Stability of non-conservative elastic structures under additional kinematics constraints

In this paper, the specific effect of additional constraints on the stability of undamped non-conservative elastic systems is studied. The stability of constrained elastic system is compared to the stability of the unconstrained system, through the incorporation of Lagrange multipliers. It is theoretically shown that the second-order work criterion, dealing with the symmetric part of the stiffness matrix corresponds to an optimization criterion with respect to the kinematics constraints. More specifically, the vanishing of the second-order work criterion corresponds to the critical kinematics constraint, which can be interpreted as an instability direction when the material stability analysis is considered (typically in the field of soil mechanics). The approach is illustrated for a two-degrees-of-freedom generalised Ziegler’s column subjected to different constraints. We show that a particular kinematics constraint can stabilize or destabilize a non-conservative system. However, for all kinematics constraints, there necessarily exists a constraint which destabilizes the non-conservative system. The constraint associated to the lowest critical load is associated with the second-order criterion. Excluding flutter instabilities, the second-order work criterion is not only a lower bound of the stability boundary of the free system, but also the boundary of the stability domain, for all mixed perturbations based on proportional kinematics conditions.