On the stability of reinforced arches

Abstract

This paper deals with the problems of stability of circular arches supported by inextensible threads that cannot withstand compressive forces. Both ends of the thread are attached to the axis of the arch so that the distance between the attachment points cannot increase during deformation. The problems of stability and supercritical behavior of elastic systems in the presence of unilateral restrictions on the movement lead to the need to study the bifurcation points of the equations or to find the parameters for which some variational problem with restrictions on the desired functions in the form of inequalities has a non-unique solution. In the numerical study, this problem is reduced to finding and studying the bifurcation points of solutions to a nonlinear programming problem. The problem of finding bifurcation points for solutions of nonlinear programming problems is reduced to the problem of identification of conditional positive definiteness of quadratic forms on cones. There are criteria of conditional positive definiteness of quadratic forms on cones in the important special case, when the cone is a positive orthant in Euclidean space. Their application poses the necessity to calculate a large number of determinants, which is computationally extremely inefficient, although in the case of a small number of variables, they offer promise for solving Lyapunov stability problems. The paper proposes and justifies the method of searching for options to solve the problem of nonconvex quadratic programming that arises during the study of the stability of elastic unilateral restrictions on the movement. The results can be used in the design of arches and arch systems to improve their bearing capacity.