Keywords: dynamic structural analysis, moving load, spectral method, finite strips.

A moving load represents a non-conservative load on the dynamic analysis of structures. The problem is described with a partial differential equation that is characterised by a Dirac function on the right hand side. In the case of a moving mass there is a system of partial differential equations. Solution of such equations can be time consuming in the case of large structures [1].

Prior to the solution procedure, the differential equation is discretized in space and in time where there are many possibilities. Discretization in space is in most cases performed using finite elements. For long structures with one dimension more pronounced, finite strips can offer an approach that gives more acceptable solution times in a sense that one dimension is only approximately discretized [2]. The drawback of that approach is that the finite strip formulation heavily depends on the boundary conditions. Also, in the case of system of partial differential equations all equations have to be discretized in a compatible manner which is sometimes difficult to achieve. Another difficulty with finite strips in dynamic analysis is connected with the formulation of the mass matrix where it is impossible to know in advance which series components are more and which are less significant.

In this paper discretization in space is obtained using the Chebyshev spectral method. The resulting discretization in its form resembles the one from the finite strip procedure but only the stiffness matrix is replaced with spectral differential matrix operator [3]. The spectral differential operator matrix is fast to construct and is small in size. In more dimensions the Kronecker product is used to expand the operator. That type of discretization requires the introduction of some nodal points in the elements used to describe the structure. However the number of points is small since spectral methods achieve high accuracy with just a few points. On the other hand the treatment of boundary conditions is simplified and unified. Lagrange multipliers are used to introduce more complicated boundary conditions.

Discretization in time is usually performed by applying some method from the Newmark family of procedures [4]. In the examples presented a variant of the method described in [1] has been applied.

The paper presents a novel numerical approach for dynamic analysis of structures that seems to perform well when a moving load is present.

1

I. Kozar, "Security aspects of vertical actions on bridge structure", Engineering Computations, 26(1/2), 145-165, 2009.

2

M.S. Cheng, W. Li, S.E. Chediac, "Finite Strip Analysis of Bridges", E & FN Spon, London, 1996.

3

L.N. Trefethen, "Spectral methods in MATLAB", SIAM, 2000.

4

K.J. Bathe, "Finite Element Procedures", Prentice-Hall, 1996.

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