Institute of Fundamental Technological Research

A mechanical or structural element often buckles when it is compressed beyond a certain amount. Constraints may raise the value of the critical compressive load. In some cases a constraint may completely prevent the occurrence of buckling. This situation is demonstrated here by a simple two-bar model that rests on a rigid foundation and is subjected to pressure and compressive loading. If the system is ‘perfect’, buckling is not possible. However, if the foundation is not flat, or if the load is applied eccentrically, a sudden jump in deflection may occur as the load is increased.

The stability of discrete elastic systems subjected to conservative loading is governed by the potential energy, which is a function of the generalized coordinates. When dry friction is present, it affects the stability conditions and the post-critical response. Quasi-static loading is considered and the normal forces to the friction surface are assumed to be specified. General critical state conditions are obtained in terms of the potential energy and the dissipation function. Then several examples are presented to illustrate various types of behavior. It is seen that limit points, bifurcation points, and transition points (associated with slip activation or deactivation) may occur on the deformation path, and that a sudden “snap” from one state to another is a common phenomenon when the system includes dry friction.

Uni-directional buckling of an elastica with pinned ends is investigated. A rigid foundation below the beam prohibits downward deflection, and a uniform pressure acts on the beam from above. The beam weight is neglected and an axial compressive load is applied. Equilibrium paths are determined for various pressures and the effects of initial curvature and load eccentricity are examined.

A uniform formulation of sensitivity analysis for beams and plates is presented in terms of generalized stresses and strains. Both physical and geometric nonlinearities can be treated within this formulation. Next, optimal design problems for stress and deflection constraints are formulated and the relevant optimality conditions are derived using the concept of a linear adjoint structure. Finally, several numerical solutions of optimal design problems of beams are presented.