Analysis of the non linear bending and membrane stresses associated to the fundamental non-linear mode shape of fully clamped skew plates at large vibration amplitudes

Abstract

Accurate prediction of the total geometrically non-linear dynamic stress, including both the membrane and bending stresses, is of a crucial importance in the engineering design. A semi-analytical model based on Hamilton’s principle and spectral analysis has been developed recently to study the effects of large vibration amplitudes of fully clamped skew plates. The purpose of the present work was the extension of the model to the analysis of the stresses, including both the non-linear bending and membrane stresses associated to the fundamental non-linear mode shape. It was found that the non-linear frequency increases with increasing the amplitude of vibration, which corresponds to the hardening type effect due to the membrane forces induced by the large vibration amplitudes. The corresponding non-linear bending strains were obtained via the usual strains-displacement relationships, involving partial derivatives of only the transverse displacement function with respect to the space variables. To estimate the non-linear membrane stresses, without having to calculate the in-plane displacements, which would make the model much more laborious and time consuming, a simple practical engineering theory was presented here, which takes into account the contribution of the in-plane displacements u and v in an average sense, along lines parallel to the plate edges. The results show that, at large deflections, higher bending stresses occur near to the clamps, compared with those predicted by the linear theory. Numerical details are presented and comparison of the results obtained here with the ones previously ones treated in the literature shows a satisfactory agreement.