nie 2008 engineering-structures

8/14/2019 Nie 2008 Engineering-Structures

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Engineering Structures 30 (2008) 13961407

www.elsevier.com/locate/engstruct

Effective width of steelconcrete composite beam at ultimate strength state

Jian-Guo Niea, Chun-Yu Tiana, C.S. Caib,

aDepartment of Civil Engineering, Laboratory of Structural Engineering and Vibration of China Education Ministry, Tsinghua University, Beijing, 100084, ChinabDepartment of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, United States

Received 24 September 2006; received in revised form 21 May 2007; accepted 30 July 2007

Available online 10 September 2007

Abstract

In a steelconcrete composite beam section, part of the concrete slab acts as the flange of the girder in resisting the longitudinal compression.

The well-known shear-lag effect causes a non-uniform stress distribution across the width of the slab and the concept of effective width is usually

introduced in the practical design to avoid a direct analytical evaluation of this phenomenon. In the existing studies most researchers have adopted

the same definition of effective width which might induce inaccurate bending resistance of composite beam to sagging moments. In this paper, a

new definition of effective width is presented for ultimate analysis of composite beam under sagging moments. Through an experimental study

and finite element modeling, the distribution of longitudinal strain and stress across the concrete slab are examined and are expressed with some

simplified formulae. Based on these simplified formulae and some assumptions commonly used, the effective width of the concrete slab and the

depth of the compressive stress block of composite beams with varying parameters under sagging moments are analytically derived at the ultimate

strength limit. It is found that the effective width at the ultimate strength is larger than that at the serviceability stage and simplified design formulae

are correspondingly suggested for the ultimate strength design.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Steelconcrete; Composite; Effective width; Ultimate strength state; Experiment; Finite element analysis

1. Introduction

A steelconcrete composite beam consists of a concrete slab

attached to a steel girder by means of shear connectors. The

shear connectors restrain the concrete slab immediately above

the girder so that there is a non-uniform longitudinal stress

distribution across the transverse cross-section of the slab. Due

to the shear strain in the plane of the slab, the longitudinal

strain of the portion of the slab remote from the steel girder lags

behind that of the portion near the girder. This so-called shear-

lag effect causes a non-uniform stress distribution across the

width of the slab. To avoid a direct analytical evaluation of this

phenomenon, the concept of effective slab width (simply called

effective width hereafter) is usually introduced in practical

design in order to utilize a line girder analysis and beam theory

for the calculations of deflection, stress and moment resistance.

In a line girder analysis, individual girders are analysed instead

of analysing the entire bridge deck. The determination of the

Corresponding author.E-mail address: [email protected] (C.S. Cai).

effective width directly affects the computed moments, shears,

torque, and deflections for the composite section and also

affects the proportions of the steel section and the number of

shear connectors that are required.

Since the 1920s there have been many investigators

who studied the shear-lag effect in T-beam structures and

steelconcrete composite structures based on continuum

mechanics analysis, numerical method and experimental study

to develop realistic definitions of effective width. Adekola [1,

2] and Ansourian and Aust [3] studied the effective width

of composite beams using isotropic plate governing equationsin an elastic stage by numerical methods. It was found

that the effective width depends strongly on the slab panel

proportions and loading types and can only be used for

deflection and stress computations at serviceability level.

Johnson [4] studied the effective width of continuous composite

floor system at a strength limit state. Heins and Fan [5],

Elkelish and Robison [6], Amadio and Fragiacomo [7], Amadio

and Fedrigo [8] studied theoretically and experimentally the

effective width of composite beams in elastic and/or inelastic

stages. Results of these studies show that the effective width

0141-0296/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2007.07.027
http://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2007.07.027http://dx.doi.org/10.1016/j.engstruct.2007.07.027mailto:[email protected]://www.elsevier.com/locate/engstruct


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J.-G. Nie et al. / Engineering Structures 30 (2008) 13961407 1397

Notation

As tension area of steel beam section;

As compression area of steel beam section;

b width of concrete slab of composite beam;

be effective width of concrete slab of composite

beam;Ec elastic modulus of concrete;

Es elastic modulus of steel;

Et hardening modulus of steel;

f design strength of steel;

fc cylindrical compressive strength of concrete;

fcu cubic compressive strength of concrete;

ft tension strength of concrete;

fy yield strength of steel;

fu limit strength of steel;

hc height of concrete slab;

hs height of steel beam section;

L span of composite beam;Pu ultimate load of test

V shear force of the connectors between concrete

and steel;

Vu shear strength of shear studs;

zc depth between top surface of concrete and plastic

neutral axis;

zc0 depth between the plastic neutral axis and top

surface of concrete at y = 0

parameter presenting the degree of shear-lag

effect;

ratio of effective width to real width;

c compressive strain in concrete slab;

ct compressive strain on top surface of concreteslab;

curvature of concrete slab;

slip of the connectors between concrete and steel;

Poissons ratio;

height of rectangular-stress block to zc0 ratio;

c stress in concrete slab;

at the strength limit state is greater than that in the elastic

stage and can essentially be taken as the real slab width. Based

on the research results of these investigations, design codes

have adopted, in general, simplified formulae or tables for

the effective width evaluation in order to facilitate the designprocess [9,10]. These design codes use the same effective width

for both serviceability and strength limit states, thereby usually

underestimate the effective width at the strength limit state and

are too conservative for moment resistance computations.

Most previous studies have adopted the same definition of

effective width where the longitudinal stress is considered to

be constant over the effective width and the total longitudinal

force within the effective width is equal to the total force of the

actual stress distribution [1113]. However, when the effective

width from this traditional definition is used for the analysis of

composite beam sections with a simple beam theory, the total

bending moment in the concrete slab is usually different from

that based on the actual stress distribution, especially in the

strength limit state. As a result, an accurate value of resistance

to sagging moments of composite beam might not be obtained

by a simple plastic beam theory. Chiewanichakorn et al. [14]

recently proposed a different definition for the effective width

considering the through-thickness variation of stress in the

concrete slab. However, their study focuses only on compositebeams in the elastic stage, i.e., serviceability limit state. Effect

of shear lag at the strength limit state is different from that at

serviceability level.In this paper, a new definition of effective width is presented

for ultimate strength calculations of composite beams under

sagging moments using the commonly accepted rectangular-

stress block assumption. This new definition ensures that the

bending capacity of the simplified composite beam (effective

width plus block stress distribution) is the same as the

actual composite beam (actual slab width plus actual stress

distribution). Through an experimental study and finite element

analysis, the distribution of longitudinal strains and stresses

across the concrete slab are examined and expressed with somesimplified formulae. Based on the new definition and simplified

formulae, the effective width of the concrete slab and the depth

of the compressive stress block of the composite beam with

varying parameters under sagging moments are calculated.

2. New definition of effective width under sagging moment

Now consider as shown in Fig. 1 a cross-section of

composite beams under a sagging moment with a steel section

of Class 1 or 2 according to EC4 [10]. For composite beams

at the strength limit state, the resistance of section to sagging

moments, Mu , can be obtained by calculating the plastic

moment and considering a few assumptions commonly used inthe literature [15]:

(1) The tensile strength of concrete is neglected.(2) The concrete in compression resists a constant stress

of fc over a rectangular-stress block with a width of b and

depth of zc0, where b is the physical width and be = b is

the effective width of the concrete slab; zc is the compressive

stress depth from the plastic neutral axis to the top surface

of the concrete slab in general and zc0 = zc (y = 0) is the

zc value along the vertical y-axis particularly, as shown in

Fig. 1(b). Therefore, zc0 represents an equivalent depth of the

compressive stress block.

(3) The effective area of the structural steel member isstressed to its design strength f in tension or compression as

shown in Fig. 1(b).

The width and depth of the stress block are the key factors

affecting the value of Mu . In the traditional design method it

is generally assumed that zc is constant across the width of the

concrete slab; i.e., is 1. The effective width be is traditionally

obtained as

be =

hc0

b/2b/2 cdydzhc

0 c|y=0dz. (1)

Application of be from Eq. (1) will lead to a stress block

that has a total force equivalent to that based on the actual


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Fig. 3. Experimental stressstrain curve for steel materials.

Fig. 4. Stressstrain curve for steel materials used in FEM.

Table 1

Results of compression tests on concrete

Cube no. 1 2 3 4 5 6 Average

fcu (MPa) 34.9 34.7 44.0 41.9 34.7 37.2 37.9

Coupons of steel beams and concrete blocks were tested in

order to determine the stressstrain curve, Youngs modulus,

and compressive and tensile strength. For the reinforcement,

three 6-mm-diameter bars were subjected to tensile tests and the

results were averaged and plotted in Fig. 3 as the stressstrain

curve. For the steel beams, the tensile tests were performedon six specimens and the averaged stressstrain curve is also

displayed in Fig. 3. According to the experimental results a

simplified stressstrain curve shown in Fig. 4 for steel beams

and rebars is used in the finite element analyses. For the

concrete materials, the cubic compressive strength fcu was

determined through six 151515 cm3 cubes which were cast

and tested at the same time as the deck. The measured concrete

strength was shown in Table 1. The cylindrical compressive

strength fc was evaluated assuming fc = 0.8 fcu .

Each of the three longitudinal girders was subjected to

sagging moments through four-point loads ( P/4 at each point)

as shown in Fig. 2. The load was applied by three hydraulic

Fig. 5. Test model and loading frame.

jacks in series with an increment of 2 kN. During the test both

global and local quantities, such as displacements, strains of

the concrete slab and steel beams, and slip at the concrete-steel

interface were monitored. Since the test specimen is designed

as a full composite section and the slip mainly affects the

serviceability behavior of beams and its effect on ultimatestrength is insignificant [16], no detailed slip information is

presented here for the sake of brevity. The mid-span vertical

displacement reached up to 160 mm at the ultimate load Pu =

256 kN, when the collapse happened due to the crushing

on the top surface of the concrete slab. Fig. 5 shows the

deformed shape of the specimen and loading frame used for

the experiment.

Fig. 6 displays the strain distribution along half of the slab

width (with the origin at the center of the deck as shown in

Fig. 2) on the top and bottom surfaces of the concrete slab.

Such curves are displayed under different loading levels for

the mid-span section of the specimen where Pu is the ultimateload from tests. In general, the compressive strain on the top

surface of the portion of the slab remote from the steel beam

lags behind that of the portion near the beam (Fig. 6(a)), while

the tensile strain on the bottom surface of the portion of the

slab remote from the steel beam are greater than that of the

portion near the beam (Fig. 6(b)). This high tensile strain shown

in the figure indicates cracking of concrete as observed in the

experiments. For the convenience of comparison, results from

the finite element method discussed in the next section are

also plotted in the figure. Reasonable agreements between the

experimental and FEM results are clearly observed in the figure.

The load vs. mid-span vertical displacement curve of

the center longitudinal girder is plotted in Fig. 7. The

loaddisplacement relationship is nearly linear up to the load of

100 kN, beyond which a sudden reduction of stiffness occurred

due to the yielding of the steel beam. At the collapse load of

256 kN, the steel beam section at the mid-span yielded and

significantly plasticized.

4. Finite element analysis

In order to predict the distributions of the longitudinal

strains and stresses in the concrete slab of composite beams

at the ultimate strength state, a finite element analysis

through ANSYSR

(2000) was carried out considering material


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1400 J.-G. Nie et al. / Engineering Structures 30 (2008) 13961407

(a) Compressive strain on top surface of concrete slab. (b) Tensile strain on bottom surface of concrete slab.

Fig. 6. Strain distribution along slab width (b = 1200 mm, y = 01800 mm).

Fig. 7. Load vs. mid-span vertical displacement curve.

nonlinearity as shown in Fig. 8. The results from this model

were confirmed by a comparison with the experimental results.

4.1. Finite element model

The tested specimen in Fig. 2 was analysed by finite element

method. 4-node shell elements were used to mesh the steel

girders and 2-node link elements were used to mesh steel bars.

The kinematic hardening rule including Bauschinger effect and

von Mises yield criteria were used for the materials of steel bars

and beams. Multilinear stressstrain relationship of steel bars

and beams obtained from tests as shown in Fig. 4 were adopted

in the analysis. For all steel materials: Youngs modulus Es =

206,000 MPa, Et = 2000 Mpa, and Poisson ratio = 0.3;

Steel beams: fy = 295 MPa, and fu = 448 MPa; Steel bars:

fy = 380 MPa, and fu = 478 MPa.

The 8-node cubic (brick) elements for concrete material

available in ANSYS R were used for the concrete slab. The

failure surface is the modified WilliamWarnke criterion as

shown in Fig. 9 in the biaxial principal stress space and

the crushing and cracking of concrete are considered in this

element [17]. The material properties of the concrete slab used

in the analysis are: fc = 30.3 MPa, tension strength ft =

3.03 MPa, elastic modulus Ec = 30,000 MPa, and Poissons

ratio = 0.17.

Fig. 8. Finite element model.

In the ANSYS concrete model, a crack is a mechanism that

transforms the behavior from isotropic to orthotropic, where

the material stiffness normal to the crack surface becomes zero

while the full stiffness parallel to the crack is maintained. In thissmeared crack model, a smooth crack could close and all the

material stiffness in the direction normal to the crack may be

recovered. The uniaxial compressive stressstrain relationship

of concrete used in the analysis is:

0=

2

0

0

2, 0

1, 0 < cu ,

(5)

where 0 = fc, and 0 = 0.002.The shear studs were modeled by nonlinear spring elements

(shown as Combin Element in Fig. 8). Typically, the actual

loadslip curve of stud connectors was obtained by a push-out

test. Previous studies have shown that the curve is generally

nonlinear even for low stress levels. It is thus reasonable to use

a nonlinear spring in modeling the mechanical behavior of the

connectors. The constitutive relationship of the spring is given


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Fig. 9. Failure surface of concrete material in biaxial principal stress space.

by Aribert [18]:

V = Vu (1 eC1)C2 , (6)

where V and are the shear force and the shear slip of the

connector, respectively; Vu is the shear strength of the shear

studs obtained by push-out tests. The parameters C1 and C2define the shape of the curve, and the values used in this study

are C1 = 0.7 mm1

, and C2 = 0.56 [18].As discussed earlier, Fig. 6 displays the strain distribution

across half of the slab width on the top and the bottom

surfaces of the mid-span section of the concrete slab. Fig. 7

shows the load vs. mid-span vertical displacement curve of the

center longitudinal girder. The comparison of experimental and

numerical results confirmed the accuracy of the finite element

model.

4.2. Parametric analysis

In order to acquire the actual longitudinal strain and stress

distributions in the concrete slab at the ultimate strength state

for different parameters such as loading cases, beam size,and material strength, a nonlinear finite element model was

developed and analysed. This model consists of three identical

longitudinal girders, two transverse girders at the ends of the

longitudinal girders, and a concrete slab attached to the steel

girders by shear connectors as shown in Fig. 10, subjected to

a single-point load P1 at the mid-span, a two-point load P2applied at the 1/3rd points of the beam span, and a uniform

load q over the whole span length of the steel girders. For

materials, the stressstrain relationship in Fig. 4 is adopted with

various steel yield strength fy . Concrete compression strength

fc = 24 MPa, tension strength ft = 2.4 MPa, elastic modulus

Ec = 30,000 MPa, and Poissons ratio = 0.17. The degree of

shear connection is 1, i.e., full composite action is considered.

The following three series of models have been analysed:(1) Yield strength of steel beams fy = 235 MPa, L = 6 m,

b/L = 0.1, 0.2, 0.3, 0.4, or 0.5, and hc = 90 mm;(2) Yield strength of steel beams fy = 235 MPa, L = 6 m,

b/L = 0.3, and hc = 60, 75, 90, 105, or 120 mm;

(3) L = 6 m, b = 1800 mm, hc = 90 mm, and yield strength

of steel beams fy = 235, 300, 350, or 400 MPa.

The ultimate state strains of the mid-span section of the

center girder in the model were processed. Fig. 11(a)(d) show

the distributions of the compressive strains ct(y) on the top

surface of the concrete slab under different loading types, b/L

ratios, height of concrete slab, and yield strength of steel beams.

Fig. 10. Model for parametric analysis.

In all situations the compressive strains decrease from y = 0

to y = b/2 due to the shear-lag effect. The ratios b/L and

loading types have significant influence on the degree of shear

lag while other parameter, such as beam size and materialstrength have less influence on the shear-lag effect. The shear-

lag degree increases with the increase in b/L . The shear-lag

effect is more obvious under one-point load than the other

two loading types. The curved shape of the compressive strain

distribution is assumed to be parabolic and is described with a

quadratic equation as shown in the next section.The longitudinal strain distributions across the thickness of

the concrete slab at the mid-span section when fy = 235 MPa,

L = 6 m, b/L = 0.3 and hc = 90 mm, under three

loading types, are shown in Fig. 12(a)(c) from which it can

be concluded that the longitudinal strain remains linear along

the z-axis at the ultimate strength state. The curvature (y) andthe depth zc(y) between the top surface and the neutral axis of

the concrete slab, as shown in Fig. 13, can be obtained from

the strain results in Fig. 12(a)(c). While Fig. 12(a) and (c)

show that (y) remains almost constant and zc(y) decreases

from y = 0 to y = b/2 along y-axis under uniform load and

two-point loads, Fig. 12(b) shows that (y) decreases and zc(y)

almost remains constant from y = 0 to y = b/2 along y-axis

under the one-point load.

5. Analytical strain distribution across concrete slab at

ultimate strength state

Numerical results discussed earlier have verified theassumption that the longitudinal strain distribution remains

linear along the z-axis at the ultimate strength state. As it is

demonstrated below, if the compressive strain ct(y) on the

top surface and the depth zc(y) between the top surface and

the neutral axis of the concrete slab can be expressed using

simplified formulae, the compressive strain distribution in the

concrete slab can be obtained analytically, which will facilitate

an analytical solution of the effective width.According to the FEM numerical results, ct(y) can be

expressed as

ct

(y) = ct

(0)1 yb+

y2

b2 , (7)


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(a) Longitudinal strain along z-axis under load q.

(b) Longitudinal strain along z-axis under load P1.

(c) Longitudinal strain along z-axis under load P2.

Fig. 12. Strain distributions along slab depth (z-axis) for different loadings.

Fig. 13. ct

(y), (y) and zc

(y) in concrete slab.

Fig. 14. Values of under different loading types and b/L ratios.

6. Analytical effective width and depth of rectangular-

stress block

According to the uniaxial compressive stressstrain relation-

ship of concrete as shown in Eq. (5), the analytical stress dis-

tribution in the concrete slab at the mid-span of the compositebeam can be expressed as:

c(y,z) =

fcc(y,z)

0

2

c(y,z)

0

,

0 c(y,z) 0fc, 0 c(y,z) cu0, c(y,z) 0,

(11)

where 0 = 0.002, cu = 0.0033, and the tensile strength of

concrete is ignored.

At the ultimate strength state, the maximum strain at the top

surface of the concrete ct(0) = cu . By substituting Eq. (10)

and ct(0) = cu into Eq. (11) we can obtain an equation wherec(y,z) is expressed as a function of the section dimensions,

material strength, and zc0. Substituting Eq. (11) into Eq. (2)

into (4) leads to three simultaneous equations from which the

three unknowns zc0, and can be analytically solved and thus

the analytical solution of the effective width can be derived.

A series of composite beams with L = 6 m, ratios b/L =

0.10.5, hc = 90 mm, hc/ hs = 0.10.4, concrete class C30,

and yield strength of steel fy = 235 MPa, subjected to a single-

point load, a two-point load and a uniform load were analysed

by using the developed analytical approach. The steel beam

section with a variable height hs , 200 12 mm for top and

bottom flanges, and 2768 mm for web are used in all analyses.

The values ofzc0, and are solved from Eqs. (2)(4) and theresults are listed in Tables 2 and 3.

As shown in Tables 2 and 3, the effective width factor is

greater than 0.99 under various beam sizes and loading types

when b/L 0.5. In general the effective width be increases

with the increase in the actual (physical) width b. Therefore,

the be when b/L > 0.5 should be larger than the corresponding

be when b/L = 0.5. It is suggested that the be in the case of

b/L > 0.5 (this situation does not happen often) be chosen

the same value as the be in the case of b/L = 0.5, which is

on the safe side for the ultimate strength analysis. Based on

these arguments, the effective width be for the ultimate strength

design of composite beams may be obtained as


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Table 2

Results ofzc0, and with various b/L and hc / hs (under uniform load and two-point loads)

hc / hs

b/L b/L

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.1 1.000 1.000 0.999 0.996 0.993 0.285 0.886 0.910 0.882 0.852

0.2 1.000 1.000 0.998 0.996 0.993 0.886 0.942 0.914 0.883 0.8520.3 1.000 0.999 0.998 0.996 0.993 0.913 0.946 0.914 0.883 0.852

0.4 1.000 0.999 0.998 0.996 0.993 0.927 0.946 0.914 0.883 0.852

hc / hs zc0 bh fc /As f

b/L b/L

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.1 315.53 101.57 93.97 84.15 69.97 0.304 0.608 0.913 1.217 1.521

0.2 101.54 93.04 75.13 58.48 48.64 0.438 0.875 1.313 1.750 2.188

0.3 98.54 90.46 64.14 49.93 41.53 0.513 1.025 1.538 2.050 2.563

0.4 97.04 84.90 58.66 45.66 37.97 0.560 1.121 1.681 2.242 2.802

Table 3

Results ofzc0, and with various b/L and hc / hs (under one-point load)

hc / hs

b/L b/L

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.1 1.000 1.000 0.998 0.999 1.000 0.285 0.883 0.954 0.994 1.000

0.2 1.000 0.999 1.000 1.000 1.000 0.886 0.966 1.000 1.000 1.000

0.3 1.000 1.000 1.000 1.000 1.000 0.913 0.995 1.000 1.000 1.000

0.4 1.000 1.000 1.000 1.000 1.000 0.927 0.998 1.000 1.000 1.000

hc / hs zc0 bh fc/As f

b/L b/L

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.1 315.53 101.63 93.01 74.44 59.17 0.304 0.608 0.913 1.217 1.521

0.2 101.54 92.67 68.59 51.41 41.13 0.438 0.875 1.313 1.750 2.188

0.3 98.55 88.31 58.54 43.90 35.12 0.513 1.025 1.538 2.050 2.5630.4 97.05 80.43 53.53 40.14 32.12 0.560 1.121 1.681 2.242 2.802

be =

b, b/L 0.5

L/2, b/L > 0.5,(12)

where L is the span length for simply supported beam and the

distance between the points of zero bending moments under

dead load for continuous beams.

For the ultimate strength analysis AISC [9] and AASHTO

codes [19,20] adopt the same effective width as that used for

elastic analysis shown in Eq. (13) below, which is based on the

traditional definition of effective width shown in Eq. (1). Takean interior girder for example, the effective width is

be = min {b, L/4, 12ts} , (13)

where ts is the thickness of slab.

In contrast, Eq. (12) is based on the new definition of

effective width shown in Eqs. (2)(4) and the real distribution

of strain and stress at the ultimate state. Eq. (12) is more

reasonable for ultimate moment resistance calculation using

rectangular-stress block assumption.

As shown in Tables 2 and 3, the plastic neutral axis shifts

upward (with smaller zc0 values) when the ratios b/L and

hc/ hs increase. According to the position of the plastic neutral

axis, the results in Tables 2 and 3 can be distinguished into two

situations, namely (a) and (b) as follows:

(a) When bhc fc > As f (refer to the cases of normal fonts

in Tables 2 and 3), then zc0 < hc; at this situation the neutral

axis lies in the concrete slab as shown in Fig. 16(a). We then

have force equilibrium as

bzc0 fc = As f. (14)

(b) When bhc fc As f (refer to the cases of bold fonts in

Tables 2 and 3) or bhc fc As f (the italic font in Tables 2 and3, if any), then zc0 > hc; in this situation the neutral axis lies

below the concrete slab as shown in Fig. 16(b). We then have

zc0 hc. (15)

After obtaining the value of (=be/b) from Eq. (12), zc0can then be obtained with either Eq. (14) or Eq. (15). Once

both the width and depth of the stress block are known, the

moment resistance of composite beam sections at the ultimate

strength state can thus be obtained by the traditional plastic

section method specified in any design codes.

Chen et al. [21] have just published their findings from their

comprehensive study on effective width based on their NCHRP


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(a) fy = 300 MPa, hc = 90 mm, under loading q. (b) fy = 300 MPa, hc = 90 mm, under loading P1.

(c) fy = 300 MPa, hc = 90 mm, under loading P2.

Fig. 15. Comparison between ct(y) values from finite element method (FEM) and simplified formulae (SF).

Table 4

Comparison of effective width at the ultimate strength state

Cases Proposed be Proposed be in Ref. [21] AISC (AASHTO) be Comments

b/L 1/4 b b Min(b, 12ts ) Majority cases

1/4 < b/L < 1/2 b b Min(L/4, 12ts ) Some cases

b/L 1/2 L/2 b Min(L/4, 12ts ) Very rare cases

supported project [22]. The comparison between the proposed

effective width in Eq. (12), that from [21], and that specified

in the US codes (AISC and AASHTO) in Eq. (3) is listedin Table 4. The majority (perhaps 99%) practical structures

fall into the case of b/L 1/2. In this range the proposed

effective width be in the present study is exactly the same

as that of Chen et al. [21], though a different approach was

used in the two studies. By using their effective width and that

specified in AASHTO code [20], Chen et al. [21] have shown

that the difference is less than 4% in terms of ultimate capacity.

Therefore, the proposed effective width and that of the AISC

(AASHTO) are indirectly shown to be basically the same for

the case of b/L 1/2. For b/L > 1/2, while the proposed

effective width could be twice that of the AISC (AASHTO), the

practical structures rarely fall into this category (perhaps 1/2, there

is no available experimental data to directly verify the proposed

formula in the range of b/L > 1/2.

7. Discussion of the effective width

A few special notes are of worth and are mentioned below:(1) Theoretically, the obtained formulae for the effective

width and depth are only valid for the evaluation of ultimate

strength. However, the effective width is also traditionally used

for stress and deflection calculation.


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Fig. 16. Situations (a) and (b) for ultimate strength analysis (dashed rectangular means stress block).

(2) In the present study only simply supported decks

are studied, and other end boundary conditions have not

been considered. However, engineers are more concerned

about the mid-span section that is less affected by the

continuity/boundary conditions. Traditionally, for simplicity,

engineers only distinguish between positive and negative

moment sections without considering the changes of effectivewidth along the span and without considering many other

factors that may affect the effective width to different extents.

(3) The effective width depends on the level of stress and

type of loading at the section. Therefore, a more general case

of stress resultants, i.e. a case of simultaneous application

of bending and axial forces, should be analysed. However,

considering the axial force will make the problem much more

complicated since axial forces are variable. If axial force is an

important component of the section forces, then we suggest

using 3D finite element analysis directly.(4) Theoretically, effective width varies along the span

length of the composite deck. Thus, the computation of

deflections in simply supported decks or of stress resultantsin continuous beams can be a rather complex task due

to the longitudinal variation of the cross-section properties.

Using a variable effective width along the span length is too

troublesome and is not practical for routine application.(5) The proposed effective width is based on limited

finite element and experimental studies. A more meaningful

verification would be a comprehensive one that should include

many cases considering different parameters such as arbitrary

loading, which is out of the scope of the present study and

perhaps should be pursued in a separate study.

8. Conclusions

1. In the traditional definition, the effective width of concrete

slab is determined based on the equivalence of axial force

between the actual stress distribution and the simplified stress

block. In the present study, a new definition of the effective

width is presented for ultimate strength state of steelconcrete

composite beams under sagging moments. The effective width

factor , the position of neutral axis zc0, and the depth of

the rectangular-stress block zc0 are solved from a set of

simultaneous equations based on the equivalencies of both the

total axial force and the moment resistance, which ensures that

the simplified stress distribution within the effective width will

represent the actual moment resistance of the original beam.

2. Through an experimental study and finite element analysis

the distributions of longitudinal strain and stress across the

concrete slab at ultimate strength state are examined and

expressed by simplified formulae, which makes it possible to

analytically derive the effective width.

3. For composite beams at the ultimate strength state with

various loading types, , zc0 and are solved from a set ofsimultaneous equations based on the new definition of effective

width and simplified formulae of stress distributions across the

concrete slab.

4. The effective width for the ultimate strength state is found

to be nearly the same as the physical width for the cases

examined in the present study and a simplified effective width

be for composite beam sections subjected to sagging moment

is thus proposed. Simplified formulae for calculating the depth

of the rectangular-stress block zc0 are also presented for the

ultimate strength design of composite beams. Once both the

width and depth of the stress block are known, the moment

resistance of composite beam sections at the ultimate strength

state can thus be obtained by the traditional plastic section

method specified in any design codes.

Acknowledgments

The first two authors gratefully acknowledge the financial

support provided by the National Natural Science Foundation

of China (# 50438020) and the third author appreciates

the financial support from the Louisiana State University

for international travel and collaboration. The authors also

appreciate the constructive comments from the reviewers.

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