Non-linear free vibration of elliptic sector plates by a curved triangular p-element

A curved triangular p-element is developed and applied to geometrically non-linear free vibration of isotropic elliptic sector plates. The formulation takes into account shear deformation, rotary inertia, and geometric non-linearity. The element can describe the geometry of the elliptic sector plate exactly and is therefore suitable for this type of plate. The shifted Legendre orthogonal polynomials are used as enriching hierarchic shape functions. The element stiffness and mass matrices are integrated numerically using the Gauss–Legendre quadrature. The non-linear equations of free motion are obtained using the harmonic balance method and solved iteratively by the linearized updated mode method. Numerical results for the linear and non-linear frequencies of clamped elliptic sector plates are obtained. The efficiency of the proposed p-element is demonstrated through convergence studies and comparisons with published results. The effects of the ellipse eccentricity, thickness ratio, and sector angle on the backbone curves are examined. It is shown that these parameters influence the hardening behavior.