Optimal Design of a Single-Span Beam Under Deflection Constraints

An analytical solution of the initial-boundary value problem on the dynamic stress state of a structure comprising two lap bonded rods of different lengths is proposed. The problem of optimal distribution of beam height along the length of the structure under the maximum deflection constraint is solved. The beam is assumed to be statically determinable and the load is assumed to be arbitrary. The point (or points) at which the beam deflections are maximum are not known in advance and are determined in the course of solving the problem. The optimization criterion is the mass of the beam. The beam deflections are obtained as a result of solving the differential bending equation of a variable-section beam by the finite difference method. The design problem is reduced to finding the required heights of the beam in the nodal point system. Since the displacement constraints of each node are considered independently, the proposed method allows for flexible control of these constraints. The effectiveness of the method for solving optimal beam design problems in the presence of deflection constraints is demonstrated on the example of a model problem.