Nie 2008 Engineering-structures

Engineering Structures 30 (2008) 1396–1407 www.elsevier.com/locate/engstruct

Effective width of steel–concrete composite beam at ultimate strength state Jian-Guo Nie a , Chun-Yu Tian a , C.S. Cai b,∗ a Department of Civil Engineering, Laboratory of Structural Engineering and Vibration of China Education Ministry, Tsinghua University, Beijing, 100084, China b Department 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 steel–concrete 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: Steel–concrete; Composite; Effective width; Ultimate strength state; Experiment; Finite element analysis

1. Introduction A steel–concrete 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 shearlag 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). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.07.027

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 steel–concrete 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 equations in 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

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Notation As A0s b be Ec Es Et f fc f cu ft fy fu hc hs L Pu V Vu zc z c0 α β εc εct ϕ λ ν ξ σc

tension area of steel beam section; compression area of steel beam section; width of concrete slab of composite beam; effective width of concrete slab of composite beam; elastic modulus of concrete; elastic modulus of steel; hardening modulus of steel; design strength of steel; cylindrical compressive strength of concrete; cubic compressive strength of concrete; tension strength of concrete; yield strength of steel; limit strength of steel; height of concrete slab; height of steel beam section; span of composite beam; ultimate load of test shear force of the connectors between concrete and steel; shear strength of shear studs; depth between top surface of concrete and plastic neutral axis; 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; compressive strain in concrete slab; compressive strain on top surface of concrete slab; curvature of concrete slab; slip of the connectors between concrete and steel; Poisson’s ratio; height of rectangular-stress block to z c0 ratio; 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 design process [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 [11–13]. 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

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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 composite beams 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 rectangularstress 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 some simplified 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 in the literature [15]: (1) The tensile strength of concrete is neglected. (2) The concrete in compression resists a constant stress of f c over a rectangular-stress block with a width of βb and depth of ξ z c0 , where b is the physical width and be = βb is the effective width of the concrete slab; z c is the compressive stress depth from the plastic neutral axis to the top surface of the concrete slab in general and z c0 = z c (y = 0) is the z c value along the vertical y-axis particularly, as shown in Fig. 1(b). Therefore, ξ z c0 represents an equivalent depth of the compressive stress block. (3) The effective area of the structural steel member is stressed 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 z c is constant across the width of the concrete slab; i.e., ξ is 1. The effective width be is traditionally obtained as R h c R b/2 −b/2 σc dydz 0 . (1) be = R h c 0 σc | y=0 dz 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. 1. Assumption for ultimate strength analysis.

Fig. 2. Sketch of specimen and test set-up (units = mm).

stress distribution in the concrete slab, but there is no guarantee that the resultant forces will be located at the same location. Due to the shear-lag phenomenon the longitudinal stress in the concrete slab decreases from y = 0 to y = b/2 and the depth z c varies across the width of the concrete slab as shown in Fig. 1(a). Therefore, the stress block with a width of be from Eq. (1) may not have a total bending moment equivalent to that based on the actual stress distribution in the concrete slab, i.e., the accurate value of Mu may not be obtained by the traditional method when using the definition of effective width in Eq. (1). In order to ensure that the stress distribution in the concrete slab as shown in Fig. 1(a) and (b) is equivalent to ultimate strength analysis in terms of both axial force and moment, we will have to derive the effective width and the depth of the stress block by both force and moment equivalencies as: Z h c Z b/2 βbξ z c0 f c = σc (y, z)dydz (2) 0

ξ z c0 = 2

−b/2

R h c R b/2 0

−b/2 (h c − z)σc (y, z)dydz . R h c R b/2 −b/2 σc (y, z)dydz 0

(3)

Considering the force equilibrium in the entire beam section we have Z h c Z b/2 As f − A0s f = σc (y, z)dydz, (4) 0

−b/2

where As and A0s is tension and compression area of steel section, respectively. In order to predict the actual strain and stress distributions across the section, finite element method was used to analyse the composite beams with varying parameters under sagging moments. The three variables β, ξ and z c0 were then solved from Eqs. (2)–(4) after the distribution of σc in the concrete slab was obtained and the bending resistance to sagging moment of composite beams can then be obtained by a traditional plastic beam approach. To confirm the numerical results, an experimental study was also conducted and is described next. 3. Experimental study A steel–concrete composite floor model as shown in Fig. 2 was tested to investigate the shear-lag phenomenon in the concrete slab of the composite beam in both elastic and inelastic stages and the experimental results were used to verify the accuracy of the finite element model described in next section. The model consists of three identical longitudinal girders and two transverse girders at the ends of the longitudinal girders. A cast-in-place concrete slab with a height of 60 mm was connected to the girders by head studs. The experimental model with three girders represents more closely the real composite structures and can give more realistic results than traditional single-beam specimens.

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Fig. 5. Test model and loading frame. Fig. 3. Experimental stress–strain curve for steel materials.

Fig. 4. Stress–strain curve for steel materials used in FEM. Table 1 Results of compression tests on concrete Cube no.

1

2

3

4

5

6

Average

f cu (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 stress–strain curve, Young’s 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 stress–strain curve. For the steel beams, the tensile tests were performed on six specimens and the averaged stress–strain curve is also displayed in Fig. 3. According to the experimental results a simplified stress–strain 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 f cu was determined through six 15×15×15 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 f c was evaluated assuming f c = 0.8 f cu . 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

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 ultimate strength 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 ultimate load 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 load–displacement 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 R through ANSYS (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 = 0–1800 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 stress–strain relationship of steel bars and beams obtained from tests as shown in Fig. 4 were adopted in the analysis. For all steel materials: Young’s modulus E s = 206,000 MPa, E t = 2000 Mpa, and Poisson ratio ν = 0.3; Steel beams: f y = 295 MPa, and f u = 448 MPa; Steel bars: f y = 380 MPa, and f u = 478 MPa. The 8-node cubic (brick) elements for concrete material R available in ANSYS were used for the concrete slab. The failure surface is the modified William–Warnke 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: f c = 30.3 MPa, tension strength f t = 3.03 MPa, elastic modulus E c = 30,000 MPa, and Poisson’s ratio ν = 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 this smeared 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 stress–strain relationship of concrete used in the analysis is:   2 ε  ε σ − , ε ≤ ε0 2 = (5) ε0  ε0 σ0 1, ε0 < ε ≤ εcu , where σ0 = f c , and ε0 = 0.002. The shear studs were modeled by nonlinear spring elements (shown as Combin Element in Fig. 8). Typically, the actual load–slip 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 − e−C1 λ )C2 ,

(6)

where V and λ 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 C2 define the shape of the curve, and the values used in this study are C1 = 0.7 mm−1 , 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 P2 applied 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 stress–strain relationship in Fig. 4 is adopted with various steel yield strength f y . Concrete compression strength f c = 24 MPa, tension strength f t = 2.4 MPa, elastic modulus E c = 30,000 MPa, and Poisson’s ratio ν = 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 f y = 235 MPa, L = 6 m, b/L = 0.1, 0.2, 0.3, 0.4, or 0.5, and h c = 90 mm; (2) Yield strength of steel beams f y = 235 MPa, L = 6 m, b/L = 0.3, and h c = 60, 75, 90, 105, or 120 mm; (3) L = 6 m, b = 1800 mm, h c = 90 mm, and yield strength of steel beams f y = 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 εct (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 material strength have less influence on the shear-lag effect. The shearlag 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 f y = 235 MPa, L = 6 m, b/L = 0.3 and h c = 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 ϕ(y) and the depth z c (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 ϕ(y) remains almost constant and z c (y) decreases from y = 0 to y = b/2 along y-axis under uniform load and two-point loads, Fig. 12(b) shows that ϕ(y) decreases and z c (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 the assumption 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 εct (y) on the top surface and the depth z c (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, εct (y) can be expressed as   y y2 εct (y) = εct (0) 1 − α + α 2 , (7) b b

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(a) f y = 235 MPa, b/L = 0.3, h c = 90 mm, under different loading types.

(b) f y = 235 MPa, b/L = 0.1–0.5, h c = 90 mm, under loading q.

(c) f y = 235 MPa, b/L = 0.3, h c = 60–120 mm, under loading q.

(d) f y = 235–400 MPa, b/L = 0.3, h c = 90 mm, under loading q.

Fig. 11. Strain distributions on top surface of concrete across slab width (y-axis) for different parameters.

where α is a parameter representing the degree of shear-lag effect. From Eq. (7) it is derived that εct (b/2) = εct (0)(1 − 0.25α); therefore α = 4(1 − εct (b/2)/εct (0)). When the strain distribution is uniform across the slab width, then α = 0. Based on the FEM results of εct (0) and εct (b/2), the values of α under different parameters were obtained. According to the numerical results discussed earlier, the parameters h c and f y have a small influence on α. For example, when b/L = 0.3, the value of α increases by about 2% when h c increases from 60 to 120 mm, and α increases by about 4% when f y increases from 235 to 400 MPa. Therefore, the influence of f y and h c on α can be ignored. The values of α under different loading types and b/L ratios, when f y = 300 MPa and h c = 90 mm, were calculated and plotted in Fig. 14. By curve fitting the numerical results of α with b/L as the sole parameter, the α’s were derived below and are also plotted in Fig. 14 as α = 2b/L − 0.075 (for uniform load and two-point load)

(8a)

α = 7.5b/L − 0.5

(8b)

(for one-point load).

By using Eqs. (7) and (8) we can analytically obtain the values of εct (y) that are plotted in Fig. 15(a)–(c) to compare

with the finite element results under different loading types and b/L ratios, for the case of f y = 300 MPa and h c = 90 mm. The accuracy of the simplified formulae is confirmed by this comparison. As discussed earlier, the curvature φ(y) of the concrete slab remains constant from y = 0 to y = b/2 under uniform load and two-point load. By using equation φ(y) = εct (y)/z c (y) (as shown in Fig. 13) and Eq. (7), z c (y) is derived as shown in Eq. (9a) below. Meanwhile, since z c (y) remains constant from y = 0 to y = b/2 under one-point load, Eq. (9b) is adopted for this case as   y y2 z c (y) = z c0 1 − α + α 2 b b (for uniform load and two-point load) (9a) z c (y) = z c0

(for one-point load).

(9b)

From Eqs. (7)–(9) and the assumption that the longitudinal strain at the mid-span of the concrete slab remains co-linear along the z-axis, the strain at the mid-span of concrete slab εc (y, z) as a function of y and z can be expressed as: εc (y, z) =

z − h c + z c (y) εct (y). z c (y)

(10)

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

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

6. Analytical effective width and depth of rectangularstress block

(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. εct (y), ϕ(y) and z c (y) in concrete slab.

According to the uniaxial compressive stress–strain relationship of concrete as shown in Eq. (5), the analytical stress distribution in the concrete slab at the mid-span of the composite beam can be expressed as:   ε (y, z)  εc (y, z) c  2− , fc    ε0 ε0 0 > εc (y, z) > −ε0 σc (y, z) = (11)   f , −ε > ε (y, z) > −ε  c cu 0  c 0, εc (y, z) > 0, where ε0 = 0.002, εcu = 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 εct (0) = εcu . By substituting Eq. (10) and εct (0) = εcu into Eq. (11) we can obtain an equation where σc (y, z) is expressed as a function of the section dimensions, material strength, and z c0 . Substituting Eq. (11) into Eq. (2) into (4) leads to three simultaneous equations from which the three unknowns z c0 , β and ξ 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 = 0.1–0.5, h c = 90 mm, h c / h s = 0.1–0.4, concrete class C30, and yield strength of steel f y = 235 MPa, subjected to a singlepoint 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 h s , 200 × 12 mm for top and bottom flanges, and 276×8 mm for web are used in all analyses. The values of z c0 , β and ξ are solved from Eqs. (2)–(4) and the results are listed in Tables 2 and 3. As shown in Tables 2 and 3, the effective width factor β 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 of z c0 , β and ξ with various b/L and h c / h s (under uniform load and two-point loads) hc / hs

β b/L 0.1 1.000 1.000 1.000 1.000

0.1 0.2 0.3 0.4 hc / hs

0.1 0.2 0.3 0.4

0.2 1.000 1.000 0.999 0.999

0.3

0.4

0.5

ξ b/L 0.1

0.999 0.998 0.998 0.998

0.996 0.996 0.996 0.996

0.993 0.993 0.993 0.993

0.285 0.886 0.913 0.927

0.886 0.942 0.946 0.946

0.910 0.914 0.914 0.914

0.882 0.883 0.883 0.883

0.852 0.852 0.852 0.852

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

z c0 b/L 0.1

0.2

0.3

0.4

0.5

βbh f c /As f b/L 0.1

315.53 101.54 98.54 97.04

101.57 93.04 90.46 84.90

93.97 75.13 64.14 58.66

84.15 58.48 49.93 45.66

69.97 48.64 41.53 37.97

0.304 0.438 0.513 0.560

0.608 0.875 1.025 1.121

0.913 1.313 1.538 1.681

1.217 1.750 2.050 2.242

1.521 2.188 2.563 2.802

0.2

0.3

0.4

0.5

Table 3 Results of z c0 , β and ξ with various b/L and h c / h s (under one-point load) hc / hs

β b/L 0.1 1.000 1.000 1.000 1.000

0.1 0.2 0.3 0.4 hc / hs

0.1 0.2 0.3 0.4

 be =

b, L/2,

0.2 1.000 0.999 1.000 1.000

0.3

0.4

0.5

ξ b/L 0.1

0.998 1.000 1.000 1.000

0.999 1.000 1.000 1.000

1.000 1.000 1.000 1.000

0.285 0.886 0.913 0.927

0.883 0.966 0.995 0.998

0.954 1.000 1.000 1.000

0.994 1.000 1.000 1.000

1.000 1.000 1.000 1.000

0.2

0.3

0.4

0.5

0.608 0.875 1.025 1.121

0.913 1.313 1.538 1.681

1.217 1.750 2.050 2.242

1.521 2.188 2.563 2.802

z c0 b/L 0.1

0.2

0.3

0.4

0.5

βbh f c /As f b/L 0.1

315.53 101.54 98.55 97.05

101.63 92.67 88.31 80.43

93.01 68.59 58.54 53.53

74.44 51.41 43.90 40.14

59.17 41.13 35.12 32.12

0.304 0.438 0.513 0.560

b/L ≤ 0.5 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). Take an 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 z c0 values) when the ratios b/L and h c / h s 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 βbh c f c > As f (refer to the cases of normal fonts in Tables 2 and 3), then z c0 < h c ; at this situation the neutral axis lies in the concrete slab as shown in Fig. 16(a). We then have force equilibrium as βbξ z c0 f c = As f.

(14)

(b) When βbh c f c ≤ As f (refer to the cases of bold fonts in Tables 2 and 3) or βbh c f c ≈ As f (the italic font in Tables 2 and 3, if any), then z c0 > h c ; in this situation the neutral axis lies below the concrete slab as shown in Fig. 16(b). We then have ξ z c0 ≈ h c .

(15)

After obtaining the value of β (=be /b) from Eq. (12), ξ z c0 can 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) f y = 300 MPa, h c = 90 mm, under loading q.

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(b) f y = 300 MPa, h c = 90 mm, under loading P1 .

(c) f y = 300 MPa, h c = 90 mm, under loading P2 . Fig. 15. Comparison between εct (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 1/4 < b/L < 1/2 b/L > 1/2

b b L/2

b b b

Min(b, 12ts ) Min(L/4, 12ts ) Min(L/4, 12ts )

Majority cases Some cases 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 listed in 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%).

However, the proposed formula is still more conservative than that of Chen et al. [21] whose proposed be is b for all girders. Again, the AISC (AASHTO) is based on elastic analysis, while the present study is based on the ultimate analysis that is supposed to be more accurate and less conservative. Due to the rareness of the practical cases in the range of b/L > 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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J.-G. Nie et al. / Engineering Structures 30 (2008) 1396–1407

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 effective width 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 resultants in 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 steel–concrete composite beams under sagging moments. The effective width factor β, the position of neutral axis z c0 , and the depth of the rectangular-stress block ξ z c0 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, β, z c0 and ξ are solved from a set of simultaneous 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 ξ z c0 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):285–9. [2] Adekola AO. The dependence of shear lag on partial interaction in composite beams. International Journal of Solids Structures 1973;10(4): 389–400. [3] Ansourian P, Aust MIE. The effective width of continuous composite beams. Civil Engineering Transitions 1983;25(1):63–9. [4] Johnson RP. Research on steel–concrete composite beams. Journal of Structural Division, ASCE 1970;96(3):445–59. [5] Heins CP, Fan HM. Effective composite beam width at ultimate load. Journal of Structural Division, ASCE 1976;102(11):2163–79. [6] Elkelish S, Robison H. Effective widths of composite beams with ribbed metal desk. Canadian Journal of Civil Engineering 1986;13(2):66–75.

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[15] Johnson RP. Composite structure of steel and concrete, 2nd ed. vol. 1. London: Blackwell Scientific Publications; 1994. [16] Nie JG, Cai CS. Steel–concrete composite beams considering shear slip effect. Journal of Structural Engineering, ASCE 2003;129(4):495–506. [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. 395–407. [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 width criteria for composite bridge girders. Journal of Bridge Engineering, ASCE 2007;12(3):325–38. [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.