Genetic algorithm-based optimization of fiber orientation in VAT composite plates for maximum thermal buckling temperature

Abstract

This study presents an optimization framework to enhance the thermal buckling temperature of a variable angle tow (VAT) laminated composite plate. A finite element model is formulated using first-order shear deformation theory and eight-nodes iso-parametric elements. The developed model is validated by comparing the results of the present model with the existing benchmark solution. A genetic algorithm is employed to optimize the fibre’s path using MATLAB. The effect of geometric and material characteristics, such fiber orientation, aspect ratio, boundary conditions, material anisotropy, and thermal expansion coefficients is extensively examined. The results show that finite element-based optimization can improve the critical buckling temperature by up to 111.76%. These insights are valuable for the efficient structural design of variable-angle tows composite plates, offering potential benefits for various engineering applications.

Introduction

Composite material consists of two or more distinct constituents combined at a macroscopic level, resulting in enhanced characteristics such as a high strength-to-weight ratio.^1,2 Common composite systems are carbon fiber-reinforced composite, glass fiber-reinforced composite, carbon nanotube reinforced composites, and metal- or ceramic-matrix composites. Compared to traditional metals, laminated fiber-reinforced composites offer superior weight efficiency, specific strength, and stiffness. ³ These materials typically contain straight, unidirectional fibers with uniform orientation within each ply. This leads to a constant stiffness of each layer, limiting the use of typical composite materials under specific conditions, such as those containing holes. ⁴ With gradual advances in Automated Fiber Placement (AFP) technology, ⁵ VAT composites exhibit superior mechanical properties and design capabilities compared to constant-stiffness composites (CSC) with straight fibers. Structures made up of layers of fiber-reinforced composite materials with different fiber orientations within each layer are referred to as Variable Angle Tow (VAT) composite laminates. Lozano et al. ⁶ publish a review paper to discuss the various manufacturing aspects of the VAT composites. In conventional laminates, the fibers in each layer of are usually aligned in one direction, whereas the fibers of VAT laminates are arranged at various angles within the same layer. It results in improved mechanical properties. The fiber orientation of VAT laminates can change within a single layer either continuously or discretely. Automated Fiber Placement (AFP) machines is capable of precisely control the insertion of fibers at varying angles. In VAT laminates, by varying the fiber orientation within each layer can exhibit specific mechanical properties in different direction such as stiffness, strength and fatigue resistance. ⁷ The smooth variation of fiber orientations can help distribute stresses more evenly throughout the structure, reducing the chance of delamination and improving overall durability. This will allow researcher to optimize the performance of the laminate for specific loading conditions and design requirements.

For structural parts used in high-speed aircraft, rockets, and spacecraft, where thermal loads are generated by aerodynamic and solar radiation heating as well as nuclear reactors and chemical plants, which are frequently exposed to high temperatures throughout their service lives. ⁸ The top and bottom surfaces of skin panels have quite different temperatures, which can create significant thermal gradients that cause substantial deflections and compressive strains. When the temperature reaches certain critical values, these effects can cause buckling in the panels. It results in structural failure if the temperature continues to rise beyond the critical point. Therefore, understanding and analysing the thermal buckling behaviour of composite laminates under thermal loads has become the major design criteria for efficient and optimal usage of these materials in structural applications. ⁹ Mondal et al. ¹⁰ investigates the impact of hygro-thermal effects on the free vibration and flutter characteristics of a smart variable stiffness composite laminate. Yan and Zhang ¹¹ used modified legendre expansions to predict quadratic through-thickness displacement distributions in 1D CUF models for aerothermal-elastic stability analysis of curvilinear fiber laminates. Gong et al. ¹² developed a theoretical model to study the aero thermoelastic flutter of VAT composite plates on elastic foundations in supersonic flow, validating it with numerical calculations. Polit et al. ¹³ studied an extended finite element model with 3D structural buckling responses of constant- and variable-stiffness composite curved beams. Lal and Markad ¹⁴ investigated the thermal buckling of SMA embedded sandwich structure. Evran ¹⁵ computed the buckling behaviour of laminated composite for different orientation and found a better orientation of laminated composite. Theoretically and experimentally examined the buckling behaviour of carbon nanotube reinforced composite by Raphael et al. ¹⁶

When the structure loses its elastic stability due to a rise in thermal stresses, thermal buckling takes place. Over the years, thermal buckling investigations have been conducted on straight fiber composite or conventional composite plates. Chen and Chen ¹⁷ investigated the thermal buckling behaviour of laminated composite plates under uniform temperature changes using Galerkin method. Thangaratnam et al. ⁸ first attempt to determine critical buckling temperature of a straight fiber composite plate using FEM for various boundary conditions and the laminates. Following this work, several other analyses were conducted using FEM to look at thermal buckling. Among them, Shiau et al. ¹⁸ emphasized on how the material’s mechanical and thermal property ratios influenced the critical buckling temperatures. In order to determine how various factors affect buckling at both uniform and non-uniform temperatures, Chen et al. ¹⁹ used the thermalelastic Mindlin plate theory. Chen and Chen ²⁰ also explored the impact of temperature-dependent mechanical properties on thermal buckling behaviour using the FEM.

In recent years, researchers have concentrated on the thermal buckling study of VAT plates. Some of the latest studies in this area include: Li et al. ⁷ conducted a thermal buckling analysis of VAT laminates under uniform temperature using the first-order shear deformation theory (FSDT). Zhang et al. ²¹ explored a finite integral transform approach for the new exact thermal buckling analysis of fully rotationally restrained composite rectangular thin plates. Khaniki et al. ²² used the differential quadrature method to find the buckling response of a non-uniform beam with nonlocal strain gradient. Nguyen and Phung ²³ study the bending, vibration and buckling behaviour of the functionally graded composite plate. Liang et al. ²⁴ investigated a reduced-order solution for the thermo-elastic geometrically non-linear response of simply supported thin-walled structures subjected to pure thermal loading. Bracaglia et al. ²⁵ proposed a study on the thermal buckling of Variable Angle Tow (VAT) composite plates using higher-order theories. Manikam et al. ²⁶ employ FEM and FSDT to illustrate how various laminations and material factors affect the critical buckling temperature. Duran et al. ²⁷ investigated the buckling responses by varying the fiber orientation. Venkatachari et al. ²⁸ and Oliveri et al. ²⁹ studied the thermo-mechanical buckling and postbuckling characteristics of VAT plates. However, these researches did not evaluate the impact of temperature-dependent material properties. Abdalla et al. ³⁰ studied buckling capacity of composite laminates with curved fiber as compared to straight fibers. Nie and Chen ³¹ study the thermomechanical buckling behaviour of VAT composite laminates with elastically constrained edges and provide an analytical solution. Dash et al. ³² establish a semi-analytical model to study the instability properties of VSLC plates and shallow cylindrical shell panels under non-uniform thermomechanical stresses. Chandrakar et al. ³³ using an FSDT-based FE model and an RBFN surrogate, it analyzes thermal buckling, damage effects, and failure probabilities for different laminations and boundary conditions.

Numerous studies have been conducted on optimizing laminated structures for thermal buckling analysis with constant stiffness by various authors, such as: Kamarian et al. ³⁴ employed the firefly algorithm to optimize the stacking sequence of composite plates under thermal buckling conditions. Singha et al. ³⁵ utilized a GA to enhance the buckling temperature of laminated composites, considering ply orientation angles and plate thickness as design parameters. Spallino and Thierauf ³⁶ also adopted a GA-based approach to maximize thermal buckling stress in such laminates. To improve the critical thermal buckling capacity, Topal and Uzman ³⁷ applied the Modified Feasible Direction (MFD) method for stacking sequence optimization and examined the influence of aspect ratio, thermal expansion mismatch, material anisotropy, and boundary conditions. Despite extensive research on optimization for VAT composites, most studies^38–42 have primarily addressed buckling due to mechanical loading. IJsselmuiden et al. ⁴³ optimized VAT laminates for maximum buckling load by manipulating lamination parameters. Guenanou and Houmat ⁴⁴ aimed to maximize the fundamental frequency through a layer-wise optimization of the stacking sequence. Belbachir et al. ⁴⁵ introduced an advanced shear deformation theory to analyse the buckling behaviour of laminated composites. Meksi et al. ⁴⁶ assessed the buckling characteristics of functionally graded composite plates under thermal conditions. Hosseini et al. ⁴⁷ explored both buckling and post buckling responses of graded composite beams exposed to thermal environments. Sager et al. ⁴⁸ applied metaheuristic techniques for multi-objective optimization of laminated composites. Kiarasi et al. ⁴⁹ analysed the buckling response of sandwich structures under hygro-thermal loading. Seghier et al. ⁵⁰ examined the buckling behaviour of steel beams using finite element methods and later optimized the results using least squares and metaheuristic algorithms.” The VAT Plate’s thermal buckling optimization has not received much research. Among these for VAT fiber composite plates with gap/overlap free design, Zhou et al. ⁵¹ developed an optimization problem aiming to maximize the thermal buckling temperature. The objective of Duran et al. ²⁷ investigation was to find out which material model could withstand the critical buckling temperature. In a symmetric linear orthotropic laminate, Acar et al. ⁵² looks at the best spatially changing fiber paths. The result reveals that when uniform thermal loads are applied, the critical buckling temperature increases. Zhao et al. ⁵³ investigated the thermal buckling analysis and optimization of stiffened composite panels with variable angle two (VAT) laminates and curvilinear stiffeners. Duran et al. ²⁷ used the FE approach and classical lamination theory to determine the critical buckling temperature of VAT laminates.

The available literature on thermal buckling of VAT Plate is limited. Given the limited research on the thermal buckling of VAT laminates, further research in this area is essential, particularly focusing on the optimal design of VAT plates. Most existing studies on the optimal design of composite plates have focused on constant-stiffness laminates with straight fibers. Even in the case of VAT plates, optimization has predominantly been performed under mechanical loading. To the best of the authors’ knowledge, there is currently no published research addressing the optimal design of VAT composite plates subjected to thermal loads, particularly considering different aspect ratios, boundary conditions, degrees of material anisotropy, and thermal expansion properties. This study explores the influence of these factors on the optimal design. VAT laminates, due to their variable stiffness, introduce additional design parameters, making the optimization process more complex and demanding advanced techniques. Among the most effective approaches for optimizing VAT plates under thermal loading is the GA.^54,55 In light of this, the current work proposes a novel finite element-based optimization framework utilizing GA to determine the optimal curvilinear fiber paths in VAT composite structures.

Theory and formulation

Variable angle tow plate fiber orientation

A VAT composite plate is considered, with its length, width, and thickness represented by a, b, and h, respectively. In general, fiber orientation in VAT plate varies across the surface and can be a function of the x and y coordinates. In this study fiber orientation varies linearly with only x, from ${\theta }_0$ at the plate’s center (x = 0) to the ${\theta }_1$ at the distance x = ${}_{-}{}^{+}\hspace{0.17em}a\hspace{0.17em}/2\hspace{0.17em}$ (edges of plate).

The fiber orientation (θ) at a distance x from the fiber’s origin is represented by the following equation ⁵⁵

$\left(\theta \left(x\right)\right)=\frac{-2}{a}\left({\theta }_1-{\theta }_0\right)x+{\theta }_0\hspace{0.17em}for\hspace{0.17em}\frac{-a}{2}\le x\le 0$

$\left(\theta \left(x\right)\right)=\frac{2}{a}\left({\theta }_1-{\theta }_0\right)x+{\theta }_0\hspace{0.17em}for\hspace{0.17em}0\le x\le \frac{-a}{2}$

(1)

Figure 1 depicts the fiber configuration in a square VAT lamina. The fiber path rotation angle in this study is zero. The fiber orientation of single VAT composite layer represented by $\left[〈{\theta }_0/{\theta }_1〉\right]$ . Figure 2 illustrates the lamina with ⟨60°/30°⟩. For ${\theta }_0$ = ${\theta }_1$ denotes a lamina with straight fibre’s with constant stiffness.

Figure 1.

Schematics diagram of VAT laminates composite square plate.

Figure 2.

Variable Angle Tow laminates composite plate with $〈60°/30°〉$ .

Problem formulation

The objective of this work is to enhance the critical buckling enhance the critical buckling temperature of the composite plate by optimizing the fiber path in each layer. The fiber orientation in each layer of the Variable Angle Tow (VAT) laminate is considered as a design variable. Specifically, the fiber path for each layer is defined by two parameters, denoted as $〈{\theta }_0/{\theta }_1〉$ , which represent the fiber angles at the center and edges of the plate, respectively. The fiber orientation is assumed to vary smoothly within the range of 0° to 90° across the plate.

Mathematical formulation of the optimization problem

The optimization problem is formulated as follows:

Objective: Maximize the critical buckling temperature ${∆T}_{cr}$

With respect to design variables: $\left[〈{\theta }_0/{\theta }_1〉\right]$ .

$\left[〈{\theta }_0/{\theta }_1〉\right]$ is the fiber tow angle at centre and at edges.

Subject to design Constraints: $0°\le \left[〈{\theta }_0/{\theta }_1〉\right]\le 90°$ .

Manufacturing Constraint (Maximum Curvature): A constraint is imposed to keep the maximum curvature of the fibre of each lamina within the limit of 3.28 m^{-1 10}

The curvature $K\left(x\right)$ of the fiber path is given by the equation:

$K\left(x\right)=\frac{2\left({\theta }_1-{\theta }_0\right)}{a}\cos \left(\left({\theta }_1-{\theta }_0\right)\frac{x}{a/2}+{\theta }_0\right)<3.28$

Where, $a$ is the length of the plate in the fiber direction x-direction, x is the position along the plate length.

The curvature constraint is essential from a manufacturing feasibility standpoint. In VAT composites, fibers are steered along curved paths. However, excessive curvature can lead to tow wrinkling or breakage, poor bonding with the matrix, manufacturing errors or fiber misalignment. To avoid such issues, a limit is placed on the maximum allowable curvature, $K_{\mathit{max}}<3.28$ m^-1 based on empirical manufacturing capabilities of fiber placement machines.

Displacement field

FSDT is used to formulate the mathematical model for buckling analysis of plate. The displacement fields at any position from mid-plane are expressed as ⁵⁶

$u=u_0+z{\varnothing }_x$

$v=v_0+z{\varnothing }_y$

$w=w_0$

(2)

where, u ₀ , v ₀ , w ₀ represents the mid-plane displacements in x, y, and z directions. Normal rotations ${\varnothing }_x,{\varnothing }_y$ at transverse (yz and zx) planes are denoted by respectively.

The strain field of VAT composite plate is represented as:

$\left\{\epsilon \right\}=\left\{\begin{array}{l}{\mathit{\epsilon }}_x\\ {\mathit{\epsilon }}_y\\ \begin{array}{l}{\mathit{\gamma }}_{xy}\\ \begin{array}{l}{\mathit{\gamma }}_{yz}\\ {\mathit{\gamma }}_{zx}\end{array}\end{array}\end{array}\right\}=\left\{\begin{array}{c}{u,}_x\\ {u,}_y\\ \begin{array}{l}{u,}_y+{u,}_x\\ \begin{array}{l}{\varnothing }_y+{w,}_y\\ {\varnothing }_x+{w,}_x\end{array}\end{array}\end{array}\right\}=\left[B\right]\left\{q_i\right\}$

(3)

The plates in-plane strains are ${\mathit{\epsilon }}_x$ , ${\mathit{\epsilon }}_y$ , ${\mathit{\gamma }}_{xy}$ whereas transverse shear strains are ${\mathit{\gamma }}_{yz}$ , ${\mathit{\gamma }}_{zx}$ .

The strain vector in matrix form:

$\left\{\mathit{\epsilon }\right\}=\left\{{\mathit{\epsilon }}_0\right\}+z\left\{\mathit{k}\right\}\hspace{0.17em};\hspace{0.17em}\left\{\mathit{\gamma }\right\}=\left\{{\gamma }_s^0\right\}$

(4)

$\left\{{\mathit{\epsilon }}_{\mathbf{0}}\right\}=\left\{\begin{array}{c}{u}_{0,x}\\ v_{0,y}\\ u_{0,x}+v_{0,y}\end{array}\right\};\hspace{0.17em}\left\{\mathit{k}\right\}=\left\{\begin{array}{l}{\varnothing }_{x,x}\\ {\varnothing }_{y,y}\\ {\varnothing }_{x,x}+{\varnothing }_{y,y}\end{array}\right\}\hspace{0.17em};\hspace{0.17em}\left\{{\gamma }_s^0\right\}=\left\{\begin{array}{l}{\varnothing }_y+w_{0,y}\\ {\varnothing }_x+w_{0,x}\end{array}\right\}$

(5)

Where ${\mathit{\epsilon }}_0$ are the in-plane strains, ${\mathit{\gamma }}_s^0$ are the transverse shear strains and $\mathit{k}$ represents the curvatures of the middle surface of the plate, respectively.

The resulting stress-strain relationship for the k ^th layer of the VAT Composite plate having uniform temperature rise, $∆T$ is expressed as:

${\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ \begin{array}{l}{\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{zx}\end{array}\end{array}\right\}}^k={\left[\mathit{Q}\left(\mathit{x}\right)\right]}^k{\left\{\begin{array}{l}{\epsilon }_x-{\alpha }_x∆T\\ {\epsilon }_y-{\alpha }_y∆T\\ \begin{array}{c}{\gamma }_{xy}-{\alpha }_{xy}∆T\\ {\gamma }_{yz}\\ {\gamma }_{zx}\end{array}\end{array}\right\}}^k$

(6)

Where, $\mathit{\sigma }=\{$ ${\sigma }_x$ ${\sigma }_y$ ${\tau }_{xy}$ ${\tau }_{yz}$ ${\tau }_{zx}$ }^t is the stress vector, and ${\alpha }_x$ ${\alpha }_y$ ${\alpha }_{xy}$ are the thermal expansion coefficients of the k ^th layer of the VAT plate given by:

${\left\{\begin{array}{l}{\alpha }_x\left(x\right)\\ {\alpha }_y\left(x\right)\\ {\alpha }_{xy}\left(x\right)\end{array}\right\}}^k={\left[\begin{array}{ll}{m}^2& n^2\\ n^2& m^2\\ mn& mn\end{array}\right]}^k{\left\{\begin{array}{l}{\alpha }_{11}\\ {\alpha }_{22}\end{array}\right\}}^k$

(7)

m = cos $\theta$ ; n = sin $\theta$

The transformed stiffness matrix varies with respect to x. The k ^th layer changes the transformed stiffness matrix ( ${\left[\mathit{Q}\left(\mathit{x}\right)\right]}^k$ ) elements as:

${\mathit{Q}}_{11}^k\left({\theta }^k\left(x\right)\right)=L_1+L_2\hspace{0.17em}\mathit{cos}\left(2{\theta }^k\left(x\right)\right)+L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{22}^k\left({\theta }^k\left(x\right)\right)=L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)-L_2\hspace{0.17em}\mathit{cos}\left(2{\theta }^k\left(x\right)\right)+L_1$

${\mathit{Q}}_{12}^k\left({\theta }^k\left(x\right)\right)=L_4-L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{66}^k\left({\theta }^k\left(x\right)\right)=L_5-L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{16}^k\left({\theta }^k\left(x\right)\right)=L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)+\frac{1}{2}{L}_2\hspace{0.17em}\mathit{sin}\left(2{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{26}^k\left({\theta }^k\left(x\right)\right)=-L_3\hspace{0.17em}\mathit{cos}\left(4{\theta }^k\left(x\right)\right)+\frac{1}{2}{L}_2\hspace{0.17em}\mathit{sin}\left(2{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{44}^k\left({\theta }^k\left(x\right)\right)=L_7\hspace{0.17em}\mathit{cos}\left(2{\theta }^k\left(x\right)\right)+L_6$

${\mathit{Q}}_{55}^k\left({\theta }^k\left(x\right)\right)=-L_7\hspace{0.17em}\mathit{sin}\left(2{\theta }^k\left(x\right)\right)$

${\mathit{Q}}_{45}^k\left({\theta }^k\left(x\right)\right)=-L_7\hspace{0.17em}\mathit{cos}\left(2{\theta }^k\left(x\right)\right)+L_6{\mathit{Q}}_{14}^k={\mathit{Q}}_{15}^k={\mathit{Q}}_{24}^k={\mathit{Q}}_{25}^k={\mathit{Q}}_{34}^k={\mathit{Q}}_{35}^k=0$

(8)

Where,

$L_1=\frac{1}{8}\left(3R_{11}+3R_{22}+2R_{12}+4R_{66}\right),$

$L_2=\frac{1}{2}\left(R_{11}-R_{22}\right),L_3=\frac{1}{8}\left(R_{11}+R_{22}-2R_{12}-4R_{66}\right),L_4=\frac{1}{8}\left(R_{11}+R_{22}+6R_{12}-4R_{66}\right),$

$L_5=\frac{1}{8}\left(R_{11}+R_{22}-2R_{12}+4R_{66}\right),L_6=\frac{1}{2}\left(R_{44}+R_{55}\right),$

$L_7=\frac{1}{2}\left(R_{44}-R_{55}\right)\hspace{0.17em}\text{and}\hspace{0.17em}{R}_{11}=E_{11}/\left(1-{\upsilon }_{12}{\upsilon }_{21}\right),R_{22}=E_{22}/\left(1-{\upsilon }_{12}{\upsilon }_{21}\right),R_{12}=R_{21}={\upsilon }_{21}{E}_{11}/\left(1-{\upsilon }_{21}{\upsilon }_{12}\right)={\upsilon }_{12}{E}_{22}/\left(1-{\upsilon }_{21}{\upsilon }_{12}\right),R_{44}=G_{23},R_{66}=G_{12},P_{55}=G_{13}$

For change in temperature, the component of forces and moments is represented as:

$\left\{\begin{array}{l}\mathit{N}\left(x\right)\\ \mathit{M}\left(x\right)\\ {\mathit{N}}_s\left(x\right)\end{array}\right\}=\left[\begin{array}{ccc}A\left(x\right)& \mathit{B}\left(x\right)& 0\\ \mathit{B}\left(x\right)& \mathit{D}\left(x\right)& 0\\ 0& 0& {\mathit{A}}^s\left(x\right)\end{array}\right]\left\{\begin{array}{l}{\mathit{\epsilon }}_{\mathbf{0}}\\ \mathit{k}\\ \mathit{\gamma }\end{array}\right\}-\left\{\begin{array}{c}{N}^{∆T}\left(x\right)\\ M^{∆T}\left(x\right)\\ 0\end{array}\right\}$

(9)

Where $\left\{\mathit{N}\left(x\right)\right\}={\left\{N_{xx\hspace{0.17em}}\hspace{0.17em}{N}_{yy\hspace{0.17em}}\hspace{0.17em}{N}_{xy\hspace{0.17em}}\right\}}^t$ ; $\left\{\mathit{M}\left(x\right)\right\}={\left\{M_{xx\hspace{0.17em}}\hspace{0.17em}{M}_{yy\hspace{0.17em}}\hspace{0.17em}{M}_{xy\hspace{0.17em}}\right\}}^t$

$\left\{{\mathit{N}}_{\mathit{s}}\left(x\right)\right\}={\left\{N_{sx\hspace{0.17em}}\hspace{0.17em}{N}_{sy\hspace{0.17em}}\hspace{0.17em}\right\}}^t$

Where $\left[\mathit{A}\left(x\right)\right],\left[\mathit{B}\left(x\right)\right],\left[\mathit{D}\left(x\right)\right],\left[{\mathit{A}}^{\mathit{s}}\left(x\right)\right]$ are the matrices for extensional, flexural-extensional coupling, flexural, and shear stiffnesses respectively.

The coefficients of these matrices are determined as:

${\mathit{A}}_{ij}\left(x\right)={\displaystyle \sum }_{k=1}^n\left(z_k-z_{k-1}\right)Q_{ij}^k\left(x\right)$

$B_{ij}\left(x\right)=\frac{1}{2}{\displaystyle \sum }_{k=1}^n\left(z_k^2-z_{k-1}^2\right)Q_{ij}^k\left(x\right)$

${\mathit{D}}_{ij}\left(x\right)=\frac{1}{3}{\displaystyle \sum }_{k=1}^n\left(z_k^3-z_{k-1}^3\right)Q_{ij}^k\left(x\right)$

${\mathit{A}}_{ij}^s\left(x\right)=\propto {\displaystyle \sum }_{k=1}^n\left(z_k-z_{k-1}\right)Q_{ij}^k\left(x\right)$

Where i,j = 1,2,6

Here, $z_k-z_{k-1}$ are the thickness coordinate of the upper and lower coordinate of k _{th lamina} and $\propto$ is transverse shear correction factor, which is equal to 5/6.

The force and Moment are computed as:

$\left[\left\{\begin{array}{l}{N}_x^{∆T}\left(x\right)\\ N_y^{∆T}\left(y\right)\\ N_{xy}^{∆T}\left(x\right)\end{array}\right\},\left\{\begin{array}{l}{M}_x^{∆T}\left(x\right)\\ M_y^{∆T}\left(y\right)\\ M_{xy}^{∆T}\left(x\right)\end{array}\right\}\right]={\displaystyle \sum }_{k=1}^n{\int }_{-h/2}^{h/2}\left\{\left[\begin{array}{lll}{Q}_{11}\left(x\right)& Q_{12}\left(x\right)& Q_{16}\left(x\right)\\ Q_{12}\left(x\right)& Q_{22}\left(x\right)& Q_{26}\left(x\right)\\ Q_{16}\left(x\right)& Q_{26}\left(x\right)& Q_{66}\left(x\right)\end{array}\right]\left[\begin{array}{l}{\alpha }_x\left(x\right)\\ {\alpha }_y\left(x\right)\\ {\alpha }_{xy}\left(x\right)\end{array}\right]\right\}\left(1,z\right)∆Tdz$

(10)

FEM formulation

The FE model is built on an eight-node isoparametric element, each with five degrees of freedom. The quadrilateral element’s shape function in natural coordinates $\left(\xi ,\eta \right)$ are as follows:

${\mathit{N}}_i=\frac{1}{4}\left({\eta }^2+\eta {\eta }_i\right)\left({\xi }^2+\xi {\xi }_i\right),\left(i=1,2,3,4\right)$

${\mathit{N}}_i=\frac{1}{4}\left({\eta }^2+\eta {\eta }_i\right)\left(1-{\xi }^2\right),\left(i=5,7\right)$

${\mathit{N}}_i=\frac{1}{4}\left(1-{\eta }^2\right)\left({\xi }^2+\xi {\xi }_i\right),\left(i=6,8\right)$

The displacement fields are interpolated as follows:

$U_0={\displaystyle \sum }_{i=1}^8{N}_i{U}_i$

$V_0={\displaystyle \sum }_{i=1}^8{N}_i{V}_i$

$W_0={\displaystyle \sum }_{i=1}^8{N}_i{W}_i$

${\varnothing }_x={\displaystyle \sum }_{i=1}^8{N}_i{\varnothing }_{xi}$

${\varnothing }_y={\displaystyle \sum }_{i=1}^8{N}_i{\varnothing }_{yi}$

Energy equation

The total potential energy U pertaining to VAT plate for thermos-elastic analysis can be expressed as:

$U=\frac{1}{2}\int {\left\{\sigma \right\}}^T\left\{\epsilon -{\epsilon }^{\prime }\right\}dV$

(11)

Where $\epsilon$ is the elastic strain vector and ${\epsilon }^{\prime }$ is the thermal strain vector

The strain energy for the VAT plate subjected to uniform temperature is:

$U=\frac{1}{2}\iiint \left(q_i^T{B}^T{Q}^{T}B{q}_i-2q_i^T{B}^T{Q}^T{\epsilon }^{\prime }+{{\epsilon }^{\prime }}^T{Q}^T{\epsilon }^{\prime }\right)dV$

(12)

where $\left\{q_i\right\}$ , is the plate displacements and $\left[B\right]$ is strain displacement matrix.

The thermo-elastic governing equation is obtained as:

$\left[K_p\right]q_i=F_T$

(13)

Where $K_p$ = $\iiint {\mathit{B}}^{\mathit{T}}{\mathit{Q}}^T\mathit{B}\hspace{0.17em}dV$

$F_T=\iiint B^T{Q}^T{\epsilon }^{\prime }dV$

The following standard eigenvalue problem is obtained as:

$\left[K_p-{\mathrm{\lambda }}_b\left[K_G\right]\right]\left\{q\right\}=0$

(14)

$\left[K_p\right]$ is the elastic stiffness matrix for the composite plate, $\left[K_G\right]$ is the geometric stiffness due to in-plane load, ${\mathrm{\lambda }}_b$ is the buckling load factor, and the corresponding buckling mode shape is $\left\{q\right\}.$

The element stiffness matrix is $K_p^e$ and the geometric stiffness matrice for the plate is $K_G^e$

$K_p^e={\int }_{-1}^{+1}\underset{-1}{\overset{+1}{{\displaystyle \int }}}{B}^{T}DB\left|J\right|d\xi d\eta$

$\left[K_G^e\right]={\int }_{-1}^1{\int }_{-1}^1\left({\left[B^{NL}\right]}^T\left[\sigma \right]\left[B^{NL}\right]\right)\left|J\right|d\xi d\eta$

Where,

$\left[\sigma \right]=diag\left[\begin{array}{lll}h\tau & \frac{h^3}{12}\tau & \frac{h^3}{12}\tau \end{array}\right]$

$\tau =\left[\begin{array}{ll}{\sigma }_x& {\sigma }_{xy}\\ {\sigma }_{xy}& {\sigma }_y\end{array}\right]$

The composite plate’s elastic and geometric stiffness matrices are calculated using numerical integration. Gaussian quadrature is used for obtaining the integration needed for the computation of the stiffness matrix for the composite plate. The thermal buckling analysis has been modelled as a conventional eigenvalue problem. To get the critical buckling temperature ${∆T}_{cr}$ , multiply the lowest eigenvalue ${\mathrm{\lambda }}_b$ by the initial guessed value of ΔT. The critical temperature of buckling is calculated by:

${∆T}_{cr}={\mathrm{\lambda }}_b\mathrm{\Delta }\mathrm{T}$

Optimization using GA

This section describes a global finite element (FE)-based optimization technique for optimizing the critical buckling temperature of Variable Angle Tow (VAT) composite plates. A GA is employed to determine the optimal fiber tow angle by assessing several design configurations and choosing the one that results in the maximum thermal buckling temperature. Figure 3 presents an integrated optimization framework that combines Genetic Algorithm (GA) and Finite Element Method (FEM), both implemented in MATLAB, to maximize the thermal buckling resistance of composite structures. The process is divided into two main stages. In the first stage, a parametric finite element model is developed to perform thermal buckling analysis. This includes the application of thermal loads and boundary conditions, and the computation of the critical buckling temperature, which serves as the objective function for optimization. In the second stage, the GA module initiates the optimization by generating an initial population of design variables. Each design candidate is evaluated based on its fitness, which is determined through the FEM analysis. If the optimal result is not achieved, the algorithm proceeds with selection, crossover, and mutation operations to evolve the population. This iterative process continues until the convergence criteria are met. The integration of GA with FEM enables an efficient and automated approach to identify optimal design parameters that enhance the thermal stability of composite laminates.

Figure 3.

Flow chart of the finite element-based optimization.

For determining the critical buckling temperature, the first step is to build a parametric model of the FEM code using the design variables in MATLAB. The MATLAB FEM Code accepts a new set of design variables produced by the GA optimization tool with each iteration. The optimization process terminates when the best and mean fitness values converge. The finally optimized VAT plate is obtained through this iterative optimization method. In the GA optimum search, a sufficiently large population and generation are employed to prevent local optimization outcomes. To acquire convergent findings, many GA search trials were conducted with distinct beginning populations. The crossover and mutation probabilities are set at 0.8 and 0.04.

Results and discussions

The developed FE model is used to analyse the optimal VAT Plate design for the critical buckling temperature. The optimal VAT fiber orientation for each layer is determined for various plate aspect ratios, boundary conditions, variation in thermal expansion coefficient and variation in Young’s modulus. The four-layers symmetric VAT plate is identified by ${\left[{}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉\right]}_s$ which expand as first layer $+〈{\theta }_0/{\theta }_1〉,$ second layer $-〈{\theta }_0/{\theta }_1〉$ third layer $〈{\theta }_0/{\theta }_1〉$ and fourth layer $-〈{\theta }_0/{\theta }_1〉$ . Similarly, eight-layers VAT Plate is ${\left[{}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉\right]}_{2s}$ . The present FE model for a straight fiber laminate composite plate is first validated, followed by a VAT laminate composite plate. Then, the result is expanded for different lamination sequences and two different boundary conditions, like clamped on all edges and simply supported on all edges. We validated the proposed FE-based optimization using results from the literature in the section that followed. Then, we obtained optimal results for various aspect ratios. The effect of material anisotropy and the effect of thermal expansion ratio on the optimal design of the VAT plate are also studied.

Numerical validation

The finite element findings are first compared to the existing literature for straight fiber and then VAT fiber to determine the correctness of the present FE code.

Validation of straight fiber

A rectangular composite plate of dimensions length = 15 m, width = 12 m, total thickness = 0.048 m. The plate is supported along all four edges with simple supports. The material characteristics include: longitudinal modulus E₁ = 22.5 MPa, transverse modulus E ₂ = 1.17 MPa, in-plane shear modulus G ₁₂ = 0.66 MPa, and Poisson’s ratio ν ₁₂ = 0.22. The thermal expansion coefficients are α ₁ = −0.04 × 10^-6 (1/ $℃$ ) and α ₂ = 16.7 × 10^-6 (1/ $℃$ ). Table 1 shows that the critical buckling temperatures are in good agreement with those reported by Shiau et at. ¹⁸ in the literature. Due to the consideration of shear deformation, the present critical buckling temperature results are lower than the results found in the literature. Figure 4 illustrates the first critical mode shape for Angle-ply and Cross-ply Straight fiber with a corresponding load factor of 12.247 and 13.645, respectively.

Table 1.

Critical buckling temperature ( ${\mathrm{\Delta }\mathit{T}}_{\mathit{c}\mathit{r}}℃$ ) of CSC plates with straight fiber.

S.no	Laminate sequence	Shiau et at 18	Present study
1	${\left[0°/45°/-45°/90°\right]}_{\mathrm{s}}$	13.71	13.65
2	${\left[0°/90°/90°/0°\right]}_s$	12.26	12.25

Figure 4.

Thermal buckling mode shape for straight fiber. (a) Angle-ply ${\left[0°/45°/-45°/90°\right]}_{\mathrm{s}}$ (b) Cross-ply ${\left[0°/90°/90°/0°\right]}_s$ .

Validation of VAT laminated plate

A square VAT plate is taken into consideration, which has a side length of a = 0.15 m and thickness of h = 1.016 mm. A symmetric laminate design with four layers ${\left({}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉\right)}_s$ is considered with simply supported boundary conditions. The material properties of the plate are taken as the same as Duran et al. ²⁷ and presented in Table 2. First, a mesh convergence study of critical buckling temperature is undertaken, and the results are shown in Table 3. Figure 5 illustrates a 24 × 24 finite element (FE) mesh used in the numerical analysis. A finer mesh would improve precision but increase computational cost, while a coarser mesh might reduce accuracy. The 24 × 24 configuration offers sufficient resolution to accurately capture buckling behaviour without imposing excessive computational demand. The present FE results are compared with those given by Duran et al., ²⁷ Zhao et al. ⁵³ and shown in Table 4 for various materials. Table 4 demonstrate that the results are consistent with Zhao et al. ⁵³ There is a slight 1%–2% discrepancy from Refs. Duran et al. ²⁷ A noticeable discrepancy is observed between the present results and those reported by Duran et al. ²⁷ in the evaluation of in-plane stress distributions. This difference can be attributed to the fact that the elastic strain contributions were neglected in the stress calculations presented in reference 27. In contrast, the current study incorporates both thermal and elastic strains in the formulation, leading to a more accurate representation of the in-plane stress response of the composite plate under thermal loading. As a result, the present model offers improved fidelity in predicting thermo-mechanical behaviour, especially for cases where elastic deformation plays a significant role in structural performance. The thermal buckling mode shape for ${(\pm {}_{-}{}^{+}〈60.70°/32.19°〉)}_s$ VAT laminated plate is shown in Figure 6.

Table 2.

Material properties for various materials, Duran et al. ²⁷

S. No	Material	E 1	E 2	G 12	${\vartheta }_{12}$	${\alpha }_1{*10}^{-6}$	${\alpha }_2{*10}^{-6}$
1	S-Glass|epoxy	45	11.0	4.5	0.29	7.1	30
2	Graphite|epoxy	155	8.07	4.55	0.22	−0.07	30.1
3	Boron|epoxy	201	21.7	5.4	0.17	6.1	30
4	Carbon|Peek	138	8.7	5.0	0.28	−0.2	24
5	E-Glass|epoxy	41	10.04	4.3	0.28	7.0	26
6	Carbon|Polyimide	216	5.0	4.5	0.25	0.0	25
7	Kevlar|epoxy	80	5.5	2.2	0.34	−2.0	60
8	Carbon|epoxy	147	10.3	7.0	0.27	−0.9	27

Table 3.

Convergence studies of critical buckling temperature ( ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ ) for a VAT laminated plate.

S. No	Boundary condition	Mesh size
		6 × 6	12 × 12	18 × 18	24 × 24
1	SSSS	32.7071	32.1507	32.0481	32.014
2	CCCC	85.5972	77.3081	77.2228	77.2194

Figure 5.

FE meshed model of VAT Plate.

Table 4.

Comparisons of different materials critical buckling temperature ( ${\mathrm{\Delta }\mathbf{T}}_{\mathbf{cr}}℃$ ) for a VAT laminated plate for simply-supported boundary conditions.

	Material	${\left({}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉\right)}_s$	Duran et al. 27	Zhao et al. 53	Present study
1	Graphite|epoxy	$\left(60.70°/32.19°\right)$	34.26	31.99	32.01
2	E-Glass|epoxy	$\left(6.71°/58.04°\right)$	5.58	5.48	5.47
3	S-Glass|epoxy	$\left(16.12°/54.74°\right)$	5.04	4.96	4.96
4	Kevlar|epoxy	$\left(66.05°/11.73°\right)$	22.18	16.09	16.08
5	Carbon|epoxy	$\left(69.00°/-5.70°\right)$	57.79	33.80	33.77
6	Carbon|Peek	$\left(63.07°/29.50°\right)$	38.08	34.93	34.92
7	Carbon|Polyimide	$\left(56.30°/36.68°\right)$	78.28	74.89	74.93
8	Boron|epoxy	$\left(-6.57°/63.28°\right)$	7.50	7.35	7.34

Figure 6.

First and second thermal buckling mode shape for VAT Plate with laminate ${(\pm {}_{-}{}^{+}〈60.70°/32.19°〉)}_s$ (a) First thermal buckling mode shape (b) Second thermal buckling mode shape.

Numerical result

The analysis is further extended for various lamination sequences and boundary conditions. A Square VAT Plate of dimension a = 1m, b = 1m and thickness 0.008 m or (a/h = 125) with 8 layers is considered. The material considered is graphite/epoxy, which are taken from Table 2. Table 5 shows the critical buckling temperature of VAT plates for different lamination sequences and boundary conditions (SSSS & CCCC). The fiber angle distribution and boundary conditions have a substantial influence on the critical buckling temperature of the VAT plate as shown in Table 5. Also, the critical buckling temperature of VAT plates for CCCC boundary conditions is always the maximum. For Simply-supported and clamped, the first thermal buckling mode shape for VAT Plate with laminate ${\left({}_{-}{}^{+}〈30/45〉\right)}_{2s}$ as shown in Figure 7.

Table 5.

Critical thermal buckling temperature (°C) of VAT plates for various lamination sequences.

S. No	Laminates	${\mathrm{\Delta }\mathbf{T}}_{\mathbf{cr}}℃$
		SSSS	CCCC
1	${(\pm {}_{-}{}^{+}〈15/45〉)}_{2s}$	47.75	77.50
2	${(\pm {}_{-}{}^{+}〈30/45〉)}_{2s}$	51.49	99.98
3	${(\pm {}_{-}{}^{+}〈45/45〉)}_{2s}$	55.098	115.94
4	${(\pm {}_{-}{}^{+}〈60/45〉)}_{2s}$	56.68	126.14
5	${(\pm {}_{-}{}^{+}〈45/15〉)}_{2s}$	47.79	138.81
6	${(\pm {}_{-}{}^{+}〈45/30〉)}_{2s}$	54.24	130.89
7	${(\pm {}_{-}{}^{+}〈45/60〉)}_{2s}$	48.18	100.10

Figure 7.

First thermal buckling mode shape for VAT Plate with laminate ${(\pm {}_{-}{}^{+}〈30/45〉)}_{2s}$ (a) Simply-supported (b) Clamped.

Influence of laminate layers on the VAT Plate’s critical buckling temperature

In this we have considered four antisymmetric layers $\left[〈30°|0°〉,〈45°|90°〉\right]$ , square VAT Plate with dimension a = 1m and b = 1m. with h/a = 0.01 and 0.1. The first layer is $〈30°|0°〉$ and second layer is $〈45°|90°〉$ . The first four buckling temperatures for simply supported and clamped boundary conditions are presented in Table 6. The first thermal buckling mode shapes of the VAT plate for simply supported (SSSS) and clamped (CCCC) boundary conditions are illustrated in Figure 8. Figure 9 shows the variation of critical buckling temperature with respect to the aspect ratio (a/b) of a simply supported VAT laminated plate (SSSS). As observed in the plot, the critical buckling temperature decreases significantly with an increase in aspect ratio. At a square configuration (a/b = 1), the plate demonstrates the highest thermal buckling resistance. However, as the plate becomes more rectangular (increasing a/b), the critical temperature drops sharply at first and then gradually stabilizes beyond a certain aspect ratio.

Table 6.

Critical buckling temperature ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ of antisymmetric VAT Plate.

	$\left[〈30°|0°〉,〈45°|90°〉,〈30°|0°〉,〈45°|90°〉\right]$
h/a	SSSS-boundary condition				CCCC-boundary condition
	1	2	3	4	1	2	3	4
0.01	38.9609	48.1410	77.2106	105.5942	89.7576	94.6763	139.6869	155.3043
0.1	2908.0	3413.9	4781.4	5.758.7	4865.1	4891.7	6640.9	6877.8

Figure 8.

First thermal buckling mode shape of VAT Plate.

Figure 9.

Variation of critical buckling temperature with respect to plate aspect ratio (a/b = 1 to 4).

Influence of the Young’s modulus ratio (E₁/E₂)

In this section examines the effect of the Young’s modulus ratio (E₁/E₂) on the critical buckling temperature of VAT composite plates. Thin square VAT laminated plates with an aspect ratio of a/h = 125, subjected to a uniform temperature rise, are considered. The material properties considered is high modulus graphite-epoxy ¹¹ E₁₁/ E₂₂ = 40.0, ${\vartheta }_{12}$ = 0.25, G₁₂/ E₂₂ = 0.6, ${\alpha }_{11}$ = 1x ${10}^{-6}$ $℃$ ^-1, ${\alpha }_{22}/{\alpha }_{11}$ = 2, E₂₂ = 10 GPa. The critical buckling temperature ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ ) for various E₁/E₂ ratios, which vary from 3 to 40, is presented in Table 7 for two boundary conditions: all edges simply supported (SSSS) and all edges clamped (CCCC). It is observed that VAT laminated plates exhibit higher critical buckling temperatures under clamped boundary conditions. As expected, an increase in the Young’s modulus ratio (E₁/E₂) leads to a higher critical buckling temperature of the VAT composite plate, due to the corresponding increase in bending stiffness.

Table 7.

Critical buckling temperature of square laminated plates with a/h = 125.

Lamination	E1/E2	VAT plate ( ${\mathbf{\Delta }\mathrm{T}}_{\text{cr}}℃$ ) (SSSS)	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ ) (CCCC)
${(\pm {}_{-}{}^{+}〈15°/45°〉)}_{2s}$	3	71.99	173.39
	10	89.14	187.89
	20	95.10	192.70
	30	97.05	194.14
	40	97.82	194.59
${(\pm {}_{-}{}^{+}〈15°/45°〉)}_s$	3	69.34	169.11
	10	77.97	169.68
	20	79.05	166.28
	30	78.87	163.96
	40	78.51	162.28

Optimal result

In this section, GA is used to determine the optimal fiber angle distribution $\left[〈{\theta }_0/{\theta }_1〉\right]$ . of different layups for maximum critical buckling temperature. The optimization procedure is explained in Section 5. The material properties are graphite/epoxy taken from Table 2 and the ply thickness is t = 0.000,127 m. The total layers of the symmetrical VAT plates are four and eight. Firstly, we validated the proposed FE-based optimization using results from the literature. Then we obtained optimal results for various aspect ratios, the effect of material anisotropy and the effect of thermal expansion ratio. Optimal fiber angle distribution of VAT plates and its critical buckling temperature are compared with those of constant stiffness composites (CSC) with straight fibers.

Validation of optimal results

In order to demonstrate the effectiveness of method, we compare the present FEM based optimization results with the result available in literature. ²⁷ From Table 8, it is seen that the present optimization and reference 27 produces almost the same results. Figure 10 presents a typical Genetic Algorithm (GA) convergence plot, consisting of two subplots. The upper graph illustrates the progression of the fitness value over generations. It shows a rapid improvement in fitness during the initial iterations, with the values stabilizing around the best fitness of −37.682 and a mean fitness of −37.6794, indicating that the population has converged toward an optimal solution. The maximum critical buckling temperature is 55.1038 with ${(\pm {}_{-}{}^{+}〈45°\hspace{0.5em}/45°〉)}_{2s}$ orientation for constant stiffness composites (CSC) square plate

Table 8.

Comparison between the present FEM-based optimization results, and Duran et al. ²⁷ results.

S.no	Material	Optimal laminate sequence $\left[〈{\theta }_0/{\theta }_1〉\right]$	Critical buckling temperature ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )
1	Duran et al. 27 (graphite/epoxy)	$\left(60.70°/32.19°\right)$	34.26
	Present optimization	$\left(56.80°/40.01°\right)$	32.613
2	Duran et al. 27 (e-glass/epoxy)	$\left(6.70°/58.04°\right)$	5.58
	Present optimization	$\left(10.85°/57.43°\right)$	5.47

Figure 10.

GA optimization iteration history plot for objective function.

Optimal result for various aspect ratio of VAT plate

In this study, optimal results are investigated for various aspect ratios (a/b = 1,1.5.2.2.5,3) of simply-supported and clamped VAT plates with a/h = 125, where h is the total thickness of the plate. The optimum VAT plate results are then compared to a Constant Stiffness Composite (CSC) plate with straight fibers. For CSC Plate, ${(\pm {}_{-}{}^{+}〈45/45〉)}_{2s}$ laminate has been chosen as a reference because it is widely used straight fibres. The optimum results are determined for four layers ${(\pm {}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉)}_s$ and eight layers ${(\pm {}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉)}_{2s}$ of symmetrical VAT plates.

The optimal critical buckling temperatures ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ ) ) for different aspect ratios of simply-supported and clamped VAT plates with 8 layers are presented in Table 9 and Table 10, respectively. For the simply-supported case, Table 9 reveals a notable improvement in thermal buckling performance of the VAT plates compared to CSC plates. Specifically, the VAT configuration shows an increase in critical temperature by 4.31%, 36.95%, 69.14%, 88.64%, and 111.82% at aspect ratios (a/b) of 1, 1.5, 2, 2.5, and 3, respectively.

Table 9.

Optimal result for eight layers ${(\pm {}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉)}_{2s}$ simply-supported VAT Plate.

a/b	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ )	% Increase
1	58.1040	38.6323	57.477	55.1038	4.31%
1.5	70.6130	28.2741	51.57	37.6555	36.95%
2	61.8564	52.1828	50.31	29.7436	69.14%
2.5	62.54	55.19	48.20	25.5509	88.64%
3	61.1868	59.7993	48.93	23.0991	111.82%

Table 10.

Optimal result for eight layers ${(\pm {}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉)}_{2s}$ clamped VAT Plate.

a/b	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathbf{T}}_{\text{cr}}℃$ )	% Increase
1	60.4957	6.6196	151.694	115.9513	30.82%
1.5	58.1682	29.6347	118.354	85.7264	38.06%
2	54.6367	54.2068	105.327	76.5910	37.51%
2.5	54.1655	50.5499	104.011	72.8872	42.70%
3	57.1357	48.3457	102.243	71.0924	43.81%

Similarly, Table 10 highlights the behaviour under clamped boundary conditions, where the VAT plates again outperform CSC plates. The improvement in critical buckling temperature is recorded as 30.82%, 38.06%, 37.51%, 42.70%, and 43.81% at the corresponding aspect ratios. These results confirm that VAT design significantly enhances thermal stability under both boundary conditions, with especially higher gains observed as the aspect ratio increases.

Table 11 presents the critical buckling temperatures of simply-supported VAT plates with four layers across different aspect ratios. The results indicate that the VAT plate exhibits a consistent improvement over the CSC plate, with increases of 6.12%, 38.66%, 57.70%, 75.03%, and 86.12% at aspect ratios of 1, 1.5, 2, 2.5, and 3, respectively. This enhancement is due to the tailored fiber orientation in VAT laminates, which provides better resistance against thermal buckling.

Table 11.

Optimal result for four layers ${(\pm {}_{-}{}^{+}〈{\theta }_0/{\theta }_1〉)}_s$ for simply-supported plate.

a/b	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	% Increase
1	56.7622	40.1952	45.28	44.0812	6.12
1.5	63.5971	42.9582	42.46	30.5729	38.66
2	58.6682	51.1941	39.36	24.8507	57.70
2.5	61.5346	56.1941	38.29	21.9996	75.03
3	58.2236	59.3823	37.8161	20.4229	86.12

Interestingly, although the critical buckling temperatures differ significantly between VAT and CSC plates, the buckling mode shapes remain identical, indicating that the nature of deformation does not change, only the threshold at which it occurs.

Furthermore, both VAT and CSC plates show a decreasing trend in critical buckling temperature with increasing aspect ratio, as seen in Tables 9 and 11. This trend is more noticeable in shorter plates (lower aspect ratios) due to their higher structural stiffness, which allows them to resist thermal loads more effectively. As the aspect ratio increases and the plate becomes longer, the overall stiffness reduces, leading to a lesser capacity to resist buckling, and thus a lower critical buckling temperature. This explains why shorter plates exhibit higher thermal buckling resistance in both VAT and CSC configurations.

Figures 11 and 12 illustrate the comparison of critical buckling temperatures between optimal VAT (Variable Angle Tow) plates and CSC (Constant Stiffness Composite) plates under simply supported boundary conditions for eight-layer and four-layer laminates, respectively, across different aspect ratios. These results clearly demonstrate the effectiveness of VAT design in improving thermal buckling resistance over traditional CSC plates, with the benefit becoming more pronounced at higher aspect ratios and for thicker (eight-layer) laminates.

Figure 11.

Comparison of critical buckling temperature of optimal VAT Plate with reference CSC Plate for Eight layer.

Figure 12.

Comparison of critical buckling temperature of optimal VAT Plate with reference CSC Plate for four layers.

Optimal result for various material anisotropy

The influence of the material anisotropy ratio, E₁₁/ E₂₂, on the optimal design is explored for eight-layered square VAT plates. The material properties considered is high modulus graphite-epoxy Campagna et al. ⁵⁷ E₁₁/ E₂₂ = 40.0, ${\vartheta }_{12}$ = 0.25, G₁₂/ E₂₂ = 0.6, ${\alpha }_{11}$ = 1x ${10}^{-6}$ $℃$ ^-1, ${\alpha }_{22}/{\alpha }_{11}$ = 2, E₂₂ = 10 GPa. The influence of material anisotropy on optimal results for critical buckling temperature are reported in Table 12 and Table 13 for the simply-supported and clamped boundary conditions, respectively. It is seen that optimal critical temperature for various material anisotropy increases, when compare with reference CSC Plate. For various value of E₁₁/ E₂₂, optimal ${\theta }_0$ is close to 0⁰ and ${\theta }_1$ is 65⁰ to 75⁰ for simply supported plate and optimal ${\theta }_0$ is close to 0⁰ and ${\theta }_1$ is 85⁰ to 90⁰ except E₁₁/ E₂₂ = 3 for the clamped plate. The critical buckling temperature increases with an increase in the E₁₁/ E₂₂ ratio. This is because the stiffness of laminates increases with an increase in E₁₁.

Table 12.

Influence of the degree of orthotropy on optimal critical buckling temperature of a simply-supported plate.

S. No.	E11/ E22	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	% increase
1	3	0.1784	65.9478	74.9577	71.3185	5.10
2	10	0	70.1856	104.55	85.00	23.00
3	20	0.0468	72.1071	122.924	89.0111	38.09
4	30	0.2498	73.1904	132.852	90.1683	47.33
5	40	0.2420	74.0156	139.3931	90.5464	53.94

Table 13.

Influence of the degree of orthotropy on optimal critical buckling temperature of a clamped plate.

S. No.	E11/ E22	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	% increase
1	3	89.6756	5.7450	193.355	177.0672	9.198
2	10	0.2141	89.5610	215.943	185.6673	16.30
3	20	0	88.0784	250.412	187.5694	33.50
4	30	0	87.4526	271.3141	187.8696	44.41
5	40	0.1590	84.9400	285.649	187.7459	52.41

Optimal results for various thermal expansion

The effect of thermal expansion ratio, α₂/α₁, on the optimal design is investigated for eight-layered square plates. The same material used in the previous section high modulus graphite-epoxy is employed here as well. The influence of thermal expansion on the optimal critical buckling temperature is presented in Tables 14 and 15 for simply-supported and clamped plates, respectively. It is observed that for VAT Plate, critical buckling temperature increases by 92.09% for α₂/α₁ = −2, when compared with CSC Plate. However, for α₂/α₁ values greater than 50, the increment is very negligible. Overall, the critical buckling temperature increase with increasing α₂/α₁ ratio.

Table 14.

Optimal critical buckling temperature of a simply-supported plate as influenced by the α₂/α₁ ratio.

S. No.	α2/α1	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	% increase
1	−2	0.214	77.664	196.991	102.5475	92.09
2	2	0.495	73.885	138.806	90.5534	53.28
3	50	43.021	45.874	37.682	37.6751	0.018
4	100	52.252	41.532	23.574	23.4257	0.633
5	500	56.720	39.045	5.969	5.8190	2.577
6	1000	57.039	39.730	3.089	3.0003	2.95

Table 15.

Optimal critical buckling temperature of a clamped plate as influenced by the α₂/α₁ ratio.

S. No.	α2/α1	${\theta }_0$	${\theta }_1$	VAT plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	CSC plate ( ${\mathrm{\Delta }\mathrm{T}}_{\text{cr}}℃$ )	% increase
1	−2	0	86.388	458.559	212.614	115.67
2	2	0.4956	85.587	285.0966	187.7459	51.85
3	50	60.836	5.855	90.4923	78.11	15.85
4	100	59.670	5.145	58.4874	48.5682	20.42
5	500	59.1434	5.2592	15.2622	12.0644	26.50
6	1000	58.9469	5.7925	7.932	6.2204	27.51

Conclusions

This study presents an optimization framework for enhancing the thermal buckling performance of Variable Angle Tow (VAT) laminated composite plates by considering the fiber orientation within each lamina as a design variable. The critical buckling temperature was computed using finite element analysis based on First-Order Shear Deformation Theory (FSDT) with eight-node isoparametric elements. The optimized VAT configurations demonstrated significantly higher thermal buckling resistance compared to conventional Constant Stiffness Composite (CSC) laminates. Specifically, for a plate with an aspect ratio a/b = 3, the critical buckling temperature increased by 111.82% for an eight-layer laminate and by 86.12% for a four-layer laminate. The increase in critical buckling temperature in VAT laminates is due to their ability to spatially tailor stiffness and thermal expansion, resulting in a more uniform stress distribution and improved resistance to thermal buckling.

Parametric studies revealed that increasing the number of layers improves the critical buckling temperature, while a higher aspect ratio leads to a reduction. Higher aspect ratio has lower stiffness and resistance to thermal-induced buckling due to increased flexibility and slenderness. Furthermore, the thermal buckling resistance increases with a higher stiffness ratio (E₁₁/E₂₂) and decreases with a higher thermal expansion ratio (α₂/α₁). These findings highlight the effectiveness of fiber path optimization in tailoring the mechanical and thermal responses of laminated composites.

The enhanced performance of VAT plates is attributed to their ability to spatially vary stiffness and thermal properties, enabling improved stress distribution and delayed thermal buckling. The results provide valuable insights for the structural design of high-performance composite components subjected to thermal loads, particularly in aerospace and energy sectors.

Future research should address experimental validation of the optimized configurations, integration of fiber curvature manufacturing constraints, and extension to multi-objective design problems that consider both mechanical and thermal loads. Additionally, the incorporation of advanced manufacturing techniques such as automated fiber placement (AFP) will be essential to realize the full potential of optimized VAT laminates in practical applications.