ABSTRACT
A beam is a simple but very common and important structural component. A huge volume of earlier research works have focused on analysis of beams, which is still one of the most essential topics in mechanical engineering training. The finite element method (FEM) is now the mainstream method for analysis of all kinds of problems involving beams [1,2]. In recent years, meshfree methods have also been applied to analyze beams, such as the EFG method for modal analyses of Euler-Bernoulli beams and Kirchhoff plates [3], point interpolation methods (PIMs) [4], MLPG for thin beams [6], and local PIMs (LPIM and LRPIM) for both thin and thick beams [7]. As a beam is one-dimensional (1D) spatially, PIM works perfectly well, and there is no
singular issue in the moment matrix, as long as there are no duplicate nodes. Using PIM shape functions and the Galerkin weak form, discrete system equations can be established easily. In fact the procedure is almost the same as FEM for beams, if a Hermite interpolation is used. The primary difference is that FEMuses only the nodes of the element to create shape functions, but PIM may use nodes beyond the integration cells. When the GS-Galerkin weakened-weak (W2) form is used, we need only the first-order consistence for the assumed functions, as predicted in Chapter 3. We will materialize this by formulating the NS-PIM for the fourth-order differential equations that is capable to produce upper-bound solutions. This chapter deals only with straight beams governed by the Euler-Bernoulli beam
theory. We consider only the bending deformation, and it is assumed that the beam is planar, meaning it deforms only within the x-y plane.
