A finite element method for analyzing the free vibration of tapered functionally graded porous (FGMP) arches with point masses

This study examines the linear free vibration of a linearly tapered circular arch made of functionally graded porous material and carrying a concentrated mass at an arbitrary curvilinear location. Both uniform and non-uniform through-thickness porosity distributions are considered. A conventional two-node Euler-Bernoulli finite-element model is used to combine tapering, material gradation, porosity-dependent section properties, and direct nodal assembly of the attached mass; no enriched or generalized finite-element formulation is introduced. Because transverse shear deformation and rotary inertia are neglected, the model is intended for slender arches. Hamilton's principle yields a standard symmetric generalized eigenvalue problem, which is solved with a direct eigensolver. The formulation is assessed through mesh convergence, limiting cases, and comparison with a published clamped-clamped FGM arch benchmark. Over the range studied, the natural frequencies decrease as either the taper coefficient or the power-law index increases. The response to the attached mass depends on the modal displacement at its location, with the largest effect near an antinode and a negligible effect near a node. Porosity causes smaller, model-dependent changes: uniform porosity lowers the frequencies, whereas the adopted non-uniform distribution may produce a slight increase because the pores are concentrated near the neutral axis. In all cases, the porosity-induced shifts remain within a few percent and should not be regarded as a universal design rule.