Geometrically Nonlinear Free Vibration of Composite Skewed Stiffened Elliptic Paraboloidal Panels

Abstract

In the literature, free vibration studies of skewed elliptical paraboloidal panels with composite stiffeners are missing, especially using geometric nonlinearity. This paper fills that deficiency. It proposes a C0 finite-element formulation that models the skewed panels using eight-noded doubly curved elements and stiffeners by three-noded elements. Geometrically nonlinear strains, Lagrange's equation of motion, and Hamilton's principle are combined for the governing equation. The transverse shear strains are taken as constant with proper correction factors. The numerical code is prepared using a Fortran compiler and implemented through a Dell Precision 5860 Tower workstation. Closed-form solutions and experimental results act as benchmark to check correctness of the proposed code. The fundamental frequencies and mode shapes are obtained for varying boundary conditions, skew angles, laminations, ratios of radii of curvatures, and side-to-thickness ratios. The orientation, number, and depth of stiffeners are also varied. The findings indicated that the performance improves as the skewness of panels increases. The clamped panels offer the best performance for (45°/-45°)s composite. In order to maximize the frequencies of these panels, the radius of curvature along y-axis should be half of that along x-axis, side-to-thickness ratio should be 75 and panels must be skewed to 35°. Additionally, the panels must have five stiffeners along the x-axis and six stiffeners along the y-axis. Those stiffeners should be located on the concave side of the panels and have a depth thrice the panel thickness. The first four modes of vibration should be considered for a comprehensive vibration study. Practical Applications The body panels of aircraft and spacecraft are fabricated using doubly curved panels. The wings, ailerons, elevator, flaps, rotor blades, and doors of landing gear are the practical examples. These parts are generally made of laminated composites to have a lightweight panel. The lightweight panels may undergo instability issues under dynamic excitations. The concave region of such curved panels can be stiffened to address the instability problem. This study focuses on a nonlinear finite-element model that simulates the composite doubly curved panels with stiffeners. The dynamic performances in terms of fundamental frequencies and mode shapes of stiffened curved panels are studied for varying parameters such as boundary conditions, laminations, radii of curvatures, and skew angles. The dimensions, alignment, and thicknesses of the composite stiffeners are also varied. The relative performance study reported in this paper leads to a set of design guidelines that maximizes the performances within a given quantity of material consumption. © 2025 Elsevier B.V., All rights reserved.