Optimal design of elastic beams under multiple design constraints

Abstract

This paper discusses the optimization of elastic beams under multiple load conditions and self-weight subject to stress and displacement constraints as well as limits on the cross-sectional area and its rate of spatial change (“Niordson constraint”). The general formulation allows for the effect of both bending moments and shear forces on the stresses and deflections. The proposed method is based on static-kinematic optimality criteria which have been successfully used in optimal plastic design. In the above approach, the Lagrangian of the equilibrium condition is regarded as an “associated” (or “Pragerian”) displacement field. The general theory is then illustrated with the example of a built-in beam subjected to stress and Niordson constraints; the statical redundancy of the beam provides a (zero) displacement constraint. Allowance is also made for the cost of clamping moments. It is found that, in general, some segments of the beams are “understressed” and the associated displacement field contains concentrated rotations (“curvature impulses”). Moreover, the solution of this example is found to take on a surprising number of different forms. A beam example with allowance for self-weight will be discussed in Part II of this study.