A hierarchical finite element for geometrically non-linear vibration of doubly curved, moderately thick isotropic shallow shells

A p-version, hierarchical nite element for doubly curved, moderately thick, isotropic shallow shells is derived and geometrically non-linear free vibrations of panels with rectangular planform are investigated. The geometrical non-linearity is due to large displacements, and the e ects of the rotatory inertia and transverse shear are considered. The time domain equations of motion are obtained by applying the principle of virtual work and the d’Alembert’s principle. These equations are mapped to the frequency domain by the harmonic balance method, and are nally solved by a predictor–corrector method. The convergence properties of the element proposed and the in uence of several parameters on the dynamic response are studied. These parameters are the shell’s thickness, the width-to-length ratio, the curvatureto- width ratio and the ratio between curvature radii. The rst and higher order modes are analysed. Some results are compared with results published or calculated using a commercial nite element package. It is demonstrated that with the proposed element low-dimensional, accurate models are obtained