Optimum Design System for Steel Cable-Stayed Bridges Dealing with Shape, Sizing Variables and Cable Prestresses

Abstract: A general-purpose, rigorous and efficient optimum design system for steel cable-stayed bridges is developed, in which not only can the cable anchor positions on the main girder and pylon, and the crosssectional dimensions of the member elements be dealt with as design variables, but also the pseudo-loads applied to the cables. A powerful two-stage optimum design method is proposed to determine the optimum values of design variables for the cost minimization problem under stress constraints. At the first-stage optimization process, the cable arrangement and sizing variables are optimized by using the approximate concept and dual method with mixed direct/inverse design variables. Then the optimum values of pseudoloads, which induce the optimum prestresses into the cables, and the optimum sizing variables are determined so as to minimize the total cost of the bridge further by utilizing the sensitivities with respect to the pseudo-loads and a modified linear programming algorithm. The rigorousness, eficiency and practical usefulness of the proposed optimum design system are demonstrated by giving numerical design examples and the investigations of the optimum solutions at various design conditions. The significances of dealing with cable anchor positions and pseudo-loads as design variables are also emphasized.     

基于离散变量的桁架结构截面优化及软件实现

以杆件横截面为设计变量,以结构重量为目标函数,采用基于离散变量的序列两级优化方法对桁架结构进行优化设计;在优化设计过程中采用了0阶修正,并且利用PCL语言在有限元软件MSC．NASTRAN和MSC．PATRAN上开发出了解决含应力约束和位移约束的桁架结构离散截面优化问题的程序。     

Structural optimization of hollow-section steel trusses by differential evolution algorithm

This paper deals with the weight minimization of planar steel trusses by adopting a differential evolution-based algorithm. Square hollow sections are considered. The design optimization refers to size, shape and topology. The design variables are represented by the geometrical dimensions of the cross sections of the different components of the truss, directly involving the size of the structure, and by some geometrical parameters affecting the outer shape of the truss. The topology is included in the optimization search in a particular way, since the designer at different runs of the algorithm can change the number of bays keeping constant the total length of the truss, to successively choose the best optimal solution. The minimum weight optimum design is posed as a single-objective optimization problem subject to constraints formulated in accordance with the current Eurocode 3. The optimal solution is obtained by a Differential Evolutionary (DE) algorithm. In the DE algorithm, a particular combination of mutation and crossover operators is adopted in order to achieve the best solutions and a specific way for dealing with constraints is introduced. The effectiveness of the proposed approach is shown with reference to two case-studies. The analysis results prove the versatility of the optimizer algorithm with regard to the three optimization categories of sizing, shape, topology as well as its high computational performances and its efficacy for practical applications. In particular useful practical indications concerning the geometrical dimensions of the various involved structural elements can be deduced by the optimal solutions: in a truss girder the cross section of the top chord should be bigger than the one of the bottom chord as well as diagonals should be characterized by smaller cross sections with respect to the top and bottom chords in order to simultaneously optimize the weight and ensure an optimal structural behaviour.     

基于拟满应力遗传算法的桁架结构形状优化设计

以拟满应力法和改进遗传算法为基础,提出了一种可以解决具有连续变量和离散变量的桁架结构形状优化问题的拟满应力遗传算法。该算法既充分利用了遗传算法全局寻优能力强的特点,也发挥了力学准则法局部寻优长处,具有很高的搜索效率。用拟满应力遗传算法解决15杆桁架结构形状优化问题的结果表明,这是一种解决具有连续、离散混合变量的桁架结构优化设计问题的很有效方法。     

钢框架结构优化设计的多集约束和自动构件分组

对于结构优化问题,如钢框架结构的用钢量最小化,设计尺寸取决于构件横截面面积。构件横截面面积可从一些可行性表格中查得,如美国钢结构协会(AISC)提供的表格。构件的横截面尺寸bf,tf,d和tw(可为分散的或连续的)能独立确定。讨论关于设计尺寸变量的框架结构优化问题,提出一种获得理想构件分组法则。此法则考虑了制作、检查、装配和焊接的优势,而这些因素在传统预算里并没有被考虑。运用先前提出的适应性损失法,增强结构优化中的机械约束。     

Multiple cardinality constraints and automatic member grouping in the optimal design of steel framed structures

For structural optimization problems, such as the weight minimization of steel framed structures, the sizing design variables are often defined as the cross-sectional areas of the members, which are to be chosen from commercially available tables such as those provided by the American Institute of Steel Construction. Alternatively, the cross-section dimensions, b_f, t_f, d and t_w (which may be discrete or continuous) can be defined independently for each profile. This paper discusses the structural optimization problem of framed structures involving sizing design variables where a special genetic algorithm encoding is proposed in order to establish a strategy to discover ideal member grouping of members. Advantages in fabrication, checking, assembling, and welding, which are usually not explicitly included in the cost function, are thus expected. The adaptive penalty method (APM) previously developed by the authors is applied to enforce all other mechanical constraints considered in the structural optimization problems discussed in this paper.     

Genetic Algorithms in Topologic Design of Grillage Structures

The paper describes the use of genetic algorithms in determining the optimal layout and sizing of two-dimensional (2D) and three-dimensional (3D) grillage structures for stress, displacement, and element buckling constraints. The design space for this problem is highly nonconvex and not readily amenable to traditional methods of nonlinear programming. The approach develops an optimal topology from a set of predefined structural universe so as to satisfy kinematic stability requirements and other constraints on structural response. A two-level genetic algorithm–based search is used, wherein the kinematic stability constraints are imposed at one level, followed by the treatment of stress and displacement constraints at a second level of optimization. Since genetic algorithms search for an optimal design from a discrete set of alternatives in the design space, their adaptation in the topologic design problem is natural and is governed only by issues related to computational efficiency. Strategies designed to alleviate the computational requirements of a genetic algorithm–based search are discussed in the paper.     

A particle swarm ant colony optimization for truss structures with discrete variables

In this paper, a particle swarm optimizer with passive congregation (PSOPC), ant colony optimization (ACO) and harmony search scheme (HS) are combined to reach to an efficient algorithm, called discrete heuristic particle swarm ant colony optimization (DHPSACO). This method is then employed to optimize truss structures with discrete variables. The DHPSACO applies a PSOPC for global optimization and the ant colony approach for local search, similar to its continuous version. The problem-specific constraints are handled using a modified feasible-based mechanism, and the harmony search scheme is employed to deal with variable constraints. Some design examples are tested using the new method and their results are compared to those of PSO, PSOPC and HPSO algorithms to demonstrate the effectiveness of the present method.     

微分演化算法在桁架形状优化中的应用

为了获得全局最优和解决具有应力约束、几何约束以及局部稳定性约束的桁架形状优化问题中2类不同设计变量耦合给优化带来的困难,将1种新型智能优化算法——微分演化(Differential Evolution,DE)应用于桁架结构的形状优化问题中。给出了考虑节点坐标和截面面积两类不同性质的设计变量的桁架结构优化的数学模型,并对几个经典的桁架结构进行优化,将所得结果与其他优化算法结果进行了比较。数值结果表明了DE算法具有良好的收敛性和稳定性,可以有效地进行桁架结构的形状优化设计。     

考虑结构整体稳定性的单层网壳优化设计

提出了把近似满应力设计方法、弧长法和相对差商法结合起来的改进的离散变量两级优化方法,给出了同时考虑应力约束、位移约束和结构整体稳定性约束的优化算法。在考虑几何非线性的基础上,研究了单层网壳结构以结构质量最轻为目标函数的优化问题,编制了相应的计算程序,并通过算例进行了分析验证。     

Structural optimization of thin-walled tubular trusses using a virtual strain energy density approach

The present paper deals with the weight minimization of tubular trusses subjected to multiple loads under size, stress and buckling constraints. The applied optimization procedure is based on a virtual strain energy density approach developed by the first two authors, already tested in plane and space truss structures. The key point of the method is the activation of at least one of the imposed displacement constraints. In case where such limitations are absent, a dummy displacement constraint is introduced instead, which iteratively sustains corrections until convergence is achieved within the desirable tolerance. The efficiency and practicability of the proposed method was tested in typical cases of tubular truss structures. For reasons of comparison, the same cases were also optimized using Sequential Quadratic Programming (SQP), which is a powerful mathematical programming optimization method. The results revealed that the proposed method performs very well in terms of convergence, of required number of iterations and of optimum tracing, while the value of the introduced dummy displacement constraint has insignificant effect on the optimization procedure.     

满应力齿行法中松弛系数对桁架结构优化的影响

用M ATLAB编制了优化程序,实现带位移约束的桁架满应力优化设计方法,在迭代过程中引入松弛系数,并通过算例分析松弛系数对优化结果的影响及其原因.     

索承网壳结构的截面优化

根据索承网壳结构静力、稳定性分析特点,从适合索承网壳结构工程应用的角度出发,在考虑结构几何非线性基础上,提出用改进的离散变量两级优化方法来研究选定索力情况下的截面优化问题。该方法适用于同时具有应力约束、稳定性约束和位移约束的离散变量结构优化问题的计算。最后通过算例,验证该方法的正确性和有效性。     

基于改进双向渐进结构优化法的桁架结构拓扑优化

为了将双向渐进结构优化法应用于桁架结构优化设计中,结合能量原理和满应力设计准则,推导了以结构最小应变能为目标函数的优化计算公式,提出了可以应用于桁架结构优化的桁架-双向渐进结构优化(T-BESO)法.T-BESO法以杆件截面面积为设计变量,以结构应变能为目标函数,以应力约束和满应力设计准则为约束条件.在T-BESO法中...     

Design optimization of truss structures with continuous and discrete variables by hybrid of biogeography‐based optimization and differential evolution methods

This paper presents a hybrid BBO‐DE algorithm by hybridizing biogeography‐based optimization (BBO) and differential evolution (DE) methods for optimum design of truss structures with continuous and discrete variables. In BBO‐DE, the migration operator of BBO method serves as a local exploiter mechanism during the search process. Besides, DE has a role of the global exploration by performing multiple search directions in the search space to preserve more diversity in the population. By embedding of DE algorithm in BBO method as a mutation mechanism, the balance between the exploration and exploitation abilities is further improved. The comparative results with some of the most recently developed methods demonstrate the fast convergence properties of the proposed algorithm and confirm its effectiveness to solve optimum design problems of truss structures with continuous and discrete variables.     

Optimal Layout of Bridge Trusses by Genetic Algorithms

In this paper we present an approach to the layout and shape-optimization problem of bridge truss structures using genetic algorithms. The objective is to find an optimal layout design that will have minimum weight or material volume, subject to performance constraints related to member stresses, joint displacements, and member buckling. An automated two-stage optimization search process, which integrates structural analysis by finite-element method, genetic algorithms, and cognitive topology patterns (domain knowledge), is developed to solve the optimal problem. Two examples concerning bridge truss structure are investigated to demonstrate the effectiveness of the proposed method in solving these layout-optimization problems.     

混合变量桁架结构的拟分层遗传算法优化设计

混合变量桁架结构形状优化中采用并行计算思想,在单台计算机上实现了一种类似分层遗传算法的拟分层遗传算法。该算法能够产生更加平等的竞争机会,提供更多的优良个体,提高了种群多样性,同时不用人为的控制信息交换,再加上多层分级控制,一定程度上避免了标准遗传算法容易出现的"早熟"现象,加快了收敛速度,具有很高的搜索效率。用拟分层遗传算法解决25空间杆桁架结构形状优化问题的结果表明,这是一种解决具有连续、离散混合变量的桁架结构优化设计问题的很有效方法。     

Optimal design of steel frames under seismic loading using two meta-heuristic algorithms

In this paper, optimal design of steel frames is performed under seismic loading. The variables of the problem are taken as the cross-sectional areas of the members. These variables are considered as discrete, and are selected from a list of existing cross sections. Here, the charged system search and improved harmony search algorithms are utilized for optimization. For optimal design of steel frames in the first phase a time history analysis with the relative lateral displacement constraints is performed, and in the second phase a simultaneous dynamic–static analysis with the relative displacement and stress constraints is utilized using two meta-heuristic algorithms. Moment frames and their shear frame counterparts are considered, and their performances are compared for optimal design. In the case of moment frames, apart from the columns, the cross sections of the beams are also considered as design variables. The results indicate a good performance of the optimized moment frame and show that considering the effect of both drift and stress constraints, instead of only drift constraints, one obtains a better design. These results also show the suitability of the charged system search algorithm for optimal design of frames under seismic loading, as an extremely nonlinear problem.     

Structural dynamic optimal design based on dynamic reliability

In this work, dimension and shape optimization of structures under stochastic process excitation is addressed in the context of element or system dynamic reliability constraints, where the structural gross mass is taken to be the objective function. Firstly, based on the dynamic response analysis of truss structures under stochastic process loads, the dynamic reliability constraints are developed and simplified, and the normalization of design variables is discussed to avoid some variables being drowned by others during optimization due to their different dimensions and orders of magnitude. The optimal models of dimension and shape with element or system dynamic reliability constraints are then presented. Two numerical examples are finally used to illustrate the results of different optimal designs, which demonstrate that the efficiency to solve the structural optimization with dynamic reliability constraints can be significantly improved if the design variables and their initial values are selected properly.     

Optimal shakedown design of metal structures under stiffness and stability constraints

Optimal shakedown design of metal frame and truss-like structures is considered in this paper. Strength, stiffness and stability (only for trusses) constraints are included in non-linear mathematical models of volume minimization problems of structures. Stiffness conditions are realized by the restriction of structure deflections or nodal displacements, which can vary non-monotonically during the adaptation process. Determination of displacements is especially complicated if the variable repeated load is defined by variation bounds, which are not related to time. For trusses, stability constraints are related to recommendations of Eurocode 3 (EC3). Numerical examples concerning the calculation of frame and truss structures are presented. The results are valid for the small displacement assumptions.