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Abstract:
This paper investigates the dynamic behaviour of a thick Gao beam subjected to high-velocity inertial moving loads and substantial axial compressive forces. This approach addresses a significant gap by combining geometric nonlinearity, transverse shear deformation, and complete inertial effects (including Coriolis and centrifugal forces).
The mathematical model consists of two strongly coupled nonlinear hyperbolic partial differential equations, solved using the finite element method with space-time integration. Key findings include: (1) nonlinearity produces substantial geometric stiffening effects, with deflections decreasing by factors of 2-3 compared to linear models; (2) supercritical axial compression induces snap-through phenomena and bifurcation between equilibrium states; (3) under combined compression and moving loads, beam deflections are primarily governed by axial force magnitude rather than load weight, with multiple passages producing non-repeating response patterns; (4) maximum accelerations occur neither at mid-span nor at support entry, remaining relatively insensitive to transit velocity. The results indicate a strong detuning between the beam oscillations and the load transition cycles, especially at higher velocities. These findings have important implications for railway bridge design and structures experiencing simultaneous thermal stresses and dynamic vehicular loads, where simplified linear models may significantly underestimate dynamic effects.

Keywords:
Structural dynamics, Extended Gao beam, Moving mass, Inertial load, Non-linear dynamics, Shear deformation

Abstract:
This work derives and numerically investigates a corrected stationary formulation of the extended 2D Gao beam. Starting from the model introduced by Dyniewicz, Shillor and Bajer (Meccanica, 2024), the governing equations are rederived under a simplified loading configuration. The resulting formulation corrects inconsistencies in the previous rotational equilibrium equation and, unlike the previous formulation, reduces exactly to the classical Gao beam when the shear deformations vanish. The extended 2D Gao beam combines the nonlinear features of the classical Gao beam with an independent rotation of the cross sections, as in Timoshenko beam theory, and is therefore suitable for describing moderately thick beams undergoing buckling. The full stationary model consists of three highly nonlinear coupled differential equations supplemented by appropriate boundary conditions. Its variational formulation is derived from the potential energy functional, and the solutions are its stationary points. A reduced model consisting of two coupled equations is then constructed and investigated. Based on its variational formulation, a Finite Element Method (FEM) algorithm is developed and implemented. Five numerical examples illustrate the behavior of the corrected model and identify the loading and parameter regimes in which its predictions are similar to or different from those of the classical Gao beam. The proposed formulation may provide a useful modeling tool for moderately thick beam-like structures, including those in micro-electromechanical systems (MEMS).

Keywords:
Extended static gao beam, 2D beam, Shear, Variational form, Simulation