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Engineering Structures 30 (2008) 13961407
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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/engstruct8/14/2019 Nie 2008 Engineering-Structures
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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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(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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1406 J.-G. Nie et al. / Engineering Structures 30 (2008) 13961407
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.
References
[1] Adekola AO. Effective widths of composite beams of steel and concrete.
Structural Engineer 1968;46(9):2859.
[2] Adekola AO. The dependence of shear lag on partial interaction in
composite beams. International Journal of Solids Structures 1973;10(4):
389400.
[3] Ansourian P, Aust MIE. The effective width of continuous composite
beams. Civil Engineering Transitions 1983;25(1):639.
[4] Johnson RP. Research on steelconcrete composite beams. Journal of
Structural Division, ASCE 1970;96(3):44559.
[5] Heins CP, Fan HM. Effective composite beam width at ultimate load.
Journal of Structural Division, ASCE 1976;102(11):216379.
[6] Elkelish S, Robison H. Effective widths of composite beams with ribbed
metal desk. Canadian Journal of Civil Engineering 1986;13(2):6675.
8/14/2019 Nie 2008 Engineering-Structures
12/12
J.-G. Nie et al. / Engineering Structures 30 (2008) 13961407 1407
[7] Amadio C, Fragiacomo M. Effective width evaluation for steelconcrete
composite beams. Journal of Constructional Steel Research 2002;58(3):
37388.
[8] Amadio C, Fedrigo C. Experimental evaluation of Effective width in
steelconcrete composite beams. Journal of Constructional SteelResearch
2004;60(2):199220.
[9] AISC. Load & resistance factor design, Volume 1, Part 5: Composite
design. 1998.
[10] CEN. 1994. Commission of the European communities. ENV 1994-1-1.
Eurocode 4-Design of composite steel and concrete structures-Part 1-1:
General rules and rules for buildings, Bruxelles.
[11] Song QG, Scordelis AC. Formulas for shear-lag effect of T-, and I-,
and box beams. Journal of Structural Engineering, ASCE 1990;116(5):
130618.
[12] Song QG, Scordelis AC. Shear-lag analysis of T-, I-, and box beams.
Journal of Structural Engineering, ASCE 1990;116(5):1290305.
[13] Elhelbawey M, Fu CC, Sahin MA, Schelling DR. Determination of slab
participation from weigh-in-motion bridge testing. Journal of Bridge
Engineering, ASCE 1999;4(3):16573.
[14] Chiewanichakorn M, Aref AJ, Chen SS, Ahn II S. Effective flange
width for steelconcrete composite bridge girder. Journal of Structural
Engineering, ASCE 2004;130(2):201630.
[15] Johnson RP. Composite structure of steel and concrete, 2nd ed. vol. 1.
London: Blackwell Scientific Publications; 1994.
[16] Nie JG, Cai CS. Steelconcrete composite beams considering shear slip
effect. Journal of Structural Engineering, ASCE 2003;129(4):495506.
[17] ANSYS Inc. ANSYS theory reference. 2000.
[18] Aribert JM. Slip and uplift measurements along the steel and concrete
interface of various types of composite beams. In: Proceedings of the
international workshop on needs in testing metals: Testing of metals for
structures. London: E. &FN Spon; 1992. p. 395407.
[19] AASHTO. Standard specification for highway bridges. Washington (DC):
American Association of State Highway and Transportation Officials,
AASHTO; 2002.
[20] AASHTO. LRFD bridge design specifications. Washington (DC):
American Association of State Highway and Transportation Officials,
AASHTO; 2002.
[21] Chen SS, Aref AJ, Chiewanichakorn M, Ahn II S. Proposed effective
widthcriteria for compositebridge girders. Journal of Bridge Engineering,
ASCE 2007;12(3):32538.
[22] Chen SS, Aref AJ, Ahn I-S, Chiewanichakorn M, Carpenter JA,
Nottis A et al. Effective slab width for composite steel bridge members
NCHRP Report 543. Washington (DC): Transportation Research Board;
2005.