Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions
Introduction
The use of laminated composites for design of aerospace structures allows the stiffness, strength and flexibility to be controlled in different directions. In conventional laminates, the fibre angles are kept constant within a lamina which results in constant stiffness properties across the planform of the plate and limited tailorability options. Two commonly reported approaches to improve the structural response by allowing in-plane tailorability of composites are (1) adding patches of additional layers with different fibre orientations and (2) varying the fibre orientation angles over the planform of the plate. Biggers et al. [1], [2] employed piecewise uniform redistribution of layers with specified orientations to create beneficial stiffness distributions across the planform of the plate. This approach resulted in better buckling performance of composite plates and used finite element models to compute the critical buckling load. Hyer et al. [3], [4] initiated the use of curvilinear fibre paths aligned along the principal directions of the stress fields for improving the buckling resistance of composite plates with a hole. They used finite element (FE) analysis to model the buckling problem and then determined the optimal fibre distribution in each element to achieve better buckling performance. Nagendra et al. [5] used non-uniform rational B-splines for designing improved fibre variations in the planform of the plate and performed FE analysis to optimise the fibre design based on buckling load and natural frequency. Leissa and Martin [6] performed vibration and buckling analysis of composite plates with a varying fibre distribution using the Ritz method. Gurdal and Olmedo [7] varied the stiffness properties by introducing a linear fibre variation definition along the length of the composite laminate and employed an iterative collocation numerical technique to solve the in-plane response of a VAT panel governed by a system of coupled equilibrium equations expressed in terms of displacements. Senocak and Tanriover [8] used the Galerkin method with polynomial trial functions to solve the in-plane response of composite plate with varying fibre content and linearly varying fibre orientations. The existing models for in-plane analysis of VAT plates are based on displacement formulation which requires an iterative strategy to achieve the desired accuracy. This approach is computationally expensive due to the number of iterations involved and also application of pure stress or mixed boundary conditions is cumbersome. Gurdal et al. [9] studied the buckling response of variable stiffness panels by allowing stiffness variation both along and perpendicular to the loading direction. They employed a Rayleigh–Ritz (RR) method to compute the buckling coefficient which did not include the effect of flexural-twist coefficients D16, D26. Alhajahmad et al. [10] used variable stiffness to study the pressure pillowing problem of fuselage skin. They used a Rayleigh–Ritz method to perform nonlinear analysis and designed plates with optimal fibre paths for maximum failure load. Weaver et al. [12] designed and manufactured variable stiffness panels using an embroidery based process. Their FE results showed a similar buckling performance of VAT panels when compared to quasi-isotropic laminates, but exhibited superior post-buckling behaviour. Numerical studies on post-buckling analysis of VAT plates typically use FE modelling [11] and analytical formulations have yet to be developed to study the effect of tow steering on post-buckling strength of VAT plates. Abdalla et al. [13] employed FE models to design variable stiffness panels for maximising natural frequency using lamination parameters. Setoodeh et al. [14] used a reciprocal approximation technique to optimise variable stiffness panels for maximum buckling load. They employed a conforming bilinear finite element for buckling analysis of VAT panels. In addition, various higher order plate finite element formulations were used along with optimisation techniques to design VAT panels for better structural performance [15], [16]. Most of the work reported for analysis of VAT panels use FE methods and require many elements for accurate solutions. Furthermore, when the FE method is coupled with optimisation algorithms, the analysis may become computationally expensive. To overcome this drawback, new methods are required which are fast, accurate, general and easily integrable with optimisation algorithms for design of VAT panels. As an alternative approach to variational methods, the Differential Quadrature Method (DQM) is investigated in this work for structural analysis of VAT panels. DQM is an efficient and straight forward discretization technique for obtaining accurate numerical solution with less computational effort.
DQM introduced by Bellman and Casti [17] is based on the assumption that the partial derivatives of a function in one direction can be expressed as a linear combination of functional values at all the grid points along that direction. DQM require fewer grid points to compute solutions with reasonable accuracy when compared to FE analysis [18]. DQM has been successfully applied to solve various problems in structural mechanics [20] and the results demonstrate the potential as an attractive numerical technique. The key issue in the application of DQM is to determine the weighting coefficients and non-uniform grid distribution for approximating the partial derivatives of the function. Shu [19] developed a generalised DQM approach for handling various structural problems more effectively and demonstrated the accuracy of the method. Sherbourne and Pandey [21] studied the effect of grid distribution in buckling analysis of anisotropic composite plates and reported that using non-uniform grids gives better results compared to uniform grids. Darvizeh et al. [22] compared the performance of DQM with the Rayleigh–Ritz method for buckling analysis of composite plates and the study showed the accuracy and reliability of DQM. Furthermore, because the level of discretization in DQM is at the level of the differential operator rather than the independent variable, w, as in RR method, it is expected that DQM may be able to capture more accurate buckling loads for the same degrees of freedom. Wang et al. [23] studied the buckling of thin plates subjected to nonlinearly distributed edge compression loads using DQM. Recently, a collocation method similar to DQM, which uses radial basis functions (RBFs) to approximate functions, has been successfully applied to solve the partial differential equations. Ferreira et al. [24] applied the RBF collocation method to higher order plate theories based on Carrera’s [25] unified formulation to perform the buckling analysis of constant stiffness composite plates. Rodrigues et al. [26] proposed a local radial basis function-based differential quadrature (LRBFDQ) method to solve the buckling problem of thick laminated plates. The LRBFDQ method combines the good approximation of derivatives by differential quadrature (DQ) in a local mesh-free framework provided by RBFs.
In this paper, numerical methodology based on the DQM is developed for buckling analysis of VAT panels modelled using classical laminated plate theory (CLPT). However, for thick VAT plates CLPT overestimates the buckling loads and higher order plate models based on shear deformation theories [27], zig-zag theories [28] and equivalent single layer variational models [29] should be considered. The new aspect of the present work is the use of Airy’s stress function to perform the prebuckling analysis of anisotropic VAT plates which considerably reduces the problem size and when coupled with DQM requires less computational effort than FE method. The generality of the formulation helps in efficient modelling of pure stress and mixed in-plane boundary conditions applied to the VAT plate. Also, the governing differential equation for buckling analysis of VAT composite plate includes the effect of flexural-twist coupling coefficients. The buckling performance of VAT panels with linearly varying fibre orientations, in the planform, under different boundary conditions was then studied using DQM and the results were validated using the FE method. The stability and robustness of DQM in computing the buckling performance of VAT panels were also analysed.
The remainder of this work is organized as follows. In the next section, the numerical aspects of DQM such as choice of test functions, grid point distributions and approaches to apply boundary conditions are discussed. The concept of variable stiffness is introduced in Section 3 and the prebuckling analysis formulation of VAT panels using DQM is presented in Section 4. The DQM formulation for buckling analysis of VAT panels under different boundary conditions are discussed in Section 5. In Section 6, several numerical examples of VAT panels are presented to demonstrate the accuracy of the method and close with a few concluding remarks in Section 7.
Section snippets
Differential quadrature method
Differential quadrature is a numerical discretization technique for approximating the partial derivatives of a function with respect to a spatial variable, using a weighted linear combination of function values at some intermediate points in that variable. For example, the nth order partial derivative of a function f(x) at the ith discrete point is approximated bywhere xi is the set of discrete points in the x direction; and is the weighting coefficients of
Variable angle tow panels
The concept of VAT placement provides the designer with a wider design space for tailoring composite laminated structures for enhanced structural performance under prescribed loading conditions. Potential benefits may be achieved using VAT placement, for example, by blending (minimising) stiffness variations between structural components (e.g. stiffener to skin) to reduce inter-laminar stresses. Also, the in-plane fibre orientation and local thickness distribution can be tailored to reduce the
Prebuckling analysis
For symmetric VAT composite plates, the mid-plane strains (ϵ0) in terms of stress resultants can be expressed usingwhere A∗(x, y) is the in-plane compliance matrix [32]. The Airy’s stress function and the strain compatibility condition was used to solve the in-plane response of symmetric VAT panels. A stress function Ω [33] is introduced such that
The
Buckling analysis
The moment equilibrium equation for symmetrical VAT plate is given by,where Mx, My, Mxy are the moment distributions and q is the load applied in z direction. The moment distributions are related to the mid-plane curvatures by the following relation,where D is the laminate bending stiffness matrix and the curvatures are given by
Results and discussion
The prebuckling and buckling results of VAT panels subjected to different loading conditions are presented in this section. For the numerical simulation, the material properties for each lamina are chosen as, E1 = 181 GPa, E2 = 10.27 GPa, G12 = 7.17 GPa, ν12 = 0.28 with thickness t = 1.272 × 10−4 m. In order to validate the DQM results, FE modelling of the VAT panels was carried out using ABAQUS. The S4 shell element was chosen for discretization of the VAT plate structure and appropriate mesh density was
Conclusion
Differential quadrature methodology based on Airy’s stress function approach was proposed to solve the in-plane analysis of VAT plates subjected to cosine distributed compressive loads and uniform axial compression. The in-plane analysis results obtained using DQM matches very well with FE results and show the ability of DQM to model stress/displacement boundary conditions effectively with less computational effort. Buckling analysis of rectangular VAT plates with simply supported, clamped and
Acknowledgements
The authors thank EPSRC, Airbus and GKN for supporting the work carried out under the ABBSTRACT2 project.
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