Optimal Elastic Design with Constrained Taper: Prescribed Deflections

Abstract

This paper concerns a general theory for elastic structures with prescribed deflections and constrained ``taper'' (= rate of spatial change of the cross-sectional area). The proposed theory is applied to minimum weight beams with a linear relationship between the flexural stiffness and the cross-sectional area. This application is then illustrated through an example of a cantilever beam with constrained taper, having a prescribed slope at its free end. As this structure is statically determinate, the ``associated'' displacement field (that is, the Lagrangian for the equilibrium condition) is not employed directly in obtaining the soptimal solution. However, it can be used for checking the validity of the solution through a dual formulation. In this example, the optimal length of the section with constrained taper is given by another Lagrangian. In addition to verifying them through the dual formula, the results are checked by another independent method (differential calculus for an assumed class of solutions). Finally, it is pointed out that in optimizing statically indeterminate elastic structures, the solution can be obtained directly by considering the associated (Pragerian) displacement field.