Closed-form formulations of 2-D higher-order shear deformation theories (HOSTs) for the free vibration analysis of simply supported cross-ply laminated composite and sandwich doubly curved shells are presented. The formulation includes the Sanders theory for doubly curved shells. Two of the HOSTs account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of transverse shear deformation. In these developments, a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique and by solving the eigenvalue equations. Numerical results are presented for the natural frequencies of laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plates and shells are compared with the available 3-D elasticity and analytical solutions and it is believed that the solutions for sandwich laminates using various higher-order displacement models will serve as the benchmark in future.
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