This paper focuses on the analytical procedure for determining novel exact expressions for internal forces and displacements of hemi-ellipsoidal shells formed as an axisymmetric shell of revolution under uniformly distributed load such as imposed loads. The simplest form of expressions for solutions is derived based on the linear membrane theory under symmetrical loading that is formed as a shell of revolution. The results have been validated and have a good consistency with numerical solutions from the finite-element method (FEM) which are derived based on the principle of virtual work and differential geometry. The obtained analytical exact solution is only valid for small displacements or if the response does not exceed the linearity limit. In cases of large displacements, geometrical nonlinear finite-element analysis is recommended to determine the solution. The linearity limit determination is demonstrated, and the effects of shells’ geometry, thickness, and magnitude of applied loads are presented. Additionally, the linear buckling analysis has been performed. The study found that the size ratio, thickness, and support condition have a significant effect on the critical load of the first mode, and the hemispherical shells have the highest buckling resistance due to the geometry.

This paper focuses on free vibration of hemi-ellipsoidal shells with the consideration of the bending rigidity and nonlinear terms in strain energy. The appropriate form of the energy functional is formulated based on the principle of virtual work and the fundamental form of surfaces. Natural frequencies and their corresponding mode shapes are determined using the modified direct iteration method. The obtained results, which show a close agreement with previous research, are compared with those obtained based on the membrane theory. The effect of the support condition, thickness, size ratio, and volume constraint condition on frequency parameters and mode shapes is demonstrated. With the bending rigidity, shell thickness has a significant impact on the frequency, especially in higher vibration modes and in shells with a considerable thickness but the frequency parameter converges to that determined by using the membrane theory while the reference radius-to-thickness ratio is increasing. In addition, accounting for the bending rigidity solves the issue of determining natural frequencies and mode shapes of the shells using the membrane theory without the volume constraint condition. The obtained results also indicate that the free vibration analysis with bending is essential for the hemi-ellipsoidal shell with a base radius-to-thickness ratio of less than 100, which gives over 2.84% difference compared with that of the shell derived by membrane theory, and this allows engineers to perform the analysis in more applications.