Development of a 1D complex cross-section beam model and the local differential quadrature method for static and dynamic analysis

Abstract

A strong-form general one-dimensional (1D) higher-order prismatic beam model is proposed for the mechanical analysis of straight beams with arbitrary cross-sections. This model accommodates complex cross-sections by interpolating displacement fields at cross-sectional grid points. It introduces two cross-sectional property matrices to accurately describe both the complex geometry and material properties. To enhance numerical efficiency, a local differential quadrature method (LDQM) is refined from the global differential quadrature method. The LDQM incorporates a local bandwidth parameter, optimizing stability and memory usage for numerical differential approximations. However, the optimal selection criteria for this bandwidth parameter remain to be fully established, as excessive bandwidth values may insignificantly improve accuracy while substantially increasing computational costs. Using the LDQM, the proposed 1D model analyzes displacements and stresses in both static and dynamic (transient and steady-state) problems for beams with complex cross-sections. The accuracy of the 1D model and LDQM is verified through extensive computational comparisons with three-dimensional (3D) solid and plate models. Additionally, a comprehensive convergence analysis is conducted for natural frequencies and static displacements. The convergence curves highlight the advantages of the proposed 1D model over conventional 3D solid and plate finite element models. It should be noted that the cross-sectional mesh discretization strategy requires careful consideration—our numerical experiments demonstrate that overly refined meshes, particularly higher-order elements in slender T-beams, often yield diminishing returns in accuracy improvement relative to their added computational burden. Finally, a static stress analysis assesses the performance of the LDQM in terms of stress smoothness. The numerical results reveal that the LDQM avoids the common issues typically encountered with traditional finite element methods.