In-plane Nonlinear Behaviour of Circular Pinned Arches With Elastic Restraints Under Thermal Loading

This paper considers the nonlinear in-plane behaviour of a circular arch subjected to thermal loading only. The arch is pinned at its ends, with the pins being on roller supports attached to longitudinal elastic springs that model an elastic foundation, or the restraint provided by adjacent members in a structural assemblage. By using a nonlinear formulation of the strain-displacement relationship, the principle of virtual work is used to produce the differential equations of in-plane equilibrium, as well as the statical boundary conditions that govern the structural behaviour under thermal loading. These equations are solved to produce the equilibrium equations in closed form. The possibility of thermal buckling of the arch is addressed by considering an adjacent buckled equilibrium configuration, and the differential equilibrium equations for this buckled state are also derived from the principle of virtual work. It is shown that unless the arch is flat, in which case it replicates a straight column, thermal buckling of the arch in the plane of its curvature cannot occur, and the arch deflects transversely without bound in the elastic range as the temperature increases. The nonlinear behaviour of a flat arch (with a small included angle) is similar to that of a column with a small initial geometric imperfection under axial loading, while the nonlinearity and magnitude of the deflections decrease with an increase of the included angle at a given temperature. By using the closed form solutions for the problem, the influence of the stiffness of the elastic spring supports is considered, as is the attainment of temperature-dependent first yielding of a steel arch.