A Smoothed Point Interpolation Method for the Timoshenko Beam

This paper aims to extend the meshless smoothed point interpolation methods (SPIMs) to the analysis of the Timoshenko beam problem. These methods are based on the concepts of smoothing domains and weakened-weak (W2) form; their use is made possible by the extension of the weakened-weak form that they are based on to the case of the Timoshenko beam. The provided numerical simulations emphasize that, by changing the number of nodes used to build the shape functions at each interest point, it is possible to obtain a lower-bound or an upper-bound approximation to the analytical solution of the beam problem. This property is further exploited with the concept of αPIM shape function that, blending together the different bounds to the analytical solution, allows to improve the convergence. The proposed formulation is naturally locking-free, i.e., no additional treatment is necessary to avoid the spurious stiffer behavior that commonly occurs in FEM simulations of shear-deformable beams.