Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged arches

This paper investigates the in-plane linear dynamic behaviour of multi-stepped and multi-damaged circular arches under different boundary conditions. Cracked cross-sections are modelled as massless elastic rotational hinges. In damaged configuration, cracks can be located both at the interface between two adjacent portions as well as inside the portion itself. For each arch portion bounded by two cracks, the differential equations of motion have been established considering axial extension, transverse shear effects and rotatory inertia. The equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement, tangential displacement and rotation. Analytical and numerical solutions for multi-stepped arches, in undamaged as well as in damaged configurations, are proposed. The analytical solution is based on the Euler characteristic exponent procedure involving the roots of characteristic polynomials, while the numerical method is focused on the Generalized Differential Quadrature (GDQ) method and the Generalized Differential Quadrature Element (GDQE) technique. Numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi-stepped and multi-damaged arches with different boundary conditions. Finally, convergence and stability characteristics of the GDQE procedure are investigated. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good.