Collapse behavior of non-uniform shallow arch under a concentrated load for fixed and pinned boundary conditions

Abstract

The collapse behavior of a non-uniform circular shallow arch is analytically studied for both fixed-fixed and pinned-pinned boundary conditions. The non-uniformity is characterized by three piecewise constant stiffness regions. The equilibrium equations are derived based on the least potential energy principle and the resulting solutions are presented by proper non-dimensionalization in a form independent of total length of the whole arch by identifying two modified slenderness parameters. Detailed parametric study of both stiffer center case and stiffer end case has been conducted to investigate the influence of various geometric parameters and distinctly different collapse behaviors are found for different boundary conditions. Some criteria are derived to identify the snap-through buckling. Moreover two limiting cases of extreme non-uniformity(rigid end case and rigid center case) are studied via the augmented potential energy with introduced Lagrangian multipliers. Through an asymptotical analysis, a finite modified slenderness parameter is shown to suffice to ensure the occurrence of snap-through buckling for rigid end case with pinned boundary conditions when the dimension of center region is approaching to zero. It is shown rigorously that the same conclusion holds even when there is some rotational restraint at the pinned ends. For rigid center case, closed-formed criteria are derived for snap-through buckling. Finally analysis has been carried out on the case where one part of the arch has large bending rigidity but moderate stretching rigidity. This paper aims to enhance the understanding of collapse behavior of non-uniform shallow arch under a central concentrated load for various boundary conditions.