Static, stability and free vibration analysis of arches using a new differential transformation-based arch element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved arch element including the effects of shear deformation and extensibility of the neutral axis, considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved arch element. Since the element stiffness, mass and geometric stiffness matrices for an arch element are developed based on the exact displacement shape functions, the arch element developed here is more accurate than the elements developed in the past. By adopting the finite-element method the static, stability and free and forced vibrations of axially functionally graded tapered arches including shear deformation and rotary inertia are studied by solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal arches (both forms – prime and quadratic) with hinged–hinged, hinged–clamped and clamped–clamped and clamped–free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross-section are studied. The lowest four natural frequencies are calculated and compared with the published results.