Divergence instability of kinematically constrained Hencky chains: Analytic results and asymptotic behavior

This paper investigates the stability of any n degree-of-freedom Hencky chain subjected to a full follower force at its tip and under kinematic constraints. We specifically solve this discrete problem and focus on its asymptotic behavior when n tends towards infinity and compare it with its analogous continuous formulation, also referred to as the kinematically constrained Beck's column. The divergence instability load of this non-conservative discrete system under general kinematic constraints is ruled by the so-called second-order work criterion, which involves the symmetric part of the stiffness matrix. The application of the second-order work criterion to the discrete repetitive system allows to find the exact divergence load pn under kinematic constraints, whatever the size of the discrete system. The critical value pn is obtained as the smallest positive root of explicit transcendent equations for both the generic (an not equal 0) and the singular (an=0) cases, these two cases being illustrated by different values (respectively 2C and C) of the stiffness of the elastic torsion spring at the basis of the discrete system. The corresponding destabilizing kinematic constraint is also calculated. Asymptotic derivations show the convergence of the dimensionless buckling load towards the continuous one, equal to pi(2), and give the corresponding destabilizing kinematic constraint of the Beck's column.