Abstract
The work is aimed at the construction of an algorithm for studying the equilibrium states of a reinforced plate near critical points, using the first (cubic terms) nonlinear terms of the potential energy expansion. Using geometrically nonlinear analysis of displacement, deformation and stress fields, the Eigenforms of buckling were calculated and bifurcation solutions and solutions for equilibrium curves with limit points were constructed depending on the initial imperfections.
Highlights
Interest in the work of compressed reinforced plates appeared long ago, only starting with the work of Koiter and Kuiken [1], Koiter and Pignataro [2], van der Neut [3], Tvergaard [4], Hunt [5] and at a later time Manevich [6,7,8], systems of equilibrium equations were obtained taking into account geometric nonlinearity, which make it possible to analyze the bearing capacity of the mentioned plate either taking into account the total deflection, or taking into account the interaction of this deflection with local waveforms in the ribs or in the plate
He isolated a regular T-shaped fragment, which was investigated in more detail; some of the results obtained by Tvergaard were used by Hunt [5] to construct the bifurcation surface of the homeoclinic bifurcation point corresponding to the catastrophe of the hyperbolic umbilic
Important results, using equations of the first and second geometrically nonlinear approximations, were obtained in the 80s by Manevich [7,8]. He was able to establish that, limiting himself to the first approximation, it is possible to obtain acceptable estimates of the bearing capacity of a compressed reinforced plate if the critical loads of the wave formation in its elements are close to or exceed the critical buckling load according to the general deflection scheme
Summary
Interest in the work of compressed reinforced plates appeared long ago, only starting with the work of Koiter and Kuiken [1], Koiter and Pignataro [2], van der Neut [3], Tvergaard [4], Hunt [5] and at a later time Manevich [6,7,8], systems of equilibrium equations were obtained taking into account geometric nonlinearity, which make it possible to analyze the bearing capacity of the mentioned plate either taking into account the total deflection, or taking into account the interaction of this deflection with local waveforms in the ribs or in the plate. Important results, using equations of the first and second geometrically nonlinear approximations (taking into account cubic and quartic terms in the expansion of potential energy), were obtained in the 80s by Manevich [7,8] He was able to establish that, limiting himself to the first approximation (only cubic terms are taken into account), it is possible to obtain acceptable estimates of the bearing capacity of a compressed reinforced plate if the critical loads of the wave formation in its elements are close to or exceed the critical buckling load according to the general deflection scheme. One obtains a simplified version of the equilibrium equations:
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