Buckling analysis of geometrically nonlinear curved beams

Abstract

The equation of motion of curved beams is derived in polar coordinate system which represents exactly the geometry of the beam. The displacements of the beam in radial and circumferential directions are expressed by assuming Bernoulli–Euler’s theory. The nonlinear strain–displacement relations are obtained from the Green–Lagrange strain tensor written in cylindrical coordinate system, but only the components related with radial and circumferential displacements are used. The equation of motion is derived by the principle of virtual work and it is discretized into a system of ordinary differential equations by Ritz method. Static analysis is performed in parametrical domain, assuming the magnitude of the applied force as parameter, and stability of the solution is determined. The nonlinear system of equations is solved by Newton–Raphson’s method. Prediction for the next point from the force–displacement curve is defined by the arc-length continuation method. Bifurcation points are found and the corresponding secondary branches with the deformed shapes are obtained and presented.