Linear analytical study of large deflection of simply supported and elastically-bossed plate under pretension

A simplified linear problem of large deflection of a pre-tensioned and elastically bossed plate due to lateral load is studied. von Karman's plate theory of large deflection is employed and extended to a simply supported and symmetrically-layered case with an elastic boss. The governing equations reduce to a modified Bessel equation for the lateral slope, by neglecting the arising nonlinear terms. The analytical solutions were developed by imposing the end support condition and the interface continuity, leading to modified Bessel functions of both the first and the second kinds, in contrast to only the first kind for a flat plate. The recurrence relations of the modified Bessel functions were then employed to derive the lateral deflection and curvature. The central deflection of the bossed plate is further evaluated to study the transition behavior. The approach was implemented with typical silicon-based materials commonly used for a sensing device, for various radial sizes and thicknesses of the center boss and a wide range of pretension, as a parametric study. For a monolithic plate with a narrow and shallow boss under a slight pretension, the solutions agree well with those of classical plate approach under free pretension, thus the presented approach is validated. For typically bossed plates, no edge effect arises around the support end and the geometrical responses show that, the plate mode in low pretension may shift to a membrane mode in a comparatively high pretension. Compared to the available clamped-ended problem, the present study reveals a relatively lower pretension for the plate to change to a membrane mode. The center boss geometry shows an apparent influence upon the structural responses but is limited to a low pretension regime. In this case, a bigger radial size for the center boss may slightly extend the magnitude of pretension for the bossed plate to retain a plate behavior. For a pretension beyond a moderate threshold magnitude, the responses tend to be unified, implying that the pretension effect dominates those of the center boss.