Limit Load Analysis of a Spring-Supported Shallow Arch of Variable Thickness Given by a Power-Law, Exponential, or Logarithmic Formula

Thin shallow arches may become unstable under transverse loading as the built-up internal compressive forces reach a limiting value beyond which the structure undergoes a sudden large displacement towards a new stable configuration. This phenomenon could be both desirable (in toggle switches) and disastrous (collapse of a dome or truss). Hence, it is important to carry out the so-called snap-or limit-load analysis to reveal the factors influence the phenomenon and to give guidelines in designing structures to behave favorably. Although energy methods are a common means of this analysis, the phenomenon could also be analyzed by considering the equations governing the displacement of the arch and by monitoring the load-displacement characteristic of the structure until it reaches the limit point. This is the procedure adopted here. Researches on the subject mostly consider constant thickness arches with common pin-ended or fixed supports. Here the thickness is varied along the arch in three forms: power-law, exponential, and logarithmic. The supports are considered to be nonrigid fixed; i.e., pinned ends are equipped with torsional springs with constant stiffness. By changing these stiffnesses, various combinations of pinned and/or fixed states, or an intermediate state at each end could be developed. By considering the analytical solution for the transverse displacement for the general power-law case, the limit-load is obtained by numerical solution of the limit-load condition, which is a highly complicated function of the derived displacement field. Several parameter studies, such as that of the effects of shallowness and slenderness of arch and spring stiffnesses on the critical load, are carried out. The results are verified by those of simpler cases available in the literature, as well as those from a finite-element approach.