Study on the Deflection Calculation of a Steel Truss Web–Concrete Composite Beam Under Pre-Stress

Abstract

The aim of this study is to establish an accurate calculation method for the deflection caused by the effect of pre-stress in a steel truss web–concrete composite girder bridge based on the energy variational principle, considering the influence of shear deformation and the shear lag effect of the steel truss web member on the accuracy of the deflection calculation. The pre-stress effect is determined by the equivalent load method, and the deflection analytical solution for a composite girder bridge under straight-line, broken-line, and curve pre-stressing tendon arrangements is established. The reliability of the formula is verified using ANSYS 2022 finite element numerical simulation. At the same time, the influence of shear deformation, the shear lag effect, and their combined (dual) effect on the deflection calculation accuracy is analyzed under different linear pre-stressed reinforcement arrangements and comprehensive arrangements of pre-stressed reinforcement. The analysis of the example shows that the analytical solution for the deflection of the steel truss web–concrete composite beam, when considering only the shear deformation and the dual effect, is more consistent with the finite element numerical solution. The shear deformation of the steel truss web member under the eccentric straight-line arrangement alone does not cause additional deflection, and the additional deflection caused by the shear lag effect can be ignored. The influence of shear deformation on deflection is higher than that of the shear lag effect. The contribution ratio of the additional deflection caused by the dual effect is greater than 14%, and the influence of the dual effect on deflection is more obvious under a broken-line arrangement. Under the comprehensive arrangement of pre-stressing tendons, the contribution rate of shear deformation to the total deflection is about 3.5 times that of shear lag. Compared with the deflection value of the primary beam, the mid-span deflection is increased by 3.0%, 11.0%, and 13.9% when only considering the shear lag effect, only considering shear deformation, and considering the dual effect, respectively. Therefore, shear deformation and the shear lag effect should be considered when calculating the camber of a steel truss web–concrete composite girder bridge to improve the calculation accuracy.

1. Introduction

Steel truss web–concrete combined girder bridges have the advantages of light deadweight, good wind resistance, and beautiful modeling, meaning that they have been widely used in domestic and international bridge construction in recent years; such a structure is shown in Figure 1 [1]. In order to improve the force performance of the combined girder bridge, the stability of the structure, and the stiffness of the girder body, it is usually arranged with linear, curved, and folded internal and external pre-stressing tendons [2].

To date, domestic and international scholars have conducted in-depth research and exploration on the full-bridge bearing capacity [3], joint structure [1,4,5,6,7], bending resistance [7,8,9], torsional distortion effect [10,11,12], and shear lag effect [13,14,15,16,17] of steel truss web–concrete composite girder bridges by means of testing, finite element, and analytical methods. In terms of deflection research, in Reference [18], the steel truss web–concrete composite girder bridge was considered equivalent to a thin-walled box girder. The deflection calculation formula was established using the displacement mode and solution method of the composite girder, and the applicability of the formula was verified through finite element analysis. In Reference [19], the deflection calculation formula of a steel truss web–concrete composite girder bridge was derived using the energy variational method, considering the dual effects of shear hysteresis and shear deformation, and the deflection calculation formula of a high-span steel truss web–concrete composite girder bridge was studied. The influence of the height–span ratio and horizontal inclination angle of the steel truss web on the deflection of a composite girder bridge was also studied. In Reference [20], the deflection equation of a steel truss web–concrete composite girder bridge under different load conditions was derived using Timoshenko beam theory and the energy decomposition principle, and the ratio of the shear lag effect to the additional deflection contribution caused by shear deformation was analyzed. In Reference [21], the influence of web member deformation on beam deflection was considered, and the deflection displacement of the composite box girder was obtained using the energy variational method. Based on the finite beam segment method, the stiffness matrix and node load of the beam segment analysis element can be derived, and the influence of structural parameters on the additional deflection is studied.

It can be seen that the above research only considers the two conventional loading modes of concentrated load and uniform load when establishing an analytical solution for deflection, and there has been no research on calculation methods for the upward deformation caused by pre-stressed tendons inside and outside the composite beam. However, References [22,23] found that the shear deformation and shear lag effect will have different degrees of influence on the deformation calculation in the calculation of deflection caused by pre-stress for box girder structures and corrugated steel web structures. The structural characteristics of a steel truss web–concrete composite girder bridge are similar to those of a box girder, and the shear lag effect and shear deformation are also contributing factors. When the beam body deforms upward under the action of pre-stress, it will also be affected by both factors. If the deformation calculation is not accurate, it may cause the inverted arch to be too large, resulting in adverse conditions [24]. It is important to consider the effect of pre-stress in the design of a new bridge; controlling the deformation caused by the pre-stressed tension after the completion of the bridge or the reinforcement of an old bridge with an external pre-stressed beam is of great significance for accurate calculation of the deflection caused by pre-stress.

Therefore, based on the principle of energy splitting, considering the shear deformation and shear lag effect of the steel truss web, in combination with the pre-stressing equivalent load method, this study establishes a deflection calculation method for a steel truss web–concrete composite girder bridge with a diagonal arrangement of the web in straight-line, folded-line, and curved-line formations of pre-stressing beams, and verifies the reliability of the analytical method through finite element numerical solution of the example. At the same time, the distribution of deflection along the girder length direction and the contribution ratio of additional deflection to the total deflection caused by shear deformation, the shear lag effect, and their combined (dual) effect are compared and analyzed, respectively, for this composite girder bridge under different linear and integrated beam deployment conditions, in order to provide references for the construction control and deformation calculation of steel truss web–concrete composite girder bridges under the action of pre-stressing force.

2. Derivation of the Deflection Differential Equation

2.1. Underlying Assumption

(1)

The deformation of the top and bottom plates of the steel truss web-type concrete composite girder bridge conforms to the “assumption of the proposed plane”;

(2)

The bending moments generated by pre-stressing are borne by the concrete top and bottom plates, the steel truss web only bears the shear force, and the web that extends into the interior of the combined girder wing plate is ignored;

(3)

The out-of-plane shear deformation and the vertical deformation of the wing plate are assumed to be zero;

(4)

The concrete and pre-stressing tendons of the combined girder bridge are considered as elastic work;

(5)

The longitudinal bending stiffness of the steel truss web is very small—it is assumed that the bending strain energy of the steel truss web is zero, and only the shear strain energy of the web is considered [10].

2.2. Equivalent Beam Conversion

In order to facilitate the calculation, a steel truss web–concrete composite girder bridge with a diagonal arrangement of web bars is taken as an example, and the discontinuous steel truss web bars are converted to equivalent steel webs based on the principle of equal shear deformation according to the literature [25]. Assuming that the length of the diagonal web is $l$, its vertical height and horizontal projection length are $h$ and $b$, respectively, and the vertical angle of the steel truss web is $\theta$, the equivalent steel web plate with equal width and height can be converted.

From Figure 2, it can be seen that the steel truss web bar is under the action of shear force $V$. Assuming that the displacement of node $N$ with respect to node $M$ is ${\eta }_1$, then ${\eta }_1$ is as follows.

$${\eta }_1={\displaystyle \frac{Vl}{E_0{A}_0\sin {\theta }^2}}={\displaystyle \frac{V{l}^3}{E_0{A}_0{h}^2}}$$

(1)

where $E_0$ is the modulus of elasticity of steel, and $A_0$ is the cross-sectional area of the steel truss web.

The equivalent steel web produces a shear angle of $\gamma =\tan \gamma ={\eta }_2/b$ for the same magnitude of shear force $V$. The shear deformation ${\eta }_2$ between nodes $NM$ is

$${\eta }_2=b\gamma ={\displaystyle \frac{bV}{G_s{A}_s}}={\displaystyle \frac{bV}{G_{s}h{t}_w}}$$

(2)

In this equation, $G_s$ is the steel shear modulus; $A_s$ is the equivalent cross-sectional area of the steel web, $A_s=h{t}_w$; and $t_w$ is the equivalent thickness of the steel web.

According to the principle of equal shear deformation, that is, ${\eta }_1={\eta }_2$, the equivalent steel web thickness formula is as follows:

$$t_w={\displaystyle \frac{E_0}{G_s}}{\displaystyle \frac{bh{A}_0}{l^3}}=2{\displaystyle \frac{\left(1+{\mu }_s\right)bh{A}_0}{l^3}}$$

(3)

where ${\mu }_s$ is the Poisson’s ratio of the steel.

After the steel truss web bars have been equivalently converted, the equivalent beam cross-sectional dimensions are assumed, as shown in Figure 3. In the figure, the width of the top and bottom plates of the combined girder bridge is $2b$, the width of the cantilever plate is $\zeta b$, the distance of the upper edge of the upper wing plate from the central axis of its cross-section shape is $Z_u$, the distance of the lower edge of the lower wing plate from the central axis of its cross-section shape is $Z_b$, the thickness of the top plate is $t_{u1}$, the thickness of the cantilever plate is $t_{u2}$, the thickness of the bottom plate is $t_b$, and the thickness of the converted steel web plate is $t_w$.

In accordance with the above basic assumptions, two generalized functions are introduced to analyze the deformation of the combined beam under vertical bending effects—namely, the vertical deflection $\omega (x)$ and longitudinal displacement $u(x,y)$ of the beam—taking into account the effect of the shear deformation of the converted steel web.

$$u(x,y)=Z_i[{\omega }^{\prime }(x)-\alpha (x)]+f(y,z)U(x)$$

(4)

where $Z_i$ is the vertical coordinate of the combined beam, and $-Z_U$ is taken for the upper wing plate and $Z_b$ for the lower wing plate. ${\omega }^{\prime }(x)$ is the beam vertical deflection of the corresponding angle; $\alpha (x)$ is the combination of beam section shear deformation caused by the angle and the value of $\alpha (x)={\omega }^{\prime }(x)-\phi =V(x)/(G_s{A}_s)$, where $\phi$ is the combination of beams due to the bending moment caused by the bending angle; $U(x)$ is the maximum difference between the top- and bottom-plate shear deformation; $V(x)$ is the combination of beam cross-section shear; $A_s$ is the conversion of the effective shear area of the steel web plate; and $f(y,z)$ is the warping displacement function.

2.3. Warp Displacement Function

Reference [26] proved theoretically that the quadratic parabola is a reasonable warping displacement function in the analysis of the shear lag effect of box beams, with the expression

$$f(y,z)=\left\{\begin{array}{l}-Z_u\left[1-{\left({\displaystyle \frac{y}{b}}\right)}^2\right]\mathrm{roof}\mathrm{plate}\\ -Z_u\left[1-{\left({\displaystyle \frac{\zeta b+b-y}{\zeta b}}\right)}^2\right]\mathrm{cantilever}\mathrm{plate}\\ Z_b\left[1-{\left({\displaystyle \frac{y}{b}}\right)}^2\right]\mathrm{base}\mathrm{plate}\end{array}\right.$$

(5)

The longitudinal positive strain ${\epsilon }_{x1}$, ${\epsilon }_{x2}$, ${\epsilon }_{x3}$ and transverse shear strain ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ of the top, cantilever, and bottom plates of the combined beam are obtained from the longitudinal displacement at any point of the section:

$$\begin{array}{l}{\epsilon }_{xi}={\displaystyle \frac{\partial u(x,y)}{\partial x}}\\ =Z_i[{\omega }^{″}(x)-{\alpha }^{\prime }(x)]+f(y,z)U^{\prime }(x)\end{array}$$

(6)

$${\gamma }_{xi}={\displaystyle \frac{\partial u(x,y)}{\partial y}}=f^{\prime }(y,z)U(x)$$

(7)

2.4. Derivation of Deflection Control Differential Equations Using the Energy Variational Method

According to the principle of minimum potential energy, when the structure is in equilibrium, the total potential energy of its system under the action of external forces becomes zero in the first order, that is,

$$\delta \prod =\delta (\overline{V}-\overline{W})=0$$

(8)

where $\overline{V}$ is the strain energy of the beam, and $\overline{W}$ is the potential energy of the external force on the beam.

According to the principle of energy differentiation, the potential energy of the external force when the beam is bent can be obtained as follows:

$$\begin{array}{l}\overline{W}=-{\displaystyle {\int }_0^{l}M(x)\left[{\omega }^{″}(x)-{\alpha }^{\prime }(x)\right]}dx-{\displaystyle {\int }_0^{l}V(x)\alpha (x)dx}\\ =-{\displaystyle {\int }_0^{l}M(x){\phi }^{\prime }}dx-{\displaystyle {\int }_0^{l}V(x)\left[{\omega }^{\prime }(x)-\phi (x)\right]dx}\end{array}$$

(9)

The deformation potential energy $\overline{V}$ of the beam consists of the top-, cantilever-, and bottom-plate strain energy and the converted steel web shear strain energy together, i.e.,

$$\overline{V}={\overline{V}}_1+{\overline{V}}_2+{\overline{V}}_3+{\overline{V}}_w$$

(10)

$${\overline{V}}_1=2\left[{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{\displaystyle {\int }_0^b{t}_{u1}\left(E_c{\epsilon }_{x1}^2+G_c{\gamma }_1^2\right)}}dydx\right]$$

(11)

$${\overline{V}}_2=2\left[{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{\displaystyle {\int }_b^{b+\varsigma b}{t}_{u2}\left[E_c{\epsilon }_{x2}^2+G_c{\gamma }_2^2\right]}}dydx\right]$$

(12)

$${\overline{V}}_3==2\left[{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{\displaystyle {\int }_0^b{t}_b\left[E_c{\epsilon }_{x3}^2+G_c{\gamma }_3^2\right]}}dydx\right]$$

(13)

$${\overline{V}}_w=2\left[{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{G}_s{A}_s{\alpha }^2(x)}dx\right]$$

(14)

where $E_c$ is the modulus of elasticity of the top and bottom slabs of concrete for composite girder bridges, and $G_c$ is the shear modulus of the top and bottom slabs of concrete for composite girder bridges.

Equations (9) and (10) are brought into Equation (8) for the variational operation and simplified by using the partition integral formula, which can be organized to obtain the control differential equations based on the variational principle:

$$\left\{\begin{array}{l}-2V(x)-E_{c}I{\phi }^{″}(x)-{\displaystyle \frac{2}{3}}{E}_{c}I{U}^{″}(x)\\ +2G_s{A}_s(\phi (x)-{\omega }^{\prime }(x))=0\\ -V^{\prime }(x)+2G_s{A}_s({\phi }^{\prime }(x)-{\omega }^{″}(x))=0\\ -{\displaystyle \frac{2}{3}}{E}_{c}I{\phi }^{″}(x)-{\displaystyle \frac{8}{15}}{E}_{c}I{U}^{″}(x)+{\displaystyle \frac{4}{3}}{G}_c{I}_{s}U(x)=0\end{array}\right.$$

(15)

where the boundary conditions are:

$$\left\{\begin{array}{l}\left[\left(M(x)+E_{c}I{\phi }^{\prime }(x)+{\displaystyle \frac{2}{3}}{E}_{c}I{U}^{\prime }(x)\right)\delta \phi \right]{{\displaystyle |}}_0^l=0\\ \left[\left(V(x)+2G_s{A}_s{\omega }^{\prime }(x)-2G_s{A}_s\phi (x)\right)\delta \omega \right]{{\displaystyle |}}_0^l=0\\ \left[\left({\displaystyle \frac{2}{3}}{E}_{c}I{\phi }^{\prime }(x)+{\displaystyle \frac{8}{15}}{E}_{c}I{U}^{\prime }(x)\right)\delta U\right]{{\displaystyle |}}_0^l=0\end{array}\right.$$

(16)

where $I_1=2b{Z}_u^2{t}_{u1}$, $I_2=2(\varsigma b)Z_u^2{t}_{u2}$, $I_3=2b{Z}_b^2{t}_b$, $I=I_1+I_2+I_3$, and $I_s={\displaystyle \frac{I_1}{b^2}}+{\displaystyle \frac{I_2}{{(\varsigma b)}^2}}+{\displaystyle \frac{I_3}{b^2}}$.

Collating Equations (15) and (16) yields

$${\phi }^{″}(x)=-{\displaystyle \frac{V(x)}{E_{c}I}}-{\displaystyle \frac{2}{3}}{U}^{″}(x)$$

(17)

$$U^{″}(x)-k^{2}U(x)={\displaystyle \frac{15}{2E_{c}I}}V(x)$$

(18)

where $k=\sqrt{15{\displaystyle \frac{G_c{I}_s}{E_{c}I}}}$.

2.5. Unified Expression for Deflection Calculation

The deflection of a steel truss web–concrete composite girder bridge at any position $(x,y)$ after considering the shear lag effect and the shear deformation of the steel truss web bar is calculated as a unified expression:

$$\omega =-{\displaystyle \int \left[{\displaystyle \int \frac{M(x)}{E_{c}I}dx}\right]}dx-{\displaystyle \int \left[{\displaystyle \int \frac{V^{\prime }(x)}{2G_s{A}_s}dx}\right]}dx-{\displaystyle \frac{2}{3}}{\displaystyle \int U(x)dx+C_{1}x+}{C}_2$$

(19)

where the first term is the deflection obtained for the primary beam, the second term is the additional deflection due to the shear deformation of the converted steel web, and the third term is the additional deflection due to the shear lag effect.

3. Answers to Deflection Due to Pre-Stressing

3.1. Pre-Stressing Equivalent Load

In the calculation of pre-stressed concrete structures, the equivalent load method proposed by R.B.B. Moorman converts the action of the pre-stressing tendons on the member at the anchorage end and in the span into equivalent loads [27]. The equivalent load at the anchorage end is decomposed into horizontal pre-axial force, vertical pre-shear force, and pre-bending force. The in-span equivalent load is determined by the shape of the mating bundle: a folded mating bundle is equivalent to a concentrated force, and a curved mating bundle is equivalent to a uniform load.

The equivalent load at the anchorage end is only the pre-bending force that causes vertical deflection of the beam, and the equivalent loads within the span all cause bending deformation of the beam, so the deflection calculation method of the beam under the eccentric straight-line, folded-line, and curved-line beams is investigated.

3.2. Eccentric Linear Cloth Bundle

As shown in Figure 4, the linear pre-stressing bundle arranged in the girder body has an effective pre-stressing force of $N_y$ and an eccentricity of $e$, and its pre-stressing equivalent load acts on the girder end of the centralized bending moment $M_0$. Under this equivalent load, the shear force of the combined girder bridge is 0, meaning that the stress solution considering the shear lag effect cannot be obtained directly, so it is transformed into the superposition of the deflection solution when only the girder end is subjected to the centralized bending moment on the left and right sides [27,28].

At this time, the beam is a purely curved beam, with bending and no shear, so the additional deflection caused by shear deformation is 0. According to the boundary condition $\omega (x){|}_{x=0}=0$, $\omega (x){|}_{x=l}=0$, the left and right sides of the beam when the beam end is subjected to the role of the concentrated bending moment, the deflection solution superposition can be obtained in a straight line under the beam of the steel truss web–concrete composite girder bridge deflection calculation expression for

$$\begin{array}{l}\omega ={\displaystyle \frac{N_{y}e}{2E_{c}I}}(x^2-lx)-{\displaystyle \frac{5N_{y}e}{E_{c}I{k}^2}}\left(1-chkx+{\displaystyle \frac{shkx}{shkl}}(chkl-1)\right)\\ ={\omega }_1+{\omega }_3\end{array}$$

(20)

When the eccentricity distance is 0 (i.e., the pre-stressing tendons are in a straight center bundle), the beam exhibits only axial deformation.

3.3. Folded Wire Bundle in the Span

As shown in Figure 5, the folded pre-stressing bundle arranged in the beam body is anchored at the beam end at the form axis, the angle with the form axis is $\beta$, its effective pre-stressing force is $N_y$, the eccentricity at the folded corner is $e$, the distance between the folded corner and the end of the beam is $a$, and its pre-stressing equivalent load is the vertical concentrated force at the folded corner $P_0$.

According to the loading form and the boundary condition ${\left.\omega (x)\right|}_{x=0}=0$, ${\left.\left[{\omega }^{\prime }(x)-\alpha (x)\right]\right|}_{x=l/2}=0$, the expression of the cross-sectional deflection of the steel truss web–concrete composite girder bridge can be obtained under the folding-line bundle in the span:

When $0\le x\le a$,

$$\begin{array}{l}\omega (x)=\left({\displaystyle \frac{N_{y}e{x}^3}{6E_{c}Ia}}-{\displaystyle \frac{N_{y}e{l}^{2}x}{8E_{c}Ia}}\right)+\left(-{\displaystyle \frac{N_{y}ex}{2G_s{A}_{s}a}}\right)+\left(-{\displaystyle \frac{5N_{y}el}{ab{E}_{c}I{k}^2}}\left[{\displaystyle \frac{shk(l-a)}{shkl}}\left(ch{\displaystyle \frac{kl}{2}}x-{\displaystyle \frac{shkx}{k}}\right)\right]\right)\\ ={\omega }_1+{\omega }_2+{\omega }_3\end{array}$$

(21)

3.4. In-Span Curve Bundles

As shown in Figure 6, the curved pre-stressing bundle arranged in the girder body is anchored at the girder end at the form axis, and the angle with the form axis is $\beta$, with an effective pre-stressing force of $N_y$ and an eccentricity of $e$ in the span, and its pre-stressing-equivalent load is approximated as a vertical homogeneous load $q_0$.

According to the load form and boundary conditions ${\left.\omega (x)\right|}_{x=0}=0$, ${\left.\omega (x)\right|}_{x=l}=0$, the deflection calculation expression can be obtained as

$$\begin{array}{l}\omega (x)=\left[-{\displaystyle \frac{N_{y}e}{3E_{c}I{l}^2}}\left(x^4-2l{x}^3+l^{3}x\right)\right]+\left[-{\displaystyle \frac{2N_{y}e}{G_s{A}_s{l}^2}}\left(lx-x^2\right)\right]\\ +\left[-{\displaystyle \frac{60N_{y}e}{E_{c}I{l}^2{k}^4}}\left(chkx+{\displaystyle \frac{1-chkl}{shkl}}shkx+{\displaystyle \frac{k^2\left(xl-x^2\right)}{2}}-1\right)\right]\\ ={\omega }_1+{\omega }_2+{\omega }_3\end{array}$$

(22)

In actual engineering, when the steel truss web–concrete composite girder bridge adopts an in-span fold or curve arrangement of pre-stressing tendons, the anchorage end is usually in the form of eccentric anchorage. At this time, the equivalent load is the superposition of Figure 4 and Figure 5 or Figure 6, and the corresponding deflection solution is the superposition of Equations (20)–(22) of the deflection solution obtained in the small-deformation condition [26].

4. Mathematical Example

4.1. Engineering Background and Finite Element Modeling

In this paper, the Jiangshan Bridge of the Nanjing Bypass Highway is taken as the engineering case study. Its superstructure adopts a two-span (35 m + 35 m), single-box, single-compartment, equal-section, steel truss web–pre-stressed concrete composite girder structure. The steel truss web is made of Q345C grade steel pipes, diagonally arranged, with a horizontal inclination angle of about 1.95 m and section spacing of 1.95 m [15]. The combined girder bridge combines in vivo and ex vivo pre-stressing beam deployment methods. The actual bridge of the project is shown in Figure 7, and one of the spans is selected as the simple supported system to be analyzed. The transverse and longitudinal structural dimensions are shown in Figure 8.

The finite element model of the steel truss web–concrete composite girder bridge is established using the finite element software ANSYS 2022. The concrete top, bottom plate, and cantilever plate are simulated using the solid element SOLID185, and the steel truss web is simulated using the beam element BEAM188, without considering the relative slip between the steel tube and concrete. The elastic modulus of concrete E = 3.45 × 10⁴ MPa, and Poisson’s ratio = 0.2; the elastic modulus of steel e = 2.06 × 10⁵ MPa, and Poisson’s ratio = 0.3. The rigid connection between the steel truss web and the concrete flange is realized by establishing the constraint equation and the coupling degree of freedom. The pre-stressing effect is simulated by applying the equivalent load, and the bending moment of the beam end is applied by using the equivalent equilibrium concentrated force. At the same time, through grid sensitivity analysis and applying different loads to the model, the internal force and displacement of the section are compared with the theoretical value to verify the reliability of the model. The finite element model is shown in Figure 9 [16,19,21].

4.2. Deflection Analysis Under Separate Bundles with Different Line Shapes

When calculating the deflection distribution along the longitudinal girder direction of the combined girder bridge with different alignments of individual bundles using the formulas presented in this paper, it is assumed that the single pre-stressing tendons are stranded with the $15{\varphi }^{s}15.2$ strand, the standard value of the tensile strength is $f_{pk}=1860 \mathrm{MPa}$, and the effective pre-stressing force is taken to be 0.6 $f_{pk}$.

The cross-sectional geometric properties and the converted steel web parameters are shown in

Table 1. Cross-sectional geometric properties and converted steel web parameters.

tw/m	h/m	As/m2	k/m−1	Is/m4	I/m4
3.63 × 10−2	2.30	4.42 × 10-2	1.27	1.19	4.41

, and the deflection distribution along the longitudinal girder direction is shown in Figure 10, which is obtained by considering the effects of different linear individual bundles.

According to Figure 10, it can be seen that the deflection analytical solution of the equivalent beam considering the dual effect and the analytical solution considering only the shear deformation match well with the finite element solution, and the absolute value of the relative error with the finite element solution is within 6.0% and 3.0%, which verifies the reliability of this paper’s computational method. Under the eccentric straight-line beam, the shear deformation does not cause any additional deflection, and the analytical solution considering only shear hysteresis is closer to the finite element solution. Under the in-span folded-line and the in-span curve beam, the analytical solution of deflection considering the dual effect is higher than the finite element solution, and the analytical solution considering only the shear deformation is in better agreement with the finite element solution. The reason for this is related to the type of cell selected for finite element modeling.

4.3. Analysis of Each Effect on the Deflection Under Different Linear Individual Fabric Bundles

In order to quantitatively examine the degree of influence of shear deformation, the shear lag effect, and the dual effect on the deflection of a steel truss web–concrete composite girder bridge under pre-stressing, the influence coefficient is introduced and defined as follows:

$${\chi }_1={\displaystyle \frac{{\omega }_2}{\omega (x)}}$$

(23)

$${\chi }_2={\displaystyle \frac{{\omega }_3}{\omega (x)}}$$

(24)

$${\chi }_f={\displaystyle \frac{{\omega }_2+{\omega }_3}{\omega (x)}}$$

(25)

where ${\chi }_1$ is the contribution of additional deflection caused by the shear deformation of the steel truss web to its total deflection, ${\chi }_2$ is the contribution of additional deflection caused by the shear lag effect to its total deflection, and ${\chi }_f$ is the contribution of additional deflection to its total deflection caused by the dual effect.

Table 2. Deflection due to pre-stressing in the mid-span section.

Linear	Total Deflection/mm	Euler Beam/mm	${\mathit{\chi }}_{\mathbf{1}}$/%	${\mathit{\chi }}_{\mathbf{2}}$/%	${\mathit{\chi }}_{\mathit{f}}$/%
Straight line	−2.94	−2.94	0	0.05	0.05
Folding line	−2.30	−1.96	13.64	3.68	17.32
Curve	−2.58	−2.22	11.00	3.10	14.10

provides the total deflection of the mid-span section; primary beam deflection; and ${\chi }_1$, ${\chi }_2$, ${\chi }_f$ under different linear individual bundles. From Table 2, it can be seen that under the eccentric straight-line fabrication, the shear deformation does not contribute to the deflection, and the shear lag effect contributes only 0.05%, which is negligible. Under the in-span bending-line and in-span curve fabric bundle, ${\chi }_1$ is greater than 10%, while ${\chi }_2$ is less than 4%, and ${\chi }_f$ is more than 14%. The degree of shear deformation on the deflection is higher than the shear lag effect, and the dual effect on the deflection is not negligible; if only the shear deformation or shear lag effect is taken into consideration in the calculations, the value of the upward deflection caused by pre-stressing will be underestimated to varying degrees.

4.4. Deflection Analysis Under the Integrated Fabric Bundle

In order to analyze the deflection distribution under the integrated bundle, and at the same time to compare the influence of each effect on the accuracy of the beam deflection calculation, four bundles of pre-stressing tendons are arranged along the beam, numbered N1, N2 (straight line), N3 (folded line), and N4 (curved line), and the specifications of the pre-stressing bundles and the values of the effective pre-stressing stresses are the same as those in the case of the individual bundles. The arrangement of the pre-stressing bundles in the mid-span and girder-end anchorage parts is shown in Figure 11a, the arrangement along the girder body is shown in Figure 11b, and the deflection distribution of the combined girder bridge under the comprehensive bundle deployment using the formulas in this paper and finite element simulation is shown in Figure 12.

As can be seen from Figure 12, under the integrated fabrication of four bundles of pre-stressing tendons, the deflection analytical solution obtained by considering the dual effect and shear deformation only is basically consistent with the deflection distribution trend of the finite element solution; the deflection analytical solution considering the shear lag effect only has a larger error relative to the finite element solution; and the deflection analytical solution considering the dual effect is more biased towards safety.

4.5. Analysis of the Influence of Each Effect on the Deflection Under the Integrated Fabric Bundle

In order to further analyze the influence of each effect on the total deflection of a typical cross-section under the integrated fabrication, the total deflections at the l/8, l/4, and l/2 cross-sections; the primary beam deflections; and ${\chi }_1$, ${\chi }_2$ are examined. The distribution of ${\chi }_1$, ${\chi }_2$ along the longitudinal direction of the beam is provided in Figure 12.

As shown in

Table 3. Proportion of deflection of each part caused by pre-stress of a typical section.

Placement	Total Deflection/mm	Euler Beam/mm	${\mathit{\chi }}_{\mathbf{1}}$/%	${\mathit{\chi }}_{\mathbf{2}}$/%	${\mathit{\chi }}_{\mathit{f}}$/%
l/8	−2.08	−1.84	9.74	2.71	12.45
l/4	−3.83	−3.35	9.70	2.72	12.43
l/2	−5.48	−4.72	11.00	3.00	13.93

and Figure 13, the total deflection in each typical section of the beam is mainly contributed by the theoretical deflection of the primary beam under the comprehensive beam fabrication, and the contribution ratio of the additional deflection caused by each part is basically the same at l/8 and l/4, while it is higher closer to the l/2 part. The total deflection at each typical section is more than 12%, around 10%, and less than 5%, which shows that the dual effect has the greatest influence on the accuracy of the deflection calculation, followed by the shear lag effect, where the shear lag effect is the smallest; the contribution ratio of the shear lag effect to the total deflection is about 3.5 times that of the shear lag effect only and the equivalent beam width and height. It can be seen that the dual effect has the greatest influence on the accuracy of deflection calculation, with shear deformation being the second largest and the shear lag effect the smallest, because the width and height of the equivalent beam are relatively small. The contribution of shear deformation to the total deflection is about 3.5 times that of the shear lag effect, and the accuracy of mid-span deflection is improved by 3.0%, 11.0%, and 13.9% when only considering the shear lag effect, only considering shear deformation, and only considering the dual effect, respectively. In order to make the calculation easier, the mid-span deflection of the equivalent beam in the calculation can be corrected by multiplying the deflection of the primary beam by 1.16 times. For other beams of different sizes, further studies are needed to determine the values of the coefficients.

5. Conclusions

In this paper, considering the influence of shear deformation of the steel truss web and the shear lag effect on the deflection calculation, a method for calculation of the deflection of a steel truss web–concrete composite girder bridge under pre-stressing equivalent load was established based on the energy variational method. Numerical simulation was utilized to validate the reliability of the proposed method. The following main conclusions can be drawn through comparative analysis of the calculation examples:

(1)

Based on the formula in this paper, the analytical solution of the upward deformation of the beam under different linear pre-stressed arrangements was in good agreement with the finite element numerical solution, which proves the accuracy and applicability of the proposed formula and supplements the loading mode of the deflection calculation of the composite beam bridge.

(2)

When the equivalent beam is separately restrained under the eccentric straight line, the shear deformation and shear lag effect do not contribute to the total deflection, and the actual deflection of the beam can be directly taken as the theoretical deflection value of the primary beam. The calculation accuracy of shear deformation is higher than that of the shear lag effect under the mid-span fold line and the mid-span curve alone, and their dual effect under the fold-line bundle has a more significant influence on the deflection. This is because the mid-span fold line is concentrated at the position of the equivalent load in the mid-span, which should be given attention in the calculation.

(3)

In the equivalent beam under the comprehensive arrangement of pre-stressed tendons, the influence of the double effect on mid-span deflection is higher than that of the l/8 and l/4 sections. The contribution rate of shear deformation to the total deflection is about 3.5 times that of the shear lag effect. Only considering shear deformation, only considering the shear lag effect, and only considering their dual effect can improve the accuracy of beam deflection calculation to varying degrees.

(4)

The calculation method proposed in this paper can be applied for the preliminary design of this kind of composite-structure, deformation-control work after the completion of the bridge, and the use of external pre-stressing tendons to strengthen old bridges. The calculation accuracy meets the engineering requirements. The verification of the proposed formula in a real bridge or scale model tests is expected to be addressed in a future study.