On the Masur Paradox

It was shown by Barnett, Shield, and Prager in the sixties that in optimal elastic beam design for a deflection constraint, the continuously variable stiffness is proportional to the square root of the product of the real and virtual beam moments. In the case of a deflection constraint at a single point, the virtual moments equilibrate a unit load at that point. As Masur pointed out in the early seventies, the above optimality condition implies that the real and virtual beam moments must everywhere have the same sign. Tn view of the fact that the corresponding real and adjoint displacement fields must be kinematically admissible, Masur justifiably concluded that a solution satisfying all the above requirements may not exist for some problems. An explanation of this apparent paradox, involving an unusual type of singularity in the optimal solution, is given in this paper. It is also shown that mathematical programming (MP) methods do not yield a close approximation of the exact solution for the class of problems considered, while iterative optimality criteria (OC) methods fully confirm the above findings