Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method

Abstract

Fiber-reinforced and laminated composite materials, widely used in engineering applications, may develop periodic curvature during manufacturing due to technological requirements. Given such curvatures in widely used composites, static and dynamic analyses of plates and shells under loads, along with related stability issues, have been extensively investigated. However, studies focusing specifically on the static analysis of such materials remain limited. Composite materials with structural curvature exhibit complex mechanical behavior, making their analysis particularly challenging. Predicting their mechanical response is crucial in engineering. In response to this need, the present study conducts a static analysis of plates made of periodically curved composite materials using the Navier method. The plate equations were derived based on the Kirchhoff–Love plate theory within the framework of the Continuum Theory proposed by Akbarov and Guz’. Using the Navier method, deflection, stress, and moment distributions were obtained at every point of the plate. Numerical results were computed using MATLAB. After verifying the convergence and accuracy of the developed MATLAB code by comparing it with existing solutions for rectangular homogeneous isotropic and laminated composite plates, results were obtained for periodically curved plates. This study offers valuable insights that may guide future research, as it employs the Navier method to provide an analytical solution framework. This study contributes to the limited literature with a novel evaluation of the static analysis of composite plates with periodic curvature.

1. Introduction

Due to their widespread use in industries such as aerospace, shipbuilding, aviation, automotive, and medicine, composite materials have become a subject of interest in various scientific disciplines, including chemistry and mechanics. To enhance their strength or to meet specific functional requirements, these materials can be manufactured with periodic curvature. In fiber-reinforced and laminated composite materials, structural curvatures may arise either during the design stage or as a result of manufacturing processes. These geometric imperfections can significantly affect the mechanical properties of the material, including stiffness, strength, and stability [1,2,3,4,5]. As a result, the deflection, buckling, and vibration behavior of such materials under various static and dynamic loads has become a significant topic of research [1,2,3,4,5,6].

Akbarov and Guz’ [1,6,7] derived the continuity equations for composite materials with curvature and presented both theoretical and numerical results for different types of matrix and fiber materials. Yahnioğlu [8], using the Continuum Theory developed by Akbarov and Guz’, investigated two- and three-dimensional boundary value problems of curved composite materials through the finite element method and provided numerical examples. Kutug [9,10] conducted analyses on the natural vibration and stability behavior of composite beams and plates with curvature based on the Continuum Theory of Akbarov and Guz’ and employed the third-order shear deformation plate theory.

Yahnioğlu [11] carried out a three-dimensional stress analysis of multilayered composite plates with small-scale periodic curvatures by employing the Continuum Theory of Akbarov and Guz’ and the semi-analytical finite element method. The results highlighted the significant impact of structural curving on the stress distribution in such plates.

Akbarov and Guz’ [1] investigated the problems of stability loss, nonlinear vibrations, and failure under compressive loads in composite structures with material curvature. They emphasized that the influence of curvature on stress distribution constitutes a significant area for future research.

Youzera, Ali, Meftah, Tounsi, and Hussain [12] investigated the parametric nonlinear vibration behavior of three-layer symmetric laminated plates using the harmonic balance method and the Galerkin approach. They analyzed the frequency response curves for various materials and geometric configurations of the core layer.

Draiche and Tounsi [13] performed the static bending and free vibration analysis of cross-ply laminated composite spherical shells by employing an equivalent single-layer shell displacement model. For this purpose, they developed a new hyperbolic shear deformation theory (RHSDT) based on the equivalent single-layer shell displacement model. The governing equations were derived using Hamilton’s principle and validated through the Navier solution method. Additionally, the study examined the impact of geometric parameters on the bending and free vibration responses of supported laminated spherical shells.

Studies on the dynamic behavior of laminated sandwich composite shells indicate that there is a need for advanced analytical approaches in this field. Attia, Berrabah, Bourada, Bousahla, Tounsi, Salem, Khedher, and Le [14] investigated the free vibration behavior of laminated sandwich composite shells using both analytical methods and the finite element method, highlighting the importance of combining theoretical and numerical approaches in this area.

Demiriz and Akbarov [15] investigated the natural and forced vibration behavior of composite plates with periodic curvature in two directions within the framework of the three-dimensional theory of elasticity. The plates were clamped on all edges and subjected to uniformly distributed time-varying normal loads on their upper surfaces. Numerical results obtained via three-dimensional finite element modeling revealed that the local maxima of the thickness-direction normal stress significantly increased due to the structural curvature, emphasizing the necessity of considering these effects when predicting the mechanical performance, such as the adhesion strength, of composite plates with two-directional periodic curvature.

Zamanov [16] investigated the effect of periodic structural curvature on the stress distribution of thick rectangular composite plates subjected to forced vibration, using the exact three-dimensional theory of elasticity. The study revealed that even low-frequency dynamic loading leads to significantly higher stress concentrations due to the curvature effects.

In a related study, Zamanov [17] analyzed the natural vibration behavior of rectangular composite plates with a periodically curved structure using the three-dimensional elasticity theory. His findings further highlighted the significant influence of structural curvature on the vibration characteristics of such plates.

Akbarov and Yahnioğlu [18] investigated the stress distribution in multilayered composite plates exhibiting local structural damage in the form of curved fiber layers. By employing three-dimensional elasticity theory and a semi-analytical finite element method, they demonstrated that such local curvatures, which may arise during manufacturing, significantly affect the internal stress fields and must be accounted for through accurate constitutive models based on continuum mechanics.

Akbarov, Maksudov, Panakhov, and Seifullayev [19] investigated crack problems in composite materials with periodically curved layers by employing the exact three-dimensional elasticity theory within the framework of a piecewise-homogeneous body model. Their study highlights a specific fracture mechanism observed in unidirectional composites under uniaxial loading, where the material splits along the loading direction, an effect not seen in homogeneous materials. The authors emphasize that even slight curvatures in the reinforcing elements can significantly influence the internal stress distribution and act as a trigger for crack formation. To address this, they developed a modeling approach that incorporates the presence of periodically curved layers and allows for a more accurate investigation of fracture behavior. The proposed method considers cracks as pre-existing along the most vulnerable regions of the matrix, parallel to the external load, offering deeper insight into the mechanical response of such complex composite systems.

While the modeling in [19] assumes periodically curved reinforcing layers, it has been observed that, in practical applications, the curvature of the reinforcing elements may not always follow a periodic pattern. Instead, local curvatures can arise due to manufacturing imperfections or design irregularities. In this context, Akbarov [20] extended the existing approach to analyze the fracture behavior of composites with locally curved reinforcing layers, using the same piecewise-homogeneous body model and exact elasticity theory. His findings highlight the importance of incorporating such irregularities in fracture modeling, especially when standard macroscopic criteria fail to capture the underlying failure mechanisms.

Huang, Chen, and Dai [21] proposed a two-scale asymptotic homogenization method for composite Kirchhoff plates with in-plane periodicity. By simplifying a 3D periodic plate problem into a fourth-order elliptic PDE, they developed an efficient approach to capture both macro- and microscale behaviors. Their method, validated through static and dynamic analyses, demonstrated high accuracy in predicting the mechanical response of thin periodic plates.

Qatu [22] presented the first known natural frequencies for laminated composite angle-ply triangular and trapezoidal plates with completely free boundaries using the Ritz method. The effectiveness of the approach was validated through convergence studies conducted on plates with various arbitrary geometries.

Aydoğdu [23] proposed a higher-order shear deformation theory derived from 3D elasticity solutions using an inverse method. The theory satisfies the stress boundary conditions exactly and provides highly accurate results for bending and stress analysis compared to classical five-degree-of-freedom models, offering improvements, particularly for laminated composite plates.

Mohite and Upadhyay [24] introduced a region-by-region modeling strategy for laminated composite plates, which enables the combination of equivalent, intermediate, and layerwise models within different zones of the plate domain. Instead of applying a single model throughout the structure, their approach allows the assignment of different models with varying orders in the thickness direction to specific areas of interest. This strategy offers a balance between computational efficiency and local accuracy, especially in complex regions with unsymmetric laminae, cut-outs, local damage, or material transitions. Their numerical results demonstrated that the method accurately captures local stresses and displacements while maintaining a low computational cost.

Moleiro, Mota Soares C.M., Mota Soares C.A. and Reddy [25] proposed a mixed finite element model based on the least-squares variational principle for the static analysis of laminated composite plates. The formulation adopts the first-order shear deformation theory, considering generalized displacements and stress resultants as independent variables. High-order C⁰ Lagrange interpolation functions are employed to enhance accuracy while maintaining computational efficiency. The numerical results, obtained for various boundary conditions and side-to-thickness ratios, demonstrate the effectiveness of the proposed method and its insensitivity to shear-locking when using high-order interpolation functions.

Predicting the mechanical behavior of composite plates with periodic curvature in structural analyses is of great significance, especially for advanced engineering applications, due to their geometric complexity. In response to this need, the present study investigates the static behavior of rectangular plates with periodic curvature, using the Navier method within the framework of the Continuum Theory of Akbarov and Guz’, based on the Kirchhoff–Love plate theory. While the literature offers numerous studies on the mechanical behavior of homogeneous isotropic or classically laminated composite materials [12,13,14], systematic static analyses focusing on deflection, bending moments, and stress distribution in periodically curved composite plates remain limited. Although both the Navier method and Kirchhoff–Love plate theory are well-established analytical tools, their integration with periodically curved geometries in laminated composite plates remains unexplored. While the Kirchhoff–Love plate theory remains a reliable analytical tool for thin plate behavior, its neglect for transverse shear deformation can limit its accuracy in the analysis of moderately thick and thick plates. Recent studies have addressed these limitations by proposing refined theoretical models and advanced numerical methods that incorporate shear deformation and thermomechanical coupling in layered or functionally graded structures, as seen in [26,27].

This study thoroughly examines the influence of material curvature on deflection, bending moments, and stress distribution. It also aims to address the research gap proposed by Akbarov and Guz’ [1] by numerically evaluating the effects of periodic curvature on the static response. Plates made of homogeneous isotropic, laminated composite, and periodically curved laminated materials were analyzed through coding carried out in the MATLAB 2024a environment. The results obtained for the homogeneous isotropic and laminated composite plates were compared with those from the literature and ANSYS 2024 R2 analyses, and the accuracy of the formulated equations and the implemented code was verified. Thus, the reliability of the numerical results obtained for the periodically curved laminated plates was established, as further illustrated by the results shown in Figure 1.

It is also worth noting that the theoretical foundation and modeling strategies adopted in this study were shaped by classical references [28,29], which provided substantial conceptual support.

2. Materials and Methods

The geometric boundaries of the plate in the Ox₁x₂x₃ coordinate system are defined as:

$$-{\displaystyle \frac{h}{2}}\le x_2\le {\displaystyle \frac{h}{2}} 0\le x_1\le l_1 0\le x_3\le l_3$$

(1)

As illustrated in Figure 1.

The equations of displacements, according to the Kirchhoff–Love plate theory, are expressed in Equations (2)–(4). This theory is based on the assumption that the normals to the mid-surface remain straight and perpendicular after deformation. Therefore, it is applicable only to thin plates where transverse shear effects are negligible [30].

$${\mathrm{u}}_1\left({\mathrm{x}}_1,{\mathrm{x}}_2{,x}_3\right)=u\left({\mathrm{x}}_1,{\mathrm{x}}_3\right)-x_2{w,}_1$$

(2)

$$u_2\left(x_1,x_2,x_3\right)=w\left(x_1,x_3\right)$$

(3)

$$u_3\left(x_1,x_2{,x}_3\right)=v\left(x_1,x_3\right)-x_2{w,}_3$$

(4)

where u₁, u₂, and u₃ are the displacements in the x₁, x₂, and x₃ directions, respectively. Similarly, u, w, and v represent the displacements of the mid-surface (or mid-plane) of the plate in the same x₁, x₂, and x₃ directions. The notation used herein is expressed in Equation (5), the stress–strain relationships are given by Equation (6), and the strain–displacement expressions are presented in Equation (7).

$${\left(\dots \right),}_{\mathrm{i}}={\displaystyle \frac{𝜕\left(\dots \right)}{{𝜕x}_{\mathrm{i}}}} i=\mathrm{1,3}.$$

(5)

$${\sigma }_{\mathrm{i}\mathrm{j}}=C_{\mathrm{i}\mathrm{j}\mathrm{k}\mathrm{l}}{\epsilon }_{\mathrm{k}\mathrm{l}} i,j,k,l=\text{1,2,3}.$$

(6)

$${\epsilon }_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{𝜕u_{\mathrm{i}}}{x_{\mathrm{j}}}}+{\displaystyle \frac{𝜕u_j}{x_{\mathrm{i}}}}\right) \mathrm{i},\mathrm{j}=\text{1,2,3}.$$

(7)

For a composite plate, Equation (6) can be rewritten in the form given in Equation (8).

$${\sigma }_i=A_{\mathrm{i}\mathrm{j}}{\epsilon }_{\mathrm{j} }i,j=\text{1,2,3,4,5,6}.$$

(8)

The expanded form of Equation (8) is given as follows:

$$\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\right]=\left[\begin{array}{c}{A}_{11} \\ A_{21} \\ A_{31} \\ A_{41}\\ A_{51} \\ A_{61} \end{array}\begin{array}{c}{A}_{12} \\ A_{22} \\ A_{32}\\ A_{42}\\ A_{52}\\ A_{62}\end{array}\begin{array}{c}{A}_{13} \\ A_{23 }\\ A_{33}\\ A_{43}\\ A_{53}\\ A_{63}\end{array}\begin{array}{c}{A}_{14} \\ A_{24} \\ A_{34}\\ A_{44}\\ A_{54}\\ A_{64}\end{array}\begin{array}{c}{A}_{15} \\ A_{25} \\ A_{35}\\ A_{45}\\ A_{55}\\ A_{65}\end{array}\begin{array}{c}{A}_{16}\\ A_{26}\\ A_{36}\\ A_{46}\\ A_{56}\\ A_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\right]$$

(9)

The following notation is expressed in Equation (9), as shown in Equation (10):

$$\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\right]=\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right],....\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\right]=\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right]$$

(10)

In the present study, A₁₅ and A₃₅ are taken as zero due to the presence of periodic curvature only in the x₁ direction [3].

Substituting Equations (2)–(4) into Equation (7) and Equation (9)

$$\left[\begin{array}{c}{\sigma }_1\\ {\begin{array}{c}\sigma \end{array}}_3\\ {\sigma }_5\end{array}\right]=\left[\begin{array}{c}{\begin{array}{c}A\end{array}}_{11} \\ \\ A_{31} \\ \\ 0 \\ \end{array} \begin{array}{c}{\begin{array}{c}A\end{array}}_{13} \\ \\ A_{33} \\ \\ 0 \\ \end{array}\begin{array}{c}0 \\ \\ 0 \\ \\ A_{55} \\ \end{array}\right].\left[\begin{array}{c}{\epsilon }_{11}^0-x_2{w}_{,11}\\ {\epsilon }_{33}^0-x_2{w}_{,33}\\ {\epsilon }_{13}^0-{2x}_2{w}_{,13} \end{array}\right]$$

(11)

It is obtained where ${\epsilon }_{ij}^0$ denotes the strains at the mid-surface of the plate. It should be noted that ${\epsilon }_{12}= {\epsilon }_{23}= {\epsilon }_{22}=0$, and the periodic curvature is introduced only along the $x_1$ direction [7]. This assumption is based on the classical Kirchhoff–Love plate theory, which adopts the plane stress condition for thin plates and neglects transverse shear deformations [30].

For composite materials with periodic curvature along the x₁ direction, the stiffness coefficients A_ij (x₁) are expressed as functions of the curvature parameters [8,10]. As described in Ref. [7], the material constants in Equation (11) can be expanded as

$$\begin{gathered} A_{11}\left(x_1\right)=A_{11}^0{\varnothing }_1^4\left(x_1\right)+2A_{12}^0{\varnothing }_1^2\left(x_1\right){\varnothing }_2^2\left(x_1\right)+ \\ {+A}_{22}^0{\varnothing }_2^4\left(x_1\right)+4A_{66}^0{\varnothing }_1^2\left(x_1\right){\varnothing }_2^2\left(x_1\right), \\ {A}_{13}\left(x_1\right)=A_{13}^0{\varnothing }_1^2\left(x_1\right)+A_{23}^0{\varnothing }_2^2\left(x_1\right), \\ {A}_{33}\left(x_1\right)=A_{33}^0, \\ {A}_{55}\left(x_1\right)=A_{55}^0{\varnothing }_1^2\left(x_1\right)+A_{44}^0{\varnothing }_2^2\left(x_1\right). \end{gathered}$$

(12)

Related to Equation (12), the material constants A₁₁⁰, A₃₃⁰, A₁₂⁰, A₂₃⁰, A₂₂⁰, A₆₆⁰, A₅₅⁰, A₁₃⁰, and A₄₄⁰ correspond to the normalized material constants for the composite plate without periodic curvatures [7]:

$$\begin{gathered} A_{11}^0=A_{33}^0={\mu }_1{\eta }_1+{\mu }_2{\eta }_2+\left({\mu }_1+{\eta }_1\right){\eta }_1 +\left({\mu }_2+{\eta }_2\right){\eta }_2- {\displaystyle \frac{{\left({\lambda }_1-{\lambda }_2\right)}^2}{\left({\lambda }_1+{2\mu }_1\right){\eta }_2+\left({\lambda }_2+{2\mu }_2\right){\eta }_1}} , \\ {A}_{12}^0=A_{23}^0={\lambda }_1{\eta }_1+{\lambda }_2{\eta }_2-\left({\lambda }_1-{\lambda }_2\right){\eta }_1{\eta }_2{\displaystyle \frac{\left({\lambda }_1+2{\mu }_1\right)\left({\lambda }_2+2{\mu }_2\right)}{\left({\lambda }_1+{2\mu }_1\right){\eta }_2+\left({\lambda }_2+{2\mu }_2\right){\eta }_1 }}, \\ { A}_{22}^0=\left({\lambda }_1+2{\mu }_1\right){\eta }_1+\left({\lambda }_2+2{\mu }_2\right){\eta }_2-{\eta }_1{\eta }_2{\displaystyle \frac{{\left[\left({\lambda }_1+2{\mu }_1\right)-\left({\lambda }_2+2{\mu }_2\right)\right]}^2}{\left({\lambda }_1+{2\mu }_1\right){\eta }_2+\left({\lambda }_2+{2\mu }_2\right){\eta }_1}}, \\ {A}_{66}^0=A_{55}^0={\displaystyle \frac{{\mu }_1{\mu }_2}{{\mu }_1{\eta }_2+{\mu }_2{\eta }_1}}, \\ {A}_{13}^0=\left({\lambda }_1+{\mu }_1\right){\eta }_1+\left({\lambda }_2+{\mu }_2\right){\eta }_2-{\eta }_1{\eta }_2{\displaystyle \frac{{\left({\lambda }_1-{\lambda }_2\right)}^2}{\left({\lambda }_1+2{\mu }_1\right){\eta }_2+\left({\lambda }_2+2{\mu }_2\right){\eta }_1}}- \\ -{\eta }_1{\mu }_1-{\eta }_2{\mu }_2, \\ {A}_{44}^0={\mu }_1{\eta }_1+{\mu }_2{\eta }_{2 }. \end{gathered}$$

(13)

λ_i and μ_i denote the Lamé constants and η_i volume fractions of the matrix and reinforcement materials, respectively, where the index i = 1 refers to the matrix material and i = 2 to the reinforcement material.

In Equation (13),

$$\begin{gathered} {\varphi }_1={\displaystyle \frac{1}{\sqrt{1+{\frac{dF\left(x_1\right)}{d{x}_1}}^2}}} \\ {\varphi }_2={\varphi }_1{\displaystyle \frac{dF\left(x_1\right)}{d{x}_1}} \end{gathered}$$

(14)

where the function F (x₁) in Equation (14) can be expressed as shown in Equation (15).

$$\begin{gathered} F\left(x_1\right)=\epsilon f\left(x_1\right) \\ \epsilon ={\displaystyle \frac{\overline{h}}{\Lambda }} \\ f\left(x_1\right)=\mathit{sin}\left({\displaystyle \frac{\pi x_1}{\Lambda }}+\delta \right) \end{gathered}$$

(15)

Here, Λ, $\bar{\mathrm{h}}$, and δ represent the wavelength, amplitude, and phase difference of the periodic curvature, respectively, as also illustrated in Figure 1 and Figure 2.

According to the three-dimensional elasticity theory, the equilibrium equations can be expressed as given in Equation (16).

$${\sigma }_{\mathrm{i}\mathrm{j},\mathrm{j}}+F_{\mathrm{i}}=0, i,j=\text{1,2,3}.$$

(16)

Integrating the stresses σ_ij through the thickness of the plate gives

$${\mathrm{T}}_{ij}={\int }_{-h/2}^{h/2}{\sigma }_{ij}{dx}_2 \mathrm{i},\mathrm{j}=\text{1,2,3}.$$

(17)

Similarly, by multiplying the stress components σ_ij by x₂ and integrating through the plate thickness, the bending moments M_ij can be obtained, as expressed in Equation (18). These resultant quantities, namely T_ij and M_ij, are expressed per unit width of the plate.

$${\mathrm{M}}_{ij}={\int }_{-h/2}^{h/2}{\sigma }_{ij}{x_{2}dx}_2 \mathrm{i},\mathrm{j}=\mathrm{1,2,3}.$$

(18)

Taking into account the condition σ₂₂ = 0 and integrating Equation (16) through the thickness of the plate,

$$N_{\mathrm{12,1}}+N_{\mathrm{23,3}}=P(x_1,x_3)$$

(19)

is obtained, where P(x₁,x₃) is the distributed vertical load on the plate surface. The moments obtained from Equation (18),

$$M_{\mathrm{11,1}}+M_{\mathrm{13,3}}-N_{12}=0$$

(20)

$$M_{\mathrm{13,1}}+M_{\mathrm{33,3}}-N_{23}=0$$

(21)

are obtained. Differentiating Equation (20) with respect to x₁ and Equation (21) with respect to x₃ and substituting the resultant equations in Equation (19) gives the following equilibrium equation for a plate:

$$M_{11,11}+2M_{13,13}+M_{\mathrm{33,33}}=P(x_1,x_3)$$

(22)

where, from (18),

$$\left[\begin{array}{c}{M}_{11}\\ {\begin{array}{c}M\end{array}}_{33}\\ M_{13}\end{array}\right]=\left[\begin{array}{c}{\begin{array}{c}A\end{array}}_{11} \\ \\ A_{13} \\ \\ 0 \\ \end{array} \begin{array}{c}{\begin{array}{c}A\end{array}}_{13} \\ \\ A_{33} \\ \\ 0 \\ \end{array}\begin{array}{c}0 \\ \\ 0 \\ \\ A_{55} \\ \end{array}\right].\left[\begin{array}{c}-{\displaystyle \frac{h^3}{12}}{w}_{,11}\\ -{\displaystyle \frac{h^3}{12}}{w}_{,33}\\ -{\displaystyle \frac{h^3}{6}}{w}_{,13} \end{array}\right]$$

(23)

By substituting Equation (23) into Equation (22), the governing equation of the plate becomes

$$\begin{gathered} -{\displaystyle \frac{h^3}{12}}{\left[A_{11}\left(x_1\right)w_{,11}+{A_{13}\left({\mathrm{x}}_1\right)w}_{,33}\right]}_{,11}-{\displaystyle \frac{h^3}{3}}{\left[A_{55}(x_1{)w}_{,13}\right]}_{,13}- \\ -{\displaystyle \frac{ h^3}{12}}{\left[A_{13}\left(x_1\right)w_{,11}+A_{33}\left(x_1\right)w_{,33}\right]}_{,33}=P(x_1,x_3) \end{gathered}$$

(24)

Accordingly, the plate equilibrium equation is obtained. The boundary conditions for the simply supported plate on all four edges are

$$\begin{gathered} x_1=0,{\mathrm{l}}_1; w=0, w_{,11}=0 \\ x_3=0,{\mathrm{l}}_3; w=0, w_{,33}=0. \end{gathered}$$

(25)

The deflection expression that satisfies the boundary conditions of Equation (24) is given as

$$w(x_1,x_3)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{w}_{mn} \sin {\displaystyle \frac{m\pi x_1}{l_1}}\sin {\displaystyle \frac{n\pi x_3}{l_3}}$$

(26)

According to the Navier method, the distributed load P(x₁, x₃) is expressed in the form of a series [30] as

$$P\left(x_1,x_3\right)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{P}_{\mathrm{m}\mathrm{n}} \mathit{sin}{\displaystyle \frac{m\pi x_1}{l_1}}\mathit{sin}{\displaystyle \frac{n\pi x_3}{l_3}} .$$

(27)

$$P_{\mathrm{m}\mathrm{n}}={\displaystyle \frac{16p_0}{{\mathsf{\pi }}^{2}mn}} m,n=\text{1,3,5}\dots$$

(28)

Substituting Equations (26) and (27) into Equation (24),

$$\begin{array}{cc}\hfill {\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \sum _{n=1}^{\infty }}{w}_{mn}{\displaystyle \frac{h^3}{12}}& \left[\right(A_{11,11}{\left({\displaystyle \frac{m\pi }{l_1}}\right)}^2-A_{11}{\left({\displaystyle \frac{m\pi }{l_1}}\right)}^4+A_{\mathrm{13,11}}{\left({\displaystyle \frac{n\pi }{l_3}}\right)}^2\hfill \\ & -2A_{13}{\left({\displaystyle \frac{m\pi }{l_1}}\right)}^2{\left({\displaystyle \frac{n\pi }{l_3}}\right)}^2-4A_{55}{\left({\displaystyle \frac{m\pi }{l_1}}\right)}^2{\left({\displaystyle \frac{n\pi }{l_3}}\right)}^2\hfill \\ & -A_{33}{\left({\displaystyle \frac{n\pi }{l_3}}\right)}^4)sin{\displaystyle \frac{m\pi x_1}{l_1}}sin{\displaystyle \frac{n\pi x_3}{l_3}}\hfill \\ & +(2A_{11,1}{\left({\displaystyle \frac{m\pi }{l_1}}\right)}^3+2A_{\mathrm{13,1}}\left({\displaystyle \frac{m\pi }{l_1}}\right){\left({\displaystyle \frac{n\pi }{l_3}}\right)}^2\hfill \\ & +4A_{\mathrm{55,1}}\left({\displaystyle \frac{m\pi }{l_1}}\right){\left({\displaystyle \frac{n\pi }{l_3}}\right)}^2)cos{\displaystyle \frac{m\pi x_1}{l_1}}sin{\displaystyle \frac{n\pi x_3}{l_3}}]\hfill \\ & ={\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \sum _{n=1}^{\infty }}{P}_{mn} \mathit{sin}{\displaystyle \frac{m\pi x_1}{l_1}}\mathit{sin}{\displaystyle \frac{m\pi x_3}{l_3}} .\hfill \\ & m,n=\text{1,3,5}\dots \hfill \end{array}$$

(29)

Here, the coefficients w_mn are determined for each pair of m and n by utilizing the known values of P_mn. In this study, only the expression in Equation (28) is used for the case of a uniformly distributed load. For other loading conditions, new numerical results can be obtained from Equation (29) by employing the expressions given by Szilard [30].

Equation (29) is first solved for w_mn, and the obtained coefficients are then used in Equation (26) to calculate the deflection at the desired points. Consequently, the moment and stress values follow from Equation (23) and Equation (11), respectively. For a homogeneous isotropic plate, Equation (29) has constant coefficients; thus, the w_mn coefficients can be easily determined as functions of P_mn. However, in the case of a composite plate, the dependence of these coefficients on x₁ transforms the governing equation into a variable–coefficient form. Under such circumstances, the use of a numerical solution method, such as the Galerkin approach, becomes indispensable. Because the numerical results presented below were obtained within this framework, they are semi-analytical.

3. Results and Discussion

The plate, subject to the loading and boundary conditions illustrated in Figure 1, was analyzed with l₁ = 1 m, l₃ = 2 m, and a plate thickness of h = 0.1 m, where the applied load is defined as P (x₁, x₃) = 10,000 N/m². Solutions were obtained for three different material configurations: homogeneous isotropic, laminated composite, and curved laminated composite. For the composite case, the matrix material was assigned Young’s modulus of E₁ = 20 GPa and Poisson’s ratio ν₁ = 0.3, while the reinforcement material was assigned E₂ = 200 GPa and ν₂=0.3. The volume fractions of both components were set to η₁ = η₂ = 0.5. In the homogeneous isotropic case, the material properties of the matrix were used.

The deflection values and the corresponding relative errors (e_r) for the homogeneous isotropic plate at the coordinates x₁ = l₁/2, x₃ = l₃/2, obtained from the literature [30], the ANSYS model, and the MATLAB code developed in this study, are presented in

Table 1. Midpoint deflection values for an isotropic plate.

Midpoint Deflection (mm)
[30]	ANSYS	MATLAB
−0.05515	−0.05696 er = 2.81%	−0.05536 er = 0.38%

. It should be noted that in Ref. [30], a direct numerical value for the deflection is not provided; instead, an analytical formula is given for homogeneous isotropic rectangular plates under uniform loading. In this study, the deflection value was computed using that formula and then compared with the result obtained from the developed MATLAB code.

The analytical formula for the maximum deflection, wmax, for a homogeneous isotropic plate under uniform loading can be expressed as follows:

$$w_{max}= {\displaystyle \frac{0.0101p_0{a}^4}{D}}$$

(30)

where a is the plate’s span in the x₁ direction, and D is the flexural rigidity of the plate defined as

$$D={\displaystyle \frac{E{h}^3}{12(1-{\upsilon }^2)}}$$

(31)

This formula was used to compute the deflection value for comparison with the MATLAB code result. The ANSYS simulations were conducted using SHELL181 elements for both homogeneous isotropic and laminated composite plates. For the composite plate, the ACP (Pre) module was used to define the layered configuration. In both models, a mesh with 800 elements and 861 nodes was used to ensure consistency in comparison. In addition, a convergence check was performed by increasing the number of Fourier terms from 5 to 10 and 15, and the difference between 10 and 15 terms was found to be less than 0.001%. Therefore, the number of terms was not increased further, and the result presented in Table 1 corresponds to the case with five terms.

Figure 3 shows the deflection values of the homogeneous isotropic plate obtained from the ANSYS program.

The moment values M₁₁ and M₃₃ for the homogeneous isotropic case, obtained from the literature [30] and the MATLAB code developed in this study, are presented in

Table 2. M₁₁ and M₃₃ moment values for the case of an isotropic plate.

[30] ${\mathit{M}}_{11}$ (kN.m)	MATLAB ${\mathit{M}}_{11}$ (kN.m)	[30] ${\mathit{M}}_{33}$ (kN.m)	MATLAB ${\mathit{M}}_{33}$ (kN.m)
1.0250	1.032 er = 0.68%	0.4754	0.4745 er = 0.18%

. The values presented in this table were evaluated at the center of the plate (x₁ = l₁/2, x₃ = l₃/2).

Similarly, the stress values σ₁₁ and σ₃₃ obtained from the ANSYS model for the homogeneous isotropic case are compared with those obtained from the present study in

Table 3. σ₁₁ and σ₃₃ stress values for the case of isotropic plate.

ANSYS ${\mathit{\sigma }}_{11}$(MPa)	MATLAB ${\mathit{\sigma }}_{11}$ (MPa)	ANSYS ${\mathit{\sigma }}_{33}$ (MPa)	MATLAB ${\mathit{\sigma }}_{33}$ (MPa)
0.60834	0.61393 er = 0.92%	0.27885	0.28473 er = 2.11%

. The values presented in this table were evaluated at the center of the plate (x₁ = l₁/2, x₃ = l₃/2)

The deflection, moment, and stress values of homogeneous isotropic rectangular plates, obtained by the Navier method based on the Kirchhoff–Love plate theory, are found to be in good agreement with the results presented by Szilard [30] and those obtained from the ANSYS model.

For the laminated composite plate, the numerical values and corresponding relative errors (e_r) obtained from both the ANSYS model and the MATLAB code developed within the scope of this study at the coordinates x₁ = l₁/2 and x₃ = l₃/2 are presented in

Table 4. Midpoint deflection values for laminated composite plate.

ANSYS Deflection (mm)	MATLAB Deflection (mm)
−0.01242	−0.01271 er = 2.33%

. Since no analytical formula is available in the literature for systems with the same material properties as those used in the laminated composite case, only the results obtained from the ANSYS model and the developed MATLAB code are compared in Table 4 and

Table 5. ${\sigma }_{11}$ and ${\sigma }_{33}$ stress values for the case of a laminated composite plate.

ANSYS ${\mathit{\sigma }}_{11}$(MPa)	MATLAB ${\mathit{\sigma }}_{11}$ (MPa)	ANSYS ${\mathit{\sigma }}_{33}$ (MPa)	MATLAB ${\mathit{\sigma }}_{33}$ (MPa)
0.7148	0.7242 er = 1.31%	0.3262	0.3012 er = 7.69%

.

Figure 4 illustrates the deflection distribution obtained from the ANSYS analysis of the laminated composite plate.

The σ₁₁ and σ₃₃ stress values obtained from the ANSYS model for the laminated composite plate are compared with those obtained in this study in Table 5. The values presented in this table were evaluated at the center of the plate (x₁ = l₁/2, x₃ = l₃/2).

The deflection, moment, and stress results calculated for laminated composite rectangular plates using the Navier method based on the Kirchhoff–Love plate theory with MATLAB were found to be in close agreement with those obtained from ANSYS and reported in [30], with acceptable relative errors.

In this study, the static analysis of a laminated composite material with periodic curvature in a single direction (x₁) was conducted by varying the ε parameter. Since the plate geometry must satisfy the condition

$$\Lambda \ll h$$

(32)

The deflection values at the plate’s midpoint were computed up to the maximum allowable value of ε (i.e., ε = 0.009), and these results are presented in

Table 6. Midpoint deflection (×10⁻³ mm) of a composite plate with periodic curvature. For E_2/E₁ = 10 under varying ε and l₁/l₃ ratios.

	${\mathit{l}}_1/{\mathit{l}}_3$	0.1	0.5	1.0	2.0
$\mathit{\epsilon }$
0.000		−14.690	−12.710	−5.743	−0.7942
0.001		−14.710	−12.720	−5.746	−0.7943
0.002		−14.770	−12.770	−5.756	−0.7945
0.003		−14.880	−12.840	−5.772	−0.7949
0.004		−15.030	−12.950	−5.794	−0.7955
0.005		−15.240	−13.100	−5.824	−0.7963
0.006		−15.500	−13.280	−5.861	−0.7972
0.007		−15.830	−13.510	−5.907	−0.7983
0.008		−16.230	−13.790	−5.961	−0.7996
0.009		−16.730	−14.130	−6.025	−0.8011

.

Figure 5 illustrates the effect of varying the curvature parameter on the midpoint deflection for four different l₁/l₃ ratios, given that E₂/E₁ = 10. It can be observed that as the curvature increases, the amount of deflection increases significantly. Figure 5 illustrates the effect of varying the curvature parameter on the midpoint deflection for four different l₁/l₃ ratios, given that E₂/E₁ = 10. It can be observed that as the curvature increases, the amount of deflection increases significantly. Moreover, plates with lower l₁/l₃ ratios (for example, 0.1 and 0.5) exhibit higher sensitivity to curvature because the load transfer is concentrated along the shorter edge, which amplifies the effect of curvature on bending stiffness. In contrast, for wider plates (l₁/l₃ = 2), the load is distributed over a larger area, and the impact of curvature on global deflection becomes negligible.

Practical context: In this study, l₁ = 1 m was chosen so that the curvature parameter epsilon could be directly linked to typical manufacturing tolerances. Following Ref. [8], the wavelength-to-length ratio was set to Λ/l₁ = 1/16, which reflects realistic fiber waviness patterns. With this setup, the peak waviness amplitude is approximately 0.0625 × ε meters (about 62,500 × epsilon microns). Therefore, epsilon = 0.001, 0.005, and 0.009 correspond to about 63 μm, 313 μm, and 563 μm of fiber waviness over a 1 m layup, respectively. These magnitudes help practitioners assess when curvature effects become critical in thin laminated plates.

The values of bending moments M₁₁/M₃₃ at the midpoint of the laminated plate with periodic curvature are presented in

Table 7. $M_{11}$ and $M_{33}$ moment values for a laminated composite plate with periodic curvature ($E_2/E_1=10$; $\mathsf{\epsilon },l_1/l_3$: parametric).

${\mathit{M}}_{11}/{\mathit{M}}_{33}$  $\left({10}^{-2}\mathbf{k}\mathbf{N}.\mathbf{m}\right)$
	${\mathit{l}}_1/{\mathit{l}}_3$	0.1	0.5	1.0	2.0
$\mathit{\epsilon }$
0.000		134.5500	120.7100	63.2392	12.5752
		37.4866	50.1994	63.2189	30.1792
0.001		134.6900	120.8000	63.2510	12.5730
		37.5208	50.2534	63.2493	30.1815
0.002		135.0900	121.0900	63.2866	12.5665
		37.5335	50.4166	63.3408	30.1883
0.003		135.7700	121.5700	63.3472	12.5556
		37.8249	50.6935	63.4951	30.1996
0.004		136.7600	122.2600	63.4343	12.5403
		38.1010	51.0911	63.7147	30.2157
0.005		138.0800	123.1800	63.5506	12.5207
		38.4704	51.6202	64.0035	30.2366
0.006		139.7900	124.3600	63.6992	12.4967
		38.9449	52.2956	64.3665	30.2626
0.007		141.9300	125.8400	63.8845	12.4683
		39.5408	53.1370	64.8102	30.2940
0.008		144.5900	127.6500	64.1115	12.4356
		40.2799	54.1701	65.3427	30.3398
0.009		147.8700	129.8700	64.3869	12.3985
		41.1912	55.4288	65.9741	30.3740

.

Figure 6 illustrates the distributions of M₁₁ and M₃₃ moments for E₂/E₁ = 10 as the curvature parameter increases. It is observed that an increase in curvature leads to a noticeable rise in both moment components.

For l₁ = 1 m and l₃ = 0.5 m (i.e., l₁/l₃ = 2), the plate exhibits a beam-like behavior due to its short span along the x₃ direction, despite being simply supported on all edges. This geometry slightly redistributes the bending moments, with a minor relaxation of M₁₁ at the midpoint and a small increase in M₃₃. However, the overall change remains negligible (below 1.5%) and does not affect the general trends or conclusions.

Table 8. σ₁₁ and σ₃₃ stress values for a laminated composite plate with periodic curvature. ($E_2/E_1=10$; $\epsilon , l_1/l_3$: parametric).

${\mathit{\sigma }}_{11}/{\mathit{\sigma }}_{33}\left({10}^{-2} \mathbf{M}\mathbf{P}\mathbf{a}\right)$
	${\mathit{l}}_1/{\mathit{l}}_3$	0.1	0.5	1.0	2.0
$\mathit{\epsilon }$
0.000		80.7320	72.4250	37.9440	7.5451
		22.4900	30.1200	37.9310	18.1080
0.001		80.8110	72.4810	37.9510	7.5438
		22.5120	30.1520	37.9500	18.1090
0.002		81.0530	72.6100	37.9720	7.5399
		22.5800	30.2500	38.0040	18.1130
0.003		81.4640	72.9400	38.0080	7.53330
		22.6950	30.4160	38.0970	18.1200
0.004		82.0570	73.3550	38.0610	7.5242
		22.8610	30.6550	38.2290	18.1290
0.005		82.8510	73.9090	38.1300	7.51240
		23.0820	30.9720	38.4020	18.1420
0.006		83.8720	74.6170	38.2200	7.4980
		23.3670	31.3770	38.6200	18.1580
0.007		85.1570	75.5020	38.3310	7.48100
		23.7240	31.8820	38.8860	18.1760
0.008		86.7530	76.5900	38.4670	7.46140
		24.1680	32.5020	39.2060	18.1990
0.009		88.7250	77.9200	38.6320	7.43910
		24.7150	33.2570	39.5840	18.2240

presents the values of the stresses σ₁₁|^+h/2 and σ₃₃ |^+h/2 obtained for the laminated composite plate with periodic curvature.

As illustrated in Figure 7, the increase in the curvature parameter not only amplifies the deflection but also leads to a noticeable rise in bending moments and internal stress distributions. This suggests that the impact of curvature extends beyond deflection alone, significantly influencing both the load-bearing capacity and stress safety of the plate.

In addition to the observed increases in deflection and internal forces, it is noted that the structural response becomes more sensitive to curvature variations, particularly in plates with lower l₁/l₃ ratios. This is attributed to the fact that geometric changes caused by periodic curvature can significantly influence the stiffness and internal force distribution of the plate. As the curvature increases, these changes become more prominent, potentially compromising structural performance and reliability in practical applications.

3.1. MATLAB Implementation

The analytical solution was implemented in MATLAB using the Navier method, which is based on the Galerkin approach and combined with Gauss–Legendre quadrature for accurate evaluation of integrals. The algorithm starts by defining the geometry, material constants, and loading parameters. Periodic curvature functions and the associated material stiffness coefficients A_ij (x₁) are computed symbolically, and their derivatives are evaluated within the integration domain. Harmonic terms (m, n) are used to construct the load matrices P_mn, while the transformation matrices are assembled through Gauss integration of the governing equations. The coefficients wmn are then solved, from which the deflection w (x₁, x₃), bending moments M₁₁, M₃₃, and in-plane stresses σ₁₁, σ₃₃ are determined through structured loops.

3.2. Limitations

This study focuses on laminated rectangular plates with small-amplitude, one-dimensional periodic curvature (Λ << h, ε ≤ 0.009) under simply-supported boundary conditions. The analysis assumes linear elasticity and small deflections, neglecting geometric nonlinearity. Moreover, the formulation is based on the classical Kirchhoff–Love plate theory, where transverse shear deformations are not considered.

4. Conclusions

• An increase in the curvature parameter ε leads to a noticeable rise in deflection for all considered l₁/l₃ ratios.
• The curvature parameter ε becomes more effective as the l₁/l₃ ratio decreases.
• As the curvature parameter ε increases, the bending moments M₁₁ and M₃₃ increase correspondingly with the deflection for all l₁/l₃ ratios.
• As the curvature parameter ε increases, the stress components σ₁₁ and σ₃₃ also increase for all l₁/l₃ ratios.

This pioneering study enables the prediction of the engineering behavior of rectangular plates with periodic curvatures in their material structure. Since the curvature within the composite material affects the stress distribution, it may lead to material failure [8]. In this regard, this study offers a significant contribution to assessing design tolerances of composite structures and anticipating performance-related risks in advance.