Deformation of Arches: Elastic Buckling Behavior

In-plane buckling, in the Euler sense, of slender, incompressible arches under funicular loading is investigated. A differential equation, applicable to arches of any shape, is derived. An analytical solution appears to exist only for the particular case of uniform circular arches. A strain-energy method using the lagrangian multiplier technique to handle geometric restrictions is developed. An infinite series solution is presented for uniform circular arches, and the results obtained are identical to the differential equation solution. Having established that the energy method gives exact results for circular arches, it is then used to derive an approximate solution for uniform parabolic arches using a finite series approach. Numerical results are given for both the buckling loads and buckling modes. Convergence tests are included showing that as few as three or four terms give solutions accurate to within 1%.