A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem

Abstract

Abstract Owing to its particular characteristics, the direct discretization of the Dirac-delta function is not feasible when point discretization methods like the differential quadrature method (DQM) are applied. A way for overcoming this difficulty is to approximate (or regularize) the Dirac-delta function with simple mathematical functions. By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. On the other hand, it is possible to combine the DQM with the integral quadrature method (IQM) to handle the Dirac-delta function. Alternatively, one may use another definition of the Dirac-delta function that the derivative of the Heaviside function, H(x), is the Dirac-delta function, δ(x), in the distribution sense, namely, dH(x)/dx = δ(x). This approach has been referred in the literature as the direct projection approach. It has been shown that although this approach yields highly oscillatory approximation of the Dirac-delta function, it can yield a non-oscillatory approximation of the solution. In this paper, we first present a modified direct projection approach that eliminates such difficulty (oscillatory approximation of the Dirac-delta function). We then demonstrate the applicability and reliability of the proposed method by applying it to some moving load problems of beams and rectangular plates.

Keywords

• DQM
• Dirac-delta function
• Heaviside function
• modified direct projection approach
• moving load problem
• beams
• rectangular plates