Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells

The hierarchical trigonometric Ritz formulation (HTRF) developed in the framework of the Carrera unified formulation (CUF), for the first time, is extended to shell structures in order to cope with the free vibration response of doubly-curved anisotropic laminated composite shells. The HTRF is the outcome of the combination of advanced shell theories hierarchically generated via the CUF with the trigonometric Ritz method. It is based on so-called Ritz fundamental primary nuclei obtained by virtue of the principle of virtual displacements (PVD). The PVD is further used to derive the governing differential equations and natural boundary conditions. Donnell–Mushtari’s shallow shell-type equations are given as a particular condition. Several shell geometries accounting for thin and thick shallow cylindrical and spherical shells, deep cylindrical shells and hollow circular cylindrical shells, with cross-ply and angle-ply staking sequences are investigated. CUF-based refined shell models are assessed by comparison with the 3D elasticity solution. Convergence and accuracy of the presented formulation are examined. The effects of significant parameters such as stacking sequence, length-to-thickness ratio and radius-to-length ratio on the circular frequency parameters are discussed.