Stochastic Deflection of Axially Loaded Timoshenko Beam-Columns under Random Non-Stationary Dynamic Impact across Different End Conditions for Bridge Piers

Abstract

Traditional structural assessments rely on deterministic Euler-Bernoulli formulations; however, this research derives a time-dependent Timoshenko beam-column model that explicitly accounts for cross-sectional rotatory inertia, transverse shear deformation, and second-order geometric non-linearities caused by persistent superstructure axial loads and accidental impact loads. To capture the unpredictable nature of real-world vehicular or accidental impacts, the lateral collision force is mathematically represented as a non-stationary stochastic process using a filtered Poisson model with lognormal peak amplitudes. To address this research gap in higher-order stochastic dynamics, the investigation is organized into four methodological phases: an analytical framework deriving coupled Timoshenko PDEs with P-Δ effects and filtered Poisson stochastic impact forces; spatial modal analysis; a comparative slenderness ratio (λ) and shear deformation study contrasting Euler-Bernoulli and Timoshenko mechanics; and closed-form expressions for time-dependent dynamic deflection obtained via Duhamel convolution integrals. The resulting models were evaluated across three core outcomes: convergence studies confirming that a 10-mode summation achieves numerical stability (99.66% convergence); comparative slenderness evaluations showing that classical theory underestimates peak deflections by up to 18.59% in squat piers (λ<25); and a sensitivity analysis showing that as persistent axial loads approach critical limits (P/P_cr = 0.65), P-Δ magnification increases lateral deflections by up to 42.3%, with the degree of amplification strongly dependent on the end-boundary conditions. The fully restrained Fixed–Fixed pier configuration shows the greatest resistance to lateral drift, followed by the mixed hinge–fixed and fixed–hinge combinations in second and third place, respectively. The fully articulated Hinged–Hinged pier, in contrast, shows the lowest resistance and the highest vulnerability to lateral drift, triggering dynamic bifurcation and structural instability due to the complete absence of rotational restraint. The derived closed-form solutions give structural engineers a reliability-based design tool for optimizing the impact safety of sustainable bridge infrastructure.