An investigation concerning optimal design of solid elastic plates

Abstract

We consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory. Assuming the material volume and plate domain to be given, we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account. On the basis of a convenient tensorial formulation of the problem, where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity, we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained.These numerical results enable us to investigate the optimization problem itself in terms of its major parameters, particularly the maximum and minimum values specified for the plate thickness. For problems associated with large ratios between these constraint values, plate designs with significant integral stiffeners are obtained. We find however, that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities. In order to determine a global optimal solution associated with given thickness constraint values, it is therefore necessary to change the optimal design formulation. With implications for a number of similar two-dimensional optimization problems, our results offer valuable indications of the lines along which such changes should be performed.