Traditionally, a trial-and-error procedure is carried out to design cross-sectional areas of stay cables in cable-stayed bridges. This design process is monotonous, expensive, time-consuming, and incapable of finding the optimum design solution. The aim of this study is to develop a robust design optimization technique in order to achieve the minimum cross-sectional areas of stay cables. The developed optimization technique integrates finite element method, B-spline curves, and genetic algorithm. The capability and efficiency of the proposed optimization technique is tested and assessed by applying it to a practical sized cable-stayed bridge.
Because of their aesthetic appeal, ease of erection, efficient utilization of structural materials, and other notable advantages, cable-stayed bridges have found wide applications all over the world in the last few decades [1]. Bridges of this type have recently entered a new era with main spans exceeding a value of 1000 m. In modern long-span cable-stayed bridges, such as the Sutong Bridge in China (2088 m), a large number of stay cables would be required in order to achieve reasonable distribution of bending moments along the bridge deck. The unit cost of stay cables is relatively high compared to other construction materials; therefore, there is a need for the development of an optimization technique to determine the minimum cost of stay cables in cable-stayed bridges.
In the current practice, the design process of stay cables is performed in two stages. The first stage involves the determination of initial post-tensioning cable forces, which are evaluated corresponding to zero vertical deflections of the deck and zero horizontal deflections of the pylons’ tops under only self-weight of the bridge. These forces are required to determine the initial configuration of the bridge. In the second stage, the cross-sectional areas of stay cables are determined under the combined effect of self-weight, initial post-tensioning cable forces, and live load cases. To date, this design stage is based on a trial-and-error procedure, which depends on the designer’s experience and skills [2] and [3]. A set of cross-sectional areas of stay cables is first assumed. Structural analysis for the bridge is then carried out in order to obtain the bridge deflections and stresses. If the deflections and stresses satisfy the requirements imposed by design codes, the assumed cross-sectional areas of stay cables are adopted. Otherwise, the cross-sectional areas are modified and the structural process is repeated until all the design criteria are met. The previous iterative design procedure is expensive, tedious, and time consuming. Moreover, it does not guarantee that the final solution will be the best of all the possible design solutions that satisfy the requirements of design codes
There have been many studies concerning the determination of the optimum post-tensioning cable forces under self-weight [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and [15]; however, there have been only a few attempts to determine the optimum cross-sectional areas of stay cables under self-weight, initial post-tensioning cable forces, and live load cases. One of the first attempts was conducted by [16]. In their study, a convex scalar function was used to minimize the cost of a box-girder deck cable-stayed bridge. The proposed function combines dimensions of the cross-sections of the bridge and post-tensioning cable forces. This method is very sensitive to the constraints, which should be imposed very cautiously to obtain a practical output [8]. In the research done by [3], the optimization module implemented in MATLB (fmincon), together, with the commercial finite element software, ABAQUS, are employed to evaluate the minimum cost of stay cables for cable stayed bridges.
It should be noted that the two previous studies are based on direct search optimization techniques. The drawback of these direct techniques is that they begin the search procedure with a guess solution, which is often chosen randomly in the search space. If the guess solution is not chosen close enough to the global minimum solution, the optimization technique will be trapped in local minima. As a result, the final solutions of these previous studies may not be the global minimum [3]. On the other hand, the cross-sectional areas of stay cables are considered as discrete design variables in both studies. With the increase in the number of stay cables, the number of design variables becomes quite large leading to potential numerical problems in the optimization technique. In addition, the increase in the number of stay cables makes the final distribution of the cross-sectional areas of stay cables non-smooth. Hence, the resulting values from these methods may be impractical in such cases.
The objective of the current study is to present a powerful optimization design technique in order to achieve the optimum cross-sectional areas of stay cables, which is directly proportional to the cost of the material. The proposed study focuses on the second design stage, where self-weight, initial post-tensioning cable forces, and live load cases are applied to the bridge. The proposed optimization technique involves interaction between three numerical schemes: finite element method (FEM), B-spline curves, and Real Coded Genetic Algorithm (RCGA). The novelty of this combined technique lies in the adoption of the B-spline curves to represent the distribution of cross-sectional areas of stay cables along the bridge length, which significantly reduces the number of design variables. In addition, RCGA, which is a global optimization method, is capable of finding the global optimal solution.
The remainder of the paper is organized as follows. In the next section, the geometry, finite element modeling, and design loads of the bridge chosen for the study are described. In Section 3, a description of the design variables, design constraints, objective function, and optimization technique is presented. A detailed description of the optimization design technique that involves a combination between the FEM, B-spline curves, and RCGA is presented in Section 4. In Section 5, detailed presentation and discussion of the numerical optimization results are given. Finally, Section 6 presents the main conclusions drawn from the study.
In this study, an optimization design technique that combines finite element method (FEM), B-spline curves, and Real Coded Genetic Algorithm (RCGA) is developed. The introduced technique is utilized to predict the optimum cross-sectional areas of stay cables in cable-stayed bridges under the combined effect of self-weight of the bridge, initial post-tensioning cable forces, and live load cases. One of the advantages of this technique is that the B-spline curve is used to represent the distribution of cross-sectional areas of stay cables along the bridge length, which significantly reduces the number of design variables. In addition, the RCGA, which is a global search technique, is employed to reach a global optimal solution of the weight function for stay cables. Several analyses have been carried out by applying the developed optimization technique to a practical sized cable-stayed bridge. The following conclusions could be drawn from the results:
1.
The proposed optimization technique can be applied as a powerful tool for the design of stay cables in cable-stayed bridges.
2.
The distribution of optimized cross-sectional areas of stay cables along the bridge length shows that the back-stay cables have cross-sectional areas larger than the rest of the cables. Those back-stay cables are used to balance large overturning moments on the pylon by acting as longitudinal bracing for the whole bridge.
3.
The stay cables close to the bridge’s center plane have large cross-sectional areas in order to satisfy deflection constraint of the deck and the stress constraints of the cables.
4.
Although the objective function, weight of the steel in stay cables (C), is simple and monotonic, the optimization problem exhibits several local optima. This can be traced back to the high redundancy of cable-stayed bridges and the interaction of the constraints with the objective function.
5.
Direct search methods are not able to find an optimum solution; therefore, a global optimization technique is required to find the global optimum solution for the weight function of stay cables.
6.
Optimizing shapes of the cable area functions instead of the cross-sectional areas of stay cables significantly reduces the number of the design variables, which decreases the computational time required to find the optimum solution.
7.
Compared with classical optimization techniques, where the number of independent design variables depends upon the number of stay cables, the proposed optimization technique leads to 32% reduction in the weight of steel in stay cables.
8.
The number of the design variables is unrelated to the number of stay cables. Therefore, the proposed technique is very efficient for modern long-span cable-stayed bridges having a large number of stay cables.
9.
The sag nonlinearity in the inclined cable stays has a slight effect on the determination of the optimum cross-sectional areas of stay cables in cable-stayed bridges.
