Spectral Element Analysis of Bars, Beams, and Levy Plates

Abstract

This thesis is primarily concerned with the development and coding of a Levy-type spectral plate element to analyze the harmonic response of simply supported plates in the mid to high frequency range. The development includes the governing PDE, displacement field, shape function, and dynamic stiffness matrix. A two DOF spectral Love bar element and both a four DOF spectral Euler-Bernoulli and a four DOF spectral Timoshenko beam element are also developed to gain insight into the performance of spectral elements.

A cantilever beam example is used to show how incorporating eigenfunctions for the dynamic governing PDE into the displacement field enables spectral beam elements to represent the structural dynamics exactly. A simply supported curved beam example is used to show that spectral beam elements can converge the effects of curved geometry with up to a 50% reduction in the number of elements when compared to conventional FE. The curved beam example is also used to show that the rotatory inertia and shear deformation, from Timoshenko's beam theory, can result in up to a 28% shift in natural frequency over the first three bending modes.

Finally, a simply supported Levy-plate model is used to show that the spectral Levy-type plate element converges the dynamic solution with up to three orders of magnitude fewer DOF then the conventional FE plate formulation. A simply-supported plate example problem is used to illustrate how the coefficients of the Fourier series expansion can be used as edge DOF for the spectral Levy-plate element. The Levy-plate element development gives insight to future research developing a general spectral plate element.