PSI - Issue 78

Filippo Dringoli et al. / Procedia Structural Integrity 78 (2026) 395–403

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1. Introduction Dynamic instability refers to the phenomenon in which the seismic response of a structure transitions suddenly from oscillatory motion to continuous, unbounded displacement in one direction. Because this limit state typically leads to total structural collapse, accurately identifying and characterizing it is crucial for performance-based seismic design. A defining characteristic of this instability is its abrupt onset. As demonstrated by research conducted by Husid (1967), Jennings and Husid (1968), Takizawa and Jennings (1980), and Bernal (1990) on simple structural systems, as well as by Bernal (1992a, 1992b, 1998) on more complex, multistory models, the gravitational effect on inelastic seismic response generally remains minor except within a narrow range of structural strengths. Within this critical strength range, seismic responses can grow without limit. This phenomenon is illustrated clearly in Figure 1, which presents the maximum displacement response at the roof of a twenty-story building model subjected to progressively increasing ground motion amplitudes. The comparison between first-order and second-order nonlinear dynamic analyses, as depicted in the figure, highlights that the influence of P-Delta effects on maximum displacement remains negligible, except near the critical strength threshold, where instability initiates.

Fig. 1. Maximum displacement response at roof level in a 20-story model of a steel frame.

In this work, we focus on tall buildings, since they can reach statically unstable configurations without forming a kinematic mechanism, as these are most susceptible to dynamic instability. That is to say, while plasticity configurations associated with kinematic mechanisms are always unstable, instability can also arise from other distributions of plasticity. We define a structure as stable under a given excitation if small increases in the input magnitude lead to correspondingly small variations in the response. To assess static stability, one can examine the eigenvalues [λ] of the effective tangent stiffness matrix [K e ]. Specifically, since these eigenvalues represent generalized measures of stiffness along the directions of their associated eigenvectors [ ϕ], static stability requires that all values in [λ] remain positive. A more intuitive way to express this is that a structure remains stable if transitioning between any two equilibrium states necessitates an external force performing positive work. However, statically unstable configurations can arise during dynamic motion without necessarily leading to collapse, meaning that the presence of negative eigenvalues is a necessary but not a sufficient condition for dynamic instability. A mathematical analysis of how inertia generates transient stabilizing forces that enable the system to persist through statically unstable configurations can be found in Bernal (1998). In multi-story structures, design constraints ensure that the second-order elastic stiffness maintains a relatively high minimum eigenvalue, well above zero. However, during intense seismic excitation, the onset of plasticity leads to a reduction in stiffness, the extent of which depends significantly on the spatial distribution of plasticization (i.e., plastic hinges). Certain distributions may result in negative eigenvalues, indicating statically unstable configurations. This paper examines variations in the effective stiffness matrix and its corresponding minimum eigenvalue as a function of different plastic hinge locations within a frame structure. The aim is to identify hinge configurations most