EVOLUTIONARY OPTIMIZATION OF BRACED STEEL FRAMEWORKS. FOR TALL BUILDINGS USING A HYBRID OC-GA METHOD

C.-M. Chan and K.-M. Wong

Department of Civil Engineering,

The Hong Kong University of Science and Technology, Hong Kong, P. R. China

Abstract

Having many attractive advantages, genetic algorithms (GAs) have been applied to many design optimization problems. However, the practical application of GAs to realistic tall building design is still rather limited, since GAs require a large number of structural reanalyses and perform poorly in local searching. While the Optimality Criteria (OC) method can be applied effectively to the element sizing optimization of tall buildings, there is no guarantee that the OC method can always lead to the global optimum. In this paper, the so-called hybrid OC-GA method is presented to fully exploit the merits of both OC and GA for topology and element sizing optimization of braced tall steel frameworks. While the GA is particularly useful in the global exploration for optimal topologies, the OC technique serves as an efficient local optimizer for resizing elements of selected topologies. The effect of population size and the importance of the local OC search operator have been investigated. The applicability and efficiency of the hybrid OC-GA method were tested with two braced steel building examples. Results indicate that the incorporation of the OC operator into the GA has remarkably improved the efficiency and robustness of the evolutionary algorithm and thus make the hybrid method particularly useful for topology optimization of practical tall building structures involving a large number of structural elements and the use of numerous structural forms.

1. Introduction

For the past few decades, genetic algorithms (GAs) first developed by Holland (1975) have gained wide popularity and demonstrated their advantages over the conventional gradient-based optimization techniques. GAs are stochastic search methods, which mimic the principle of the survival of the fittest in natural selection. Due to their generality, GAs have been applied to a wide range of design problems especially those with discrete sizing variables, geometrical and topological variables. Unlike conventional optimization techniques, GAs are able to explore simultaneously the entire design space with a population of designs and therefore is capable of seeking for the global optimum. GAs can be applied directly and conveniently to structural design problems with both discrete and continuous design variables.

GAs can be regarded as a type of zero-order method, which requires numerous functional evaluations for achieving solution convergence. Consequently, as the scale and complexity of building structures increase, the required computational effort also increases, thus making GAs prohibitively difficult in solving practical large design problems. Another shortcoming of the GA approach is its lack of precision in searching for the definitive global optimum point. Since GAs are stochastic techniques, they are incapable of determining the precise global optimum and converge generally only to a near­optimum point.

One effective approach for the element sizing optimization of building structures has been based on the Optimality Criteria (OC) method, which has been shown to suit particularly well for tall building design with many design variables (Chan, 2001; Chan 2004). In the OC method, a set of necessary optimality criteria for the optimal design is first derived and a recursive algorithm is then applied to resize the element sizing design variables to indirectly satisfy the optimality criteria. For the lateral stiffness design of tall buildings, the OC method generally converges quite rapidly in a few design

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M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 205-214.

© 2006 Springer. Printed in the Netherlands.

cycles indicating a weak dependence of the computational efficiency on the number of design variables. Although the OC method can be remarkably efficient in sizing optimization, they cannot be applied to topological optimization problems with addition and removal of discrete structural elements. Furthermore, there is no guarantee that the OC method can always lead to the global optimum (Kirsch 1993).

To overcome the problems associated with GAs while maintaining their merits, hybrid methods incorporating local search techniques into GAs, which provide great flexibility for hybridization, were proposed. Sakamoto and Oda (1993) proposed a hybrid method comprising a genetic algorithm and the generalized OC method to optimize both the layout and cross sectional area of simple trusses with nine nodes. Element sizing designs were refined locally by a proportional scaling in their proposed algorithm. Yeh (1999) inserted the fully stressed OC method to GAs for optimising element sizes of truss structures subject to stress and displacement constraints. The best design amongst a population size of 60 individuals in any generation was optimised by the fully stressed OC method. The results suggested that the hybrid GA method is more superior to a pure GA in terms of both quality of the optimal design and convergence behaviour. Chan et al. (2003) developed a hybrid method which combines GA with a rigorously derived OC technique. The method has the advantage over the conventional GAs and is capable of solving element sizing design problems of practical tall building structures in which the conventional OC method has encountered the problem of achieving solution convergence. Espinoza et al. (2005) demonstrated that a hybrid GA method with local search algorithm required significant reduction in the number of function evaluations for obtaining the optimal design when compared to simple GAs alone in solving a groundwater remediation problem. Fawaz et al. (2005) presented an evolutionary algorithm with a globally stochastic but locally heuristic search strategy. Considerably fewer computational operations have been found in the shape optimization of a simple 18-bar truss with stress constraints.

In this paper, the so-called hybrid OC-GA method is further extended to both topological and element sizing optimization of skeletal steel building frameworks. A local search operator based on a rigorously derived OC technique is developed and embedded in the framework of a GA. While the GA is used to explore the entire design space and generate improved topologies, the OC operator is applied as an efficient local optimizer for element resizing of selected topologies. The hybrid OC-GA method works in concert with the global GA searching method and the local OC optimizer to provide better optimal topological and element sizing design of building frameworks than that GA could provide alone. Different rates of the local OC search operator have been investigated for different population sizes with the aim of determining the most appropriate value of the probability of an OC operation for the topological and element sizing optimization of tall steel frameworks using the proposed hybrid OC-GA method.

2. General Design Problem Formulation for steel frameworks

Consider a general steel building framework having initially i = 1,2,…,# elements (or element fabrication groups), the minimum cost design the optimal topology and element sizing design in terms of material cost can be formulated as:

N

Minimize: Subject to:

W(t,,Ai) = Z ^ *t, • A,

i=1

d = 8j – 83-і < dU

J hj J

(j = 1,2,.

., M)

(la)

(lb)

– 1}i * 0

(i = 1,2,.

., N

(1c)

4 є 4 = (ац, a,^..^ащ)

(i = 1,2,.

., N

(1d)

tt = 0 or 1

(i = 1,2,.

., N

(1e)

Eq. (1a) defines the material cost of the building framework in which wt denotes the unit material cost per unit cross sectional area of element i and the design variables ti, Ai represent the Boolean variable and the cross sectional area of the element, respectively. The Boolean variable t defines the presence or absence of the element i. If tt = 1, then the element i exists; otherwise tt = 0 and the element is removed from the structural model. The element sizing variable Ai is selected from a specified set of discrete commercial sections A, as given in Eq, (1d), where nt represents the number of available discrete sections for the ith element. Eq. (1b) defines the set of j = 1,2,..,M serviceability lateral drift criteria, where j and j are the lateral deflections of two adjacent floor levels j and j-1; hj is the corresponding jth story height; dj, and djU are the drift ratio and its corresponding allowable limit. Eq. (1c) defines a set of element strength constraints, where ob and of are the individual element stress and its allowable limit. Since the design of tall buildings is predominately controlled by serviceability lateral stiffness requirements, element strength design constraints are secondary design considerations that only a small number of them are critical to a tall building design.