PSI - Issue 78

Tahir Ahmad et al. / Procedia Structural Integrity 78 (2026) 631–638

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Fig. 5. Model of the Steel Storage Rack.

(a)

(b) (d) Fig. 6. Typical Sections: (a) Column upright; (b) Base; (c) Brace; (d) Beam. (c)

3.2. Nonlinear static (Pushdown) and dynamic analysis Nonlinear push-down analysis was performed for two column removal scenarios (namely S 1 and S 2 ): one from a bay without additional braces and another with them (Fig. 7). A displacement-controlled approach was applied, incrementally amplified gravity loads with structural response evaluated by increasing vertical displacement. According to the latest versions of both GSA (2016) and DoD (2024) guidelines, the following gravity loads were applied: LF ×(1.2 DL +0.5 LL ) for floor areas away from the damage scenario, and  N × LF ×(1.2 DL +0.5 LL ) for floor areas above the damage scenario. Many empirical formulas have been proposed in the literature for the dynamic amplification factor (McKay et al. , 2012; Tsai et al. , 2009; Mashhadi, 2017; Ferraioli, 2016, 2018, 2019a-c). However, these approaches often fail to deliver consistent accuracy and efficiency across all operational scenarios. Early design guidelines for progressive collapse recommended using a dynamic amplification factor ( Ω N ) of 2.0. This value was based on the assumption that, under elastic behavior, the maximum dynamic displacement could reach twice the static displacement in the event of a critical column removal. More recent editions of the GSA (2016) and DoD (2024) guidelines have adopted a more refined formulation developed by McKay et al. (2012), which accounts for structural system type, the nature of structural response, and the plastic rotation capacity of the connections. For cold-formed steel systems, a dynamic increase factor of 2.0 is still generally recommended. The analysis incorporated P-Delta and large-displacement effects for a realistic assessment of collapse resistance. To capture the transient effects of sudden column removal, a nonlinear dynamic (ND) analysis was also conducted using the load combination LF ×(1.2 DL +0.5 LL ). This involved a three-step procedure: To apply gravity loads to the intact structure, recording column end forces, then replacing the column with equivalent reaction forces ( – N , – V , – M ), applying amplified gravity loads ramped over 1 second and held for 25 seconds (longer due to rack flexibility for stabilization) and finally, removing reaction forces, simulating sudden column loss and capturing inertia and force redistribution. The resulting pushdown curves (Load factor vs vertical displacement in the location of the removed column) are shown in Fig. 8. The nonlinear static analysis of the two column removal scenarios reveals distinct structural responses. In Scenario S 1 , a gradual load redistribution occurs, characterized by out-of-plane deformation of the beams under large displacement conditions. This behavior activates the typical stiffening observed in the pushdown curve due to catenary effects. The first brace connection fails at around 3 mm, then the first beam connection yields at around 7 mm, and finally, the beam connection fails at 25 mm, reflecting a resilient response. In contrast, Scenario S 2 - which included additional bracing in the adjacent bays - exhibited early and simultaneous yielding of the first beam and the initial brace connection at around 6 mm of displacement. The dynamic response (Fig. 9) further highlighted this difference: S 1 experienced a larger vertical displacement (27 mm) with prolonged oscillations due to effective energy dissipation, whereas S 2 showed a rapid, limited displacement (2 mm) and quick stabilization, confirming its stiffness-driven, brittle collapse.