Stability of fixed-fixed shallow arches under arbitrary radial and vertical forces

In the present paper the planar stability of fixed-fixed shallow circular arches is investigated. The arches are made of linearly elastic, functionally graded material and are subject to a concentrated radial or vertical dead force at an arbitrary position. To describe the behaviour, the one-dimensional Euler-Bernoulli kinematic hypothesis is used. The effect of the bending moment on the membrane strain is incorporated into the model. The related coupled differential equations of the problem are derived from the principle of virtual work. Exact solutions are found both for the pre- and post-bucking displacements. Closed-form analytical solution is given for the buckling load when the load is radial while for vertical force, the solution is numerical. It is found that for fixed-fixed members, only limit point buckling is possible. Such shallow arches are not sensitive to small imperfections in the load position or in the load direction. It turns out that the material behaviour and geometry have significant effects on the behaviour and buckling load. If the load is placed far enough from the crown point, the load bearing abilities become better than for crown-load. Comparisons with an analytical literature model and commercial finite element software confirm the validity of the new findings.