Nonlinear static and dynamic thermo-mechanical buckling responses of incline stiffened functionally graded graphene platelet-reinforced composite panels with porous core

Abstract

A semi-analytical model is developed in this study to investigate the nonlinear static and dynamic thermo-mechanical buckling behavior of functionally graded graphene platelet-reinforced composite (FG-GPLRC) panels with a porous core and inclined FG-GPLRC stiffeners. The panels are assumed to rest on a Pasternak elastic foundation and are subjected to axial compression and thermal loading. The formulation is based on the higher-order shear deformation theory (HSDT), incorporating geometrical nonlinearities described by the von Kármán strain-displacement relation. Three types of complex curvatures, including cylindrical, parabolic, and sinusoid, are considered in the panel geometries. An enhanced smeared stiffener technique is proposed to model the effect of inclined stiffeners, and the nonlinear stress function is approximated using a Galerkin approach. The governing motion equations are derived via the Lagrangian formalism, and numerical simulations are performed using the Runge-Kutta method to capture the time-dependent responses. The critical dynamic buckling loads are determined according to the Budiansky-Roth criterion. The results highlight the combined influences of stiffener, porosity distribution, curvature geometry, and foundation stiffness on the overall buckling performance of FG-GPLRC panels.

Introduction

Structural elements such as plates and cylindrical panels are extensively employed in modern engineering applications due to their versatile load-bearing capabilities. Consequently, these structures have been the focus of numerous investigations concerning static and dynamic stability, as well as vibration characteristics.

Numerous investigations were conducted on functionally graded materials (FGMs) and sandwich FGM structures with plates, cylindrical panels and shells, spherical shells, conical shells, and doubly curved shallow shells, focusing on linear and nonlinear stability, dynamic and vibration responses.^1–7 These studies employed various theoretical models, such as zigzag-elasticity plate theory, classical plate and shell theories, first-order and higher-order shear deformation theories (FSDT and HSDT), and used analytical and semi-analytical approaches. A general study of the dynamic responses of three-directional FGM beam, plate, and cuboidal solid structures with elastic restraints was carried out using three-dimensional elasticity theory and energy method. ⁸ In the nanoscale, the buckling and vibration problems were investigated for FGM nanobeams,^9,10 and for FGM nanoplates. ¹¹ Furthermore, the thermo-mechanical buckling and free vibration behavior of FGM plates and cylindrical shells with orthogonal and spiral stiffeners was analyzed using HSDT and Donnell shell theory in conjunction with the Galerkin methods and the improved smeared stiffener techniques.^12–14

Beyond conventional FGMs, increasing attention has been directed toward advanced nanocomposites, including functionally graded carbon nanotube-reinforced composites (FG-CNTRCs), functionally graded graphene-reinforced composites (FG-GRCs), and functionally graded graphene platelet-reinforced composites (FG-GPLRCs). These materials offer superior mechanical, thermal, and multifunctional properties, making them highly attractive for next-generation structural applications.

Nonlinear mechanical and thermal buckling and postbuckling behaviors of FG-CNTRC plates were explored using the perturbation technique within the framework of HSDT.^15,16 By using the dynamic relaxation method and incremental load technique, the buckling behavior of FG-CNTRC plates were investigated. ¹⁷ Thermal postbuckling responses of FG-CNTRC plates with tangentially restrained edges were analyzed through the first-order shear deformation theory (FSDT) in conjunction with Galerkin method. ¹⁸ Additionally, the vibration responses of FG-CNTRC plates under hygrothermal effects were studied using the Galerkin method and the harmonic balance method. ¹⁹ For FG-CNTRC cylindrical panels, the mechanical buckling and vibration, thermal buckling and hygrothermal postbuckling responses were mentioned with elastically restrained edges, ²⁰ with nonlinear elastic foundation, ²¹ and with nonlinear viscoelastic foundations. ²² The FG-CNTRC hemispherical shells were mentioned in the asymmetric and axisymmetric buckling problems with temperature-dependent material properties under thermal loads. ²³ Moreover, CNT-reinforced composites can be further enhanced by incorporating shape memory alloy (SMA) fibers, ²⁴ resulting in a novel class of hybrid composites (SMA/CNT/polymer) that exhibit superior thermomechanical characteristics and hold strong potential for applications in smart and thermally loaded structures. ²⁵ Boron nitride nanotubes (BNNTs) have also emerged as promising nanoreinforcements in polymer matrices, enabling the development of advanced composite materials with exceptional thermal stability, chemical inertness, and electrical insulation properties. ²⁵

For FG-GRC cylindrical panels, both nonlinear thermal postbuckling and vibration behaviors on elastic foundations have been investigated using HSDT in combination with perturbation methods.^26,27 Thermal postbuckling and free vibration analyses were also conducted through the isogeometric finite element approach to obtain the responses under thermal loading.^28,29 Additionally, the nonlinear static and dynamic buckling and vibration performance of FG-GRC cylindrical panels and complexly curved panels reinforced with FG-GRC stiffeners was examined, considering the influence of elastic foundations and integrated piezoelectric layers. These studies employ HSDT along with an enhanced smeared stiffener technique to model stiffener–shell interactions.^30,31

A wide range of studies examined the mechanical behaviors of sandwich FG-GPLRC and FG-GPLRC plates and curved panels, including buckling, postbuckling, bending, and vibration responses under various loading and boundary conditions. Nonlinear static bending, buckling, vibration, and dynamic responses of FG-GPLRC plates, cylindrical, and doubly curved panels were analyzed using Navier’s solutions, Runge-Kutta integration, and Hamilton’s principle.^32–34 The vibrational response of FG-GPLRC skew plates and doubly curved panels was explored through the FSDT in combination with Chebyshev polynomials, the Ritz method, and the Sanders-type kinematic assumptions.^35,36 The multi-objective design optimizations and nonlinear dynamic buckling behavior of porous FG-GPLRC plates with and without magneto-electro-elastic sheets were also studied under various biaxial impacts.^37,38 The vibration behavior of FG-GPLRC tapered plate and cylindrical panel subjected to the transverse excitation was investigated based on the FSDT, Lagrange’s formulation and the Chebyshev-Ritz method. ³⁹ Nonlinear mechanical vibration and dynamic buckling of FG-GPLRC panels stiffened by inclined FG-GPLRC stiffeners were mentioned by applying the HSDT and energy method. ⁴⁰ By using the FSDT and novel four-unknown refined theories, the free vibration behavior of rotating FG-GPLRC cylindrical shells and toroidal shell segments were investigated.^41,42 Moreover, graphene origami-enabled metamaterials have recently emerged as architected materials with programmable mechanical properties, offering superior tunability compared to conventional graphene-reinforced composites. ⁴³

Sandwich structures are widely employed in engineering applications due to their excellent stiffness-to-weight ratio and superior energy absorption capabilities.^{1,3,10,17,44,45} The structural performance of plates and curved panels under mechanical loading can be substantially enhanced through the integration of stiffeners. While numerous studies have focused on stiffened configurations with orthogonal reinforcement,^30,31,42 the implementation of inclined stiffeners^12,13,40 has recently garnered attention due to their superior stabilizing effect. Existing numerical evidence indicates that inclined stiffeners provide more effective suppression of buckling and postbuckling deformations compared to their orthogonal counterparts.

Despite these promising advantages, the application of inclined stiffeners in advanced FG-GPLRC panels, especially those with porous cores, remains underexplored, motivating the present investigation. Analytical and semi-analytical methods have long been recognized for their reliability and superior accuracy in structural analysis. Recently, these methods have been effectively integrated with machine learning algorithms to predict the bending and buckling behavior of structures. ⁴⁶ Additionally, composite panels with complex curvature are increasingly employed in advanced structural applications, especially where architectural and functional constraints are very important; the literature lacks analytical and semi-analytical studies addressing their nonlinear thermo-mechanical buckling behavior.^31,40

A semi-analytical solution for investigating the nonlinear static and dynamic thermo-mechanical buckling of cylindrical, sinusoid, and parabolic FG-GPLRC panels reinforced with orthogonal and inclined stiffeners is presented in this paper. The formulation is based on the HSDT and incorporates an improved smeared stiffener technique adapted for inclined FG-GPLRC stiffeners. To address the geometric complexity of the panels, a nonlinear stress function is approximated using a Galerkin procedure. The governing nonlinear equations of motion are derived through Hamilton’s principle and the Euler-Lagrange equations. The Runge-Kutta method is employed to compute time-dependent responses; and by using the Budiansky-Roth criterion, the critical dynamic buckling loads can be obtained. The findings emphasize the considerable impact of inclined stiffeners, porosity properties, graphene distribution, and shell curvature on the nonlinear buckling performance of FG-GPLRC panels.

Geometry and composition of FG-GPLRC panels with porous core and stiffener

Figure 1 presents the schematic representations of sandwich panels composed of FG-GPLRC facesheets and a porous core, subjected to axial compression and thermal load. The panel geometry is characterized by the total thickness $h$ , core thickness $h_{co}$ , straight edge length $a$ , and curved edge length $b$ , with mid-surface rise $h_z$ and radius of curvature $R\left(y\right)$ .

Figure 1.

Geometrical configuration and coordinate system of three types of FG-GPLRC panels with porous cores, reinforced by orthogonal or inclined stiffeners.

A Cartesian coordinate system $Oxyz$ is approximately employed under the assumption of shallow curvature. Orthogonal stiffeners are aligned with the principal $x$ - and $y$ -axes, while inclined stiffeners are placed at an angle $\mathrm{\iota }$ relative to the $x$ -axis. The stiffener dimensions, with height $h_{st}$ , width $b_{st}$ , and spacing $d_{st}$ , are consistently defined for both stiffener types.

The shape functions and radius are analytically derived for parabolic panels to represent the curvature profile, as

$z_p=\left(\frac{y^2}{b}-y\right)\frac{4h_z}{b},R_p=\frac{{\left[16h_z^2{\left(b-2y\right)}^2+b^4\right]}^{1.5}}{8b^4{h}_z},$

(1)

and for sinusoid panels

$z_s=-h_z\hspace{0.17em}\sin \left(\frac{\mathrm{\pi }y}{b}\right),R_s=\frac{{\left[b^2+h_z^2{\mathrm{\pi }}^2\hspace{0.17em}{\cos }^2\left(y\mathrm{\pi }/b\right)\right]}^{1.5}}{b{h}_z{\mathrm{\pi }}^2\hspace{0.17em}\sin \left(y\mathrm{\pi }/b\right)}.$

(2)

The distribution of graphene platelets (GPLs) within the FG-GPLRC layers follows five distinct patterns: uniform (U) type, X-type, O-type, V-type, and Λ-type. ⁴¹ These distributions are applied to both the top and bottom composite facesheets, while the core is porous. For the stiffeners, the GPL distribution is selected such that the GPL mass fraction at the interface between the stiffeners and the facesheets is equal, aiming to minimize stress concentration at the bonded regions. Accordingly, five structural models are defined based on the GPL distribution through the panel thickness: U/PC/U-U, X/PC/X-X, O/PC/O-O, V/PC/Λ-V, and Λ/PC/V-Λ, where “PC” denotes the porous core.

The effective Young’s modulus of the FG-GPLRC facesheets and stiffeners is estimated using the extended Halpin–Tsai homogenization scheme, which accounts for the reinforcement efficiency of dispersed GPLs, presented as ⁴¹

$E_{GP}=\frac{3\left(1+{\mathrm{\kappa }}_1^{GP}{\mathrm{\mu }}_1^{GP}{V}_{GP}^{*}\right)}{8\left(1-{\mathrm{\mu }}_1^{GP}{V}_{GP}^{*}\right)}{E}_M+\frac{5\left(1+{\mathrm{\kappa }}_2^{GP}{\mathrm{\mu }}_2^{GP}{V}_{GP}^{*}\right)}{8\left(1-{\mathrm{\mu }}_2^{GP}{V}_{GP}^{*}\right)}{E}_M,$

(3)

where

${\mathrm{\mu }}_1^{GP}=\frac{E_G/E_M-1}{E_G/E_M+{\mathrm{\kappa }}_1^{GP}},{\mathrm{\mu }}_2^{GP}=\frac{E_G/E_M-1}{E_G/E_M+{\mathrm{\kappa }}_2^{GP}},{\mathrm{\kappa }}_1^{GP}=2a_{GP}/t_{GP},{\mathrm{\kappa }}_2^{GP}=2b_{GP}/t_{GP},$

with $E_M$ and $E_G$ denote the Young’s moduli of the isotropic matrix and graphene platelets, respectively. The geometrical characteristics of the GPLs are represented by their length $a_{GP},$ , width $b_{GP},$ and thickness $t_{GP}$ .

The effective volume fraction of GPLs, $V_{GP}^{*}$ , is computed as

$V_{GP}^{*}=\frac{W_{GP}^{*}}{W_{GP}^{*}+\left({\mathrm{\rho }}_G/{\mathrm{\rho }}_M\right)\left(1-W_{GP}^{*}\right)},$

(4)

where ${\mathrm{\rho }}_M$ and ${\mathrm{\rho }}_G$ are the mass densities of the matrix and GPLs, respectively, and $W_{GP}^{*}$ denotes the GPL mass fraction, which varies through the panel thickness depending on the selected distribution pattern, defined by

• +) The upper layer $\left(\frac{-h}{2}\le z\le \frac{-h_{co}}{2}\right)$

$‐\hspace{0.17em}\mathrm{U}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=W_{GP}$

(5)

$‐\hspace{0.17em}\mathrm{X}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\left|\frac{8z+2h+2h_{co}}{h-h_{co}}\right|\right)W_{GP}$

(6)

$‐\hspace{0.17em}\mathrm{O}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(2-\left|\frac{8z+2h+2h_{co}}{h-h_{co}}\right|\right)W_{GP}$

(7)

$‐\hspace{0.17em}\mathrm{V}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{-4z-2h_{co}}{h-h_{co}}\right)W_{GP}$

(8)

$‐\hspace{0.17em}\mathrm{\Lambda }\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{4z+2h}{h-h_{co}}\right)W_{GP}$

(9)

• +) The lower layer $\left(\frac{h_{co}}{2}\le z\le \frac{h}{2}\right)$

$‐\hspace{0.17em}\mathrm{U}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=W_{GP}$

(10)

$‐\hspace{0.17em}\mathrm{X}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\left|\frac{8z-2h-2h_{co}}{h-h_{co}}\right|\right)W_{GP}$

(11)

$‐\hspace{0.17em}\mathrm{O}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(2-\left|\frac{8z-2h-2h_{co}}{h-h_{co}}\right|\right)W_{GP}$

(12)

$‐\hspace{0.17em}\mathrm{V}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{-4z+2h}{h-h_{co}}\right)W_{GP}$

(13)

$‐\hspace{0.17em}\mathrm{\Lambda }\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{4z-2h_{co}}{h-h_{co}}\right)W_{GP}$

(14)

• +) The stiffener $\left(\frac{h}{2}\le z\le \frac{h}{2}+h_{st}\right)$

$‐\hspace{0.17em}\mathrm{U}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=W_{GP}$

(15)

$‐\hspace{0.17em}\mathrm{X}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\left|\frac{-4z+2h}{h_{st}}+2\right|\right)W_{GP}$

(16)

$‐\hspace{0.17em}\mathrm{O}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(2-\left|\frac{-4z+2h}{h_{st}}+2\right|\right)W_{GP}$

(17)

$‐\hspace{0.17em}\mathrm{V}\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{-2z+h}{h_{st}}+2\right)W_{GP}$

(18)

$‐\hspace{0.17em}\mathrm{\Lambda }\hspace{0.17em}\text{distribution}:W_{GP}^{*}=\left(\frac{2z-h}{h_{st}}\right)W_{GP}$

(19)

where the average GPL mass fraction is $W_{GP}$ .

The effective Poisson’s ratio ${\mathrm{\nu }}_{GP}$ , thermal expansion coefficient ${\mathrm{\alpha }}_{GP}$ , and density ${\mathrm{\rho }}_{GP}$ of the panel layers are determined using the rule of mixtures, as

$\begin{array}{l}{\mathrm{\nu }}_{GP}={\mathrm{\nu }}_M\left(1-V_{GP}^{*}\right)+{\mathrm{\nu }}_G{V}_{GP}^{*},\\ {\mathrm{\alpha }}_{GP}={\mathrm{\alpha }}_M\left(1-V_{GP}^{*}\right)+{\mathrm{\alpha }}_G{V}_{GP}^{*},\\ {\mathrm{\rho }}_{GP}={\mathrm{\rho }}_M\left(1-V_{GP}^{*}\right)+{\mathrm{\rho }}_G{V}_{GP}^{*}.\end{array}$

(20)

where ${\mathrm{\nu }}_M,{\mathrm{\alpha }}_M$ and ${\mathrm{\nu }}_G,{\mathrm{\alpha }}_G$ are the Poisson ratio, and thermal expansion coefficient of matrix and GPL, respectively.

The core layer is assumed to be composed of the same base material as the matrix in the FG-GPLRC layers. Its effective mechanical properties, including Young’s modulus $E_{co}$ , coefficient of thermal expansion ${\mathrm{\alpha }}_{co}$ , and Poisson’s ratio ${\mathrm{\rho }}_{co}$ , are estimated as functions of the porosity coefficient $e$ ( $0\le e\le 1$ ), as

$\{\begin{array}{l}{E}_{co}=E_M\left[1-e\hspace{0.17em}\cos \left(\mathrm{\pi }z/h_{co}\right)\right]\\ {\mathrm{\alpha }}_{co}={\mathrm{\alpha }}_M\\ {\mathrm{\rho }}_{co}={\mathrm{\rho }}_M\left[1-\left(1-\sqrt{1-e}\right)\cos \left(\mathrm{\pi }z/h_{co}\right)\right]\end{array},-\frac{h_{co}}{2}<z<\frac{h_{co}}{2},$

(21)

The constitutive relations for both the FG-GPLRC facesheets and the porous core are established using Hooke’s law, incorporating thermal strain effects, presented as

$\begin{array}{l}\left[\begin{array}{l}{\mathrm{\sigma }}_x\\ {\mathrm{\sigma }}_y\\ {\mathrm{\sigma }}_{xy}\end{array}\right]=\left[\begin{array}{lll}{\widehat{Q}}_{11}& {\widehat{Q}}_{12}& 0\\ {\widehat{Q}}_{12}& {\widehat{Q}}_{22}& 0\\ 0& 0& {\widehat{Q}}_{66}\end{array}\right]\left[\begin{array}{l}{\mathrm{\epsilon }}_x-\mathrm{\alpha }\left(z\right)\mathrm{\Delta }T\\ {\mathrm{\epsilon }}_y-\mathrm{\alpha }\left(z\right)\mathrm{\Delta }T\\ {\mathrm{\gamma }}_{xy}\end{array}\right],\\ {\mathrm{\sigma }}_{xz}={\widehat{Q}}_{44}{\mathrm{\gamma }}_{xz},{\mathrm{\sigma }}_{yz}={\widehat{Q}}_{55}{\mathrm{\gamma }}_{yz},\end{array}$

(22)

where the temperature difference $\mathrm{\Delta }T$ reflects the change from the stress-free reference state to the deformed configuration, ${\mathrm{\sigma }}_x,{\mathrm{\sigma }}_y,{\mathrm{\sigma }}_{xy},{\mathrm{\sigma }}_{xz},{\mathrm{\sigma }}_{yz}$ are the stress components, ${\mathrm{\epsilon }}_x,{\mathrm{\epsilon }}_y,{\mathrm{\gamma }}_{xy},{\mathrm{\gamma }}_{xz},{\mathrm{\gamma }}_{yz}$ are the strain components, and the associated reduced stiffness coefficients ${\widehat{Q}}_{ij}$ are determined as

${\widehat{Q}}_{11}={\widehat{Q}}_{22}=\frac{E\left(z\right)}{1-\mathrm{\nu }{\left(z\right)}^2},{\widehat{Q}}_{12}=\frac{\mathrm{\nu }\left(z\right)E\left(z\right)}{1-\mathrm{\nu }{\left(z\right)}^2},{\widehat{Q}}_{44}={\widehat{Q}}_{55}={\widehat{Q}}_{66}=\frac{E\left(z\right)}{2\left[1+\mathrm{\nu }\left(z\right)\right]}.$

Fundamental formulations

The present study adopts the HSDT to account for both transverse shear effects and geometrical nonlinearities arising from large deflections, based on the von Kármán assumptions.

The displacement field $\widehat{u}=\widehat{u}\left(x,y,z,t\right),\widehat{v}=\widehat{v}\left(x,y,z,t\right),\widehat{w}=\widehat{w}\left(x,y,z,t\right)$ of a point in the panel is expressed as follows ¹

$\left\{\begin{array}{l}\widehat{u}\\ \widehat{v}\\ \widehat{w}\end{array}\right\}=\left\{\begin{array}{l}u\\ v\\ w\end{array}\right\}+z\left\{\begin{array}{l}{\mathrm{\varphi }}_x\\ {\mathrm{\varphi }}_y\\ 0\end{array}\right\}-\frac{4z^3}{3h^2}\left\{\begin{array}{l}{\mathrm{\varphi }}_x+w_{,x}\\ {\mathrm{\varphi }}_y+w_{,y}\\ 0\end{array}\right\},$

(23)

where $u=u\left(x,y,t\right),v=v\left(x,y,t\right)$ and $w=w\left(x,y,t\right)$ are the displacements and deflection at the mid-surface; ${\mathrm{\varphi }}_x={\mathrm{\varphi }}_x\left(x,y,t\right)$ and ${\mathrm{\varphi }}_y={\mathrm{\varphi }}_y\left(x,y,t\right)$ are the rotations at the mid-surface, respectively.

The strain expressions with respect to panel thickness ${\mathrm{\epsilon }}_x,{\mathrm{\epsilon }}_y,{\mathrm{\gamma }}_{xy},{\mathrm{\gamma }}_{xz},{\mathrm{\gamma }}_{yz}$ can be derived as¹

$\left\{\begin{array}{l}{\mathrm{\epsilon }}_x\\ {\mathrm{\epsilon }}_y\\ {\mathrm{\gamma }}_{xy}\end{array}\right\}=\left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{0x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{0y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0xy}^{*}\end{array}\right\}+z\left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{1x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{1y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{1xy}^{*}\end{array}\right\}+z^3\left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{3x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{3y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{3xy}^{*}\end{array}\right\},\left\{\begin{array}{l}{\mathrm{\gamma }}_{xz}\\ {\mathrm{\gamma }}_{yz}\end{array}\right\}=\left\{\begin{array}{l}{\widehat{\mathrm{\gamma }}}_{0xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0yz}^{*}\end{array}\right\}+z^2\left\{\begin{array}{l}{\widehat{\mathrm{\gamma }}}_{2xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{2yz}^{*}\end{array}\right\},$

(24)

Additionally, initial geometric imperfections are incorporated into the displacement field to reflect more realistic structural behavior. Accordingly, the nonlinear strain–displacement relations for FG-GPLRC panels with shallow curvature can be formulated as follows ³⁰

$\begin{array}{l}\left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{0x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{0y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0xy}^{*}\end{array}\right\}=\left\{\begin{array}{l}{u}_{,x}+w_{,x}^2/2+w_{,x}{\overline{w}}_{,x}\\ v_{,y}+w_{,y}^2/2+w_{,y}{\overline{w}}_{,y}-w/R\left(y\right)\\ v_{,x}+u_{,y}+w_{,x}{w}_{,y}+w_{,x}{\overline{w}}_{,y}+w_{,y}{\overline{w}}_{,x}\end{array}\right\},\\ \left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{1x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{1y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{1xy}^{*}\end{array}\right\}=\left\{\begin{array}{l}{\mathrm{\varphi }}_{x,x}\\ {\mathrm{\varphi }}_{y,y}\\ {\mathrm{\varphi }}_{y,x}+{\mathrm{\varphi }}_{x,y}\end{array}\right\},\left\{\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{3x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{3y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{3xy}^{*}\end{array}\right\}=-\mathrm{\lambda }\left\{\begin{array}{l}{w}_{,xx}+{\mathrm{\varphi }}_{x,x}\\ w_{,yy}+{\mathrm{\varphi }}_{y,y}\\ {\mathrm{\varphi }}_{y,x}+{\mathrm{\varphi }}_{x,y}+2w_{,xy}\end{array}\right\},\\ \left\{\begin{array}{l}{\widehat{\mathrm{\gamma }}}_{0xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0yz}^{*}\end{array}\right\}=\left\{\begin{array}{l}{\mathrm{\varphi }}_x+w_{,x}\\ {\mathrm{\varphi }}_y+w_{,y}\end{array}\right\},\left\{\begin{array}{l}{\widehat{\mathrm{\gamma }}}_{2xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{2yz}^{*}\end{array}\right\}=-3\mathrm{\lambda }\left\{\begin{array}{l}{\mathrm{\varphi }}_x+w_{,x}\\ {\mathrm{\varphi }}_y+w_{,y}\end{array}\right\},\mathrm{\lambda }=\frac{4}{3h^2}.\end{array}$

(25)

where $\overline{w}$ is the initial imperfection.

To account for the stiffening effect of inclined FG-GPLRC stiffeners, an improved smeared stiffener technique, originally proposed by Lekhnitskii and subsequently extended, is employed. This technique models the stiffeners as anisotropic beams embedded within the composite panel and distributes their mechanical contributions continuously over the panel surface. By incorporating coordinate transformation, the inclined orientation of the stiffeners relative to the global axes is taken into account, enabling representation of the full stiffness components of stiffeners. The expressions relating the internal force and moment resultants to the strains, rotations, and deflections are established for the stiffened panels by integrating the contributions from both the panel skin and the stiffeners within the framework of HSDT, as

$\left\{\begin{array}{l}{N}_x\\ N_y\\ N_{xy}\\ M_x\\ M_y\\ \begin{array}{l}{M}_{xy}\\ T_x\\ T_y\\ T_{xy}\end{array}\end{array}\right\}=\left[\begin{array}{lllllllll}{\overline{L}}_{11}& {\overline{L}}_{12}& 0& {\overline{H}}_{11}& {\overline{H}}_{12}& 0& {\overline{G}}_{11}& {\overline{G}}_{12}& 0\\ {\overline{L}}_{12}& {\overline{L}}_{22}& 0& {\overline{H}}_{12}& {\overline{H}}_{22}& 0& {\overline{G}}_{12}& {\overline{G}}_{22}& 0\\ 0& 0& {\overline{L}}_{66}& 0& 0& {\overline{H}}_{66}& 0& 0& {\overline{G}}_{66}\\ {\overline{H}}_{11}& {\overline{H}}_{12}& 0& {\overline{F}}_{11}& {\overline{F}}_{12}& 0& {\overline{A}}_{11}& {\overline{A}}_{12}& 0\\ {\overline{H}}_{12}& {\overline{H}}_{22}& 0& {\overline{F}}_{12}& {\overline{F}}_{22}& 0& {\overline{A}}_{12}& {\overline{A}}_{22}& 0\\ 0& 0& {\overline{H}}_{66}& 0& 0& {\overline{F}}_{66}& 0& 0& {\overline{A}}_{66}\\ {\overline{G}}_{11}& {\overline{G}}_{12}& 0& {\overline{A}}_{11}& {\overline{A}}_{12}& 0& {\overline{D}}_{11}& {\overline{D}}_{12}& 0\\ {\overline{G}}_{12}& {\overline{G}}_{22}& 0& {\overline{A}}_{12}& {\overline{A}}_{22}& 0& {\overline{D}}_{12}& {\overline{D}}_{22}& 0\\ 0& 0& {\overline{G}}_{66}& 0& 0& {\overline{A}}_{66}& 0& 0& {\overline{D}}_{66}\end{array}\right]\left[\begin{array}{l}{\widehat{\mathrm{\epsilon }}}_{0x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{0y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0xy}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{1x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{1y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{1xy}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{3x}^{*}\\ {\widehat{\mathrm{\epsilon }}}_{3y}^{*}\\ {\widehat{\mathrm{\gamma }}}_{3xy}^{*}\end{array}\right]-\left[\begin{array}{l}{\widehat{\mathrm{\Phi }}}_{1x}\\ {\widehat{\mathrm{\Phi }}}_{1y}\\ 0\\ {\widehat{\mathrm{\Phi }}}_{2x}\\ {\widehat{\mathrm{\Phi }}}_{2y}\\ 0\\ {\widehat{\mathrm{\Phi }}}_{4x}\\ {\widehat{\mathrm{\Phi }}}_{4y}\\ 0\end{array}\right]\mathrm{\Delta }T,$

(26)

where $N_x,N_y,N_{xy}$ are the extension forces, $M_x,M_y,M_{xy}$ are the moments, and the higher-order moments are denoted by $T_x,T_y,T_{xy}$ . The thermal extension forces, thermal moments, and thermal higher-order moments are denoted by ${\widehat{\mathrm{\Phi }}}_{1x},{\widehat{\mathrm{\Phi }}}_{1y}$ , ${\widehat{\mathrm{\Phi }}}_{2x},{\widehat{\mathrm{\Phi }}}_{2y}$ , and ${\widehat{\mathrm{\Phi }}}_{4x},{\widehat{\mathrm{\Phi }}}_{4y}$ .

The expressions for the transverse shear forces, including both the conventional and higher-order components ( $Q_x,Q_y$ and $S_x,S_y$ ), can be formulated as follows

$\left[\begin{array}{l}{Q}_x\\ Q_y\\ S_x\\ S_y\end{array}\right]=\left[\begin{array}{llll}{\overline{C}}_{44}& 0& {\overline{B}}_{44}& 0\\ 0& {\overline{C}}_{55}& 0& {\overline{B}}_{55}\\ {\overline{B}}_{44}& 0& {\overline{E}}_{44}& 0\\ 0& {\overline{B}}_{55}& 0& {\overline{E}}_{55}\end{array}\right]\left[\begin{array}{l}{\widehat{\mathrm{\gamma }}}_{0xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{0yz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{2xz}^{*}\\ {\widehat{\mathrm{\gamma }}}_{2yz}^{*}\end{array}\right].$

(27)

The stiffness coefficients and thermally induced force resultants appearing in equations (26) and (27) are expressed in the following forms

$\begin{array}{l}{\overline{L}}_{11}={\overline{L}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{L}}_{12}={\overline{L}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{L}}_{22}={\overline{L}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{L}}_{66}={\overline{L}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{L}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\end{array}$

${\overline{H}}_{11}={\overline{H}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{H}}_{12}={\overline{H}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),$

${\overline{H}}_{22}={\overline{H}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{H}}_{66}={\overline{H}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{H}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),$

$\begin{array}{l}{\overline{G}}_{11}={\overline{G}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{G}}_{12}={\overline{G}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{G}}_{22}={\overline{G}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{G}}_{66}={\overline{G}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{G}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\end{array}$

$\begin{array}{l}{\overline{F}}_{11}={\overline{F}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{F}}_{12}={\overline{F}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{F}}_{22}={\overline{F}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{F}}_{66}={\overline{F}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{F}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\end{array}$

$\begin{array}{l}{\overline{A}}_{11}={\overline{A}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{A}}_{12}={\overline{A}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{A}}_{22}={\overline{A}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{A}}_{66}={\overline{A}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{A}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\end{array}$

$\begin{array}{l}{\overline{D}}_{11}={\overline{D}}_{11}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{in}\hspace{0.17em}{\cos }^4\left(\mathrm{\iota }\right),{\overline{D}}_{12}={\overline{D}}_{12}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{D}}_{22}={\overline{D}}_{22}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{in}\hspace{0.17em}{\sin }^4\left(\mathrm{\iota }\right),{\overline{D}}_{66}={\overline{D}}_{66}^p+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{D}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right){\sin }^2\left(\mathrm{\iota }\right),\end{array}$

$\begin{array}{l}{\overline{C}}_{44}={\overline{C}}_{44}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{C}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{C}}_{st}^{in}\hspace{0.17em}{\sin }^2\left(\mathrm{\iota }\right),{\overline{B}}_{44}={\overline{B}}_{44}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{B}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{B}}_{st}^{in}\hspace{0.17em}{\sin }^2\left(\mathrm{\iota }\right),\\ {\overline{E}}_{44}={\overline{E}}_{44}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{E}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{E}}_{st}^{in}\hspace{0.17em}{\sin }^2\left(\mathrm{\iota }\right),{\overline{C}}_{55}={\overline{C}}_{55}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{C}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{C}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right),\\ {\overline{B}}_{55}={\overline{B}}_{55}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{B}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{B}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right),{\overline{E}}_{55}={\overline{E}}_{55}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\overline{E}}_{st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\overline{E}}_{st}^{in}\hspace{0.17em}{\cos }^2\left(\mathrm{\iota }\right),\end{array}$

$\begin{array}{l}{\widehat{\mathrm{\Phi }}}_{1x}={\widehat{\mathrm{\Phi }}}_{1x}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{1st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{1st}^{in}\left[{\cos }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],\\ {\widehat{\mathrm{\Phi }}}_{2x}={\widehat{\mathrm{\Phi }}}_{2x}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{2st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{2st}^{in}\left[{\cos }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],\\ {\widehat{\mathrm{\Phi }}}_{4x}={\widehat{\mathrm{\Phi }}}_{4x}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{4st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{4st}^{in}\left[{\cos }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],\\ {\widehat{\mathrm{\Phi }}}_{1y}={\widehat{\mathrm{\Phi }}}_{1y}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{1st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{1st}^{in}\left[{\sin }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],\\ {\widehat{\mathrm{\Phi }}}_{2y}={\widehat{\mathrm{\Phi }}}_{2y}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{2st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{2st}^{in}\left[{\sin }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],\end{array}$

${\widehat{\mathrm{\Phi }}}_{4y}={\widehat{\mathrm{\Phi }}}_{4y}^p+{\mathrm{\phi }}_1\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{4st}^{or}+2{\mathrm{\phi }}_2\frac{b_{st}}{d_{st}}{\widehat{\mathrm{\Phi }}}_{4st}^{in}\left[{\sin }^6\left(\mathrm{\iota }\right)+{\sin }^2\left(\mathrm{\iota }\right){\cos }^2\left(\mathrm{\iota }\right)\right],$

where, the superscripts “ $p$ ”, “ $or$ ”, and “ $in$ ” are respectively used to denote the panel, orthogonal stiffener, and inclined stiffener. The binary parameters ${\mathrm{\phi }}_1$ and ${\mathrm{\phi }}_2$ are introduced to indicate the stiffener configuration: ${\mathrm{\phi }}_1$ = 0 and ${\mathrm{\phi }}_2$ = 0 define the unstiffened case; ${\mathrm{\phi }}_1$ = 0 and ${\mathrm{\phi }}_2$ = 1 represent inclined stiffener; ${\mathrm{\phi }}_1$ = 1 and ${\mathrm{\phi }}_2$ = 0 correspond to orthogonal stiffeners, and

$\begin{array}{l}\left({\overline{L}}_{ij}^p,{\overline{H}}_{ij}^p,{\overline{F}}_{ij}^p,{\overline{G}}_{ij}^p,{\overline{A}}_{ij}^p,{\overline{D}}_{ij}^p\right)={\int }_{-h/2}^{h/2}{\widehat{Q}}_{ij}\left(1,z,z^2,z^3,z^4,z^6\right)dz,\left(i,j=1,2,6\right),\\ \left({\overline{C}}_{ij}^p,{\overline{B}}_{ij}^p,{\overline{E}}_{ij}^p\right)={\int }_{-h/2}^{h/2}{\widehat{Q}}_{ij}\left(1,z^2,z^4\right)dz,\left(i,j=4,5\right),\end{array}$

$\begin{array}{l}\left({\widehat{\mathrm{\Phi }}}_{1x}^p,{\widehat{\mathrm{\Phi }}}_{2x}^p,{\widehat{\mathrm{\Phi }}}_{4x}^p\right)={\int }_{-h/2}^{h/2}\mathrm{\alpha }\left(z\right)\left({\widehat{Q}}_{12}+{\widehat{Q}}_{11}\right)\left(1,z,z^3\right)dz,\\ \left({\widehat{\mathrm{\Phi }}}_{1y}^p,{\widehat{\mathrm{\Phi }}}_{2y}^p,{\widehat{\mathrm{\Phi }}}_{4y}^p\right)={\int }_{-h/2}^{h/2}\mathrm{\alpha }\left(z\right)\left({\widehat{Q}}_{12}+{\widehat{Q}}_{22}\right)\left(1,z,z^3\right)dz,\end{array}$

$\begin{array}{l}\left(L_{ij},H_{ij},F_{ij},G_{ij},A_{ij},D_{ij}\right)={\int }_{h/2}^{h/2+h_{st}}{\widehat{Q}}_{ij}\left(1,z,z^2,z^3,z^4,z^6\right)dz,\left(i,j=1,2,6\right),\\ \left({\overline{C}}_{st}^i,{\overline{B}}_{st}^i,{\overline{E}}_{st}^i\right)={\int }_{h/2}^{h/2+h_{st}}{\widehat{Q}}_{44}\left(1,z^2,z^4\right)dz,\left(i=or,in\right),\\ \left({\widehat{\mathrm{\Phi }}}_{1st}^i,{\widehat{\mathrm{\Phi }}}_{2st}^i,{\widehat{\mathrm{\Phi }}}_{4st}^i\right)={\int }_{h/2}^{h/2+h_{st}}\mathrm{\alpha }\left(z\right)\left({\widehat{Q}}_{12}+{\widehat{Q}}_{11}\right)\left(1,z,z^3\right)dz,\left(i=or,in\right).\end{array}$

$\begin{array}{l}\left[\begin{array}{lll}{\overline{L}}_{st}^i& {\overline{H}}_{st}^i& {\overline{G}}_{st}^i\\ {\overline{H}}_{st}^i& {\overline{F}}_{st}^i& {\overline{A}}_{st}^i\\ {\overline{G}}_{st}^i& {\overline{A}}_{st}^i& {\overline{D}}_{st}^i\end{array}\right]=\left[\begin{array}{lll}{L}_{11}& H_{11}& G_{11}\\ H_{11}& F_{11}& A_{11}\\ G_{11}& A_{11}& D_{11}\end{array}\right]-\left[\begin{array}{llllll}{L}_{12}& 0& H_{12}& 0& G_{12}& 0\\ H_{12}& 0& F_{12}& 0& A_{12}& 0\\ G_{12}& 0& A_{12}& 0& D_{12}& 0\end{array}\right]\\ \times {\left[\begin{array}{llllll}{L}_{22}& 0& H_{22}& 0& G_{22}& 0\\ 0& L_{66}& 0& H_{66}& 0& G_{66}\\ H_{22}& 0& F_{22}& 0& A_{22}& 0\\ 0& H_{66}& 0& F_{66}& 0& A_{66}\\ G_{22}& 0& A_{22}& 0& G_{66}& 0\\ 0& D_{22}& 0& A_{66}& 0& D_{66}\end{array}\right]}^{-1}\left[\begin{array}{lll}{L}_{12}& H_{12}& G_{12}\\ 0& 0& 0\\ H_{12}& F_{12}& A_{12}\\ 0& 0& 0\\ G_{12}& A_{12}& D_{12}\\ 0& 0& 0\end{array}\right],\left(i=or,in\right)\end{array}$

The compatibility condition, incorporating the nonlinear strain–displacement relationships, is formulated based on equation (25) as follows

${\widehat{\mathrm{\epsilon }}}_{0x,yy}^{*}+{\widehat{\mathrm{\epsilon }}}_{0y,xx}^{*}-{\widehat{\mathrm{\gamma }}}_{0xy,xy}^{*}=-{\overline{w}}_{,xx}{w}_{,yy}+2w_{,xy}{\overline{w}}_{,xy}+w_{,xy}^2-w_{,xx}{w}_{,yy}-w_{,xx}{\overline{w}}_{,yy}-\frac{w_{,xx}}{R\left(y\right)}.$

(28)

The proposed stress function $\mathrm{\varsigma }\left(x,y\right)$ is constructed to satisfy the following three fundamental conditions, as

$N_{xy}=-{\mathrm{\varsigma }}_{,xy},N_x={\mathrm{\varsigma }}_{,yy},N_y={\mathrm{\varsigma }}_{,xx}.$

(29)

By using equations (26) and (29), the strain compatibility condition given in equation (27) can be reformulated as follows

$\begin{array}{l}{\overline{L}}_{22}^{*}{\mathrm{\varsigma }}_{,xxxx}+\left({\overline{L}}_{66}^{*}+2{\overline{L}}_{12}^{*}\right){\mathrm{\varsigma }}_{,xxyy}+{\overline{L}}_{11}^{*}{\mathrm{\varsigma }}_{,yyyy}+\left({\overline{H}}_{21}^{*}-\mathrm{\lambda }{\overline{G}}_{21}^{*}\right){\mathrm{\varphi }}_{x,xxx}\\ +\left(\mathrm{\lambda }{\overline{G}}_{66}^{*}-\mathrm{\lambda }{\overline{G}}_{11}^{*}+{\overline{H}}_{11}^{*}-{\overline{H}}_{66}^{*}\right){\mathrm{\varphi }}_{x,xyy}+\left(\mathrm{\lambda }{\overline{G}}_{66}^{*}-\mathrm{\lambda }{\overline{G}}_{22}^{*}+{\overline{H}}_{22}^{*}-{\overline{H}}_{66}^{*}\right){\mathrm{\varphi }}_{y,xxy}\\ +\left({\overline{H}}_{12}^{*}-\mathrm{\lambda }{\overline{G}}_{12}^{*}\right){\mathrm{\varphi }}_{y,yyy}-\mathrm{\lambda }{\overline{G}}_{21}^{*}{w}_{,xxxx}-\mathrm{\lambda }\left({\overline{G}}_{11}^{*}+{\overline{G}}_{22}^{*}-2{\overline{G}}_{66}^{*}\right)w_{,xxyy}\\ -\mathrm{\lambda }{\overline{G}}_{12}^{*}{w}_{,yyyy}+w_{,xx}{\overline{w}}_{,yy}-w_{,xy}^2+w_{,xx}{w}_{,yy}+{\overline{w}}_{,xx}{w}_{,yy}-2w_{,xy}{\overline{w}}_{,xy}+w_{,xx}/R\left(y\right)=0,\end{array}$

(30)

where

$\begin{array}{l}{\overline{L}}_{11}^{*}={\overline{L}}_{22}/\left({\overline{L}}_{11}{\overline{L}}_{22}-{\overline{L}}_{12}^2\right),{\overline{L}}_{12}^{*}=-{\overline{L}}_{12}/\left({\overline{L}}_{11}{\overline{L}}_{22}-{\overline{L}}_{12}^2\right),{\overline{L}}_{22}^{*}={\overline{L}}_{11}/\left({\overline{L}}_{11}{\overline{L}}_{22}-{\overline{L}}_{12}^2\right),\\ {\overline{L}}_{66}^{*}=1/{\overline{L}}_{66},{\overline{H}}_{11}^{*}={\overline{L}}_{11}^{*}{\overline{H}}_{11}+{\overline{L}}_{12}^{*}{\overline{H}}_{12},{\overline{H}}_{12}^{*}={\overline{L}}_{11}^{*}{\overline{H}}_{12}+{\overline{L}}_{12}^{*}{\overline{H}}_{22},{\overline{H}}_{21}^{*}={\overline{L}}_{22}^{*}{\overline{H}}_{12}+{\overline{L}}_{12}^{*}{\overline{H}}_{11},\\ {\overline{H}}_{22}^{*}={\overline{L}}_{22}^{*}{\overline{H}}_{22}+{\overline{L}}_{12}^{*}{\overline{H}}_{12},{\overline{H}}_{66}^{*}={\overline{L}}_{66}^{*}{\overline{H}}_{66},{\overline{G}}_{11}^{*}={\overline{L}}_{11}^{*}{\overline{G}}_{11}+{\overline{L}}_{12}^{*}{\overline{G}}_{12},{\overline{G}}_{12}^{*}={\overline{L}}_{11}^{*}{\overline{G}}_{12}+{\overline{L}}_{12}^{*}{\overline{G}}_{22},\\ {\overline{G}}_{21}^{*}={\overline{L}}_{22}^{*}{\overline{G}}_{12}+{\overline{L}}_{12}^{*}{\overline{G}}_{11},{\overline{G}}_{22}^{*}={\overline{L}}_{22}^{*}{\overline{G}}_{22}+{\overline{L}}_{12}^{*}{\overline{G}}_{12},{\overline{G}}_{66}^{*}={\overline{L}}_{66}^{*}{\overline{G}}_{66}.\end{array}$

Approximate stress function, and energy method

In the present investigation, three distinct panel geometries, including cylindrical, sinusoid, and parabolic panels, are analyzed under four different boundary conditions.

The first boundary condition assumes that all four edges of the panel are simply supported and free to move tangentially, herein referred to as the 4F configuration, as

$\begin{array}{l}{N}_{0x}=N_x=-h{P}_x,M_x=0,T_x=0,N_{xy}=0,w=0,{\mathrm{\varphi }}_y=0,at\hspace{0.5em}x=0,a,\\ N_y=N_{0y}=0,T_y=0,M_y=0,N_{xy}=0,w=0,{\mathrm{\varphi }}_x=0,at\hspace{0.5em}y=0,b.\end{array}$

(31)

The second boundary condition, denoted as 2F2I, considers the panel to be simply supported along all four edges. In this configuration, the edges at $x=0,a$ are free to move in-plane, while the edges at $y=0,b$ are restrained from in-plane movement. This condition is mathematically described as

$\begin{array}{l}{N}_{0x}=N_x=-h{P}_x,M_x=0,T_x=0,N_{xy}=0,w=0,{\mathrm{\varphi }}_y=0,at\hspace{0.5em}x=0,a,\\ N_y=N_{0y},T_y=0,M_y=0,v=0,w=0,{\mathrm{\varphi }}_x=0,at\hspace{0.5em}y=0,b.\end{array}$

(32)

The third boundary condition, referred to as 2I2F, assumes that the panel is simply supported along all edges. In this case, the edges at $y=0,b$ are free to move in-plane displacement, whereas the edges at $x=0,a$ are in-plane constrained. This boundary setting can be represented as follows

$\begin{array}{l}{N}_x=N_{0x},T_x=0,M_x=0,w=0,u=0,{\mathrm{\varphi }}_y=0,at\hspace{0.5em}x=0,a,\\ N_{0y}=N_y=0,T_y=0,M_y=0,N_{xy}=0,w=0,{\mathrm{\varphi }}_x=0,at\hspace{0.5em}y=0,b.\end{array}$

(33)

Finally, the fourth boundary condition, denoted as 4I, considers all four edges of the panel to be simply supported and fully restrained from in-plane movement. This condition represents the most constrained configuration and is defined as follows

$\begin{array}{l}{N}_x=N_{0x},T_x=0,M_x=0,u=0,w=0,{\mathrm{\varphi }}_y=0,at\hspace{0.5em}x=0,a,\\ N_y=N_{0y},T_y=0,M_y=0,v=0,w=0,{\mathrm{\varphi }}_x=0,at\hspace{0.5em}y=0,b.\end{array}$

(34)

These are common boundary conditions widely used to model real-world structural components. Their application depends on the specific design of the edge supports and the level of approximation acceptable in the simulation.

To fulfill the requirements imposed by the four types of boundary conditions, the approximate expressions for deflection, rotational components, and initial geometric imperfections are assumed in the following forms ³⁰

$\begin{array}{l}w=W\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\alpha }}}_{1}x\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y,\overline{w}=\mathrm{\eta }h\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\alpha }}}_{1}x\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y,\\ {\mathrm{\varphi }}_x={\mathrm{\Phi }}_x\hspace{0.17em}\cos \hspace{0.17em}{\overline{\mathrm{\alpha }}}_{1}x\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y,{\mathrm{\varphi }}_y={\mathrm{\Phi }}_y\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\alpha }}}_{1}x\hspace{0.17em}\cos \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y,\end{array}$

(35)

where $\mathrm{\eta }$ is the imperfection size, numbers of half-wave in straight and curved axes are defined by $m$ and $n$ , and ${\overline{\mathrm{\alpha }}}_1=m\mathrm{\pi }/a,{\overline{\mathrm{\beta }}}_1=n\mathrm{\pi }/b$ ; $W$ and ${\mathrm{\Phi }}_x,{\mathrm{\Phi }}_y$ are the amplitudes of deflection, and rotations, respectively.

Owing to the complex curvature functions associated with parabolic and sinusoid panels, directly obtaining an exact stress function from the strain compatibility equation (30) becomes analytically difficult. To address this challenge, an approximate stress function is adopted, considering both linear and nonlinear terms in a manner analogous to the exact formulation used for cylindrical panels

$\mathrm{\varsigma }=\frac{1}{2}{N}_{0y}{x}^2+\frac{1}{2}{N}_{0x}{y}^2+{\mathrm{\varsigma }}_1\hspace{0.17em}\cos \hspace{0.17em}2{\overline{\mathrm{\alpha }}}_{1}x+{\mathrm{\varsigma }}_2\hspace{0.17em}\cos \hspace{0.17em}2{\overline{\mathrm{\beta }}}_{1}y+{\mathrm{\varsigma }}_3\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\alpha }}}_{1}x\hspace{0.17em}\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y.$

(36)

By substituting equations (35) and (36) into the compatibility condition given in equation (30), and subsequently applying the Galerkin procedure, the coefficients of the approximated stress function are determined as follows

$\begin{array}{l}{\mathrm{\varsigma }}_1={\overline{J}}_{11}W\left(2\mathrm{\eta }h+W\right)+{\overline{J}}_{12}W,{\mathrm{\varsigma }}_2={\overline{J}}_{21}W\left(2\mathrm{\eta }h+W\right)+{\overline{J}}_{22}W,\\ {\mathrm{\varsigma }}_3={\overline{J}}_{31}{\mathrm{\Phi }}_x+{\overline{J}}_{32}{\mathrm{\Phi }}_y+{\overline{J}}_{33}W,\end{array}$

(37)

where

$\begin{array}{l}{\overline{J}}_{11}=\frac{{\overline{\mathrm{\beta }}}_1^2}{32{\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}},{\overline{J}}_{21}=\frac{{\overline{\mathrm{\alpha }}}_1^2}{32{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{11}^{*}},{\overline{\mathrm{\zeta }}}_m={\left(-1\right)}^m-1,{\overline{\mathrm{\zeta }}}_n={\left(-1\right)}^n-1,\\ {\overline{J}}_{12}=\frac{8\hspace{1em}{\overline{\mathrm{\zeta }}}_m^2{\overline{\mathrm{\zeta }}}_n^2\left[{\overline{R}}_{2\left(c,p,s\right)}-{\overline{R}}_{3\left(c,p,s\right)}\right]-12\hspace{1em}{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\hspace{1em}ab{\overline{\mathrm{\zeta }}}_m\hspace{1em}{\overline{\mathrm{\zeta }}}_n\hspace{1em}{\overline{R}}_{1\left(c,p,s\right)}-9\hspace{1em}{\overline{R}}_{2\left(c,p,s\right)}\hspace{1em}{a}^2{b}^2{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2}{72\hspace{1em}{a}^3{b}^3{\overline{\mathrm{\alpha }}}_1^6{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{22}^{*}-128\hspace{1em}ab{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}\hspace{1em}{\overline{\mathrm{\zeta }}}_m^2{\overline{\mathrm{\zeta }}}_n^2},\\ {\overline{J}}_{22}=\frac{-8\hspace{1em}{\overline{\mathrm{\zeta }}}_m^2{\overline{\mathrm{\zeta }}}_n^2\left[{\overline{R}}_{2\left(c,p,s\right)}-{\overline{R}}_{3\left(c,p,s\right)}\right]-12\hspace{1em}{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_{1}ab\hspace{1em}{\overline{\mathrm{\zeta }}}_m\hspace{1em}{\overline{\mathrm{\zeta }}}_n\hspace{1em}{\overline{R}}_{1\left(c,p,s\right)}-9\hspace{1em}{\overline{R}}_{3\left(c,p,s\right)}\hspace{1em}{a}^2{b}^2{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2}{72\hspace{1em}{a}^3{b}^3{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^6{\overline{L}}_{11}^{*}-128\hspace{1em}ab{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}\hspace{1em}{\overline{\mathrm{\zeta }}}_m^2{\overline{\mathrm{\zeta }}}_n^2},\\ {\overline{J}}_{31}=-\frac{{\overline{\mathrm{\alpha }}}_1\hspace{1em}\left\{\left[\left({\overline{G}}_{66}^{*}-{\overline{G}}_{11}^{*}\right)\mathrm{\lambda }+{\overline{H}}_{11}^{*}-{\overline{H}}_{66}^{*}\right]{\overline{\mathrm{\beta }}}_1^2+{\overline{\mathrm{\alpha }}}_1^2\left(-\mathrm{\lambda }\hspace{1em}{\overline{G}}_{21}^{*}+{\overline{H}}_{21}^{*}\right)\right\}}{{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}},\end{array}$

${\overline{J}}_{32}=-\frac{{\overline{\mathrm{\beta }}}_1\hspace{1em}\left\{\left[\left({\overline{G}}_{66}^{*}-{\overline{G}}_{22}^{*}\right)\mathrm{\lambda }+{\overline{H}}_{22}^{*}-{\overline{H}}_{66}^{*}\right]{\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^2\left(-\mathrm{\lambda }\hspace{1em}{\overline{G}}_{12}^{*}+{\overline{H}}_{12}^{*}\right)\right\}}{{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}},$

$\begin{array}{l}{\overline{J}}_{33}=\frac{\mathrm{\lambda }\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4{\overline{G}}_{21}^{*}+\mathrm{\lambda }{\overline{\mathrm{\beta }}}_1^2\left({\overline{G}}_{11}^{*}+{\overline{G}}_{22}^{*}-2\hspace{1em}{\overline{G}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2+\mathrm{\lambda }\hspace{1em}{\overline{\mathrm{\beta }}}_1^4{\overline{G}}_{12}^{*}}{{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}}\\ -\frac{{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\hspace{1em}\left\{36a\hspace{1em}b{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1{\overline{R}}_{1\left(c,p,s\right)}+24\hspace{1em}\left[{\overline{R}}_{3\left(c,p,s\right)}+{\overline{R}}_{2\left(c,p,s\right)}\right]{\overline{\mathrm{\zeta }}}_m\hspace{1em}{\overline{\mathrm{\zeta }}}_n\right\}}{\left[{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right)+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}\right]\left(9\hspace{1em}{a}^2{b}^2{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2-16\hspace{1em}{\overline{\mathrm{\zeta }}}_m^2{\overline{\mathrm{\zeta }}}_n^2\right)},\\ {\overline{R}}_{1c}=\frac{-{\overline{\mathrm{\alpha }}}_1^{2}ab{h}_z}{2\left[h_z^2+{\left(b/2\right)}^2\right]},{\overline{R}}_{2c}={\overline{R}}_{3c}=\frac{2h_z{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\zeta }}}_m\hspace{1em}{\overline{\mathrm{\zeta }}}_n}{3{\overline{\mathrm{\beta }}}_1\left[h_z^2+{\left(b/2\right)}^2\right]},\\ {\overline{R}}_{1\left(p,s\right)}=-\frac{1}{2}{\overline{\mathrm{\alpha }}}_1^{2}a{\int }_0^b\frac{{\sin }^2\hspace{0.5em}{\overline{\mathrm{\beta }}}_{1}y}{R_{\left(p,s\right)}}\hspace{0.17em}dy,{\overline{R}}_{2p}=-\frac{1}{3}{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\zeta }}}_m{\int }_0^b\frac{\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y}{R_{\left(p,s\right)}}\hspace{0.17em}dy,{\overline{R}}_{3p}={\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\zeta }}}_m{\int }_0^b\frac{\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y\hspace{0.17em}\cos \hspace{0.17em}2{\overline{\mathrm{\beta }}}_{1}y}{R_{\left(p,s\right)}}\hspace{0.17em}dy.\end{array}$

For the considered boundary conditions involving immovable edges, the corresponding constraints are imposed in an averaged sense and can be expressed as follows

$\begin{array}{l}{\int }_0^b{\int }_0^a{u}_{,x}dxdy={\overline{k}}_{41}W+{\overline{k}}_{42}{\mathrm{\Phi }}_x+{\overline{k}}_{43}{\mathrm{\Phi }}_y+{\overline{k}}_{44}W\left(2\mathrm{\eta }h+W\right)\\ +{\overline{k}}_{45}{N}_{0x}+{\overline{k}}_{46}{N}_{0y}+{\overline{k}}_{47}\mathrm{\Delta }T=0,\end{array}$

(38)

$\begin{array}{l}{\int }_0^b{\int }_0^a{v}_{,y}dxdy={\overline{k}}_{51}W+{\overline{k}}_{52}{\mathrm{\Phi }}_x+{\overline{k}}_{53}{\mathrm{\Phi }}_y+{\overline{k}}_{54}W\left(2\mathrm{\eta }h+W\right)\\ +{\overline{k}}_{55}{N}_{0x}+{\overline{k}}_{56}{N}_{0y}+{\overline{k}}_{57}\mathrm{\Delta }T=0,\end{array}$

(39)

where

$\begin{array}{l}{\overline{k}}_{41}=-\left[{\overline{\mathrm{\alpha }}}_1^2\left({\overline{J}}_{33}{\overline{L}}_{12}^{*}-\mathrm{\lambda }{\overline{G}}_{11}^{*}\right)-{\overline{\mathrm{\beta }}}_1^2\left(\mathrm{\lambda }{\overline{G}}_{12}^{*}-{\overline{J}}_{33}\hspace{1em}{\overline{L}}_{11}^{*}\right)\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{42}=\left[{\overline{J}}_{31}{\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{12}^{*}+\left({\overline{H}}_{11}^{*}-\mathrm{\lambda }{\overline{G}}_{11}^{*}\right){\overline{\mathrm{\alpha }}}_1+{\overline{\mathrm{\beta }}}_1^2{\overline{J}}_{31}{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{43}=\left[{\overline{J}}_{32}{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{11}^{*}+\left({\overline{H}}_{12}^{*}-\mathrm{\lambda }{\overline{G}}_{12}^{*}\right){\overline{\mathrm{\beta }}}_1+{\overline{J}}_{32}{\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{12}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{44}=-ab{\overline{\mathrm{\alpha }}}_1^2/8,{\overline{k}}_{45}={\overline{L}}_{11}^{*}ab,{\overline{k}}_{46}={\overline{L}}_{12}^{*}\hspace{1em}ab,{\overline{k}}_{47}=\left({\overline{L}}_{11}^{*}{\widehat{\mathrm{\Phi }}}_{1x}+{\overline{L}}_{12}^{*}{\widehat{\mathrm{\Phi }}}_{1y}\right)ab,\\ {\overline{k}}_{51}=-\left({\overline{J}}_{33}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}+{\overline{J}}_{33}{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}-{\overline{\mathrm{\alpha }}}_1^2\mathrm{\lambda }{\overline{G}}_{21}^{*}-{\overline{\mathrm{\beta }}}_1^2\mathrm{\lambda }{\overline{G}}_{22}^{*}\right){\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right)+{\overline{R}}_{4\left(c,p,s\right)},\\ {\overline{k}}_{52}=-\left[{\overline{\mathrm{\alpha }}}_1^2{\overline{J}}_{31}{\overline{L}}_{22}^{*}+\left(\mathrm{\lambda }{\overline{G}}_{21}^{*}-{\overline{H}}_{21}^{*}\right){\overline{\mathrm{\alpha }}}_1+{\overline{\mathrm{\beta }}}_1^2{\overline{J}}_{31}{\overline{L}}_{12}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{53}=-\left[{\overline{J}}_{32}{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}+\left(\mathrm{\lambda }{\overline{G}}_{22}^{*}-{\overline{H}}_{22}^{*}\right){\overline{\mathrm{\beta }}}_1+{\overline{J}}_{32}{\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{54}=-a{\overline{\mathrm{\beta }}}_1^{2}b/8,{\overline{k}}_{55}=ba{\overline{L}}_{12}^{*},{\overline{k}}_{56}=ba{\overline{L}}_{22}^{*},{\overline{k}}_{57}=ab\left({\overline{L}}_{12}^{*}{\widehat{\mathrm{\Phi }}}_{1x}+{\overline{L}}_{22}^{*}{\widehat{\mathrm{\Phi }}}_{1y}\right),\end{array}$

${\overline{R}}_{4c}=\frac{2h_z{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n}{\left[h_z^2+{\left(b/2\right)}^2\right]{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1},{\overline{R}}_{4\left(p,s\right)}=\frac{{\overline{\mathrm{\zeta }}}_m}{{\overline{\mathrm{\alpha }}}_1}{\int }_0^b\frac{\sin \hspace{0.17em}{\overline{\mathrm{\beta }}}_{1}y}{R_{\left(p,s\right)}}\hspace{0.17em}dy.$

The total energy of the panel system ${\widehat{\mathrm{\Xi }}}_{Total}$ is composed of the kinetic energy, the strain energy, and the work done from external loads and interactions with the elastic foundation, and is expressed as follows

$\begin{array}{l}{\widehat{\mathrm{\Xi }}}_{Total}={\widehat{\mathrm{\Xi }}}_t-{\widehat{\mathrm{\Xi }}}_{in}+{\widehat{\mathrm{\Xi }}}_{ext}=\frac{1}{2}{\int }_{-h/2}^{h/2}{\int }_0^b{\int }_0^a\mathrm{\rho }\left(z\right){\left(w_{,t}+{\overline{w}}_{,t}\right)}^{2}dxdydz\\ -\frac{1}{2}{\int }_{-h/2}^{h/2}{\int }_0^b{\int }_0^a\left[{\mathrm{\sigma }}_{xz}{\mathrm{\gamma }}_{xz}+{\mathrm{\sigma }}_{xy}{\mathrm{\gamma }}_{xy}+{\mathrm{\sigma }}_x\left({\mathrm{\epsilon }}_x-\mathrm{\alpha }\left(z\right)\mathrm{\Delta }T\right)+{\mathrm{\sigma }}_y\left({\mathrm{\epsilon }}_y-\mathrm{\alpha }\left(z\right)\mathrm{\Delta }T\right)+{\mathrm{\sigma }}_{yz}{\mathrm{\gamma }}_{yz}\right]dxdydz\\ +N_{0x}{\int }_0^b{\int }_0^a{u}_{,x}dxdy-{\int }_0^b{\int }_0^a\left\{\frac{1}{2}\left[K_{1}w-K_2\left(w_{,xx}+w_{,yy}\right)\right]w\right\}dxdy.\end{array}$

(40)

with $K_1$ (N/m³) and $K_2$ (N/m) are the two parameters of foundation.

The potential function associated with structural damping is formulated using the Rayleigh dissipation function. The dissipation function $d_v$ can be expressed as

$d_v=\frac{1}{2}{\int }_0^b{\int }_0^{a}c{w}_{,t}^{2}dxdy.$

(41)

By applying the Euler–Lagrange equations in conjunction with the Rayleigh dissipation function, the governing equations of motion for the system are derived and can be expressed as

$\begin{array}{l}\frac{d}{dt}\left(\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial \stackrel{.}{W}}\right)-\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial W}+\frac{\partial d_v}{\partial \stackrel{.}{W}}=0,\\ \frac{d}{dt}\left(\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial {\stackrel{.}{\mathrm{\Phi }}}_x}\right)-\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial {\mathrm{\Phi }}_x}=0,\\ \frac{d}{dt}\left(\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial {\stackrel{.}{\mathrm{\Phi }}}_y}\right)-\frac{\partial {\widehat{\mathrm{\Xi }}}_{Total}}{\partial {\mathrm{\Phi }}_y}=0,\end{array}$

(42)

leads to

$\begin{array}{l}{\overline{k}}_{11}\hspace{1em}W+{\overline{k}}_{12}\hspace{1em}{\mathrm{\Phi }}_x+{\overline{k}}_{13}\hspace{1em}{\mathrm{\Phi }}_y+{\overline{k}}_{14}\hspace{1em}{\mathrm{\Phi }}_x\hspace{1em}\left(W+\mathrm{\eta }h\right)+{\overline{k}}_{15}\hspace{1em}{\mathrm{\Phi }}_y\hspace{1em}\left(W+\mathrm{\eta }h\right)+{\overline{k}}_{16}\hspace{1em}W\left(W+4/3\mathrm{\eta }h\right)\\ +{\overline{k}}_{17}\hspace{1em}W\left(W+2\mathrm{\eta }h\right)\left(W+\mathrm{\eta }h\right)+{\overline{k}}_{18}\hspace{1em}{N}_{0x}+{\overline{k}}_{19}\hspace{1em}{N}_{0x}\hspace{1em}\left(W+\mathrm{\eta }h\right)+{\overline{k}}_{110}\hspace{1em}{N}_{0y}\\ -{\overline{k}}_{113}\mathrm{\Delta }T+{\overline{k}}_{115}\hspace{1em}\stackrel{..}{W}+{\overline{k}}_{116}\hspace{1em}c\stackrel{.}{W}=0,\end{array}$

(43)

${\overline{k}}_{12}W+{\overline{k}}_{22}{\mathrm{\Phi }}_x+{\overline{k}}_{23}{\mathrm{\Phi }}_y+{\overline{k}}_{24}W\left(W+2\hspace{1em}\mathrm{\eta }h\right)+{\overline{k}}_{25}{N}_{0x}+{\overline{k}}_{26}{N}_{0y}-{\overline{k}}_{27}\mathrm{\Delta }T=0,$

(44)

${\overline{k}}_{13}W+{\overline{k}}_{23}{\mathrm{\Phi }}_x+{\overline{k}}_{33}{\mathrm{\Phi }}_y+{\overline{k}}_{34}W\left(W+2\hspace{1em}\mathrm{\eta }h\right)+{\overline{k}}_{35}{N}_{0x}+{\overline{k}}_{36}{N}_{0y}-{\overline{k}}_{37}\mathrm{\Delta }T=0,$

(45)

where

${\overline{k}}_{11}=\frac{ab}{4}\left\{\begin{array}{l}\left(32{\overline{L}}_{22}^{*}{\overline{J}}_{12}^2+{\overline{L}}_{22}^{*}{\overline{J}}_{33}^2+{\mathrm{\lambda }}^2{\overline{D}}_{11}^{*}\right){\overline{\mathrm{\alpha }}}_1^4\\ +\left[\left({\overline{D}}_{12}^{*}+{\overline{D}}_{21}^{*}+4{\overline{D}}_{66}^{*}\right){\mathrm{\lambda }}^2+{\overline{J}}_{33}^2\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right)\right]{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\\ +\left(9{\overline{E}}_{44}{\mathrm{\lambda }}^2-6{\overline{B}}_{44}\mathrm{\lambda }+{\overline{C}}_{44}\right){\overline{\mathrm{\alpha }}}_1^2+\left(32{\overline{L}}_{11}^{*}{\overline{J}}_{22}^2+{\overline{L}}_{11}^{*}{\overline{J}}_{33}^2+{\mathrm{\lambda }}^2{\overline{D}}_{22}^{*}\right){\overline{\mathrm{\beta }}}_1^4\\ +\left(9{\overline{E}}_{55}{\mathrm{\lambda }}^2-6{\overline{B}}_{55}\mathrm{\lambda }+{\overline{C}}_{55}\right){\overline{\mathrm{\beta }}}_1^2+K_1+K_2\left({\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^2\right)\\ -32\left[{\overline{J}}_{12}{\overline{L}}_{22}^{*}{\overline{\mathrm{\alpha }}}_1^4+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}\left({\overline{J}}_{12}+{\overline{J}}_{22}\right)+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{22}{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n{\overline{J}}_{33}/\left(3ab{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right)\end{array}\right\},$

${\overline{k}}_{12}=\frac{ab}{8}\left\{\begin{array}{l}2{\overline{J}}_{31}\hspace{1em}{\overline{J}}_{33}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+\mathrm{\lambda }\hspace{1em}\left(2\mathrm{\lambda }\hspace{1em}{\overline{D}}_{11}^{*}-{\overline{A}}_{11}^{*}-A_{11}^{*}\right){\overline{\mathrm{\alpha }}}_1^3+2{\overline{J}}_{31}\hspace{1em}{\overline{J}}_{33}{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}\\ +\left[\left({\overline{D}}_{12}^{*}+{\overline{D}}_{21}^{*}+4\hspace{1em}{\overline{D}}_{66}^{*}\right){\overline{\mathrm{\beta }}}_1^2{\mathrm{\lambda }}^2+\left(-A_{21}^{*}-2A_{66}^{*}-{\overline{A}}_{12}^{*}-2{\overline{A}}_{66}^{*}\right){\overline{\mathrm{\beta }}}_1^2\mathrm{\lambda }\right]{\overline{\mathrm{\alpha }}}_1\\ +\left(18\hspace{1em}{\mathrm{\lambda }}^2{\overline{E}}_{44}-12{\overline{B}}_{44}\mathrm{\lambda }+2{\overline{C}}_{44}\right){\overline{\mathrm{\alpha }}}_1+{\overline{\mathrm{\beta }}}_1^2\left(4{\overline{L}}_{12}^{*}+2{\overline{L}}_{66}^{*}\right){\overline{J}}_{33}\hspace{1em}{\overline{J}}_{31}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^2\\ -32\left[{\overline{J}}_{12}{\overline{L}}_{22}^{*}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{12}+{\overline{J}}_{22}\right)\hspace{1em}{\overline{L}}_{12}^{*}+{\overline{J}}_{22}\hspace{1em}{\overline{\mathrm{\beta }}}_1^4\hspace{1em}{\overline{L}}_{12}^{*}\right]{\overline{J}}_{31}\hspace{1em}{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left(3ab{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right)\end{array}\right\},$

${\overline{k}}_{13}=\frac{ab}{8}\left\{\begin{array}{l}2\hspace{1em}{\overline{J}}_{33}\hspace{1em}{\overline{L}}_{11}^{*}\hspace{1em}{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{32}+\mathrm{\lambda }\hspace{1em}\left(2\mathrm{\lambda }\hspace{1em}{\overline{D}}_{22}^{*}-{\overline{A}}_{22}^{*}-A_{22}^{*}\right){\overline{\mathrm{\beta }}}_1^3+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2{\overline{J}}_{32}\\ \hspace{1em}\left(4{\overline{L}}_{12}^{*}+2{\overline{L}}_{66}^{*}\right){\overline{J}}_{33}\\ +2\hspace{1em}{\overline{L}}_{22}^{*}\hspace{1em}{\overline{J}}_{33}\hspace{1em}{\overline{J}}_{32}{\overline{\mathrm{\beta }}}_1^4+\left[\begin{array}{l}\left({\overline{D}}_{12}^{*}+{\overline{D}}_{21}^{*}+4\hspace{1em}{\overline{D}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2{\mathrm{\lambda }}^2\\ +\left(-{\overline{A}}_{21}^{*}-2\hspace{1em}{\overline{A}}_{66}^{*}-A_{12}^{*}-2\hspace{1em}{A}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2\mathrm{\lambda }\end{array}\right]{\overline{\mathrm{\beta }}}_1\\ +\left(18{\mathrm{\lambda }}^2\hspace{1em}{\overline{E}}_{55}-12\hspace{1em}\mathrm{\lambda }{\overline{B}}_{55}+2\hspace{1em}{\overline{C}}_{55}\right){\overline{\mathrm{\beta }}}_1\\ -32\left[{\overline{J}}_{12}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{L}}_{12}^{*}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{12}+{\overline{J}}_{22}\right)+{\overline{J}}_{22}\hspace{1em}{\mathrm{\beta }}_1^4{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_n{\overline{\mathrm{\zeta }}}_m\hspace{1em}{\overline{J}}_{32}/\left(3ab{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right)\end{array}\right\},$

${\overline{k}}_{14}=-8\left[{\overline{J}}_{11}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{11}+{\overline{J}}_{21}\right){\overline{L}}_{12}^{*}+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{21}{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n\hspace{0.17em}{\overline{J}}_{31}/\left(3{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),$

${\overline{k}}_{15}=-8\left[{\overline{J}}_{11}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{11}+{\overline{J}}_{21}\right){\overline{L}}_{12}^{*}+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{21}{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n{\overline{J}}_{32}/\left(3{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),$

$\begin{array}{l}{\overline{k}}_{16}=-4\left[{\overline{J}}_{11}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{11}+{\overline{J}}_{21}\right){\overline{L}}_{12}^{*}+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{21}{\overline{L}}_{11}^{*}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n{\overline{J}}_{33}/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right)\\ +24ab\left({\overline{J}}_{11}{\overline{J}}_{12}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{J}}_{21}{\overline{J}}_{22}{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}\right),\end{array}$

$\begin{array}{l}{\overline{k}}_{17}=16ba\left({\overline{J}}_{11}^2{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{J}}_{21}^2{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}\right),{\overline{k}}_{18}=\mathrm{\lambda }\left({\overline{\mathrm{\alpha }}}_1^2{\overline{G}}_{11}^{*}+{\overline{\mathrm{\beta }}}_1^2{\overline{G}}_{12}^{*}\right){\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{19}={\overline{\mathrm{\alpha }}}_1^{2}ab/4,{\overline{k}}_{110}=-{\overline{J}}_{33}\left({\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}\right){\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\end{array}$

${\overline{k}}_{113}=-\left[\left({\overline{G}}_{11}^{*}\hspace{1em}{\widehat{\mathrm{\Phi }}}_{1x}+{\overline{G}}_{21}^{*}{\widehat{\mathrm{\Phi }}}_{1y}-{\widehat{\mathrm{\Phi }}}_{4x}\right){\overline{\mathrm{\alpha }}}_1^2+{\overline{\mathrm{\beta }}}_1^2\left({\overline{G}}_{12}^{*}\hspace{1em}{\widehat{\mathrm{\Phi }}}_{1x}+{\overline{G}}_{22}^{*}\hspace{1em}{\widehat{\mathrm{\Phi }}}_{1y}-{\widehat{\mathrm{\Phi }}}_{4y}\right)\right]\mathrm{\lambda }{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),$

${\overline{k}}_{115}=\frac{ab}{4}\left[{\int }_{-h/2}^{h/2}\mathrm{\rho }\left(z\right)dz+2\frac{b_{st}}{d_{st}}{\int }_{h/2}^{h/2+h_{st}}\mathrm{\rho }\left(z\right)dz\right],{\overline{k}}_{116}=ab/4,$

${\overline{k}}_{22}=\frac{ab}{4}\left\{\begin{array}{l}\left[{\overline{L}}_{22}^{*}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left(2\hspace{1em}{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right)\right]{\overline{J}}_{31}^2+9\hspace{1em}{\overline{E}}_{44}\hspace{1em}{\mathrm{\lambda }}^2-6\hspace{1em}{\overline{B}}_{44}\hspace{1em}\mathrm{\lambda }+{\overline{C}}_{44}\\ +\left[{\mathrm{\lambda }}^2{\overline{D}}_{11}^{*}+\left(-{\overline{A}}_{11}^{*}-A_{11}^{*}\right)\mathrm{\lambda }+{\overline{F}}_{11}^{*}\right]{\overline{\mathrm{\alpha }}}_1^2+\left[{\mathrm{\lambda }}^2{\overline{D}}_{66}^{*}+\left(-{\overline{A}}_{66}^{*}-A_{66}^{*}\right)\mathrm{\lambda }+{\overline{F}}_{66}^{*}\right]{\overline{\mathrm{\beta }}}_1^2\end{array}\right\},$

${\overline{k}}_{23}=\frac{ab}{4}\left\{\begin{array}{l}\left[{\overline{L}}_{22}^{*}{\overline{\mathrm{\alpha }}}_1^4+\left(2{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right){\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2+{\overline{L}}_{11}^{*}{\overline{\mathrm{\beta }}}_1^4\right]{\overline{J}}_{32}{\overline{J}}_{31}\\ +\frac{1}{2}{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\left[\begin{array}{l}\left({\overline{D}}_{12}^{*}+{\overline{D}}_{21}^{*}+2{\overline{D}}_{66}^{*}\right){\mathrm{\lambda }}^2+{\overline{F}}_{21}^{*}+{\overline{F}}_{12}^{*}+2{\overline{F}}_{66}^{*}\\ +\left(-{\overline{A}}_{12}^{*}-{\overline{A}}_{21}^{*}-2{\overline{A}}_{66}^{*}-A_{12}^{*}-A_{21}^{*}-2A_{66}^{*}\right)\mathrm{\lambda }\end{array}\right]\end{array}\right\},$

$\begin{array}{l}{\overline{k}}_{24}=-4\left({\overline{J}}_{11}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{11}+{\overline{J}}_{21}\right){\overline{L}}_{12}^{*}+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{21}{\overline{L}}_{11}^{*}\right){\overline{J}}_{31}{\overline{\mathrm{\zeta }}}_n{\overline{\mathrm{\zeta }}}_m/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{25}=-\left({\overline{H}}_{11}^{*}-{\overline{G}}_{11}^{*}\mathrm{\lambda }\right){\overline{\mathrm{\zeta }}}_n{\overline{\mathrm{\zeta }}}_m/{\overline{\mathrm{\beta }}}_1,{\overline{k}}_{26}=-\left({\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}\right){\overline{J}}_{31}{\overline{\mathrm{\zeta }}}_n{\overline{\mathrm{\zeta }}}_m/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\end{array}$

${\overline{k}}_{27}=\left[\left(-{\overline{G}}_{11}^{*}{\widehat{\mathrm{\Phi }}}_{1x}-{\overline{G}}_{21}^{*}{\widehat{\mathrm{\Phi }}}_{1y}+{\widehat{\mathrm{\Phi }}}_{4x}\right)\mathrm{\lambda }+{\overline{H}}_{21}^{*}{\widehat{\mathrm{\Phi }}}_{1y}+{\overline{H}}_{11}^{*}{\widehat{\mathrm{\Phi }}}_{1x}-{\widehat{\mathrm{\Phi }}}_{2y}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/{\overline{\mathrm{\beta }}}_1,$

${\overline{k}}_{33}=\frac{ab}{4}\left\{\begin{array}{l}\left[{\overline{L}}_{22}^{*}\hspace{1em}{\overline{\mathrm{\alpha }}}_1^4+{\overline{\mathrm{\beta }}}_1^4{\overline{L}}_{11}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left(2\hspace{1em}{\overline{L}}_{12}^{*}+{\overline{L}}_{66}^{*}\right)\right]{\overline{J}}_{32}^2+9\hspace{1em}{\overline{E}}_{55}\hspace{1em}{\mathrm{\lambda }}^2-6\hspace{1em}{\overline{B}}_{55}\hspace{1em}\mathrm{\lambda }+{\overline{C}}_{55}\\ +\left[{\mathrm{\lambda }}^2{\overline{D}}_{22}^{*}+\left(-{\overline{A}}_{22}^{*}-A_{22}^{*}\right)\mathrm{\lambda }+{\overline{F}}_{22}^{*}\right]{\overline{\mathrm{\beta }}}_1^2+\left[{\mathrm{\lambda }}^2{\overline{D}}_{66}^{*}+\left(-{\overline{A}}_{66}^{*}-A_{66}^{*}\right)\mathrm{\lambda }+{\overline{F}}_{66}^{*}\right]{\overline{\mathrm{\alpha }}}_1^2\end{array}\right\},$

$\begin{array}{l}{\overline{k}}_{34}=-4\left[{\overline{J}}_{11}{\overline{\mathrm{\alpha }}}_1^4{\overline{L}}_{22}^{*}+{\overline{\mathrm{\alpha }}}_1^2{\overline{\mathrm{\beta }}}_1^2\left({\overline{J}}_{11}+{\overline{J}}_{21}\right){\overline{L}}_{12}^{*}+{\overline{\mathrm{\beta }}}_1^4{\overline{J}}_{21}{\overline{L}}_{11}^{*}\right]{\overline{J}}_{32}{\overline{\mathrm{\zeta }}}_n{\overline{\mathrm{\zeta }}}_m/\left(3{\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{35}=\left({\overline{G}}_{12}^{*}\mathrm{\lambda }-{\overline{H}}_{12}^{*}\right){\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/{\overline{\mathrm{\alpha }}}_1,{\overline{k}}_{36}=-{\overline{J}}_{32}\left({\overline{\mathrm{\alpha }}}_1^2{\overline{L}}_{22}^{*}+{\overline{\mathrm{\beta }}}_1^2{\overline{L}}_{12}^{*}\right){\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/\left({\overline{\mathrm{\alpha }}}_1{\overline{\mathrm{\beta }}}_1\right),\\ {\overline{k}}_{37}=\left[\left(-{\overline{G}}_{12}^{*}{\widehat{\mathrm{\Phi }}}_{1x}-{\overline{G}}_{22}^{*}{\widehat{\mathrm{\Phi }}}_{1y}+{\widehat{\mathrm{\Phi }}}_{4y}\right)\mathrm{\lambda }+{\overline{H}}_{22}^{*}{\widehat{\mathrm{\Phi }}}_{1y}+{\overline{H}}_{12}^{*}{\widehat{\mathrm{\Phi }}}_{1x}-{\widehat{\mathrm{\Phi }}}_{2y}\right]{\overline{\mathrm{\zeta }}}_m{\overline{\mathrm{\zeta }}}_n/{\overline{\mathrm{\alpha }}}_1,\end{array}$

$\begin{array}{l}{\overline{F}}_{11}^{*}=-{\overline{H}}_{11}{\overline{H}}_{11}^{*}-{\overline{H}}_{12}{\overline{H}}_{21}^{*}+{\overline{F}}_{11},{\overline{F}}_{12}^{*}=-{\overline{H}}_{11}{\overline{H}}_{12}^{*}-{\overline{H}}_{12}{\overline{H}}_{22}^{*}+{\overline{F}}_{12},\\ {\overline{F}}_{66}^{*}=-{\overline{H}}_{66}{\overline{H}}_{66}^{*}+{\overline{F}}_{66},{\overline{F}}_{21}^{*}=-{\overline{H}}_{12}{\overline{H}}_{11}^{*}-{\overline{H}}_{22}{\overline{H}}_{21}^{*}+{\overline{F}}_{12},{\overline{F}}_{22}^{*}=-{\overline{H}}_{12}{\overline{H}}_{12}^{*}-{\overline{H}}_{22}{\overline{H}}_{22}^{*}+{\overline{F}}_{22},\end{array}$

$\begin{array}{l}{\overline{A}}_{11}^{*}=-{\overline{H}}_{11}{\overline{G}}_{11}^{*}-{\overline{H}}_{12}{\overline{G}}_{21}^{*}+{\overline{A}}_{11},{\overline{A}}_{12}^{*}=-{\overline{H}}_{11}{\overline{G}}_{12}^{*}-{\overline{H}}_{12}{\overline{G}}_{22}^{*}+{\overline{A}}_{12},\\ {\overline{A}}_{66}^{*}=-{\overline{H}}_{66}{\overline{G}}_{66}^{*}+{\overline{A}}_{66},{\overline{A}}_{21}^{*}=-{\overline{H}}_{12}{\overline{G}}_{11}^{*}-{\overline{H}}_{22}{\overline{G}}_{21}^{*}+{\overline{A}}_{12},{\overline{A}}_{22}^{*}=-{\overline{H}}_{12}{\overline{G}}_{12}^{*}-{\overline{H}}_{22}{\overline{G}}_{22}^{*}+{\overline{A}}_{22},\end{array}$

$\begin{array}{l}{A}_{11}^{*}=-{\overline{G}}_{11}{\overline{H}}_{11}^{*}-{\overline{G}}_{12}{\overline{H}}_{21}^{*}+{\overline{A}}_{11},A_{12}^{*}=-{\overline{G}}_{11}{\overline{H}}_{12}^{*}-{\overline{G}}_{12}{\overline{H}}_{22}^{*}+{\overline{A}}_{12},\\ A_{66}^{*}=-{\overline{G}}_{66}{\overline{H}}_{66}^{*}+{\overline{A}}_{66},A_{21}^{*}=-{\overline{G}}_{12}{\overline{H}}_{11}^{*}-{\overline{G}}_{22}{\overline{H}}_{21}^{*}+{\overline{A}}_{12},A_{22}^{*}=-{\overline{G}}_{12}{\overline{H}}_{12}^{*}-{\overline{G}}_{22}{\overline{H}}_{22}^{*}+{\overline{A}}_{22},\end{array}$

$\begin{array}{l}{\overline{D}}_{11}^{*}=-{\overline{G}}_{11}{\overline{G}}_{11}^{*}-{\overline{G}}_{12}{\overline{G}}_{21}^{*}+{\overline{D}}_{11},{\overline{D}}_{12}^{*}=-{\overline{G}}_{11}{\overline{G}}_{12}^{*}-{\overline{G}}_{12}{\overline{G}}_{22}^{*}+{\overline{D}}_{12},\\ {\overline{D}}_{66}^{*}=-{\overline{G}}_{66}{\overline{G}}_{66}^{*}+{\overline{D}}_{66},{\overline{D}}_{21}^{*}=-{\overline{G}}_{12}{\overline{G}}_{11}^{*}-{\overline{G}}_{22}{\overline{G}}_{21}^{*}+{\overline{D}}_{12},{\overline{D}}_{22}^{*}=-{\overline{G}}_{12}{\overline{G}}_{12}^{*}-{\overline{G}}_{22}{\overline{G}}_{22}^{*}+{\overline{D}}_{22}.\end{array}$

In the case of panels subjected to axial compression, only two boundary conditions: 4F and 2F2I, are considered. For the 4F condition, the governing equations of motion are formulated by combining equations (43)–(45) with $N_{0y}=0$ and $N_{0x}=-P_{x}h$ . In the 2F2I case, the governing equations are obtained by incorporating equations (43)–(45) along with equation (39) and the axial load condition $N_{0x}=-P_{x}h$ , obtained by

$\begin{array}{l}\left({\mathrm{\vartheta }}_1{\overline{o}}_{11}+{\mathrm{\vartheta }}_2{\overline{o}}_{21}\right)W+\left({\mathrm{\vartheta }}_1{\overline{o}}_{12}+{\mathrm{\vartheta }}_2{\overline{o}}_{22}\right)\mathrm{\Delta }T\hspace{1em}\left(\mathrm{\eta }h+W\right)+\left({\mathrm{\vartheta }}_1{\overline{o}}_{13}+{\mathrm{\vartheta }}_2{\overline{o}}_{23}\right)\hspace{1em}\left(\hspace{1em}W+\mathrm{\eta }h\right)W\\ +\left({\mathrm{\vartheta }}_1{\overline{o}}_{14}+{\mathrm{\vartheta }}_2{\overline{o}}_{24}\right)\hspace{1em}\left(W+2\hspace{1em}\mathrm{\eta }h\right)W+\left({\mathrm{\vartheta }}_1{\overline{o}}_{15}+{\mathrm{\vartheta }}_2{\overline{o}}_{25}\right)\hspace{1em}\left(2\hspace{1em}\mathrm{\eta }h+W\right)\left(W+\mathrm{\eta }h\right)W\\ -\left({\mathrm{\vartheta }}_1{\overline{o}}_{16}+{\mathrm{\vartheta }}_2{\overline{o}}_{26}\right)P_{x}h-\left({\mathrm{\vartheta }}_1{\overline{o}}_{17}+{\mathrm{\vartheta }}_2{\overline{o}}_{27}\right)P_{x}h\left(W+\mathrm{\eta }h\right)+\left({\mathrm{\vartheta }}_1{\overline{o}}_{110}+{\mathrm{\vartheta }}_2{\overline{o}}_{28}\right)\mathrm{\Delta }T\\ +{\overline{k}}_{16}\hspace{1em}W\hspace{1em}\left(W+4\mathrm{\eta }h/3\hspace{1em}\right)+{\overline{k}}_{115}\hspace{1em}\stackrel{..}{W}+{\overline{k}}_{116}\hspace{1em}c\stackrel{.}{W}=0,\end{array}$

(46)

where, ${\mathrm{\vartheta }}_1=1,{\mathrm{\vartheta }}_2=0$ are applied for the 4F case, and ${\mathrm{\vartheta }}_1=0,{\mathrm{\vartheta }}_2=1$ are applied for the case of 2F2I, and

$\begin{array}{l}{\overline{o}}_{11}={\overline{k}}_{12}{\overline{T}}_{14}+{\overline{k}}_{13}{\overline{T}}_{24}+{\overline{k}}_{11},{\overline{o}}_{12}={\overline{k}}_{14}{\overline{T}}_{15}+{\overline{k}}_{15}{\overline{T}}_{25},{\overline{o}}_{13}={\overline{k}}_{14}{\overline{T}}_{14}+{\overline{k}}_{15}{\overline{T}}_{24},\\ {\overline{o}}_{14}={\overline{k}}_{12}{\overline{T}}_{11}+{\overline{k}}_{15}{\overline{T}}_{24},{\overline{o}}_{15}={\overline{k}}_{14}{\overline{T}}_{11}+{\overline{k}}_{15}{\overline{T}}_{21}+{\overline{k}}_{17},{\overline{o}}_{16}={\overline{k}}_{12}{\overline{T}}_{13}+{\overline{k}}_{13}{\overline{T}}_{23}+{\overline{k}}_{18},\\ {\overline{o}}_{17}={\overline{k}}_{14}{\overline{T}}_{13}+{\overline{k}}_{15}{\overline{T}}_{23}+{\overline{k}}_{19},{\overline{o}}_{18}={\overline{k}}_{12}{\overline{T}}_{12}+{\overline{k}}_{13}{\overline{T}}_{22}+{\overline{k}}_{110},{\overline{o}}_{19}={\overline{k}}_{14}{\overline{T}}_{12}+{\overline{k}}_{15}{\overline{T}}_{22},\\ {\overline{o}}_{110}={\overline{k}}_{12}{\overline{T}}_{15}+{\overline{k}}_{13}{\overline{T}}_{25}-{\overline{k}}_{113},\\ {\overline{o}}_{21}={\overline{T}}_{32}{\overline{o}}_{18}+{\overline{o}}_{11},{\overline{o}}_{22}={\overline{T}}_{34}{\overline{o}}_{19}+{\overline{o}}_{12},{\overline{o}}_{23}={\overline{T}}_{32}{\overline{o}}_{19}+{\overline{o}}_{13},{\overline{o}}_{24}={\overline{T}}_{31}{\overline{o}}_{18}+{\overline{o}}_{14},\\ {\overline{o}}_{25}={\overline{T}}_{31}{\overline{o}}_{19}+{\overline{o}}_{15},{\overline{o}}_{26}={\overline{T}}_{33}{\overline{o}}_{18}+{\overline{o}}_{16},{\overline{o}}_{27}={\overline{T}}_{33}{\overline{o}}_{19}+{\overline{o}}_{17},{\overline{o}}_{28}={\overline{T}}_{34}{\overline{o}}_{18}+{\overline{o}}_{110},\end{array}$

${\overline{T}}_{11}=\frac{{\overline{k}}_{33}\hspace{1em}{\overline{k}}_{24}-{\overline{k}}_{34}\hspace{1em}{\overline{k}}_{23}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{12}=-\frac{{\overline{k}}_{23}\hspace{1em}{\overline{k}}_{36}-{\overline{k}}_{33}\hspace{1em}{\overline{k}}_{26}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{13}=-\frac{{\overline{k}}_{35}\hspace{1em}{\overline{k}}_{23}-{\overline{k}}_{33}\hspace{1em}{\overline{k}}_{25}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},$

${\overline{T}}_{14}=-\frac{-{\overline{k}}_{33}{\overline{k}}_{12}+{\overline{k}}_{13}\hspace{1em}{\overline{k}}_{23}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{15}=\frac{{\overline{k}}_{37}\hspace{1em}{\overline{k}}_{23}-{\overline{k}}_{33}\hspace{1em}{\overline{k}}_{27}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},$

${\overline{T}}_{21}=-\frac{-{\overline{k}}_{34}\hspace{1em}{\overline{k}}_{22}+{\overline{k}}_{23}\hspace{1em}{\overline{k}}_{24}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{22}=\frac{{\overline{k}}_{36}\hspace{1em}{\overline{k}}_{22}-{\overline{k}}_{23}\hspace{1em}{\overline{k}}_{26}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{23}=\frac{-{\overline{k}}_{23}\hspace{1em}{\overline{k}}_{25}+{\overline{k}}_{35}\hspace{1em}{\overline{k}}_{23}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},$

${\overline{T}}_{24}=\frac{-{\overline{k}}_{12}\hspace{1em}{\overline{k}}_{23}+{\overline{k}}_{13}{\overline{k}}_{22}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},{\overline{T}}_{25}=\frac{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{37}+{\overline{k}}_{23}\hspace{1em}{\overline{k}}_{27}}{-{\overline{k}}_{22}\hspace{1em}{\overline{k}}_{33}+{\overline{k}}_{23}^2},$

$\begin{array}{l}{\overline{T}}_{31}=-\frac{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{11}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{21}+{\overline{k}}_{54}}{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{12}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{22}+{\overline{k}}_{56}},{\overline{T}}_{32}=-\frac{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{14}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{24}+{\overline{k}}_{51}}{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{12}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{22}+{\overline{k}}_{56}},\\ {\overline{T}}_{33}=-\frac{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{13}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{23}+{\overline{k}}_{55}}{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{12}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{22}+{\overline{k}}_{56}},{\overline{T}}_{34}=-\frac{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{15}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{25}+{\overline{k}}_{57}}{{\overline{k}}_{52}\hspace{1em}{\overline{T}}_{12}+{\overline{k}}_{53}\hspace{1em}{\overline{T}}_{22}+{\overline{k}}_{56}}.\end{array}$

In the case of panels subjected to uniform thermal loading, two boundary conditions, 2I2F and 4I, are considered. For the 2I2F condition, the governing equations of motion are constructed by combining equations (38), and (43)–(45), together with the condition $N_{0y}=0$ . In the case of the 4I case, the governing equations are formulated by incorporating equations (38), (39), and (43)–(45), presented as

$\begin{array}{l}\left({\mathrm{\vartheta }}_3{\overline{o}}_{31}+{\mathrm{\vartheta }}_4{\overline{o}}_{41}\right)W+\left({\mathrm{\vartheta }}_3{\overline{o}}_{32}+{\mathrm{\vartheta }}_4{\overline{o}}_{42}\right)\mathrm{\Delta }T\hspace{1em}\left(\hspace{1em}\mathrm{\eta }h+W\right)+\left({\mathrm{\vartheta }}_3{\overline{o}}_{33}+{\mathrm{\vartheta }}_4{\overline{o}}_{43}\right)\hspace{1em}\left(\hspace{1em}W+\mathrm{\eta }h\right)W\\ +\left({\mathrm{\vartheta }}_3{\overline{o}}_{34}+{\mathrm{\vartheta }}_4{\overline{o}}_{44}\right)W\hspace{1em}\left(2\hspace{1em}\mathrm{\eta }h+W\right)+\left({\mathrm{\vartheta }}_3{\overline{o}}_{35}+{\mathrm{\vartheta }}_4{\overline{o}}_{45}\right)\hspace{1em}W\hspace{1em}\left(2\hspace{1em}\mathrm{\eta }h+W\right)\left(W+\mathrm{\eta }h\right)\\ +\left({\mathrm{\vartheta }}_3{\overline{o}}_{36}+{\mathrm{\vartheta }}_4{\overline{o}}_{46}\right)\mathrm{\Delta }T+{\overline{k}}_{16}\hspace{1em}W\hspace{1em}\left(W+4\mathrm{\eta }h/3\hspace{1em}\right)+{\overline{k}}_{115}\hspace{1em}\ddot{W}+{\overline{k}}_{116}\hspace{1em}c\stackrel{.}{W}=0,\end{array}$

(47)

where, ${\mathrm{\vartheta }}_3=1,{\mathrm{\vartheta }}_4=0$ for the case of the 2I2F condition, ${\mathrm{\vartheta }}_3=0,{\mathrm{\vartheta }}_4=1$ for the case of 4I, and

$\begin{array}{l}{\overline{o}}_{31}={\overline{o}}_{11}+{\overline{T}}_{42}{\overline{o}}_{16},{\overline{o}}_{32}={\overline{o}}_{12}+{\overline{T}}_{44}{\overline{o}}_{17},{\overline{o}}_{33}={\overline{o}}_{13}+{\overline{T}}_{42}{\overline{o}}_{17},{\overline{o}}_{34}={\overline{o}}_{14}+{\overline{T}}_{41}{\overline{o}}_{16},\\ {\overline{o}}_{35}={\overline{o}}_{15}+{\overline{T}}_{41}{\overline{o}}_{17},{\overline{o}}_{36}={\overline{o}}_{110}+{\overline{T}}_{44}{\overline{o}}_{16},\\ {\overline{o}}_{41}={\overline{o}}_{11}-\frac{{\overline{o}}_{16}\left({\overline{T}}_{32}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{42}\right)+{\overline{o}}_{18}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{42}+{\overline{T}}_{32}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\\ {\overline{o}}_{42}={\overline{o}}_{12}-\frac{{\overline{o}}_{17}\hspace{1em}\left({\overline{T}}_{34}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{44}\right)+{\overline{o}}_{19}\hspace{1em}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{44}+{\overline{T}}_{34}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\\ {\overline{o}}_{43}={\overline{o}}_{13}-\frac{{\overline{o}}_{17}\hspace{1em}\left({\overline{T}}_{32}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{42}\right)+{\overline{o}}_{19}\hspace{1em}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{42}+{\overline{T}}_{32}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\end{array}$

$\begin{array}{l}{\overline{o}}_{44}={\overline{o}}_{14}-\frac{{\overline{o}}_{16}\left({\overline{T}}_{31}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{41}\right)+{\overline{o}}_{18}\hspace{1em}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{41}+{\overline{T}}_{31}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\\ {\overline{o}}_{45}={\overline{o}}_{15}-\frac{{\overline{o}}_{17}\hspace{1em}\left({\overline{T}}_{31}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{41}\right)+{\overline{o}}_{19}\hspace{1em}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{41}+{\overline{T}}_{31}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\end{array}$

$\begin{array}{l}{\overline{o}}_{46}={\overline{o}}_{110}-\frac{{\overline{o}}_{16}\left({\overline{T}}_{34}\hspace{1em}{\overline{T}}_{43}+{\overline{T}}_{44}\right)+{\overline{o}}_{18}\hspace{1em}\left({\overline{T}}_{33}\hspace{1em}{\overline{T}}_{44}+{\overline{T}}_{34}\right)}{{\overline{T}}_{33}\hspace{1em}{\overline{T}}_{43}-1},\\ {\overline{T}}_{41}=-\frac{{\overline{T}}_{11}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{21}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{44}}{{\overline{T}}_{13}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{23}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{45}},{\overline{T}}_{42}=-\frac{{\overline{T}}_{14}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{24}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{41}}{{\overline{T}}_{13}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{23}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{45}},\\ {\overline{T}}_{43}=-\frac{{\overline{T}}_{12}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{22}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{46}}{{\overline{T}}_{13}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{23}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{45}},{\overline{T}}_{44}=-\frac{{\overline{T}}_{15}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{25}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{47}}{{\overline{T}}_{13}\hspace{1em}{\overline{k}}_{42}+{\overline{T}}_{23}\hspace{1em}{\overline{k}}_{43}+{\overline{k}}_{45}}.\end{array}$

Equations (46) and (47) serve as the basis for evaluating the dynamic buckling response of the panels. As can be seen that the coefficients of equations (46) and (47) are explicitly expressed in the above expressions, and they can be determined using the simple calculation tools. In this analysis, time-dependent linear loading profiles are assumed for both axial compression and thermal loading, expressed as $P_x=P_{0}t$ and $\mathrm{\Delta }T={\mathrm{\Delta }}_{0}t$ , where $P_0$ (Pa/s) and ${\mathrm{\Delta }}_0$ (K/s) represent the respective loading speeds. The resulting time-dependent responses are computed using the Runge–Kutta numerical integration method.

For the static analysis, the inertial and damping components in equations (46) and (47) are neglected. Consequently, the postbuckling solutions corresponding to the axial compression and thermal loading are derived as follows

$\begin{array}{l}{P}_x=[\left({\mathrm{\vartheta }}_1{\overline{o}}_{11}+{\mathrm{\vartheta }}_2{\overline{o}}_{21}\right)\hspace{1em}{W}^{*}+\left({\mathrm{\vartheta }}_1{\overline{o}}_{12}+{\mathrm{\vartheta }}_2{\overline{o}}_{22}\right)\hspace{1em}\mathrm{\Delta }T\left(W^{*}+\mathrm{\eta }\right)+\left({\mathrm{\vartheta }}_1{\overline{o}}_{13}+{\mathrm{\vartheta }}_2{\overline{o}}_{23}\right)\hspace{1em}{W}^{*}h\left(W^{*}+\mathrm{\eta }\right)\\ +{\overline{k}}_{16}\hspace{1em}{W}^{*}h\left(W^{*}+\frac{4}{3}\mathrm{\eta }\right)+\left({\mathrm{\vartheta }}_1{\overline{o}}_{15}+{\mathrm{\vartheta }}_2{\overline{o}}_{25}\right)\hspace{1em}{W}^{*}{h}^2\left(W^{*}+2\mathrm{\eta }\right)\left(W^{*}+\mathrm{\eta }\right)+\frac{\mathrm{\Delta }T}{h}\left({\mathrm{\vartheta }}_1{\overline{o}}_{110}+{\mathrm{\vartheta }}_2{\overline{o}}_{28}\right)\\ +\left({\mathrm{\vartheta }}_1{\overline{o}}_{14}+{\mathrm{\vartheta }}_2{\overline{o}}_{24}\right)\hspace{1em}{W}^{*}h\left(W^{*}+2\mathrm{\eta }\right)]/\left[\left({\mathrm{\vartheta }}_1{\overline{o}}_{16}+{\mathrm{\vartheta }}_2{\overline{o}}_{26}\right)+\left({\mathrm{\vartheta }}_1{\overline{o}}_{17}+{\mathrm{\vartheta }}_2{\overline{o}}_{27}\right)\hspace{1em}h\hspace{1em}\left(W^{*}+\mathrm{\eta }\right)\right],\end{array}$

(48)

$\begin{array}{l}\mathrm{\Delta }T=\frac{-1}{\frac{{\mathrm{\vartheta }}_3{\overline{o}}_{36}}{h}+\frac{{\mathrm{\vartheta }}_4{\overline{o}}_{46}}{h}+\left({\mathrm{\vartheta }}_3{\overline{o}}_{32}+{\mathrm{\vartheta }}_4{\overline{o}}_{42}\right)\hspace{1em}\left(\hspace{1em}\mathrm{\eta }+W^{*}\right)}[\left({\mathrm{\vartheta }}_3{\overline{o}}_{33}+{\mathrm{\vartheta }}_4{\overline{o}}_{43}\right)W^{*}h\hspace{1em}\left(\hspace{1em}\mathrm{\eta }+W^{*}\right)\\ +\left({\mathrm{\vartheta }}_3{\overline{o}}_{31}+{\mathrm{\vartheta }}_4{\overline{o}}_{41}\right)W^{*}+\left({\mathrm{\vartheta }}_3{\overline{o}}_{35}+{\mathrm{\vartheta }}_4{\overline{o}}_{45}\right)\hspace{1em}{W}^{*}{h}^2\hspace{1em}\left(2\mathrm{\eta }+W^{*}\right)\left(W^{*}+\mathrm{\eta }\right)\\ +\left({\mathrm{\vartheta }}_3{\overline{o}}_{34}+{\mathrm{\vartheta }}_4{\overline{o}}_{44}\right)W^{*}h\hspace{1em}\left(2\hspace{1em}\mathrm{\eta }+W^{*}\right)+{\overline{k}}_{16}\hspace{1em}{W}^{*}h\hspace{1em}\left(W^{*}+4\mathrm{\eta }/3\hspace{1em}\right)],\end{array}$

(49)

where $W^{*}=W/h.$

By omitting the inertial, damping, imperfection, and nonlinear terms, and considering the limiting case $W^{*}\to 0$ in equations (48) and (49), the critical buckling load under axial compression for 4F panels is derived from equation (48) as follows

$P_x^{cr}=\frac{{\overline{o}}_{11}+{\overline{o}}_{12}\mathrm{\Delta }T}{{\overline{o}}_{17}h}.$

(50)

The expression for the critical buckling temperature corresponding to the 2I2F panel configuration is derived from equation (49) as follows

$\mathrm{\Delta }{T}_{cr}=-\frac{{\overline{o}}_{31}}{{\overline{o}}_{32}}.$

(51)

Numerical examples and discussions

Table 1 presents a comparison between the present results and those reported in the literature^47,48 for the critical buckling load of unstiffened SUS304 cylindrical panels. The comparison is conducted under identical material properties, boundary conditions, and geometrical parameters. As shown, the critical load values obtained by the proposed semi-analytical model are in excellent agreement with the reference results, with relative errors remaining within a very small range.

Table 1.

Comparisons of critical buckling loads ${\overline{P}}_x^{cr}=P_x^{cr}bh$ (MN) of unstiffened SUS304 cylindrical panels between the present results and previous results ( $b$ = 0.3 m, $\mathrm{\Delta }T$ = 0 K).

$a/R_c$	$b/h$	References
0.5	30	Shen et al. 47	4.9565 (1,1) a
		Thang et al. 48	5.207
		Present	5.232 (1,1)
1.0	30	Shen et al. 47	10.2899 (3,1)
		Thang et al. 48	10.414
		Present	10.374 (3,1)
0.5	0.6	Shen et al. 47	1.2968 (1,3)
		Thang et al. 48	1.301
		Present	1.308 (3,1)

^aBuckling modes $\left(m,n\right)$

In this section, a series of numerical simulations is carried out to investigate the nonlinear static and dynamic buckling behavior of FG-GPLRC panels with porous core and inclined stiffeners under axial compression and thermal loading. The material properties employed in the analysis are adopted from Wang et al., ³³ where the GPLs are dispersed in a copper matrix. The properties of GPL and copper are consistently applied for both the coating layers and the stiffeners.

Table 2 summarizes the influence of panel geometry, porosity coefficient $e$ , and axial loading speed $P_0$ on both static and dynamic critical buckling loads. It is evident that sinusoid panels exhibit the highest critical buckling load, followed by parabolic and cylindrical panels for both static and dynamic cases. As the porosity coefficient $e$ increases, a clear reduction in both static and dynamic critical buckling loads is observed across all panel types. This degradation in buckling performance is attributed to the reduction in effective stiffness caused by the presence of pores within the core layer. Moreover, dynamic buckling loads are consistently larger than their static counterparts, and the difference becomes more pronounced as the axial loading speed $P_0$ increases. This result highlights the sensitivity of the loading speed of the dynamic stability behavior and confirms the necessity of considering dynamic effects in design under rapidly applied compressive loads.

Table 2.

Static and dynamic critical buckling load $P_x^{cr}$ (MPa) of unstiffened FG-GPLRC panel with porous core, (U/PC/U, 4F, $h$ = 0.02 m, $h_{co}=0.7h$ , $a=b=20h$ , $h_z$ = 0.042 m, $\mathrm{\eta }$ = 0, c = 0 N.s/m³, $\mathrm{\Delta }T$ = 0 K, $W_{GP}$ = 0.5%, $m$ = 1, $n$ = 1, $K_1$ = 10 MN/m³, $K_2$ = 0.1 MN/m).

$e$	Type	Static	Dynamic
			$P_0$ = 109(Pa/s)	$P_0$ = 2 × 109(Pa/s)
0.2	Sinusoid panel	3518.60	3542.40	3561.00
	Parabolic panel	3514.42	3540.00	3556.20
	Cylindrical panel	3513.65	3538.00	3556.00
0.4	Sinusoid panel	3296.54	3319.80	3335.40
	Parabolic panel	3292.71	3317.40	3332.40
	Cylindrical panel	3292.01	3317.00	3332.00
0.6	Sinusoid panel	3073.67	3099.00	3112.20
	Parabolic panel	3070.21	3094.20	3109.20
	Cylindrical panel	3069.57	3093.60	3109.00

Table 3 investigates the critical thermal buckling temperatures of FG-GPLRC panels with different geometrical forms, porosity coefficients $e$ , and thermal loading speed ${\mathrm{\Delta }}_0$ . Similar to the axial compression case, sinusoid panels display the highest thermal stability, followed by parabolic and cylindrical configurations. The porosity effect is again evident, with an increase in $e$ leading to a noticeable increase in critical buckling temperature. The presence of pores lowers the overall thermal stiffness of the panel; contrary, it causes the thermal expansion coefficient of the panel to decrease. The opposite effect of these two parameters makes it more stable under increased thermal loads. Furthermore, a comparison between static and dynamic thermal buckling results shows that dynamic critical temperatures are consistently larger, and the difference widens as ${\mathrm{\Delta }}_0$ increases.

Table 3.

Static and dynamic thermal buckling loads $\mathrm{\Delta }{T}_{cr}$ (K) of unstiffened FG-GPLRC panel with porous core, (U/PC/U, 2I2F, $h$ = 0.02 m, $h_{co}=0.7h$ , $a=b=30h$ , $h_z=1.5h$ , $\mathrm{\eta }$ = 0, c = 0 N.s/m³, $P_x$ = 0 Pa, $W_{GP}$ = 0.5%, $m$ = 1, $n$ = 1, $K_1$ = 10 MN/m³, $K_2$ = 0.1 MN/m).

$e$	Type	Static	Dynamic
			${\mathrm{\Delta }}_0$ = 100 (K/s)	${\mathrm{\Delta }}_0$ = 200 (K/s)
0.2	Sinusoid panel	533.77	539.67	542.98
	Parabolic panel	533.14	538.92	542.28
	Cylindrical panel	533.08	538.86	542.22
0.4	Sinusoid panel	556.81	562.79	566.46
	Parabolic panel	556.10	562.12	565.73
	Cylindrical panel	556.03	562.06	565.67
0.6	Sinusoid panel	584.86	590.72	594.49
	Parabolic panel	584.05	589.96	593.67
	Cylindrical panel	583.98	589.89	593.60

Figure 2 presents the static postbuckling responses of FG-GPLRC panels with porous cores and inclined stiffeners under axial compression, emphasizing the influence of stiffener characteristics, panel geometry, GPL distribution, and porosity. In Figure 2(a), a clear enhancement in postbuckling performance is observed for panels reinforced with inclined stiffeners compared to those with orthogonal stiffeners, indicating the superior effectiveness of inclined stiffeners in redistributing axial loads and delaying structural instability. Figure 2(b) shows that increasing the stiffener height results in a significant improvement in postbuckling load-carrying capacity due to the rise in stiffness. The effect of stiffener angle is examined in Figure 2(c), the most effective stiffener angle obtained in this investigation is about $\mathrm{\pi }/3.6$ , for both perfect and imperfect panel cases. Figure 2(d) investigates the role of panel geometry, where sinusoid panels consistently exhibit the highest buckling strength, followed by parabolic and cylindrical panels. The influence of GPL distribution patterns is depicted in Figure 2(e), with the Λ/PC/V-Λ type configurations yielding the highest postbuckling resistance due to their enhanced reinforcement near the surface layers. Lastly, Figure 2(f) demonstrates the detrimental impact of porosity, as an increase in the porosity coefficient $e$ leads to a substantial decrease in the postbuckling capacity, which is expected given that the introduction of voids within the core region diminishes the effective stiffness of the panel.

Figure 2.

Static compressive postbuckling curves of FG-GPLRC panels stiffened by inclined stiffeners ( $h=0.02$ m, $h_{co}=0.7h$ , $n=1,$ $b_{st}=h,$ $d_{st}=a/8$ , $K_1=10$ MN/m³, $K_2=0.1$ MN/m).

Figure 3 illustrates the static thermal postbuckling behavior of FG-GPLRC panels reinforced with inclined stiffeners, highlighting the influence of stiffener configuration, stiffener angle, stiffener height and stiffener spacing. As shown in Figure 3(a), inclined stiffeners significantly improve the thermal postbuckling resistance compared to orthogonal configurations, confirming their superior ability to counteract thermally induced deformations through better directional stiffness alignment. Figure 3(b) shows the complexity in the postbuckling behavior of panels with different stiffener angles. While the postbuckling strength is best with a stiffener angle $\mathrm{\pi }/3$ in the small deflection region, in the large deflection region, a stiffener angle $\mathrm{\pi }/4$ shows superiority. The effect of stiffener spacing on the postbuckling behavior of the panel is illustrated in Figure 3(c). Naturally, as the stiffener spacing increases, the load-carrying capacity of the panel decreases. In Figure 3(d), the increase in stiffener height enhances the panel’s resistance to thermal loads, owing to the greater bending stiffness provided by taller stiffeners. These results collectively demonstrate the crucial role of stiffener design in optimizing the thermal stability of FG-GPLRC structures with porous core.

Figure 3.

Static thermal postbuckling curves of FG-GPLRC panels stiffened by inclined stiffeners ( $h=0.02$ m, $h_{co}=0.7h$ , $a=b=30h,$ $m=1,$ $n=1,$ $h_{st}=1.5h,$ $W_{GP}=0.5\%,$ $K_1=10$ MN/m³, $K_2=0.1$ MN/m).

Figure 4 depicts the dynamic buckling responses of FG-GPLRC panels with porous cores and inclined stiffeners subjected to axial compression, examining the effects of stiffener configuration, GPL mass fraction, porosity coefficient, and boundary condition type under dynamic loading conditions. Effects of stiffener configuration on the postbuckling performance are illustrated in Figure 4(a). Similar to the static case, the superiority of inclined stiffeners over orthogonal stiffeners can be observed. In addition, the phenomenon of sudden buckling phenomenon is only observed with unstiffened panels. Figure 4(b) evaluates the role of GPL mass fraction on dynamic buckling performance. As the mass fraction of GPL increases, the postbuckling performance improves significantly, especially in the large deflection region. Figure 4(c) demonstrates the adverse effect of porosity on dynamic buckling resistance. As the porosity coefficient increases, the structural stiffness degrades significantly, resulting in a sharp decrease in critical dynamic loads. Effects of boundary conditions on the postbuckling behavior of the panel are illustrated in Figure 4(d). As can be observed, the inclined stiffeners and 2F2I cases present the superior load-carrying capacity of panels.

Figure 4.

Dynamic compressive postbuckling curves of FG-GPLRC panels stiffened by inclined stiffeners ( $h=0.02$ m, $h_{co}=0.7h$ , $h_z=h,$ $m=1,$ $n=1,$ $h_{st}=1.5h,$ $b_{st}=h,$ $d_{st}=a/8$ , $\mathrm{\iota }=\mathrm{\pi }/4$ , $K_1=10$ MN/m³, $K_2=0.1$ MN/m, $P_0=1$ GPa/s).

Figure 5 presents the dynamic thermal buckling responses of FG-GPLRC panels reinforced with inclined stiffeners, focusing on the effects of stiffener configurations, GPL mass fractions, porosity coefficients, and boundary conditions under time-dependent thermal loading. The superior performance of inclined stiffeners on the dynamic response of the panel compared to orthogonal and unstiffened stiffeners can be observed in Figure 5(a). As seen in Figure 5(b), the dynamic response curve decreases with greater GPL mass fraction, indicating that GPLs provide enhanced resistance under rapid heating conditions. Figure 5(c) shows that increasing the porosity coefficient considerably increases the dynamic buckling resistance, indicating a pronounced sensitivity of porous-core structures to thermal instabilities when subjected to rapidly rising temperatures. Finally, the different behavior in the dynamic response curve of the panels with different boundary conditions can be observed in Figure 5(d).

Figure 5.

Dynamic thermal postbuckling curves of FG-GPLRC panels stiffened by inclined stiffeners ( $h=0.02$ m, $h_{co}=0.7h$ , $a=b=30h,$ $m=1,$ $n=1,$ $h_{st}=1.5h,$ $d_{st}=a/4$ , $\mathrm{\iota }=\mathrm{\pi }/4$ , $K_1=10$ MN/m³, $K_2=0.1$ MN/m, ${\mathrm{\Delta }}_0=100$ K/s).

Conclusion

This study developed a semi-analytical framework to investigate the nonlinear static and dynamic buckling behaviors of FG-GPLRC panels with porous core reinforced with inclined stiffeners under axial and thermal loads. The HSDT, improved smeared stiffener technique, and approximate nonlinear stress function were integrated to address complex panel geometries, including cylindrical, sinusoid, and parabolic forms. The key conclusions drawn from the numerical results are as follows:

(1) Inclined FG-GPLRC stiffeners significantly enhance both static and dynamic buckling capacities compared to orthogonal stiffeners, in all investigation conditions.

(2) Among the considered panel types, sinusoid panels consistently demonstrate the highest buckling resistance compared with parabolic and cylindrical panels.

(3) Stiffener height and GPL distribution pattern play significant roles in postbuckling performance.

(4) Increasing porosity markedly reduces the buckling resistance in mechanical load cases, and oppositely for thermal load cases.

(5) Dynamic effects lead to an increase in critical buckling loads compared to static responses, and this increase becomes more significant with increasing loading speed, underscoring the necessity of considering loading speed in design.