A theoretical model for investigating shear lag in composite cable-stayed bridges

Abstract

The slab of the composite girder is usually very wide in composite cable-stayed bridges, and the main girder has an obvious shear lag. There is an axial force in the main girder due to cable forces, which changes the normal stress distribution of the composite girder and affects the shear lag. To investigate the shear lag in the twin I-shaped composite girder (TICG) of cable-stayed bridges, analytical solutions of TICGs under bending moment and axial force were derived by introducing the additional deflection into the longitudinal displacement function. A shear lag coefficient calculation method of the TICG based on additional deflection was proposed. Experiments with three load cases were conducted to simulate the main girder in cable-stayed bridges. And the stress, deflection, and shear lag coefficient obtained from the theoretical method considering additional deflection (TMAD) were verified by the experimental and finite element results. A generalized verification of a composite girder from existing references was made, indicating that the proposed method could provide more accurate results for the shear lag effect.

Graphical abstract

Keywords

cable-stayed bridge / composite girder / shear lag / energy method / additional deflection

1 Introduction

The slab width of the composite girder in modern cable-stayed bridges is usually very wide. As one girder forms in composite cable-stayed bridges, the twin I-shaped composite girders (TICGs) have uneven normal stress distribution and significant shear lag under dead and external loads. Besides, axial forces generated by stay cables on the main girder make the shear lag different from that of the simply supported or continuous girder because the axial force changes the normal stress distribution pattern in the composite girder. The investigation of the shear lag is essential in designing the cable-stayed bridge.

Because of the advantages of simple analysis and reliable calculation, the energy variational method was widely used to calculate the analytical solution of the girders considering shear lag. The generalized displacement differential equations considering the shear lag were established for rectangular concrete box girders without cantilever flanges by energy method, and this is the first time to introduce the definitions of effective width and shear lag from aircraft to engineering structures. Since the 1980s, the energy method has been applied to analyze the shear lag effect in concrete box girders with cantilever flanges [1–3]. The results proved that the cantilever flange influences normal stress distribution patterns and the axial equilibrium of the warping stress in the flanges is no longer satisfied with the cantilever flange. To keep axial equilibrium, an analytical model adding uniform displacement was proposed by Ni and Qian [4], and the computational precision of shear lag is improved for concrete box girders after considering axial equilibrium. Considering the different shear lag warping degrees of the inner flange, cantilever flange, and bottom and top flanges in concrete box girders, some scholars have studied the warping displacement forms, and concluded that different warpage functions would affect the calculation results accuracy in the shear lag [5,6]. Therefore, it is essential to construct a proper warping displacement function for each part of the girders. In the theoretical studies of the shear lag mentioned above, the physical meaning of the generalized displacement for the shear lag is unclear. Zhang et al. [6,7] and Yang et al. [8] proposed that the shear lag deformation is expressed by additional deflection, which clarifies the physical meaning of shear lag. The total deflection and additional deflection of concrete box girders considering shear lag were obtained. The shear lag coefficient calculated by additional deflection can better represent the distribution and variation pattern of the shear lag in concrete box girders.

The composite girder differs from the concrete girder because of the steel−concrete interface slip. The interface slip will influence the normal stress distribution of the slabs and flanges. Due to the decreased stiffness caused by the interface slips, the deformation of the composite girder increased under the load actions. So it is essential to take into account the effect of interface slip between steel girders and concrete slabs. According to the variational principle, the shear lag of concrete slabs can be calculated by introducing the cross-section angle function to consider the interface slip [9,10]. When the composite box girder cross-section is not biaxial symmetry, the warping stress in the slab obtained from the warping displacement function cannot constitute the axial equilibrium in the composite girder. Then, Li and Nie [11] and Zhang and Lin [12] proposed a shear lag analysis model of composite girders considering axial equilibrium, and the governing differential equations and analytical solutions under vertical load were established. Zhu et al. [13] improved the one-dimensional finite element model (FEM) of the decks, and the shear lag of the composite deck under the negative and positive bending moments was studied. In selecting the warping displacement function of the concrete slab, the warping displacement function for twin-girder composite decks with a cosine function was developed by Lezgy-Nazargah et al. [14]. However, the above studies only investigated the shear lag effect under the vertical load, not considering axial force. Some studies have shown that axial forces significantly affect the shear lag effect [15,16]. Li and Nie [11] and Dezi et al. [15] studied the analytical solution of twin-girder composite decks considering the shear lag effect under axial force. Wang et al. [17] conducted the model tests under the vertical load and axial force, and a finite segment method was proposed to analyze the shear lag in composite cable-stayed bridges.

At present, the influence of additional deflection is only considered in calculating the shear lag effect in concrete girders of the simple girder system. There is no research on the shear lag of main girders considering additional deflection for composite cable-stayed bridges. In previous studies, the theoretical analysis of composite girders considering the shear lag has the following deficiencies. One is that the physical meaning of the shear lag generalized displacement needs to be clarified and cannot be measured directly. Second, the traditional shear lag calculation method uses the ratio of the normal stress considering the shear lag to the normal stress of the elementary beam theory to define the shear lag coefficient, leading to an overestimation of shear lag coefficients in slabs or flanges when sections with low-stress levels, and incorrect determining the normal stress distribution. For this reason, additional deflection needs to be introduced in composite girders to represent the shear lag effect.

In this paper, the additional deflection was introduced into the composite girder analytical solution considering shear lag for the first time. The longitudinal displacement functions are constructed by the slip function, axial equilibrium function, and different warping displacement functions. The theoretical solutions of the normal stress, total and additional deflections of the TICG under bending moment, axial and shear forces were obtained. An experimental study was carried out to investigate the shear lag effect in the TICG of the cable-stayed bridge. The stress and deflection of the main girder obtained by the theoretical method considering additional deflection (TMAD) are compared with the test and the FEM results, and analytical solution [11]. Finally, the reasonableness of the TMAD is verified by comparing it with numerical methods and experimental results in Ref. [18].

2 Shear lag analysis model considering additional deflection

In this paper, the theoretical model refers to the model by Zhang [16] while introducing the additional deflection studied by Zhang et al. [6,7] and Yang et al. [8]. Different from their studies, additional deflection is introduced into composite girders for the first time, and the additional deflection is applied to the longitudinal displacement function to calculate the analytical solutions for the warpage function, slip function, etc. The longitudinal displacement mode of shear lag is expressed by the warpage and the warpage strength functions through the variable separation method. The closed solutions of the stress and deflection can be obtained by establishing the governing differential equations.

2.1 Beam−column model and basic assumptions

In the cable-stayed bridge, internal forces of the main girder segment are displayed in Fig.1, where q is the uniform load; Q(x) indicates the vertical shear force. In the beam−column model, the stress in the main girder is [19]:

(1){\sigma }_{\text{a}} =\frac{N_0}{A}\pm \frac{M}{W}={\sigma }_{{\text{N}}_0}\pm {\sigma }_{\text{M}} ,

where N₀ is the axial force, M is the bending moment, A is the cross-sectional area of the composite girder, W is the section resistance moment of the composite girder, and {\sigma }_{{\text{N}}_0} and {\sigma }_{\text{M}} are the average compressive and bending stress obtained from the elementary beam theory.

The mechanical behavior of the main girder in the cable-stayed bridge can be decomposed as beam acting only and axial acting only. This paper separately calculated the composite girder under the beam action and axial force.

In the rectangular coordinate system, the x-axis is parallel to the girder’s longitudinal direction before deformation, and the cross-section of the main girder is symmetric about the coordinate plane x−z.

Assuming that the structure is in the elastic state, the displacement in the composite girder section obeys the linear superposition principle [20]. The shear strain and normal strain of the composite girder are considered and ignore the out-of-plane strain [21].

2.2 Analytical solution of additional deflection of shear lag under beam action

2.2.1 Fundamental equations

As shown in Fig.2, the cross-section of the TICG is symmetric about the z-axis and asymmetric with the y-axis. Hence, the axial equilibrium needs to be considered. The interface slip model of TICG is shown in Fig.2(a), where O_{\mathrm{s}} and O_{\mathrm{s}}^{\prime } are the center points of the steel girder before and after the slip, respectively. O_{\mathrm{c}} and O_{\mathrm{c}}^{\prime } are the center points of the concrete slab before and after the slip, respectively. O_{\mathrm{c}\mathrm{s}} and O_{\mathrm{c}\mathrm{s}}^{\prime } are the center points of the composite girder before and after the slip, respectively. h_co is the height between O_c and O_cs. h_so is the height between O_s and O_cs. In Fig.2(b), h_c is the concrete slab thickness. {\sigma }_x is the normal stress of the concrete slab.

The shear lag leads to the total deflection w(x) (Eq. (2)) of the composite girder being greater than the deflection calculated by elementary beam theory. Assuming that the additional deflection w_a(x) is part of the w(x) exceeding w_e(x) considering the shear lag effect:

(2)w(x)=w_{\text{e}}(x )+w_{\mathrm{a}}( x),

where w(x) is the total deflection of the main girder, w_e(x) is the deflection calculated by the elementary beam theory, and w_a(x) is the additional deflection of the composite girder.

The total longitudinal displacement u(x, y,\mathit{\text{z}}) at any point of the TICG is composed of the warping longitudinal displacement u_i^{\text{w}}( x,y,\mathit{\text{z}}), the longitudinal displacement u_i^{\mathrm{l}} (x,y ,\mathit{\text{z}}), and the interface slip u_i^{\text{slip}}(x,y ,\mathit{\text{z}}):

(3)u(x, y,\mathit{\text{z}})=u_i^{\mathrm{l}}(x,y,\mathit{\text{z}})+u_i^{\text{w}}(x,y,\mathit{\text{z}})+u_i^{\text{slip}}(x,y,\mathit{\text{z}}),

(4)u_i^{\text{slip}}(x)=k_i\zeta (x).

The longitudinal displacement in the concrete slab is expressed as:

(5)u_{\text{c}} =-\mathit{\text{z}}(w_{\text{e}}^{\prime }(x) +w_{\mathrm{a}}^{\prime }(x))- f(x)\omega (y)+ k_{\text{c}} \zeta (x),

where z is the distance from a point on the composite girder to the neutral axis of the converted section, \omega (y) is the warpage function of concrete slab cross-section, f(x) is the strength function of \omega (y) along the longitudinal direction, \zeta (x) is the interface slip function, and k_{\text{c}}=A_{\text{c}}/ (A_{\text{c}} +n{A}_{\text{s}}).

(6)\begin{array}{rl}& \omega (y)=\\ & \{\begin{array}{ll}-{\alpha }_1\left(1-{\displaystyle \frac{y^2}{b_1^2}}\right)+D,& \text{concrete inner slab between webs},\\ -{\alpha }_2\left(1-{\displaystyle \frac{y^2}{b_2^2}}\right)+D,& \text{concrete cantilever slab},\\ D,& \text{steel girder},\end{array}\end{array}

where D is the axial equilibrium function, {\alpha }_1 and {\alpha }_2 are the shear lag warpage amplitude of the concrete slab inner and cantilever parts, respectively; b₁, b₂ are the width of the cantilever and inner part of the concrete slab, respectively, i = 1, 2.

The longitudinal displacement of the steel girder is expressed as:

(7)u_{\text{s}} =-\mathit{\text{z}}(w_{\text{e}}^{\prime }(x) +w_{\mathrm{a}}^{\prime }(x))- f(x)\omega (y)+ k_{\text{s}} \zeta (x),

where k_{\text{s}} =-n{A}_{\text{s}} /(A_{\text{c}}+n{A}_{\text{s}} ), A_c and A_s are the cross-sectional area of the concrete slab and the steel girder, respectively; n=E_{\text{s}} /E_{\text{c}}, E_s and E_c are the elastic modulus of the steel girder and concrete slab, respectively.

The strain energy density [22] of the concrete slab is:

(8){\chi }_{\text{c}} =\frac{1}{2}(E{\epsilon }_{\mathrm{c}x}^2+ G{\gamma }_{\mathrm{c}xy}^2 )=\frac{1}{2}\left\{E{\left(\frac{\mathrm{\partial }{u}_{\text{c}}}{\mathrm{\partial }x}\right)}^2+G{\left(\frac{\mathrm{\partial }{u}_{\text{c}}}{\mathrm{\partial }y}\right)}^2\right\},

(9){\epsilon }_{\text{c}xi}=-\mathit{\text{z}}(w_{\text{e}}^{″}+w_{\text{a}}^{″}) -f^{\prime }({\alpha }_1(1- \frac{y^2}{b_i^2} )+D)+k_{\text{c}}{\zeta }^{\prime },

(10){\gamma }_{\mathrm{c}xyi}={\alpha }_{i}f\frac{2y}{b_i^2},

where {\epsilon }_{\mathrm{c}xi} and {\gamma }_{\mathrm{c}xy} are the tensile strain and the shear strain of the concrete slab, respectively, and i = 1, 2 represents the cantilever and inner part of the concrete slab.

The strain energy density [22] of the steel girder is:

(11){\chi }_{\text{s}} =\frac{1}{2}E{\epsilon }_{\mathrm{s}x}^2=\frac{1}{2}E{\left(\frac{\mathrm{\partial }{u}_{\text{s}}}{\mathrm{\partial }x}\right)}^2,

(12)\begin{array}{rl}{\epsilon }_{\text{s}xi}=& -\mathit{\text{z}}(w_{\text{e}}^{″}+w_{\text{a}}^{″})-f^{\prime }D+k_{\text{s}}{\zeta }^{\prime }{\epsilon }_{\text{c}xi}\\ =& -\mathit{\text{z}}(w_{\text{e}}^{″}+w_{\text{a}}^{″})-f^{\prime }({\alpha }_1\left(1-\frac{y^2}{b_i^2}\right)+D)+k_{\text{c}}{\zeta }^{\prime },\end{array}

where {\epsilon }_{\text{s}xi} is the tensile strain of steel girder, i = 1, 2, 3 stand for the web, top, and bottom flanges, respectively.

Using the minimum potential energy principle, the total potential energy of TICG under uniform load is:

(13)\begin{array}{rl}\text{Γ}=& {\displaystyle \sum _{i=1}^2{\text{Γ}}_{\text{c}i}+{\displaystyle \sum _{i=1}^3{\text{Γ}}_{\text{s}i}}}+{\text{Γ}}_{\text{ld}}+{\text{Γ}}_{\text{slip}}\\ =& P_1{\displaystyle {\int }_0^L{\zeta }^{\prime 2}}\text{d}x+P_2{\displaystyle {\int }_0^L{w}_{\text{e}}^{″2}}\text{d}x+P_2{\displaystyle {\int }_0^L{w}_{\text{a}}^{″2}}\text{d}x\\ & +2P_2{\displaystyle {\int }_0^L{w}_{\text{e}}^{″}{w}_{\text{a}}^{″}}\text{d}x+P_3{\displaystyle {\int }_0^L{f}^{\prime 2}}\text{d}x\\ & +P_4{\displaystyle {\int }_0^L{w}_{\text{e}}^{″}{f}^{\prime }}\text{d}x+P_4{\displaystyle {\int }_0^L{w}_{\text{a}}^{″}{f}^{\prime }}\text{d}x+P_5{\displaystyle {\int }_0^L{\zeta }^{\prime }{f}^{\prime }}\text{d}x\\ & -P_6{\displaystyle {\int }_0^L{\zeta }^{\prime }{w}_{\text{e}}^{″}}\text{d}x-P_6{\displaystyle {\int }_0^L{\zeta }^{\prime }{w}_{\text{a}}^{″}}\text{d}x\\ & +P_7{\displaystyle {\int }_0^L{f}^2}\text{d}x+{\displaystyle {\int }_0^{L}M(x)}{w}_{\text{a}}^{″}\text{d}x\\ & +{\displaystyle {\int }_0^{L}M(x)}{w}_{\text{e}}^{″}\text{d}x+\frac{1}{2}{\displaystyle {\int }_0^L{k}_i}{\zeta }^2\text{d}x,\end{array}

where P_1=k_i^2(E_{\mathrm{c}} A_{\mathrm{c}i}+E_{\mathrm{s}} A_{\mathrm{s}i}), P_2=E_{\mathrm{c}}{I}_{\mathrm{c}i}+E_{\mathrm{s}}{I}_{\mathrm{s}i}, P_3= E_{\mathrm{c}}{A}_{\mathrm{c}i}\left(D^2-{\displaystyle \frac{2}{3}}{\alpha }_{i}D+{\displaystyle \frac{23}{15}}{\alpha }_i^2\right)+E_{\mathrm{s}} A_{\mathrm{s}i}{D}^2, P_4=E_{\mathrm{c}}{S}_{\mathrm{c}i}(D-{\alpha }_i/3)+ E_{\mathrm{s}} S_{\mathrm{s}i}D, P_5=2k_i{E}_{\mathrm{c}} A_{\mathrm{c}i}(D-{\alpha }_i/3)+ 2k_i{E}_{\mathrm{s}}{A}_{\mathrm{s}i}D, P_6= E_{\mathrm{s}} S_{\mathrm{s}i}D, P_7=G_{\mathrm{c}}{A}_{\mathrm{c}i}{\displaystyle \frac{{\alpha }_i^2}{3b_i^3}}, {\mathrm{\Gamma }}_{\mathrm{l}\mathrm{d}} is the load potential energy, {\mathrm{\Gamma }}_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}} is the slip potential energy, {\mathrm{\Gamma }}_{\mathrm{c}i} and {\mathrm{\Gamma }}_{\mathrm{s}i} are the potential energy of the concrete slab and steel girder, respectively, I_{\mathrm{c}i} is the moment of inertia of the concrete slab’s inner or cantilever part to its neutral axis, I_{\mathrm{s}i} is the moment of inertia in the steel girder (i = 1, 2, 3) and S_{\mathrm{c}i} is the concrete slab’s first moment of area, M(x) is the bending moment of the main girder.

The variation of the total potential energy of the composite girder is:

(14)\begin{array}{rl}\delta \text{Γ}=& 2P_1{\displaystyle {\int }_0^L{\zeta }^{\prime }}\mathrm{d}x\delta {\zeta }^{\prime }+2P_2{\displaystyle {\int }_0^L{w}_{\mathrm{e}}^{″}}\mathrm{d}x\delta w_{\mathrm{e}}^{″}\\ & +2P_2{\displaystyle {\int }_0^L{w}_{\mathrm{a}}^{″}}\mathrm{d}x\delta w_{\mathrm{a}}^{″}+2P_2{\displaystyle {\int }_0^L{w}_{\mathrm{a}}^{″}}\mathrm{d}x\delta w_{\mathrm{e}}^{″}\\ & +2P_2{\displaystyle {\int }_0^L{w}_{\mathrm{e}}^{″}}\mathrm{d}x\delta w_{\mathrm{a}}^{″}+2P_3{\displaystyle {\int }_0^L{f}^{\prime }}\mathrm{d}x\delta f^{\prime }\\ & +P_4{\displaystyle {\int }_0^L{w}_{\mathrm{e}}^{″}}\mathrm{d}x\delta f^{\prime }+P_4{\displaystyle {\int }_0^L{f}^{\prime }}\mathrm{d}x\delta w_{\mathrm{e}}^{″}\\ & +P_4{\displaystyle {\int }_0^L{w}_{\mathrm{a}}^{″}}\mathrm{d}x\delta f^{\prime }+P_4{\displaystyle {\int }_0^L{f}^{\prime }}\mathrm{d}x\delta w_{\mathrm{s}}^{″}\\ & +P_5{\displaystyle {\int }_0^L{\zeta }^{\prime }}\mathrm{d}x\delta f^{\prime }+P_5{\displaystyle {\int }_0^L{f}^{\prime }}\mathrm{d}x\delta {\zeta }^{\prime }\\ & -P_6{\displaystyle {\int }_0^L{\zeta }^{\prime }}\mathrm{d}x\delta w_{\mathrm{e}}^{″}-P_6{\displaystyle {\int }_0^L{w}_{\mathrm{e}}^{″}}\mathrm{d}x\delta {\zeta }^{\prime }\\ & -P_6{\displaystyle {\int }_0^L{\zeta }^{\prime }}\mathrm{d}x\delta w_{\mathrm{a}}^{″}-P_6{\displaystyle {\int }_0^L{w}_{\mathrm{s}}^{″}}\mathrm{d}x\delta {\zeta }^{\prime }\\ & +2P_7{\displaystyle {\int }_0^L}f\mathrm{d}x\delta f+{\displaystyle {\int }_0^{L}M(x)\mathrm{d}x\delta }{w}_{\mathrm{a}}^{″}\\ & +{\displaystyle {\int }_0^{L}M(x)\mathrm{d}x\delta }{w}_{\mathrm{e}}^{″}+k_i{\displaystyle {\int }_0^L\zeta }\mathrm{d}x\delta \zeta .\end{array}

Based on the first variational principle \delta \mathrm{\Gamma }=0, the governing differential equations are obtained:

(15)-2P_1{\zeta }^{″}+P_6{w}_{\text{e}}^{(3)}+ P_6{w}_{\text{a}}^{(3)}- P_5{f}^{″}+k_i\zeta =0,

(16)-2P_2{w}_{\text{e}}^{(3)}-2P_2{w}_{\text{a}}^{(3)}-P_4{f}^{}+P_6{\zeta }^{″}-M^{\prime }(x) =0,

(17)-2P_3{f}^{″}-P_4{w}_{\text{e}}^{(3)}- P_4{w}_{\text{a}}^{(3)}- P_5{\zeta }^{″}+2P_{7}f =0.

Boundary conditions:

(18)\left(\text{2}{P}_1{\zeta }^{\prime }+P_5{f}^{\prime }-P_6{w}_{\text{e}}^{″}-P_6{w}_{\text{a}}^{″}\right)\delta {\zeta }^{″}| {}_0^L= 0,

(19)\left(2P_3{f}^{\prime }+P_4{w}_{\text{e}}^{″}+P_4{w}_{\text{a}}^{″}+P_5{\zeta }^{\prime }\right) \delta f|{}_0^L=0,

(20)\left(2P_2{w}_{\text{e}}^{″}+2P_2{w}_{\text{a}}^{″}+P_4{f}^{\prime }-P_6{\zeta }^{\prime }+M(x)\right)\delta w_{\text{e}}^{\prime }|{}_0^L=0,

(21)\left(2P_2{w}_{\text{a}}^{″}+2P_2{w}_{\text{e}}^{″}+P_4{f}^{\prime }-P_6{\zeta }^{\prime }+M(x)\right)\delta w_{\text{a}}^{\prime }|{}_0^L=0.

By Eqs. (15)–(17), a fourth-order differential equation is obtained for the shear lag strength function f :

(22)f^{(4)}+ P_1^{\mathrm{♯}}{f}^{(2)}+P_2^{\mathrm{♯}}f=P_3^{\mathrm{♯}}{M}^{\prime }(x).

The higher solution differential Eq. (22) is solved:

(23)f=C_1\text{sh}{k}_{1}x+C_2\text{ch}{k}_{1}x+ C_3\text{sh}{k}_{2}x+C_4\text{ch} k_{2}x+C_{5}x,

where C_i is the boundary condition coefficient, and k_i is the characteristic root.

When the interface slip is not considered in the calculation, \zeta (x) =0, the governing differential equations reduce to:

(24)(P_6+P_4)w_{\text{e}}^{(3)}+(P_6+P_4)w_{\text{s}}^{(3)}+(2P_3-P_5)f^{″}=0,

(25)-2P_2{w}_{\text{e}}^{(3)}- 2P_2{w}_{\text{s}}^{(3)}- P_4{f}^{}- M^{\prime }(x)=0,

(26)\left(2P_3{f}^{\prime }+P_4{w}_{\text{e}}^{″}+P_4{w}_{\text{s}}^{″}\right)\delta f|{}_0^L= 0.

The analytical solution is:

(27)f=C_1\mathrm{s}\mathrm{h}{k}_{1}x+C_2\mathrm{c}\mathrm{h}{k}_{2}x+f^{\ast },

where C_i needs to be determined by specific boundary conditions, k_i is the characteristic root of the characteristic equation, and f^{\ast } is the special solution only related to M^{\prime }(x).

2.2.2 Closed-form solutions

For the TICG under the beam action (Fig.3), the boundary conditions are as follows:

(28)w_{\mathrm{a}}|{}_{x=0}=0,w_{\mathrm{a}}|{}_{x=L}=0,w_{\mathrm{e}}|{}_{x=0}=0,w_{\mathrm{e}}|{}_{x=L}=0.

When the girder is subjected to uniform load q, the additional deflection is:

(29)w_{\mathrm{a}}=\frac{qt}{E_{\text{s}}{I}_0{k}_2^2}\left[-\frac{\text{ch}{k}_2(L-2x)}{k_2^2\text{ch}{k}_{2}L}-\frac{Lx-x^2}{2}\right],

where I₀ is the moment of inertia in the composite girder and t=2P_1{P}_3- {\alpha }_i+{\alpha }_i{\displaystyle \frac{y^2}{b_i^2}}+D.

According to the elementary beam theory, the theoretical deflection is shown in Eq. (30):

(30)w_{\text{e}} =\frac{q}{E_{\text{s}}{I}_0}\left(-\frac{x^4}{24}+\frac{L{x}^3}{12}-\frac{L^{3}x}{24}\right).

Similarly, the additional deflection formula of the TICG subjected to concentrated force P (Fig.3) can be obtained as follows:

(31)w_{\text{a1}}=\frac{Pt}{E_{\text{s}}{I}_0{k}_2^3}\left(\frac{\text{sh}{k}_2(x-a)}{\text{sh}{k}_{2}L}\text{sh}{k}_{2}x-k_2\frac{x-a}{L}x\right),x\in \left[0,a\right] ,

(32)\begin{array}{rl}& w_{\text{e1}}=\frac{P}{E_{\text{s}}{I}_0}\left[\frac{(x-a)x^3}{6L}+\left(a^2-\frac{2a^3}{3L}-\frac{a^2(x-a)}{2L}-\frac{aL}{3}\right)x\right],\\ & x\in \left[0,a\right],\end{array}

(33)w_{\text{a2}}=\frac{Pt}{E_{\text{s}}{I}_0{k}_2^3}\left[\frac{\text{sh}{k}_{2}a\cdot \text{sh}{k}_2(L-x)}{\text{sh}{k}_{2}L}-\frac{a}{L}{k}_2(x-L)\right], x\in (a,L],

(34)\begin{array}{rl}{w}_{\text{e2}}=& \frac{P}{E_{\text{s}}{I}_0}[-\frac{a{x}^3}{6L}+\frac{a{x}^2}{2}-\left(\frac{aL}{3}+\frac{a^3}{6L}\right)x\\ & +\frac{a^3}{2}-\frac{a^4}{3L}-\frac{a^3(x-a)}{3L}],x\in (a,L].\end{array}

In girders, the expression of the shear lag coefficient is [22,23]:

(35){\sigma }_{\text{e}} =\frac{{\int }_{-b/2}^{b/2}{\sigma }_x\text{d}y}{b},

(36)\eta \text{=}\frac{{\sigma }_x}{{\sigma }_{\text{e}}},

where b is the actual geometric width of the concrete slab, \eta is the shear lag coefficient, {\sigma }_{\text{e}} is the normal stress of the concrete slab obtained from elementary beam theory, and {\sigma }_x is the concrete slab’s normal stress.

From Eq. (36), it is observed that the stresses of different points in the concrete slab are often various under the shear lag effect. Thus, many different shear lag coefficients exist in a concrete slab cross-section. Therefore, the shear lag coefficients obtained by the traditional calculation method only indicate the magnitude of the actual stress at a section point that exceeds the elementary beam theoretical value and cannot represent the uneven degree of stress distribution and stress level in the whole section.

This paper defines the shear lag coefficient using the additional deflection to reflect the shear lag degree on the entire cross-section. According to different boundary conditions, the shear lag coefficient of TICGs on account of additional deflection is written as:

(37)\eta \text{=}\frac{w}{w_{\text{e}}}=\frac{w_{\mathrm{a}}+w_{\text{e}}}{w_{\text{e}}}.

When the composite girder is under uniform load with simply supported, the expression of the shear lag coefficient is as Eq. (38):

(38)\eta =1+\frac{24t\text{ch}[r_2(L-2x)]}{r_2^4\text{ch}[r_{2}L(x^4-2L{x}^3-L^{3}x)]}+\frac{24t(L-x)}{r_2^{2}L(x^3-2L{x}^2-L^3)}.

When the composite girder is under concentrated load with simply supported, the expression of shear lag coefficient based on deflection can be obtained:

(39)\begin{array}{rl}\eta =& 1+\frac{6tL}{(x-a)k_2^3{x}^3+(6L{a}^2-a^3-3a^{2}x-2a{L}^2)k_2^{3}x}\\ & \cdot \left[\frac{\text{sh}{k}_2(x-a)}{\text{sh}{k}_{2}L}\text{sh}{k}_{2}x-k_2\frac{x-a}{L}x\right],\end{array}

(40)\begin{array}{rl}\eta =& 1-\frac{6tL}{k_2^{3}a\left[x^3+3L{x}^2-(2L^2+a^2)x+a^2(3L-2x)\right]}\\ & \cdot \left[\frac{\text{sh}{k}_{2}a\cdot \text{sh}{k}_2(L-x)}{\text{sh}{k}_{2}L}-\frac{a}{L}{k}_2(x-L)\right].\end{array}

For a cantilevered TICG (Fig.4), the boundary conditions are as follows:

(41)w_{\mathrm{a}}|{}_{x=0}=0,w_{\mathrm{a}}^{\prime }|{}_{x=0}=0,w_{\text{a}}^{″}|{}_{x=0}=0, w_{\mathrm{e}}|{}_{x=0}=0.

When the cantilevered girder is subjected to uniform load q (Fig.4), the additional deflection is:

(42)\begin{array}{rl}{w}_{\mathrm{a}}=& -\frac{qt}{E_{\text{s}}{I}_0{k}_2^2}[-\frac{L{k}_2\text{sh}kL+1}{k_2^2\text{ch}{k}_{2}L}\mathrm{c}\mathrm{h}kx\\ & -\frac{L{k}_2\text{sh}kL+1}{k_2^2\text{ch}{k}_{2}L}+\frac{L}{k_2}\text{sh}kx+\frac{Lx-x^2}{2}].\end{array}

According to the elementary beam theory, the bending moment of the composite girder is M（ x） =-q(L-x^2)/2, so the theoretical deflection is shown in Eq. (43):

(43)w_{\text{e}} =\frac{q}{E_{\text{s}}{I}_0}\left(\frac{x^4}{24}-\frac{L{x}^3}{6}-\frac{L^2{x}^2}{4}\right).

The expression of the shear lag coefficient based on additional deflection can be obtained:

(44)\begin{array}{rl}\eta =& 1+\frac{24tL\text{sh}{k}_{2}x}{(x^2-4Lx+6l^2)k_2^3{x}^3}+\frac{12t(x-2L)}{(x^2-4Lx+6l^2)k_2^{2}x}\\ & +\frac{24t(L{k}_2\text{sh}{k}_2+1)(\text{ch}{k}_{2}x-1)}{(x^2-4Lx+6l^2)k_2^4{x}^2}.\end{array}

2.3 Analytical solution of shear lag under axial force

In the cable-stayed bridge, cables are usually anchored at the bottom surface of the top flange of steel girders or concrete slabs, and the horizontal component of the stay cable does not act directly at the location of the section center, which will cause additional bending moments. The horizontal component force of cables (N_A) is decomposed into the axial force (N₀) acting on the section centroid and bending moment (M_A) (Fig.5). This paper deduced the shear lag effect under vertical load earlier, and now the shear lag effect under the axial force of TICG is deduced.

There is no vertical and shear deflection in girders under the axial force, so the vertical deflection of the longitudinal bridge under axial force is zero. The axial natural displacement [16] is:

(45)u_0=\frac{N_{\text{A}}}{E_{\text{s}}{A}_0}x,

where u_0 is the longitudinal displacement under axial force, N_A is the axial force, and A₀ is the converted cross-sectional area of the composite girder.

The longitudinal displacement of any point on the concrete slab is expressed as:

(46)u_{\text{c}} =u_0-f(x) \omega (y)+k_{\text{c}}\zeta (x).

The longitudinal displacement in the steel girder is expressed as:

(47)u_{\text{s}} =u_0-f(x) \omega (y)+k_{\text{s}}\zeta (x).

The concrete slab’s strain energy is:

(48){\epsilon }_{\mathrm{N}\mathrm{c}xi}=f^{\prime }(D-{\alpha }_i(1-\frac{y^2}{b_i^2})) +k_{\text{c}}{\xi }^{\prime }-\frac{N_{\text{A}}}{E_{\text{s}}{A}_0},

(49){\gamma }_{\mathrm{N}\mathrm{c}xyi}={\alpha }_{i}f\frac{2y}{b_i^2},

where {\epsilon }_{\mathrm{N}\mathrm{c}xi} is the normal strain, and {\gamma }_{\mathrm{N}\mathrm{c}xyi} is the shear strain.

The strain energy of the steel girder is:

(50){\epsilon }_{\mathrm{N}\mathrm{s}xi}=f^{\prime }D+k_{\text{s}} {\xi }^{\prime }-\frac{N_{\text{A}}}{E_{\text{s}}{A}_0}.

The expression of the total potential energy of TICG under axial load is:

(51)\begin{array}{rl}\mathrm{\Gamma }=& \sum _{i=1}^2{\mathrm{\Gamma }}_{\text{c}i}+\sum _{i=1}^3{\mathrm{\Gamma }}_{\text{s}i}+{\mathrm{\Gamma }}_{\mathrm{l}\mathrm{d}}+{\mathrm{\Gamma }}_{\mathrm{s}\mathrm{l}}=K_1{\int }_0^L{{\zeta }^{\prime }}^2\text{d}x\\ & +K_2{\int }_0^L\frac{N_{\text{A}}^2}{E_{\text{s}}^2{A}_0^2}\text{d}x+K_3{\int }_0^L{f}^{{}^{\prime }2}\text{d}x\\ & +K_4{\int }_0^L\frac{N_{\text{A}}}{E_{\text{s}}{A}_0}{f}^{\prime }\text{d}x+K_5{\int }_0^L{\zeta }^{\prime }{f}^{\prime }\text{d}x\\ & -K_6{\int }_0^L{\zeta }^{\prime }\frac{N_{\text{A}}}{E_{\text{s}}{A}_0}\text{d}x+K_7{\int }_0^L{f}^2\text{d}x\\ & +\frac{1}{2}{\int }_0^L{k}_i{\zeta }^2\text{d}x+N_{\text{A}}k\left[\zeta (0)-\zeta (L)\right]\\ & +N_{\text{A}}D\left[f(0)-f(L)\right]-\frac{N_{\text{A}}^2}{E_{\text{s}}{A}_0}L,\end{array}

where K_1=E_{\text{s}}{\displaystyle \sum _{i=1}^4{A}_i}, K_2= E_{\text{s}} {\displaystyle \sum _{i=1}^4{k}_i^2{A}_i}, K_3=E_{\text{s}}{\displaystyle \sum _{i=1}^4{A}_i\cdot } \left(D^2-{\displaystyle \frac{4}{3}}{\alpha }_{i}D+{\displaystyle \frac{8}{15}}{\alpha }_i^2\right), K_4= -2E_{\text{s}}{\displaystyle \sum _{i=1}^4\left({\displaystyle \frac{2}{3}}{\alpha }_i-D\right)k_i{A}_i}, K_5= 2E_{\text{s}} {\displaystyle \sum _{i=1}^4{k}_i{A}_i=0}, K_6=-2 E_{\text{s}} {\displaystyle \sum _{i=1}^4\left({\displaystyle \frac{2}{3}}{\alpha }_i-D\right)A_i}, and K_7={\displaystyle \frac{4G_{\text{s}}}{3}}\cdot {\displaystyle \sum _{i=1}^4{\displaystyle \frac{{\alpha }_i{A}_i}{b_i^2}}}.

The governing differential equations are:

(52)k_i\zeta -2K_1{\zeta }^{″}-K_5{f}^{″}\text{=}0,

(53)2K_{7}f-2K_3{f}^{″}-K_5{\zeta }^{″}\text{=}0 .

The analytical solution is:

(54)\zeta \text{=}{C}_1\text{sh}rx+C_2\text{ch} rx,

(55)f=C_1\text{sh}{k}_{1}x+C_2\text{ch}{k}_{1}x+ C_3\text{sh}{k}_{2}x,

where C_i is determined by boundary conditions (i = 1, 2, 3) and k_i is the characteristic root.

In cable-stayed bridges, the bending moments in the concrete slab under dead load would be the same as the bending moment for a continuous girder under the uniform load [24]. According to the principle of equivalent bending moment, the TMAD can be used to calculate the stress of the main girder in composite cable-stayed bridges. Using the TMAD, the main girder of the composite cable-stayed bridge is only needed to establish a one-dimensional girder with beam element, and select the concerned main girder section to study. Extract the internal forces at both ends of the girder segment containing the concerned section. And substitute the internal forces into the analytical solution method to obtain the normal stresses in the cross-section considering the shear lag effect.

3 Theoretical method validation

3.1 Test specimen and procedure

To verify the accuracy of the TMAD, comparative experimental studies were conducted. Wang et al. [17,25] conducted an experimental study of a TICG. The test girder was a main girder scaled model in Xiazhang Bridge, and consisted of a concrete slab, five cross beams, and twin girders. The steel girder and cross beams steel was Q345 steel, and the reinforcement in the slab was HRB235 steel. The concrete slab was C40 concrete. The length of the test girder was 6300 mm, and the with of the concrete slab was 2400 mm. The calculated span L was 6000 mm, as shown in Fig.6. The test girder was under vertical and axial loads. The axial load is applied by tensioning the prestressed steel strands and vertical loads by a hydraulic jack. The setup and layout of measuring points are shown in Fig.6. The following three load cases were used to compare the analytical results.

In load case 1, the test girder with a 6000 mm calculated span (L = 6000 mm) was subjected to mid-span load (P = 360 kN) and axial load (N = 600 kN). A two-span continuous girder was designed with mid-span loading (P = 180 kN) and axial load (N = 600 kN) in load case 3. Load case 3 represents the main girder with negative and positive bending moment regions with two cable spacing. The test girder was transformed into a two-span continuous girder by putting the rubber bearings under the lower flange of the mid-span of two main steel girders. The standard bearings were used to simulate the cables vertical supports and simulation of axial force component of stay cables with prestressing tensioned at the TICG ends. The test girder’s mechanical behavior is similar to those in main girders of cable-stayed bridges. For comparative analysis, the test girder with a 6000 mm calculated span was only subjected to mid-span loading (P = 360 kN) in load case 2. The loading forms under three load cases are shown in Fig.6(c).

3.2 Finite element model

To verify the accuracy of the TMAD, the FEM was built based on the test girder geometries using ABAQUS. The main girders, transverse beams, stiffening ribs, anchor blocks, concrete slab, and studs were modeled by brick elements (C3D8R). The reinforcement bars and prestressing tendons were defined by three-dimensional truss elements (T3D2). In the elaborate element model, a multilinear model is used to simulate the steel girder. To determine the mesh size, a mesh sensitivity analysis was conducted, and a mesh size of 30 mm was used. The elastic modulus of steel E_s = 2.06 × 10⁵ MPa, Poisson’s ratio υ_s is 0.3. The yield strength is 388.3 MPa, and the ultimate strength is 553.4 MPa [11]. The concrete elastic modulus E_c = 3.23 × 10⁴ MPa, Poisson’s ratio υ_c is 0.2, and concrete compression strength f_c = 32.1 MPa (f_c = 0.76f_cu). The constitutive relationship of concrete used the recommended rules in Ref. [26]. The hard contact simulated the interaction between the concrete slab and the steel girder. The tangential interaction adopts penalty friction, and the friction coefficient is taken as 0.4 [27]. According to the actual boundary conditions, fixed hinge supports are arranged at one end, while sliding hinges are supported at the leading end. The FEM is shown in Fig.7.

3.3 Comparison of test results in three load cases

The shear lag effect was represented in the unevenness degree of the normal stress distribution of the girder. To verify the applicability of the TMAD, the normal stress in concrete slabs of three load cases was analyzed by combining the results of the test, TMAD, and FEM.

In Fig.8, the abscissa represents the transverse position of the test girder cross-section. x is the space to the test girder end. L_t = L/2 is the calculated span of the continuous girder. The normal stress results of the TMAD, FEM, and the test are compared in Tab.1 and Fig.8. As shown in Tab.1 and Fig.8, there are differences in the normal stresses between the results from the TMAD, FEM, and the test. Still, their normal stress distributions are consistent, which indicates that the laws of the shear lag effect expressed by those three results are the same. In detail, the deviation between the FEM and the test results is 1.1%–15.3%, with an average deviation of 8.7%. The difference between the TMAD and the test results is 0.5%–17.6% with an average difference of 6.2%. The deviation between the TMAD and the FEM results is 0%–7.3%, with an average deviation of 3.2%. The normal stress calculated by the TMAD agrees well with the FEM and test results, and the applicability of the TMAD was validated.

The strain of the main steel girder obtained by experimental, FEM, and TMAD is illustrated in Fig.9. And show that the results of theoretical and FEM in this paper fit well with the test results.

The deflection of the test girder was calculated to verify the accuracy of the TMAD. The comparison results are shown in Fig.10. As shown in Fig.10, the deflection calculated by the TMAD could well match the FEM and test results, with the maximum deviation at 3.0%.

4 Discussion

4.1 Example 1

4.1.1 Comparison with the existing method

Taking the test girder in Subsection 3.1 as the analysis object, the analytical solution based on the energy method in Ref. [11]. was compared with the TMAD. The analytical solution of Ref. [11]. does not consider the additional deflection. Besides, the longitudinal displacement function u(x, y,\mathit{\text{z}}) and the warpage function \omega (y) used in this paper differed from those in Ref. [11]. The normal stress of the concrete slab was calculated by FEM, analytical solution in Ref. [11], and TMAD was compared.

Tab.2 shows that the distribution of the normal stresses in concrete slabs calculated by several methods is consistent. Both the L/4 section and the mid-span section exhibit the positive shear lag effect law. The deviations in the results of the normal stresses obtained with TMAD and analytical solution [11] from the FEM were illustrated in Tab.3. It can be seen that the deviation of the results between TMAD and FEM ranges from 0% to 7.3%, with an average deviation of 2.7%. The deviation of the results between the analytical solution [11] and FEM ranges from 3.0% to 15.1%, with an average deviation of 8.8%. The normal stress obtained by the TMAD fits well with the FEM and test results than the analytical solution [11], and the calculation accuracy is improved.

4.1.2 Analysis of shear lag coefficient based on additional deflection

Taking the test girder as the analysis object, numerical research on the changing boundary conditions was conducted. Since the cantilever construction method was often used for constructing main girders of cable-stayed bridges, the shear lag of the cantilever boundary condition was further studied. The shear lag coefficient of the test girder under uniform load and axial force was calculated. The uniform load q = 40 kN/m and axial force N = 60 kN. The comparison results are shown in Fig.11.

The shear lag coefficients of girders with fixed-bearing, fixed-end, and mobile-bearing were calculated by Eq. (37) to analyze the effect of different boundaries on the shear lag effect. At the same time, the shear lag coefficient results calculated by Eq. (37) and the traditional calculation formula Eq. (36) [19,23] were compared. As seen in Fig.11(a) that by the TMAD, the shear lag at the fixed end is more obvious due to the large bending moment and deformation. The shear lag at the restrained end of the simply supported girder is less than that at the cantilever girder fixed end. This is because the stronger the constraint, the higher the bending moment and deformation of the section constrained; the greater the unevenness of the normal stress distribution in the section, and the greater the shear lag effect. The effect of different forms of restraint on the shear lag is reflected by using the additional deflection to calculate the shear lag coefficient.

Fig.11(b)–Fig.11(d) indicate that there are different shear lag coefficients at different points of the same section of the concrete slab obtained by Eq. (36) [19,23], and the shear lag coefficient has other or even opposite variation laws in sections along the longitudinal bridge direction. From the line of web position of the concrete slab, the shear lag coefficient of the girder segment near the zero moment zone is very large by the traditional calculation formula Eq. (36) [19,23], but the real law is that the shear lag in the cantilever end and the zero moment zone are small by Eq. (37) as shown in Fig.11(c) and Fig.11(d). The traditional calculation method Eq. (36) [19,23] has the wrong judgment on the shear lag effect of the section near the zero bending moment zone.

4.1.3 Influence of the interface slip

To study the effect of the interface slip on composite girders, this paper derives the analytical solution of TICG without considering slip when \zeta (x)=0, see Eq. (27). At the same time, the concrete slab and the steel girder are connected by common nodes to establish the FEM without considering the slip. Taking load case 1 as an example, the analytical solution and FEM results with and without slipping are compared, and the comparison results of normal stress in the concrete slab, composite girder deflection, and the normal stress of steel girder upper flange are obtained.

In Fig.12(a) and Fig.12(b), the effective width coefficient (b_e/b) represents the shear lag effect of the section, where b_e is the effective width of the concrete slab, and b is the geometric width of the concrete slab. It can be seen from Fig.12 that the slip influences the stress and deflection of the composite girder. The stress of concrete slab without considering slip is higher than that of considering slip, an average increase of 4.1%, but the effective width coefficient is basically unchanged. The stress of steel girder without considering slip is reduced than that of considering slip, 7.0% reduction on average. The deflection of the composite girder without considering slip is smaller than that of considering slip.

4.2 Example 2

For comparison reasons, the numerical and experimental study of the shear lag on a composite girder was conducted by Zhu et al. [18] and Li [28] was used as an example. The length of the test girder was 5.2 m, the calculated span was 5 m, and the slab width was 2.8 m. The concrete slab was C50 concrete. The girder was Q345C steel. The center spacing of the web was 2 m. Load case 1 and load case 5 were selected to verify the TMAD. The test girder dimensions and loading cases are shown in Fig.13.

The shear lag indicates the non-uniformity distribution degree of the normal stress. Therefore, the normal strain obtained by the TMAD and finite element results in comparison with the values retrieved from Ref. [18], as shown in Fig.14 and Tab.4. The result indicates that the error of the normal strain between the TMAD and test results [18] may reach 0%–78.4% with an average difference of 19.4%. In comparison, the error of normal strain between the numerical method [18] and test results [18] may reach 0%–87.8% with an average difference of 22.5%; the error is reduced while using the TMAD. The deviation of the TMAD from the FEM is 0.3%–5.7%, with an average difference of 3.0%. The deviation of the TMAD from the numerical method [18] is 0%–16%, with an average difference of 4.0%. Besides, the reason for the large deviation between the three methods and the test results [18] is that the analytical calculation model is an approximate method, and the test data will be affected by support, loading, local stress behavior, inaccurate gauge distance, and other factors, so there will be variation. Fig.14(a) provides that under the concentrated load, the normal stress distribution in the mid-span section is more uneven, and there is an obvious positive shear lag. And the shear lag in the mid-span section under uniform load is smaller than that under the concentrated load, as displayed in Fig.14(b).

Three methods are used to calculate the deflection of the main girder under two load cases for comparison, as shown in Fig.15. It is shown that the deflection calculated by the TMAD is consistent with the test value [18] and FEM result, which verifies the reliability of the TMAD. The additional deflection in the mid-span is the largest in the two load cases. The ratio of the additional deflection to the total deflection reaches a maximum of 10.8% in the mid-span.

5 Conclusions

This paper proposes an improved theoretical model of the shear lag in composite girders under bending moment and axial force. The shear lag is directly expressed by introducing the additional deflection as generalized displacement. The analytical solution of the normal stress and deflection was obtained considering the interface slip and different warping functions. An experimental study was conducted on a TICG to simulate the main girder of cable-stayed bridges. Besides, the TMAD has been verified by experiments, FEM, and other scholars’ methods and tests. The main conclusions are drawn as follows.

1) For TICG under the bending moment and axial force, the theoretical solutions of the interface slip function, warpage strength function, and the additional deflection were derived. The analytical solutions of the normal stress and deflection were obtained from the TICG.

2) Compared with experimental studies, FEM, and analytical methods [11], the applicability of the TMAD was verified. The accuracy of the TMAD in predicting the shear lag effect of TICG was improved by about 6% compared to others.

3) Compared shear lag coefficients at different boundaries, it is demonstrated that the shear lag coefficient obtained by the additional deflection is more accurate than that calculated by the traditional method in the girder segment near the zero moment zone. The stronger the support constraint, the more obvious the shear lag effect near the support section.

4) By comparing the experiments and numerical method in Ref. [18], the calculation deviation between the TMAD and test results [18] is 3.1%. By calculating additional deflection, the ratio of additional deflection to total deflection can reach a maximum of 10.8% in the mid-span and needs attention in the design.

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