Nonlinear deformation and stability of geometrically exact elastic arches

In the present paper a plane round double-hinged arch under the potential dead load is investigated. To describe the stress-strain state and the equilibrium stability the geometrically exact theory is used. According to this theory every point of the bar has two translational degrees of freedom and one rotational, which is independent from the previous two. To solve the problem no displacements are simplified and all the stiffnesses are used: axial, shear and bending. Exact nonlinear differential equations are found for the static problem. A variational definition for the problem is defined as finding a stationary point of Lagrange functional. The match of the differential and variational formulations is shown. Exact stability equations accounting non-linear geometric deformations in pre-buckling state were worked out. The problem of the equilibrium stability of the round arch under the potential dead load was solved using the obtained equations regarding all the bar’s stiffnesses. The characteristic transcendental equation and its asymptotic solution as simple formulas, suitable for practical application, were worked out. The comparison of described solution which regards all the bar’s stiffnesses and classical solution, based on bending stiffness, was made.