Keywords: finite spectral element, differential quadrature, spectral method, natural boundary conditions.

The paper produces the associated stiffness matrices from the non-symmetric differential matrices of the differential quadrature and the spectral methods for composite structures. For specific differential matrices, the error decreases exponentially with the increasing order of the differentiation matrix or the number of discrete points. So far, only single domain problems have been solved except for simple frame problems and no provision has been given to couple with finite elements or other similar methods. It is the purpose of the paper to form very accurate super-elements in terms of only boundary unknowns to be coupled with general finite elements using differential matrices so that commercial packages can be developed for engineering analyses. The paper will concentrate on linear one, two and three-dimensional composite elasticity, beam and plate problems. The main idea is that, in order to form the stiffness matrix, one must employ the variational consistence natural boundary conditions so that the generalized displacements and forces are dual to each other satisfying the reciprocal theorem. We shall develop the theory in rectangular domains initially and in other shapes using transformation. Thirteen numerical examples are given.

Example 1 uses the differentiation matrix of Trefethen [1] to study the effect of slenderness ratio on the natural frequencies. Example 2 studies the effect of slenderness ratio on the stiffness matrix. Example 3 gives the convergent rate to the natural modes with respect to the differentiation order. Example 4 finds the frequency-buckling compression interaction diagram. Example 5 investigates the multi-parameter buckling of a cantilever column. Example 6 studies a non-uniform beam. Example 7 verifies the natural vibration of a square plate in plane stress. Example 8 considers a clamp-free beam analyzed as a two-dimensional plane stress problem. Example 9 gives an example that cannot be done by the method of differential quadrature of a continuous L plate. Example 10 considers a trapezoidal cantilever plan stress plate. Example 11 finds the natural modes of the union of a solid offset on the top of another solid with a common edge along the z-axis. Example 12 studies a cantilever hexagonal prism and Example 13 consider the free vibration of a quadrilateral plate.

1

L.N. Trefethen, "Spectral Methods in MATLAB", SIAM, Philadelphia PA, USA, 2000.

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