Fourier p-elements for curved beam vibrations

Abstract

Several Fourier p-elements for in-plane vibration of thin and thick curved beams are presented. Fourier trigonometric functions are used as enriching functions to avoid the ill-conditioning problems associated with high order polynomials. The element matrices are analytically integrated in closed form. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly improved. Furthermore, the elements with enriching shape functions can avoid membrane and shear locking. The vibration of a thin ring, whose exact solutions are available, is analyzed by the present elements. The present elements can compute accurately high natural modes as the higher mode shapes synchronize with the Fourier functions nicely. The free vibration analysis of a number of hinged circular arches with various subtended angles and the tapered cantilever arches having uniform and non-uniform cross-section is carried out as numerical examples. The condition numbers for polynomial p-elements and Fourier p-elements are compared to show the superior numerical stability of the method.