Optimal Plastic Design of Shallow Curved Beams with Step-wise Varying Cross Sections*

The problem of designing minimum weight thin curved beams with piece-wise constant thickness is studied. The material of the beam is assumed to be rigid-plastic that obeys the yield condition and associated flow law. The influence of geometric changes that occur in the post-yield stage are taken into account. The weight minimization is performed under conditions where the maximum deflection of the beam coincides with the deflection of a reference beam of constant thickness. Necessary optimality conditions are derived with the aid of variational methods from optimal control theory. Numerical results are presented for shallow circular beams with two, three, and four steps in thickness.