A unified Ritz framework for free vibration analysis of shells

Abstract

This paper presents a unified Ritz-based framework for the free vibration analysis of shells, integrating symbolic and numerical computation to automatically evaluate shell geometry from its position vector. Unlike many existing Ritz-based models that are restricted to specific shell geometries, the proposed approach overcomes these case-specific limitations by automating the derivation of differential geometry parameters, enabling the analysis of a broad range of shapes within a single formulation. It employs enriched trigonometric sets to handle various boundary conditions. The modular framework allows comparison of different kinematic models, where a seven-parameter formulation with Enhanced Assumed Strain effectively mitigates locking effects in moderately thick shells. A key advantage of this methodology is its high computational efficiency, achieving converged results with significantly fewer degrees of freedom compared to high-fidelity 3D finite element models. While current applications are focused on linear problems within regular computational domains, the formulation is conceived to enable future extensions to complex nonlinear analyses. The formulation robustness is demonstrated by numerical examples on functionally graded and laminated composites, showing strong agreement with 3D finite element models.