H. Rahamia, A. Kavehb,_, Y. Gholipoura
a Engineering Optimization Research Group, University of Tehran, Tehran, Iran
b Centre for Excellence for Fundamental Studies in Structural Engineering, Iran Universityof Science and Technology, Narmak, Tehran-16, Iran
Received 25 April 2007; received in revised form 1 January 2008; accepted 15 January 2008
Available online 10 March 2008
1. Introduction
For size/geometry optimization of structures with fixed topology, it becomes necessary to optimize structural crosssections and geometry simultaneously. For such optimization,usually large numbers of design variables will be encountered consisting of cross-sectional areas and nodal coordinates, thus resulting in design spaces with large dimensions. Selecting the cross-sectional areas from a list of profiles leads to a discrete design space, and due to the constraints on member stresses, buckling stresses, and nodal displacements, the possibility of being trapped in a local optimum increases.Goldberg is one of the pioneers in developing the Genetic algorithm [1]. Early papers on structural optimization using GA are due to Goldberg and Samtani [2], Jenkins [3], Adeli and Cheng [4] and Rajeev and Krishnamoorthy [5]. Many others
have published papers improving the results and increasing the speed of GA in the last decade.In the process of optimizing the geometry (shape) of a structure by the Genetic Algorithm (GA), if minimizing the structural weight is taken as the objective, by altering the geometry of the primary structure and increasing the dimension of the design space, the optimization may lead to local optima.Analysis of structures by the force method is well established by Argyris and Kelsey [6]. Further developments are due to Herderson [7], Cassell et al. [8], Denke [9],Felippa [10], and Kaveh [11] among many others. A comprehensive list of references can be found in the review paper of Kaveh [12].Energy methods are the most important approaches for the linear and nonlinear analyses of structures. In this article, an energy method and the force method are used for minimizing static indeterminacy of the structure S. The efficiency of the present approach is illustrated through six examples of different configurations
2. An introduction to energy methods
the following a brief introduction is provided to different energy methods. In general, the stress–strain relationship can be expressed in the form " = f (_ ) or _ = g("). Then the strain energy, complementary energy, and the total potential energy can be calculated as:
（1）
（2）
where U is the strain energy, Uc is the complementary energy, V is the total potential energy, P is the vector of the external loads and u is the vector of joint displacements.
According to Castigliano’s first theorem, for an elastic (linear or nonlinear) system, the potential energy in stable equilibrium is minimum. Similarly according to the second theorem, the complementary potential energy is minimum for a system of internal forces which satisfies the compatibility.In general, U corresponds to the stiffness method and Uccorresponds to the flexibility approach. In the first case one
looks for the displacements and in the latter case we look for redundant forces. Since in a statically indeterminate structure,after calculating the redundant leads, the remaining member the weight of a structure employing genetic algorithm as a powerful optimization technique. Since energy is a scalar entity,therefore it is used as a suitable objective function in this algorithm.The main idea proposed in this paper is the manner in which the input variables are reduced. Using the force method is of a considerable help in handling this problem. One of the difficulties with the application of GA for optimization of a structure is that for each chromosome in each generation, the calculations should be performed by an analyzer. When both geometry and topology are considered, for each chromosome the stiffness matrix and the matrices involved in the force method will be changed and therefore the inversion of such matrices in the GA paradigm will require many generations, increasing the computational time and reducing the convergence rate. In the present approach, there is no need to find the inverse of matrices, and only the addition of (S) variables to GA is required, where (S) is the degree of forces can easily be obtained, hence using Uc, i.e. the flexibility method, corresponds to smaller number of unknown. In the following the basic steps of the flexibility method based on the principle of minimum work is described.
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