I-section model

Dimension (mm) Per 590 mm width Per metre width (see Figure 5) of slab of slab

t~ tb~ ~ b~ 123 208 t~ 6.026 tb= 10.214 ~ h~ 52 52 bbf 554 939

b' 545 924

In order to overcome this slight discrepancy, the estimates above for dimensions be, tw and bbl have been multiplied by (1.678/1.708) = 0.9824 to arrive at the nominal profile dimensions given in Table 1.

Constitutive relationship for steel sheeting The stress- strain curve for the steel sheeting can normally be assumed to be a straight line from zero-stress to the yield strength fsy.sh at a slope equal to the modulus of elasticity E,, strain assumed to increase thereafter at constant stress ignoring any strain-hardening, i.e. the steel exhibits linear-elastic plastic behaviour. For design purposes Es can be taken as 200 GPa.

The high-tensile steel strip used to produce Bondek II has a nominal yield strength f,y.sh of 550 MPa, does not strain-harden, and exhibits a flat yield plateau before frac- turing in tension. When tested at a slow strain-rate the resulting stress-strain curve can be slightly rounded as it approaches the yield plateau, as shown in Figure 7 for a

48o

Basu

1 6 0 ~ 8 0 TTesl

Ol I I

..,7

0 2500 50(]0 7500 10(~0 12500 15000 17500

T ~ . ~ e Strain (p~)

Figure 7 Example of stress-strain curve for Bondek II steel sheeting

case with tbm = 1.00 mm, in which case the form of the constitutive relationship (at least at high stresses) is assumed to be 9

f s t / f s y . s h = loge [(1 + ef3)/(l+ef3(l-e,%,/fsy.sh))]//3 (3)

where

f~, steel tensile stress /3 coefficient es, steel tensile strain corresponding to fs,

For the example shown in Figure 7, it can be seen that the curve derived for/3 = 5 is a close fit to the test result. The effect that rounding of the stress-strain curve has on the M,p-T relationship of typical under-reinforced and over- reinforced cross-sections has been theoretically studied for the Bondek II profile and found to be insignificant for design purposes 4. Therefore, linear-elastic, plastic behaviour of the steel has been assumed while deriving the M.p- T relationships used in later analyses.

Constitutive relationship for concrete. The compressive stress-strain relationship adopted for normal-density concrete is that recommended in the commentary to the Aus- tralian Concrete Structures Standard AS 3600 l°a~ and described by the Comit6 Europren du Brton (CEB) ~2 as follows

fJ fo = (al - a2 ec)ec/(1 + a3ec) (4)

where

fe concrete compressive stress (MPa) ee concrete compressive strain fo strength parameter determined such that the maximum

value of fc (which occurs at ec = 0.0022) equals f~. (MPa) (see equation (5))

al 39 0 0 0 ( f o -1- 7 ) -0.953

a2 20 6600 a 3 65 600 (fo + 10)-I.o85 _ 850

The numerical analysis is not continued when the strain ec in the extreme fibre at the top face of the slab exceeds a



value of 0.18877 ~, + 7) I).953 which corresponds to a stress f of zero in the top face.

It can be shown that f , can be estimated as

f , = -0 .3563 + 1.0268f~ for 10 ~ f~ < 20 (5a)

= 0.0095 + !.0099.f~ for 20 --< f~. --< 50 (5b)

Subdivision of cross-section elements The concrete and steel elements should be subdivided into a sufficient number of horizontal layers to achieve accurate results. In Fig- ure 8, the steel flanges are also shown to be subdivided vertically, which in particular allows the effects of transverse residual stress patterns developed during roll-forming to be studied.

The tensile strength of the concrete is conservatively ignored in the analysis.

Slip strain e,~ Slip strain e,t can be expressed as a linear function of cross-section curvature X such that (see Fig- ure 8)

e,./= 8(D,, - h,. + hp)x (6)

where

6 slip factor, such that 6 = 0 and 6 = 1 correspond to complete and zero interaction, respectively, noting that for some cross-sections with complete shear connection it is possible that 6 > 0 (e.g. see Figure 9(a))

hp depth of centroid of profiled steel sheeting below the tops of the ribs

The analysis is performed by keeping the value of 6 constant noting that 0 --< $ -< 1.

Sheeting residual and initial stresses Residual and initial longitudinal stresses locked in the sheeting before the slab acts compositely primarily arise from two sources: firstly, while the profile is being roll-formed, unpredictable resultant residual tensile and compressive stresses are induced in the sheeting which vary across the width of the sheeting; and secondly, while the sheeting acts as formwork supporting the wet concrete immediately before the concrete hard- ens, initial flexural stresses develop in the sheeting which vary according to the magnitude and distribution of load, the support conditions, the position in the span, etc. Design values for these latter stresses can normally be adequately estimated theoretically.

The effects that these residual and initial stresses have on the M,p-T relationship of typical under-reinforced and over-reinforced cross-sections has been theoretically studied for the Bondek II profile and found to be insignificant for design purposes 4. Therefore they will not be considered further here, and these stresses have been assumed to equal zero while deriving the M,~,-T relationships used in later analyses•

v I

i i i i i

h r ~ . ,

I , i \

~ X

1

Tens ion • C o m p r e s s i o n

S h e e t i n g Cen t ro id

Figure8 Strain distr ibut ion and subdiv is ion of cross-section elements

Typical M,p-T relationships Two typical M,p-T relationships are shown in Figures 9(a) and 9(b) for an under- reinforced and an over-reinforced cross-section, respectively. They have been derived for slabs incorporating Bon- dek II sheeting assuming th,,, = 1.00 mm and f,,,j, = 550 MPa. For the under-reinforced slab Dc~ = 200 mm and f / = 32 MPa, while for the over-reinforced slab D,~ = 90 mm and f~ = 25 MPa.

As can be seen in Figure 9, the M,p-T relationships have been derived by conducting the moment-curvature analysis for values of 6 from 0 to 1 in increments of 0.1, and fitting straight lines between the resulting points. The following general comments are made with respect to these curves.

• By assuming the sheeting to be compact in positive bending, the plastic moment capacity Mp of the sheeting is developed when T= 0 (i.e. for 1.00 mm Bondek II, Mp = 13.85 kNm m -~)

• The maximum resultant tensile force T developed in the sheeting cannot exceed f~y.~,h A p / b = 922.9 kN m -~ for 1.00 mm Bondek II (see Figure 9(a))

• Their shape is concave downward, and therefore a straight line connecting any two points on one of the curves yields a conservative approximation to the curve (see section on alternative &Mup-T relationships for design)

Rectangular stress block theory Rectangular stress block theory may be used to calculate the moment capacity of

140- E

Z 120-

~ o o

.--- 80-

60-

40-

2o-

a

0 5 0.4 0.6 to 0.0

0 . 7 ~

~ M o f n e n ~ a t t ~ ¢ Ar~ysis

- - Simple-Plastic RSB Them~

I t I 4 I I I t 100 200 300 400 500 600 700 800 900 tO00

Resultant Tensi le Force T ( k N / m )

5O

~" 45

40

35 ~° 25 2

~ ~o

E 10

0.3 0 0

0.7

0.8

0.9

Moment-C~vat~/u~alysis

Simple-l~astic RSB Theory

O I I I * I I 4 100 200 300 400 5O0 600 7~1 8OO

b Resultant Tensi le Force T ( k N / m )

Figure 9 Typical M,p--T re lat ionships (Bondek II sheeting, to,, = 1.00 mm). (a) under-re inforced cross-sect ion (Des = 200 mm, fc = 32 MPa); (b) over- re inforced cross-section (De8 = 90 ram, fc = 25 MPa)


Partial shear connection design of composite slabs: M. Patrick and R. Q. Bridge

cross-sections of composite slabs which exhibit either complete or partial shear connection.

The calculation principles assumed for simple-plastic rectangular stress block theory are stated as follows (see Figure 10), and only sheeting and concrete have been assumed to be present.

• The concrete has zero tensile strength • A uniform compressive stress (equal to 0.85f~. in Figure

10) develops in the top of the concrete slab • The compressive force in the concrete cannot exceed the

total longitudinal shear force which can be transferred by the shear connection between the sheeting and the concrete at ultimate load

• The portion of the sheeting in tension is stressed uniformly to the yield s t r e s s fsy.sh

• Any portion of the sheeting in compression is stressed uniformly to the yield stress fsy.sh

• The resultant tensile force in the sheeting equals the compressive force in the concrete

• The effects of vertical shear on the stress distribution can be ignored

Although the equations for calculating moment capacity M,p corresponding to resultant tensile force T are simple in principle to derive, many different situations arise for a cross-section like that shown in Figure 5 (see below). This makes it essential that designers have computer software to perform the task if they are to use the method in practice. It is beyond the scope of this paper to present these equations here.

approach is to treat the sheeting as a line element with its cross-sectional area Ap concentrated at its centroid, and to then use the normal flexural strength theory for reinforced- concrete members which assumes a linear variation of strain with depth as shown in Figure 11. In the rectangular stress block theory of AS 3600 the following assumptions apply

• Plane sections normal to the longitudinal axis remain plane after bending

• The maximum strain e, in the extreme compressive fibre of the concrete is 0.003

• A uniform compressive stress of 0.85f'~ acts on the concrete to a depth of Tk, d from the extreme compressive fibre, where T = [0.85-0.007(f'~-28)] within the limits 0.65 and 0.85

Consider an under-reinforced cross-section with complete shear connection and the neutral axis in the cover slab, i.e. kud <- (Dcs-hr). Applying the principles stated in the preceding subsection, it would be assumed that the full tensile capacity of the sheeting Ap fsy.sh can be developed and that

k,d = Apfsy.sh](0.85 f'c Tb) (7)

whereby

M. = Ap f, y.,h d(1 - 0.5 7k,) (8)

Cross-sections with complete shear connection Three locations of the plastic neutral axis need to be considered in practice for cross-sections with complete shear connection, viz.: in the concrete cover slab; in the top flange of the sheeting ribs; or in the web of the sheeting ribs (see Figure 10). The first of these situations is for under- reinforced cross-sections, while the latter two are for over- reinforced cross-sections. Once formulated, the equations can be used to calculate the point on the M,p-T curve corresponding to flexural capacity M, and maximum resultant tensile force T,, e.g. M, = 154.7 kNm m -1 and T, = 922.9 kN m-1 for the under-reinforced cross-section examined in Figure 9(a).

The principles stated in the preceding subsection assume that the strain gradient across the plastic neutral axis is infi- nite, i.e. strain compatibility is ignored. An alternative

However, strain compatibility shows that the strain in the top fibre of the sheeting flange cannot be yielded in tension unless k,d is less than the following limiting value (calculated assuming Es = 200 GPa)

( k u d ' ) t i m = [600/(600 + f,y.sh)] (Des -- hr) (9)

For the under-reinforced cross-section examined in Figure 9(a), equation (7) gives k,d= 41.3 mm which is less than ( k , , d ) l i m = 76.2 mm calculated using equation (9). Therefore it is valid to use equation (8), whereby if it is assumed that d = (Des - hr + hp) = (De, - 15.46) mm (i.e. hp = 38.54 mm and hr = 54 mm for 1.00 mm Bondek II), then M, = 154.7 kNm m -1 which is naturally identical to the previous value calculated using rectangular stress block theory ignoring strain compatibility.

hp

Sheetin 8 Centroid

Figure 10 Simple-plastic rectangular stress block theory

m

m

fsy.sh A;/b [ .

I

Engng Struct. 1994, Volume 16, Number 5 3 5 5


H £U =0,003 ~._~0.85f¢

T , , : / ~L_~

~" i Es t T~ fsy.sh Ap /b

Sheeting Centroid

Figure 11 Rectangular stress block theory with strain compatibility

For the over-reinforced slab cross-section examined in Figure 9(b), equation (7) gives k.d = 51.1mm > (D~. - h.) which confirms that the equation cannot be used in this case. Furthermore, since d=90-15 .46 = 74.54 mm, ku = 0.686. The neutral axis parameter for balanced conditions k.. is given by

k.~ = e . l ( e . + fsy.s,dED (lO)

and therefore k.b = 0.522 < 0.686 confirming that the cross-section is over-reinforced. It follows that the neutral axis parameter k. can be calculated, assuming all the steel to be concentrated at its centroid at depth d, and all the steel to be elastic (i.e. no partial plasticity), as

k. = (u 2 + 2 u ) 1/2 - U (11)

where

u = 0.6 e. E, Ap/(Tf'~ b d) (12)

and finally

M. = 0.85 Yf'~ b cF k.(1 - 0.5 yk.) (13)

Using equations (11)-(13), k. = 0.571 and M. = 43.9 kNm m - I. This compares to values shown in Figure 9(b) of 49.6 and 49.3 kNm m - l calculated using moment curvature analysis and simple-plastic rectangular stress block theory, respectively, Therefore, the rectangular stress block theory accounting for strain compatibility and assuming concentrated steel is slightly more conservative.

Cross-sections with partial shear connection Once the extreme point (T., M.) corresponding to complete shear connection has been computed, other points (T, M.p) can be calculated along the M.p-T curve such that Mup < M. and T < T.. This can be approached by systematically varying T between zero and T., for example in equal fractions of T. (see below). In cases involving partial shear connection and applying the simple-plastic theory, the plastic neutral axis can only fall in the steel section, and it is necessary to consider when it is either in the top flange, the web, or the bottom flange of the sheeting.

The points determined using the simple-plastic rectangular stress block theory are shown on the M.p-T relationships in Figures 9(a) and (b) to which the following comments apply.

• Ten points were generated for each curve, i . e . T . / 9 between each consecutive pair

• For the under-reinforced case in Figure 9(a),the points effectively fall exactly on the curve generated using the more exact moment-curvature analysis

• The same is true for the over-reinforced case in Figure 9(b), except that the rectangular stress block theory slightly over-estimates the value of 7". corresponding to complete shear connection

• In both examples there is almost exact agreement between the values of M. calculated using the two the- ories, viz.: 156.1 versus 154.7 kNm m -~ in Figure 9(a) and 49.6 versus 49.3 kNm m l in Figure 9(b) for the moment-curvature and rectangular stress block the- ories, respectively.

Therefore, for the two examples cited, the simple-plastic rectangular stress block theory generally predicts the M.p- T relationship well.

Alternative C~Mup-T relationships for design It will be assumed here that in design it is the design moment capacity ckM.p (including ~bM. and ~pMp) which must be calculated as a function of the resultant tensile force T. Nominal material properties and dimensions of the sheeting and slab concrete are used to calculate the nominal moment capacity M.p for a particular value of T, and the M.p value is then multiplied by the capacity factor ~b to give the design value.

The M~p-T c u r v e for the under-reinforced Bondek II slab of Figure 9(a) has been converted into the design curve shown in Figure 12. Since nominal material properties and dimensions of the sheeting and slab concrete were used to calculate the M.p-T curve in Figure 9(a), it has merely been

u =123.8 kNm/m

125

, I <o,Too.o.

75 ~- Simple-Pl~tic ~ 7 - ~J " "<ii) 1

..~ 25 " : " / i . : ~ / " / T --922.9 kN /m

/ 0 100 200 30o ,too 5o0 60o oo 60 I

L/(~M p -11 A kNr~m Resultalrlt Tensile Fot~'e T (kN/m)

Figure 12 Alternative ~bMo~T relationships for design



necessary to multiply the M.. values by ~b. A value of ~b = 0.8 has been adopted for this example since this is the value used in AS 3600 for situations when k. -< 0.4,

Although the more complete curve shown in Figure 12 (which was derived using simple-plastic rectangular stress block theory) can be used in design, at the expense of some loss of accuracy the calculations can be simplified by approximating the curve by one of the following.

• Line (i), drawn between the origin and the point (T., ~bM.) corresponding to complete shear connection

• Line (ii), drawn between the points (0, c~Me) and (T., ~bM.)

• Line (iii), which includes the points (0, ~bMp) and (T~, ~bM~) along with an intermediate point such as at T = 0.5T., i.e. (0.57"., M~p, 05)

However, line (i) is generally too conservative and its use is not recommended except for deep slabs with c~M e < < ~bM.. Line (ii) provides a reasonable fit (see the section on the worked example) which improves as Des increases, but the value of ckM. must be calculated. This can be readily done using simple-plastic rectangular stress block theory by approximating the sheeting as an I-section as shown in Fig- ure 5, and this only needs to be performed once for each base metal thickness of any particular profiled sheet. Finally, line (iii) is a very good approximation to the full theoretical curve, and simple-plastic theory is the most direct way of calculating a point like (0.5T., M..,o.5), although this does require the necessary equations to have been formulated for the partial shear connection cases.

Limiting moment capacity curve According to equation (la), when the frictional resistance /JR. develops at the end of a slab as load is applied, the maximum resultant tensile force T which can develop in the sheeting at cross-sections adjacent to the support increases (at least until fsy.sh Ae/b is reached). Therefore, the moment capacity of each cross-section can be affected by the loading conditions.

The concept of a limiting moment capacity curve has been developed for this situation when strength is affected by loading, and when it is necessary to compute the magnitude of the failure loads TM, rather than just checking whether a slab has adequate strength to support loads of known magnitude. The curve is established for a particular load case by assuming each cross-section at regular intervals along the slab is critical (i.e. M* = M.p, or ~bM.p in design).

An example of a limiting design moment capacity curve is shown in Figure 13. Its shape is affected by the loading pattern, the c~M..-T relationship, and the longitudinal slip resistance of the profile, i.e. the values of Hrib/b r and i~. The curve has been derived for a uniformly loaded, simply supported Bondek II slab with the dpM.p-T curve shown in Figure 12 (approximated by line (ii)), and assuming Lc = 0. With f'c = 32 MPa and tbm = 1.00 mm, it follows from the section on the shear connection performance from the slip block test that nrib/b r - - 650 kPa while/x = 0.8. Because the limiting design moment capacity curve is symmetrical about the centre of the slab, it is only shown for 0 <--x <-L/2. For cross-sections close to the ends, very large loads with correspondingly large end reactions R. (R* for design) and therefore frictional resistance/zR.(/xR* for design) must develop for these cross-sections to be critical.

125

I00

75

~ 50 ".G

I I I I 500 I(XX) 1500 2(X)0 2500

D i s t a n c e x from End Support ( r a m )

Figure 13 Example of l imiting design moment capacity curve

Therefore, at cross-sections adjacent to the end of the slab the limiting design moment capacity curve rises sharply, and very close to the ends reaches a maximum possible value of d~M.. Frictional resistance has less influence at cross-sections further away from the ends, and the shape of the curve is affected more by the shape of the ~ M u p - T relationship. Eventually, at cross-sections sufficiently far away from the ends, the cross-sections may theoretically attain their design flexural capacity ~bM. irrespective of the magnitude of the design support reaction R*.

Determination of critical cross-sections and ultimate load Once the limiting moment capacity curve has been calculated, the position(s) of the critical cross-section(s) and the magnitude of the ultimate load to cause failure can be determined. A critical cross-section is defined herein as that where the applied bending moment diagram touches the limiting (design) moment capacity curve, that is M* = fbMup, all other cross-sections having M* < c~M.p. The

bending moment diagram for the uniformly loaded Bondek II slab (with a span L assumed to equal 5000 mm) has been superimposed on the limiting design moment capacity curve in Figure 13 (see Figure 14). The curves touch at the midspan cross-section, which is therefore the critical cross-section where it can be seen that M* equals the design flexural capacity ~bM.. The maximum design load in this example is therefore governed by the flexural capacity of the midspan cross-section rather than the longitudinal slip resistance of the profile.

Moment capacity curve corresponding to maximum design load Once the maximum design load has been determined, the design moment capacity qbM.p of each cross-section corre-

1 2 5

72

. ~ 5 0 "

2 5 -

Limiting Design Moment • . . . , -" " " " " "" "- . . . . . . . .

C a p a c i l y C u r v e • • . . . - "" • .'"

~ . ~ ~ D e s i g n Moment Capacity C u ~ e

• , " * C r i t i c a l / Cioss-scction

/ • ( x = 2 5 0 0 ) / . "

I I I I I I I

25(I 5 0 0 7 5 0 1000 1250 1500 1 7 5 0 2 0 0 0 2 2 5 0 2 5 0 0

Distance x from End Support ( m m )

Figure 14 Critical cross-section at intersection of l imiting design moment capacity curve and bending moment diagram, including design moment capacity curve



sponding to this loading condition can be calculated, although this is not an essential step in the analysis process. The first step is to calculate the design end reaction R*. The resultant tensile force T which can develop at typical cross-sections distance x from the end can then be calculated directly using equation l(a) (replacing R, by R* for design), and the corresponding design moment capacity 95M,p of each cross-section (see the heavily-dashed line in Figure 14) can be determined knowing the 95M, p-T relationship. The same procedure can be followed when it is being checked whether or not a slab has adequate strength to support loads of known magnitude (e.g. see 'Checks' in the section on change in loading pattern).

It should be noted that in situations when the loading pattern is asymmetrical about the centre of the slab, the procedure described in the preceding sections needs to be followed considering the slab as a whole.

Worked example The new partial shear connection strength theory presented in the previous section can readily take account of many factors which affect the ultimate strength of composite slabs. The Bondek II slab (tb,, = 1.00 mm, Des = 200 mm, L = 5000 mm, Lc = 0 mm, f~ = 32 MPa) studied previously will be used in a design example to demonstrate this. In the context of this example, the ultimate strength of the slab will be considered to be defined by the design moment capacity 95Mup of the critical cross-section.

It will be shown, for example, that the ultimate strength of the slab would be altered by a change in the loading pattern, a reduction in H,b/b, a change in the sheeting support conditions, addition of cantilever portions, and a reduction in the shear span Ls.

The Steel Deck Institute (SDI) in America has recently published a composite slab design handbook 14 which includes a newly proposed method for ultimate strength design. It is stated in the handbook that the design method does not apply 'to minimally connected or unconnected composite single span slabs, without pour stops or other end restraints'. Therefore, some longitudinal slip resistance must always be available from end anchorage for the design method to be applicable, although the exact amount is not specified in the handbook. The spans are assumed to be simply supported, and when designing slabs without welded studs acting as end anchors, the following assumptions are made 14,~5.

• The design moment capacity of the cross-section under maximum positive bending can be calculated using modular ratio theory, which is based on linear elastic analysis and assumes the following: a no-slip condition exists between the sheeting and the concrete, i.e. complete interaction and complete shear connection; longitudinal strains are linearly distributed in the section; and the concrete carries no tensile stress.

• The tensile stress in the extreme bottom fibre of the sheeting does not exceed 0.6f,.y_,h under working load after taking stage construction effects into account. (In the SDI handbook it is assumed that the dead load of the concrete does not act on the composite section, and therefore the working load is taken as the live load.) The compressive stress in the extreme top fibre of the slab does not exceed 0.45f'~, this additional condition being specified in the SDI handbook. (The SDI handbook also

states that the resulting live load capacity may be increased by 10% if welded-wire fabric is present in the slab, which approximately equates to a 10% increase in the permissible stresses.) According to these assumptions, the moment capacity of the critical cross-section of a simply-supported composite slab is not affected by any of the factors mentioned above which are to be studied in the following sections. The implications of this in regard to the assumed strength of the end anchorage will be studied throughout the course of the example.

For the Bondek II slab being studied, it can be shown that the maximum live-load bending moment is 67.2 kNm m-~ (which was governed by the concrete limit rather than the steel limit, which otherwise would have given a value of 83.8 kNm m i). It has been assumed that the sheeting is fully supported when the concrete is cast. Therefore, the presence of any initial stresses arising in the sheeting from the framework stage has been ignored (noting that the SDI method, being conservative, always requires some allowance to be made for this). However, no benefit has been assumed for the presence of welded-wire fabric.

When welded studs are present in 'sufficient quantity' at the ends of the sheets, the method in the SDI handbook is based on the assumption that the design flexural capacity 95M, can be developed at the maximum moment cross-section. It is further assumed that at this cross-section the full tensile capacity of the sheeting AS,,.~, can be developed. whereby M, can be calculated using equations (7) and (8) with 3' = 0.85, and then using a value of 95 = 0.85 to calculate 95M,. (Note that the SDI handbook incorrectly does not require the cross-section to be under-reinforced, which could for example be checked using equation (9)). By 'sufficient quantity', a gross assumption is made that the anchorage force T, per unit width of slab needed to be sup- plied by the studs is given by the following equation

T, =f, ysh (Ap - Aw/2 - At~f)/b (14)

where A,, = cross-sectional area of sheeting webs within width of slab b; and Abl = cross-sectional area of sheeting bottom flange within width of slab b.

The fixing of the sheeting to the steelwork supporting its ends must be designed to transmit this tensile force. For Bondek II, it follows from the section on the profile geometry and concrete section that 7",, = 260 th,, kN m with tbm in millimetres.

In the example it will be assumed that the design ultimate load W, is calculated with load factors of 1.2 and 1.6 for dead (Wd) and live (W~) loads, respectively H. The 95M.p-T curve will also be approximated by a straight line between 95Mp= 11.1 kNm m -l and ohM,= 123.8 kNm m --1 (i.e. line (ii) in Figure 12). As mentioned in the section on alternative 95M.p-T relationships for design, 95 has been taken as 0.8 compared with the SDI value of 0.85, and this will have only a small effect on the comparisons.

Change in loading pattern The Bondek II slab will be assumed to be loaded symmetri- cally about its midspan cross-section by two line loads each of magnitude 1.6Wd2 and with a shear span Ls = 1000 mm (see Figure 15). The factored dead load of 1.2Wd is assumed to equal 6.0 kPa, i.e. the density of the concrete including an allowance for reinforcement is assumed to be 2 5 k N m 3.



Slab self-weight

1,2W d = 6,0 kPa

1.6W/ / 1.6Wl

_ L,o / 2 ~'" ~ ;on:e;; tb;'~" i:00 ~ " ~' - - ~ Dcs= 200

I I

L = 5000 ,,I I)

Figure 15 2-pt Loading pattern on Bondek II slab for worked example

M* = t~Mup - - 100.6 kNm m -~ at critical cross-section at x = 1000 mm

¢~Mup = 123.8 kNm m - I at midspan cross-section

M* = 107.35 kNm m - 1 at midspan cross-section

Checks

(a) At the critical cross-section, x = 1000 mm By equation l (a) (replacing R, with R* for design)

T = 6 5 0 x ( 1 0 0 0 + 0 ) / 1 0 0 0 + 0 .8 × 103.6

= 732.9 kN m -1

Assuming a straight-line fit between ~bMp = 11.1 kNm m -1 and ~bM, = 123.8 kNm m -1,

m

i

Limiting Design Moment Capacity Curve

125

/ ~ Design Moment / Capacity Curve

/ . . , Or~p

M

Critical Cross-s~cLion

(x=1000)

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 Distance x from End Support (mm)

Figure 16 Design of Bondek II slab in Figure 15

Firstly, however, it was shown in Figure 14 that when uniformly loaded, the Bondek II slab will develop its design flexural capacity 4~M, without any need for end anchorage. It follows that

~bM,,

1.2WdL2]8

1.6WtL2/8

WtL2/8

= 123.8 kNm m -~

= 18.75 kNm m -~

= (123.8-18.75) = 105.0 kNm m -1

= 65.6 kNm m -1

It is interesting to note that this value of l ive-load bending moment, which corresponds to flexural failure of the midspan cross-section, is almost exactly the same as that per- mitted by the SDI method ignoring the presence of welded studs, i.e. 67.2 kNm m - l .

With studs the SDI method gives ckM, = 131.5 kNm m -~ (noting again that 4, = 0.85 instead of 0.8). Correspond- ingly, WtU/8 = 70.5 kNm m -~ in which case including studs only slightly improves the live load capacity (viz. 67.2 to 70.5 kNm m - l ) , and it hardly seems worthwhile accounting for their presence.

The results obtained from analysing the two-point loading situation in Figure 15 are shown in Figure 16. It has been determined that

1.6Wd2 = 88.6 kN m -1

R* = 103.6 kN m 1

4,M ., = 4~Mp + (T/T.) (4~M. - r~Mp)

= l l . 1 + (732.9/922.9) x (123.8-11.1)

= 100.6 kNm m -~ (which is acceptable)

(b) At the midspan cross-section, x = 2500 mm By equation l (a) ,

T = 650x(2500+0) /1000 + 0.8x103.6

> 922.9 kN m -1 (=fsy.shAp/b)

dpMup = 49M.

= 123.8 kNm m -~ (which is acceptable)

> M* confirming that the section is not critical.

This example shows that at the critical cross-section, ~bM, p has been reduced from 123.8 to 100.6 kNm m -1 by the change in loading pattern. The unfactored live load component of this design moment capacity is only 55.4 kNm m -1, which is 82% of the value of 67.2 kNm m -1 given by the SDI method, and therefore the SDI method is unconservative in this case.

Finally, Figure 17 shows the situation which occurs when anchorage of 150 kN/m is added to the ends of the sheets. The capacity line lifts sufficiently for the design flexural capacity ~bM, to be reached at the critical cross-

125

~100"

75"

o

25"

Limiting Design Momen ............................. Capacity Calve , '

/ , / , ' Design Momeat

C~ity L~'¢

," M*

0 I I ) I I ~ I I P 0 250 5(10 750 11~0 1250 1500 1750 2000 2250 2500

Distance x from End Support (ram)

Figure 17 Design of Bondek II slab in Figure 15 with end anchorage of 150 kN m -1 added



section at midspan. In comparison, the SDI method requires 260 kN m ~ of anchorage capacity to be added before it can be assumed that qbM, can be developed. The SDI method is therefore overly conservative in this case.

Reduction in Hr~b/br (x 0.2)

Bondek II is a 'new generation' profiled sheet which has been specially designed to develop strong mechanical interlock in its composite mode. In this regard its performance is exceptional compared with that of a typical dovetail or trapezoidal profiled sheet, which normally only develops a small fraction of the mechanical interlock Bondek II develops ~6. The performance of a typical profiled sheet used in the situation shown in Figure 15 can be assessed by assuming H,~h/b~ is reduced to only 20% that of Bondek II (i.e. 130 kPa in this example). While studying this example, the reader should remember that mechanical interlock is not a parameter in the SDI design method.

The results of the analysis are shown in Figure 18. It has been determined that

1.6Wd2 = 18.2 kN m -j

R*= 33.2 kN m -

M* = d~M,p = 30.2 kNm m ~ at critical cross-section at x = 1000 mm

~M,p= 54.0 kNm m ~ at midspan cross-section

M*= 37.0 kNm m ~ at midspan cross-section

In this case the design moment capacity ¢~M.p of the critical cross-section has been reduced from 100.6 to 30.2 kNm m- ~ by the change in rib resistance. (Because chM,p exceeds the cracking moment, the partial shear connection strength model is still considered valid.) The unfactored live load component of this design moment capacity is only 11.4 kNm m-1, which is much less than the value of 67.2 kNm m l given by the SDI method, and therefore the method is very unconservative in this case.

Figure 19 shows the situation which occurs when anchorage of 650 kN m - ~ is added to the ends of the sheets. The capacity line lifts sufficiently for the design flexural capacity to be reached at the critical cross-section at midspan. In comparison, the SDI method would have required only 260 kN m-~ of anchorage capacity to be added before the design flexural capacity could be assumed to be developed, and again the method is very unconservative.

Moreover, recent research has shown that in composite

-'~ IOO / " Capacity Curve / /

"~ Limiting D~ign Moment " SMup / Z Capacity Curve CriticaJ

~ 75 Cro~s-sc~tion (x=2500)

50

~ ~ M*

~ 25

0 I I I I I I I ~

250 5C0 750 1000 1250 1500 1750 2000 2250 2500

Distance x from End Support ( r a m )

Figure 19 Design of Bondek II slab in Figure 15 with Hm,/b~ reduced by 80% and end anchorage of 650 kN m ~ added

slabs incorporating end-anchored profiled sheeting which develops weak mechanical interlock, longitudinal reinforcement in the form of deformed bars or welded-wire fabric should be placed in the bottom face 17. This reinforcement ensures that a regular distribution of flexural cracks forms along the member instead of a single major crack forming at midspan. This is important for serviceability and signifi- cantly reduces the effect that longitudinal slip between the sheeting and the concrete has on the load-deflection response of the member. The SDI handbook makes no men- tion of these issues.

Change in sheeting support conditions (tzR*=-O) Often the ends of the sheets only extend 50 mm or so onto the end supports while the concrete is cast over the full support width. In this case it is prudent in design to assume that the frictional resistance/xR* is zero.

The results of the analysis with /zR* = 0 are shown in Figure 20, where it can be seen that the limiting design moment capacity curve has been omitted since it is the same as the design moment capacity curve. It has been determined that

1.6Wl/2 = 78.5 kN m-

~bM,p -- 90.5 kNm m-J at critical cross-section x = 1000 mm

chM,e = 123.8 kNm m-1 at midspan cross-section

M* = 97.2 kNm m-t at midspan cross-section

In this case, the design moment capacity dpM,p of the critical cross-section has been reduced from 100.6 to 90.5 kNm

125

tOO

,~ 75

i ~ 5 0 -

2 5 -

Limiting Design Moment Capacity Curve

Critical Cr~s-gcction

/ (x=IOOO)

.......... y ...................

I I I I I b I I I

250 500 750 10OO 1250 1500 1750 20~0 2250 2500

Distan~ x from End Support (ram)

Figure 18 Design of Bondek II slab in Figure 15 with HrJb, reduced by 80%

125 / f " Design Moment

Calmelty Curie ~M~p

100 / . . . . . . . . . . . . . . E . - - . . . . . . . .

Z J" ~'~ 75 ~ ~ Critical "~ / , ' Cross-section

/ . " (x=IOOO)

/ < .-"/ 50

25 / , " /

0 I I ~ I i ~ I I I

250 500 750 IOO0 1250 1500 1750 2000 2250 2500

Dis tance x from End Support ( m m )

Figure 20 Design of Bondek II slab in Figure 15 with sheeting support condi t ions such tha t /xR* = 0

360 Engng Struct . 1994, V o l u m e 16, N u m b e r 5


m -~ by the change in sheeting support conditions. The unfactored live load component of this design moment capacity is 49.1 kNm m -~, which is 27% less than the value of 67.2 kNm m-~ given by the SDI method, and therefore without frictional resistance at the end supports the SDI method is even less conservative than before. It can be shown that by assuming /.tR* = 0, the strength of the end anchorage must increase from 150 to 220 kN m - ' for a flexural failure to occur, and the SDI method still gives an incorrect value.

Addition of cantilevers (L¢ = 230 mm)

Adding a cantilever to the free end of a slab with the sheeting continuous over the support is equivalent to adding an end anchor, except that a negative bending moment is induced at the support due to the slab self-weight.

The results of the analysis with Lc = 230 mm are shown in Figure 21. It has been determined that

1.6WJ2 = 105.2 kN m -I

R* = 121.6 kN m -1

M* = ChMup = 123.8 kNm m -~ at midspan (critical) cross-section

It follows that with the cantilevers present, the design flexural capacity ~bM~ can be developed at the midspan cross. section. This is expected from the section on the change in loading pattern where it was shown that 150kN m -~ of end anchorage was required to develop (hM,, while a 230 mm long cantilever produces 0.23x650 = 150 kN m -1 of end anchorage (cf. Figure 21 with Figure 17), and the small negative bending moment over the support has an insignificant effect. The unfactored live load component of this design component capacity is 65.8 kNm m -1, which is close to the value of 67.2 kNm m -I given by the SDI method, and therefore this is the amount of end anchorage needed in this situation for the SDI method to be valid.

Reduction in shear span (Is = 500 mm)

Reducing the shear span L~ has the effect of reducing the anchorage length of the sheeting between the end support and the loading point, while this is normally counteracted to some degree by a corresponding increase in the design support reaction R*.

The results of the analysis with L~ = 500 mm are shown in Figure 22. It has been determined that

125 "

~ 100-

75-

. ~ 50-

25

I , ' Critical c::=oo / , '

/ / , Design Moment

Capacity Curve

Note : small negative be~ding moment at supp~l #

t ,A------C~ ' : , , , , , , , 250 500 750 10G0 1250 1500 1750 2060 2250

D i s t a n c e x f r o m E n d Suppor t (mm) 2500

Figure 21 Design of Bondek II slab in Figure 15 with cantilever portions of length Lc= 230 mm added

125

~ 1oo

E

• ~ 75 _E

~ 5o

25

Limiting Design Moment / / Capacily Curve

Z Capacity Ctawe

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ / , ' Cro~-~ection (x=500) / '

/ ,

I I I I ~ I I * I

250 500 750 I0~0 1250 1500 1750 21100 2250

Distance x f r o m E n d Support (mm)

2500

Figure22 Design of Bondek II slab in Figure 15with shear span Ls reduced to 500 mm

1.6Wd2= 113.1 kN m -]

R* = 128.1 kN m -1

M* = q)M,p = 63.3 kNm m -~ at critical cross-section at x = 500 mm

~bM,p = 123.8 kNm m -1 at midspan cross-section

M* = 75.3 kNm m -1 at midspan cross-section

It follows that the design moment capacity ~bM,p of the critical cross-section has been reduced from 100.6 to 63.3 kNm m-1 by moving the position of the load. The unfactored live load component of this design moment capacity is 35.3 kNm m -1, which is only 53% of the value of 67.2 kNm m -~ given by the SDI method, and again the SDI method is very unconservative. It can be shown that the strength of the end anchorage must be 320 kN m-1 for the design flexural capacity ~bM, to be attained at the midspan cross-section, which is greater than the value of 260 kN m-1 determined using the SDI method.

The results of the worked example are summarized in Table 2, and they demonstrate that the SDI method gives erratic and often very unsafe results compared with those derived using a soundly based physical model, which achi- eves its accuracy from the slip block test and partial shear connection strength theory which has been verified by extensive testing 4,7,8.

C o n c l u s i o n s

A new method for predicting the strength of simply-supported composite slabs has been presented. The partial shear connection strength model has been proposed for slabs which incorporate profiled steel sheeting with strong longitudinal slip resistance either due to strong mechanical interlock or the presence of end anchorage devices. Analy- sis of test results has shown that in this case it is more accurate than any other models developed to date. The method is readily adaptable to the design of continuous slabs and is being developed for an Australian Standard.

A new test called the slip block test yields essential information about the shear connection performance of profiled steel sheeting for use in the model. Because different profiled sheets can exhibit vast differences in the amount of mechanical interlock they develop, each product must be

Engng Struct . 1994, V o l u m e 16, N u m b e r 5 361


Table 2 Summary of results from worked example (Bondek II slab in Figure 15)

Case

Partial Shear Connection Strength Model SDI Method

c~M,p at critical cross-section (kNm m 1)

Unfactored live End anchorage load moment to develop (~M, capacity* (M, = 154.7, 4, = 0.80) (kNm m 1) (kN m -1)

Unfactored live End anchorage load moment to develop ~M, capaci tyt (M, = 154.7, <h = 0.85) (kNm m 1) (kN m 1)

PSCSM/SDI

Ratio of unfactored live load moment capacities ->1 ~ SDI OK

udl loading 123.8 65.6 0 2-pt loading 100.6 55.4 150 Hrib/b r x 0.2 30.2 11.4 650 /LR* = 0 90.5 49.1 220 Lc = 230 mm 123.8 65.8 0 Ls = 500 mm 63.3 35.3 320

67.2 260 = 1.0 67.2 260 0.82 67.2 260 0.17 67.2 260 0.73 67.2 260 = 1.0 67.2 260 0.53

*no end blocking assumed tassumes end blocking of undefined quanti ty

tested in slip blocks before it can be used in slabs designed with the partial shear connection strength theory.

In contrast, the Steel Deck Institute in America is pro- moting a method of design which presumes that the design moment capacity of a composite slab can be computed without specifically knowing the shear connection performance of the profiled sheet. A worked example in the paper has shown that consequently the method can be expected to give erratic and often unsafe results in a range of design situations. It is stated in the SDI composite slab design handbook that the method may be used to produce tables for proprietary products 'provided that confirming tests have been done'. However, no details of the tests are given, and the worked example has shown that loading pattern for example will have a significant influence on the outcome of any such tests. Moreover, the worked example shows that typical profiled sheets may not achieve the levels of strength assumed using the model, and that very unsafe designs can result. Therefore, it is strongly recommended that the present SDI method should not be used for design.

References 1 Patrick, M. 'A new partial shear connection strength model for com-

posite slabs', Steel Constr. J. Aust. Inst. Steel Constr. 1990, 24 (3), 2-17

2 Patrick, M. and Bridge, R. Q. 'Parameters affecting the design and behaviour of composite slabs', IABSE Symposium Brussels 1990 - Mixed Structures, including New Materials, IABSE Reports, Vol 60, July 1990

3 Patrick, M. and Poh, K. W. 'Controlled test for composite slab design parameters', IABSE Symposium Brussels 1990 - Mixed Structures, including New Materials, IABSE Reports, Vol 60, July 1990

4 Patrick, M. 'Analysis of Bondek II composite slab tests to British Standard BS 5950: Part 4: 1982', BHP Research, Melbourne Labora- tories Rep. BHPR/ENG/R/92/054/PS641K, April 1992

5 Lysaght Building Industries, Bondek II Composite Slabs, BDI1-2 Design Manual for Masonry Wall and Steel-Frame Construction, Part A - Composite Design Manual', Ref. No. BDII-2, August 1991

6 Patrick, M. and Bridge, R. Q. 'Design of composite slabs for vertical shear', Proc. Engineering Foundation Composite Construction H Conf., Missouri, June 1992 and Proc. Pac~c Structural Steel 3 Conf. Tokyo, October 1992

7 Patrick, M. 'Testing and design of Bondek II composite slabs for vertical shear', Steel Constr. J. Aust. Inst. Steel Constr. 1993, 27 (2), 2-26.

8 Patrick, M. 'Slip block test results for Bondek II profiled steel sheeting', BHP Research, Melbourne Laboratories Rep. BHPRML/PS64/91/002, June 1991

9 Basu, A. K. 'Computation of failure loads of composite columns', Proc. lnstn Civ. Engrs 1967, 36, 557-578

10 Standards Australia, Concrete structures, AS 3600-1988 (including AS 3600/Amendment 1/1990-06-04)

11 Standards Australia, 'Concrete structures Commentary (Supplement to AS 3600-1988), AS 3600 Supplement 1990

12 Comit6 Europren du B&on (CEB), 'Deformability of concrete structures - basic assumptions', Bulletin D'Information, No. 90, 1973

13 Patrick, M, 'Proposed Bondek II composite slab tests to British Stan- dard BS 5950: Part 4: 1982', BHP Research, Melbourne Laboratories Rep. BHPRML/PS64/91/029, January 1991

14 Heagler, R. B., Luttrell, L. D. and Easterling, W. S. Composite Deck Design Handbook, Steel Deck Institute, Ohio, (lst edn) 1991

15 Easterling, W. S. and Young, C. S. 'Strength of composite slabs', J. Struct. Engng, ASCE 1992, 118 (9), 2370-2389

16 Patrick, M. 'The slip block test - experience with some overseas profiles (Part A)', BHP Melbourne Research Laboratories Rep. MRL/PS64/90/021, June 1990

17 Patrick, M. 'Long-spanning composite members with steel decking', Proc. Tenth Inter. Specialty Conf. on Cold-Formed Steel Structures, St. Louis, MO, USA, October 1990, pp. 81-102


partial shear connection design of composite slabs

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