A necessary condition for collapse during response to a strong earthquake excitation is that the effective tangent stiffness matrix ceases to be positive definite at some point in time. Loss of positive definitiveness does not necessarily imply that collapse will occur, however, because the velocity pattern at the time the effective stiffness loses positiveness does not generally coincide with the velocity distribution associated with the unstable mode and the resistance to the transition from one shape to the other - which is provided by inertia (and to a lesser extent damping) - gives a temporary stabilizing effect. A negative eigenvalue in the effective tangent stiffness is thus necessary, but it is not sufficient for instability to occur during dynamic response.  

Constraints that must be satisfied in the design of multistory structures ensure that the lowest eigenvalue of the second-order elastic stiffness is always substantial. During strong ground motion, however, plasticity develops, the lowest eigenvalue decreases, and statically unstable configurations can be reached, opening the possibility for collapse. To determine if a building with a given plasticity distribution is or is not statically stable it is necessary to determine how plasticity affects the tangent stiffness. In this study we assume that any degradation in the post-yield stiffness resulting from cyclic loading is compensated by strain hardening so that both effects can be discarded. With these assumptions we’re able to examine the stability of a given distribution of plasticity without the need to associate it with a specific response history or strain hardening. For practicality a lumped plasticity model is used. As noted, while plasticity configurations associated with kinematic mechanisms are always unstable, unstable configurations can be reached with plasticity distributions that do not render the system a mechanism.

The theme explored in this work can be put forth by considering the situation where one is interested in increasing the scaling of a ground motion needed to induce instability of a building by some factor a. The objective can be attained by increasing the yield strength of all members by a but one suspects this is not an efficient solution. Indeed, one expects that a more efficient alternative would be to distribute the increases in strength strategically so that the distribution of plasticity that governs shifts from the original to a more favorable one. In this work we refer to the foregoing scheme as the use of “local modifications”. We focus on the case of buildings that are tall enough to reach statically unstable configurations without the kinematic mechanisms as these are the ones that where dynamic instability is more likely to become a relevant issue. We judge the effectiveness of local modifications by tracking how the eigenvalue of the effective tangent stiffness changes and, after settling on a modification, compute the intensity of the ground excitation that leads to collapse using nonlinear second order time history analyses to confirm the anticipated effectiveness.