Large Deflections, Loss of Stability and Over-Critical Behavior of Sloping Panels and Arches of Variable Thickness on an Elastic Base

The problem of large deflections of a flat arch in its plane (or an infinitely long panel) loaded with transverse loads is considered. A variational approach has been applied to solve it. The resolving nonlinear equations are reduced to finding the deflection and longitudinal force, which is considered constant along the length of the arch due to its flatness. An approximate solution method is proposed by decomposing the displacements into a Fourier series. The peculiarity of the approach used is that in order to pass the limit points, it is not necessary to use special algorithms such as the continuation method by parameter. It also allows you to trace the process of supercritical deformation of the arch. The proposed approach allows us to consider problems for arches of variable thicknesses, located on elastic supports, on an elastic base with a variable bed coefficient and various loads. Therefore, it is also convenient in problems of finding, for example, the optimal thickness distribution under restrictions on critical loads, on rigidity stiffness and on maximum compression or tension stresses. The results of numerical calculations are presented. The convergence is studied of the solution depending on the number of terms of the series, into which the desired deflection decomposes. A good agreement with the known analytical results is obtained earlier by solving the equilibrium equations of the arch element and panel in the case of simple types of loading. At the same time, even in more complex cases of loading and supercritical bending of an arch without an elastic base, the results are obtained when holding three, four and five members of the Fourier series, the maximum deflections and critical loads differed by no more than 2.5%. On the basis of numerical experiments, the features of the arch behavior caused by the rearrangement of geometry during loading are revealed. The processes of stability loss of the arch and its supercritical behavior are investigated. The effect of symmetric deformation in the case of kinematic loading is found, namely, it is revealed that at a certain value of the concentrated force, an infinite number of equilibrium forms are possible. Arches on an elastic base, as well as with variable thickness are considered (the results are presented in the form of load–displacement diagrams and in the form of pictures of deformed arch shapes). An interesting effect has been revealed based on numerical experiments. It turned out that the increase in thickness when moving away from the supports does not change the nature of the load–displacement diagram, i.e., with some load, a slap occurs. On the contrary, a decrease in thickness when moving away from the supports leads to a flattening of the diagram, and after a certain thickness value in the center, its further decrease leads to the fact that the arch does not occur.