Finite elements based on Jacobi shape functions for the free vibration analysis of beams, plates, and shells

This article proposes the use of Jacobi polynomials as shape functions for the free vibration analysis of beam, plate, and shell structures. Jacobi polynomials, indicated as P(γ,θ)p, belong to the family of classical orthogonal polynomials, and depend on two scalar parameters γ and θ, with p being the polynomial order. The Jacobi-like shape functions are built in the context of the Carrera unified formulation, which permits the expression of displacement kinematics in a hierarchical form. In this manner, it is possible to adopt several classical to complex higher-order theories with ease. Particular attention is focused on the attenuation and the correction of the shear locking. The results have been compared with analytical results from the literature. For the plate benchmark, analytical results are introduced as the reference results in this article for the first time using the closed form of CUF. Beams, plates, and shells with different thicknesses have been considered. It is demonstrated that the parameters γ and θ are not influential for the calculations.