Abstract
This paper presents closed form solutions to the classical beam elasticity differential equation in order to effectively model the displacement of standard aerodynamic geometries used throughout a number of industries. The models assume that the components are constructed from in-plane generally anisotropic (though shown to be quasi-isotropic) composite materials. Exact solutions for the displacement and strains for elliptical and FX66-S-196 and NACA 63-621 aerofoil approximations thin wall composite material shell structures, with and without a stiffening rib (shear-web), are presented for the first time. Each of the models developed is rigorously validated via numerical (Runge-Kutta) solutions of an identical differential equation used to derive the analytical models presented. The resulting calculated displacement and material strain fields are shown to be in excellent agreement with simulations using the ANSYS and CATIA commercial finite element (FE) codes as well as experimental data evident in the literature. One major implication of the theoretical treatment is that these solutions can now be used in design codes to limit the required displacement and strains in similar components used in the aerospace and most notably renewable energy sectors.
Highlights
There is great deal of interest in the design of structurally efficient aerodynamic structures constructed of advanced composite materials
The results of the finite element (FE) displacement calculations from the ellipise, with an outside major axis of 300 mm, an outside minor axis of 89 mm, and shell of thickness 5 mm cantilever of length 4.2 m (These dimensions being analogous to the forthcoming analysis of the NACA 63-621 aerofoil section as detailed in [17]) benchmark case are shown in Figure 7 with an applied load of 8.5 kN
Best agreement is being observed between the Runge-Kutta and anti-derivative shell formulations which themselves are in close parity with the Runge-Kutta finite difference (FD) solution to the solid formulation with an average difference of just over 0.08% between the calculations
Summary
There is great deal of interest in the design of structurally efficient aerodynamic structures constructed of advanced composite materials. The consequence is that the polymeric material is made a couple of orders of magnitude stronger in addition to significantly increasing the displacement at failure when compared to the original ceramics employed as the reinforcement. It is known that most composite material wind-turbine blades employed in the industry fail via a combination of strain and displacement modes, where fibre-glass is concerned; it is widely accepted [1] that such materials fail via maximum stain. In order to reduce both the cost of physical testing and time prohibitive computationally expensive simulations in the design phase, the need for accurate and effective modelling of the displacement and strain fields in materials which are used to construct wind turbine blades, under standard testing conditions, has very recently become of utmost importance
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