Mechanics of tapered axially functionally graded shallow arches

This paper investigates the forced nonlinear mechanics of a tapered shallow arch made of an axially functionally graded (AFG) material. A nonlinear model for a clamped-clamped boundary is developed based on the nonlinear strain-displacement relation in the framework of Hamilton’s principle. To this end, first, the exponential distributions along the length of the shallow arch are formulated for the material properties such as the mass density, moduli of elasticity, Poisson’s ratio, as well as the geometric characteristics such as the cross-sectional area and the second moment of area of the AFG tapered shallow arch. Second, the expressions for the kinetic and the potential energies of the AFG tapered shallow arch, as well as the work of the external excitation force, are constructed and balanced via Hamilton’s principle. Third, the nonlinear non-uniform equations of motion are discretised via the Galerkin method retaining an adequate number of symmetric and asymmetric modes. The asymmetric nature of the system with respect to the axial coordinate, due to the non-uniform mechanical properties and geometry along the length, is addressed through extensive numerical simulations. The influence of internal resonances and internal energy transfers on the dynamic response of the system is investigated. Moreover, the effect of the gradient index as well as the curvature amplitude on the vibration characteristics of the tapered AFG shallow arch is investigated.