The Stability Problems for Bars and Plates

The stability problems for bars and plates are considered. Variation formulations are used for the stability problems. The positive definiteness of the potential energy functional is studied. The transition from the three-dimensional stability problem to the corresponding problems for bars and plates is executed. Concepts of displacements through the section of bar and plate thickness are used for geometrically nonlinear problems. These concepts are derived from the assumptions of the vanishing of the strain through the cross-section plane of the bar or the thickness of the plate. The second variations of non-linear deformations are calculated. The integration through the cross-section of the bar and plate thickness was made, using known formulas for the efforts and the equilibrium equation. The stability functionals for bars and plates are obtained. Comparing to the results known before is conducted. A solution to the test problem for the centrally compressed cantilever beam with a cross section of Pi, which was modeled by plates, is given.