Geometrically nonlinear fundamental frequencies, mode shapes and optimization aids for laminated composite rhombic stiffened hyperbolic paraboloids

Abstract

This paper solves the free vibration of rhombic stiffened hyperbolic paraboloids which is missing in the literature. An isoparametric C0 continuous finite element code is proposed, combining eight-noded elements for the panels, three-noded elements for the stiffeners, geometrically nonlinear strains and the first order shear deformation theory. Hamilton's principle based on Lagrange's equation of motion is used to derive the governing equation of motion, which is solved using a subspace iteration algorithm. Benchmark problems are solved to confirm the correctness of the proposed approach. The parametric variations include cross and angle-ply laminations of graphite/epoxy and glass/epoxy composites, clamped and simply supported boundary conditions, and different side tilts (ϕ) of the rhombic panels. The findings indicate that rhombic panels perform superior to rectangular ones, and for ϕ = 35° the frequencies are maximum. The frequencies of graphite/epoxy panels are higher than those of the glass/epoxy ones. The optimal laminations are 45°/-45°/-45°/45° for clamped and 45°/-45°/45° for simply supported boundary conditions. Mode shapes are found to vary with boundary condition, material properties, stiffener arrangements, and ϕ values. Finally, the vibration study provides optimization guidelines to assist practicing engineers in economically increasing the frequencies of bare panels within the practical limits of stiffener depth and quantity. © 2026 Elsevier Ltd