A New Method for Finding the Shape of the Main Cable in the Special Cable Plane

Abstract

The main cable and suspender of the spatial special-shaped suspension bridge are in the state of spatial stress; the structural stress is more complex than that of the parallel cable plane and finding the shape of the main cable of the spatial special-shaped cable plane is more difficult. In order to solve the problem of finding the shape of the main cable of the special-shaped suspension bridge, a new calculation model and algorithm of the main cable are proposed in this paper. The new calculation model adds constraints on the transverse bridge direction coordinates of the midspan on the basis of the original calculation model. The new calculation model can timely correct the calculation errors in all directions of the control node during the main cable shape finding process and improve the calculation accuracy. The new algorithm is a hybrid algorithm. The algorithm first uses the modified quantum genetic algorithm to solve and calculate the cable end force close to the real value, and then uses the cable end force obtained by the modified quantum genetic algorithm as the initial value to iterate through the modified least squares method. In this paper, a single cable plane curved suspension bridge is taken as the research background, and the differences of different calculation models in the shape-finding calculation of the main cable of the spatial special-shaped cable plane are compared. The results show that the proposed model is more stable in the calculation process, and the proposed algorithm has high accuracy and strong adaptability.

1. Introduction

In recent years, a variety of new bridge structures have emerged in the world, and the spatial special-shaped suspension bridge is one of them [1]. Different from the parallel cable plane, the main cable and suspender of the spatial special-shaped cable plane are under spatial stress, and the stress of the overall structure is more complex [2]. The main cable is the main load-bearing component of the suspension bridge, and the main cable shape is one of the main size parameters to be considered in the design of the suspension bridge. Therefore, the reasonable main cable shape has always been the focus of research workers.

According to the main cable alignment, suspension bridges can be divided into plane cable plane suspension bridges and space cable plane suspension bridges [3]. Spatial cable plane suspension bridges can also be subdivided into spatial cable plane suspension bridges and spatial special-shaped cable plane suspension bridges [4]. At present, the calculation theory and method of the main cable alignment of suspension bridges with parallel cable planes have been relatively mature, and some valuable results have been achieved. In terms of theory, Tang [5] put forward the calculation model of finding the main cable shape in the parallel cable plane on the basis of analyzing the main cable stress in the parallel cable plane. Brotton [6] introduced the finite element method into the calculation of suspension structures and proposed the basic process of finding the shape of the main cable in the parallel cable plane. In recent years, there have also been some valuable advances in computational methods. Wei Zhaolin [7] applied the heuristic algorithm to form a finding calculation of the main cable. Luo [8] proposed an optimization algorithm considering the influence of cable clamps. Song [9] proposed a new form-finding finite element analysis method for the main cable. Suspension bridges with a spatial cable plane are a common structural form of the spatial cable plane. The main cables of the plane suspension bridge with spatial cables are in the state of spatial stress, and the suspenders are in the state of plane stress. The stress of the structure is more complex than that of the suspension bridge with a parallel cable plane. In recent years, the research of the space main cable has made some progress. In terms of the calculation theory of space main cable, Luo [10] analyzed the force on the main cable of the space cable plane and proposed the calculation model of space main cable bridge alignment. In terms of the calculation method of the space main cable, Wang [11] summarized the forces on the space main cable and proposed the Steffens Newton method with adjustable parameters. Zheng [12] found that the existing space main cable calculation models rely on high-precision iterative initial values, and the traditional influence matrix method faces the problem of nonconvergence in some cases. To solve this problem, he proposed the modified Marquardt least squares method. Tian [13] proposed a nonlinear GRG method. Zhang [14] proposed a form-finding method considering the thermal effect of the main cable. Zhu [15] proposed a finite element method to solve the shape of main cable with the Euler method. Ma [16] proposed a finite element method with unknown parameters of cable node coordinates and element internal forces. Zhou [17] proposed a form-finding algorithm considering cable saddle friction. The research on finding the shape of the main cable of a spatial special-shaped suspension bridge has also made some progress in recent years. Li Libin [18] proposed a modeling method that uses the Midas modeling assistant to repeatedly modify the cable inclination. Wu Xingyue [2], based on the catenary theory, proposed an analytical expression for the completed alignment of the main cable with a spatial special-shaped cable plane considering the coupling between the main cable and the suspender. The main cable and suspender of the spatial special-shaped cable plane are in the spatial stress state, and the structural stress is more complex than that of the spatial slant cable plane. At present, although some achievements have been made in the research of spatial special-shaped suspension bridges, there are still many problems that need to be further studied, such as the new calculation method of finding the shape of the main cable on the spatial special-shaped cable plane, the optimization of the results of spatial special-shaped suspension bridges, etc.

In order to solve the problem of finding the shape of the main cable of the spatial special-shaped cable plane suspension bridge, this paper first analyzes the force of the spatial special-shaped cable plane suspension bridge and puts forward a new calculation model for finding the shape of the main cable of the spatial special-shaped cable plane suspension bridge—the four-coordinate calculation model. The new calculation model takes into account the transverse direction error of the midspan, which is more in line with the engineering practice. Secondly, this paper proposes a new form-finding method for space main cable—the hybrid quantum genetic algorithm. The algorithm first performs interval search by the modified quantum genetic algorithm [19] to obtain a high-precision iterative initial value, and then performs iterative calculation by the modified least squares method [20] until it meets the requirements. Finally, a suspension bridge with space special-shaped cable plane is taken as an example to study. The results show that the calculation model proposed in this paper is more stable, and the algorithm proposed has the characteristics of fast calculation speed, high calculation accuracy and strong adaptability.

2. Force Analysis of Spatial Special-Shaped Cable Plane

2.1. Balance Equation of Space Main Cable

The main cable of the suspension bridge with a space special-shaped cable plane is only subjected to the self weight between the suspension points, so the main cable meets the spatial catenary equation, but the components of the cable along the bridge direction are no longer equal. Wu Xingyue [2] has given the calculation diagram of a single cable.

$$\begin{array}{l}{L}_i=\frac{L_u{H}_i}{E_m{A}_m}+\frac{H_i}{g_m}\left[{\sinh }^{-1}\left(\frac{V_i}{H_i}\right)-{\sinh }^{-1}\left(\frac{V_i-L_{\mathrm{u}}{g}_m}{H_i}\right)\right]\\ h_i=\frac{1}{g_m}\left[\sqrt{H_i{}^2-V_i{}^2}-\sqrt{H_i{}^2-{\left(V_i-L_{\mathrm{u}}{g}_m\right)}^2}\right]+\frac{L_u}{E_m{A}_m}\left(V_i-\frac{L_{\mathrm{u}}{g}_m}{2}\right)\\ H_i=\sqrt{H_{xi}{}^2+H_{zi}{}^2}\\ L_i=\sqrt{d_{xi}{}^2+d_{zi}{}^2}=d_x\frac{H_{xi}}{H_{zi}}\end{array}$$

(1)

where L_u is the stress-free length of the main cable; g_m, E_m and A_m are the dead load concentration, elastic modulus and cross-sectional area of main cable; d_xi, d_zi and h_i are the transverse, longitudinal and vertical lengths of the main cable of section i, respectively; H_xi, H_zi and V_i are the transverse, longitudinal and vertical components of the main cable of section i, respectively; and H_i is the horizontal resultant force of the main cable in section i. See Figure 1.

2.2. Force Transfer Equation of Space Suspender

The suspender is an important part of the space shaped cable plane, and the clear force transmission process of the suspender is the key to finding the shape of the main cable of the space shaped cable plane. Based on the force transfer equation of the inclined plane suspender, the recurrence relationship between the force of the main cable section on the special plane of the space cable and the force of the main cable end is proposed (Equation (2)). By analyzing the force transfer equation of the suspender, it can be found that the transverse bridge component H_xi in different sections of the main cable changes significantly under the action of the suspender transverse bridge component P_xi. Due to the existence of the transverse bridge force component of the suspender, it is necessary to calculate the transverse bridge force component in the middle of the span during the calculation of the finished bridge alignment of the spatial special-shaped cable plane, which increases the calculation difficulty of finding the main cable shape of the spatial special-shaped cable plane.

$$\left\{\begin{array}{ll}{H}_{xi}& =H_{xi-1}-P_{xi}=H_{x0}-{\displaystyle \sum _{i=1}^n{P}_{xi}}\\ H_{zi}& =H_{zi-1}-P_{zi}=H_{z0}-{\displaystyle \sum _{i=1}^n{P}_{zi}}\\ V_i& =V_{i-1}-g_m{L}_{mi}{}_{-1}-P_{vi}-g_n{L}_{ni}\\ & =V_0-g_m{\displaystyle \sum _{i=1}^{m-1}{L}_{mi}-g_n{\displaystyle \sum _{i=1}^n{L}_{ni}-{\displaystyle \sum _{i=1}^n{P}_{vi}}}}\end{array}\right.$$

(2)

where g_m is the dead weight of the sling; L_ni is the unstressed length of the ith hanger; and P_xi, P_zi and P_vi are the along the bridge, the transverse bridge, and the vertical components of the suspender of section i, respectively, at the lower end. The specific meanings of other letters are the same as those above.

3. New Calculation Model

A numerical calculation model is the key to solve the problem. Luo Xiheng [10] proposed the calculation model of the space main cable (Equation (3)) based on the analysis of the stress of the space main cable.

The calculation model of the space main cable proposed by Luo Xiheng is:

$$\left\{\begin{array}{l}{f}_1\left(H_{x0},H_{z0},V_0\right):ey1={\displaystyle \sum _{i=1}^n{h}_i-\mathsf{\Delta }y1}\begin{array}{cccc}& & & \end{array}\\ f_2\left(H_{x0},H_{z0},V_0\right):ey2={\displaystyle \sum _{i=1}^m{h}_i-\mathsf{\Delta }y2}\begin{array}{cccc}& & & \end{array}\\ f_3\left(H_{x0},H_{z0},V_0\right):ez4={\displaystyle \sum _{i=1}^m{z}_i-\mathsf{\Delta }z4}\begin{array}{cccc}& & & \end{array}\end{array}\right.$$

(3)

At present, there is a lack of efficient numerical calculation models for the shape-finding calculation of the main cable in the space special-shaped cable plane. On the basis of Luo Xiheng, this paper proposes a calculation model (Equation (4)) suitable for finding the shape of the main cable of the spatial special-shaped suspension bridge, which takes into account the influence of the calculation error of the midspan transverse direction. The new calculation model considers both the vertical rise span ratio and the transverse rise span ratio of the spatial main cable, which is more consistent with the actual situation of the spatial special-shaped cable plane.

The calculation model of the space main cable proposed in this paper is:

$$\left\{\begin{array}{l}{f}_1\left(H_{x0},H_{z0},V_0\right):ey1={\displaystyle \sum _{i=1}^n{h}_i-\mathsf{\Delta }y1}\begin{array}{cccc}& & & \end{array}\\ f_2\left(H_{x0},H_{z0},V_0\right):ey2={\displaystyle \sum _{i=1}^m{h}_i-\mathsf{\Delta }y2}\begin{array}{cccc}& & & \end{array}\\ f_3\left(H_{x0},H_{z0},V_0\right):ez3={\displaystyle \sum _{i=1}^n{z}_i-\mathsf{\Delta }z3}\begin{array}{cccc}& & & \end{array}\\ f_4\left(H_{x0},H_{z0},V_0\right):ez4={\displaystyle \sum _{i=1}^m{z}_i-\mathsf{\Delta }z4}\begin{array}{cccc}& & & \end{array}\end{array}\right.$$

(4)

where ey1 is the vertical calculation error at the right end of the main cable, ey2 is the vertical calculation error in the middle of the main cable span, ez3 is the calculation error in the transverse direction of the right end of the main cable and ez4 is the transverse calculation error in the middle of the main cable span. m and n are the total number of segments of the main cable and the number of segments from the left end of the main cable to the midspan control point, respectively. H_i is the vertical displacement of the i-end main cable, and z_i is the transverse bridge displacement of the i-end main cable. Δy1 is the design vertical displacement of the left and right ends of the main cable, Δy2 is the design vertical displacement from the left end of the main cable to the midspan, Δz1 is the design transverse bridge displacement of the left and right ends of the main cable and Δy2 is the design transverse bridge displacement from the left end of the main cable to the midspan.

4. Modified Hybrid Quantum Genetic Algorithm

By analyzing the new calculation model, it can be found that the number of equations of the new calculation model is greater than the number of unknown quantities, so the model is an overdetermined equation group. The traditional Newton iteration method cannot solve the overdetermined equation problem. In order to solve the problem that the new calculation model is difficult to calculate, this paper proposes a new calculation method for the main cable of the spatial special-shaped cable plane—the improved hybrid quantum genetic algorithm (HQGA). The algorithm consists of two parts. First, it relies on the powerful search ability of quantum genetic algorithm to conduct interval search on the basis of parabola theory to obtain the cable end force close to the true value. Second, it relies on the modified least squares method to conduct local refinement analysis on the basis of the quantum genetic algorithm.

4.1. Modified Quantum Genetic Algorithm

The quantum genetic algorithm is an algorithm developed on the basis of the classical genetic algorithm. This algorithm uses quantum computing to improve the classical genetic algorithm’s shortcomings of relying on the selection of initial population and its convergence rate being not guaranteed. Quantum coherence makes quantum computing have strong parallel processing ability. In recent years, the quantum genetic algorithm has contributed to valuable research achievements in solving nonlinear equations [21,22,23]. In this paper, the quantum genetic algorithm is introduced into the form-finding calculation of the main cable on the space irregular cable plane, and the corresponding fitness function is proposed. Aiming at the problem that the rotation angle of the classical quantum revolving door is fixed, this paper makes a modest improvement to better adapt to the shape-finding calculation of the main cable.

4.1.1. The Fitness Function Proposed in This Paper

The fitness function is key for the intelligent optimization algorithm, and the appropriate fitness function can accelerate the evolution speed of quantum genetic algorithm. Wei Zhaolin proposed the fitness function (Equation (5)) of the parallel cable plane during the form-finding of the main cable of the parallel cable plane. In this paper, the fitness function (Equation (6)) suitable for finding the shape of the main cable on the special space cable plane is proposed, which selects the maximum error of the four control nodes. The fitness function can more truly reflect the size of individual error and reduce the calculation of the optimization algorithm.

The fitness function of the parallel cable plane M = (M₁² + M₂²)^1/2

$$\left\{\begin{array}{l}{M}_1=\left|1/\left({\displaystyle \sum _{i=1}^m{h}_i-\mathsf{\Delta }y1}\right)\right|\\ M_2=\left|1/\left({\displaystyle \sum _{i=1}^n{h}_i-\mathsf{\Delta }y2}\right)\right|\end{array}\right.$$

(5)

The spatial cable plane fitness function M: M = min (M₁, M₂, M₃, M₄) is proposed in this paper.

$$\left\{\begin{array}{l}{M}_1=\left|1/\left({\displaystyle \sum _{i=1}^m{h}_i-\mathsf{\Delta }y1}\right)\right|\\ M_2=\left|1/\left({\displaystyle \sum _{i=1}^n{h}_i-\mathsf{\Delta }y2}\right)\right|\\ M_3=\left|1/\left({\displaystyle \sum _{i=1}^m{z}_i-\mathsf{\Delta }z1}\right)\right|\\ M_4=\left|1/\left({\displaystyle \sum _{i=1}^n{z}_i-\mathsf{\Delta }z2}\right)\right|\end{array}\right.$$

(6)

The meanings of the equation letters are the same as in the preceding text.

4.1.2. Improvement of Quantum Rotary Gate

The quantum revolving gate is the main operator of population evolution [24,25], and, generally, the rotation angle of the quantum revolving gate Δθ_i is determined in advance.

Table 1. Classical Quantum Revolving Gate.

${\mathit{x}}_{\mathit{i}}$	${\mathit{b}}_{\mathit{i}}$	$\mathit{f}\left(\mathit{x}\right)\ge \mathit{f}\left(\mathit{b}\right)$	$\mathbf{\Delta }{\mathit{\theta }}_{\mathit{i}}$	$\mathit{s}\left({\mathit{\alpha }}_{\mathit{i}}{\mathit{\beta }}_{\mathit{i}}\right)$
				${\mathit{\alpha }}_{\mathit{i}}{\mathit{\beta }}_{\mathit{i}}>0$	${\mathit{\alpha }}_{\mathit{i}}{\mathit{\beta }}_{\mathit{i}}<0$	${\mathit{\alpha }}_{\mathit{i}}=0$	${\mathit{\beta }}_{\mathit{i}}=0$
0	0	False	0	0	0	0	0
0	0	True	0	0	0	0	0
0	1	False	ε	+1	−1	0	±1
0	1	True	ε	−1	+1	±1	0
1	0	False	ε	−1	+1	±1	0
1	0	True	ε	+1	−1	0	±1
1	1	False	0	0	0	0	0
1	1	True	0	0	0	0	0

shows the adjustment strategy of the rotation angle of the classical quantum genetic algorithm.

In the table, x_i is the i-th locus of the current individual, b_i is the i-th locus of the current optimal individual, f(x) is the fitness value of the current individual, f(b) is the fitness value of the current optimal individual, Δθ_i is the rotation angle and s(α_iβ_i) is the direction of rotation.

The quantum rotation angle of the classical quantum genetic algorithm Δθ_i is a fixed value during calculation ε. In the actual calculation process, people found that if the selection of quantum rotation angle ε is too small, the number of calculations increases significantly. If the selection of quantum rotation angle ε is too big, it may lead to the evolution of individuals exceeding the optimal solution.

In order to speed up the calculation in this paper, the rotation angle Δθ_i of the classical quantum rotary gate was moderately optimized according to the following rules. When the fitness value of the individual to be updated differs greatly from the fitness value of the best individual in the current generation, the rotation angle ε will be higher than the normal value, and the specific rotation angle will be determined according to the deviation degree. When the difference between the fitness of the individual to be updated and the fitness of the best individual in the current generation is small, the rotation angle ε will be lower than the normal value, and the specific rotation angle will be determined according to the deviation degree.

4.2. Modified Least Squares Method

The Gauss–Newton iteration method is one of the least squares methods, which has unique advantages in solving nonlinear overdetermined equations [26,27]. This paper introduces the modified least squares method to the calculation of finding the main cable shape of spatial special-shaped cable planes. In view of the difficulty in calculating the Jacobian matrix, this paper uses the influence matrix to replace the Jacobian matrix approximately. In order to avoid generating singular matrices in the calculation process, the Levenberg–Marquardt algorithm was used to modify the least squares method. In order to ensure that the objective function of each iteration is lowered, the step size is searched in three dimensions.

The specific process of the modified least squares method is as follows:

• Determination of objective function

The objective function of the least squares method depends on the calculation of the control node. Referring to the calculation function of the least squares method of the space cable plane proposed by Zheng Jiujian [10], this paper proposes an objective function suitable for the calculation model of four coordinate constraints ϕ(x):

$$\varphi (x)={\displaystyle \sum _{i=1}^4{f}_i^2(x)}$$

where f(x) is calculated according to the calculation model (Equation (4)) proposed in this paper.

2.

Calculation of Jacobian Matrix

The calculation of the Jacobian matrix is the key to conduct the modified least squares method. Because there is no explicit expression for the calculation model of the space main cable, it is difficult to solve the partial derivative. This paper refers to the influence matrix method and uses the influence matrix to replace the Jacobian matrix approximately. The specific calculation of the influence matrix method is as follows:

Take H_x0 = H_x0 + 1, H_z0 = H_z0, V₀ = V₀; H_x0 = H_x0, H_z0 = H_z0 + 1, V₀ = V₀; H_x0 = H_x0, H_z0 = H_z0, V₀ = V₀ + 1;

Substitute them into Equation (4) and calculate the corresponding calculation error to obtain the influence matrix A

$$A=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\\ A_{41}& A_{42}& A_{43}\end{array}\right]$$

3.

Calculate the best step size

In order to avoid the problem of convergence failure due to the singularity of ATkAk in the calculation process, this paper uses the Levenberg–Marquardt algorithm to modify it.

$$d^{(k)}=-{(A_k^T{A}_k+\alpha I)}^{-1}{A}_k^T{f}^k$$

In order to ensure that the value of the objective function decreases in each iteration, a one-dimensional search is conducted from x(k) to obtain the best step size λd^(k)

$$\varphi (x^{(k)}+{\lambda }_k{d}^{(k)})=\underset{\lambda }{\min }\varphi (x^{(k)}+\lambda d^{(k)})$$

4.

Correct the initial value

$$x^{(k+1)}=x^{(k)}+{\lambda }_k{d}^{(k)}$$

The initial value shall be corrected continuously, and the iteration shall be repeated until convergence.

5. Analysis of the Main Cable Curve of Spatial Special-Shaped Suspension Bridge

In order to study the differences between different calculation models and different algorithms in the calculation process of spatial special-shaped main cables, this paper takes a single cable plane curved suspension bridge as an example for calculation. This bridge is a double-tower, three-span steel truss spatial special-shaped suspension bridge, and the bridge span is 61 m + 168 m + 61 m. The midspan vertical rise span ratio is 1/8, and the transverse rise span ratio is 1/14. There are 48 suspenders in the whole bridge. The distance between the two suspenders along the bridge is 6m, and the vertical force at the lower end of the suspender on the middle span of the main cable is 135KN-267KN. The stiffening beam is a steel truss beam with a height of 2.4 m, a width of 5 m and a longitudinal slope of 2.2%. The elevation and plan of the bridge are shown in Figure 2, and the material properties of the main cable and suspender are shown in

Table 2. Material Characteristics.

Component	No Stress Concentration/(N/m)	Area/(m2)	Elastic Modulus/Gpa
Main cable	1361.2376	1.7671 × 10−2	195
Sling	96.6672	1.2566 × 10−3	195

.

5.1. Differences between Different Calculation Models

In order to study the difference between the four-coordinate constraint model proposed in this paper and the three-coordinate calculation model proposed by Luo Xiheng, this paper selects several representative classical algorithms in the main cable shape-finding process and an improved heuristic algorithm with strong nonlinear computing capability. See

Table 3. Algorithms under different calculation models.

Calculation Model	Influence Matrix Method	Modified Least Squares Method	Quantum Genetic Algorithm	Improved Particle Swarm Optimization	Quantum Genetic Algorithm + Influence Matrix Method	Improved Particle Swarm Optimization + Influence Matrix Method	Quantum Genetic Algorithm + Least Squares Method	Improved Particle Swarm Optimization + Least Squares Method
Three-coordinate calculation model	A	B	D	F	H	J	M	N
Four-coordinate meter	nothing	C	E	G	nothing	nothing	I	K

for the specific representation. For comparability, the convergence criteria of the two methods are uniformly taken as max(|ey1|, |ey2|, |ez3|, |ez4|) < eps, where eps = 10⁻³ m. This paper refers to the parabola theory proposed by Wang Xiaoming to estimate the space special-shaped main cable (H_x0, H_z0, V₀) = (4800 kN, 1180 kN, 2400 kN). This paper uses the estimated value as the initial point of iteration to carry out the iterative calculation of the influence matrix method and the modified least squares method. By comparing the forces of the spatial main cable and the spatial special-shaped main cable, it is found that the vertical force and the transverse force of the two bridge types have little difference, while the stress along the bridge direction has a large difference. Based on the parabola theory, the transverse and vertical force components are magnified 1.15 times, and the longitudinal force components are magnified 1.6 times. Therefore, the initial search interval of the heuristic algorithm proposed in this paper is H_x0 = (4080–5520 kN), H_z0 = (472–1888 kN) and V₀ = (2040–2760 kN).

Figure 3 shows the calculation results using the traditional calculation model, and Figure 4 shows the calculation results using the new calculation model. The calculation results in Figure 3 show that the calculation results of the existing calculation models have no obvious regularity, and the calculation error is large. Among them, the calculation results of the influence matrix method and those of Zheng [12] have the same regularity. They both become larger first and then tend to a larger error. The reason for this phenomenon is that the traditional Newton-like iterative method has weak computing power for strong nonlinear problems and cannot obtain correct results.

The calculation results in Figure 4 show that the calculation results of the calculation model proposed in this paper have an obvious downward trend, and the hybrid heuristic algorithm is obviously better than the single heuristic optimization algorithm. This phenomenon occurs because the heuristic algorithm has relatively weak optimization ability in the later stage and is not easy to jump out of the matrix optimal solution. However, the hybrid heuristic algorithm uses the modified least squares method in the later stage of the calculation, and the modified least squares method has a strong ability to search the local optimal solution.

5.2. Calculation Differences of Different Algorithms

In order to further study the differences of different algorithms under the four-coordinate computing model, this paper selects the hybrid algorithm of the improved quantum genetic algorithm and modified least squares method, which performs better in algorithm 4.1, and the hybrid algorithm of the improved particle swarm optimization and least squares method. The difference between different algorithms is studied by changing the initial interval. In order to avoid contingency, in each basic case, the two algorithms are calculated three times, respectively.

The calculation results in Figure 5 show that both algorithms (

Table 4. Two working conditions.

Search Range	I	II
Along bridge magnification	1.15	1.30
Transverse bridge magnification	1.60	2.20
Vertical magnification	1.15	1.30

) can obtain better results. In the same case, the calculation results of the hybrid quantum genetic algorithm are better than those of the hybrid particle swarm optimization algorithm. This phenomenon occurs because the quantum genetic algorithm has a stronger ability to jump out of the local optimal solution.

5.3. Research on the Adaptability of Hybrid Quantum Genetic Algorithm

In order to verify the adaptability of further research on the hybrid quantum genetic algorithm, this paper considers a variety of computing conditions on the basis of 5.2, and studies the adaptability of the hybrid quantum genetic algorithm by studying the performance of the hybrid quantum genetic algorithm in different environments.

The calculation results in Figure 6 show that the number of calculations required by the hybrid quantum genetic algorithm increases significantly with the increase in the initial search range (

Table 5. Multiple working conditions.

Search Range	I	II	III	IV	V	VI	VII	VIII
Along bridge magnification	1.15	1.30	1.30	1.15	1.15	1.30	1.30	1.15
Transverse bridge magnification	1.60	2.20	1.60	2.20	1.60	2.20	1.6	2.20
Vertical magnification	1.15	1.30	1.15	1.15	1.30	1.15	1.30	1.30

). Moreover, Figure 6 shows that the hybrid quantum genetic algorithm is more sensitive to the change of the initial search interval of the transverse bridge force. This phenomenon occurs because the number of excellent individuals in the initial population decreases as the search interval becomes larger, which affects the evolution speed.

In order to further verify the calculation accuracy of the hybrid quantum genetic algorithm, we used the professional bridge finite element analysis software Midas to verify the calculation results. In this paper, the Midas suspension bridge refined balance analysis module was used to perform refined balance analysis on the calculation results of the hybrid quantum genetic algorithm to verify the correctness of the algorithm proposed in this paper. Figure 7 shows the finite element model of the bridge, and Figure 8 shows the calculation error between the hybrid quantum genetic algorithm and the finite element method.

By analyzing Figure 8, it can be found that the errors of the two methods are large near the midspan, but the maximum error is less than 0.06%. By observing the calculation results in Figure 8, it can be concluded that the calculation results of the hybrid quantum genetic algorithm are basically consistent with those of the finite element method.

6. Conclusions

In this paper, the traditional calculation model of the main cable of spatial suspension bridges is improved, and a new calculation model and algorithm of the main cable of spatial suspension bridges with irregular cable planes are proposed. The new calculation model is more suitable for finding the shape of main cables of suspension bridges with special cable planes. This is because the new calculation model takes into account the calculation error of the midspan transverse direction, which can timely correct the calculation error of each direction of the control node in the calculation process, avoid the influence of the calculation error of the unstable midspan transverse direction and improve the stability of the calculation model. At the same time, the new calculation model also provides a strong support for quantitative analysis of the influence of the transverse vertical span ratio on the structure. The new algorithm has higher accuracy. This is because the new algorithm is a hybrid algorithm, which gives full play to the advantages of the traditional Newton algorithm and heuristic algorithm. The new algorithm can provide reference for the design of similar algorithms.