In this paper, a general, strain-consistent, third-order displacement field of shells is proposed. On this basis, the nonlinear theory of composite laminated shells including third-order transverse shear deformation and rotary inertia is developed. Hamilton’s variational principle is used to derive the nonlinear static and dynamic governing differential equations and the corresponding boundary conditions of composite laminated anisotropic general shells under the orthogonal curvilinear coordinates. And the form of spherical shells is degenerated. The model accounts for the cubic variations of the in-plane displacements through the thickness (thus, does not require any shear correction coefficients), stretching of transverse normals, the von karman nonlinear strain-displacement relations, initial geometric imperfection, rotary inertia, elastic foundations and viscous damping in order to simulate the real structures. The corresponding constitutive equations of composite laminated shells are derived. This paper uses the global orthogonal interior collocation method in space domain and the unconditionally stable Newmark-Beta and Houbolt numerical integration schemes in time domain to comprehensively investigate dynamic axisymmetric behavior of laminated cylindrically orthotropic composite shallow spherical shells subjected to suddenly applied loads. The sufficient condition for dynamic buckling, from the energy transfer consideration, is defined as the smallest load for which an unbounded motion is initiated at one generalized displacement, and the pressure corresponding to a sudden jump in the maximum volume change in the time history of the shell structure is taken as dynamic buckling pressure herein. For the sake of comparison, the results of first-order transverse shear and classical shell theories are presented in this paper. Results of detailed parametric imperfection, the total number of layers, lamination schemes, viscous damping, loading type and loading duration, the loaded area, loading rates, Winkler-Pasternak elastic foundations, opening semi-angle of circular hole, aritifical viscosity of difference scheme, boundary conditions, rotary inertia, transverse shear deformation, h/R (thickness/radius ratio of shells) and dynamic buckling criterion on the buckling loads. Previously the researches on shallow spherical shells were based on shallow shell assumptions. But in this paper, these assumptions are abandoned, thus the theoretical model can be used to study shallow spherical shells, as well as hemi-spherical shells and complete spherical shells. The semi-analytical solutions offered by this paper will provide bench-mark numerical results to test the accuracy of numerical methods. This paper clears up the confused ideas about the different effects of transverse shear deformation on the shallow and deep spherical shells in references.

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