Technical Paper Volume 1 Issue 3
School of Mechanical Engineering, Jimma University, Ethiopia
Correspondence: Rajesh Kumar, School of Mechanical Engineering, JIT, Jimma University, P.O. Box378, Jimma, Ethiopia, Tel 251909462675
Received: May 11, 2017  Published: October 9, 2017
Citation: Kumar R. Effects of elastic foundations on flexural response of shear deformable laminated plates subjected to transverse uniform lateral pressure uncertain system environment and hygrothermomechanical loading. Aeron Aero Open Access J. 2017;1(3):84101. DOI: 10.15406/aaoaj.2017.01.00012
In this paper, the effect of elastic foundations on flexural response of shear deformable laminated plates subjected to transverse uniform lateral pressure under uncertain system environment and hygrothermomechanical loading using MATLAB [R2010a] code for micromechanical model approach is investigated. A C0 finite element method in conjunction with the first order perturbation technique extended by authors for plate subjected to lateral loading is employed to find out the second order response statistics (expected mean and coefficient of variations) of the transverse deflection of the plate. Plate material properties and elastic foundation parameters are taken as basic random variables. The plate is analysed for plate thickness ratios, aspect ratios, boundary conditions, lamina layup, fiber volume fractions, load deflections and environmental conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection by comparison with the results available in literatures and standard deviation results with independent Monte Carlo simulation before data generation.
Keywords: stochastic bending response, finite element method, uncertain system properties, elastic foundations
${A}_{i}{}_{j},\text{}{B}_{i}{}_{j},\text{}etc$ 
Laminate stiffnesses 
a, b 
Plate length and breadth 
h 
Thickness of the plate 
${E}_{f},\text{}{E}_{m}$ 
Elastic moduli of fiber and matrix, respectively. 
${G}_{f},{\text{G}}_{m}$ 
Shear moduli of fiber and matrix, respectively. 
${v}_{f},{\text{v}}_{m}$ 
Poisson’s ratio of fiber and matrix, respectively. 
${V}_{f},{\text{V}}_{m}$ 
Volume fraction of fiber and matrix, respectively. 
$\alpha f,\text{}\alpha m$ 
Coefficient of thermal expansion of fiber and matrix, respectively. 
b_{i} 
Basic random material properties 
E_{11}, E_{22} 
Longitudinal and Transverse elastic moduli 
G_{12}, G_{13}, G_{23} 
Shear moduli 
K_{l}, 
Linear bending stiffness matrix 
K_{g} 
Thermal geometric stiffness matrix 
D 
Elastic stiffness matrices 
${M}_{\alpha \beta},{m}_{\alpha \beta}$ 
Mass and inertia matrices 
ne, n 
Number of elements, number of layers in the laminated plate 
Nx, Ny, Nxy 
Inplane thermal buckling loads 
nn 
Number of nodes per element 
N_{i} 
Shape function of ith node 
${\overline{C}}_{{}^{{}^{{}^{p}}}ijkl}$ 
Reduced elastic material constants 
$f,\text{}{\left\{f\right\}}^{\left(e\right)}$ 
Vector of unknown displacements, displacement vector of eth element 
u, v, w 
Displacements of a point on the mid plane of plate 
${\overline{u}}_{1},\text{\hspace{0.17em}}{\overline{u}}_{2},\text{\hspace{0.17em}}{\overline{u}}_{3},$ 
Displacement of a point (x, y, z) 
${\overline{\sigma}}_{ij},{\overline{\epsilon}}_{i}{}_{j}$ 
Stress vector, Strain vector 
$\psi y,\text{}\psi x$ 
Rotations of normal to mid plane about the x and y axis respectively 
θ_{x}, θ_{y}, θ_{k} 
Two slopes and angle of fiber orientation wrt xaxis for kth layer 
x, y, z 
Cartesian coordinates 
$\begin{array}{l}\rho ,\text{}\lambda ,\text{}Var(.)\hfill \end{array}$ 
Mass density, eigenvalue, variance 
$\omega ,\varpi $ 
Fundamental frequency and its dimensionless form 
RV_{s} 

DT, DC, 
Difference in temperatures and moistures 
${\alpha}_{1},\text{}{\alpha}_{2},{\beta}_{1},{\beta}_{2}$ 
Thermal expansion and hygroscopic coefficients along x and y direction, respectively. 
Laminated composite plates are often subjected to combination of lateral pressure and hygrothermomechanical loading. The plates are more advantageous over plates made of conventional materials and they are more hygrothermally and mechanically stable than plates made of conventional metals. The capability to predict the structural response and enable a better understanding and characterization of the actual behavior of laminated composite plates resting on elastic foundations in terms of structural response when subjected to combined load is of prime interest for structural analysis. Composite structures have inherent dispersion in system properties due to lack of strict quality control and the characteristics of the large parameters involved with the manufacturing and fabrication process. The variation in the system properties of the composite materials necessitates the inclusion of randomness of system properties in the analysis; otherwise predicted response may differ significantly rendering the structures unsafe. For reliable and safe design, especially for sensitive engineering applications in thermal environments. Accurate prediction of system behavior of composite structures in the presence of uncertainties in the system properties favors a probabilistic analysis rather than analytical approach by modeling their properties as basic random variables.
Stochastic micromechanical modeling investigation yields more accurate system behavior and proved to be superior technique for design compared to stochastic macromechanical modeling investigations. Material properties, geometric properties, foundation stiffness parameters, fiber orientations, lamina layup sequence design and curing are of prime importance and these parameters must be considered in accurate prediction of uncertain system behavior of composites. Thus there is importance of uncertainties accountability in the responses. Laminated composite plates are widely used in aerospace, submarines, automotive industries, nuclear structures and in various structural components such as beams, thin and thick plates, shells, panels etc.
A considerable literature is available on the static response of geometrically linear and nonlinear composite laminated plates under various thermal, hygrothermal and mechanical loads or combination of both. Notably among them are Shen,^{1} Huang & Tauchert,^{2} Sen,^{3} Lin et al.^{4} Shen.^{5} Whitney et al. ^{6} studied the effect of environment on the elastic response of layered composite elates. Adam et al.^{7} Lee et al.^{8} Sai Ram and Sinha.^{9} Patel et al.^{10} Shen^{11,12} Nonlinear bending of shear deformable laminated plates under lateral pressure and thermal loading and resting on elastic foundations and hygrothermal effects on the nonlinear bending of shear deformable laminated plates.
Salim et al.^{13} examined the effect of randomness in material properties (like elastic modulus Poisson’s ratios etc.,) on the response statistics of a composite plate subjected to static loading using classical plate theory (CLT) in conjunction with first order perturbation techniques (FOPT).^{1417}
A little literature is available on stochastic analysis for macromechanocal and micromechanical model investigation.^{1830} Keeping in mind the above aspect, to the best of the authors’ knowledge, there is no literature covering effects of elastic foundations on flexural response of shear deformable laminated plates subjected to transverse uniform lateral pressure under uncertain system environment and hygrothermomechanical loading,
However, no work is available dealing with effects of elastic foundations on flexural response of shear deformable laminated plates subjected to transverse uniform lateral pressure uncertain system environment and hygrothermomechanical loading to the best of author’s knowledge.
In the present investigation, effects of elastic foundations on flexural response of shear deformable laminated plates subjected to transverse uniform lateral pressure uncertain system environment and hygrothermomechanical loading in the presence of small random variation in the system properties, the transverse shear strain using higher order shear deformation theory (HSDT) with vonKarman sense is studied by using stochastic analysis for micromechanical model. The C^{0} finite element method is employed to determine the second order statistics (mean and standard deviation) of flexural response parameter of laminated composite plates with uniform constant temperature (U.T). The numerical illustrations are concerned with flexural response behavior under different sets of thermomaterial properties, stacking sequence, fiber volume fractions, plate thickness ratios, aspect ratios, different boundary conditions, and foundation stiffness, coefficient of hygroscopic expansions and coefficients of thermal expansions. It is observed that small amount of random variations in above mentioned parameters of the composite plate significantly affect the flexural response especially at micromechanical investigation.
Consider geometry of laminated composite rectangular plate of length a, width b, and thickness h, which consists of Nplies located in three dimensional Cartesian coordinate system (X, Y, Z) where X and Y plane passes through the middle of the plate thickness with its origin placed at the corner of the plate as shown in Figure 1. Let $\left(\overline{u},\text{\hspace{0.17em}}\overline{v},\text{\hspace{0.17em}}\overline{w}\right)$ be the displacements parallel to the (X, Y, Z) axes, respectively. The thickness coordinate Z of the top and bottom surfaces of any k_{th} layer are denoted by Z_{(k1)} and Z_{(k)} respectively. The fiber of the k_{th} layer is oriented with fiber angle θ_{k} to the X axis. The plate is assumed to be subjected to uniformly distributed transverse static load is defined as $q\left(x,y\right)={q}_{o}$
The plate is assumed to attach to the foundation so that no separation takes place in the process of deformation.^{1,2} The interaction between the plate and the supporting foundation follows the two parameter model (Pasternaktype) as:
$P={K}_{1}w{K}_{2}{\nabla}^{2}w$
Where P is the foundation reaction per unit area, and ${\nabla}^{2}={\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}$ is Laplace differential operator and K_{1} and K_{2} are the Winkler and Pasternak foundation stiffness. This model is simply known as Winkler type when ${K}_{2}=0$.^{1}
In the present study, the assumed displacement field is based on the Reddy’s^{22,23} higher order shear deformation theory [1996], which requires C^{1} continuous element approximation. In order to avoid the usual difficulties associated with these elements the displacement model has been slightly modified to make the suitability for C^{0} continuous element [1997]. In modified form, the derivatives of outofplane displacement are themselves considered as separate degree of freedom (DOFs). Thus five DOFs with C^{1} continuity are transformed into seven DOFs with C^{0} due to conformity with the HSDT. In this process, the artificial constraints are imposed which should be enforced variationally through a penalty approach.
In the present study, the assumed displacement field is based on the Reddy’s^{22,23} higher order shear deformation theory, which requires C^{1} continuous element approximation. In order to avoid the usual difficulties associated with these elements the displacement model has been slightly modified to make the suitability for C^{0} continuous element in modified form, the derivatives of outofplane displacement are themselves considered as separate degree of freedom (DOFs). Thus five DOFs with C^{1} continuity are transformed into seven DOFs with C^{0} due to conformity with the HSDT. In this process, the artificial constraints are imposed which should be enforced variationally through a penalty approach. However, the literature^{26} demonstrates results using C^{0} can be obtained. The modified displacement field along the X, Y, and Z directions for an arbitrary composite laminated plate is now written as
$\begin{array}{l}\overline{u}=u+{f}_{1}(z){\psi}_{x}+{f}_{2}(z){\varphi}_{x};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \overline{v}=v+{f}_{1}(z){\psi}_{y}+{f}_{2}(z){\varphi}_{y};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \overline{w}=w;\end{array}$ (1)
where $\overline{u}$ ,$\overline{v}$ and $\overline{w}$ denote the displacements of a point along the (X, Y, Z) coordinates axes: u, v, and w are corresponding displacements of a point on the mid plane, ${\varphi}_{x}=w{,}_{x}$ and${\varphi}_{y}=w{,}_{x}$ and ${\psi}_{x}$ , ${\psi}_{y}$ are the rotations of normal to the mid plane about the yaxis and xaxis respectively. The function ${f}_{1}\left(z\right)$ and ${f}_{2}\left(z\right)$ can be written as
${f}_{1}\left(z\right)={C}_{1}z{C}_{2}{z}^{3}$ ; ${f}_{2}\left(z\right)={C}_{4}{z}^{3}$ with${C}_{1}=1,\text{}\text{\hspace{0.05em}}\text{\hspace{0.17em}}{C}_{2}={C}_{4}=4{h}^{2}/3$ .
The displacement vector for the modified C^{0} continuous model is denoted as
$\left\{\Lambda \right\}={\left[\begin{array}{ccccccc}u& v& w& {\varphi}_{y}& {\varphi}_{x}& {\psi}_{y}& {\psi}_{x}\end{array}\right]}^{T}$ , (2)
where, comma (,) denotes partial differential.
For the structures considered here, the relevant strain vector consisting of strains in terms of midplane deformation, rotation of normal and higher order terms associated with the displacement for k_{th} layer are written as
$\left\{\epsilon \right\}=\left\{{\epsilon}_{l}\right\}\left\{{\overline{\epsilon}}_{HT}\right\}$ (3)
where$\left\{{\epsilon}_{l}\right\}$ and$\left\{{\overline{\epsilon}}_{HT}\right\}$ are the linear and hygrothermal strain vector, respectively. Nonlinear analysis is given in.^{25}
Using Eq. (3) the linear strain vector can be obtained using linear strain displacement relations which can be written as
$\left\{{\epsilon}_{l}\right\}=\left\{\begin{array}{c}{\epsilon}_{P}^{L}\\ 0\end{array}\right\}+\left\{\begin{array}{c}z{\epsilon}_{b}^{L}\\ {\epsilon}_{s}\end{array}\right\}+\left\{\begin{array}{c}0\\ {z}^{2}{\epsilon}_{s}^{*}\end{array}\right\}+\left\{\begin{array}{c}{z}^{3}{\epsilon}^{*}\\ 0\end{array}\right\}$ (4)
The hygrothermal strain vector$\left\{{\overline{\epsilon}}_{t}\right\}$ is represented as
$\left\{{\overline{\epsilon}}_{HT}\right\}=\left\{\begin{array}{c}{\overline{\epsilon}}_{x}\\ {\overline{\epsilon}}_{y}\\ {\overline{\epsilon}}_{xy}\\ {\overline{\epsilon}}_{yz}\\ {\overline{\epsilon}}_{zx}\end{array}\right\}=\Delta T\left\{\begin{array}{c}{\alpha}_{1}\\ {\alpha}_{2}\\ {\alpha}_{12}\\ 0\\ 0\end{array}\right\}+\Delta C\left\{\begin{array}{c}{\beta}_{1}\\ {\beta}_{2}\\ {\beta}_{12}\\ 0\\ 0\end{array}\right\}$ (5)
Stress–strain relation
The constitutive law of thermoelasticity for the materials under considerations relates the stresses with strains in a plane stress state for the k_{th} lamina oriented as an arbitrary angle with respect to reference axis for the orthotropic layers is given by.^{27}
${\left\{\sigma \right\}}_{k}={\left[\overline{Q}\right]}_{k}{\left\{\epsilon \right\}}_{k}$
or
${\left\{\begin{array}{c}{\sigma}_{x}\\ {\sigma}_{y}\\ {\sigma}_{xy}\\ {\sigma}_{yz}\\ {\sigma}_{xz}\end{array}\right\}}_{k}={\left[\begin{array}{ccccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}& 0& 0\\ {\overline{Q}}_{12}& {\overline{Q}}_{22}& {\overline{Q}}_{26}& 0& 0\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}& 0& 0\\ 0& 0& 0& {\overline{Q}}_{44}& {\overline{Q}}_{45}\\ 0& 0& 0& {\overline{Q}}_{45}& {\overline{Q}}_{55}\end{array}\right]}_{k}\left\{\left\{{\epsilon}_{l}\right\}\left\{{\overline{\epsilon}}_{HT}\right\}\right\}$ (6)
where, ${\left\{{\overline{Q}}_{ij}\right\}}_{k}$ , ${\left\{\sigma \right\}}_{k}$ and ${\left\{\epsilon \right\}}_{k}$ are transformed stiffness matrix, stress and strain vectors of the k_{th} lamina, respectively,
Strain energy of the plate
The strain energy ${\Pi}_{\text{SE}}$ of the laminated composite plates can be expressed as
${\Pi}_{\text{SE}}=\frac{1}{2}{\displaystyle {\int}_{}{}_{V}\left\{\epsilon \right\}{}^{T}\left[\sigma \right]dV}$ . (7)
Strain energy due to hygrothermal stresses
The strain energy $({\Pi}_{TH})$ storage by hygrothermal load (uniform and transverse change in temperature and moisture across the thickness) is written as
$\begin{array}{l}{\Pi}_{TH}=\frac{1}{2}{\displaystyle {\int}_{A}\left[{N}_{x}{\left(w{,}_{x}\right)}^{2}+{N}_{y}{\left(w{,}_{y}\right)}^{2}+2{N}_{xy}\left(w{,}_{x}\right)\left(w{,}_{y}\right)\right]}\text{\hspace{0.17em}}dA\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}{{\displaystyle {\int}_{A}\left\{\begin{array}{c}w{,}_{x}\\ w{,}_{y}\end{array}\right\}}}^{T}\left[\begin{array}{cc}{N}_{x}& {N}_{xy}\\ {N}_{xy}& {N}_{y}\end{array}\right]\left\{\begin{array}{c}w{,}_{x}\\ w{,}_{y}\end{array}\right\}dA\end{array}$ (8)
where, N_{x}, N_{y} and N_{xy} are prebuckling thermal stresses.
Strain energy due to elastic foundations
The strain energy $({\Pi}_{3})$ due to elastic foundation having foundation layers can be written a
$\begin{array}{l}{\Pi}_{f}=\frac{1}{2}{\displaystyle {\int}_{A}\left\{{K}_{1}{\left(w{,}_{x}\right)}^{2}+{K}_{2}\left[{\left(w{,}_{x}\right)}^{2}+{\left(w{,}_{x}\right)}^{2}\right]\right\}}\text{\hspace{0.17em}}dA\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}{{\displaystyle {\int}_{A}\left\{\begin{array}{c}\begin{array}{l}w\\ w{,}_{x}\end{array}\\ w{,}_{y}\end{array}\right\}}}^{T}\left[\begin{array}{ccc}{K}_{1}& 0& 0\\ 0& {K}_{2}& 0\\ 0& 0& {K}_{2}\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}w\\ w{,}_{x}\end{array}\\ w{,}_{y}\end{array}\right\}dA\end{array}$ (9)
The potential energy due to work done by external mechanical loading of intensity $q\left(x,y\right)$ is given by
$W={W}_{q}={\displaystyle {\int}_{A}q\left(x,y\right)\text{}w}\text{\hspace{0.17em}}dA$ (10)
where, q(x, y) is the intensity of distributed transverse static load which is defined as
$q\left(x,y\right)=\frac{Q{E}_{22}{h}^{3}}{{b}^{4}}$ (11)
In the present study, a C^{0} ninenoded isoparametric finite element with 7 degree of freedoms (DOFs) per node as described earlier by $\left\{\Delta \right\}$ is employed. For this type of element, the displacement vector and the element geometry are expressed as
$\left\{\Lambda \right\}={\displaystyle \sum _{i=1}^{NN}{\phi}_{i}}{\left\{\Lambda \right\}}_{i};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x={\displaystyle \sum _{i=1}^{NN}{\phi}_{i}}{x}_{i};$ and $y={\displaystyle \sum _{i=1}^{NN}{\phi}_{i}}{y}_{i}$ (12)
where ${\phi}_{i}$ is the interpolation function for the i^{th} node, ${\left\{\Lambda \right\}}_{i}$ is the vector of unknown displacements for the i^{th} node, NN is the number of nodes per element and x_{i} and y_{i} are cartesian coordinate of the i^{th} node.
${\Pi}_{HT}={\displaystyle \sum _{e=1}^{NE}\left[\frac{1}{2}{\left\{{\Lambda}^{\left(e\right)}\right\}}^{T}\left[{K}^{{}^{\left(e\right)}}\right]{\left\{\Lambda \right\}}^{\left(e\right)}{\left\{{\Lambda}^{\left(e\right)}\right\}}^{T}\left[{F}_{HT}^{{}^{\left(e\right)}}\right]\right]}=\frac{1}{2}{\left\{q\right\}}^{T}\left[K\right]\left\{q\right\}{\left\{q\right\}}^{T}\left[{F}^{HT}\right]$ (13)
Work done due to external transverse load
Using finite element model (Eq. (10), Equation (9) may be written as
$W={\displaystyle \sum _{e=1}^{NE}{W}^{\left\{e\right\}}}$ where ${W}^{\left\{e\right\}}={\displaystyle {\int}_{{A}^{\left(e\right)}}{\left\{\Lambda \right\}}^{\left(e\right)}{}^{T}{\left\{{P}_{M}\right\}}^{\left(e\right)}}\text{\hspace{0.17em}}dA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left\{q\right\}}^{\left(e\right)}{}^{T}{\left\{{P}_{M}\right\}}^{\left(e\right)}$ (14)
Governing equations
The governing equation for the bending analysis can be derived using Variational principle, which is generalization of the principle of virtual displacement.^{23} For the bending analysis, the minimization of first variation of total potential energy $\Pi \text{}\left({\Pi}_{SE}+\text{}{\Pi}_{HT}+\text{}{\Pi}_{f}\text{}W\right)$ with respect to displacement vector is given by
$\delta \left({\Pi}_{SE}+\text{}{\Pi}_{HT}+\text{}{\Pi}_{f}\text{}W\right)\text{}=\text{}0$ (15)
$\left[K\right]\left\{W\right\}=\left\{{F}_{HT}\right\},$ (16)
with $\left\{{F}_{HT}{}^{}\right\}=\left\{{P}_{M}\right\}+\left\{{P}_{HT}\right\}$
where [K], {W}, {P_{M}} and {P_{H}_{T}} are global linear stiffness matrix, response vector, mechanical and hygrothermal force vector, respectively.
The stiffness matrixes [K], displacement vector {W} and force vector [F^{HT}] are random in nature, being dependent on the system properties. Therefore the eigenvalues and eigenvectors also become random. In deterministic environment, the solution of Eq. (16) can be obtained using conventional method.
In the present study, our aim is to find the second order statistics of ${W}_{i}^{R}$ when the second order statistics of primary RV^{s} ${b}_{i}^{R}$ are known. Any random variable can be expressed as the sum of its mean and the zero mean random variable which is expressed .The expression only up to the firstorder terms and neglecting the second and higherorder terms are given as random variable RV^{R} =mean (RV^{s})+ zeromean random variable (RV^{R})
The operating random variables in the present case are defined as:
${b}_{i}^{R}={b}_{i}^{d}+{b}_{i}^{r};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{K}_{sij}^{R}={K}_{sij}^{d}+{K}_{sij}^{r};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{i}^{R}={W}_{i}^{d}+{W}_{i}^{r}$
We can express the above relations in the form:
${b}_{i}^{R}={b}_{i}^{d}+\in {b}_{i}^{r};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{K}_{sij}^{R}={K}_{sij}^{d}+\in {K}_{sij}^{r};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{i}^{R}={W}_{i}^{d}+\in {W}_{i}^{r}$
where $\in $ is a scaling parameter, and is small in magnitude. We consider a class of problems where the zeromean random variation is very small as compared to the mean part of random variables. i.e., $R{V}^{d}>>\in R{V}^{r}$ . Using the Taylor series expansion and neglecting the second and higherorder terms since first order approximation is sufficient to yield results with desired accuracy with low variability which is the cases in most of the sensitive application.
The governing equation (16) can be written in the most general form as:^{24,2830}
$\left[{K}^{R}\right]\left\{{W}^{R}\right\}=\left\{{F}_{HT}^{R}\right\}=\left\{{F}^{R}\right\}$ (17)
Zeroth order perturbation equation $\left({\in}^{0}\right):\left[{K}^{d}\right]\left\{{W}^{d}\right\}=\left\{{F}^{d}\right\}$ (18)
First order perturbation equation $\left({\in}^{1}\right):\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{K}^{d}\right]\left\{{W}^{r}\right\}+\left[{K}^{r}\right]\left\{{W}^{d}\right\}=\left\{{F}^{r}\right\}$ (19)
$W={W}^{d}+\left\{\frac{\partial {W}^{d}}{\partial {b}_{l}^{r}}\right\}{b}_{l}^{r}$ and var$\left(W\right)=E{\left[{\displaystyle \sum _{l}\frac{\partial {W}^{d}}{\partial {b}_{l}{}^{R}}{b}_{l}{}^{r}}\right]}^{2}$ (20)
Where E [ ] and var (.) are the expectation and variance respectively. The variance can further be written as.^{28}
var $\left(W\right)={\displaystyle \sum _{l}^{N}{\displaystyle \sum _{l}^{N}diag}}\left[\frac{\partial {W}^{d}}{\partial {b}_{l}{}^{R}}{\left(\frac{\partial {W}^{d}}{\partial {b}_{l}{}^{R}}\right)}^{T}\right]E\left({b}_{l}{}^{r},{b}_{l}{}^{r}\right)$ (21)
var$\left(W\right)=\left(\frac{\partial {W}^{d}}{\partial {b}_{l}{}^{R}}\right)\left[{\sigma}_{b}\right]\left[\rho \right]\left[{\sigma}_{b}\right]{\left(\frac{\partial {W}^{d}}{\partial {b}_{l}{}^{R}}\right)}^{T}$ (22)
Eq. (22) expresses the covariance of the deflection in terms of standard deviations (SD) of random variables b_{i} (i=1, 2,…, R) and correlation coefficients.
A nine noded Lagrange isoparamatric element, with 63 DOFs per element for the present HSDT model has been used for discretizing the laminate. Based on convergence study conducted a (8 × 8) mesh has been used throughout the study. In the all problem considered, the individual layers are taken be equal thickness.
The results are presented taking COV of the system property equal to 0.10.^{31} However, the scattering of system can be taken by allowing the COV to vary from 0 to 20% and the presented results would be sufficient to extrapolate the results keeping in mind the limitation of FOPT. The basic random system variables such as E_{1}, E_{2}, G_{12}, G_{13}, G_{23}, υ_{12}, α_{1}, α_{2}, k_{1}, k_{2} and Q are sequenced and defined as
${b}_{1}={E}_{11},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}={E}_{22},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{3}={G}_{12},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{4}={G}_{13},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{5}={G}_{23},\text{\hspace{0.17em}}{b}_{6}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\alpha}_{1},\text{\hspace{0.17em}}\text{}{b}_{7}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\alpha}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{8}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\beta}_{2},\text{\hspace{0.17em}}{b}_{9}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{10}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{2},{b}_{11}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}Q$
The following dimensionless linear transverse mean central deflection has been used in this study.
${W}_{0l}={W}_{l}/h$ , ${k}_{1}={K}_{1}{b}^{4}/{E}_{22}^{d}{h}^{3}$ ; ${k}_{2}={K}_{2}{b}^{2}/{E}_{22}^{d}{h}^{3}$ ,
in which W_{i}, K_{1} and K_{1} dimensional mean transverse central deflection, dimensionless Winkler and Pasternak foundation stiffness parameters respectively.
In the present study,
various combination of boundary edge support conditions namely, simply supported (S1 and S2), clamped and combination of clamped and simply supported have been used and shown in.Figure 2
The plate geometry used is characterized by aspect ratios (a/b) = 1and 2, side to thickness ratios (a/h) = 20, 30, 40, 50, 60, 80 and 100. The following material properties are used for computation Shen:^{12}
Validation study for mean transverse central deflection
Table 1 compares the hygrothermal effects on the linear and nonlinear bending behavior of a (±45^{0})_{2T} laminated square plate, dimensionless load deflection (Q) where (Q= q b^{4}/E_{22}h^{4}=100, 150, 200), fiber volume fraction (V_{f} =0.6), plate thickness ratio (a/h =10), simple support SSSS (S2) boundary conditions and under environmental conditions. It is noticed that present [HSDT] result for mean hygrothermal deflection are in good agreement with the deterministic results of.^{12} Figure 3 compares the hygrothermal effects on the linear and nonlinear bending behavior of a (±45^{0})_{2T} laminated square plate, dimensionless load deflection (Q) where (Q= q b^{4}/E_{22}h^{4}=100, 150, 200), fiber volume fraction (V_{f} =0.6), plate thickness ratio (a/h =10), simple support SSSS (S2) boundary conditions and under environmental conditions. It is noticed that present [HSDT] result for mean hygrothermal deflection are in good agreement with the deterministic results of.^{12}
(Q) 
NonDimensional Hygrothermal Bending Load 


Shen ^{12} 
Present[HSDT] 
Shen ^{12} 
Present[ HSDT] 

$\Delta T={0}^{0}C,\text{}\Delta C=0\%$ 
$\Delta T={0}^{0}C,\text{}\Delta C=0\%$ 
$\Delta T=300,\text{}\Delta C=3\%$ 
$\Delta T=300,\text{}\Delta C=3\%$ 

Nonlinear 
Linear 
Nonlinear 
Nonlinear 
Linear 
Nonlinear 

100 
0.6887 
1.328 
0.6952 
0.6261 
0.9377 
0.6222 
150 
0.8857 
1.5801 
0.883 
0.7126 
1.1067 
0.7264 
200 
0.9909 
1.7325 
0.9989 
0.777 
1.1837 
0.768 
Table 1 Comparison of Hygrothermal Effects on the Linear and Nonlinear Bending Behavior of a (±45^{0})_{2T} Laminated Square Plate, Load Deflection (Q) where Q= q b^{4}/E_{22}h^{4}, Fiber Volume Fraction (V_{f}) =0.6, Plate Thickness Ratio (a/h) =10, Simple Support SSSS (S2) Boundary Conditions under Environmental Conditions
Figure 3 Validation study of dimensionless transverse central deflection of angleply square laminated composite plate subjected to linearly varying temperature.
Validation study for random transverse central deflection
Validation study for random hygrothermal central deflection of material properties (E22), plate thickness ratio (a/h) =30, aspect ratio (a/b) =1, rise in temperature$\left(\Delta T={200}^{0}C\right)$ , rise in moisture concentration$\left(\Delta C=2\%\right)$ , simple support SSSS (S2), fiber volume fraction (V_{f}=0.6) , dimensionless load deflection (Q =100), angle ply antisymmetric (±45^{0})_{2T} laminated composite plate resting on Winkler (k_{1}=100, k_{2}=00) and Pasternak( k_{1}=100, k_{2}=10) elastic foundations is shown in Figure 4. It is seen that present FOPT results are satisfactory with the MCS results.
Figure 4 Validation of present SFEM results with independent MCS results of laminate composite square plates resting on Winkler and Pasternak elastic foundations, subjected hygrothermal lateral loading having SSSS (S2) support condition for only one system properties E_{22}.
In (Table 2a) (Table 2b), it is observed that as increases the lateral pressure, the mean transverse central deflection increases and corresponding coefficient of variation decreases. It is noticed that expected mean transverse central deflection (W_{0l}) value of individual random variables of hygrothermal deflection decreases for Winkler elastic foundation and it further decreases for Pasternak elastic foundation whereas COV of hygrothermal deflection increases on rise in temperature and moisture concentration as shown in Figure 5.
(bi) 
(k_{1}=100, k_{2}=00) 


$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{100}^{0}C,\text{}\Delta C=0.01$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T={300}^{0}C,\text{}\Delta C=0.03$  
COV, Wl, 
COV, Wl, 
COV, Wl, 
COV, Wl, 

${E}_{11}(i=1)$ 
(0.0351) 2.14e07 
(0.0312) 6.15e06 
(0.0273) 1.83e05 
(0.0234) 3.87e05 
${E}_{22}(i=2)$ 
0.0024 
5.00E04 
3.27E04 
3.11E04 
${G}_{12}(i=3)$ 
5.06E07 
1.31E06 
3.35E06 
5.76E06 
${G}_{13}(i=4)$ 
1.81E05 
2.12E05 
2.26E05 
2.15E05 
${G}_{23}(i=5)$ 
9.08E06 
1.06E05 
1.13E05 
1.07E05 
${V}_{12}(i=6)$ 
7.19E05 
7.19E06 
4.77E06 
7.14E06 
${\alpha}_{11}(i=7)$ 
1.53E11 
8.77E11 
1.58E10 
1.86E10 
${\alpha}_{22}(i=8)$ 
4.05E12 
2.10E10 
6.75E10 
1.40E09 
${\beta}_{2}\left(i=9\right)$ 
7.42E10 
4.11E08 
1.31E07 
2.73E07 
${k}_{1}\left(i=10\right)$ 
6.76E06 
7.00E06 
7.32E06 
7.77E06 
${k}_{2}\left(i=11\right)$ 
0 
0 
0 
0 
$Q\left(i=12\right)$ 
2.52E04 
2.61E04 
2.73E04 
2.88E04 
Table 2(A) Effects of the Variation of Individual Random System Properties b_{i}, [{(i =1 to 12), = 0.10] on the Dimensionless Expected Mean (W_{0l}) and Coefficient of Variation (W_{l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) Elastic Foundations, Subjected to Uniform Constant Temperature and Moisture (U.T), inplane Biaxial Compression, Plate Thickness Ratio (a/h=20), with Simple Support S2 Boundary Conditions. The Dimensionless Mean Hygrothermal Deflections are given in Brackets. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
(bi) 
(k_{1}=100,k_{2}=10) 


$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{100}^{0}C,\text{}\Delta C=0.01$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T={300}^{0}C,\text{}\Delta C=0.03$ 

COV, Wl, 
COV, Wl, 
COV, Wl, 
SD/Mean, Wl, 

${E}_{11}(i=1)$ 
(0.0330) 1.07e05 
(0.0291)3.26e06 
(0.0253) 8.63e06 
COV, Wl, 
${E}_{22}(i=2)$ 
0.0012 
3.63E04 
2.98E04 
3.01E04 
${G}_{12}(i=3)$ 
1.03E07 
1.18E06 
2.68E06 
4.48E06 
${G}_{13}(i=4)$ 
7.42E06 
6.09E06 
2.43E06 
4.57E06 
${G}_{23}(i=5)$ 
3.71E06 
3.04E06 
1.21E06 
2.28E06 
${V}_{12}(i=6)$ 
0.0017 
1.08E04 
1.20E05 
2.51E07 
${\alpha}_{11}(i=7)$ 
2.03E10 
3.10E10 
4.21E10 
5.30E10 
${\alpha}_{22}(i=8)$ 
2.17E09 
1.81E09 
1.39E09 
8.93E10 
${\beta}_{2}\left(i=9\right)$ 
4.16E07 
3.47E07 
5.82E06 
1.67E07 
${k}_{1}\left(i=10\right)$ 
5.63E06 
5.70E06 
5.82E06 
5.98E06 
${k}_{2}\left(i=11\right)$ 
5.29E06 
5.59E06 
5.79E06 
5.85E06 
$Q\left(i=12\right)$ 
2.52E04 
2.64E04 
2.78E04 
2.93E04 
Table 2(B) Effects of the Variation of Individual Random System Properties b_{i}, [{(i =1 to 12), = 0.10] on the Dimensionless Expected Mean (W_{0l}) and Coefficient of Variation (W_{l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Pasternak (k_{1}=100, k_{2}=10) Elastic Foundations, Subjected to Uniform Constant Temperature and Moisture (U.T), inplane Biaxial Compression, Plate Thickness Ratio (a/h=20), with Simple Support S2 Boundary Conditions. The Dimensionless Mean Hygrothermal Deflections are given in Brackets. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
For the same lateral pressure and temperature distribution, it is seen that the COV of transverse central deflection becomes more important as the plate thickness decreases i.e., a/h increases (Table 3). It is seen that on variations of thickness ratio the mean (W_{0l}) hygrothermal deflection increases whereas COV of hygrothermal deflection decreases with different combinations of input random variables when laminated composite plates resting on Winkler and for Pasternak elastic foundations hygrothermal deflection decreases further with different combinations of input random variables in Figure 6.
a/h 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

5 
0.0126 
0.0097 
0.0304 
0.0261 
10 
0.0278 
0.0228 
0.0314 
0.0268 
30 
0.0377 
0.0285 
0.0338 
0.0229 
50 
0.0411 
0.0332 
0.0351 
0.0227 
100 
0.0449 
0.0256 
0.0364 
0.0175 
Table 3 Effects of Plate Thickness Ratios (a/h) with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
In Table 4, it is noticed that on increase of aspect ratio the mean (W_{0l}) hygrothermal central deflection value increases for Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) elastic foundations whereas COV of hygrothermal deflection decreases for all different combinations of input random variables, it is more dominant for Pasternak elastic foundations as shown in Figure 7.
a/b 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

0.5 
0.0027 
0.0024 
0.0024 
0.0019 
1 
0.0395 
0.0308 
0.0345 
0.0223 
1.5 
0.1947 
0.1314 
0.1562 
0.1049 
2 
0.5555 
0.3814 
0.4088 
0.29 
Table 4 Effects of Aspect Ratios (a/b) with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=40). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
In Table 5 it is noticed that combined simple support and clamped support CSCS have significance effects on mean (W_{0l}) hygrothermal deflection with different combinations of input random variables under environmental conditions. However the COV of hygrothermal deflection also varies accordingly under given environmental conditions and different combinations of input random variables for square composite plates resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) elastic foundations, mean and COV are significant for$\Delta T\text{}={200}^{0}C,\text{}\Delta C=0.02$ Pasternak(k_{1}=100,k_{2}=10) elastic foundations as shown in Figure 8.
BCs 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

SSSS (S1) 
0.0426 
0.0346 
0.0358 
0.0232 
SSSS (S2) 
0.0426 
0.0345 
0.0357 
0.0231 
CCCC 
0.0423 
0.0411 
0.0353 
0.0358 
CSCS 
0.0408 
0.0686 
0.0391 
0.0777 
Table 5 Effects of Support Conditions with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=60). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
In Table 6, it is observed that on change of lamina layup the mean (W_{0l}) of hygrothermal central deflection decreases significantly for cross ply symmetric plate. The COV of hygrothermal central deflection also decreases for different combinations of input random variables. It is significant to note for cross ply symmetric plates with $\Delta T\text{}={200}^{0}C,\text{}\Delta C=0.02$ and Pasternak elastic foundations as shown in Figure 9.
Layup 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

(±45^{0})_{2T} 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 
(±45^{0})_{S} 
0.0335 
0.0268 
0.0302 
0.0239 
[0^{0}/90^{0}]_{2T} 
0.0226 
0.0178 
0.0202 
0.0156 
[0^{0}/90^{0}]_{ S} 
0.0107 
0.0088 
0.0101 
0.0082 
Table 6 Effects of LayUp with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=50). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection Q=100, Fiber Volume Fraction (V_{f}=0.6)
In Table 7, it is seen that on increasing load deflection the mean (W_{0l}) hygrothermal central deflection increases in given environmental conditions and different combinations of input random variables .The central deflection value and COV of hygrothermal central deflection decreases in similar conditions and important to note for plates with $\Delta T\text{}={200}^{0}C,\text{}\Delta C=0.02$ and Pasternak elastic foundations as shown in Figure 10.
(Q) 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

100 
0.0377 
0.0285 
0.0338 
0.0229 
150 
0.0563 
0.0476 
0.0501 
0.0392 
200 
0.0698 
0.063 
0.0616 
0.0523 
Table 7 Effects of Load Deflections (Q) with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=30). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Fiber Volume Fraction (V_{f}=0.6)
In Table 8, it is noticed that on varying fiber matrix volume fraction the mean (W_{0l}) hygrothermal central deflection increases in given environmental conditions and different combinations of input random variables whereas the value of COV of hygrothermal central deflection also varies in similar conditions. It is to be noted for Pasternak elastic foundations and plates with $\Delta T\text{}={200}^{0}C,\text{}\Delta C=0.02$ as shown in Figure 11.
(V_{f}) 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 


Mean, W_{0l} 
Mean, W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 
$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.00$ 
$\Delta T=\text{}{200}^{0}C,\text{}\Delta C=0.02$ 

0.50 
0.0381 
0.0289 
0.0332 
0.0209 
0.55 
0.0385 
0.0296 
0.0336 
0.0213 
0.60 
0.0395 
0.0308 
0.0345 
0.0223 
0.65 
0.0413 
0.0327 
0.0361 
0.024 
0.70 
0.044 
0.0355 
0.0385 
0.0265 
Table 8 Effects of Fibre Volume Fractions (V_{f}) with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=40). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection (Q)=100
In Table 9, it is noticed that on increasing temperature and moisture the mean (W_{0l}) hygrothermal central deflection decreases in given environmental conditions and different combinations of input random variables whereas the value of COV of hygrothermal central deflection increases in similar conditions as noticed for plates with $\Delta T\text{}={200}^{0}C,\text{}\Delta C=0.02$ and Pasternak (k_{1}=100,k_{2}=10) elastic foundations as shown in Figure 12.
Environmental conditions 
(k_{1}=100, k_{2}=00) 
(k_{1}=100, k_{2}=10) 

W_{0l} 
W_{0l} 

$\Delta T=\text{}{0}^{0}C,\text{}\Delta C=0.0$ 
0.0533 
0.0488 
$\Delta T=\text{10}{0}^{0}C,\text{}\Delta C=0.01$ 
0.0486 
0.0438 
$\Delta T=\text{20}{0}^{0}C,\text{}\Delta C=0.02$ 
0.0436 
0.0388 
$\Delta T=\text{30}{0}^{0}C,\text{}\Delta C=0.03$ 
0.0384 
0.0338 
$\Delta T=\text{40}{0}^{0}C,\text{}\Delta C=0.04$ 
0.033 
0.0288 
$\Delta T=\text{50}{0}^{0}C,\text{}\Delta C=0.05$ 
0.0276 
0.0238 
Table 9 Effects of Temperature and Moisture Rise $(\Delta T,\text{}\Delta C)$ with Random Input Variables b_{i}, [{(i =1 to 9), (7..9), (10,11) and (12)} = 0.10] on the Dimensionless Expected Mean (W_{0l}) of Hygrothermomechanically Induced Central Deflection of Angle Ply (±45^{0})_{2T} Square Laminated Composite Plates Resting on Winkler (k_{1}=100,k_{2}=00) and Pasternak(k_{1}=100,k_{2}=10) Elastic Foundations, Plate Thickness Ratios (a/h=20). Subjected to Uniform Constant Temperature (U.T), and Inplane Biaxial Compression with Simple Support S2 Boundary Conditions. Load Deflection (Q)=150, Fiber Volume Fraction (V_{f}=0.6)
A C^{0} SFEM probabilistic procedure is adopted to compute the second order statistics of transverse central deflection of geometrically linear laminated composite plate in the framework of HSDT with randomness in material properties, coefficients of thermal expansion and coefficients of hygroscopic expansion, elastic foundation parameters and lateral loading.
Among the different system properties studied, the elastic moduli, elastic foundation parameters, lateral loading and environmental conditions have dominant effect on the COV of the transverse deflection when compared to other system properties subjected to uniform and linearly varying temperature distribution. In order to assess the effects of temperature and moisture on the bending behavior of shear deformable laminated plates, a theoretical analysis is developed based on a micro mechanical model.
The COV of the hygrothermomechanically induced transverse central deflection of the plate increases as distribution in lateral pressure increases, this bring out importance of considering hygrothermomechanical loading along with lateral pressure from design point of view specially in aerospace and other sensitive application where reliability of the components is important. Tight controls of these properties are therefore required for high reliability of the plate design. The flexural response of the laminated composite plate deteriorates considerably with the increase in temperature and moisture concentration and this hygrothermal environment becomes more detrimental as the working temperature reaches higher temperature.
$\left({A}_{ij},{B}_{ij},{D}_{ij},{E}_{ij},{F}_{ij},{H}_{ij}\right)={\displaystyle {\int}_{h/2}^{h/2}{Q}_{ij}\left(1,z,{z}^{2},{z}^{3},{z}^{4},{z}^{6}\right)}dz$
; (i,j=1,2,6)
$\left({A}_{ij},{D}_{ij},{F}_{ij}\right)={\displaystyle {\int}_{h/2}^{h/2}{Q}_{ij}\left(1,{z}^{2},{z}^{4}\right)}dz$
; (i,j=4,5)
$\left[{K}_{b}\right]={\displaystyle \sum _{i=1}^{n}{\displaystyle {\int}_{{A}^{\left(e\right)}}{\left[{B}_{b}{}^{\left(e\right)}\right]}^{T}}}\left[{D}_{b}\right]\left[{B}_{b}{}^{\left(e\right)}\right]dA$
;
$\left[{K}_{s}\right]={\displaystyle \sum _{i=1}^{n}{\displaystyle {\int}_{{A}^{\left(e\right)}}{\left[{B}_{s}{}^{\left(e\right)}\right]}^{T}}}\left[{D}_{s}\right]\left[{B}_{s}{}^{\left(e\right)}\right]dA$
${\left[{K}_{G}\right]}^{e}={\displaystyle {\int}_{A}{\left[{B}_{NL}\right]}^{T}\left\{\phi \right\}dA}={\displaystyle {\int}_{A}{\left[G\right]}^{T}\left[\phi \right]\left[G\right]dA}$
$\left[{K}_{f}\right]=\frac{1}{2}{\displaystyle {\int}_{A}{\left[{B}_{f}\right]}^{T}\left[{D}_{f}\right]\left[{B}_{f}\right]dA}$
$\left\{q\right\}={\displaystyle \sum _{e=1}^{NE}{\left\{\Lambda \right\}}^{\left(e\right)}}$
$\left[{F}^{T}\right]={\displaystyle \sum _{i=1}^{n}{\displaystyle {\int}_{{A}^{\left(e\right)}}\left[{\left[{B}_{1i}{}^{\left(e\right)}\right]}^{T}\left[{N}^{T}\right]+{\left[{B}_{b1i}{}^{\left(e\right)}\right]}^{T}\left[{M}^{T}\right]+{\left[{B}_{b2i}{}^{\left(e\right)}\right]}^{T}\left[{P}^{T}\right]\right]dA}}$
.
where
$\left[{D}_{b}\right]=\left[\begin{array}{ccccccc}{\phi}_{i}{,}_{x}& 0& 0& 0& 0& 0& 0\\ {\phi}_{i}{,}_{y}& 0& 0& 0& 0& 0& 0\\ 0& {\phi}_{i}{,}_{x}& 0& 0& 0& 0& 0\\ 0& {\phi}_{i}{,}_{y}& 0& 0& 0& 0& 0\\ 0& 0& {\phi}_{i}{,}_{x}& 0& 0& 0& 0\\ 0& 0& {\phi}_{i}{,}_{y}& 0& 0& 0& 0\\ 0& 0& 0& {C}_{1}{\phi}_{i}{,}_{x}& 0& 0& 0\\ 0& 0& 0& 0& {C}_{1}{\phi}_{i}{,}_{y}& 0& 0\\ 0& 0& 0& {C}_{1}{\phi}_{i}{,}_{y}& {C}_{1}{\phi}_{i}{,}_{x}& 0& 0\\ 0& 0& 0& {C}_{2}{\phi}_{i}{,}_{x}& 0& {C}_{2}{\phi}_{i}{,}_{x}& 0\\ 0& 0& 0& 0& {C}_{2}{\phi}_{i}{,}_{y}& 0& {C}_{2}{\phi}_{i}{,}_{y}\\ 0& 0& 0& {C}_{2}{\phi}_{i}{,}_{y}& {C}_{2}{\phi}_{i}{,}_{x}& {C}_{2}{\phi}_{i}{,}_{y}& {C}_{2}{\phi}_{i}{,}_{x}\end{array}{\phi}_{i}{,}_{x}\right]\left\{q\right\}$ ,
$\left[{D}_{s}\right]=\left[\begin{array}{ccccccc}0& 0& {\phi}_{i}{,}_{x}& 1& 0& 0& 0\\ 0& 0& {\phi}_{i}{,}_{x}& 0& 1& 0& 0\\ 0& 0& 0& 3& 0& 3& 0\\ 0& 0& 0& 0& 3& 0& 3\end{array}\right]\left\{q\right\}$
$\left[{B}_{gi}\right]=\left[\begin{array}{ccccccc}0& 0& {\phi}_{i}{,}_{x}& 0& 0& 0& 0\\ 0& 0& {\phi}_{i}{,}_{y}& 0& 0& 0& 0\end{array}\right]$
, $\left[{N}_{0}\right]=\left[\begin{array}{cc}{N}_{x}& {N}_{xy}\\ {N}_{xy}& {N}_{y}\end{array}\right]$
,
${\overline{C}}_{ijkl}=\left[\begin{array}{ccccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}& 0& 0\\ {\overline{Q}}_{12}& {\overline{Q}}_{22}& {\overline{Q}}_{26}& 0& 0\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}& 0& 0\\ 0& 0& 0& {\overline{Q}}_{44}& {\overline{Q}}_{45}\\ 0& 0& 0& {\overline{Q}}_{45}& {\overline{Q}}_{55}\end{array}\right]$
$\begin{array}{l}{\overline{Q}}_{12}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{Q}}_{21}{\mathrm{cos}}^{4}\alpha +2({Q}_{12}+2{Q}_{66}){\mathrm{cos}}^{2}\alpha {\mathrm{sin}}^{2}\alpha +{Q}_{22}{\mathrm{sin}}^{4}\alpha \\ {\overline{Q}}_{12}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{Q}}_{21}=({Q}_{11}+{Q}_{22}4{Q}_{66}){\mathrm{cos}}^{2}\alpha {\mathrm{sin}}^{2}\alpha +{Q}_{12}({\mathrm{cos}}^{4}\alpha +{\mathrm{sin}}^{4}\alpha )\\ {\overline{Q}}_{16}=({Q}_{11}{Q}_{12}2{Q}_{66})\mathrm{sin}\alpha {\mathrm{cos}}^{3}\alpha +({Q}_{12}{Q}_{22}2{Q}_{66}){\mathrm{sin}}^{3}\alpha \mathrm{cos}\alpha \\ {\overline{Q}}_{22}={Q}_{11}{\mathrm{sin}}^{4}\alpha +2({Q}_{12}+2{Q}_{66}){\mathrm{cos}}^{2}\alpha {\mathrm{sin}}^{2}\alpha +{Q}_{22}{\mathrm{cos}}^{4}\alpha \\ {\overline{Q}}_{26}=({Q}_{11}{Q}_{12}2{Q}_{66}){\mathrm{sin}}^{3}\alpha \mathrm{cos}\alpha +({Q}_{12}{Q}_{22}2{Q}_{66})\mathrm{sin}\alpha {\mathrm{cos}}^{3}\alpha \\ {\overline{Q}}_{26}=({Q}_{11}+{Q}_{22}2{Q}_{12}2{Q}_{66}){\mathrm{cos}}^{2}\alpha {\mathrm{sin}}^{2}\alpha +{Q}_{66}({\mathrm{cos}}^{4}\alpha +{\mathrm{sin}}^{4}\alpha )\\ {\overline{Q}}_{44}={Q}_{44}{\mathrm{cos}}^{2}\alpha +{Q}_{55}{\mathrm{sin}}^{2}\alpha \\ {\overline{Q}}_{45}=({Q}_{55}{Q}_{44})\mathrm{sin}\alpha \mathrm{cos}\alpha {Q}_{54}\\ {\overline{Q}}_{55}={Q}_{55}{\mathrm{cos}}^{2}\alpha +{Q}_{44}{\mathrm{sin}}^{2}\alpha \end{array}$
Where
$\begin{array}{l}{Q}_{11}=\frac{{E}_{11}}{(1{v}_{12}{v}_{21})}\text{\hspace{0.17em}},{Q}_{12}=\frac{{v}_{12}{E}_{22}}{(1{v}_{12}{v}_{21})}\text{\hspace{0.17em}},=\frac{{v}_{21}{E}_{11}}{(1{v}_{12}{v}_{21})}\text{\hspace{0.17em}}={Q}_{21},\\ {Q}_{22}=\frac{{E}_{22}}{(1{v}_{12}{v}_{21})}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{Q}_{66}={G}_{12},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{44}={G}_{13},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{55}={G}_{12},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{21}=\frac{{v}_{12}{E}_{22}}{{E}_{11}}\text{\hspace{0.17em}}\end{array}$
None.
Author declares that there is no conflict of interest.
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