Finite Element Analysis of Plate and Shell Vibrations

In practical engineering applications, plate and shell structures are typically analyzed using two primary theoretical approaches based on Kirchhoff and Hencky assumptions. The classical thin plate/shell theory derived from Kirchhoff assumptions neglects shear deformation effects, making it particularly suitable for analyzing thin plate/shell structures. Conversely, the thick plate/shell theory based on Hencky assumptions incorporates shear deformation effects, providing accurate results for moderately thick plate/shell structures.

Unlike Hencky's approach, Kirchhoff plate/shell governing equations involve higher-order partial differential equations, requiring C1-continuous interpolation functions for displacement fields in finite element implementation - presenting significant challenges in element formulation. Therefore, this program implements a finite element model using Hencky theory, where the Hencky assumption maintains that normals to the mid-surface remain straight after deformation but rotate about the x and y axes by certain angles, numerically distinct from the first derivative of deflection. The program also incorporates Reissner-Mindlin plate theory, which provides enhanced shear deformation modeling capabilities. In code implementation, this involves specialized element formulation with rotational degrees of freedom and shear correction factors to ensure numerical stability and accuracy.

These theoretical frameworks and assumptions provide engineers and scientists with powerful tools for analyzing and designing plate/shell structures, enabling effective handling of various thicknesses and geometries while deepening understanding of mechanical principles governing plate/shell behavior. The computational implementation includes algorithms for stiffness matrix assembly, eigenvalue extraction for vibration analysis, and numerical integration techniques suitable for different element types.