Post-Buckling, Limit Point, and Bifurcation Analyses of Shallow Nano-Arches by Generalized Displacement Control and Finite Difference Considering Small-Scale Effects

Post-buckling of shallow nano-arches is examined numerically in this study. The small-scale effect is taken into account by using the nonlocal theory. The variational formulation is employed to derive the equilibrium equations of the arch based on the Euler–Bernoulli beam hypothesis. Moderate rotations are considered by including the von Karman nonlinear strains. The governing equations are discretized by the finite difference method and are solved iteratively by the generalized displacement control algorithm. In the buckling analysis, the effects of different factors, such as load distribution, initial height, arch span, and nonlocal parameter, on the buckling loads are investigated. The behavioral analysis of the arch with respect to its initial height follows and detailed analyses for the limit point and bifurcation buckling are presented. It is concluded that the value of nonlocal parameters can influence the arch model in two ways: apparently changing its initial rise and switching its buckling mechanism. There is also a comparison between the use of secant and tangent stiffness moduli for tracing the equilibrium paths.