Three-dimensional vibration analysis of a cantilevered parallelepiped: Exact and approximate solutions

Abstract

The paper investigates the hypothetical assumption of neglecting transverse normal stress in vibration analysis for cantilevered thick plates and rectangular parallelepiped. The analysis solves the three-dimensional elasticity energy functional including, as well as excluding, transverse normal stress and obtains free vibration solutions for a cantilevered parallelepiped. Although it is widely accepted, the omission of transverse normal stress is well justified in Kirchhoff–Love thin-plate theory and higher-order thick-plate models; the transverse normal stress effects and thickness extent to which the thick-plate models apply as the thickness increases are practically unknown. The inconsistency of assuming constant transverse normal displacement through thickness for thick-plate models is also addressed. The paper concludes that for a rectangular parallelepiped with thickness exceeding a certain limit, there is considerable discrepancy if transverse normal stress is neglected.