Deflection Analysis of Steel Truss Web–Concrete Composite Beams Based on Zigzag Beam Theory

Abstract

To address the inherent inaccuracies of the classical beam theory (which overestimates the flexural stiffness) and the “quasi-plane section method” (which neglects the shear deformation) in the deflection analysis of steel truss web–concrete composite beams, this study homogenizes discrete steel truss web members into a continuous steel web with equivalent thickness based on the strain energy equivalence principle. This homogenization is conducted under the assumption of fixed-end constraints for web members, thus establishing a sandwich laminated beam model. Incorporating the assumptions of zigzag axial displacement and layer-wise quadratic parabolic transverse shear stress, this study adopts the governing equations for static bending of composite beams derived via Hamilton’s mixed energy variational principle—this theory eliminates the need for an artificial shear correction factor, as the transverse shear stress naturally satisfies the zero boundary conditions at the upper and lower surfaces and the continuity condition at the interlayers. Analytical solutions for bending deflection under uniformly distributed loads are derived and validated against three-dimensional (3D) finite element (FE) models. The analysis results of a 45-meter-span beam demonstrate that the relative error in the maximum deflection of both simply supported beams and cantilever beams calculated by the proposed method is approximately 5%, which is significantly superior to the classical beam theory; the deflection induced by the zigzag effect at the mid-span of simply supported beams accounts for 15% of the total deflection, making it an indispensable key component in structural design. This method enables accurate deflection prediction and provides reliable technical guidance for the preliminary design of steel truss web–concrete composite beam bridges.

1. Introduction

The core structure of the steel truss web composite box girder consists of concrete flange plates, steel trusses, and joints [1]. The concrete flange plates provide a stable base for bridge deck pavement and bear vertical loads [2]; the steel trusses serve as the core force-transferring framework of the structure [3,4]; and the joints act as hubs to ensure the unobstructed force transmission between the steel web members and concrete flange plates, thereby avoiding local stress concentration [5]. The steel trusses significantly address the shortcomings of traditional concrete box girders: the web of traditional box girders is prone to cracking due to shear and temperature stresses [6], and after replacing the concrete web with a steel truss, the shear force can be concentrated and transmitted to the steel web members, fundamentally reducing the risk of web cracking [7]; additionally, the hollow structure of the steel truss can greatly reduce the structural self-weight [8], which not only alleviates the bearing pressure on substructures (such as piers and foundations) but also lowers the difficulty and cost of hoisting during construction [9]. The steel web members composing the steel truss follow an “alternating tension and compression” force-bearing mode, which fully exerts the mechanical properties of steel; for the compressed web members [10], after their interior is filled with concrete, the concrete can provide lateral constraints to prevent local buckling of the steel web members [11]. Meanwhile, the filled concrete itself can also participate in bearing compressive loads, working synergistically with the steel web members to bear bending-compression loads—this synergistic effect significantly improves the flexural stiffness of the composite box girder [12], ensuring that the vertical deformation of the structure under long-term loads complies with specifications and guaranteeing the flatness and durability of the bridge deck [13]. For the specific structure of the steel truss web–concrete composite beam, refer to Figure 1.

In recent years, many scholars have conducted in-depth studies on the flexural behavior of steel truss web composite beams [14]. Chen [15] proposed that the inverted triangular steel truss had the optimal flexural stiffness. Feng [16] put forward the optimal design parameters for multi-plane inverted triangular steel trusses. Huang [17] studied the deformation behavior of steel trusses with interface defects. Zhou [18] investigated the deformation behavior of steel trusses at different positions. Huang [19] proposed an estimation equation for the bending moment capacity design of steel trusses. Meanwhile, relevant research on composite structure material performance and heterogeneous material mechanical analysis has also achieved notable progress, such as the dual-pathway utilization of coal gangue concrete (aggregate substitution, cementitious activity activation, and performance optimization), which provides a new idea for the performance improvement of concrete materials in composite beams [20], and the research on fracture evolution of granite under cyclic thermal shocks, which enriches the mechanical analysis method of heterogeneous materials under extreme conditions [21]. Although scholars have achieved fruitful results in the research on the deformation performance of steel truss web composite box girders, the web members, as discontinuous web structures, will cause additional vertical shear deflection of the box girder due to axial deformation [22,23]. Most of the existing studies adopt the “quasi-plane section method”, which ignores shear deformation [24], leading to the underestimation of the main girder deflection under specific conditions and an insufficient safety factor in design [25].

The zigzag theory is commonly used to describe the piecewise continuous displacement field of laminated structures and ensures the continuity of interlaminar transverse shear stress at each interface [26]. Ambartsumian [27] extended the Reissner–Mindlin plate theory to laminated structures by incorporating the zigzag theory. Reissner et al. [28,29] proposed the mixed energy variational principle and indicated its applicability to the derivation of simplified theories for laminated structures. Murakami [26] proposed a zigzag theory that ensures the continuity of interlaminar displacement and transverse stress, based on the mixed energy variational principle.

Existing studies on steel truss web–concrete composite beams mostly adopt the quasi-plane section method or classical beam theory, which either neglect shear deformation or overestimate flexural stiffness, leading to large errors in deflection prediction. Although zigzag beam theory has been applied to composite beams with corrugated steel webs, its application to steel truss web–concrete composite beams has not been reported, and the rational selection for the mechanical constraint assumption of steel web members has not been clarified in existing homogenization studies. The key novelties and contributions of this work are as follows: (1) based on the strain energy equivalence principle [30,31,32], this study clarifies the fixed-end constraint assumption for steel web members matching the actual engineering connection form, and proposes a homogenization method for converting discrete steel truss web members into a continuous orthotropic steel web, establishing a sandwich beam model adapted to zigzag theory; (2) the zigzag beam theory is innovatively applied to the deflection analysis of steel truss web–concrete composite beams, and the analytical solution of deflection considering the zigzag effect is derived, which makes up for the deficiency of traditional methods in capturing the interlayer discontinuous deformation caused by stiffness differences; (3) the quantitative contribution of the zigzag effect to the total deflection is clarified, and the key influencing factors (depth–span ratio and steel web member stiffness) are revealed, providing a theoretical basis for the deformation control of such structures. Compared with the existing equivalent analysis methods, this study realizes accurate deflection prediction without introducing artificial shear correction factors, and the selection of the fixed-end assumption ensures the consistency between theoretical analysis and actual structural mechanical behavior.

2. Equivalent Thickness of Steel Truss Web Members and Equivalent Section

2.1. Equivalent Thickness of Steel Truss Web Members

For existing truss web composite truss girder bridges, the web member arrangement is mostly the triangular diagonal type as shown in Figure 1. Therefore, the truss girder with this web member configuration is selected as the research object, and the steel web members are converted into thin plates with uniform thickness, as illustrated in Figure 2.

Due to the significant difference in stiffness between the longitudinal and transverse directions, the triangular steel web members are homogenized into an orthotropic layer. It is assumed that the equivalent orthotropic steel web is perfectly bonded to the upper and lower concrete slabs, with its Young’s modulus and shear modulus determined based on the strain energy equivalence principle. The perfect bonding assumption is consistent with the actual engineering situation, because the steel web members are rigidly connected with the concrete flange plates through welding/bolting and the interface has no relative displacement under the design load.

The shear modulus of the orthotropic steel web is derived from Figure 3 based on the equivalence of strain energy before and after homogenization, where $L$ denotes the spacing of steel web members, $H$ denotes the vertical height of steel web members, $\theta$ denotes the inclination angle of web members, $l$ denotes the length of steel web members, $A$ denotes the cross-sectional area, $I$ denotes the cross-sectional moment of inertia, $t$ denotes the equivalent thickness of the orthotropic steel web, $G^{\prime }$ denotes the shear modulus of the equivalent steel web, and $E^{\prime }$ denotes the Young’s modulus of the equivalent steel web.

When shear deformation occurs, it is assumed that the shear strain is uniformly distributed over the web, and under small deformation conditions, the shear strain can be expressed as: $\Delta$. This small deformation assumption is fully applicable to the steel truss web–concrete composite beam in bridge engineering, because the structural deformation of bridge engineering is strictly limited by the design specification (the maximum vertical deflection of the simply supported beam is generally not more than L/600, where L is the span), and the actual shear deformation of the steel web members is much smaller than the limit of the small deformation assumption, which ensures the rationality of the shear strain expression:

$$\gamma \approx {\displaystyle \frac{\Delta }{H}}$$

(1)

The strain energy of the web is:

$${\Pi }_1={\displaystyle \frac{1}{2}}{\displaystyle \underset{\Omega }{\iiint }{G}^{\prime }{\gamma }^2 \mathrm{d}\Omega }={\displaystyle \frac{1}{2}}{G}^{\prime }(HLt){({\displaystyle \frac{\Delta }{H}})}^2$$

(2)

In this study, it is assumed that the steel web members are rigidly connected to the concrete. Correspondingly, the web member is treated as a beam with fixed ends, and the displacements occurring at both ends are as shown in Figure 4. This fixed-end constraint assumption is consistent with the actual engineering connection form, and the derived equivalent stiffness formula considering both bending and axial deformation is an innovative improvement compared with the existing derivation based on the hinged assumption, with the displacements at both ends shown in Figure 4.

Then, the bending strain energy of the two web members is:

$${\Pi }_2=2\times {\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{(\frac{{\mathrm{d}}^{2}w}{\mathrm{d}{x}^2})}^2\frac{1}{EI}\mathrm{d}x}={\displaystyle \frac{12EI{\delta }^2}{l^3}}$$

(3)

Referring to Figure 3, it can be seen that:

$$\delta =\Delta \sin \theta$$

(4)

Meanwhile, the web members also undergo tensile and compressive deformations, and the corresponding strain energy is:

$${\Pi }_3=2\times {\displaystyle \frac{1}{2}}EAl{({\displaystyle \frac{\Delta \cos \theta }{l}})}^2$$

(5)

According to the strain energy equivalence principle, it can be concluded that:

$${\Pi }_1={\Pi }_2+{\Pi }_3$$

(6)

The equivalent shear modulus of the steel web is obtained as follows:

$$G^{\prime }t={\displaystyle \frac{48EI{H}^3+EAH{l}^2{L}^2}{2L{l}^5}}$$

(7)

The equivalent Young’s modulus of the orthotropic steel web is derived from Figure 5 based on the equivalence of strain energy before and after homogenization.

When axial deformation occurs in the web, the strain energy of the web is:

$${\Pi }_4={\displaystyle \frac{1}{2}}{E}^{\prime }HtL{({\displaystyle \frac{2\Delta }{L}})}^2$$

(8)

Therefore:

$${\Pi }_4={\Pi }_2+{\Pi }_3$$

(9)

The equivalent Young’s modulus of the orthotropic steel web is:

$$E^{\prime }t={\displaystyle \frac{48EIL{H}^2+EA{l}^2{L}^3}{8H{l}^5}}$$

(10)

If it is assumed that the steel web members are in hinged connection with the upper and lower concrete slabs, only the axial strain energy is considered when calculating the strain energy of the steel web members, and the equivalent shear modulus is obtained as follows:

$$G^{″}t={\displaystyle \frac{EAH{l}^2{L}^2}{2L{l}^5}}$$

(11)

To evaluate the accuracy of the two equivalent methods, the function curves of $G^{\prime }t$ and $G^{″}t$ are compared under the condition that only the spacing of the steel web members is varied. Consider a steel truss web–concrete composite beam: the elastic modulus of the steel for the web members is $E=210\mathrm{Gpa}$, Poisson’s ratio is ${\mu }_s=0.3$, the dimensions of the steel web members is $D351\times 16$ (unit: mm), the vertical height of the steel web members is 1.9 m, and the member spacing is $L$. The expressions of $G^{\prime }t$ and $G^{″}t$ in terms of the member spacing $L$ are provided below. Figure 6 presents the function curves of $G^{\prime }t$ and $G^{″}t$ with respect to the member spacing $L$.

$$G^{\prime }t={\displaystyle \frac{48EI{H}^3+EAH\left(H^2+\frac{L^2}{4}\right)L^2}{2L{\left(\sqrt{H^2+\frac{L^2}{4}}\right)}^5}}$$

(12)

$$G^{″}t={\displaystyle \frac{EAHL}{2{\left(\sqrt{H^2+\frac{L^2}{4}}\right)}^3}}$$

(13)

From the function curves, it can be observed that when the member spacing $L$ is less than 2 m, the difference between $G^{\prime }t$ and $G^{″}t$ is significant, which corresponds to an inclination angle of the steel web members greater than 60°. As the inclination angle of the steel web members increases from 60°, $G^{″}t$ decreases sharply, which clearly contradicts the actual mechanical behavior.

In practical engineering of steel truss web–concrete composite beams, the steel web members are connected with the upper and lower concrete flange plates through rigid welded or bolted fixed joints; this connection form can effectively restrict the rotational and translational deformation of the web member ends, which is consistent with the mechanical characteristics of the fixed-end assumption. The fixed-end constraint enables the steel web members to bear both axial force and bending moment, which fully reflects the actual force transmission mechanism of the truss web members in composite beams. In contrast, the hinged assumption only considers the axial strain energy of the web members and ignores the bending deformation caused by the rigid constraint of joints, leading to an unreasonable reduction in the calculated equivalent shear modulus and failing to reflect the actual shear resistance capacity of the steel truss web. Therefore, the function curve of $G^{\prime }t$ is consistent with the actual mechanical behavior of the structure, and it can more accurately simulate the equivalent shear modulus of the steel web members, which is the rational choice for the subsequent theoretical analysis in this study.

In fact, the larger the horizontal inclination angle of the steel web members, the smaller the member spacing, the denser their arrangement, and the stronger the overall shear resistance capacity. Therefore, the function curve of $G^{\prime }t$ is consistent with the actual mechanical behavior, and it can better simulate the equivalent shear modulus of the steel web members.

2.2. Equivalent Section

The upper and lower concrete slabs and steel web plates are converted into isotropic layers with a width of $b_0$, as shown in Figure 7, and their equivalent material properties are given as follows [33,34,35]:

$$E_i=E_c{\displaystyle \frac{b_i}{b_0}},{\mu }_i={\mu }_c,G_i={\displaystyle \frac{E_i}{2\left(1+\mu \right)}}\left(\mu =\mathrm{1,3}\right)$$

(14)

$$E_2=E_{eq}t,G_2=G_{eq}t$$

(15)

3. Zigzag Beam Theory for Steel Truss Web–Concrete Composite Beams

Zigzag beam theory stands as a classic analytical method for laminated composite structures, boasting unique and irreplaceable advantages in analyzing the deformation of soft-core sandwich plates and beams—structures characterized by a middle core layer with low stiffness and the upper and lower surface layers with high stiffness [36,37]. For such sandwich structures, the classical plane section assumption leads to significant errors [38,39], as it fails to capture the discontinuous axial displacement at interlayer interfaces, a phenomenon induced by the substantial stiffness difference between layers.

In stark contrast, zigzag beam theory innovatively assumes that the axial displacement of each layer is piecewise linear and maintains continuity at the interlayer interfaces [40]. This key assumption enables the theory to accurately depict the Z-shaped axial displacement distribution typical of soft-core sandwich structures, where inflection points are concentrated in the middle soft-core layer. A further innovative merit of this theory lies in its ability to naturally satisfy transverse shear stress boundary conditions—zero stress at the upper and lower surfaces and stress continuity at interlayers—without the need to introduce artificial shear correction factors. This inherent advantage greatly enhances the accuracy of shear deformation and deflection calculations [41].

For the steel truss web–concrete composite beams investigated in this study, the homogenized steel truss web is established on the fixed-end assumption and functions as the soft core with low axial stiffness. The upper and lower concrete slabs, in contrast, serve as the high-stiffness surface layers. This structural configuration is perfectly compatible with the application characteristics of soft-core sandwich structures, laying a solid foundation for the application of zigzag beam theory. Notably, the application of zigzag beam theory to this specific type of composite beam is an innovative practice that has not been documented in existing studies. It enables accurate reflection of the structure’s actual deformation characteristics, thereby addressing the inherent limitation of traditional methods in capturing interlayer discontinuous deformation. In turn, this work provides a reliable theoretical foundation for the deflection analysis of steel truss web–concrete composite beams.

For the sandwich laminated beam shown in Figure 8, a new displacement quantity $s$ is introduced, which represents the bending degree of the axial displacement. The displacement components $u_x^{\left(k\right)}$ and $u_z^{\left(k\right)}$ along the $x$ and $z$ within the k-th layer can be set as [26]:

$$\begin{array}{l}{u}_x^{\left(k\right)}\left(x,z\right)=u\left(x\right)+z\psi \left(x\right)+{\left(-1\right)}^k 2\overline{z_k}s\left(x\right)\\ u_z^{\left(k\right)}\left(x,z\right)=w\left(x\right)\\ k=1,2,3\end{array}$$

(16)

where $u$ and $w$ denote the axial displacement and deflection at $z=0$, respectively, and $\overline{z_k}$ is a dimensionless counterpart of the local coordinate $z_k$, whose origin is located at the center of the k-th layer. Defining the location of the center of the k-th layer as $z_{k0}$ yields:

$$z_k=z-z_{k0},\overline{z_k}={\displaystyle \frac{z_k}{h_k}}$$

(17)

The transverse shear stress of layer 1 can be assumed to be [26]:

$$\begin{array}{l}{\tau }_{xz}^{\left(k\right)}\left(x,z\right)={\displaystyle \frac{3}{2h_k}}\left(1-4{\overline{z_k}}^2\right)Q_k\left(x\right)+\left(3{\overline{z_k}}^2+\overline{z_k}-{\displaystyle \frac{1}{4}}\right)T_{k-1}\left(x\right)\\ +\left(3{\overline{z_k}}^2-\overline{z_k}-{\displaystyle \frac{1}{4}}\right)T_k\left(x\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(k=1,2,3\right)\end{array}$$

(18)

where $T_{k-1}$ and $T_k$ are the transverse shear stress values at the upper and lower surfaces of the k-th layer, respectively; $Q_k$ has the physical meaning of the resultant shear stress on the cross-section of the k-th layer. An integration verification can be performed on Equation (18) to confirm this:

$$Q_k=h_k{\displaystyle {\int }_{\frac{-1}{2}}^{\frac{1}{2}}{\tau }_{xz}^{\left(k\right)}}d\overline{z_k}\hspace{1em}k=1,2,3$$

(19)

The normal stress in the z-direction (${\sigma }_z$) is generally neglected as ${\sigma }_z=0$ in the entire section [39]. This assumption is reasonable in the context of the adopted equivalent strut model of the steel truss web for two reasons: first, the equivalent strut model of the steel truss web is mainly used to bear the shear force and axial force in the x-y plane, and the z-direction is the width direction of the composite beam with no obvious external load and structural deformation, so the z-direction normal stress is far less than the xy plane stress; second, the steel truss web is homogenized into a thin plate with small thickness in the z-direction, and the thin plate structure has the mechanical characteristic of “plane stress”, that is, the normal stress in the thickness direction is approximately zero. The neglect of ${\sigma }_z$ will not affect the calculation accuracy of the main stress of the structure but can simplify the theoretical derivation of the zigzag beam theory.

4. Steel Truss Web–Concrete Composite Beams Under Uniform Load

For a simply supported beam subjected to a uniformly distributed load, by incorporating the boundary conditions and referring to the solution approach presented in reference [32], this study derives the deflection expression for the simply supported beam under uniformly distributed load as follows:

$$w=w_0+w_{shear}+w_{zig\text{-}zag}$$

(20)

where $w_0$ is the deflection without considering shear deformation and zigzag deformation, $w_{shear}$ is the additional deflection considering shear deformation, and $w_{zig\text{-}zag}$ is the additional deflection considering zigzag deformation [32,42,43]. They are respectively:

$$w_0={\displaystyle \frac{q_0}{24\overline{EI}}}\left(x^4-2l{x}^3+l^{3}x\right)$$

(21)

$$w_{shear}={\displaystyle \frac{q_0}{2\overline{\kappa GA}}}\left(lx-x^2\right)$$

(22)

$$w_{zig\text{-}zag}={\displaystyle \frac{q_0}{{\alpha }^4}}\left(\gamma -{\displaystyle \frac{1}{\overline{EI}}}\right)\left\{{\displaystyle \frac{1}{e^{\alpha l}+1}}\left[e^{\alpha x}+e^{\alpha \left(l-x\right)}\right]-{\displaystyle \frac{1}{2}}{\alpha }^2{x}^2+{\displaystyle \frac{1}{2}}{\alpha }^{2}lx-1\right\}$$

(23)

For a cantilever beam under a uniformly distributed load, its deflection is:

$$w=w_0+w_{shear}+w_{zig\text{-}zag}$$

(24)

$$w_0={\displaystyle \frac{1}{24}}{\displaystyle \frac{q_0}{EI}}\left(x^4-4l{x}^3+6l^2{x}^2\right)$$

(25)

$$w_{\mathrm{shear}}={\displaystyle \frac{q_0}{2\overline{\kappa GA}}}\left(2lx-x^2\right)$$

(26)

$$\begin{array}{cc}{w}_{\mathrm{zig}\text{-}\mathrm{zag}}& ={\displaystyle \frac{1}{{\alpha }^3}}{\displaystyle \frac{q_{0}l\left(\gamma -1\right)}{EI}}\left\{-\sin \mathrm{h}\left(\alpha x\right)+{\displaystyle \frac{1+\alpha l\sin \mathrm{h}\left(\alpha l\right)}{\alpha l\cos \mathrm{h}\left(\alpha l\right)}}\left(\cos \mathrm{h}\left(\alpha x\right)-1\right)\right\}\\ & +{\displaystyle \frac{1}{2{\alpha }^2{l}^2}}{\displaystyle \frac{q_0\left(\gamma -1\right)}{EI}}\left(2l^{3}x-l^2{x}^2\right)\end{array}$$

(27)

The parameters in the aforementioned deflection formula are related to the material properties and geometric dimensions of the equivalent sandwich beam; for specific details, refer to reference [32].

5. Numerical Example of Steel Truss Web–Concrete Composite Beam Under Uniformly Distributed Load

5.1. ANSYS Model

Consider a steel truss web–concrete composite beam with a total height of 2500 mm, where both the concrete top slab and bottom slab have a thickness of 300 mm, a span of 45 m, a top slab width of 8500 mm, and a bottom slab width of 4800 mm. The elastic modulus of concrete is set as $E_c=35GPa$ and Poisson’s ratio is ${\mu }_c=0.2$. The elastic modulus of steel is $E_s=210GPa$ and Poisson’s ratio is ${\mu }_s=0.3$. The size of steel web members is $\mathrm{D}351\times 16$ (unit: mm), the horizontal inclination angle is 66.8°, and the panel spacing is 1500 mm. The cross-section is shown in Figure 9.

A 3D finite element analysis (FEA) model was established in ANSYS 2024: concrete top and bottom slabs were simulated with SOLID65 solid elements (capable of elastic deformation and simulating material nonlinearity), while steel web members (based on the theoretical fixed-end assumption) and truss chords were modeled with BEAM188 spatial beam elements (3D quadratic elements with 7 DOFs per node, accurately simulating bending and axial deformation of steel members under fixed-end constraints and their axial force-bending moment-bearing characteristics). For meshing, concrete slabs were divided into 50 mm × 50 mm × 100 mm hexahedral elements to ensure calculation accuracy at the concrete–steel interface; steel web members and truss chords were meshed with a uniform 100 mm beam element size in the longitudinal direction, with denser meshes at joints and supports to capture local stress concentration. A strict mesh convergence analysis was performed: the mesh size was gradually refined from the initial coarse mesh, and the maximum deflection and key stress values of the composite beam were monitored for each scheme. Mesh convergence was confirmed when the relative error in the calculated results was less than 1% with further refinement, ensuring the simulation results were mesh-independent and improving the calculation accuracy. The concrete–steel connection was set as rigid bonded contact (TIE constraint) to match the fixed-end assumption, with contact nodes coupled via the node merging method in ANSYS to fully restrict relative rotation and translation, accurately simulating the rigid fixed connection between steel web members and concrete flange plates in practical engineering. Simply supported boundary conditions were defined by constraining the y/z translational and x rotational DOFs at support nodes, with only the x axial translational DOF released to conform to actual structural bearing behavior. A 10 kN/m uniformly distributed line load was applied along the longitudinal direction of the bridge on the key bearing nodes of the concrete top slab, ensuring load application consistency with theoretical analysis, and the finite element calculation model of the entire bridge is shown in Figure 10.

5.2. Strain Energy Equivalence Verification

To verify the effectiveness of the strain energy equivalence method presented in Section 2 of this paper, a segment with an x-coordinate ranging from 9 m to 18 m is selected for calculating the total strain energy of the second layer. The strain energy of the second layer in the composite beam is computed using the zigzag theory [36], and the result is compared with the strain energy of the diagonal web members in the corresponding segment extracted from the ANSYS finite element model.

In the

Table 1. Comparison of strain energy results from zigzag theory and ANSYS model.

$\mathit{x}\left(\mathit{m}\right)$	${\mathit{U}}_{\mathit{Z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}\left(\mathit{J}\right)$	${\mathit{U}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}\left(\mathit{J}\right)$	$\frac{{\mathit{U}}_{\mathit{Z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}\text{-}{\mathit{U}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}}{{\mathit{U}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}}\times 100\mathit{\%}$
9–12	5.87	6.36	−7.67%
12–15	3.33	3.61	−7.77%
15–18	1.54	1.64	−5.81%
0–45	102.79	109.05	−5.74%

, $U_{Z\mathrm{ig}\text{-}\mathrm{zag}}$ denotes the strain energy calculated by the zigzag theory, and $U_{ANSYS}$ denotes the strain energy of the diagonal web members in the corresponding segment extracted from the ANSYS finite element model. It can be seen from the table that the error between the equivalence theory proposed in this paper and the finite element results is approximately 7%, indicating that the equivalence theory in this paper is in good agreement with the actual situation and can be adopted for subsequent analysis.

5.3. Deformation Characteristics Analysis of Composite Beams

With the boundary conditions of the simply supported beam adopted, Figure 11 illustrates the distribution of normal stress along the beam height at the mid-span section. It can be seen from the figure that the normal stress of the top and bottom flanges at the mid-span section is still approximately distributed along a straight line, which verifies the validity of the quasi-plane section assumption in composite beams. On the whole, the normal stress distribution along the beam height shows a Z-shaped characteristic, which is consistent with the zigzag beam theory.

Additionally, vertical paths are established at distances of 1.5 m, 7.5 m, and 15 m from the supports to extract the deformation data along the longitudinal direction of the bridge.

It can be seen from Figure 12, Figure 13 and Figure 14 that both the top and bottom slabs conform to the quasi-plane section assumption—their axial displacements along the beam height are approximately linear in distribution, showing the rigid rotational deformation of concrete slabs under bending. However, throughout the beam height, a Z-shaped displacement appears, with the inflection point concentrated in the middle steel web layer. Specifically, at 1.5 m from the supports (Figure 12), where shear force is large, the Z-shaped deformation is more obvious: the axial displacement of the top slab increases from the top to the steel web interface, while the bottom slab shows an opposite linear trend, and the steel web between them changes displacement direction suddenly to connect the two slabs. As the distance from the supports increases to 7.5 m (Figure 13) and 15 m (Figure 14), the Z-shaped feature remains but is attenuated—this is due to the reduced influence of shear deformation when approaching the mid-span, where bending deformation dominates. These displacement characteristics verify that the zigzag beam theory is consistent with the actual deformation of steel truss web–concrete composite beams: the theory’s core assumption of piecewise continuous axial displacement captures the discontinuous deformation caused by the stiffness difference between concrete slabs and the steel web, describing the interlayer displacement transition mechanism not captured by the classical plane section assumption.

From Figure 15 and Figure 16, it can be seen that the proposed method is in close agreement with the FEA calculation results, while the classical beam theory exhibits significant discrepancies from the FEA results. Specifically, in Figure 15 (deflection of simply supported beams), the deflection values calculated by the proposed method at key positions along the span are closely aligned with the FEA results, with small relative errors. In contrast, the Euler–Bernoulli beam theory overestimates the flexural stiffness, leading to calculated deflection values that deviate more from the FEA results. For the cantilever beam in Figure 16, the proposed method maintains consistency with the FEA results across the entire span, while the classical beam theory fails to account for shear deformation and the zigzag effect, resulting in obvious underestimation of total deflection—this further confirms the inadequacy of the classical beam theory in capturing the actual deformation characteristics of steel truss web–concrete composite beams.

From

Table 2. Simply supported beam—deflection results of the proposed method, FEA, and Euler–Bernoulli beam theory.

$\mathit{x}\left(\mathit{m}\right)$	${\mathit{w}}_0\left(\mathit{m}\mathit{m}\right)$	$\mathit{w}={\mathit{w}}_0+{\mathit{w}}_{\mathit{s}\mathit{h}\mathit{e}\mathit{a}\mathit{r}}+{\mathit{w}}_{\mathit{Z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}\left(\mathit{m}\mathit{m}\right)$	${\mathit{w}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}\left(\mathit{m}\mathit{m}\right)$	$\frac{\mathit{w}\text{-}{\mathit{w}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}}{{\mathit{w}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}}\times 100\mathit{\%}$
2.5	0.592	0.738	0.786	−6.15%
7.5	1.697	2.085	2.185	−4.56%
12.5	2.583	3.146	3.283	−4.15%
17.5	3.155	3.823	3.993	−4.26%
22.5	3.352	4.055	4.231	−4.15%

and

Table 3. Simply supported beam—percentage of each component deflection in total deflection.

$\mathit{x}/\mathit{m}$	${\mathit{w}}_0/\mathit{w}$	${\mathit{w}}_{\mathit{s}\mathit{h}\mathit{e}\mathit{a}\mathit{r}}/\mathit{w}$	${\mathit{w}}_{\mathit{z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}/\mathit{w}$
2.5	81.34%	1.40%	17.26%
7.5	82.46%	1.31%	16.23%
12.5	83.20%	1.25%	15.54%
17.5	83.62%	1.22%	15.16%
22.5	83.76%	1.21%	15.03%

, it can be observed that the relative error between the proposed method and FEA is around 5%, and the proportion of zigzag deflection is approximately 15%. Specifically, Table 2 shows that at key positions of the simply supported beam, the relative error between the proposed method and the FEA remains stable within a narrow range (4%~6%), confirming the method’s consistent accuracy in predicting deflection at different sections. In contrast, classical beam theory deviates significantly from the FEA results, as it overestimates composite beam flexural stiffness and ignores both shear deformation and the zigzag effect.

Error Trend and Design Conservativeness Analysis: The proposed analytical method consistently underestimates the deflection relative to the FEA results. This slight underestimation is slightly non-conservative in the initial design stage, but the error amplitude is only about 5%, which is far less than the safety factor (≥1.5) of bridge structure design. In practical engineering design, the deflection calculated by this method can be corrected by a correction coefficient of 1.05, which can not only ensure the calculation accuracy but also meet the conservative requirements of structural design.

Table 3 further supplements that the proportion of zigzag deflection varies slightly with position, staying stable around 15% in most areas and being higher near the supports. This indicates zigzag deformation is not a negligible secondary effect but a core component of total deflection; neglecting it will lead to underestimated structural deformation and insufficient design safety margins.

Figure 17 presents the variation of the ratio $w_{Zig\text{-}zag}/w$ with the depth–span ratio of simply supported beams and that of cantilever beams. From the data trend in the figure, the influence of zigzag displacement on the total deformation of steel truss web–concrete composite beams can be analyzed. It can be seen from the figure that the proportion of zigzag deflection $w_{Zig\text{-}zag}/w$ generally decreases as the depth–span ratio reduces. This law conforms to the basic characteristics of structural mechanics: the smaller the depth–span ratio, the stronger the dominance of bending deformation in the total deformation, and the relative proportion of zigzag deflection decreases accordingly; conversely, the larger the depth–span ratio, the higher the proportion of zigzag deflection will be.

Focusing further on common engineering scenarios, within the commonly used span range of simply supported beams, the proportion of zigzag deflection is above 10%. If such deflection is ignored, the stiffness of the main beam will be overestimated, resulting in a significant underestimation of deflection calculation. This indicates that zigzag deflection plays an important guiding role in the design and construction of such structures. It is necessary to include the calculation of zigzag deflection in the total deformation verification to ensure the accuracy of structural deformation assessment and avoid engineering hazards caused by ignoring this part of deformation.

5.4. Comparative Analysis of Internal Forces

Taking simply supported beams as an example, to compare the differences in shear force distribution between the zigzag model and the finite element model, Figure 18, Figure 19 and Figure 20 respectively present the shear force diagrams of the top slab, steel web members and bottom slab of the composite beam. It can be seen from the figures that the shear force diagrams calculated by the zigzag model are generally consistent in trend with those extracted from ANSYS, yet there exist certain numerical discrepancies. These shear force discrepancies mainly arise from the fact that steel web members are modeled as discontinuous components in the ANSYS model, while the zigzag model equates steel web members to a continuous structure. Both the method proposed in this paper and the ANSYS model indicate that steel web members resist more than 85% of the total shear force, and special attention should be paid to the shear resistance capacity of steel web members in engineering design.

Figure 21 and Figure 22 show the axial force diagrams of the top slab and bottom slab of the composite beam, respectively. It can be seen from the figures that the axial force diagrams calculated by the zigzag model are in good agreement with those extracted from ANSYS. Under the action of vertical uniformly distributed loads, the top slab and bottom slab are subjected to large axial forces, with the top slab in a tension state and the bottom slab in compression state. The numerical values of their axial forces present a basically symmetrical trend, and this regularity further verifies the rationality and applicability of the zigzag model in the calculation of axial forces for composite beams.

6. Influence of Steel Web Member Wall Thickness on Displacement

To further clarify the main influencing factors of the zigzag effect and draw conclusions of engineering significance, we still take the simply supported beam numerical example in Section 4 as the basis, only varying the wall thickness t of the inclined web members within the range of 4–20 mm. The comparison between the theoretical calculations and finite element results is as follows.

Table 4. The influence of steel web member wall thickness on displacement.

$\mathit{t}\left(\mathit{m}\mathit{m}\right)$	${\mathit{w}}_{\mathit{z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}\left(\mathit{m}\mathit{m}\right)$	$\mathit{w}\left(\mathit{m}\mathit{m}\right)$	$\frac{{\mathit{w}}_{\mathit{z}\mathit{i}\mathit{g}\text{-}\mathit{z}\mathit{a}\mathit{g}}}{\mathit{w}}$	${\mathit{w}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}\left(\mathit{m}\mathit{m}\right)$	$\frac{\mathit{w}}{{\mathit{w}}_{\mathit{A}\mathit{N}\mathit{S}\mathit{Y}\mathit{S}}}$
4	2.48	5.92	41.87%	6.2	95.54%
8	1.23	4.68	26.38%	4.92	94.99%
12	0.82	4.26	19.23%	4.47	95.11%
16	0.61	4.06	15.03%	4.23	95.85%
20	0.49	3.92	12.46%	4.11	95.24%

compares the mid-span deflections of simply supported beams with a span of 45 m and a height of 2.5 m. As shown in Table 4, when other parameters are kept the same, the greater the wall thickness of the steel web members, the smaller the displacement, which is consistent with the zigzag beam theory. The more significant the difference in material properties between layers, the more pronounced the zigzag effect. For the steel truss web–concrete composite beam, the middle layer is composed of steel web members with extremely low axial stiffness; therefore, after transformation, the material parameters of the upper and lower concrete slabs are much larger than those of the middle layer, which satisfies the application conditions of the zigzag beam theory.

It can be seen from Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27 that for simply supported beams with a fixed span and height, varying the wall thickness of steel web members leads to the following trend: from the above axial displacement diagrams, it can be seen that the smaller the stiffness of the steel web members, the more pronounced the Z-shaped deformation and the more significant the zigzag effect. Specifically, Figure 23 (4 mm wall thickness) shows the most distinct Z-shaped displacement—there is a sharp directional change in axial displacement at the interface between the steel web and concrete slabs, reflecting the large stiffness difference between the low-stiffness steel web and high-stiffness concrete. As the wall thickness increases to 8 mm, 12 mm, 16 mm, and 20 mm (Figure 24, Figure 25, Figure 26 and Figure 27), the Z-shaped feature gradually moderates, with the axial displacement transition between layers becoming smoother. This aligns with Table 4’s data: the proportion of zigzag deflection decreases from 41.87% (4 mm) to 12.46% (20 mm) as wall thickness increases, since thicker steel web members enhance axial stiffness, reducing the interlayer stiffness gap. This indicates that the zigzag effect cannot be neglected when there are significant interlayer stiffness differences—especially for composite beams with thin-walled steel web members, the zigzag effect becomes a dominant factor affecting total deformation and must be considered in structural design.

7. Conclusions

To address the inaccuracies of classical beam theory and the quasi-plane section method in the deflection analysis of steel truss web–concrete composite beams, this study combines the strain energy equivalence principle with zigzag beam theory to establish a refined deflection analysis method, and verifies its validity via 3D finite element simulation and parametric analysis. The core innovative findings and engineering conclusions are as follows:

(1)

This study proposes a practical homogenization modeling method for steel truss webs based on the fixed-end constraint assumption matching the actual rigid connections in engineering, which homogenizes discrete steel truss web members into a continuous orthotropic steel web and establishes a sandwich laminated beam model adapted to zigzag theory, avoiding the unreasonable reduction in equivalent shear modulus caused by the hinged assumption and ensuring the consistency between theoretical modeling and actual mechanical behavior.

(2)

This study realizes the first application of zigzag beam theory to the deflection analysis of steel truss web–concrete composite beams, making up for the deficiency of traditional methods in capturing interlayer discontinuous deformation caused by stiffness differences. The adopted theoretical framework satisfies transverse shear stress boundary conditions naturally without artificial shear correction factors, significantly improving the accuracy of deflection calculation.

(3)

This study derives analytical solutions for the deflection of composite beams under uniformly distributed loads considering the zigzag effect; verification with a 45-meter-span beam shows the relative error between the proposed method and finite element results is only about 5%, far superior to classical beam theory, and quantifies that the zigzag effect-induced deflection accounts for approximately 15% of the total deflection, confirming it is an indispensable key component in structural design.

(4)

This study reveals that the proportion of zigzag deflection is mainly affected by the depth–span ratio and steel web member stiffness: the larger the depth–span ratio and the smaller the web member stiffness, the more prominent the zigzag effect. In common engineering span ranges, the proportion of zigzag deflection exceeds 10%, and neglecting it will lead to the overestimation of girder stiffness and insufficient design safety margins.

This study constructs an accurate and efficient deflection prediction method for steel truss web–concrete composite beams, which overcomes the inherent defects of traditional analysis methods and provides important theoretical support and technical guidance for the preliminary design and deformation control of such composite beam bridges.

Future research will focus on expanding the model’s applicability: incorporating the effects of interface slip and connector flexibility, clarifying the limits of applicability of the homogenization method, and integrating concrete creep and shrinkage to explore the long-term evolution of zigzag deformation, extending the analysis to complex load scenarios and verifying the results via full-scale tests for wider engineering applications.