A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads

Abstract

There are many physical and mechanical phenomena that can be well described by means of the Dirac-delta function. For instance, the bending and vibration behavior of structures under concentrated loads, impulsive loading, moving loads, and impact loading; and thermoelastic vibration behavior of structures under heat source points can be mathematically modeled by means of the Dirac-delta function. It is well known that such phenomena or problems can be easily handled by weak form based methods such as the Ritz and finite element methods. However, the strong form based methods such as the finite difference and differential quadrature methods may encounter some difficulties in mathematical modeling and treatment of the Dirac-delta function. This is mainly caused by the fact that the properties of the Dirac-delta function are in the form of integrals and not in the form of derivatives. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature method in which such type of problems can be easily and accurately modeled. Its accuracy and reliability are demonstrated through the static and dynamic analysis of beams and rectangular plates under concentrated loads. This paper also presents a simple differential quadrature formulation for the analysis of rectangular plates. The proposed formulation first reduces the original plate problem to two simple beam problems. Each beam problem in then discretized using the differential quadrature method (DQM) in a simple manner. Compared with the conventional DQM, the proposed DQM is superior since its implementation and programming are easier and simpler.