Keywords: differential quadrature, differential quadrature element method, Chebyshev DQ model, Timoshenko beam, two-parameter elastic foundation, weighting coefficients.

The method of differential quadrature (DQ) approximates a partial derivative of a variable function with respect to a coordinate at a node as a weighted linear sum of the function values at all nodes along that coordinate direction [1]. The DQ method has been used to solve many problems. Because only problems having simple regular domains and under simple external environments can be solved by using DQ, the application of this method is very limited.

The author has proposed a differential element method (DQEM) for solving a generic engineering or scientific problem having an arbitrary domain configuration [2]. Like the finite element method (FEM), in this method, the analysis domain of a problem is first separated into a certain number of subdomains or elements. Then the DQ or GDQ discretization is carried out on an element-basis. The governing differential or partial differential equations defined on the elements, the transition conditions on inter-element boundaries, and the boundary conditions on the analysis domain boundary are in computable algebraic forms after the DQ or GDQ discretization. By assembling all discrete fundamental equations an overall algebraic system can be obtained which is used to solve the problem.

The development of DQEM solution of Timoshenko beam structures resting on a two-parameter elastic foundation was carried out. The DQEM uses Chebyshev DQ model to the element-basis discretization [3]. Numerical results solved by the developed numerical algorithms are presented. The convergence of the developed DQEM analysis models is efficient.

1

Bellman R.E., Casti J., "Differential Quadrature and Long-term Integration", J. Math. Anal. Appls., Vol. 34, pp. 235-238, 1971. doi:10.1016/0022-247X(71)90110-7

2

Chen C.N., "A Differential Quadrature Element Method", Proceedings of the First International Conference on Engineering Computation and Computer Simulation, Changsha, China, 25-34, 1995.

3

Chen C.N., "The Timoshenko Beam Model of the Differential Quadrature Element Method", Comput. Mech., Vol. 24, pp. 65-69, 1999. doi:10.1007/s004660050438

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