A modified differential quadrature procedure for numerical solution of moving load problem

Abstract

The differential quadrature method is a powerful numerical method for the solution of partial differential equations that arise in various fields of engineering, mathematics, and physics. It is easy to use and also straightforward to implement. However, similar to the conventional point discretization methods like the collocation and finite difference methods, the differential quadrature method has some difficulty in solving differential equations involving singular functions like the Dirac-delta function. This is due to complexities introduced by the singular functions to the discretization process of the problem region. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature procedure in which such singular functions are simply handled. The mixed scheme can be easily applied to the problems in which the location of the singular point coincides with one of the differential quadrature grid points. However, for problems in which such condition is not fulfilled (i.e. for the case of arbitrary arranged grid points), especially for moving load class of problems, the coupled approach may fail to produce accurate solutions. To solve this drawback, we also introduce two simple approximations and show that they can yield accurate results. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the moving load problems of Euler–Bernoulli and Timoshenko beams. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.