PSI - Issue 78

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

397

critical to stability. These analyses will help determine the most vulnerable areas of the structure, specifically those that contribute most to the reduction of the minimum eigenvalue during plasticization. To focus on plastic hinges layout only, we assume that any degradation in the post-yield stiffness resulting from cyclic loading is compensated by strain hardening. This allows us to discard both effects and focus on plasticity distributions, independently of response history. The study first evaluates the effectiveness of local strengthening modifications by tracking changes in the minimum eigenvalue; then, it confirms results with nonlinear second-order time history analyses, computing the intensity of the ground excitation that leads to collapse in the structural configuration with local modifications. These analyses are performed numerically on a case study of a real high-rise steel frame, based on its actual structural design. Based on the findings, the study proposes the implementation of targeted and effective structural strengthening interventions, focusing on the most critical elements, as a cost-effective way to withstand and/or delay instability during high-intensity seismic excitations, increasing the number of plastic hinges required before the structure reaches an unstable configuration. . 2. How plasticity influences Static Instability (5-Story, 20-Story frames) As inelasticity progresses, the eigenvalues of the stiffness matrix gradually decrease, ultimately leading to structural instability when the lowest eigenvalue transitions from a positive to a negative value. To characterize and model this inelastic behavior, this study adopts a concentrated plasticity approach, representing inelastic deformations through plastic hinges. Simplified models are analyzed and subsequently employed to evaluate the variation of the effective stiffness matrix as a function of axial loads through the introduction of plastic hinges within the frame. Figure 2b presents the variation of eigenvalues as plastic hinges progressively form in a single-span, five-story frame with a pinned base. The formation process of plastic hinges is halted once a negative eigenvalue is reached, indicating structural instability. The graph illustrates the evolution of eigenvalues for two different plasticization scenarios. In the first configuration, Figure 2a, the process is accelerated: the plasticization of the columns (highlighted in red) induces a local kinematic mechanism, leading to structural instability with only six plastic hinges. In contrast, in the second configuration, Figure 2c, a negative eigenvalue, corresponding to an unstable configuration, occurs when a global kinematic mechanism forms, involving all the beams and requiring ten plastic hinges.

Fig. 2. (a) Local kinematic mechanism; (b) Lowest Eigenvalue due to distribution of plastic hinges for a 5-story; (c) Global kinematic mechanism

In this case, considering a five-story structure, the effects of axial loads are relatively low. As a result, the unstable mechanism coincides with either the local or global kinematic mechanism. However, it is important to note that if the progression of the smallest eigenvalue were analyzed as the number of plastic hinges increased in a twenty-story structure, it would become evident that, under higher vertical loads, an unstable configuration could emerge even in the absence of a global mechanism (Figure 3c).