MINIMUM-MAXIMUM DEFLECTION AND STRESS DESIGN OF BEAMS UNDER HARMONIC EXCITATION BY MATHEMATICAL PROGRAMMING

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Abstract
A computational solution procedure is formulated to determine the optimal distribution of the cross-sectional area of a beam subjected to harmonically oscillating loads of the same forcing frequency. The objective function to be minimized is the maximum dynamic deflection or stress of the beam. The cross-sectional area is approximated by splines of order zero or one, and the values of the splines at the knots serve as design variables. The design algorithm consists of successive stages of analysis and optimization. The analysis for a given cross-sectional shape is carried out by iteratively solving an equivalent integral equation formulation of the problem. The optimization stage is carried out by using a quasi-Newton minimization routine. The numerical results for statically determinate and statically indeterminate beams indicate that the maximum deflection and normal stress of optimally designed beams are considerably smaller than those of. uniform beams. The approxtmate result obtained by means of the procedure presented below show good agreement with the exact results available in the case of static loading.