Symmetric free vibration of a cracked, quasi-continuous, masonry arch

The paper investigates the natural periods of symmetrically cracked semi-circular masonry arches. The cracks are present due to previous actions, additionally new cracks do not appear and the depth of the cracks do not change during motion, which assumptions are acceptable, if the amplitude of the vibration is relatively small. During vibration the states of the cracks can change from open to closed. The motion of the structures consists of states with linear vibrations. In our work we deal with non-breathing and breathing cracks. We speak about non-breathing crack, if the state of the crack do not change during vibration. If the cracks can change their states during vibration, then we refer to the cracks as breathing cracks. A new approximate approach is presented, which is capable of handling the changes in the state of the cracks in the case of breathing cracks. The motion is periodic if during a half-period the kinetic energy vanishes twice: at the beginning and at the end of a half-period. Between the two extremes the boundary and continuity conditions change, which result in the changing of the modal coordinates in various states of the cracks. In our solutions some parasitic results may appear as well, which are sorted out during analysis. In the paper the position of a crack and the depth of the cracks are under examination. The goal of the work is to analyse the linear and the nonlinear normal modes and natural periods of cracked masonry arches, because these fundamental periods can refer to the magnitude of the degradation in the system, which may help to monitor the safety of structures.