Research on the Cable Force Optimization of the Precise Closure of Steel Truss Arch Bridges Based on Stress-Free State Control

Abstract

During the construction of large-span steel truss arch bridges, challenges such as complex control calculations, frequent adjustments of the cantilever structure, and deviations in the closure state often arise in the process of the assembly and closure of arch ribs. Based on the stress-free state control theory, this paper proposes a precise assembly control method for steel truss arch bridges, which takes the minimization of structural deformation energy and the maintenance of the stress-free dimensions of the closure wedge as the control objectives. By establishing a mathematical relationship between temporary buckle cables and the spatial position of the closure section, as well as adopting the influence matrix method and the quadprog function to determine the optimal parameters of temporary buckle cables (i.e., size, position, and orientation) conforming to actual construction constraints, the automatic approaching of bridge alignment to the target alignment can be achieved. Combined with the practical engineering case of Muping Xiangjiang River Bridge, a numerical calculation study of the precise assembly and closure of steel truss arch bridges was conducted. The calculated results demonstrate that, under the specified construction scheme, the proposed method can determine the optimal combination for temporary buckle cable tension. Considering the actual construction risk and the economic cost, the precise matching of closure joints can be achieved by selectively trimming the size of the closure wedge by a minimal amount. The calculated maximum stress of the structural rods in the construction process is 42% of the allowable value of steel, verifying the feasibility and practicality of the proposed method. The precise assembly method of steel truss arch bridges based on stress-free state control can significantly provide guidance and reference for the design and construction of bridges of this type.

1. Introduction

Steel truss arch bridges are increasingly being applied in practical engineering due to their advantages, such as their light dead weight, strong load-bearing capacity, large span capability, high level of prefabrication, and the ability to arch in place [1,2]. As a highly hyperstatic structure, the construction process of steel truss arch bridges is complex, involving numerous node connections and component installations. The closure is the final step in the transformation of the bridge into a complete structural system [3,4,5]. The precise closure of steel truss arch bridges ensures the overall load-bearing capacity and safety of the structure, improving construction efficiency and shortening the construction period. For large-span steel truss arch bridges, the structure is in a maximum cantilever state before closure, during which the geometry of the closure joint is influenced by various uncertain factors arising from structural geometric nonlinearity. In this state, even minor closure deviations can lead to significant secondary stresses within the structure. Therefore, the accurate closure of the main arch is the most critical and important step in the construction process of steel truss arch bridges [6,7,8].

Currently, the main closure methods for steel truss arch bridges are natural closure and forced closure. The natural closure method [9,10,11,12] involves the continuous observation of the relative position and temperature of the bridge closure joint to determine control parameters, such as the matching length, placement temperature, and closure temperature of the closure segment. The feeding beam and closure procedures are carried out separately using the temperature difference method. Once the closure is complete, the temporary constraints between the tower and the beam are removed, thereby finalizing the transformation of the structural system. However, this method has a high dependence on temperature, and the matching of the closure segment can alter the geometric dimensions of the steel beam in its designed state, increasing the difficulty of component manufacturing.

The forced closure method [13,14,15] is a widely adopted approach that mainly involves the sensitivity analysis of various parameters at the bridge closure joint to determine closure adjustment measures. This method also actively eliminates the positioning deviations at the closure joint, thereby achieving the stress-free closure of the steel beam. Zhao et al. [16] employed a construction sequence of closing first before tensioning the cable force during the closure of the main beam of the Qingzhou Channel Bridge on the Hong Kong–Zhuhai–Macao Bridge. For the adjacent spans, a top-pushing and matching–cutting method was utilized for closure, while for the middle span, the closure segment was directly matched according to the actual dimensions of the closure joint. While meeting the precision requirements for closure, the cutting method can alter the stress-free geometric dimensions of the closure segment. Chen et al. [17] used a plan involving overall factory assembly and on-site lifting and installation for construction, where one end of the closure beam segment was supported by a side tower pier bracket, utilizing jacks to precisely adjust the position of the steel box beam. The other end of the closure beam segment was supported by a jacking bracket system, allowing for precise adjustments in pushing, lifting, and rotating the steel box beam, thus addressing the closure angle and elevation issues of the box beam. This resulted in the stress-free closure of the bridge closure beam segment. However, this method presents high construction difficulty and equipment requirements. Hu et al. [18] investigated the real-time changes in the structural temperature field, which cause uneven temperature distributions in the arch ribs, as well as causing the temperature difference between the cables and the arch ribs. By analyzing the displacements on both sides of the closure joint under the influence of sunlight, the authors determined the optimal window for controlling the closure of large-span steel truss arch ribs. Following an accurate analysis of the temperature distribution pattern of the truss arch ribs, the temperature-induced stresses and displacements of structure are calculated; then, the optimal closure time can be predicted. Dan et al. [19] conducted an analysis on the challenges related to double strut closure for steel truss cable-stayed bridges, examining the effects of different closure sequences on the displacement of the closure segment. By selecting appropriate closure schemes and sequences for the double struts, as well as determining the optimal adjustment parameters for the closure rods, the precise alignment of the closure rods can be achieved. While this method allows for the forced closure of steel truss cable-stayed bridges, it can potentially introduce installation errors that could induce additional stresses in the rods. Yi et al. [20] considered the construction process and the width of the box beam welds, analyzing the impact of temperature changes on the status of the closure joints. For the middle span of the cable-stayed bridge, a cutting and pushing closure scheme was adopted. This method observed the spatial position of the closure joint and the geometric characteristics of the closure segments using a conventional construction procedure. Then, bridge closure can be achieved by lifting and moving back a specific displacement on one side of the main beam through various measures such as cable force adjustments, weight additions, and counterweight. However, the construction process entails certain risks, especially when the theoretical values for the cable status significantly deviate from the actual values before closure, which can lead to changes in the status of the closure joint. Jia et al. [21] employed a cable-stayed suspension method for the installation of steel pipe arch ribs based on the “quiet and stable” control principle and the theory of the stress-free state method. By utilizing an iterative approach to calculate the tension in the cables during the assembly of the arch ribs, an optimal structural linear design was successfully achieved. Wang et al. [22] focused on the issue of the geometric deviations of the ribs at the closure point within a double-layer rigid suspension three-truss-reinforced continuous steel truss bridge framework. They employed a temporary supporting device on the pier top to keep the closure node position stress-free and subsequently adjusted the longitudinal displacement of the steel beam using the temperature difference method to meet the closure requirements. This method achieved zero deviation in bridge closure; however, its applicability is constrained by the construction method of the steel beam, requiring additional equipment and imposing high demands on construction processes and techniques.

To achieve the precise closure of steel truss arch bridges and to reduce the construction difficulties and risks, we propose a precise closure method for steel truss arch bridges based on stress-free state control. Firstly, we take the minimization of structural deformation energy and the maintenance of the stress-free dimensions of the closure wedge as the control objectives. The mathematical relationship between the temporary buckle cables and the spatial position of the closure section is established through the influence matrix, and the optimization model of the temporary buckle cables in each construction stage is constructed. Then, we set the displacement constraints of the closure and used the quadprog function in the quadratic programming algorithm to optimize the solution in order to determine the cable force that meets the precise closure. Therefore, the main contributions of this work are as follows.

• We propose a new precise closure method, which takes the effectively unchanged stress-free dimensions of the closure wedge and the minimum bending energy of the structure as the control objectives, for steel truss arch bridges. Compared with traditional forced closure, this method has simple and clear control objectives, effectively reducing the amount of cutting required for the closed section and simplifying the construction process.
• We carry out precise closure numerical calculations in relation to Muping Xiangjiang River Bridge to verify the correctness and feasibility of the method. This provides a reference for the design and construction of similar types of bridges.
• Under the specified construction scheme, the proposed method can determine the optimal tensioning combination of temporary buckle cables, including the size, position, and orientation.
• Depending on the actual construction risks and economic costs, we offer the option to selectively cut the size of the closure wedge by a minor amount. This scheme enables the precise closure of steel truss arch bridges while reducing economic costs.

The remainder of this paper is organized as follows. Section 2 systematically presents the theoretical approach to arch formation control based on the proposed accurate control of stress-free parameters. Section 3 describes the construction of a cable force optimization model based on the influence matrix method and quadprog functions. Finally, Section 4 concludes this paper.

2. Stress-Free Parameter-Based Precise Control Method for Arch Formation

2.1. Stress-Free Parameter Control Theory

The stress-free state parameters of a structural member refer to the relevant geometric parameters after all stresses in the member unit have been removed, i.e., the stress-free length and stress-free curvature of the member unit. According to the stress-free state control theory, for a given structural system, when the structural loads and boundary conditions remain constant, the internal forces and the alignment of the bridge upon completion will correspond to the design state as long as the stress-free lengths and curvatures of the various members in the installation state during the construction process are consistent with those in the stress-free state [23]. This indicates that the final state of the bridge is determined by the stress-free parameters of the members and is independent of the installation sequence or any load changes that occur during the construction process. This theory has been thoroughly validated in the field of cable-stayed bridges [24,25], and this paper extends its application to the construction of steel truss arch bridges.

The arch rib segments of steel truss arch bridges are relatively lightweight, generally being processed and manufactured in the factory according to their stress-free shape. During construction, the arch rib segments are positioned and installed using cable hoisting, as well as utilizing cable-stayed components and buckle-hanging. The segments are then joined together using internal flange connections or welding, and the arch segments naturally satisfy the stress-free state during installation. Consequently, during the cantilevering process, there is no requirement for a forced control target for the alignment of the arch; therefore, it is sufficient to meet the safety requirements of bridge construction. For the precise assembly of steel truss arch bridges controlled by the stress-free state, keeping the stress-free dimensions of the closure wedge in the closure segment essentially unchanged is taken to be the control target. To ensure that the spatial position of the closure joint accurately matches the closure wedge, various measures—such as applying counterweights, adjusting cable forces, and pulling top forces—are implemented, as shown in Figure 1.

The determination of closure control targets is crucial for the bridge state following arch formation. During the closure of the main arch, it is essential to ensure that the stress-free dimensions of the closure wedge remain essentially unchanged. This allows for the installation of the closure wedge with precise geometric dimensions that correspond to stress-free alignment (i.e., the manufactured alignment) while simultaneously satisfying the control targets for stress-free length and curvature. This method avoids the need for on-site secondary cutting of the closure wedge. After the closure of the main arch, its internal forces and alignment will match those of the main arch constructed using the one-time landing technique, thereby achieving a high degree of match between the final bridge alignment and the design objective.

2.2. Arch Rib Construction Control Method

The large-span steel truss arch bridge is a four-member truss structural system, where a stress-free state closure entails the precise installation of the closure wedge according to the stress-free dimensions specified in the manufactured alignment. Compared to the traditional full-support scaffolding construction method, which requires the construction of a massive scaffolding system to balance the self-weight of the arch ribs and maintain the stress-free assembly state for the bridge structure throughout the whole construction process, this proposed method significantly reduces the construction difficulty, risk, and economic cost. For the construction characteristics of large-span steel truss arch bridges, the cable-stayed hanging method demonstrates significant technical advantages. Temporary buckle cables can dynamically adjust and control the internal forces within the main truss structure, effectively preventing excessive internal forces in the steel beams during cantilever construction. Additionally, these temporary buckle cables help control the structural alignment, ensuring that the designated space for the closure joint at the mid-span cantilever end meets the geometric control requirements. Therefore, the stress-free dimensions of the closure wedge are maintained during the closure of the main arch.

In accordance with the actual construction conditions, a certain number of temporary buckle cables are selected for tensioning, as shown in Figure 2. During the maximum cantilever state of the bridge, the tensioned temporary buckle cables T and applied temporary loads F work together to adjust the geometric dimensions of the closure joint, facilitating the precise installation of the closure wedge. It should be noted that increasing the number of tensioned temporary buckle cables results in a more uniform distribution of forces within the structure, as well as higher economic costs.

To address the problem of parameter coupling caused by traditional one-time cable adjustment, a phased multi-objective combined optimization strategy is proposed. A multi-stage optimization model is established with the control targets of minimizing structural deformation energy $U={\displaystyle \int \left(M{(x)}^2/2EI\right)}\mathrm{d}x$ during each construction phase, as well as maintaining the stress-free dimensions of the closure wedge. By using a multi-objective optimization algorithm, the optimal configuration scheme for parameters T and F is collaboratively solved to achieve the high-precision assembly control of the arch rib segments. The assembly process is illustrated in Figure 3. This method reasonably controls the structural deformation energy during various construction stages of the arch rib’s suspended assembly, which is beneficial for ensuring structural internal force safety, rational alignment, and the precise closure of the steel truss arch bridge during the construction process. Moreover, this approach helps prevent excessive structural deformation in the early construction phase, which could make the adjustment of the closure joint displacement difficult during the closure stage.

The minimization of structural deformation energy for the entire bridge involves more than just seeking the energy extremum in a single stage. It focuses on minimizing the bending deformation energy at each stage while maintaining the stress-free dimensions of the closure wedge. The tension of temporary buckle cables follows the principle of “tensioning means locking”, permitting only adjustments to cable tension values after the temporary buckle cables have been tensioned. In this process, the spatial positions and orientation parameters of temporary buckle cables remain fixed. The optimization of tensioning with four temporary buckle cables is demonstrated in Figure 4 and Figure 5.

When the steel beam is installed in the designated position, the first batch of temporary buckle cables T₁ and T₂ are tensioned. Taking U_1min = f (T₁, T₂) as the initial control objective, the parameters T₁ and T₂ are further optimized. Subsequently, the second batch of temporary buckle cables T₃ and T₄ are introduced as the steel beam continues to be erected. With U_2min = f (T₁, T₂, T₃, T₄) as the new control objective, only the values of T₁ and T₂ are adjusted to optimize the parameters of the newly tensioned buckle cables (i.e., T₃ and T₄). Once the steel beam reaches the maximum cantilevering closure stage, temporary loads F₁ and F₂ are introduced. Using U_3min = f (T₁, T₂, T₃, T₄, F₁, F₂) as the final control objective, with $‖\Delta L_{\mathrm{close}}‖\le {\epsilon }_L$ (where ${\epsilon }_L$ represents the allowable deviation for the closure joint) as a constraint, the overall optimal parameters are determined, thereby facilitating accurate matching for closure at the mid-span.

For the specified construction method, the quantity and sequence of tensioning temporary buckle cables and loads are determined by comprehensively considering the construction environment and economic costs, as illustrated in Figure 5. By constructing a collaborative control optimization model that accounts for structural deformation energy, temporary buckle cables, and load parameters, as well as the accuracy of the stress-free dimensions of the closure wedge at each construction stage, the relevant parameters (size, position, and orientation) of T and F are globally optimized. This proposed method effectively facilitates the precise stress-free closure of steel truss arch bridges, enabling the alignment of the completed bridge to automatically approach the target alignment.

3. Cable Force Optimization Based on the Influence Matrix Method

3.1. Determination of Control Objectives for Closure

The structural deformation of the closure joint at the maximum cantilevering state before the closure of the main arch of steel truss arch bridges is shown in Figure 6. Points A, B, C, and D represent the ideal geometric spatial dimensions of the closure wedge corresponding to the designed alignment, while Points A’, B’, C’, and D’ represent the actual geometric spatial dimensions of the closure wedge following the deformation of the closure joint.

$$\left\{\begin{array}{c}{d}_{\theta 1}=d_{x1}/d_{y1}\\ d_{\theta 2}=d_{x2}/d_{y2}\\ \Delta x=d_{x1}-d_{x2}\\ \Delta y=d_{y1}-d_{y2}\\ \Delta \theta =d_{\theta 1}-d_{\theta 2}\end{array}\right.$$

(1)

Here, d_x1 and d_x2 represent the horizontal displacement difference between the actual spatial position and the stress-free state position of the cantilever ends at the closure joint. Similarly, d_y1 and d_y2 denote the corresponding vertical displacement difference, while d_θ1 and d_θ2 represent the corresponding angular difference, and Δx, Δy, and Δθ represent the overall horizontal displacement, vertical displacement, and rotational deviations of the closure segment, respectively. When Δx, Δy, and Δθ are all equal to zero, the precise closure of the main arch structure can be achieved. During the closure of the main arch, measures such as adjusting the cable forces and applying counterweights are implemented to ensure that the length of the closure joint matches the stress-free length of the closure wedge and that the rotation angle of the closure joint, along with the relative elevations at both ends, are close to the stress-free state.

3.2. Optimized Calculation of Cable Forces

The current research on the optimization methods for cable forces includes the following methods: specified state optimization, unconstrained optimization, constrained optimization, and the influence matrix method [26,27]. The first three methods are mainly for conventional cable-stayed bridges. For long-span steel truss arch bridges, these optimization methods have certain limitations, such as their high computational volume and high construction difficulty. The influence matrix method can comprehensively consider various structural responses and constraints such as the cable force, displacement, and internal force, as well as can comprehensively optimize the cable force of the steel truss arch bridge, which helps to ensure the safety and rationality in the bridge state and construction process.

The influence matrix [28,29] is defined as a column array composed of influence vectors that represent the effect of unit cable force changes on the optimization objectives. By establishing a functional relationship between the cable forces and optimization objectives through the influence matrix, appropriate objective functions can be formulated and solved to derive the optimized cable force values. It is important to note that the influence matrix method is not a standalone cable force optimization technique as it requires integration with other optimization approaches. This study combined this method with the quadprog function from quadratic programming algorithms in MATLAB (R2024b) to conduct numerical studies on the optimization of closure cable forces in steel truss arch bridges.

The cable force optimization system based on the influence matrix method was established according to the principle of linear superposition [30,31,32]. Its core involves constructing the mathematical mapping relationship between mechanical parameters. The linear control equation between the adjustment vector {T} and the response vector {D} of the cable force was established as follows:

$$\left\{D\right\}=\left[C\right]\left\{T\right\}+\left\{D_0\right\}$$

(2)

where {D₀} is the structural bending moment and critical node displacement vector under initial loads (gravity and temporary loads), and the response vector {D} is a column vector composed of m independent elements at key sections or nodes in the structure, which is expressed as follows (the elements d_i(i = 1, 2, …, m) represent the structural bending moments and node displacements):

$$\left\{D\right\}={\left(\begin{array}{ccc}\begin{array}{cc}{d}_1,& d_2,\end{array}& \dots ,& d_m,\end{array}\right)}^{\mathrm{T}}$$

(3)

The adjustment vector {T} is a column vector composed of l independent elements that are adjusted to change the response vector. The elements T_i(i = 1, 2, …, l) represent the cable forces.

$$\left\{T\right\}={\left(\begin{array}{cccc}{T}_1,& T_2,& \dots ,& T_l\end{array}\right)}^{\mathrm{T}}$$

(4)

where {C} represents the influence vector, which is expressed as follows (when the jth element x_j of the adjustment vector {T} undergoes a unit change, it induces a corresponding change in the response vector {D}):

$$\left\{C_j\right\}={\left(\begin{array}{ccc}\begin{array}{cc}{c}_{j1},& c_{j2},\end{array}& \dots ,& c_{jm}\end{array}\right)}^{\mathrm{T}}$$

(5)

Similarly, when l adjustment vectors undergo a unit change, respectively, m influence vectors will be produced, which can be organized to form an influence matrix [C], as follows:

$$\left[C\right]=\left[\begin{array}{cccc}{c}_{11}& c_{12}& \dots & c_{1l}\\ c_{21}& c_{22}& \dots & c_{2l}\\ & & \dots & \\ & & \dots & \\ c_{m1}& c_{m2}& \dots & c_{ml}\end{array}\right]$$

(6)

Cable force optimization is essentially a reverse-solving process based on the influence matrix. By establishing the mathematical mapping relationship between the response vector and the adjustment vector, the optimal cable force adjustment (i.e., the adjustment vector) can be derived when the structural response parameters (i.e., response vector) reach the predetermined target values. This paper adopted the structural total bending energy at all construction stages U as the control objective for cable force optimization, with the corresponding functional analytical equation presented in Equation (7). When the value of U was minimized, the distribution of internal forces within the structure was more uniform and closer to the stress-free state.

$$U={\displaystyle \sum _{i=1}^n\frac{L_i}{4E_i{I}_i}}\left({M_{Li}}^2+{M_{Ri}}^2\right)$$

(7)

Here, n denotes the total number of beam elements in the calculated model; L_i, E_i, and I_i represent the length, elastic modulus, and bending moment of inertia of the cross-section for the ith beam element, respectively; M_Li represents the bending moment at the left end of the ith beam element; and M_Ri denotes the corresponding bending moment at the right end. Thus, the following equation can be established:

$$\left[U\right]=\left[M_{\mathrm{L}}^{\mathrm{T}}\right]\left[B\right]\left[M_{\mathrm{L}}\right]+\left[M_{\mathrm{R}}^{\mathrm{T}}\right]\left[B\right]\left[M_{\mathrm{R}}\right]$$

(8)

where [M_L] and [M_R] represent the bending moment vectors at the left and right ends of the element, respectively. Moreover, [B] is considered as the weighted matrix of the unit flexibility matrix concerning the unit bending moments as follows:

$$\left[B\right]=\left[\begin{array}{cccc}{b}_{11}& 0& \dots & 0\\ 0& b_{22}& \dots & 0\\ 0& 0& b_{33}& 0\\ 0& 0& \dots & b_{ii}\end{array}\right]$$

(9)

where $b_{ii}={\displaystyle \frac{L_i}{4E_i{I}_i}}\left(i=1,\dots ,n\right)$.

The bending moment vectors at the left and right ends of the beam element under gravity and temporary loads are denoted as {M_L0} and {M_R0}, respectively. The corresponding bending moment vector after cable adjustment can then be expressed as follows:

$$\left\{\begin{array}{c}\left[M_{\mathrm{L}}\right]=\left\{M_{\mathrm{L}0}\right\}+\left[C_{\mathrm{L}}\right]\left\{T\right\}\\ \left[M_{\mathrm{R}}\right]=\left\{M_{\mathrm{R}0}\right\}+\left[C_{\mathrm{R}}\right]\left\{T\right\}\end{array}\right.$$

(10)

where [C_L] and [C_R] represent the influence matrix of the cable force on the bending moments at the left and right ends of the beam element.

Substituting Equation (10) into Equation (8), the following is obtained:

$$\begin{array}{cc}\left[U\right]\hfill & ={\left\{M_{L0}\right\}}^{\mathrm{T}}\left[B\right]\left[C_L\right]\left\{T\right\}+{\left\{T\right\}}^{\mathrm{T}}{\left[C_L\right]}^{\mathrm{T}}\left[B\right]\left\{M_{L0}\right\}+{\left\{T\right\}}^{\mathrm{T}}{\left[C_L\right]}^{\mathrm{T}}\left[B\right]\left[C_L\right]\left\{T\right\}+\hfill \\ \hfill & {\left\{M_{R0}\right\}}^{\mathrm{T}}\left[B\right]\left[C_R\right]\left\{T\right\}+{\left\{T\right\}}^{\mathrm{T}}{\left[C_R\right]}^{\mathrm{T}}\left[B\right]\left\{M_{R0}\right\}+{\left\{T\right\}}^{\mathrm{T}}{\left[C_R\right]}^{\mathrm{T}}\left[B\right]\left[C_R\right]\left\{T\right\}+\hfill \\ \hfill & {\left\{M_{L0}\right\}}^{\mathrm{T}}\left[B\right]\left\{M_{L0}\right\}+{\left\{M_{R0}\right\}}^{\mathrm{T}}\left[B\right]\left\{M_{R0}\right\}\hfill \end{array}$$

(11)

Equation (11) can be rewritten as follows:

$$[U]={\left\{T\right\}}^{\mathrm{T}}\left[G\right]\left\{T\right\}+2{\left\{F\right\}}^{\mathrm{T}}\left\{T\right\}+P$$

(12)

Equation (12) represents the linear system for cable force optimization, where P denotes a constant vector independent of {T}. Solving this equation yields a solution vector that minimizes the structural strain energy. However, the solution only considers the deformation energy of the structure and lacks other redundant constraints. The optimization results often fail to achieve the ideal results.

According to the actual construction situation, the constraints of the proposed cable force optimization model consist of the following two points. Firstly, construction feasibility requires that the cable force remains non-negative (i.e., T > 0). Secondly, for the spatial displacement constraints at the closure joint, as shown in Equation (1), the horizontal deviation Δx can be canceled out by the pre-displacement of supports.

$$\left\{\begin{array}{l}\left\{T\right\}\ge 0\\ \left\{\mathbf{\Delta }y\right\}=\left\{\mathbf{\Delta }{y}_0\right\}+\left[C_{\Delta y}\right]\left\{T\right\}=0\\ \left\{\mathbf{\Delta }\theta \right\}=\left\{\mathbf{\Delta }{\theta }_0\right\}+\left[C_{\Delta \theta }\right]\left\{T\right\}=0\end{array}\right.$$

(13)

where {Δy₀} and {Δθ₀} are the vertical and rotational deviations under gravity and temporary loads, and [C_Δy] and [C_Δθ] are the influence matrices of the unit cable force on vertical and rotational deviations.

The mathematical model for the optimization of the cable force is obtained by combining the above objective function for the optimization of the cable force and the optimization constraints set. This paper uses the built-in quadprog function in MATLAB to solve the above quadratic programming problem. The function calling format is as follows: x = quadprog (H, f, A, b, A_eq, b_eq, lb, ub). This equation returns the vector x, which minimizes the function. The standard mathematical model for quadratic programming by this function is as follows:

$$\underset{x}{\min }{\displaystyle \frac{1}{2}}{\left\{x\right\}}^{\mathrm{T}}\left[H\right]\left\{x\right\}+{\left\{f\right\}}^{\mathrm{T}}\left\{x\right\}$$

(14)

$$\left\{\begin{array}{c}\left[A\right]\left\{x\right\}\le \left\{b\right\}\\ \begin{array}{l}\left[A_{eq}\right]\left\{x\right\}=\left\{b_{eq}\right\}\\ \left\{lb\right\}\le \left\{x\right\}\le \left\{ub\right\}\end{array}\end{array}\right.$$

(15)

where [H] is the quadratic term matrix, {f} is the primary term vector, [A] is the coefficient matrix of linear inequality constraints, {b} is the right-end vector of linear inequality constraints, [A_eq] is the coefficient matrix of linear equivalence constraints, {b_eq} is the right-end vector of linear equivalence constraints, {lb} is the vector of lower bound constraints on the independent variables, {ub} is the vector of upper bound constraints on the independent variables, and {x} is the vector to be optimized.

Substituting Equations (12) and (13) into the quadratic programming model of the function, the corresponding following specific parameter expressions are obtained:

$$\left\{\begin{array}{c}\left\{x\right\}=\left\{T\right\}\\ \left\{f\right\}=2\left\{F\right\}\\ \left[H\right]=2\left[G\right]\end{array}\right. \left\{\begin{array}{c}\left[A_{eq}\right]=\left[\begin{array}{l}{C}_{\Delta y}\\ C_{\Delta \theta }\end{array}\right]\\ \left[b_{eq}\right]=\left\{\begin{array}{l}\Delta y\\ \Delta \theta \end{array}\right\}\end{array}\right.$$

(16)

As there is no inequality constraint in the solenoid force optimization, A = [ ] and b = [ ] when calling the function. The computational model of the cable force optimization of Equation (12) is rewritten into the quadratic programming standard form, in which the constant term P can be omitted, and the quadratic programming standard form of the cable force optimization is as follows:

$$\min \left[U\right]=\underset{T}{\min }{\displaystyle \frac{1}{2}}{\left\{T\right\}}^{\mathrm{T}}\left[H\right]\left\{T\right\}+{\left\{f\right\}}^{\mathrm{T}}\left\{T\right\}$$

(17)

By utilizing the quadprog function and running the above procedure, the result of the optimization of the force can be obtained, and the solution process is shown in Figure 7.

4. Numerical Calculation Analysis

4.1. Establishment of the Cable Force Optimization Model

When evaluating the relevant design parameters of the Muping Xiangjiang River Bridge, the accuracy and feasibility of the proposed precise closure method of steel truss arch bridges based on stress-free state control were validated through numerical simulation studies. The Muping Xiangjiang River Bridge, situated in Changsha, Hunan, is a four-span continuous central-supported tied-arch steel truss bridge. The total length of the main bridge is 500 m, with span lengths of (70 + 180 + 180 + 70) m. The main spans are 2 × 180 m, and the height of the main arch is 36 m with a span-to-rise ratio of 1/5. The main beam features a composite beam structure. The transverse direction of the entire bridge consists of two main girders with a center displacement of 34.1 m. With wind bracing installed between the main girders, the main truss of the bridge adopts parallel N-shaped girders composed of upper and lower chords, as well as diagonal and vertical webs. In the completed state, the arch axis of the lower chords of the main girder follows a catenary curve, while the upper chords are designed by combining circular and curved lines. Circular curves are employed for transitions between the main arches and the side spans. The constraint system features a four-span continuous support system with fixed bearings at the central pier, as well as unidirectional or multidirectional movable bearings at the other piers. The overall design of the bridge is illustrated in Figure 8 and Figure 9.

The steel truss arches of the main bridge are assembled symmetrically, using the single cantilever installation technique, from both side piers towards the mid-span. Double cantilevers are symmetrically erected at the top of the central pier for the mid-span closure of the two main spans. The bridge employs the “arch–beam synchronization” construction method, where the steel beams are simultaneously erected during the installation process of the arch ribs. Subsequently, the arch ribs are closed first, followed by the installation of flexible tie rods, the placement of the concrete bridge deck panels, and the sectional pouring of wet joints outside the closure segment. After the installation and welding of the 300 mm long steel tie rods, which are pre-reserved at the closure joint of the bridge deck, the concrete bridge deck panels at the closure joint are installed, followed by the pouring of wet joints at the same location.

The finite element model of the steel truss arch bridge was established in MATLAB based on the design drawings and parameters of the Muping Xiangjiang Bridge, incorporating small deformation and linear elastic assumptions for cable force optimization calculations, as well as a stress analysis of primary structural members. It should be noted that the cantilever end displacements need to be measured for 24 or 48 h prior to closure. The optimal closure temperature is determined based on the measurement results. When the temperature reaches the closure temperature, the closure wedges will be installed accurately in position in order to ignore the influence of temperature change on the line shape and internal force of the joining.

In the model, the upper chord, lower chord, and web members of the main arch and side span structures are simulated using beam–column elements, with the main girder being symmetrically arranged across both spans and also being modeled with beam–column elements. The key material design parameters are presented in

Table 1. The design parameters of the main materials.

Materials	Elastic Modulus/MPa	Poisson’s Ratio	Weight/kg/m3	Allowable Combined Stress/MPa	Allowable Shear Stress/MPa	Adaptive Structure
Q420	2.06 × 105	0.3	7.85	240	140	Major arch
Q345	2.06 × 105	0.3	7.85	210	120	Web member
Q235	2.10 × 105	0.3	7.85	140	80	Beside piers

, while geometric parameters (including the section moments of inertia referenced to horizontal axes) are obtained from design documentation. The specific mechanical constraints are that the center pier is a double-column pier with fixed bearings, and the remaining piers are set as being unidirectional or multi-directional. The whole bridge contains 326 beam–column elements and 163 nodes.

4.2. Cable Force Analysis

The Muping Xiangjiang River Bridge, which is a four-span continuous central-supported tied-arch steel truss bridge, employs temporary buckle cables and horizontal cables for assembly and closure. The ultimate optimization target is to determine a set of initial tensions that is applicable for the closure stage while optimizing the cable forces locally in stages, thus avoiding multiple cable adjustments. Based on the actual construction scheme, the processes for arch rib assembly and temporary buckle cable tensioning are illustrated in Figure 10, Figure 11 and Figure 12.

During the closure, when the structure is in a maximum cantilevering state, there are four pairs of temporary buckle cables that require optimization, totaling eight variables. As the Muping Xiangjiang River Bridge is a symmetrical structure, the analysis is conducted on half of the bridge, as shown in Figure 13. The optimization is carried out for the number, position, and orientation of the buckling cables, of which T₂ and T₄ are horizontal buckle cables with a tension angle of 0. Comprehensively considering the actual construction conditions, temporary vertical loads of F = 70 t are applied at Positions A18 and A11 during the closure to prevent deflection at the cantilever ends and to avoid misalignment at the closure joint.

The objective function and constraints of the quadratic programming contain multiple influence matrices, which need to be solved when optimizing; the influence matrices of the unit cable force on the structural bending moments and displacement deviations can be output quickly with the help of MATLAB finite element software. The specific operation steps are as follows: Firstly, establish the finite element model of the whole bridge with one drop frame, apply the initial load, and view and output the structural bending moments and displacement deviations {M_L0}, {M_R0}, and [Δy₀, Δθ₀]^T = [−0.2901, −0.0268]^T. Secondly, the initial loads are removed and eight cable force optimization conditions are set for the number, size, position, and orientation of temporary buckle cables. Conditions 1 and 2 make the values of the cable force of T₁ and T₂ zero, changing the tension angle of T₃. Conditions 3 and 4 make the value of the cable force of T₁ zero, changing the tension angle of T₃. Conditions 5 and 6 change the tension angle of T₁ and T₃. Conditions 7 and 8 change the position of the tensioning nodes of T₂, T₃, and T₄. Finally, the spatial displacement constraint condition of the closure is input into the constraint condition, and the whole-bridge calculation and analysis are carried out. The influence matrix of the unit cable force on the structural bending [C_L]_163×4, [C_R]_163×4, and the influence matrix of the displacement deviation [C_Δ]_2×4 under the corresponding working conditions are viewed and output. [C_Δ] and the parts of [C_L] and [C_R] corresponding to Condition 8 are shown in

Table 2. The influence matrix of partial unit cable forces on structural buckling.

Node	{CL1}	{CL2}	{CL3}	{CL4}	{CR1}	{CR2}	{CR3}	{CR4}
A22	−6.35 × 10−4	0.00	−3.94 × 10−1	0.00	2.53 × 10−4	0.00	−6.04 × 10−1	0.00
A21	−2.26 × 10−4	0.00	3.30 × 10−1	0.00	−8.57 × 10−5	0.00	4.70 × 10−2	0.00
A20	6.28 × 10−5	0.00	−5.74 × 10−2	0.00	3.93 × 10−6	0.00	−2.58 × 10−2	0.00
A19	−1.35 × 10−6	0.00	1.50 × 10−2	0.00	2.35 × 10−6	0.00	−3.56 × 10−3	0.00
A18	−1.96 × 10−6	0.00	2.68 × 10−3	0.00	−3.86 × 10−7	0.00	4.54 × 10−4	0.00
A17	3.04 × 10−7	0.00	−4.18 × 10−4	0.00	1.67 × 10−8	0.00	−3.87 × 10−5	0.00
A16	0.00	7.62 × 10−8	0.00	1.17 × 10−5	0.00	1.06 × 10−6	0.00	1.34 × 10−4
A15	0.00	−1.17 × 10−6	0.00	−1.47 × 10−4	0.00	−1.94 × 10−7	0.00	−7.04 × 10−4
A14	0.00	−6.79 × 10−7	0.00	9.09 × 10−4	0.00	−4.01 × 10−5	0.00	−4.73 × 10−3
A13	0.00	5.61 × 10−5	0.00	7.83 × 10−3	0.00	1.29 × 10−4	0.00	1.84 × 10−2
A12	0.00	−1.28 × 10−4	0.00	−1.61 × 10−2	0.00	3.92 × 10−4	0.00	−7.65 × 10−2
A11	0.00	−8.44 × 10−4	0.00	1.34 × 10−1	0.00	−8.29 × 10−3	0.00	1.30 × 10−1
A10	0.00	1.35 × 10−2	0.00	9.93 × 10−3	0.00	2.70 × 10−2	0.00	1.34 × 10−1
A9	0.00	−2.52 × 10−2	0.00	7.91 × 10−3	0.00	−7.51 × 10−2	0.00	1.76 × 10−1
A8	0.00	1.38 × 10−1	0.00	−1.28 × 10−2	0.00	1.59 × 10−1	0.00	1.88 × 10−1

:

$${\left[C_{\Delta }\right]}_{2\times 4}=\left[\begin{array}{l}{C}_{\Delta y}\\ C_{\Delta \theta }\end{array}\right]=\left[\begin{array}{cccc}4.15\times {10}^{-8}& -2.12\times {10}^{-8}& 1.21\times {10}^{-7}& -3.99\times {10}^{-8}\\ 4.86\times {10}^{-10}& 3.23\times {10}^{-10}& 1.71\times {10}^{-9}& 6.82\times {10}^{-10}\end{array}\right]$$

Then, MATLAB was used to perform matrix operations to calculate the correlation matrix required in the optimization objective function. By utilizing the quadprog function and setting the corresponding constraint conditions, the cable force satisfying the optimization conditions can be obtained, as shown in Figure 14 and

Table 3. The effect of temporary buckle cable parameter adjustment measures on the error variation in the closure gap.

Conditions	Tensioning Node (T1–T4)	Tensioning Angles (T1–T4)/°	Δx/cm	Δy/mm	Δθ/°	U/105 J
1	A21–A9	67.25–0	7.363	4.075 × 10−2	2.539 × 10−3	2.251
2	A21–A9	83.29–0	7.376	7.622 × 10−3	2.286 × 10−3	2.054
3	A6–A21–A9	0–67.25–0	7.181	2.77 × 10−2	2.231 × 10−3	2.053
4	A6–A21–A9	0–83.29–0	7.207	8.18 × 10−3	2.260 × 10−3	2.070
5	A25–A6–A21–A9	62.73–0–67.25–0	1.334	1.82 × 10−2	0.190	1.409
6	A25–A6–A21–A9	83.25–0–83.29–0	1.1817	1.96 × 10−2	0.196	1.389
7	A25–A5–A22–A8	62.73–0–62.73–0	11.502	7.6691	3.667 × 10−3	4.475
8	A25–A6–A21–A9	62.73–0–67.25–0	7.647	4.60 × 10−2	2.590 × 10−3	2.442

.

The results demonstrate that the optimal cable force calculated using the method proposed in this paper for large-span steel truss arch bridges is highly efficient. This approach effectively eliminates the complex iterative processes required in traditional methods to meet closure accuracy during the construction process. The cable force determined under various conditions of numbers, sizes, positions, and orientations of temporary buckle cables were the smallest feasible set within the corresponding global solution. In comparison to using four pairs of temporary buckle cables, employing two or three pairs requires higher tension, which results in a greater deformation energy, as well as increased displacement deviations. Moreover, the further that the tensioning positions of horizontal cables are from the closure joint (Nodes A5 and A8), the higher the tension (with a larger deformation energy and displacement deviations) is required. Although a tensioning angle of 83° for the temporary buckle cables can reduce the deformation energy and displacement, it does not meet practical construction requirements.

4.3. Error Analysis of the Closure Joint

From the above calculations, it is evident that the vertical displacement of the closure joint Δy is close to zero, and the horizontal displacement Δx can be offset by setting a predetermined displacement at the bearings. The corresponding deformation energy and Δθ are calculated by varying the upper bound {ub} of the cable force, as shown in Figure 15. This reveals that the relative rotation angle Δθ is inversely proportional to both the maximum cable force and the structural deformation energy. To achieve a Δθ value of 0, signifying the maintenance of the stress-free dimensions of the closure wedge, the required deformation energy corresponding to Condition 8, as shown in Table 3, was set to 2.442 × 10⁵ J, with a maximum cable force T₄ of 1820 t, and a 7.647 cm predetermined bearing displacement. However, such adjustments significantly increase the construction difficulty and economic costs.

When the relative rotation angle of the closure joint reaches 0.19° (Condition 5), the maximum cable force is 1440 t, which is below the ultimate bearing capacity of 3507 t (with a safety factor exceeding 2). The structural deformation energy is 1.410 × 10⁵ J, and the predetermined bearing displacement is 1.182 cm. Due to the fact that temporary buckle cable tensioning is locked, Stage 3 (Condition 5) should be consistent with the temporary buckle cable tensioning position and the angle of Stages 1 and 2. Stages 1 and 2 are controlled by minimizing the deformation energy of the corresponding structures, respectively. In order to meet the angle and orientation requirements of Stage 3, the optimal cable force for each construction stage was determined, as shown in Figure 16. The cable forces vary constantly at different construction stages, and the corresponding constraints need to be met during the closure stage. With the calculated cable forces (Condition 5), the positive stress on the bridge rods of the left and right spans during the closure stage are calculated, as shown in Figure 17. The upper and lower chords and webs are numbered from left to right according to Figure 13, with a total of 106 rods on the left span and 58 rods on the right span. The webs bear significantly more force than the upper and lower chords, and the rods experiencing the maximum positive stresses across different types are concentrated near the tensioning points T₃ for the left span and T₄ for the right span (specifically, at A21–A20, A21–G20, G21–G20, A9–A8, A9–G10, and G10–G9). Therefore, the forces tensioned in the temporary buckle cables should not be excessively high to ensure construction safety, with the maximum tensile stress reaching 87.5 MPa, which is 42% of the allowable stress for steel (210 MPa).

An angular error analysis was conducted for the closure wedge, with its stress-free dimensions being depicted in Figure 18. When the closure joint rotates by 0.19°, with the height of the closure wedge remaining constant, a horizontal cutting of approximately 1.99 cm is required on each side. When the temperature of the closure joint reaches the design temperature, the stress-free dimensions of inserted segments according to the manufactured alignment are applied to ensure smooth installation and welding at the closure joint. Finally, the precise control of the stress-free parameters is achieved, facilitating the closure for the entire arch rib under a stress-free state.

5. Conclusions

This paper proposes a precise controlled assembly method for steel truss arch bridges based on a stress-free state. The control objectives include minimizing the structural deformation energy throughout the entire construction process and maintaining the stress-free dimensions of the closure wedge. By establishing the mathematical relationship between the displacement error of the closure joint and the relevant parameters of the temporary buckle cables, optimization models for the temporary buckle cables are established for each construction stage. Numerical calculations were conducted for the closure of the Muping Xiangjiang River Bridge based on the proposed method. The conclusions are as follows.

(1)

The effectiveness and feasibility of the proposed closure method based on the stress-free parameters of the closure wedge were validated for large-span steel truss arch bridges. The principle of maintaining the stress-free dimensions of the closure wedge provides a simple and clear control objective for the complex installation process of arch ribs. Compared to traditional forced closure methods, this approach effectively reduces the cutting volume required in the closure segment and simplifies construction processes, significantly reducing the construction risks associated with frequent adjustments in large-span cantilever structures.

(2)

Under the specified construction scheme, the proposed method can determine the optimal tensioning combination of temporary buckle cables, including the size, position, and orientation. By maintaining the stress-free dimensions of the closure wedge as the control objective, the correlation between deformation energy, cable forces, and closure joint displacement is established using the influence matrix method. A coordinated control of the structural deformation energy, temporary buckle cable parameters, and stress-free dimensions of the closure wedge was achieved.

(3)

The precise control of the complex displacement parameters at the closure joint is crucial for ensuring the accuracy of the closure. The relative rotation angle was found to be inversely correlated with both the maximum cable force and the structural deformation energy. By comprehensively considering the actual economic costs and construction challenges, minor adjustments in cutting can be selectively applied to the closure wedge dimensions to ensure smooth installation and welding at the closure joint, ultimately achieving the precise closure of steel truss arch bridges.