Nonlinear Dynamics and Stability of Shallow Spherical Caps Under Pressure Loading

Abstract

The post-buckling and nonlinear dynamic response of shallow spherical caps subjected to external pressure is analyzed. The Novozhilov's nonlinear thin shell theory is used to express the strain–displacement relations. Following the Rayleigh-Ritz method, the displacement fields are expanded using a mixed series: Legendre polynomials in the meridional direction, harmonic functions in the circumferential direction. Once the linear analysis is completed, the displacement fields are re-expanded and the nonlinear dynamic model is obtained by using the Lagrange equations. The response of clamped caps, made of isotropic and homogeneous material, is investigated. The bifurcation analyses of equilibrium points and periodic orbits are presented by using continuation techniques. Benchmark results are provided in terms of natural frequencies and critical buckling loads. The dynamic effects due to the interaction between static and dynamic pressure are investigated. Numerical results pointed out that, under particular load conditions, dynamic bifurcation results in nonnegligible asymmetric states activation in the response of the structure.