A general lower and upper bound theorem of static stability

A statically stable state of a system subjected to conservative and dissipative forces is considered as a local minimum of the sum of the potential energy and the energy dissipated from the system subject to the kinematic constraints on the system. This stability criterion is investigated by the methods of optimization under constraints. A dual mathematical program, the maximization of the complementary energy of the system subject to equilibrium contraints, is constructed. Bounds on the kinematic state space of a system and energy dissipation are introduced as inequality constraints. Lower and upper bound conditions for the loads causing instability of the system are derived. By the upper bound condition, the system is unstable if the virtual work is negative in a kinematically admissible displacement, including rigid body components. By the lower bound condition, the system is stable if the gradient vectors of the active constraints with nonzero Lagrange multipliers span the space of feasible rigid body rotations. The existence of a nonempty feasible set for the dual program is also found to ensure the stability of the system.