A new generalized approach for implementing any homogeneous and non-homogeneous boundary conditions in the generalized di erential quadrature analysis of beams

In this paper, a new way of implementing any homogeneous and nonhomogeneous boundary conditions in the Generalized Dierential Quadrature (GDQ) analysis of beams is presented. Like analytical methods in the solution of a dierential equation, this approach governs the general solution of GDQ discrete equations for the dierential equation of beams by assuming some unknown constants, and satises the boundary conditions in the general solution. Then, unknown constants are evaluated by solving the resultant algebraic equation system. Thus, the particular solution for the beam equilibrium dierential equation is obtained by the GDQ method. As described, this approach satises the boundary conditions in the general solution, so, it is referred to as SBCGS (Satisfying the Boundary Conditions in the General Solution). The SBCGS approach can satisfy any type of boundary condition exactly at boundary points with high accuracy and can easily be implemented for each type of boundary condition. So, this approach overcomes the drawbacks of previous approaches by its generality and simplicity. At the end of this paper, a comparison of the SBCGS approach, using the method of substitution of boundary conditions into governing equations (the SBCGE approach), is made by their accuracy with the analysis of beam equilibrium under lateral loading with combinations of simply supported and clamped boundary conditions. Other boundary conditions and dierent numbers of mesh point results are also discussed for the SBCGS approach only. The results indicate that although the SBCGS approach is essentially very similar to some other approaches, like SBCGE, it is an easy and powerful method for implementation of any boundary condition to the GDQ governing equations, and provides highly accurate results.