Modified Mixed Variational Principle and the State-Vector Equation for Elastic Bodies and Shells of Revolution

A modified mixed variational principle is established for a class of problems with one spatial variable as the independent variable. The specific applications are on the three-dimensional deformation of elastic bodies and the nonsymmetric deformation of shells of revolution. The possibly novel feature is the elimination in the variational formulation of the stress components which cannot be prescribed on the boundaries. The result is a form exactly analogous to classical mechanics of a dynamic system, with the equations of state exactly in the form of the canonical equations of Hamilton. With the present approach, the correct scale factors of the field variables to make the system self-adjoint are readily identified, and anisotropic materials including composites can be handled effectively. The analysis for shells of revolution is given with and without the transverse shear deformation considered.

Nonsymmetric Buckling of Plate-End Pressure Vessels

Small cylindrical pressure vessels are often constructed with a circular flat plate end closure. The end-plate undergoes large deformations under working loads. High local stresses develop at the junction between the cylinder and end-plate, causing yield under proof loading. The compressive circumferential stresses at the junction may lead to bifurcation into a nonsymmetric deformation mode. This study explores the geometrically nonlinear elastic-plastic behavior of plate-end pressure vessels. The form of the axisymmetric prebuckling path is investigated, showing the strongly stiffening nature of the response. Bifurcation of the closure into a nonsymmetric mode is then studied.

Nonsymmetric Deformation of Dome-Shaped Shells of Revolution

Bending solutions that are uniformly valid in both shallow and nonshallow regions of a dome with arbitrary meridian are determined for edge loads that vary sinusoidally in the circumferential direction. The membrane and inextensional deformation solutions are obtained in terms of a function which satisfies a simple integral equation which eliminates the usual trial-and-error method of isolating the solutions that are regular at the apex. For a specific application, curves and formulas are obtained for the stresses and deformations of a dome with rigid rings clamped to the edges under the action of axial force, side force, and tilting moment.