An Optimal Design of Beam with Variable Cross Section : Design of Cantilever by Dynamic Programming

Abstract

The problem of determining the shape of an elastic cantilever with a cross section varied stepwise is discussed. The criterion is the minimization of the strain energy of the cantilever under the constraints of given length and volume. The beam is subject to the load distributing proportionally to the power of the distance from the free end. The length of each beam segment as well as the size of its cross section is taken into consideration as the design variable. The dynamic programming procedure is used to obtain a recursive equation which is modified into an appropriate non-dimensional one to decide optimal values independently of the magnitudes of constraints (or, total volume and length) at every decision stage. Beams with a rectangular cross section of a constant width and beams with a circular cross section are treated in numerical examples. Moreover, from a viewpoint of optimization through discretization the results are compared with the analytically optimized beams with a continuously variable cross section.