Least-squares Smoothed Shape Functions for Constructing Field-Consistent Timoshenko Beam Elements

This paper presents an approach for constructing field-consistent Timoshenko beam elements using least-squares smoothed (LSS) shape functions. The variational basis for shear strain redistribution is thoroughly explained, leading to the derivation of LSS shape functions for linear, quadratic, and cubic Timoshenko beam elements. These elements are then applied to linear static analysis, bifurcation buckling analysis, and free vibration analysis of prismatic and tapered beams. Numerical tests demonstrate that the LSS-based beam elements effectively eliminate shear locking and provide accurate, reliable results. Their performance is comparable to the discrete shear gap technique but with a simpler implementation procedure. The LSS shape function approach offers a practical and efficient alternative for achieving field consistency in Timoshenko beam elements, with potential applications in enhanced finite element methods (FEMs) such as isogeometric FEM and Kriging-based FEM.