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In the present paper, the simultaneous analysis and design (SAND) concept is extended to investigate use of the alternative formulations for optimum design of frames with existing analysis programs. There is a possibility to use member forces and displacements as optimization variables. These are considered as major extensions of the formulations used previously for trusses. (Wang and Arora 2005). As noted earlier, Saka (1980) has previously used nodal displacements as optimization variables and solved the problem using the sequential linear programming (SLP) method. Major differences between that work and the present work are: (1) member forces are also treated as optimization variables in one of the formulations, (2) an existing analysis software ANSYS is used directly in the optimization process, (3) a sequential quadratic programming (SQP) algorithm is used to solve optimization problems which is more robust than the SLP method, and (4)relative performance of the formulations is studied.
Based on the foregoing discussion, two objectives of the present paper are set up as follows:
1. To present and study two alternative formulations for optimal design of planar framed structures and evaluate them by solving some example problems with known solutions
2. To investigate how the existing analysis software can be used as a black box for optimal design of structures.
With the proposed formulations, the structural equilibrium equations need not be solved explicitly during optimization iterations.These equations become equality constraints and it will be seen that all the calculations can be performed at the member level or the node level. This is a major advantage in the optimization process in terms of efficiency and numerical implementation. In addition, all the design constraints become explicit in terms of the optimization variables. Thus, no design sensitivity analysis of the problem functions is needed, which is required with the conventional formulation. This is also a major advantage of the alternative formulations, which makes it easier to use optimization methods with the existing analysis software. One drawback of the alternative formulations is that the optimization problem becomes quite large, although the problem functions are quite sparse. This feature must be exploited in the optimization process for efficiency of calculations (Gill et al. 2002_).
It is noted that a major objective of the current work is to develop and evaluate different optimization formulations using existing design examples with known solutions. Therefore, definitions of the design variables and constraints are taken from the published literature so that a direct comparison of the solutions can be made. It is realized that this definition of the design optimization problem has limitations because not all the design code constraints can be imposed for practical applications. However, this restricted problem definition can still be useful at the preliminary design stage of the structure. The alternative formulations can be applied to more practical frame design problems which will be considered in the future work..
Optimal Design Problem Statement
The optimization problem is to find a design variable vector representing member sizes to minimize a cost function, which may be volume or weight of the structure subject to the design constraints that are imposed as inequality constraints on stresses, displacements and design variables.
Design Variables
For a frame member, the design variables can be the cross sectional dimensions, i.e., depth and width of a rectangular section,radius of a solid circular section, and so on. To calculate the nodal displacements and member stresses, the cross sectional area, the moment of inertia and the section modulus of the member are required. Areas or moment of inertias of members are also popular to be chosen as primary design variables, while expressing other cross-sectional properties in terms of them by explicit nonlinear relationships. A suggested form of the relationships among the cross-sectional properties, where the member cross-sectional area A is treated as the only design variable, is (Saka 1980)