Transverse vibration of thick polygonal plates using analytically integrated trapezoidal Fourier p-element

A trapezoidal Fourier p-element for the transverse vibration analyses of thick plates is introduced. Trigonometric functions are used as enriching shape functions to avoid ill-conditioning problems associated with high order polynomials in a conventional p-element. The element matrices are analytically integrated in closed form. With the enriching Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly improved. A simply supported square thick plate is analyzed by a rectangular Fourier p-element, two trapezoidal Fourier p-elements and the conventional linear finite elements, respectively. The results show that the convergence of the present element is very fast with respect to the number of trigonometric terms and it produces higher accurate modes than the linear finite elements for the same number of degrees of freedom. The proposed element is applied to find the natural modes of the rectangular, skew and trapezoidal plates with different boundary conditions. Since one can always break a triangle into three trapezoids, the range of application of the present element is as good as the triangular element. A triangular, an irregular quadrilateral and two polygonal thick plates are analyzed by the present element. The solutions of the trapezoidal Fourier p-element are in excellent agreement with those of available published methods.