Buckling and post-buckling of parabolic arches with local damage

Most studies on cracked one-dimensional structural elements deal with their statics and free dynamics, while their stability is only given marginal consideration, especially arches. This chapter investigates buckling and post-buckling of parabolic arches with crack-like damages. The environment acts on the arch by: a vector force field, power dual of the incremental displacement of the axis; and a skew-symmetric tensor couple field, power dual of the incremental rotation of the cross-sections. The chapter presents two perturbation expansions of the finite governing equations for arches modeled as curved beams, in order to investigate their non-trivial fundamental path and their post-buckling path. Kinematics is finite, balance is in the actual configuration and only constitutive equations are supposed linear elastic. Future investigations will be about the quality of the post-buckling path and on linear vibration about non-trivial pre-stressed states, in order to detect the effect of local damages for monitoring purposes.