Keywords: optimization, weight minimization, earthquake resistant, seismic resistant, seismic analysis, equivalent static loading, optimality criteria, steel frames, sensitivity analysis.

In this paper the general formulation of optimal design of two-dimensional steel frames under seismic loading is presented. The design problem is formulated in the form of a standard optimization problem in which the weight of structure has been taken as the objective function and design requirements have been treated as the optimization constraints. The constraint set comprises all design limitations that are provided by design code, AISC-ASD. The constraints include limits on stresses, deflections, side- sways, inter-storey drifts, and upper and lower bounds on member sizes.

The cross sectional areas of members are taken as design variables and an exponential relation of the type is used to link the moment of inertia I, of any member to its cross sectional area A. Parameters a and ß are obtained via a curve fitting process for the ruled sections. The same procedure is followed to obtain the relation between section moduli S and cross section area A.

The "Equivalent Static Loading" scheme, which is used in this paper for determination of seismic loading, is described according to Iranian Seismic Code[1]. Analysis is done subject to various code-prescribed combinations of gravity and seismic loading. Upon analysis results, the design is checked and a set of potentially active constraints is selected. To that end, two margins of activity are used for the response ratios of stress and deflections. The truncated first order Taylor series expansion is then used to explicitly express the selected constraints in terms of design variables. The derivatives of constraints are obtained using proper sensitivity analysis. In the sensitivity analysis the effect of change in cross-sectional areas on gravity load is found negligible and ignored; however the variation of seismic loading due to change in design variables is considered. The optimality criteria method of optimization is briefly described and resizing formula for updating member sizes is introduced. Since resizing formula depend on dual variables (Lagrange multipliers) various methods for evaluation of these variables have been proposed in the literature. It is shown in this paper that the two frequently used methods of obtaining Lagrange multipliers, namely the "Gauss-Seidel" and "Sequential Reduced Linear Equation" methods may end up to incorrect answers. Perhaps, this is the main reason for a kind of divergence and discontinuity that sometimes occur in the design improvement in the OC method. To overcome to this difficulty, a Quadratic Programming sub-problem is proposed and incorporated within the (OC) algorithm. This modification remedies the drawbacks of current (OC) algorithms and results in monotonic reduction in the objective function and rapid convergence.

Two examples are solved to exhibit the solution procedure and the practicality of the proposed method. Example 2, clearly shows that the proposed algorithm tries to increase the first natural period of the frame, decrease the lateral forces and doubly reduce the weight of structure. Since the solution procedure in (OC) algorithm has small dependence to the number of design variables and its inherent deficiency is remedied, it can be concluded that this modified (OC) algorithm is a proper optimization engine for design of large scale structures especially those under seismic loading.

1

"Iranian Code for Seismic Resistant Design of Buildings", Building and Housing Research Centre, Tehran, Iran, 1988

2

Cheng, F.Y., "Development in Optimal Seismic Structural Design", in the Proceeding of P.R. China -U.S.A.-Japan Trilateral Symposium/ Workshop on Engineering for Multiple natural Hazard Mitigation, Beijing, China, PP. E10-1- E10-24, 1985.

3

Kramer, G. J. E. and Grierson, D. E., "Computer-Automated Design of Structures under Dynamic Loads", Journal of Computers and Structures, Vol. 32, No. 2, pp 313-325, 1989. doi:10.1016/0045-7949(89)90043-6

4

McGee, O.G. and Phan, K.F., "Adaptable Optimality Criterion Techniques for Large-Scale Space Frames with Multiple Frequency Constraints", Computers & Structures, Vol. 42, No.2 PP. 197-210, 1992. doi:10.1016/0045-7949(92)90204-D

5

Memari, A.M. and Madhkhan, M., "Optimal Design of Steel Frames Subject to Gravity and Seismic Codes' Prescribed Lateral Forces", Structural Optimization Vol. 18, PP. 56-66, 1999. doi:10.1007/s001580050068

6

Moharrami, H., and Grierson, D.E., "Computer Automated Design of Reinforced Concrete Frameworks", ASCE, Journal of Structural Engineering, Vol. 119, No. 7, PP.2036-2058, 1993. doi:10.1061/(ASCE)0733-9445(1993)119:7(2036)

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