Derivative of buckling load with respect to support locations

This paper formulates the derivative of buckling load with respect to intermediate constraint locations. These intermediate constraints include intermediate spring supports and pinned supports. The analysis is based on the generalized energy functional, which includes the product of Lagrange multipliers and boundary conditions. The results show that the derivative of buckling load with respect to the constraint position is proportional to the force between the constraint and the structure as well as to the spatial slope of the associated buckling mode at the constraint position. With the combination of this derivative formula and the Courant maximum-minimum principle, an interesting theorem on the optimal constraint position is proposed.