Study on the Main Cable Curve of Suspension Bridge Based on the Improved Particle Swarm Optimization (IPSO) Method

Abstract

Determining a reasonable main cable curve is the foundation of suspension bridge design, and the accuracy and efficiency of the curve-finding problem are key to the design of a suspension bridge. To accurately obtain the completed curve of a main cable, force equations, which are nonlinear equations, need to be solved. In this study, the improved particle swarm optimization (IPSO) algorithm with inertia weight is presented to solve these nonlinear equations. Then, taking a double-tower three-span steel-box girder suspension bridge as the research background, the accuracy and efficiency of the IPSO method in finding the main cable curve are studied and then compared with those of the N-R iteration method and the finite element method (FEM). The results show that the proposed IPSO method has a high accuracy and a fast computing speed. Furthermore, the convergence under different bridge parameters is discussed, which demonstrates that the IPSO method has a strong adaptability.

1. Introduction

A suspension bridge is one of the most advantageous bridge types for long bridges spanning over one thousand meters. The main cable is the core load-bearing component of a suspension bridge, mainly regarding tension. In the design and construction of suspension bridges, the main cable curve directly affects the blanking length of the cable, the installation location of the cable clamp, the length of the suspenders, and other parameters. In addition, the cable curve has large effects on the mechanical performance of the bridge after construction [1]. Since the main cable curve is difficult to adjust during construction, it is essential to calculate and control the curve accurately before construction to control the cable curve and to make it consistent with the design. In addition, before the construction of a suspension bridge, the alignment and internal force need to be analyzed using the forward installation and reverse dismantling method based on the alignment of the completed bridge [2]. Therefore, researchers and constructors need to pay attention to the design and calculation of the main cable curve of a suspension bridge (i.e., the main cable shape findings) to better design and construct suspension bridges and to completion their construction [3,4]. In a completed suspension bridge, the main cable curve is calculated according to three controlled parameters—the rise–span ratio, the theoretical elevation of the main cable mid-span, and the theoretical elevation of the bridge tower top—combined with the loads on the main cable and an appropriate line shape calculation theory. Obviously, according to these controlled parameters, the main cable curve results are different when calculated based on different curve calculation theories. Hence, it is very important to choose an appropriate calculation theory.

Regarding theories behind the calculation of the main cable curve, the linear calculation theory was the earliest to be presented and can be classified into two categories: the analytical method and the finite element method (FEM). The analytical method mainly includes the parabola method and the catenary method. With an in-depth study of the mechanical characteristics of suspension bridges and a large number of calculations, it was found that the main cable curve of each span of a completed suspension bridge is not an ideal parabola or catenary under the distributed load along the main cable and the concentrated load at the cable clamp. Additionally, each cable segment of the main cable between the suspenders is catenary under of its own weight, also known as segmented catenary. Tang et al. (2003) [5] proposed the segmented catenary method, which determines the cable force and main cable curve of each part according to the mechanical balance and deformation coordination conditions, and the nonlinear problem of the main cable curve was considered. Then, the main cable curve can be solved in the form of an analytical expression, which is an accurate calculation method for the main cable curve. Han et al. (2009) [6] then developed the segmented catenary method from plane to space and proposed a numerical calculation method by establishing the spatial analysis model, and with this method, the main cable curves in both the cable completion stage and the bridge completion stage can be analyzed.

At present, the segmented catenary theory has formed a relatively mature calculation system. The constraints of the mechanical balance state and the geometric relationship are key to deducing the recursive relationship between line parameters. On this basis, many achievements have been accumulated in the boundary conditions of cable saddle, in the calculation efficiency of nonlinear equations, in theoretical optimization, etc. [7,8,9,10]. However, the iterative calculation of linear deviation and internal force correction is tedious and sensitive to the selection of the initial value, and the iterative convergence speed is slow as well. Moreover, the iterative calculation cannot be completed under a certain load.

The analytical solution to calculating the main cable curve is complex and has many iterations. With the rapid development of computer technology, finite element theory can be used to solve the problem of calculating the curve, and professional finite element software such as Midas has emerged. The calculation efficiency and accuracy of FEM depend not only on a high-precision element tangent stiffness matrix, such as a truss cable element, a catenary clue element, and a saddle element, but also on the efficient numerical algorithm of nonlinear equations, such as the load increment method, the Newton–Raphson (N-R for short) method, or the hybrid method.

Brotton (1966) [11] first proposed to use nonlinear finite element theory to solve the inclined suspender problem. With a tension only main cable in mind and considering the secondary influence caused by the initial axial force and large displacement, the linear member finite element stiffness matrix was modified; the nonlinear tangent stiffness matrix was derived; and then, the N-R method was used to solve the equations in incremental form. Saafan (1966) [12] considered the initial bending moment, shear force, and the secondary effect caused by the interaction between the axial force and the bending moment but, without regard to the initial axial force and large displacement, deduced the nonlinear tangent stiffness matrix and formed a more comprehensive nonlinear finite element theory. Thus, the analysis of suspension bridges began to enter the era of finite displacement theory.

Thai et al. [13,14] constructed the element stiffness matrix based on an analytical formula of elastic catenary in three-dimensional coordinates and conducted both static and dynamic analyses of the cable structure. Furthermore, taking a multi-tower suspension bridge as research background, the main cable curve was iteratively calculated, and the influence of the dead live load ratio was studied to predict the limits of suspension bridges. Song et al. (2020) [15] proposed an improved curve-searching method for suspension bridges based on FEM, and the method involved four steps, which are hanger force determination, determination of the initial equilibrium state of the cable in the mid-span, curve finding in the side span, and examination of the whole bridge. Zhang et al. (2020) [16] put forward an analytical method for finding the main cable curve using the framework of multi-segment catenary theory and applied the method on a three-pylon suspension bridge. Zhu et al. (2021) [17] developed the method based on the nonlinear finite element approach with a Eulerian description to find the main cable curve. Combining finite element analysis, analytical formula, and an optimization algorithm, Wang et al. (2021) [18] proposed a new optimal form-finding method for spatial self-anchored hybrid cable-stayed suspension bridges. The FEM shows good calculation accuracy in the main cable curve calculation, while when solving nonlinear equations, it mostly depends on matrix iteration and complex programming. Additionally, it would often face the convergence problems of initial value selection and numerical divergence. For bridge designers, its calculation efficiency was still insufficient [19].

The particle swarm optimization (PSO) method was proposed by Kennedy and Eberhart (1995) [20], whose basic idea was to use a group of initial irregular particles to explore from disorder to order in space to obtain the optimal solution. PSO is a relatively easy-to-implement algorithm, for which the continuity of the objective function has little influence on the algorithm, and the algorithm can adapt well in various environments. Therefore, PSO is widely used in various fields. Corazza et al. (2021) [21] proposed a hybrid metaheuristic algorithm based on PSO and the dynamic penalty method, to solve complex mathematical programming problems quickly. Wu et al. (2019) [22] put forward a hybrid optimization method based on SW-PSO and the BP neural network to improve engine fuel efficiency and to reduce emissions. From the perspective of cloud computing, the PSO algorithm can be utilized to solve task scheduling problems [23]. In the application of PSO on bridges, the researchers mainly focus on the model-updating algorithm and the parameter value optimization of the control system but not on other analyses [24,25,26]. However, for suspension bridges, the main cable curve is an iterative process, and the PSO method might provide the basis of a new idea for its calculation.

In this paper, the nonlinear solution equation of the main cable of a double-tower three-span steel-box girder suspension bridge is established using the theory of the segmented catenary method, and the accuracy and convergence speed of the improved PSO (IPSO) method are verified. Section 2 introduces the segmented catenary method. Then, Section 3 presents the IPSO method and conducted the test. Section 4 takes the practical suspension bridge as a background and compares the calculation results and the convergence problem based on the IPSO method, the N-R iteration method, and FEM. Finally, the conclusions are given in Section 5.

2. Calculation Method of Main Cable Curve: Segmented Catenary Method

2.1. Single Segmented Catenary

The single segmented catenary method assumes three points. First, the main cable material conforms to Hooke’s law. Second, the main cable is an ideal flexible cable, only bearing tension. Third, the change of cross-sectional area due to the elastic deformation of the main cable is ignored. Under these assumptions, due to gravity, the main cable shape is a curve, which can only bear the forces T_A and T_B along a tangent. When the suspension bridge is constructed, there is no horizontal force acting along the main cable, so that the horizontal force of the main cable H is a constant. In this study, a segment of a single main cable is taken as the research object, and the force situation is shown in Figure 1. According to Xia (2007) [27], the calculation process of single-segment catenary equation of main cable system is shown below:

According to Newton’s law, the equilibrium equation of the segment element can be formulated as

$${\displaystyle \sum y}=0.$$

(1)

Then, Equation (1) can be written as

$$H\frac{dy}{dx}+\frac{d}{dx}\left(H\frac{dy}{dx}\right)dx+q\left(x\right)dx-H\frac{dy}{dx}=0.$$

(2)

Additionally, Equation (2) can be derived as

$$H\frac{d{y}^2}{d^{2}x}+q\left(x\right)=0.$$

(3)

2.2. Piecewise Catenary

The loading form of the main cable of suspension bridge under the weight of the hanger, a cable clamp, an anchor head, and the main girder is shown in Figure 2.

In the i-section cable, the load on the cable is equivalent to a uniformly distributed load, as shown in Figure 3.

According to Figure 3, the force can be formulated as

$$qds=q_{y}dx.$$

(4)

Substitute Equation (4) into Equation (3), we have

$$\frac{d^{2}y}{d{x}^2}=-\frac{q}{H}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}.$$

(5)

Assume $\frac{dy}{dx}=z$, and it can be written as

$$x=-\frac{H}{q}\ln \left(z+\sqrt{1+z^2}\right)+t=-\frac{H}{q}s{h}^{-1}z+t,$$

(6)

$$z=\frac{dy}{dx}=sh\left(-\frac{q}{H}x+\frac{q}{H}t\right),$$

(7)

where t represents the integral constants related to boundary conditions.

Suppose that the differential equation is in catenary form:

$$y=-\frac{H}{q}ch\left(-\frac{q}{H}x+\frac{q}{H}t\right)+m.$$

(8)

As shown in Figure 3, the segmented cable is fixed at two points A(0, 0) and B(l, c), and when the coordinates of points A and B are substituted into Equation (8), we can formulate

$$y=-\frac{H}{q}ch\left(\frac{qt}{H}\right)+m=0,$$

(9)

$$y=-\frac{H}{q}ch\left(-\frac{ql}{H}+\frac{qt}{H}\right)+m=c.$$

(10)

Then, we can obtain

$$m=\frac{H}{q}ch\left(\frac{qt}{H}\right).$$

(11)

Assume $\beta =\frac{ql}{2H}$, and we derive

$$\left(e^{-\beta }-e^{\beta }\right)\left(e^{\frac{2\beta t}{l}-\beta }-e^{\beta -\frac{2\beta t}{l}}\right)=-\frac{2cq}{H}.$$

(12)

Furthermore, it can be written as

$$sh\beta sh\left(\frac{2\beta t}{l}-\beta \right)=\frac{cq}{2H},$$

(13)

$$t=\frac{s{h}^{-1}\left(\frac{c\beta }{lsh\beta }\right)+\beta }{2\beta }l.$$

(14)

Assume $\alpha =s{h}^{-1}\left(\frac{c\beta }{lsh\beta }\right)+\beta$; then, we have

$$m=\frac{H}{q}ch\alpha .$$

(15)

Finally, the analytical solution of catenary equation is obtained as

$$y=\frac{H}{q}\left[ch\alpha -ch\left(\frac{2\beta x}{l}-\alpha \right)\right],$$

(16)

where $\alpha =s{h}^{-1}\left(\frac{c\beta }{lsh\beta }\right)+\beta$ and $\beta =\frac{ql}{2H}$.

Based on the continuous boundary requirements, the shape of the main cable can be calculated as follows and the main cable can be divided into n sections. The equation of the main cable of i-section can be written as

$$y=\frac{H}{q}\left[ch{\alpha }_i-ch\left(\frac{2{\beta }_i{x}_i}{l_i}-{\alpha }_i\right)\right],$$

(17)

where ${\alpha }_i=s{h}^{-1}\left(\frac{c_i{\beta }_i}{l_{i}sh{\beta }_i}\right)+{\beta }_i$ and ${\beta }_i=\frac{q{l}_i}{2H}$.

As for the deformation compatibility condition,

$${\displaystyle \sum _1^n{c}_i}=c.$$

(18)

Any point on the main cable should pass through a given point, and the given point of the main cable could be regarded as a special point. When passing through this point in the middle of the span, each node of the main cable should meet the boundary force balance condition as

$${H\frac{d{y}_{i-1}}{d{x}_{i-1}}|}_{x_{i-1}=l_{i-1}}-{H\frac{d{y}_i}{d{x}_i}|}_{x_i=0}=p_{i-1},$$

(19)

Hence, the iterative calculation formulas can be established according to Equations (16)–(19). The detailed iteration format is as follows [28]:

As q, l, and f (the rise of the main span) are known, we then assume

$$H_0=\frac{q{l}^2}{8f},$$

(20)

$$p_0=\frac{ql}{2}.$$

(21)

Then,

$$H_{0}sh{\alpha }_i=p_i.$$

(22)

Setting i = 1, then,

$${\beta }_1=\frac{ql}{2H_0},$$

(23)

$$c_1=\frac{H}{q}\left[ch{\alpha }_1-ch\left(2{\beta }_1-{\alpha }_1\right)\right],$$

(24)

$${H\frac{d{y}_1}{d{x}_1}|}_{x_1=l_1}=-Hsh\left(2{\beta }_1-{\alpha }_1\right).$$

(25)

Combining with Equation (19), the next segments ${\alpha }_2$ and ${\beta }_2$ are established, and then, $c_2$, $c_3$, … $c_n$ are obtained in turn. Using deformation compatibility conditions,

$$\left|{\displaystyle \sum _1^n{c}_i-c}\right|\le \epsilon ,$$

(26)

it is determined whether Equation (26) is true. If not true, the value of p_i is adjusted to reiterate from Equation (22) again. If true, it continues to calculate whether to pass the control point. If it does not pass, H₀ is changed and reiterated from Equation (22). Until it passes, the calculation is over.

The iterative process can be regarded by solving nonlinear equations:

$$\mathbf{R}={\left[x-{\displaystyle \sum _1^n{x}_i},y-{\displaystyle \sum _1^n{y}_i}\right]}^T=\mathrm{0}.$$

(27)

3. Principle of Particle Swarm Optimization

3.1. Basic Particle Swarm Optimization

The basic PSO algorithm is a swarm intelligence algorithm based on a biological population system and solves computational problems by simulating biological population behavior. It is supposed that there is only one piece of food in an area and that birds would search for food in the area. However, no birds know where the food is, but only the distance between themselves and the food. According to these characteristics, those birds transmit information to each other in the search process, and the simplest and most effective way to find food is to search within the surrounding area of the bird closest to the food. Regarding every bird in the search space as a particle, the particle is the potential solution in the PSO method. A single particle position is adjusted according to a previous one, and the optimal solution is obtained through iterations. In the early stage, the PSO method was used to deal with optimization problems, which is widely used in topology, size, and shape optimization of structures [29].

The basic PSO method adopts velocity to represent the positional change of each particle in the population range, and each particle records the best position it reaches. In the iteration, the particles change their position by tracking two optimal solutions, which are the individual optimal location and the global optimal location. The individual optimal position is the optimal solution found by the particle itself, while the global optimal location is the optimal solution currently found by the population.

Suppose that there is a group of particles in D-dimensional space moving at a certain speed. The state of particle i at time t is set as follows:

For position,

$$x_i^t={\left(x_{i1}^t,x_{i2}^t,\dots ,x_{id}^t\right)}^T,$$

(28)

where $x_{id}^t\in \left[L_d,U_d\right]$. L_d and U_d stand for the upper and lower limits of the search space, respectively.

For velocity,

$$v_i^t={\left(v_{i1}^t,v_{i2}^t,\dots ,v_{id}^t\right)}^T,$$

(29)

where $v_{id}^t\in \left[v_{\min }{}_{,d},v_{\max ,}{}_d\right]$. v_min,d and v_max,d stand for the minimum and maximum speeds, respectively.

For the individual optimal position,

$$p_i^t={\left(p_{i1}^t,p_{i2}^t,\dots p_{iD}^t\right)}^T,$$

(30)

For the global optimal location,

$$p_g^t={\left(p_{g1}^t,p_{g2}^t,\dots p_{gD}^t\right)}^T,$$

(31)

where 1 ≤ d ≤ D, 1 ≤ i ≤ M.

Hence, at time t + 1, the particle would be updated by

$$v_{id}^{t+1}=v_{id}^t+c_1{r}_1\left(p_{id}^t-x_{id}^t\right)+c_2{r}_2\left(p_{gd}^t-x_{gd}^t\right),$$

(32)

$$x_{id}^{t+1}=x_{{}_{id}}^t+v_{{}_{id}}^{t+1},$$

(33)

where r₁ and r₂ represent random numbers uniformly distributed in (0, 1) interval. c₁ and c₂ represent learning factors, and c₁ = c₂ = 2 in general.

A basic flowchart of the PSO method is shown in Figure 4.

3.2. Improved PSO (IPSO) Method

The principle of the basic PSO method is relatively simple as it has few adjustable parameters. In order to expand the application of the PSO method, many researchers have expanded the range of PSO, in which introducing the inertia weight ω is one of the widely used optimization algorithms. By introducing inertia weight, the global and local optimization capabilities of basic PSO are coordinated.

We modified the velocity equation Equation (32) in the basic PSO algorithm into Equation (34), and the inertia weight ω can then be used to determine the response of particle to the current velocity.

$$v_{id}^{t+1}=\omega v_{id}^t+c_1{r}_1\left(p_{id}^t-x_{id}^t\right)+c_2{r}_2\left(p_{gd}^t-x_{gd}^t\right),$$

(34)

Therefore, whether IPSO has good exploration and development abilities depends on the selected value of inertia weight ω.

Currently, there are two strategies to determine the inertia weight ω. One is the nonlinear method, in which the inertia weight ω first increases and then decreases, written as follows:

$$\omega \left(t\right)=\{\begin{array}{l}1\times \frac{t}{T}+0.40\le \frac{t}{T}\le 0.5\\ -1\times \frac{t}{T}+1.40\le \frac{t}{T}\le 0.5\end{array},$$

(35)

This strategy combines the advantages of decreasing and increasing strategies to improve the performance of the algorithm.

The other one is the linear method proposed by Shi and Eberhart (1998) [30], and the inertia weight ω can be obtained by

$$\omega ={\omega }_{\max }-\frac{t}{t_{\max }}\left({\omega }_{\max }-{\omega }_{\min }\right),$$

(36)

where ω_max and ω_min are the maximum and minimum values of the inertia weight ω. t and t_max are the t-step iteration and maximum step iterations.

The IPSO method with added inertia weight value is widely used, and the inertia weight value would also have different effects on the calculation results. When the inertia weight ω is large, it benefits the global search. Although the convergence speed is fast, it is not easy to obtain an exact solution. On the other side, when the inertia weight ω is small, it is conducive to local searches and obtains more accurate values, but it becomes easy to be trapped in a local search. In this paper, the IPSO method with inertia weight is used as the algorithm for solving the optimization equations. Due to the stochastic nature of the IPSO algorithm, no safe conclusions can be derived through one optimization run. Thus, 50 independent runs should be performed and the mean value should be derived. Compared with the standard PSO method, the IPSO method has faster optimization times and fewer iterations, indicating that IPSO has faster convergence speed and better performance.

3.3. Application Method of IPSO in Calculation of Main Cable Profile

The shape-finding problem of suspension bridges is transformed into an optimization problem using IPSO algorithm, and its expression is as follows:

$$\underset{x}{\min }f\left(\mathbf{x}\right)\mathbf{x}={\left[H_0,P_0\right]}^T\left({\mathbf{x}}_l<\mathbf{x}<{\mathbf{x}}_u\right),$$

(37)

where x is the design variable; ${\mathbf{x}}_l$ and ${\mathbf{x}}_u$ are the upper and lower limits of the design variable x, that is, the search range of the solution space; ${\mathbf{x}}_l={[0,0]}^T$; ${\mathbf{x}}_u={[{10}^5,{10}^5]}^T$; and f(x) is the objective function, which returns a scalar value to be minimized.

Combined with Equation (27), the objective function of the IPSO algorithm used in the form-finding analysis can adopt the L2-norm, which is expressed as follows. According to the expression, the optimal value of f(x) is 0.

$$f\left(\mathbf{x}\right)=\sqrt{{\left(x-{\displaystyle \sum _1^n{x}_i}\right)}^2+{\left(y-{\displaystyle \sum _1^n{y}_i}\right)}^2},$$

(38)

4. Analysis of Main Cable Curve of Suspension Bridge

4.1. Engineering Background

The Yangtze River Bridge is a double-tower three-span steel-box girder suspension bridge with an arrangement of (290 + 1160 + 402) m. The rise–span ratio of the main cable in the midspan is 1:9, the distance between two main cables is 26.5 m, and the hangers are only set in the mid-span. The stiffening girder is a steel box with a height of 2.8 m and a width of 34.7 m. The dead load is 6.103 t/m. The weight of the main cable is 25.199 kN/m, and the weight of the hanger is 0.034 kN/m. The rise and area of the main cable are f = 128.89 m and 0.321 m², respectively. The bridge layout is shown in Figure 5.

4.2. Calculation Accuracy

The IPSO method in Section 3 was introduced in the calculation of Equations (16)–(19). In the iteration, the spacing between hangers, the natural load of the bridge, and other initial parameters were taken as known quantities, and the completed coordinate value of the main cable was taken as the unknown quantity. The learning factors were all set as 2, the maximum iteration step number was 5000, ${\omega }_{\max }$ = 0.9, ${\omega }_{\min }$ = 0.4, and the convergence criterion was $f\left(\mathbf{x}\right)<1.0\times {10}^{-6}$.

Due to focus of the paper, only the results of the control points on the main span of the bridge in the completed state are listed in

Table 1. Curve coordinates of the main cable bridge by the IPSO method.

Point Number	Position	x(m)	z(m)
1	South side of the top of the tower	0.0	211.625
2	1/16 point of main span (1)	72.5	181.204
3	1/8 point of main span (1)	145.0	154.931
4	1/16 point of main span (2)	217.5	132.809
5	1/4 point of main span (1)	290.0	114.751
6	1/16 point of main span (3)	362.5	100.737
7	1/8 point of main span (2)	435.0	90.737
8	1/16 point of main span (4)	507.5	84.742
9	Mid-span of main span	580.0	82.732
10	1/16 point of main span (5)	652.5	84.736
11	1/8 point of main span (3)	725.0	90.737
12	1/16 point of main span (6)	797.5	100.736
13	1/4 point of main span (2)	870.0	114.751
14	1/16 point of main span (7)	942.5	132.807
15	1/8 point of main span (4)	1015.0	154.930
16	1/16 point of main span (8)	1087.5	181.200
17	North side of the top of the tower	1160.0	211.625

, where the results are the average values after 50 calculation steps performed by the software MATLAB.

(1)

Results compared with N-R iterative method

In practical engineering, designers often use the N-R method to find the shape of the main cable of a suspension bridge. Therefore, in this section, the traditional N-R method is used to iteratively solve Equations (16)–(19); the method was presented in detail by Qin and Xia (2013) [31]. A comparison of the main cable curves between the proposed IPSO method and N-R iterative method is shown in Figure 6, and the differences are presented in Figure 7. The maximum difference between the two results is less than 1 cm, which can be regarded as meeting the engineering demand.

In order to verify the calculation accuracy of the IPSO method further, based on practical engineering, the three additional cases listed in

Table 2. Four cases.

Case Number	Rise–Span Ratio	Area of Main Cable	Hanger Force
O (Practical engineering)	1/9	0.321 m2	Fdesign
Ⅰ	1/11	0.321 m2	Fdesign
Ⅱ	1/9	0.321 m2	1.1 × Fdesign
Ⅲ	1/9	0.350 m2	Fdesign

were compared for their rise–span ratios, hanger forces, and area of main cable. In Table 2, F_design stands for the design force of the hanger.

It can be seen from

Table 3. Curve coordinates of the main cable bridge by the IPSO method and the N-R iterative method for four cases.

Point Number	Position	x(m)	z(m)		Difference I (%)	z(m)		Difference Ⅱ (%)	z(m)		Difference Ⅲ (%)
			PSO Ⅰ	N-R Ⅰ		PSO Ⅱ	N-R Ⅱ		PSO Ⅲ	N-R Ⅲ
1	South side of the top of the tower	0.0	213.625	213.625	0.000	211.761	211.760	0.000	208.094	208.094	0.000
2	1/16 point of main span (1)	72.5	186.862	186.859	0.002	181.339	181.337	0.001	181.328	181.328	0.000
3	1/8 point of main span (1)	145.0	165.428	165.429	−0.001	155.138	155.139	−0.001	155.123	155.125	−0.001
4	1/16 point of main span (2)	217.5	147.316	147.318	−0.001	133.003	133.004	−0.001	132.990	132.991	−0.001
5	1/4 point of main span (1)	290.0	132.506	132.507	−0.001	114.904	114.906	−0.002	114.895	114.896	−0.001
6	1/16 point of main span (3)	362.5	120.987	120.990	−0.002	100.834	100.837	−0.003	100.826	100.830	−0.004
7	1/8 point of main span (2)	435.0	112.759	112.764	−0.004	90.784	90.789	−0.006	90.778	90.785	−0.008
8	1/16 point of main span (4)	507.5	107.820	107.824	−0.004	84.752	84.756	−0.005	84.749	84.755	−0.007
9	Mid-span of main span	580.0	106.164	106.170	−0.006	82.728	82.736	−0.010	82.729	82.736	−0.008
10	1/16 point of main span (5)	652.5	107.819	107.824	−0.005	84.749	84.756	−0.008	84.751	84.755	−0.005
11	1/8 point of main span (3)	725.0	112.761	112.764	−0.003	90.783	90.789	−0.007	90.779	90.785	−0.007
12	1/16 point of main span (6)	797.5	120.989	120.990	−0.001	100.833	100.837	−0.004	100.826	100.830	−0.004
13	1/4 point of main span (2)	870.0	132.504	132.506	−0.002	114.904	114.906	−0.002	114.893	114.895	−0.002
14	1/16 point of main span (7)	942.5	147.315	147.318	−0.002	133.003	133.004	−0.001	132.988	132.990	−0.002
15	1/8 point of main span (4)	1015.0	165.428	165.429	−0.001	155.137	155.139	−0.001	155.124	155.125	−0.001
16	1/16 point of main span (8)	1087.5	186.858	186.859	−0.001	181.336	181.337	−0.001	181.327	181.328	−0.001
17	North side of the top of the tower	1160.0	213.625	213.625	0.000	211.761	211.760	0.000	208.094	208.094	0.000

that the results obtained by the IPSO and N-R iteration methods are basically consistent. From Figure 7 and Table 3, under these four cases, the maximum calculated difference between the IPSO method, and the N-R iteration method is 0.008%, and the difference ranges from −0.008% to 0.003%. It can be considered that the results produced by the IPSO method are basically consistent with those of the N-R iteration method.

(2)

Results compared with FEM

In order to further illustrate the accuracy of the IPSO method, a three-dimensional finite element model was established by using the professional finite element software BNLAS. The BNLAS software was developed by Prof. Shen’s team around 2000 [28,32,33]. Currently, BNLAS has been used by more than 40 design corporations in China and has become the main software for design and analysis of suspension bridges in China. The BNLAS software contains two subsystems: (1) the spatial static and dynamic nonlinear analysis system of bridge structure; (2) the design and construction calculation system of the main cable system of suspension bridges. In this paper, the second subsystem was used to calculate the ideal curve of the main cable after completion, the curve of the empty cable and the theoretical pre-deviation of saddle, and the unstressed blanking length of the cable.

The main cable was simulated by the spatial cable element, the hanger was simulated by the membrane element, and the stiffening girder and bridge tower were simulated by the beam element. The finite element software adopts nonlinear linear element theory, takes the improved incremental iterative method as the nonlinear iterative scheme, and takes the infinite norm of the unbalance force and relative displacement error as the iterative convergence checking criterion. Therefore, the software can automatically take into account the influence of geometric nonlinearity without manual input. The completed finite element model of the Yangtze River Bridge is shown in Figure 8. A comparison of the difference in curves between the IPSO method and FEM is shown in Figure 9.

The maximum difference is 0.009% at Point 9, and the difference ranges from −0.009% to 0.002%. It is supposed that the results produced by IPSO are basically consistent with those by FEM. The solution to the nonlinear equations in this paper are correct; the solution accuracy can also meet the practical requirements; and thus, the solution can be properly applied in future engineering examples.

IPSO is an analytical method in which the results need to be verified by FEM. The unstressed length of each cable and the coordinate results of each node can be obtained via the analytical method. From these results, the finite element model can be established and the displacement of each point can be obtained through the nonlinear finite element solution. If the error is small and meets the accuracy requirements, the calculation result of the analytical method can be considered correct.

In this paper, the accuracy of the results produced by the IPSO method is compared with that of N-R and FEM, and the error is proven to be small, which shows that the accuracy of the IPSO calculation can meet the demands.

4.3. Calculation Efficiency

As for the N-R iteration method, there is a great correlation between the iterative convergence speed and the selected initial value, and the calculation is quite different under different initial value conditions. Since the initial value of IPSO is generated randomly, the convergence of IPSO does not depend on the selection of an initial value. Hence, under different initial value conditions, the calculation efficiency of IPSO is almost the same.

The computing time of the four cases are shown in

Table 4. Computing time between the IPSO and N-R iteration methods for four cases.

Case	Computing Time (s)
	IPSO Method	N-R Iteration Method
O	35	50
Ⅰ	37	53
Ⅱ	32	60
Ⅲ	34	49

.

It can be seen that, under the four cases, the computing time for the two methods is relatively stable. As the cases compared in the paper are common cases in practical engineering, there are many experiences in selecting the initial values of N-R iteration method, which can basically ensure convergence. In addition, in these cases, the time consumption of the IPSO method is shorter than that of yjr traditional N-R iteration method. Under the same accuracy, it is obvious that the efficiency of IPSO method is better than that of the N-R iteration method.

4.4. Calculation Convergence

To study the calculation convergence further, different initial horizontal forces were adopted to simulate different initial conditions on the basis of Case O. Additionally, the initial horizontal force in Case O was set as H_design. The results of this convergence are shown in

Table 5. Convergence between the IPSO and N-R iteration methods for different hanger forces.

Initial Horizontal Force	Computing Time (s)
	IPSO Method	N-R Iteration Method
1.5 × Hdesign	Yes	Yes
2.0 × Hdesign	Yes	Yes
5.0 × Hdesign	Yes	No
10.0 × Hdesign	Yes	No

.

It can be seen that the convergence of the N-R iteration method depends on the selection of an initial horizontal force, and inappropriate initial values cause difficulty during calculation. As the initial value of the IPSO method is randomly generated, its convergence does not depend on the given initial value. In the search range, it can ensure the convergence of calculations.

5. Conclusions

In this study, aiming to find an efficient solution to calculating the main cable curve of a double-tower three-span steel-box girder suspension bridge, we established a nonlinear solution equation based on the theory of the segmented catenary method and the equations were formulated based on the IPSO method to obtain the node coordinates of the main cable curve. The results were compared with the traditional N-R iteration method and FEM. The following main conclusions were made:

1.

As for the calculation accuracy, the proposed IPSO method shows similar accuracy when solving the multi-segment catenary nonlinear equations compared with the traditional N-R iteration method and FEM.

2.

The N-R iteration method might take a long time and have a strong dependence on the initial value. Ensuring an equivalent solution accuracy, the proposed method based on the IPSO method shows a faster convergence speed. Moreover, when the case is more complicated, the proposed method more easily converges.

3.

In the current work, the curve-finding problem is only a two-dimensional problem. This is because, in most three-dimensional suspension bridges, the main cable itself is a two-dimensional curve, which can be easily transformed into a two-dimensional situation through coordinate transformation. Therefore, two-dimensional problems are statistically and comparatively analyzed to evaluate the convergence of IPSO. However, with the rapid development of bridges with long spans, there are many other engineering problems that are three-dimensional problems and cannot be simplified. In the future, statistical analyses and mathematical research on bridge engineering based on the IPSO method will be considered.