Some structurally nonlinear problems of stability of elastic systems under one-side-displacement constraints

The problems of stability of elastic systems with unilateral displacement constraints are considered. These problems belong to the contact problems of the theory of elasticity with an unknown region of active interaction of structural elements. Since the mathematical formalization of such problems involves inequalities and non-differentiable functions, they are structurally nonlinear. If the load exceeds a critical value, the elastic system can go to the adjoining state of equilibrium. As a rule, small disturbances lead to large changes in the system state, including the loss of bearing capacity. Unlike the classical case, we need to find and explore the bifurcation points of nonsmooth equations or nonlinear programming problems. The problem of stability of a rod, whose deflection is limited on one side by a rigid barrier, with the boundary conditions of free edge is solved analytically. An analytical solution is also obtained for the problem of stability of rings under the action of central forces or external normal pressure and backed by threads that cannot withstand compressive forces. The axisymmetric problem of stability of a toroidal shell filled with elastic filler loaded by external normal pressure under the assumption that the shell may depart from the filler is solved numerically.