Buckling and postbuckling of variable angle tow composite plates under in-plane shear loading International Journal of Solids and Structures

A geometrically nonlinear analysis of symmetric variable angle tow (VAT) composite plates under in-plane shear is investigated. The nonlinear von Karman governing differential equations are derived for postbuckling analysis of symmetric VAT plate structures which are subsequently solved using the differ- ential quadrature method. The effect of in-plane extension-shear coupling on the buckling and postbuckling performance of VAT composite plates is investigated. The buckling and postbuckling behaviour of VAT plates under positive and negative shear is studied for different VAT ﬁbre orientations, aspect ratios, combined axial compression and their performance is compared with that of straight ﬁbre composites. It is shown that there can be enhanced shear buckling and postbuckling performance for both displace-ment-control and load-control and that the underpinning driving mechanics are different for each. (cid:2) 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Stability analysis of variable angle tow (VAT) composites under compression load has been studied extensively and response has been shown to have superior structural performance over conventional straight fibre composites (Hyer and Lee, 1991;Gurdal and Olmedo, 1993;Gurdal et al., 2008). In this work, the buckling and postbuckling behaviour of VAT plates under in-plane shear is investigated. The concept of tow steering provides more freedom to design light-weight composite structures with improved structural performance when compared to traditional straight fibre designs. Little work has been reported on the stability analysis of VAT plates under shear load. Biggers and Fageau (1994) studied the concept of stiffness tailoring for improving the shear buckling performance of composite plates by redistributing the layups with certain fibre orientations across the planform of the plate. Their study showed a 50% improvement of shear buckling load over straight fibre composites by redistributing ±45 plies along the diagonal directions. Waldhart (1996) used the Rayleigh-Ritz method to study the buckling performance of tow steered VAT plates under uniform end-shortening and in-plane shear load. The effects of extensional-shear coupling ðA 16 ; A 26 Þ were not considered in their shear buckling study. Nemeth (1997) performed a parametric study on the buckling behaviour of long symmetrical composite plates under shear and reported the effects of membrane anisotropy are more important for shear loaded plates than compression or in-plane bending. Weaver (2004) studied the elastic tailoring of long composite laminates using both flexural and membrane anisotropy and quantified their effects on positive/negative shear buckling behaviour. Wu et al. (2012) studied the buckling performance of VAT plates under compression, shear and combined loading using energy methods. Their study investigated the effect of extensional-shear coupling ðA 16 ; A 26 Þ and bend-twist coupling ðD 16 ; D 26 Þ on the buckling behaviour of VAT plates. Lopes et al. (2010) and Gomes et al. (2013) studied the buckling and postbuckling failure response of variable stiffness composites with cut-outs under compression and shear loading, respectively. They used finite element analysis to model the failure of VAT plates which requires significant computational effort. Rahman et al. (2011) studied the postbuckling response of VAT plates using a perturbation approach, coupled with finite element modelling, to generate a reduced-order model for computation of postbuckling coefficients to predict the postbuckling stiffness of VAT plates. Wu et al. (2013) studied the postbuckling performance of VAT plates under axial compression with linear fibre angle variation for different in-plane boundary conditions and proposed different measures to quantify the postbuckling performance of VAT plates.
Numerous studies on VAT plates rely on finite element (FE) modelling for analysis and design of these structures. As a consequence of variable stiffness coefficients, prebuckling stress distributions can be highly nonlinear (spatially) in-plane, even for uniform loading, and not immediately intuitive (Gurdal and Olmedo, 1993V). Furthermore, it is not obvious whether an FE mesh which is converged for prebuckling analysis will also be converged for buckling or for subsequent postbuckling analyses. A further limitation of FE analysis is that rather than modelling continuous fibre paths, the fibre angle distribution is treated as piecewise constant within each element, leading to spurious stress and strain residuals (i.e. noise) in coarse meshes. Therefore, a strong need for developing semi-analytical models that complements finite element analysis for modelling of VAT panel is required. In the present work, numerical methodology based on the differential quadrature method (DQM) is developed for buckling and postbuckling analysis of VAT panels under in-plane shear load. In prior works, the authors have successfully applied DQM for evaluation of buckling and postbuckling behaviour of VAT plates under compression for different plate boundary conditions . DQM, as a numerical tool, has been shown to be accurate and require less degrees of freedom than FE for solving the buckling and postbuckling problem of VAT panels. Once simple geometries in FE analysis have been validated by DQM models, then the designer can proceed with increased confidence to more complicated geometries and loads. More importantly still, is the physical insight gained in stress redistribution tailoring, and the ability to massage buckling phenomena to be more benign.
In the present work, the underlying mechanics behind the improvement of shear buckling and postbuckling behaviour of VAT plates with linear fibre angle variation is studied. The effect of in-plane extension-shear coupling on the buckling and postbuckling performance of VAT composite plates is investigated for different in-plane boundary conditions. Furthermore, the effect of direction of the applied shear on the postbuckling behaviour of VAT plates under compression is also discussed.

Differential quadrature method
In the differential quadrature method, the derivative of a function, with respect to a space variable at a given discrete grid point, is approximated as a weighted linear sum of the function values at all of the grid points in the entire domain of that variable (Bellman and Casti, 1971). The nth order partial derivative of a function f ðxÞ at the ith discrete point is approximated by where x i = set of discrete points in the x direction; and A ðnÞ ij are the weighting coefficients of the nth derivative and repeated index j indicates summation from 1 to N x . The partial derivatives of a function f ðx; yÞ in matrix form are given by, where P; Q ; R; S with subscripts x; y are the DQM weighting coefficient matrices for the first, second, third, and fourth order partial derivatives with respect to x and y directions, respectively. The unknown function f is expressed in matrix form along the twodimensional grid, as shown in Fig. 1 and superscript T represents the transpose of the matrix. The domain grid points refer to the points where the governing partial differential equations are expressed in DQM form and the boundary grid points refer to the points where multiple boundary conditions are applied (Fig. 1). The information regarding the grid distribution for computation of weighting coefficient matrices and modelling multiple boundary conditions are explained, in detail, in the textbook by Shu (2000).

Postbuckling analysis of VAT panels
In symmetric VAT panels, stiffness (A; D matrices) varies with x-y coordinates and the constitutive equation in partial inverse form is given by, where N; M are the stress and moment resultants, A Ã ¼ A À1 is the compliance matrix and D is the bending stiffness matrix. The nonlinear midplane strains 0 and curvatures j are defined as 0 where u; v; w are the displacements and w 0 is the initial imperfection function. A stress function X is introduced such that the stress resultants are defined by, The compatibility condition in terms of mid-plane strains in a plane stress condition is given by (Whitney, 1987) 0 x;yy þ 0 y;xx À 0 xy;xy ¼ w 2 ;xy À w ;xx w ;yy þ 2w ;xy w 0;xy À w ;xx w 0;xx À w ;yy w 0;xx :  The differential equation of transverse motion that governs the postbuckling analysis of a symmetrical VAT plate is given by, where M x ; M y ; M xy are the moment distributions and q is the load applied in z direction (Whitney, 1987 Thus, Eqs. (7) and (9) represent coupled fourth order nonlinear elliptic partial differential equation in terms of stress function X and transverse deflection w with variable coefficients for postbuckling analysis of VAT composite plates. The stress function boundary conditions are given by X ;yy j x¼0;a ¼ 0; X ;xx j y¼0;b ¼ 0; X ;xy j x¼0;a;y¼0;b ¼ P xy ; Xj x¼0;y¼0 ¼ X ;x j x¼0;y¼0 ¼ X ;y j x¼0;y¼0 ¼ 0: where P xy is the applied shear load. The boundary conditions expressed in terms of X represent uniform shear applied all along the edges of the plate. The simply supported plate boundary conditions are given by, x ¼ 0; a; w ¼ 0; M x ¼ ÀD 11 ðx; yÞw ;xx À D 12 ðx; yÞw ;yy À 2D 16 ðx; yÞw ;xy ¼ 0 M y ¼ ÀD 12 ðx; yÞw ;xx À D 22 ðx; yÞw ;yy À 2D 26 ðx; yÞw ;xy ¼ 0: ð11Þ Chen et al. (2000) used DQM to solve the geometrically nonlinear bending problem of isotropic and orthotropic rectangular plates. Taheri and Moradi (2000) applied DQM to perform postbuckling analysis of straight fibre composites and used an arclength approach to solve the nonlinear algebraic equations. In their works, the nonlinear GDEs were written in terms of displacements (u; v; w) and DQM was subsequently applied to solve them. Whilst studying the nonlinear bending of orthotropic plates, the Hadamard product (), Kronecker product () and SJT product (}) of matrices were used by Chen et al. (Chen et al., 2000) to simplify the DQM form of governing differential equations. Consider matrices A; B of size m Â n, the Hadamard product A B is a matrix of the same dimensions with elements given by Similarly, given a m Â n matrix A and a p Â q matrix B, the Kronecker product A B is a matrix of size mp Â nq given by The SJT product (}) between a matrix A of size m Â n and a vectorṽ of size m Â 1 results in a matrix of same size as A and is expressed as where I is the identity matrix. Chu (2009) applied a similar direct matrix product, equivalent to Chen's approach, to solve nonlinear integro-differential equations. In the present work, the Kronecker, Hadamard and SJT matrix products are applied to the coupled nonlinear postbuckling equations (Eqs. (7) and (9)) and the DQM form of these expressions are given by, whereX;w;w 0 ;q;Ã Ã ij ;D ij ði; j ¼ 1; 2; 6Þ are vectors generated by stacking the columns of the corresponding matrices X; w; w 0 ; q; A Ã ij ; D ij ;J ¼ ½1; 1; . . . ; 1 1ÂN is a row vector, N ¼ N x þ N y represents the total number of grid points in the two-dimensional domain and N x ; N y represent the number of grid points along the x and y directions. The size of the identity matrices I x ; I y depends on N x and N y , respectively. Eqs. (15) and (16) are further simplified into matrix forms given by L 1X ¼ ðL 2w Þ ðL 2w Þ À ðL 3w Þ ðL 4w Þ þ 2ðL 2w Þ ðL 2w0 Þ À ðL 3w Þ ðL 4w0 Þ À ðL 4w Þ ðL 3w0 Þ; L 5w À ðL 6X Þ ðL 3 ðw þw 0 ÞÞ þ 2ðL 7X Þ ðL 2 ðw þw 0 ÞÞ À ðL 8X Þ ðL 4 ðw þw 0 ÞÞ þq ¼ 0: L 5 ¼ ðD 11 JÞ ðS x I x Þ þ ð4D 16 JÞ ðR x P y Þ þ ð2ðD 12 þ 2D 66 Þ JÞ ðQ x Q y Þ þ ð4D 26 JÞ ðR x P y Þ þ ðD 22 JÞ ðI y S y Þ þð2ðD 11;x þD 16;y Þ JÞ ðR x I x Þ þ ðð6D 16;x þ 2D 12;y þ 4D 66;y Þ JÞ ðQ x P y Þ þ ðð2D 12;x þ 4D 66;x þ 6D 26;y Þ JÞ ðP x Q y Þ þ ð2ðD 26;x þD 22;y Þ JÞ ðI y R y Þ þ ððD 11;xx þ 2D 16;xy þD 12;yy Þ JÞ ðQ x I x Þ þ ðð2D 16;xx þ 4D 66;xy þ 2D 26;yy Þ JÞ ðP x P y Þ þ ððD 12;xx þ 2D 26;xy þD 22;yy Þ JÞ ðI y Q y Þ; where is the Kronecker matrix product and is the Hadamard matrix product and all the matrix operators are of size N Â N. Various methods have been reported previously for implementation of multiple boundary conditions along the plate edges using DQM (Shu, 2000). In the current work, the direct substitution method proposed by Shu andDu (1997, 1999) has been used to implement the different combination of plate boundary conditions. Using this approach, the boundary conditions for stress function and transverse displacement were applied to the boundary grid points in the DQM domain. The values at the boundary grid points were expressed in terms of unknown domain grid point values. The modified matrices are reduced to size N d Â N d where N d represent the total number of domain points and the modified DQM equations are given by, which represent the nonlinear algebraic DQM equations as a function of transverse displacement (w d ) and were solved using a Newton-Raphson algorithm. The Jacobian of these equations with respect tow d is obtained using the SJT matrix product as, The SJT product allows computation of the Jacobian for discretized nonlinear partial differential equations similar to calculation of derivative of a single variable scalar function. The SJT approach facilitates fast and accurate evaluation of the Jacobian matrix and the Newton-Raphson iteration approach is used to determine the nonlinear displacement field The Newton-Raphson algorithm ensures quadratic convergence and requires few iterations to converge for each load step applied in the nonlinear postbuckling regime.

Problem definition
The VAT plates considered are symmetrically laminated and the material properties for each lamina are given by E 1 = 181 GPa, E 2 = 10.27 GPa, G 12 = 7.17 GPa, m 12 = 0.28 with lamina thickness t = 1.272 mm and number of laminae, n = 8. The VAT plate with linear angle variation along the x direction is given by where / is the angle of rotation, T 0 is the fibre orientation angle at the panel center x ¼ 0, and T 1 is the fiber orientation angle at the panel ends x ¼ AEa=2 (see Fig. 2). The non-uniform grid distribution given by the Chebyshev-Gauss-Labotto points are used for the computation of weighting matrices and is given by where N is the number of grid points. In order to validate the DQM results, finite element modelling of the VAT panels was carried out using ABAQUS. The S4 shell element was chosen for discretization of the VAT plate structure. To achieve good accuracy, mesh sizes of 40 Â 40; 90 Â 30 were selected for plates with aspect ratio 1 and 3, respectively. Using the linear fibre angle definition, fibre orientation was evaluated at the centroid of each element. The material properties for elements were then defined using the fibre orientation information. Prior to the buckling analysis, in-plane analysis of the VAT laminates under shear was carried out to compute the stress resultant distributions. The in-plane analysis results were then used in the buckling analysis for evaluating the critical shear buckling load efficient. The postbuckling of VAT plates was then performed using buckling analysis results. The imperfection function required for nonlinear FE analysis was chosen to be the first buckling mode shape. The imperfection magnitude was taken to be one percent of the plate thickness and the arc length parameters required for Riks analysis were adapted to the particular VAT configuration that was studied. All plates considered in this study were considered to be simply supported.
Buckling results are normalised with respect to that of a homogeneous quasi-isotropic (QI) laminate. The laminates ½45= À 45=0=90 s ; ½90=0=45= À 45 s are commonly termed QI, yet they contain different amounts of flexural anisotropy (Diaconu and Weaver, 2006). To nullify the effects of flexural anisotropy we have chosen a layup of 48 layers comprising 0; 90; AE45 fibre orientations. Alternatively, the homogeneous QI laminate with equivalent Young's modulus E iso , Poisson's ratio m iso and bending stiffness D iso are given by (Pandey and Sherbourne, 1993;Weaver and Nemeth, 2007), where U 1 ; U 2 ; U 4 are material invariants (Jones, 1998). The critical buckling load N xycr is normalised with respect to the critical buckling state N iso xycr of a homogeneous QI laminate. In the numerical study, the effects of in-plane extension-shear coupling A 16 ; A 26 on the shear buckling and postbuckling performance of VAT composite plates are investigated.

Buckling analysis under in-plane shear load
The buckling problem of VAT plates was solved by neglecting the nonlinear terms in Eq. (9) and DQM was then applied to solve the resulting governing differential equations. Initially, the effect of direction of shear load on the buckling behaviour of unidirectional composite plates with different fibre angle orientations and aspect ratios was studied. The DQM simulation was carried out using N x ¼ N y ¼ 19 grid points for square plate (a = b=0.254 m, thickness = 1 mm) and N x ¼ N y ¼ 21 grid points for a rectangular plate with aspect ratio = 3 (a = 0.762 m, b = 0.254 m, thickness = 1 mm). The normalised buckling load obtained using DQM is shown in Fig. 3. The buckling loads obtained under negative shear are higher than positive shear and this behaviour can be attributed to the alignment of compressive force in the fibre direction by the applied negative shear load. For square and rectangular plates, the 45°and 60°layup respectively, show higher shear buckling performance compared to all other fibre orientations.
Next, the buckling performance of VAT plates under shear load was studied for different linearly varying fibre angle distributions. The DQM grid size was chosen to be N x ¼ N y ¼ 19 for a square plate (a = b=0.254 m, thickness = 1 mm) and N x ¼ N y ¼ 21 for a plate with aspect ratio = 3 (a = 0.762 m, b = 0.254 m, thickness = 1 mm) based on convergence studies. The normalised shear buckling load evaluated using DQM is shown in Fig. 4 for VAT plates with / ¼ 0 and various values of T 0 ; T 1 . In the case of square plates, the straight fibre layup ½45; À45 2s shows higher buckling load for both positive and negative shear when compared to all other VAT layups. For composite plates with aspect ratio = 3, the straight fibre layup ½60; À60 2s exhibits high buckling load for both positive and negative shear. Fig. 5 shows the buckling load results for VAT plates with / ¼ 45 and different values of T 0 ; T 1 . For square plates under negative shear, the straight fibre layup ½45 8 shows higher buckling coefficient of 1.91 compared to all VAT plates. This is mainly because the laminate ½45 8 is unbalanced and the finite extensional-shear coupling stiffness coefficients A 16 ; A 26 introduce N x ; N y distributions which enhances the negative shear buckling performance. However, for rectangular plates, the VAT layups (45 AE h45j0i 2s ; 45 AE h50j0i 2s ) shows high buckling load compared to all other layups. The improvement in buckling performance of VAT plates under constant shear load is not as significant as observed under axial compression .
Furthermore, we studied the effect of A 16 ; A 26 coefficients on the shear buckling performance by choosing symmetric unbalanced laminates containing both straight fibre and VAT layups. For the numerical study, the square plate with layup configuration ½45 2 ; 45 AE hT 0 jT 1 i s was considered and the normalised buckling load results are shown in Fig. 6. The buckling results for negative shear clearly shows that many layups attain higher values than straight fibre designs, whereas the results under positive shear are not as high compared to negative shear. The reason is due to the compressive component of the applied negative shear load acting along the 45°straight fibre direction. In addition, the added 45 layers to the VAT layups make the laminate unbalanced and introduces non-zero extensional-shear coupling stiffness coefficient A 16 ; A 26 distributions. These A 16 ; A 26 distributions result in secondary stress resultant states in the plate which aids the buckling resistance in the negative shear direction. If the straight fibre and VAT layups are rotated by 90°, the buckling performance under positive shear will be higher than straight fibre layups, but result The shear buckling performance of rectangular composite layups ½60 2 ; 60 AE hT 0 jT 1 i s (aspect ratio = 3) are shown in Fig. 8. The layup ½60 2 ; 60 AE h90j10i s under negative shear has higher buckling load compared to other composite layups. The buckling mode shape of the layup ½60 2 ; 60 AE h90j10i s was evaluated using DQM and FE modelling and the results are shown in Fig. 9. Furthermore, the A 16 ; A 26 stiffness coefficient distributions of the VAT plates play a critical part in the improvement of shear buckling performance. The computation of shear buckling load will be erroneous, if the effects of in-plane extension-shear coupling coefficients A 16 ; A 26 are ignored and results in a lower buckling load. Thus, we conclude from this numerical study, the improvement in buckling performance under shear load can be achieved by tailoring the A 16 ; A 26 distributions by designing symmetric unbalanced hybridised straight fibre-VAT laminates.
where a ¼ sin À1 2Dx a À Á ; D y ¼ a 2 1 À cosa ð Þand D x is the applied displacement magnitude. As the boundary conditions are specified solely in terms of displacements, DQM was applied to solve the in-plane coupled partial differential equations expressed in terms of displacements instead of the stress function based differential equation. Details of the GDEs expressed in terms of displacement u; v; w for solving the prebuckling, buckling and postbuckling problem of symmetric VAT plates are given in the Appendix. This approach was taken due to the difficulty in applying the displacement based conditions in terms of stress function, as they are expressed using integral expressions which are nonlocal boundary conditions and pose additional problems to satisfy them accurately at the boundary grid points. The DQM procedure discussed in the work of Groh and Weaver (2014) was used here to solve the prebuckling problem expressed in terms of in-plane displacements. The resulting DQM algebraic equations were solved for shear displacement boundary conditions and results in non-uniform stress resultant distributions for VAT plates. To determine the average shear buckling load applied to the VAT panel, it is essential to examine the contribution of each stress resultant in satisfying the specified displacement boundary conditions. For the edges to be straight, the stress resultants N x ; N y are nonzero along the edges and have  T1   T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Negative shear, Aspect ratio=1  T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Positive shear, Aspect ratio=3  T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre   45±<T0|T1>   2s Negative shear, Aspect ratio=3 (c) ( d)  Fibre angle T1   T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Positive shear, Aspect ratio=1  T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Negative shear, Aspect ratio=1 considerable magnitude which cannot be ignored when compared to N xy (Fig. 11). The non-zero stress resultants along the edges of the VAT plate result in moment distributions which have to be considered in evaluating the applied shear load. Detailed procedures for the computation of the moment due to stress resultants are explained in the work of Waldhart (1996). The moments due to N x ; N xy along edge x ¼ a=2 and moments due to N y ; N xy along edge y ¼ b=2 were used to compute the average applied shear resultant N ave xy . The moments due to N x ; N y ; N xy are defined as, Àb=2 yN x ða=2; yÞdy; where M i z is the moment about the z axis by the N x acting along the edge x ¼ a=2; M ii z is the moment about the z axis by the N y acting along the edge y ¼ b=2, and M iii z ; M iv z are the moments about the z axis created by N xy acting along the edges x ¼ a=2; y ¼ b=2, respectively. In this work, the moment in the anticlockwise direction is taken to be positive. Under positive shear deformation, M iii z is negative and M iv z is positive, but the sign of M i z ; M ii z depends on the fibre path distribution and cannot be predetermined. Therefore, the positive and negative moments are given by,  Fibre angle T1   T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Negative shear, Aspect ratio=3    The average of the moments M þ ; M À was used to compute the average shear load given by, The stress resultants obtained from the in-plane analysis of VAT plates were then used for computing the critical shear buckling load given by N xycr ¼ k cr N ave xy where k cr is the minimum eigenvalue. The buckling performance of VAT plates under shear displacement was studied for different aspect ratios. The DQM grid size was chosen to be N x ¼ N y ¼ 21 for the square plate and N x ¼ N y ¼ 25 for plate with aspect ratio = 3 based on convergence studies. For VAT plates with / ¼ 0 and various values of T 0 ; T 1 , the normalised positive and negative shear buckling loads evaluated using DQM are shown in Fig. 12. For square VAT plates, the layup 0 AE h0j45i 2s shows high positive and negative shear buckling load when compared to all other layups. For rectangular VAT plates, the layup 0 AE h90j45i 2s shows better buckling performance than other layups. The VAT laminates considered until now are balanced and there is no effect of A 16 ; A 26 stiffness coefficients on the shear buckling performance.
Next, we investigate the effects of A 16 ; A 26 stiffness coefficients on the shear buckling performance of VAT laminates AE45hT 0 jT 1 i 2s ; 45 AE hT 0 jT 1 i 2s for different aspect ratios. Both the VAT layups AE45hT 0 jT 1 i 2s ; 45 AE hT 0 jT 1 i 2s have different fibre angle definitions based on the location of the ±sign with respect to the rotation angle / (Waldhart, 1996). In the VAT layup AE45hT 0 jT 1 i 2s , a ±sign in front of the / means the reference fibre paths are rotated in equal and opposite amounts. For VAT layups AE45hT 0 jT 1 i 2s , the fibre angle definition results in a balanced laminate in certain regions and unbalanced laminate elsewhere. This introduces finite A 16 ; A 26 distributions in the VAT laminates. The normalised buckling load for VAT layups AE45hT 0 jT 1 i 2s was computed using DQM and is shown in Fig. 13 and the values are high compared to the straight fibre and VAT layups 0 AE hT 0 jT 1 i 2s . This is primarily due to the redistribution of applied shear loads from the centre of the panel towards the edge and also the nonzero A 16 ; A 26 stiffness distributions play a small role in improvement in buckling performance in either positive or negative shear direction.
laminates are unbalanced resulting in finite A 16 ; A 26 distributions and they introduce N x ; N y ; N xy distributions in the VAT laminate to keep the edges of the plate straight. These non-uniform prebuckling stress resultants are responsible for the improvement in shear buckling load coefficient and for the difference in buckling results for similar VAT layup configurations. The VAT layup 45 AE h30j80i 2s has normalised buckling load coefficient of 16.15 under negative shear and the stress resultant distributions corresponding to this layup are shown in Fig. 15 such that the interior of the plate is effectively stress-free. The DQM results validate the FE modelling solutions and the stress resultant field shows the load redistribution towards the edges of the plate. The maximum and minimum principal stress resultants for the VAT layup 45 AE h30j80i 2s are shown in Fig. 16 and show the compressive component of the applied shear being redistributed from the centre towards the plate edges. Under positive shear, the VAT layup 45 AE h20j60i 2s has normalised buckling load coefficient of 6.65 and the stress resultant distribution corresponding to this layup is shown in Fig. 17 such that a stress-free state is observed at the centre of the plate. The stress resultant distributions shown in Figs. 15 and 17 indicate the load redistribution responsible for the improvement in shear buckling performance under negative and positive shear, respectively. For rectangular VAT plates, the layup 45 AE h30j90i 2s has a high normalised buckling coefficient value of 5.56 and the corresponding mode shape computed using DQM is shown in Fig. 18. Furthermore, the A 16 ; A 26 stiffness distributions have considerable effect on the pre-buckling stress redistribution of symmetric unbalanced VAT layups and are responsible for better buckling performance in either positive or negative shear direction. We conclude, from this study, that for shear buckling under displacement boundary conditions, the redis-tribution of applied load plays a primary role in improvement of shear buckling performance.

Postbuckling analysis under in-plane shear load
The DQM methodology was extended to perform postbuckling analysis of composite plates under uniform shear load. Initially, buckling analysis of VAT plates was performed to obtain the mode shape of the critical buckling load which is subsequently used as an imperfection function for the postbuckling analysis. For square VAT plates, the number of grid points for DQM modelling was chosen to be N x ¼ N y ¼ 19 based on a convergence study for accurate evaluation of the critical buckling load and mode shape . The imperfection function magnitude (1E À5 ) for DQM was chosen to be the same as for FE modelling of the composite plate. For FE simulation, the mesh density of 40 Â 40 was selected to analyse the above problem after a mesh convergence study. Subsequently, the postbuckling behaviour of different configurations of straight fibre and VAT laminates under uniform shear was studied. The normalised shear load versus normalised maximum transverse displacement for different composite plates is shown in Fig. 19 and the DQM results match FE solutions relatively well. The variation of maximum transverse deflection of the VAT plate with respect to the applied shear load shows the improved postbuckling performance under negative shear rather than positive shear load. The postbuckling performance of all VAT layups under positive shear were not as good as the QI layup. The VAT layups 45 AE h45j30i 2s ; 45 AE h0j30i 2s exhibit higher buckling load, but lower postbuckling performance than the QI layup under negative shear. The combined straight fibre and VAT layups ½45 2 ; 45 AE h90j0i s ; ½45 2 ; 45 AE h45j0i s show higher buckling load  T1   T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Positive shear, Aspect ratio=1  Fibre angle T1   T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Negative shear, Aspect ratio=1  T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Positive shear, Aspect ratio=3  T0=0  T0=10  T0=20  T0=30  T0=40  T0=45  T0=50  T0=60  T0=70  T0=80  T0=90  Straight  fibre Negative shear, Aspect ratio=3 ±45<T0|T1> 2s (c) ( d) Fig. 13. Square VAT plate AE45hT0jT1i 2s subjected to in-plane shear displacement (a) positive shear, aspect ratio = 1 (b) negative shear, aspect ratio = 1 (c) positive shear, aspect ratio = 3 (d) negative shear, aspect ratio = 3. and improved postbuckling performance under negative shear when compared to the QI layup. DQM was then applied to study the postbuckling behaviour of rectangular plates (aspect ratio = 3) under negative shear. The number of grid points for DQM modelling was chosen to be N x ¼ N y ¼ 25 based on a convergence study. For FE modelling of rectangular VAT plates, a mesh density of 90 Â 30 was selected. The DQM and FE results for different VAT plates are shown in Fig. 20. The straight fibre layup ½60 8 and VAT layup 45 AE h45j0i 2s show better postbuckling performance compared to QI under negative shear. The postbuckling performance for combined straight fibre and VAT layups is investigated and the results are compared with VAT layups / AE hT 0 jT 1 i 2s shown in Fig. 20. The layup ½60 2 ; 60 AE h90j10i s exhibits better buckling and postbuckling performance and this improvement is significant as observed under axial compression for VAT laminates. Similar to the previously shown buckling behaviour, by tailoring the A 16 ; A 26 distributions using symmetric unbalanced layups (½45 2 ; 45 AE h90j0i s ; ½45 2 ; 45 AE h45j0i s ; ½60 2 ; 60 AE h90j10i s ) has significant effect on their postbuckling performance.

Postbuckling analysis under combined axial compression and shear
The postbuckling behaviour of VAT plates under combined axial compression and shear was studied using DQM (Fig. 24). A square VAT plate (0 AE h45j30i 2s ) with a load ratio of N xy =N x ¼ 0:5 was considered for the numerical study. The variation of maximum transverse centre deflection with increasing axial compression and shear is shown in Fig. 25. The direction of applied shear load has considerable influence on the postbuckling behaviour. The results show that negative shear improves slightly and positive shear   reduces the postbuckling performance of VAT plates. Next, a VAT plate (90 AE h0j75i 2s ) which exhibits better buckling performance compared with straight fibre composites under axial compression was considered for the numerical study. The result shows that negative shear reduces the postbuckling performance and that positive shear has little effect (Fig. 26). This arises because the effect of shear loads is reversed for angle of rotation / ¼ 90 . Thus, the results show the influence of direction of shear load on the postbuckling performance under axial compression.

Discussion
In this work, the DQM approach was successfully applied to model the buckling and postbuckling behaviour of VAT plates under uniform in-plane shear load and displacement boundary conditions. For the linear shear buckling problem, DQM required few grid points and less computational effort to achieve converged results than the FE method. Similarly, for nonlinear postbuckling analysis, DQM modelling uses few grid points, but needs more iter- ations to converge in each load step when compared to FE modeling. This problem arises due to the nonsymmetric nature of the DQM stiffness matrix and also stronger reinforcement of all the boundary conditions at the boundary grid points. Although the DQM results were comparable to FE solutions, the method is not as general as FE modelling as it cannot be applied to structures with discontinuities/complicated geometry.
For shear load boundary conditions, the benefit for buckling and postbuckling response under negative shear is only observed when unbalanced VAT laminates are used such that a biaxial tensile stress (Fig. 7) is induced in the interior of the plate. Balanced VAT layups do not exhibit improved shear buckling and postbuckling behaviour in both directions as there is no load redistribution and secondary stress condition observed. Therefore, tailoring the finite A 16 ; A 26 distributions of the hybridised unbalanced VAT laminates introduce favourable tensile stress states and improve the shear buckling and postbuckling performance in either positive or negative directions. In the case of shear displacement, balanced VAT layups exhibit improved buckling and postbuckling performance in positive and negative shear directions compared with straight fibre laminates. In addition for unbalanced VAT layups, the A 16 ; A 26 stiffness distributions has considerable effect on the buckling and postbuckling performance in either the positive or negative shear direction. The phenomenon of redistribution of the applied shear load shown in Figs. 15 and 17 is mainly responsible for the improved shear buckling and postbuckling performance of VAT plates. Thus, the phenomena of induced secondary tensile stress state at the centre of the plate and achievement of stress-free state at the centre by redistribution of applied load towards the edges of the plate are responsible for the improvement of shear buckling and postbuckling performance under shear loadcontrol and displacement-control boundary conditions, respectively. Furthermore, the buckling and postbuckling results of VAT laminates under shear displacement boundary conditions is more significant than observed under load-control.

Conclusion
In this work, the buckling and postbuckling performance of symmetric VAT composite panels with linear fibre angle variation under in-plane shear is presented. The numerical results are computed using DQM for VAT plates with different aspect ratios and they correlate well with FE analysis. The numerical study show the effect of extension-shear coupling on the buckling and postbuckling performance of VAT composite plates under different in-plane boundary conditions. The physical understanding of the mechanics responsible for the improvement of buckling and postbuckling performance of VAT plates under shear load and displacement conditions is explained. Under constant shear load boundary condition, the linear fibre angle variation of VAT layups does not exhibit improved buckling and postbuckling behaviour. But, VAT layers combined with straight fibre layers result in improved buckling and postbuckling performance under negative shear, but poor performance under positive shear. The presence of induced tensile stresses in both x; y directions is responsible for the improved shear buckling and postbuckling performance under constant shear load. In the case of shear displacement, VAT layups exhibit improved buckling and postbuckling performance compared with straight fibre laminates. The redistribution of the applied shear load is responsible for the improved shear buckling and postbuckling performance of VAT plates. For shear displacement boundary conditions, the linear fibre angle variation allows simultaneous improvement of shear buckling and postbuckling performance under negative and positive shear. Furthermore, postbuckling behaviour of VAT plates under combined axial compression and in-plane shear was studied using DQM. The results shows the effect of the applied shear can be used to increase or decrease the postbuckling performance of VAT plates under combined loading conditions.