Optimal design of elastic beams under multiple design constraints (Q1099990)

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.