Ahmed K. Sabea*
| Widad I. Majeed
© 2026 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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The present work proposes a four-variable refined theory for modelling free vibration of composite plates under thermal loading. The shape function comprises hyperbolic and polynomial terms for the first time to investigate natural frequencies in the thermal environment of cross- and angle-laminated plates. The dynamic governing equations are derived via variational framework of Hamilton’s principle by the transverse shear stress field follows a parabolic function through the plate’s depth, which yields zero traction on the free plate surfaces. A closed-form solution is conducted based on Navier’s solution with simply supported boundary conditions. The embedded shape function which was used for the first time in the free vibration under thermal loads analysis within proposed theory successfully provided less computational complexity yet convergence accuracy in predicting composite plates dynamic behavior in thermal environment compared to high order theory which involved computational challenges. Transverse shear stresses are directly modeled in the present theory, without requiring shear correction factor. The proposed theory results are consistent with other theories from the literature for both studied plates (thick and thin). Also, the influence of a range of design factors, including layer schemes, orthotropic modulus ratio, and thickness ratio, on the fundamental frequency of laminated composite plates under thermal loads have been analyzed.
natural frequency, thermal loads, laminated plates, refined plate theory, four variables, shear deformation theory
Laminate composite plates are extensively employed in modern aerospace industries, especially drones, due to their high stiffness-to-weight ratio. However, in service, temperature fluctuations can induce thermal stresses that adversely affect structural behavior, particularly by lowering natural frequencies and increasing the likelihood of vibration. Elevating the priority of researching the area of free vibration in thermal environments, such studies provide an insight into the thermo-mechanical coupling effect (vibration behavior of materials at high temperature, which differs from its behavior at room temperature), structural stability and safety, reliability, fatigue resistance, and advanced theories validation. Classical plate theory, one of the earlier theories proposed to assess laminated plates, loses its validity for multilayer composites by ignoring the critical influence of transverse shear.
Jameel and Hussien [1] investigated vibration of laminated plates influenced by thermal and mechanical or free vibration to show the effect of design variables such as geometric ratios, boundary conditions, and ply orientation angle based on classical plate theory. The study included analytical, numerical, and experimental analysis to demonstrate and confirm the effect of design parameters and the impact of thermal conditions on both natural and forced vibration. Majeed and Tayeh [2] also adopted classical plate theory to analyze the buckling stability and natural vibration characteristics of composite plates under in-plane compressive forces. Hamilton’s principle serves as the basis for establishing the governing equations, and a closed-form solution was obtained based on the Ritz method to exhibit the influence of design parameters on plate buckling and vibration. Hammed and Majeed [3] analysed free vibration of laminated thin plates resting on an elastic foundation based on classical plate theory. By applying the Ritz method and the imaginary spring technique, results were obtained, and researchers examined the effects of parameters such as (layers scheme, aspect ratio, thickness ratio, and the ratio of initial in-plane thermal load). According to the study, the influence of aspect ratio on natural frequency was between 35% and 40%. First-order shear deformation theory was proposed to overcome the physically inaccurate and overly restrictive kinematics assumption of the classical plate theory, but the theory fails to account for zero transverse shear stress on free surfaces of the plate, demanding a shear correction factor to overcome its miscalculations. Patro et al. [4] investigated the natural frequency of stiffened composite plates when subjected to thermal stresses via a finite element approach that incorporates first-order shear deformation theory in the ANSYS framework, providing numerical data that demonstrated the effect of (temperature, modulus ratio, and coefficient of thermal expansion). The results indicated that elevated temperature decreases natural frequency values. Results of the Das and Niyogi [5] study showed that increased structural stiffness correlates with an enhanced ability to withstand the combined effects of high temperature and moisture content. Wu et al. [6] used a numerical approach that combines the quadrature method with an iterative procedure, derived from Hamilton’s principle using a first-order shear deformation theory that accounts for von Karaman type geometric nonlinearity, to analyze large-amplitude vibrations Graphene-Platelet-Reinforced nanocomposite multilayer annular plates in a thermal environment. Employing a layer-based computational model, Zhai et al. [7] explored how composite sandwich plates vibrate and buckle when exposed to heat. The formulation, suitable for a range of plate thicknesses, incorporates thermal effects on material displacements. Other studies on the dynamic response of functionally graded nanoplates under free vibration were presented, considering two structural configurations. The model incorporates small-scale effects via the nonlocal elasticity theory of Eringen and uses a higher-order formulation to derive governing equations for simply supported plates, accounting for temperature-dependent material properties [8]. Ibrahim and Ghani [9] use the Rayleigh-Ritz approach to investigate the free vibration of a composite plate having general elastic fixing along its edges. By introducing sinusoidal and arbitrary continuous functional components as shape functions, the study’s numerical results showed good agreement with the literature. A study conducted by Hellal et al. [10] proposed a four-unknown shear deformation theory to investigate dynamic and buckling effects by hygro-thermal ambiance, using a functionally graded sandwich plate model supported by Winkler Pasternak elastic foundations. They added an integral term to the displacement field, thereby reducing the number of variables and the number of basic equations. Kallannavar et al. [11] used first-order deformation theory to examine the impact of temperature and moisture on bias lamination of a composite sandwich plate. The novelty of the work lies in the use of the artificial neural network technique; the model uses graphite-epoxy composite laminates as the face sheet and DYAD 606 viscoelastic material as the core. Moradi-Dastjerdi and Behdinan [12] introduced a novel intelligent sandwich structure with a five-lamina structure, a porous core, intermediate polymer/graphene nanocomposite layers, and outer piezoelectric faces. The investigation focuses on how temperature, piezoelectric effects, and an elastic foundation influence the plate’s natural frequencies. A free vibration analysis under thermal loads was conducted by Yahea and Majeed [13] for laminates with [0/90] and [±θ] ply configurations, employing trigonometric four-variable theory, applying the Hamilton principle to derive the governing equations, and using the Navier solution to evaluate the system’s undamped free-vibration frequencies, and compared to previous studies, showing good agreement. Draiche and Tounsi [14] proposed an innovative hyperbolic shear deformation theory for investigating the flexural and dynamic response of cross-ply laminated spherical shells. A refined plate theory proposed by Sadiq and Bawa [15] was used to model and evaluate the oscillatory motion of [0/90] laminated structure under initial stresses, with simply supported fixing, via Hamilton’s principle alongside computational output acquired from Navier’s solution, and validate against published findings. The work of Sahu et al. [16] investigates the vibration and dynamic transient response of hybrid laminated composite panels using a finite element model and compares it with in-house experimental data. The modal influence due to thermal loading was calculated using a higher-order displacement polynomial. Saini and Lal [17] applied first-order deformation theory to analyze the vibration of a bi-directional functionally graded circular plate under a two-dimensional thermal load, based on the energy-based Hamilton principle. The governing equations were derived; furthermore, numerical results were evaluated using the differential quadrature method. The study of Azzara et al. [18] focuses on vibration and buckling analysis of composite structures in a thermal environment using the Carrara unified formulation, employing Lagrange-like polynomials to define the motion and deformation, as well as layer-wise theories to describe the complex behavior within the composite structure. The study of Udaiwal and Sharma [19] develops a comprehensive finite element methodology to analyze the dynamic stability of layered composite and sandwich plates and beams influenced by thermal loads. The research highlights the significant influence of ply-angle, revealing that a 45° orientation maximizes non-dimensional frequency, independent of geometric ratios. Guo et al. [20] investigate the control of nonlinear and thermally coupled vibrations in flexible spacecraft solar panels. Using a finite element model of a panel with active constrained-layer damping, the research analyzes how piezoelectric patch placement, coverage, rotational speed, and damping-layer thickness affect vibration suppression. Using a strong-form collocation method using only nodal points, Kwak et al. [21] analyzed the vibration and dynamic response of laminated composite wave plates. Researchers proposed a first-order theory and derived the governing equation based on the Hamilton principle. A meshfree interpolation function that blends Tchebychev polynomials with radial basis functions, with a point interpolation framework, approximates the displacement components. Majeed and Sadiq [22] proposed, for the first time, a combination of hyperbolic and polynomial four-variable refined theory for buckling analysis of rectangular composite plates, achieving zero traction on the free transverse surfaces of a simply supported model plate. Based on the virtual displacement principle and Navier’s method, the equation of motion was derived, as well as a closed form established. Numerical results show good agreement compared to other studies. While the literature included numerous studies on the free vibration of composite plates in thermal environments, this field continues to evolve. Consequently, further research utilizing diverse theories and shape functions is required to expand the repertoire of analytical modeling approaches for diverse application needs.
To this end, the present study investigates the free vibration of laminated composite plates under thermal load using the four-variable shear deformation theory. Zero transverse shear stresses on the upper and lower surfaces are achieved by a combination of hyperbolic and polynomial shape functions proposed by the study [14]. Using the virtual displacement principle to determine the governing equations, as well as the Navier solution method of simply supported fixing, to produce a closed-form solution. The results of this study are benchmarked against previous studies and show good agreement. The analysis also examines the effect of critical design factors, including stacking sequence, orthotropic modulus ratio, and thickness ratio, on the fundamental frequency of composite plates in a thermal environment.
The problem considers a laminated composite plate in a thermal environment, with plate dimensions (a,b).
2.1 Kinematics
The formulation of the displacement field is grounded [23] as:
$u_{(x, y, z)}=u_{0(x, y)}+z\left(-\frac{\partial w^b}{\partial x}\right)+F(z)\left(-\frac{\partial w^s}{\partial x}\right)$ (1)
$v_{(x, y, z)}=v_{0(x, y)}+z\left(-\frac{\partial w^b}{\partial y}\right)+F(z)\left(-\frac{\partial w^s}{\partial y}\right)$ (2)
$w_{(x, y, z)}=w_{(x, y)}^b+w_{(x, y)}^s$ (3)
where, u₀, v₀, wb and ws are the four components of displacement, F(z) represents the shape function of the theorem, which describes the through-thickness profile of transverse shear stress, achieving zero stress at the free edges of the plate. For the present study, it was defined according to Draiche and Tounsi [14] as:
$F(z)=z-h\left(\sinh \left(\frac{z}{h}\right)\right)+\left(\left(\frac{4 z^3}{3 h^2}\right) \cosh (0.5)\right)$ (4)
2.2 Strain relations
Following Reddy [24]. The equations that relate strain to displacement linearly are written as:
$\varepsilon_{i j}=\frac{1}{2}\left(U_{i, j}+U_{j, i}\right)$ (5)
$\left\{\begin{array}{l}\varepsilon_{x x} \\ \varepsilon_{y y} \\ \gamma_{x y}\end{array}\right\}=\left\{\begin{array}{l}\varepsilon_{x x}^0 \\ \varepsilon_{y y}^0 \\ \gamma_{x y}^0\end{array}\right\}+z\left\{\begin{array}{l}\varepsilon_{x x}^1 \\ \varepsilon_{y y}^1 \\ \gamma_{x y}^1\end{array}\right\}+F(z)\left\{\begin{array}{l}\varepsilon_{x x}^2 \\ \varepsilon_{y y}^2 \\ \gamma_{x y}^2\end{array}\right\}$ (6)
$\left\{\begin{array}{l}\gamma_{y z} \\ \gamma_{x z}\end{array}\right\}=\left\{\begin{array}{l}\gamma_{y z}^0 \\ \gamma_{x z}^0\end{array}\right\}+F^{\prime}(z)\left\{\begin{array}{l}\gamma_{y z}^3 \\ \gamma_{x z}^3\end{array}\right\}$ (7)
$\left\{\begin{array}{l}\gamma_{y z} \\ \gamma_{x z}\end{array}\right\}=g(z)\left\{\begin{array}{l}\frac{\partial w^s}{\partial y} \\ \frac{\partial w^s}{\partial x}\end{array}\right\}$ (8)
$g(z)=1-F^{\prime}(z)=\cosh \left(\frac{Z}{h}\right)+\left(\frac{4 z^2}{h^2}\right) \cosh (0.5)$ (9)
$\varepsilon_{z z}=0$ (10)
2.3 Equation of motion
The governing equations are derived using the Hamilton principle [24]:
$0=\int_0^T(\delta U+\delta V-\delta K) d t$ (11)
where, δU: the virtual work of internal strains. δV: thermally induced virtual external work. δK: virtual kinetic energy.
$\delta U=\int_{\Omega_0}^K \int_{z_k}^{z_{k+1}}\left[\left\{\begin{array}{l}\sigma_{x x} \\ \sigma_{y y} \\ \sigma_{x y}\end{array}\right\}\left\{\begin{array}{l}\delta \varepsilon_{x x} \\ \delta \varepsilon_{y y} \\ \delta \gamma_{x y}\end{array}\right\}^k+\left\{\begin{array}{l}\sigma_{y z} \\ \sigma_{x z}\end{array}\right\}\left\{\begin{array}{l}\delta \gamma_{y z} \\ \delta \gamma_{x z}\end{array}\right\}^k\right] d z d x d y$ (12)
$\delta U=\int_{\Omega_0}^K\left[\begin{array}{l}\left\{\begin{array}{l}N_{x x} \\ N_{y y} \\ N_{x y}\end{array}\right\}\left\{\begin{array}{l}\delta \varepsilon_{x x}^0 \\ \delta \varepsilon_{y y}^0 \\ \delta \gamma_{x y}^0\end{array}\right\}^k+\left\{\begin{array}{l}M_{x x}^b \\ M_{y y}^b \\ M_{x y}^b\end{array}\right\}\left\{\begin{array}{l}\delta \varepsilon_{x x}^1 \\ \delta \varepsilon_{y y}^1 \\ \delta \gamma_{x y}^1\end{array}\right\} \\ +\left\{\begin{array}{l}M_{x x}^s \\ M_{y y}^s \\ M_{x y}^s\end{array}\right\}\left\{\begin{array}{l}\delta \varepsilon_{x x}^2 \\ \delta \varepsilon_{y y}^2 \\ \delta \gamma_{x y}^2\end{array}\right\}^k+\left\{\begin{array}{l}Q_y \\ Q_x\end{array}\right\}\left\{\begin{array}{l}\delta \gamma_{y z}^0 \\ \delta \gamma_{x z}^0\end{array}\right\}^k\end{array}\right] d x d y y$ (13)
$\delta U=\int_{\Omega}^k\left[\begin{array}{l}\left\{\begin{array}{l}N_{x x} \\ N_{y y} \\ N_{x y}\end{array}\right\}\left\{\begin{array}{c}\frac{\partial \delta u_0}{\partial x} \\ \frac{\partial \delta v_0}{\partial y} \\ \frac{\partial \delta u_0}{\partial y}+\frac{\partial \delta v_0}{\partial x}\end{array}\right\}^k \\ -\left\{\begin{array}{l}M_{x x}^b \\ M_{y y}^b \\ M_{x y}^b\end{array}\right\}\left\{\begin{array}{c}\frac{\partial^2 \delta w^b}{\partial x^2} \\ \frac{\partial^2 \delta w^b}{\partial y^2} \\ 2\frac{\partial^2 w^b}{\partial x \partial y}\end{array}\right\}^k-\left\{\begin{array}{c}\frac{\partial^2 \delta w^s}{\partial x^2} \\ \frac{\partial^2 \delta w^s}{\partial y^2} \\ 2 \frac{\partial^2 \delta w^s}{\partial x \partial y}\end{array}\right\}^k \\ +\left\{\begin{array}{l}Q_y \\ Q_x\end{array}\right\}\left\{\begin{array}{c}\frac{\partial \delta w^s}{\partial y} \\ \frac{\partial \delta w^s}{\partial x}\end{array}\right\}^k\end{array}\right] d x d y$ (14)
$\begin{gathered}\left(N_i, M_i^b, M_i^s\right)=\sum_{k=1}^N \int_{z_k}^{z_{k+1}} \sigma_i^k(1, z, F(z)) d z, i =(x x, y y, x y)\end{gathered}$ (15)
$\left(Q_j\right)=\sum_{k=1}^N \int_{z_k}^{z_{k+1}} \sigma_{j z}^k g(z) d z, j=(y, x)$ (16)
$\delta V=-\frac{1}{2} \int_{\Omega_0}^K\left[\left\{\begin{array}{l}N_{x x}^T \\ N_{y y}^T \\ N_{x y}^T\end{array}\right\} \delta\left\{\begin{array}{l}\left(\frac{\partial w}{\partial x}\right)^2 \\ \left(\frac{\partial w}{\partial y}\right)^2 \\ \left(\frac{\partial w}{\partial x} \frac{\partial w}{\partial y}\right)\end{array}\right\}^k\right] d x d y$ (17)
$\delta V=-\frac{1}{2} \int_{\Omega_0}^K \begin{gathered}N_{x x}^T \\ N_{y y}^T \\ N_{x y}^T \\ N_{x y}^T\end{gathered}\left[\left(\begin{array}{c}2\left(\frac{\partial w}{\partial x}\right)\left(\frac{\partial \delta w}{\partial x}\right) \\ 2\left(\frac{\partial w}{\partial y}\right)\left(\frac{\partial \delta w}{\partial y}\right) \\ \left(\frac{\partial \delta w}{\partial x} \frac{\partial w}{\partial y}\right) \\ \left(\frac{\partial w}{\partial x} \frac{\partial \delta w}{\partial y}\right)\end{array}\right)^k \right] d x d y$ (18)
where, the total deflection (w) is assumed to be the superposition of the deflections caused by bending (wb) and shear (ws) modes through the plate’s thickness [23].
$\left\{\begin{array}{l}N_{x x}^{\top} \\ N_{y y}^{\top} \\ N_{x y}^{\top}\end{array}\right\}=\sum_{k=1}^N \int_{Z_k}^{Z_{k+1}}\left[\begin{array}{lll}\bar{Q}_{11} & \bar{Q}_{12} & \bar{Q}_{16} \\ \bar{Q}_{12} & \bar{Q}_{22} & \bar{Q}_{26} \\ \bar{Q}_{16} & \bar{Q}_{26} & \bar{Q}_{66}\end{array}\right]\left\{\begin{array}{l}\alpha_{x x} \\ \alpha_{y y} \\ 2 \alpha_{x y}\end{array}\right\} \Delta T_{c r} d z$ (19)
$\delta K=\int_{\Omega_0} \int_{Z_k}^{Z_{k+1}} \rho(\dot{u} \delta \dot{u}+\dot{v} \delta \dot{v}+\dot{w} \delta \dot{w}) d z d x d y$ (20)
Substituting Eq. (14), Eq. (18) and Eq. (20) into Eq. (11) and integrating by parts to evaluate the equation of motion as:
$\delta u_0: \frac{\partial \mathrm{N}_{x x}}{\partial x}+\frac{\partial \mathrm{N}_{x y}}{\partial y}=\mathrm{I}_0\left(\ddot{u_0}\right)-\mathrm{I}_1\left(\frac{\partial \ddot{w^b}}{\partial x}\right)-\mathrm{I}_3\left(\frac{\partial \ddot{w^s}}{\partial x}\right)$ (21)
$\delta v_0: \frac{\partial \mathrm{N}_{y y}}{\partial y}+\frac{\partial \mathrm{N}_{x y}}{\partial x}=\mathrm{I}_0\left(\ddot{v_0}\right)-\mathrm{I}_1\left(\frac{\partial \ddot{w^b}}{\partial y}\right)-\mathrm{I}_3\left(\frac{\partial \ddot{w^s}}{\partial y}\right)$ (22)
$\begin{aligned} \delta w^b:\left(\frac{\partial^2 M_{x x}^b}{\partial x^2}+\frac{\partial^2 M_{y y}^b}{\partial y^2}\right. & \left.+2 \frac{\partial^2 M_{x y}^b}{\partial x \partial y}\right) +\left(N_{x x}^T \frac{\partial^2\left(w^b+w^s\right)}{\partial x^2}+N_{y y}^T \frac{\partial^2\left(w^b+w^s\right)}{\partial y^2}\right. \left.+2 N_{x y}^T \frac{\partial^2\left(w^b+w^s\right)}{\partial x \partial y}\right) \\ & =\mathrm{I}_0\left(\ddot{w^b}+\ddot{w^s}\right)+I_1\left(\frac{\partial \ddot{u_0}}{\partial x}+\frac{\partial \ddot{v_0}}{\partial y}\right) -I_2\left(\frac{\partial^2 \ddot{w^b}}{\partial x^2}+\frac{\partial^2 \ddot{w^b}}{\partial y^2}\right) -I_4\left(\frac{\partial^2 \ddot{w^s}}{\partial x^2}+\frac{\partial^2 \ddot{w^s}}{\partial y^2}\right)\end{aligned}$ (23)
$\begin{aligned} & \delta w^{\mathrm{s}}:\left(\frac{\partial^2 \mathrm{M}_{x x}^{\mathrm{s}}}{\partial x^2}+\frac{\partial^2 \mathrm{M}_{y y}^{\mathrm{s}}}{\partial y^2}+2 \frac{\partial^2 \mathrm{M}_{x y}^{\mathrm{s}}}{\partial x \partial y}+\frac{\partial \mathrm{Q}_y}{\partial y}+\frac{\partial \mathrm{Q}_x}{\partial x}\right) \\ & +\left(\mathrm{N}_{x x}^{\top} \frac{\partial^2\left(w^b+w^s\right)}{\partial x^2}+\mathrm{N}_{y y}^{\top} \frac{\partial^2\left(w^b+w^s\right)}{\partial y^2}\right. \left.+2 \mathrm{~N}_{x y}^{\top} \frac{\partial^2\left(w^b+w^s\right)}{\partial x \partial y}\right) \\ & =\mathrm{I}_0\left(\ddot{w^b}+\ddot{w^s}\right)+\mathrm{I}_3\left(\frac{\partial \ddot{\mathrm{U}_0}}{\partial x}+\frac{\partial \ddot{w_0}}{\partial y}\right)-\mathrm{I}_4\left(\frac{\partial^2 \ddot{w^b}}{\partial x^2}+\frac{\partial^2 \ddot{w^b}}{\partial y^2}\right)-\mathrm{I}_5\left(\frac{\partial^2 \ddot{w^b}}{\partial x^2}+\frac{\partial^2 \ddot{w^b}}{\partial y^2}\right)\end{aligned}$ (24)
$\begin{gathered}I_m=\int_{z_k}^{z_{k+1}} \rho\left(1, z, z^2, z F(z),(F(z))^2\right) d z, m =(1,2,3,4,5)\end{gathered}$ (25)
The plane stress reduced stiffness ($\mathrm{Q}_{i j}$) is:
$\begin{aligned} Q_{11} & =\frac{E_1}{1-v_{12} v_{21}} \\ Q_{12} & =\frac{v_{12} E_2}{1-v_{12} v_{21}} \\ Q_{22} & =\frac{E_2}{1-v_{12} v_{21}}\end{aligned}$ (26)
$\begin{aligned} & Q_{44}=G_{23} \\ & Q_{55}=G_{13} \\ & Q_{66}=G_{12}\end{aligned}$ (27)
The Stress-strain relations in (x,y,z) coordinates are:
$\left\{\begin{array}{l}\sigma_{x x} \\ \sigma_{y y} \\ \sigma_{x y}\end{array}\right\}=\left[\begin{array}{lll}\overline{\mathrm{Q}}_{11} & \overline{\mathrm{Q}}_{12} & \overline{\mathrm{Q}}_{16} \\ \overline{\mathrm{Q}}_{12} & \overline{\mathrm{Q}}_{22} & \overline{\mathrm{Q}}_{26} \\ \overline{\mathrm{Q}}_{16} & \overline{\mathrm{Q}}_{26} & \overline{\mathrm{Q}}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x} \\ \varepsilon_{y y} \\ \gamma_{x y}\end{array}\right\}$ (28)
$\begin{aligned} & \left\{\begin{array}{l}N_{x x} \\ N_{y y} \\ N_{x y}\end{array}\right\}=\left[\begin{array}{lll}\mathrm{A}_{11} & \mathrm{A}_{12} & \mathrm{A}_{16} \\ \mathrm{A}_{12} & \mathrm{A}_{22} & \mathrm{A}_{26} \\ \mathrm{A}_{16} & \mathrm{A}_{26} & \mathrm{A}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^0 \\ \varepsilon_{y y}^0 \\ \gamma_{x y}^0\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{B}_{11} & \mathrm{B}_{12} & \mathrm{B}_{16} \\ \mathrm{B}_{12} & \mathrm{B}_{22} & \mathrm{B}_{26} \\ \mathrm{B}_{16} & \mathrm{B}_{26} & \mathrm{B}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^1 \\ \varepsilon_{y y}^1 \\ \gamma_{x y}^1\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{E}_{11} & \mathrm{E}_{12} & \mathrm{E}_{16} \\ \mathrm{E}_{12} & \mathrm{E}_{22} & \mathrm{E}_{26} \\ \mathrm{E}_{16} & \mathrm{E}_{26} & \mathrm{E}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^2 \\ \varepsilon_{y y}^2 \\ \gamma_{x y}^2\end{array}\right\}\end{aligned}$ (29)
$\begin{aligned} & \left\{\begin{array}{l}M_{x x}^b \\ M_{y y}^b \\ M_{x y}^b\end{array}\right\}=\left[\begin{array}{lll}\mathrm{B}_{11} & \mathrm{B}_{12} & \mathrm{B}_{16} \\ \mathrm{B}_{12} & \mathrm{B}_{22} & \mathrm{B}_{26} \\ \mathrm{B}_{16} & \mathrm{B}_{26} & \mathrm{B}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x x}^0 \\ \varepsilon_{y y}^0 \\ \gamma_{x y}^0\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{D}_{11} & \mathrm{D}_{12} & \mathrm{D}_{16} \\ \mathrm{D}_{12} & \mathrm{D}_{22} & \mathrm{D}_{26} \\ \mathrm{D}_{16} & \mathrm{D}_{26} & \mathrm{D}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^1 \\ \varepsilon_{y y}^1 \\ \gamma_{x y}^1\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{F}_{11} & \mathrm{F}_{12} & \mathrm{F}_{16} \\ \mathrm{F}_{12} & \mathrm{F}_{22} & \mathrm{F}_{26} \\ \mathrm{F}_{16} & \mathrm{F}_{26} & \mathrm{F}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x x}^2 \\ \varepsilon_{y y}^2 \\ \gamma_{x y}^2\end{array}\right\}\end{aligned}$ (30)
$\begin{aligned} & \left\{\begin{array}{l}M_{x x}^s \\ M_{y y}^s \\ M_{x y}^s\end{array}\right\}=\left[\begin{array}{lll}\mathrm{E}_{11} & \mathrm{E}_{12} & \mathrm{E}_{16} \\ \mathrm{E}_{12} & \mathrm{E}_{22} & \mathrm{E}_{26} \\ \mathrm{E}_{16} & \mathrm{E}_{26} & \mathrm{E}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x x}^0 \\ \varepsilon_{y y}^0 \\ \gamma_{x y}^0\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{F}_{11} & \mathrm{F}_{12} & \mathrm{F}_{16} \\ \mathrm{F}_{12} & \mathrm{F}_{22} & \mathrm{F}_{26} \\ \mathrm{F}_{16} & \mathrm{F}_{26} & \mathrm{F}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x x}^1 \\ \varepsilon_{y y}^1 \\ \gamma_{x y}^1\end{array}\right\} \\ & +\left[\begin{array}{lll}\mathrm{H}_{11} & \mathrm{H}_{12} & \mathrm{H}_{16} \\ \mathrm{H}_{12} & \mathrm{H}_{22} & \mathrm{H}_{26} \\ \mathrm{H}_{16} & \mathrm{H}_{26} & \mathrm{H}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{x x}^2 \\ \varepsilon_{y y}^2 \\ \gamma_{x y}^2\end{array}\right\}\end{aligned}$ (31)
$\left\{\begin{array}{l}\mathrm{Q}_y \\ \mathrm{Q}_x\end{array}\right\}=\left[\begin{array}{ll}\mathrm{L}_{44} & \mathrm{L}_{45} \\ \mathrm{L}_{45} & \mathrm{L}_{55}\end{array}\right]\left\{\begin{array}{l}\gamma_{y z}^0 \\ \gamma_{x z}^0\end{array}\right\}$ (32)
$\left\{\begin{array}{c}A_{i j} \\ B_{i j} \\ D_{i j} \\ E_{i j} \\ F_{i j} \\ H_{i j}\end{array}\right\}=\int_{z_k}^{z_{k+1}} \bar{Q}_{i j}\left\{\begin{array}{c}1 \\ Z \\ Z^2 \\ F(z) \\ Z F(z) \\ (F(z))^2\end{array}\right\} d z,(i, j)=(1,2,6)$ (33)
$L_{i j}=\int_{z_k}^{z_{k+1}} Q_{\mathrm{ij}}(g(z))^2 d z,(i, j)=(4,5)$ (34)
2.4 Navier solution
The analytical solution was derived using Navier’s method, which required simply supported boundary conditions. Accordingly, distinct displacement expansions for cross-ply and angle-ply laminates were adopted from [23]:
at $x=0$ and $x=a: v_0=w^b=\mathrm{w}^s=M_x^b=M_x^s=0$ (35)
at $y=0$ and $y=b: u_0=w^b=\mathrm{w}^s=M_y^b=M_y^s=0$ (36)
$u_{0(x, y, t)}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} U_{m n} \cos \alpha x \sin \beta y$ (37)
$v_{0(x, y, t)}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} V_{m n} \sin \alpha x \cos \beta y$ (38)
$w_{(x, y, t)}^b=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{m n}^b \sin \alpha x \sin \beta y$ (39)
$w_{(x, y, t)}^s=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{m n}^s \sin \alpha x \sin \beta y$ (40)
at $x=0$ and $x=a: v_0=w^b=\mathrm{w}^s=M_x^b=M_x^s=0$ (41)
at $y=0$ and $y=b: u_0=w^b=\mathrm{w}^s=M_y^b=M_y^s=0$ (42)
$u_{0(x, y, t)}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} U_{m n} \sin \alpha x \cos \beta y$ (43)
$v_{0(x, y, t)}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} V_{m n} \cos \alpha x \sin \beta y$ (44)
$w_{(x, y, t)}^b=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{m n}^b \sin \alpha x \sin \beta y$ (45)
$w_{(x, y, t)}^s=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{m n}^s \sin \alpha x \sin \beta y$ (46)
Inserting Eqs. (37)-(40) and Eqs. (43)-(46) into Eqs. (21)-(24). The eigenvalue problem is established:
$\left([\mathrm{K}]-\omega^2[\mathrm{M}]\right)\{\Delta\}^{\mathrm{T}}=\{0\}$ (47)
$[\mathrm{K}]=\left[\begin{array}{cccc}\mathrm{K}_{11} & \mathrm{~K}_{12} & \mathrm{~K}_{13} & \mathrm{~K}_{14} \\ \mathrm{~K}_{12} & \mathrm{~K}_{22} & \mathrm{~K}_{23} & \mathrm{~K}_{24} \\ \mathrm{~K}_{13} & \mathrm{~K}_{23} & \mathrm{~K}_{33}-\left(\mathrm{k}_1 \alpha^2+\mathrm{k}_2 \beta^2\right) \Delta \mathrm{T} & \mathrm{K}_{34}-\left(\mathrm{k}_1 \alpha^2+\mathrm{k}_2 \beta^2\right) \Delta \mathrm{T} \\ \mathrm{K}_{14} & \mathrm{~K}_{24} & \mathrm{~K}_{34}-\left(\mathrm{k}_1 \alpha^2+\mathrm{k}_2 \beta^2\right) \Delta \mathrm{T} & \mathrm{K}_{44}-\left(\mathrm{k}_1 \alpha^2+\mathrm{k}_2 \beta^2\right) \Delta \mathrm{T}\end{array}\right]$ (48)
$[M]=\left[\begin{array}{cccc}\mathrm{I}_0 & 0 & -\mathrm{I}_1 \alpha & -\mathrm{I}_3 \alpha \\ 0 & \mathrm{I}_0 & -\mathrm{I}_1 \beta & -\mathrm{I}_3 \beta \\ -\mathrm{I}_1 \alpha & -\mathrm{I}_1 \beta & \mathrm{I}_0+\mathrm{I}_2\left(\alpha^2+\beta^2\right) & \mathrm{I}_0+\mathrm{I}_1\left(\alpha^2+\beta^2\right) \\ -\mathrm{I}_3 \alpha & -\mathrm{I}_3 \beta & \mathrm{I}_0+\mathrm{I}_4\left(\alpha^2+\beta^2\right) & \mathrm{I}_0+\mathrm{I}_5\left(\alpha^2+\beta^2\right)\end{array}\right]$ (49)
$\{\Delta\}^T=\left\{U_{m n} V_{m n} W_{m n}^b W_{m n}^s\right\}$ (50)
where, α = mπ/a and β = nπ/b and (Umn, Vmn, Wbmn, Wsmn) are coefficients to be determined [25].
Numerical results obtained by the MATLABR2024a program based on proposing a four-variable refined plate theory, a combination of hyperbolic and polynomial shape functions employed to achieve zero transverse shear stresses on the top and bottom surfaces, without the need for a shear factor [14]. Three types of material properties are used in numerical calculations, as shown below:
Material 1:
(E1/E2 = open), G12 = G13 = 0.6E2,G23 = 0.5E2,ν12 = 0.25, E2 = 1 Gpa
Material 2:
(E1/E2 = 5), G12 = G13 = 0.5E2,G23 = 0.35E2,ν12 = 0.3,E2 = 1 Gpa
Material 3:
(E1/E2 = open), G12 = G13 = 0.6E2,G23 = 0.5E2,ν12 = 0.25,E2 = 6.95 Gpa, α2/α0 = 11.4, α1/α0 = 1.14, α0 = 10−6
Material 4:
E1 = 181 Gpa, E2 = 10.3 Gpa, G12 = G13 = 7.17 Gpa, G23 = 6.21 Gpa, ν12 = 0.28, α2/α0 = 22.5, α1/α0 = 0.02, α0 = 10−6
Results of nondimensional natural frequency (ω*a2*sqr(ρ/E2h2)), which were computed with less mathematical complications compared to other shear theories like (higher order and third order shear deformation theories), showed good agreement. Table 1 shows the numerical values of natural frequencies evaluated using the present theory compared to other theories. Material 1 was used to provide a clear image of the effect of orthotropy ratios (E1/E2) on nondimensional natural frequencies for different plate laminate schemes.
Table 1. Effect of orthotropic ratio on nondimensional natural frequency for two types of cross-ply simply supported square composite plate
|
Layup |
Reference |
E1/E2 |
||||
|
3 |
10 |
20 |
30 |
40 |
||
|
(0/90)1 |
Ref. [26] |
6.258 |
6.985 |
7.675 |
8.176 |
8.563 |
|
Ref. [27] |
6.217 |
6.989 |
7.821 |
8.505 |
9.087 |
|
|
Ref. [28] |
6.149 |
6.916 |
7.692 |
8.311 |
8.826 |
|
|
Ref. [29] |
6.219 |
6.996 |
7.838 |
8.532 |
9.124 |
|
|
Ref. [13] |
6.219 |
6.997 |
7.838 |
8.532 |
9.124 |
|
|
Present |
6.217 |
6.989 |
7.821 |
8.505 |
9.088 |
|
|
(0/90)3 |
Ref. [26] |
6.610 |
8.414 |
9.840 |
10.696 |
11.273 |
|
Ref. [27] |
6.556 |
8.405 |
9.918 |
10.855 |
11.501 |
|
|
Ref. [28] |
6.492 |
8.388 |
9.927 |
10.872 |
11.519 |
|
|
Ref. [29] |
6.557 |
8.407 |
9.921 |
10.860 |
11.510 |
|
|
Ref. [13] |
6.557 |
8.407 |
9.921 |
10.860 |
11.510 |
|
|
Present |
6.556 |
8.405 |
9.918 |
10.855 |
11.501 |
|
Notes: Using material 1, thickness ratio (a/h = 5).
In Table 2, the numerical values of the nondimensional natural frequency were evaluated (based on material 1) using two antisymmetric cross-ply laminated square composite plates. Values of thickness ratio (a/h) between (4-100) to show the influence of thickness on normalized fundamental frequency of laminated plates.
Table 2. Effect of thickness ratio on nondimensional natural frequency for two types of cross-ply simply supported square composite plate
|
Layup |
Reference |
a/h |
||||
|
4 |
10 |
20 |
50 |
100 |
||
|
(0/90)1 |
Ref. [27] |
8.355 |
10.568 |
11.105 |
11.275 |
11.300 |
|
Ref. [28] |
8.035 |
10.473 |
11.078 |
11.271 |
11.299 |
|
|
Ref. [29] |
8.355 |
10.568 |
11.105 |
11.275 |
11.300 |
|
|
Ref. [13] |
8.402 |
10.581 |
11.109 |
11.276 |
11.300 |
|
|
Present |
8.355 |
10.568 |
11.105 |
11.275 |
11.300 |
|
|
(0/90)3 |
Ref. [27] |
9.988 |
15.463 |
17.377 |
18.064 |
18.170 |
|
Ref. [28] |
9.985 |
15.501 |
17.393 |
18.067 |
18.171 |
|
|
Ref. [29] |
9.988 |
15.463 |
17.377 |
18.064 |
18.170 |
|
|
Ref. [13] |
10.004 |
15.463 |
17.377 |
18.064 |
18.17 |
|
|
Present |
9.988 |
15.463 |
17.377 |
18.064 |
18.17 |
|
Notes: Using material 1, Young’s modulus ratio (E1/E2 = 40).
In Figure 1, material 3 was used to illustrates the variation of fundamental frequency (ω) with thickness ratio (a/h) range of values between (4 to 50) for a symmetric (three plies) and an antisymmetric (six plies) cross-ply laminated plate.
Figure 1. Natural frequency of symmetric and antisymmetric cross-ply plate using material 3 for various thicknesses and 40% of Tcr
By selecting material 1 for a simply supported angle-ply [45/-45] composite plate used in Table 3, nondimensional natural frequency values are evaluated for varying orthotropy and thickness values. Normalized natural frequency values demonstrated in Table 3 were calculated for thick laminated plates (thickness ratio of 4), moderately thick plates (thickness ratio of 10) and for thin plates (thickness ratio higher than 10).
Table 3. Effect of different values of orthotropic ratio (E1/E2) and thickness ratio (a/h) on nondimensional natural frequency for antisymmetric angle-ply [45/-45] square composite plate
|
E1/E2 |
a/h |
Source |
|||||
|
Ref. [30] |
Ref. [27] |
Ref. [28] |
Ref. [29] |
Ref. [13] |
Present |
||
|
10 |
4 |
7.265 |
7.347 |
7.217 |
7.347 |
7.367 |
7.347 |
|
10 |
8.989 |
8.966 |
8.932 |
8.966 |
8.971 |
8.966 |
|
|
20 |
9.327 |
9.327 |
9.317 |
9.327 |
9.328 |
9.327 |
|
|
50 |
9.438 |
9.438 |
9.436 |
9.438 |
9.438 |
9.438 |
|
|
100 |
9.512 |
9.454 |
9.454 |
9.454 |
9.454 |
9.454 |
|
|
20 |
4 |
8.049 |
8.415 |
8.119 |
8.415 |
8.460 |
8.416 |
|
10 |
10.641 |
10.715 |
10.627 |
10.715 |
10.728 |
10.715 |
|
|
20 |
11.298 |
11.277 |
11.252 |
11.277 |
11.281 |
11.277 |
|
|
50 |
11.507 |
11.455 |
11.451 |
11.455 |
11.456 |
11.455 |
|
|
100 |
11.539 |
11.482 |
11.481 |
11.482 |
11.482 |
11.482 |
|
|
30 |
4 |
8.521 |
9.175 |
8.721 |
9.175 |
9.243 |
9.176 |
|
10 |
11.893 |
12.097 |
11.946 |
12.097 |
12.118 |
12.097 |
|
|
20 |
12.842 |
12.866 |
12.821 |
12.866 |
12.872 |
12.866 |
|
|
50 |
13.157 |
13.115 |
13.108 |
13.115 |
13.116 |
13.115 |
|
|
100 |
13.204 |
13.152 |
13.151 |
13.152 |
13.153 |
13.152 |
|
|
40 |
4 |
8.843 |
9.759 |
9.161 |
9.759 |
9.850 |
9.760 |
|
10 |
12.912 |
13.263 |
13.044 |
13.263 |
13.293 |
13.263 |
|
|
20 |
14.171 |
14.246 |
14.179 |
14.246 |
14.255 |
14.246 |
|
|
50 |
14.601 |
14.572 |
14.561 |
14.572 |
14.574 |
14.572 |
|
|
100 |
14.667 |
14.621 |
14.618 |
14.621 |
14.622 |
14.621 |
|
Notes: Using material 1.
Using a simply supported symmetric cross-ply [0/90]s composite plate with its length equal to its width to examine the influence of thermal loads (thermal environment) on characteristic frequency based on the present theory and literature in Table 4. The nondimensional natural frequency values (based on material 3) were evaluated at two temperature differences (0, 100℃ for different thicknesses compared with literature.
Table 4. Dimensionless natural frequency under two sets of thermal loads for equal dimensions, symmetric [0/90]s cross-play composite plate with different thicknesses
|
a/h |
ΔT = 0℃ |
||||
|
Ref. [31] |
Ref. [32] |
Ref. [13] |
Present |
Discr. % |
|
|
50 |
18.689 |
18.7871 |
18.7382 |
18.7381 |
0.26 |
|
20 |
17.483 |
17.5231 |
17.9951 |
17.9938 |
2.69 |
|
10 |
14.702 |
14.7106 |
15.9460 |
15.9405 |
8.36 |
|
5 |
10.263 |
10.2452 |
11.7905 |
11.7711 |
14.89 |
|
a/h |
ΔT = 100℃ |
||||
|
Ref. [31] |
Ref. [32] |
Ref. [13] |
Present |
Discr. % |
|
|
50 |
16.412 |
16.2855 |
14.2340 |
14.2338 |
12.60 |
|
20 |
17.172 |
17.1715 |
17.3242 |
17.3228 |
0.88 |
|
10 |
14.592 |
14.6369 |
15.7601 |
15.7546 |
7.64 |
|
5 |
10.226 |
10.2324 |
11.7280 |
11.7085 |
14.43 |
Notes: Using material 3, Young’s modulus ratio (E1/E2 = 40).
Using a simply supported symmetric angle-ply [45/-45]2 composite plate with its length equal to its width to examine how thermal conditions effect systems resonant frequency based on the present theory and literature in Table 5. The nondimensional natural frequency values evaluated based on two temperature differences (0, 100℃) for different thicknesses are compared with literature. Material 3 were used along with orthotropy ratio of (40).
Table 5. Dimensionless natural frequency under two sets of thermal loads for equal dimensions [45/-45]2 composite plate with different thicknesses
|
a/h |
ΔT = 0℃ |
|||
|
Ref. [31] |
Ref. [13] |
Present |
Discr. % |
|
|
50 |
23.225 |
23.2220 |
23.2237 |
0.006 |
|
20 |
21.812 |
21.7982 |
21.8062 |
0.027 |
|
10 |
18.333 |
18.3062 |
18.3218 |
0.061 |
|
5 |
12.544 |
12.5295 |
12.5330 |
0.088 |
|
a/h |
ΔT = 100℃ |
|||
|
Ref. [31] |
Ref. [13] |
Present |
Discr. % |
|
|
50 |
21.477 |
19.7677 |
19.7696 |
7.95 |
|
20 |
21.524 |
21.2478 |
21.256 |
1.245 |
|
10 |
18.248 |
18.1444 |
18.1601 |
0.482 |
|
5 |
12.513 |
12.4706 |
12.4741 |
0.311 |
Notes: Using material 3, Young’s modulus ratio (E1/E2 = 40).
The unforced dynamic behavior of simply supported (45/-45)2 plates constructed from multiple bonded piles was analyzed under a range of thermal conditions (0%, 25%, 50%, 75%) of Tcr. For plates with a thickness ratio (a/h = 10) and a mass density of (1389.23 kg/m3), the numerical data obtained from the present theory and literature are outlined in Table 6.
Table 6. Dimensionless natural frequency of square, symmetric [45/-45]2 angle-play composite plate under variable values of thermal loads
|
ΔT |
Ref. [13] |
Ref. [31] |
Ref. [33] |
Present |
Discr. (%) |
|
0(Tcr) |
14.262 |
14.312 |
14.438 |
14.268 |
0.307 |
|
0.25(Tcr) |
12.351 |
12.379 |
12.504 |
12.356 |
0.183 |
|
0.5(Tcr) |
10.085 |
10.082 |
10.209 |
10.089 |
0.068 |
|
0.75(Tcr) |
7.131 |
7.075 |
7.219 |
7.134 |
0.832 |
Figure 2 shows the correlation between thickness and natural frequency of laminated composite plate for cross-ply [0/90] and angle-ply [45/-45] composite plate under the influence of a (20%) of the plate’s critical buckling temperature (based on material 3).
Figure 2. Natural frequency of cross-ply and angle-ply composite plates using material 3 for various thicknesses and 20% of Tcr
The effect of different design parameters, such as thickness ratio layers scheme, on the dimensionless fundamental frequency under four sets of thermal loads (20%, 40%, 60%, 80%) of Tcr for both cross-ply and angle-ply is illustrated in Table 7 (using material 3) and Table 8 (using material 4).
Table 7. Dimensionless natural frequency under variable sets of thermal loads for equal dimensions, different piles, cross-play composite plate with different thicknesses
|
Layup |
Tcr (℃) |
a/h |
Nondimensional Natural Frequency |
|||
|
20% (Tcr) |
40% (Tcr) |
60% (Tcr) |
80% (Tcr) |
|||
|
(0/90) |
5824.6 |
5 |
8.1294 |
7.0414 |
5.7502 |
4.0667 |
|
85.639 |
50 |
10.0847 |
8.7336 |
7.1310 |
5.0424 |
|
|
(0/90)3 |
9015.5 |
5 |
10.2872 |
8.9092 |
7.2745 |
5.1440 |
|
219.7353 |
50 |
16.1573 |
13.9926 |
11.4249 |
8.0786 |
|
|
(0/90)9 |
9376.1 |
5 |
10.5014 |
9.0947 |
7.4259 |
5.2510 |
|
234.5646 |
50 |
16.6940 |
14.4574 |
11.8044 |
8.3470 |
|
Notes: Using material 3, Young’s modulus ratio (E1/E2 = 40).
Table 8. Dimensionless natural frequency under variable sets of thermal loads for equal dimensions, different piles, and an angle-play composite plate with different thicknesses
|
Layup |
Tcr (℃) |
a/h |
Nondimensional Natural Frequency |
|||
|
20% (Tcr) |
40% (Tcr) |
60% (Tcr) |
80% (Tcr) |
|||
|
(45/-45) |
12078.67 |
5 |
8.1486 |
7.058 |
5.7637 |
4.0761 |
|
168.16 |
50 |
9.855 |
8.5344 |
6.9683 |
4.9273 |
|
|
(45/-45)3 |
18924.5 |
5 |
10.369 |
8.9802 |
7.3325 |
5.185 |
|
387.13 |
50 |
14.9551 |
12.9515 |
10.5748 |
7.4775 |
|
|
(45/-45)9 |
19684.49 |
5 |
10.587 |
9.1687 |
7.4864 |
5.2938 |
|
411.39 |
50 |
15.417 |
13.3514 |
10.9014 |
7.7084 |
|
Notes: Using material 4, Young’s modulus ratio (E1/E2 = 40).
Table 1 results demonstrate that as (E1/E2) increases, nondimensional natural frequency increases due to the rise in stiffness, the main effect comes from the increase of E1 value which directly amplifies bending stiffness via Ǭ11, in addition counter effect of lowered value of E2 allows to more shear deformation which softening the plate. As the number of layers increases, the natural frequency increases due to stiffness increase. The numerical results based on the current work showed good agreement to literature.
The present theory results demonstrated in Table 2 exhibit high convergence with other theories from literature. Also, results showed that as the thickness decreases, natural frequency decreases (the inverse values shown in Table 2 due to the effect of the (a/h) term in normalizing natural frequency), this is due to stiffness reduction which lowers natural frequency.
Numerical results in Table 3 based on the present theory shows good agreement with those of other theories. The values of the nondimensional natural frequency increase with increasing orthotropy ratio (E1/E2) due to Increased fiber-direction stiffness (E₁) enhances bending resistance, even when fibers are not aligned with plate axes. The transformation mechanics ensure some component of E₁ contributes to plate stiffness in all directions. Results also show that nondimensional natural frequency decrease with decreasing thickness due to reduced stiffness.
Table 4 shows a discrepancy of less than 13% for thin plates and a higher discrepancy of less than 15% for the two sets of thermal loading conditions because boundary conditions simulation differences.
Table 5 demonstrate a comparison of the present results [31], which shows a less than 8% discrepancy for thin plates and a higher discrepancy, less than 0.35%, for the two sets of thermal loading conditions. The results showed significant convergence.
The current findings illustrated in Table 6 correlate well with prior data, showing consistent, negligible variance. The effect of elevated temperature caused a decrease in the nondimensional natural frequency. In physical terms, the high-temperature environment softens the system, reducing its vibrational stiffness. This loss of rigidity increases dynamic flexibility, which can critically change the system’s response to operational forces.
Results showed in Table 7 and Table 8 demonstrate that as the number of layers for cross-ply and angle-ply laminates increases, the nondimensional natural frequency increases as well for the same thickness ratio and thermal load. This is because the stiffness increases. The previous tables also demonstrate the effect of thermal load on fundamental frequency, which basically reduces nondimensional natural frequency as thermal load increases (below critical buckling temperature) for both cross-ply and angle-ply if the thickness and piles don’t change, while it is greater for an angle laminated plate.as shown in Figure 2 since its stiffness is larger.
In the present paper, the four-variable refined plate theory is used for the first time to study free vibration analysis of simply supported cross-ply and angle-ply laminated composite plates under thermal loads. The displacement function, which is a combination of hyperbolic and polynomial tasks that account for zero traction stresses on free surfaces without using a shear correction factor, unlike the first-order shear deformation theory, and with less computational complexity than higher-order shear deformation theories yet maintain good agreement with all theories. The present theory reveals that the parabolic profile of transverse shear stresses is significantly modulated by ply orientation. Numerical results provide a clear image of the factors affecting the natural frequency of the plates. Basically, thermal loads significantly reduce natural frequencies by increasing the acceleration effect near the critical buckling temperature, thereby avoiding the critical temperature because buckling will occur purely due to thermal load. Also, the layer count (cross-ply or angle-ply) directly increases the plate’s stiffness, altering the frequency degradation patterns. An increase in the orthotropy ratio raises the fundamental frequency due to increased stiffness. Finally, the design margin could be increased near the predicted critical buckling temperature for specific applications to provide economic efficiency. Mechanistic analysis reveals that angle-ply configurations exhibit pronounced bending-twisting coupling, which increases fundamental frequency compared to cross-ply laminates of equivalent thickness.
|
a,b |
Plate dimensions, m |
|
Aij,Bij,Dij,Eij,Fij,Hij,Lij |
Extensional, coupling, bending, stiffnesses (N/m) |
|
E1,E2,E3 |
Young’s modulus (GPa) |
| k | Number of plate layers |
|
h |
Plate thickness (m) |
|
Mbxx,Mbyy, Mbxy |
Bending moment per unit length (N m/m) |
|
Msxx, Msyy, Msxy |
Moment per unit length due to shear (N m/m) |
|
NTxx,NTyy,NTxy |
Thermally induced membrane forces (N/m) |
|
Qy, Qx |
Transverse shear force (N) |
|
wb,ws |
Displacement in bending, shear, respectively |
|
x,y,z |
Coordinate axes |
|
Greek symbols |
|
|
εij |
Dimensionless Strain |
|
γij |
Dimensionless shear strain |
|
ρ |
Mass density kg/m3 |
|
σij(x,y,z) |
Stress components (Gpa) |
|
ν12, ν 12 |
Poison’s ratio |
|
Subscripts |
|
|
cr |
Critical |
|
mn |
Modes of plate |
|
1,2 |
Direction of fibres and transverse direction, respectively |
Stiffness matrix elements Kij for a cross-ply laminated plate:
$K_{11}=\left(A_{11} \alpha^2+A_{66} \beta^2\right), K_{12}=\left(A_{12}+A_{66}\right) \alpha \beta$
$K_{13}=-\left(B_{11} \alpha^3+\left(B_{12}+2 B_{66}\right) \alpha \beta^2\right)$,
$K_{14}=-\left(E_{11} \alpha^3+\left(E_{12}+2 E_{66}\right) \alpha \beta^2\right)$
$K_{22}=\left(A_{66} \alpha^2+A_{22} \beta^2\right)$,
$K_{23}=-\left(B_{22} \beta^3+\left(B_{12}+2 B_{66}\right) \alpha^2 \beta\right)$
$K_{24}=-\left(E_{22} \beta^3+\left(E_{12}+2 E_{66}\right) \alpha^2 \beta\right)$
$K_{33}=\left(D_{11} \alpha^4+2\left(D_{12}+2 D_{66}\right) \alpha^2 \beta^2+D_{22} \beta^4\right)$
$K_{34}=\left(F_{11} \alpha^4+2\left(F_{12}+2 F_{66}\right) \alpha^2 \beta^2+F_{22} \beta^4\right)$
$\begin{gathered}K_{44}=\left(H_{11} \alpha^4+2\left(H_{12}+2 H_{66}\right) \alpha^2 \beta^2+H_{22} \beta^4+L_{55} \alpha^2\right.\left.+L_{44} \beta^2\right)\end{gathered}$
$\begin{aligned} A_{16}=A_{26}=B_{16} & =B_{26}=D_{16}=D_{26}=E_{16}=E_{26}=F_{16} \\ & =F_{26}=H_{16}=H_{26}=L_{45}=0\end{aligned}$
Stiffness matrix elements Kij for an angle-ply laminated plate:
$K_{11}=\left(A_{11} \alpha^2+A_{66} \beta^2\right), K_{12}=\left(A_{12}+A_{66}\right) \alpha \beta$
$K_{13}=-\left(3 B_{16} \alpha^2 \beta+B_{26} \beta^3\right), K_{14}=-\left(3 A_{16} \alpha^2 \beta+E_{26} \beta^3\right)$
$K_{22}=\left(A_{66} \alpha^2+A_{22} \beta^2\right), K_{23}=-\left(B_{16} \alpha^3+3 B_{26} \alpha \beta^2\right)$
$K_{24}=-\left(E_{16} \alpha^3+3 E_{26} \alpha \beta^2\right)$,
$K_{33}=\left(D_{11} \alpha^4+2\left(D_{12}+2 D_{66}\right) \alpha^2 \beta^2+D_{22} \beta^4\right)$
$K_{34}=\left(F_{11} \alpha^4+2\left(F_{12}+2 F_{66}\right) \alpha^2 \beta^2+F_{22} \beta^4\right)$
$\begin{gathered}K_{44}=\left(H_{11} \alpha^4+2\left(H_{12}+2 H_{66}\right) \alpha^2 \beta^2+H_{22} \beta^4+L_{55} \alpha^2\right.\left.+L_{44} \beta^2\right)\end{gathered}$
$\begin{gathered}A_{16}=A_{26}=B_{11}=B_{12}=B_{22}=B_{66}=D_{16}=D_{26}=E_{11} \\ =E_{12} \\ =E_{22}=E_{66}=F_{16}=F_{26}=H_{16}=H_{26}=L_{45}=0\end{gathered}$
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