Hygrothermoelastic free vibration response of laminated composite plates resting on elastic foundations with random system properties

Abstract

This article presents the hygrothermal effects on the free vibration of laminated composite plates resting on a two-parameter Winkler and Pasternak elastic foundations with random system properties using micromechanical model. System properties such as material properties, hygroscopic expansion coefficients, thermal expansion coefficients and foundation stiffness parameters are modeled as independent basic random variables which are affected by the variation in temperature and moisture based on a micromechanical model of a laminate for accurate prediction of system behavior. A C ⁰ finite element method based on higher-order shear deformation theory has been used for deriving the standard eigenvalue problem. A Taylor series-based mean-centered first-order perturbation technique is used to find mean and standard deviation of the natural frequency subjected to uniform moisture concentration and temperature rise, plate aspect ratio, total number of plies, fiber orientations and elastic foundation parameters with different boundary support under hygrothermal environmental conditions. Typical numerical results have been validated with those available in the literature and independent Monte Carlo simulation.

Introduction

Laminated composite plates resting on elastic foundation as structural components play an important and vital role in exceptional performance extreme engineering structures as aeronautical and aerospace engineering for construction, manufacturing of aircrafts structure components, aerospace vehicles, aircraft runways and launching sites of missile and spacecraft. In construction of above mentioned structure/components, there is operational and functional requirement of inherent characteristic ability to tailor structural properties through appropriate lamination schemes for achieving high strength to weight ratio according to use. Nowadays advanced reinforced composites are being widely used in modern aerospace transportation systems and for high performance in other areas also. Plates made up of these composites are often subjected to mechanical loading in hygrothermal environment. Due to boundary constraints in composite plates, varying moisture and temperature environments besides other factors induce significant strains, thereby needs ensuing hygrothermal free vibration behavior. Therefore, it is of utmost importance to understand the hygrothermal free vibration behavior of these structural systems in order to estimate the reserve strength. The deterministic analysis is based on the assumptions that the system properties are completely determinate i.e. only the mean response is considered. The dispersion caused by the randomness in the system parameters is neglected. For reliable design, especially for sensitive application, it is important for the designer to understand and have sound knowledge of structural response. Otherwise the predicted response by deterministic analysis may differ significantly from the actual response which may be the cause of unsafe structure. Considering the above aspects to enhance accuracy in the response evaluation favors a probabilistic analysis for laminated composite by modeling their system properties as random variables (RVs).

Considerable literature is available on the vibration response of conventional structures and composite structures with randomness in system properties. Leissa and Martin ¹ have analyzed the vibration and buckling of rectangular composite plates and have studied the effects of variation in fiber spacing using classical laminate theory (CLT). Zhang and Chen ² have presented a method to estimate the standard deviation (SD) of eigenvalue and eigenvector of random multiple degree of freedom (DOF) system. Zhang and Ellingwood ³ have evaluated the effect of random material field characteristics on the instability of a simply supported beam on elastic foundation and a frame using perturbation technique. A comprehensive summary of extensive literature is available on the response analysis of the structures with deterministic material properties to random excitations. ⁴ However, the analysis of the structures with random system properties is not adequately reported in the literature. Manohar and Ibrahim ⁵ have presented excellent reviews on structural dynamic problems with parameter uncertainties. Using CLT in conjunction with First Order Perturbation Technique (FOPT), Salim et al. ⁶ have studied the static deflection, natural frequency and buckling load of composite rectangular plates with random material properties. Venini and Mariani ⁷ have investigated the eigen problem associated with the free vibrations of uncertain composite plates. The elastic moduli of the system, the stiffness of the Winkler foundation on which the plate rests and the mass density are considered to be uncertain. Given their random field-based description, a new method is presented for the computation of the second-order statistics of the eigen properties of the laminate. Yadav and Verma ⁸ have investigated the free vibration of composite circular cylindrical shells with random material properties employing CLT and FOPT for obtaining the second-order statistics natural frequencies. Singh et al. ^{9 ,10} have analyzed the composite cross-ply laminated composite plate/panel with random material properties for free vibration employing higher-order deformation theory (HSDT) with FOPT. The mean and coefficient of variation of the natural frequencies have been obtained for the first five natural frequencies using exact mean analysis approach and finite element method (FEM) with a C ⁰ element. Onkar and Yadav ¹¹ have investigated nonlinear response statistics of composite laminates with random material properties under random loading and nonlinear free vibration of laminated composite plate with random material properties. Kitipornchai et al. ¹² have studied the random free vibration response of functionally graded plates with general boundary conditions and subjected to a temperature change to obtain the second-order statistics of vibration frequencies. Tripathi et al. ¹³ have investigated the free vibration response of laminated composite conical shells with random material properties using FEM in conjunction with FOPT based on HSDT.

Most recently, the present authors investigated the effect of randomness in system properties on elastic buckling and free vibration of laminated composite plates resting on elastic foundation by assuming the system properties such as Young’s modulus, shear modulus, Poisson’s ratio and foundation parameters are modeled as random inputs variables. Lal et al. ¹⁴ applied C ⁰ linear and nonlinear FEM based on higher-order shear deformation in the presence of small deformation theory and von-Karman large deformation theory in conjunction with mean centered first-order perturbation technique to obtain the mean as well as SD of linear and nonlinear frequency response using macromechanical model. Whitney and Ashton ¹⁵ investigated the effects of environment on the elastic response of layered composite plates using deterministic FEM with macromechanical model. Chen and Chen ¹⁶ found out the vibrations of hygrothermal elastic composite plates using deterministic FEM. They investigated the different behavior of plates when exposed to moisture and temperature environments. Ram and Sinha ¹⁷ investigated the hygrothermal effects on the free vibration of laminated composite plates using deterministic finite element approach with first-order shear deformation theory. Patel et al. ¹⁸ analyzed the hygrothermal effects on the structural behavior of thick composite laminates using higher-order theory. They found that moisture and temperature significantly affect the thin plates compared to thick plates in buckling, vibration and static bending. Huang and Zheng ^19,20 studied the nonlinear vibration and dynamic response of simply supported shear deformable laminated plates on elastic foundations and in hygrothermal environments using deterministic finite element approach. Atmane ²¹ investigated the free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory. Benyoucef et al. ²² studied the bending of thick functionally graded plates resting on Winkler–Pasternak elastic foundations. Omurtag et al. ²³ investigated the free vibration analysis of Kirchhoff plates resting on elastic foundation by mixed finite element formulation based on Gateaux differential. Matsunaga ²⁴ studied the vibration and stability of thick plates on elastic foundations using shear deformation theory. Tounsi et al. ²⁵ applied a new computational method for prediction of transient hygroscopic stresses during moisture desorption in laminated composite plates with different degrees of anisotropy.

It is evident from the available literature that the studies on the free vibration response of laminated composite plates resting on elastic foundations with random system properties in hygrothermal environment using micromechanical model are not dealt by the researchers. In the present work, an attempt is made to address this problem. The laminated composite plate is analyzed using the C ⁰ linear FEM with HSDT based on von-Karman linear strain–displacement relations. FOPT approach has been employed for the first time to develop a novel probabilistic solution procedure for random linear problem with a reasonable accuracy. The approach is valid for system properties with small random variations in the system variables compared with the mean value. The condition is satisfied by most engineering materials and it hardly puts any limitation on the approach. The efficacy of the probabilistic approach is examined by comparing the results with an independent Monte Carlo simulation (MCS).

Mathematical formulation

Consider a rectangular laminated composite plate of length a, width b and total thickness h, defined in (X, Y, Z) system with x and y axes located in the middle plane and its origin placed at the corner of the plate. Let $(\bar{u},\bar{v},\bar{w})$ be the displacement parallel to the (X, Y, Z), respectively, as shown in Figure 1. The thickness coordinates Z on the top and bottom surfaces of any kth layer are denoted by Z _(k-1) and Z _(k), respectively. The fiber of the kth layer is oriented with angle θ_k to the x axis. The plate is supported by the foundation excluding any separation during the process of deformation as shown in Figure 1. The load displacement relation between the plate and the supporting foundation follows the two-parameter model (Pasternak-type) as

Figure 1.

Geometry of laminated composite plate resting on elastic foundation.

$P=K_{1}w-K_2{\mathrm{\nabla }}^{2}w\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\mathrm{\nabla }}^2=\mathrm{\partial }{,}_{x^2}+\mathrm{\partial }{,}_{y^2}$

(1)

where P is the foundation reaction per unit area, comma (,) denotes partial differential and ∇ is Laplace differential operator; K ₁ and K ₂ are normal and shear stiffness of the foundation, respectively. This model is simply known as Winkler type when K ₂ = 0. ¹⁹

Displacement field model

In the present study, the Reddy’s higher-order shear deformation theory has been employed. ²⁶ The displacement field model, after incorporating zero transverse shear stress conditions at the top and the bottom of the plate, is slightly modified, so that a C ⁰ continuous element would be sufficient. In modified form, the derivatives of out-of-plane displacement are themselves considered as separate DOF. Thus, five DOFs with C ¹ continuity are transformed into seven DOFs due to conformity with the HSDT. However, the literature ²⁷ demonstrates that without enforcing these constraints, the accurate results can be obtained. In this process, the artificial constraints are imposed which should be enforced variationally. The modified displacement field along the x, y and z directions for an arbitrary composite laminated plate is now written as. ²⁷

$\begin{array}{cc}& \bar{u}=u+f_1(z){\mathit{\psi }}_x+f_2(z){\mathit{\theta }}_x;\\ & \bar{v}=u+f_1(z){\mathit{\psi }}_y+f_2(z){\mathit{\theta }}_y;\\ & \bar{w}=w;\end{array}$

(2)

where u, v and w are corresponding displacements of a point on the mid-plane, ψ _x and ψ _y are the rotations of normal to the mid-plane about the y and x axes, respectively, where θ _x  = w, _x and θ _y  = w, _x

$f_1\mathrm{(}z\mathrm{)}=C_{1}z-C_2{z}^3;f_2\mathrm{(}z\mathrm{)}=-C_4{z}^3\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{C}_1=1,C_2=C_4=4h^2/3$

The displacement vector for the modified models is

$\{\mathrm{\Lambda }\}=[\begin{array}{ccccccc}u& v& w& {\mathit{\theta }}_y& {\mathit{\theta }}_x& {\mathit{\psi }}_y& {\mathit{\psi }}_x\end{array}{]}^T,$

(3)

where, comma (,) denotes partial differential.

Strain–displacement relations

The strain–displacements relations in von Karman sense is expressed as {∊} = {∊_l}. ²⁸

$\left\{\mathit{ϵ}\right\}=\left\{{\mathit{ϵ}}_l\right\}$

(4)

where, {∊ _l } is the linear strain vector.

Stress–strain relation

Law of thermoelasticity for the materials under consideration relates the stresses with strains in a plane stress state for the kth lamina oriented as an arbitrary angle with respect to the reference axis for an orthotropic layer is given by Reddy. ²⁹ The constitutive relationship between stress resultants and corresponding strains of laminated composite plate accounting for hygrothermal effect can be written as:

$\{\mathit{\sigma }{\}}_k=[\bar{Q}{]}_k\{\mathit{ϵ}{\}}_k$

or

${\left\{\begin{array}{c}{\mathit{\sigma }}_{xx}\\ {\mathit{\sigma }}_{yy}\\ {\mathit{\sigma }}_{xy}\\ {\mathit{\sigma }}_{yz}\\ {\mathit{\sigma }}_{xz}\end{array}\right\}}_k={\left[\begin{array}{ccccc}{\bar{Q}}_{11}& {\bar{Q}}_{12}& {\bar{Q}}_{16}& 0& 0\\ {\bar{Q}}_{12}& {\bar{Q}}_{22}& {\bar{Q}}_{26}& 0& 0\\ {\bar{Q}}_{16}& {\bar{Q}}_{26}& {\bar{Q}}_{66}& 0& 0\\ 0& 0& 0& {\bar{Q}}_{44}& {\bar{Q}}_{45}\\ 0& 0& 0& {\bar{Q}}_{45}& {\bar{Q}}_{55}\end{array}\right]}_k{\left\{\begin{array}{c}{\mathit{ϵ}}_{xx}-{\mathit{\alpha }}_{xx}\mathrm{\Delta }T-{\mathit{\beta }}_{xx}\mathrm{\Delta }C\\ {\mathit{ϵ}}_{yy}-{\mathit{\alpha }}_{yy}\mathrm{\Delta }T-{\mathit{\beta }}_{yy}\mathrm{\Delta }C\\ {\mathit{ϵ}}_{xy}-{\mathit{\alpha }}_{xy}\mathrm{\Delta }T-{\mathit{\beta }}_{xy}\mathrm{\Delta }C\\ {\mathit{ϵ}}_{yz}\\ {\mathit{ϵ}}_{zx}\end{array}\right\}}_k$

(5)

where $\{\bar{Q}{\}}_k$ , {σ} _k and {∊} _k are transformed stiffness matrix, stress and strain vectors of the kth lamina, respectively, and (α _x , α _y , α _xy ) and (β _xx , β _yy , β _xy ) are the thermal expansion and hygroscopic expansion coefficients along x, y, z direction, respectively, which can be obtained from the thermal and moisture coefficients in the longitudinal (α_l) and transverse (α _t ) directions of the fibers using transformation matrix. ▵T and ▵C are change in uniform temperature and moisture measured in terms of percentage. Substituting Eq. (3) in Eq. (5) and integrating through thickness gives a relationship between stress resultants and mid-plane strain:

$\left\{\begin{array}{c}{N}_i\\ M_i\\ P_i\end{array}\right\}=\left[\begin{array}{ccc}{A}_{ij}& B_{ij}& E_{ij}\\ B_{ij}& D_{ij}& F_{ij}\\ E_{ij}& F_{ij}& H_{ij}\end{array}\right]\left\{\begin{array}{c}{\mathit{ϵ}}_j^0\\ k_j^0\\ k_j^2\end{array}\right\}-\left\{\begin{array}{c}{N}_i^{HT}\\ M_i^{HT}\\ P_i^{HT}\end{array}\right\}(i,j=1,2,6)$

(6)

$\left\{\begin{array}{c}{Q}_2\\ Q_1\end{array}\right\}=\left[\begin{array}{cc}{A}_{4j}& D_{4j}\\ A_{5j}& D_{5j}\end{array}\right]\left\{\begin{array}{c}{\mathit{ϵ}}_j^0\\ k_j^2\end{array}\right\};\left\{\begin{array}{c}{R}_2\\ R_1\end{array}\right\}=\left[\begin{array}{cc}{D}_{4j}& F_{4j}\\ D_{5j}& F_{5j}\end{array}\right]\left\{\begin{array}{c}{\mathit{ϵ}}_j^0\\ k_j^2\end{array}\right\}(j=4,5)$

(7)

where the stress resultants per unit length are expressed in terms of mid-plane strains and curvatures:

$\begin{array}{cc}\{N_i\}& =[\begin{array}{ccc}{N}_{ix}& N_{iy}& N_{ixy}\end{array}{]}^T\\ \{M_i\}& =[\begin{array}{cc}{M}_{ix}& M_{iy}{M}_{ixy}\end{array}{]}^T\\ \{P_i\}& =[\begin{array}{ccc}{P}_{ix}& P_{iy}& P_{ixy}\end{array}{]}^T\end{array}$

(8)

The hygrothermal stress and moment resultants per unit length due to temperature and moisture change are calculated and the equivalent hygrothermal loads are defined as:

$N_i^{HT}=[N{]}^T+[N{]}^C,M_i^{HT}=[M{]}^T+[M{]}^C,P_i^{HT}=[P{]}^T+[P{]}^C$

(9)

For the hygrothermal free vibration, it is assumed that the temperature and moisture fields exhibit the linear variation through the plate thickness. For the plate subjected to uniform temperature rise (UT), it is expressed as

$T(X,Y,Z)=[T_0+C_0]$

(10)

Strain energy of the plate

The strain energy of the plate is given by

$\mathrm{\Pi }=\frac{1}{2}{\int }_v{\left\{\mathit{ϵ}\right\}}^T\left[\mathit{\sigma }\right]\mathrm{d}V$

Using Eqs. (4) and (5), the strain energy Π can be rewritten as

$\mathrm{\Pi }={\mathrm{\Pi }}_l$

(11)

where π_i is the linear strain energy, which can be expressed as

${\mathrm{\Pi }}_l=\frac{1}{2}{\int }_A{\left\{{\mathit{ϵ}}_l\right\}}^T\left[\bar{Q}\right]\left\{{\mathit{ϵ}}_l\right\}\mathrm{d}A+\frac{1}{2}{\int }_A{\left\{{\mathit{ϵ}}_l\right\}}^T\left\{\bar{Q}\right\}\mathrm{d}A+\frac{1}{2}{\int }_A\left\{\bar{Q}\right\}\left\{{\mathit{ϵ}}_l\right\}\mathrm{d}A$

Using strain vector expressed in terms of thickness coordinate, mid-plane strain vector and linear strain–displacement relations ²⁹ , Eq. (11) can be rewritten as

$\mathrm{\Pi }=\frac{1}{2}{\int }_A{\left\{{\overline{\mathit{ϵ}}}_l\right\}}^T\left[D\right]\left\{\overline{{\mathit{ϵ}}_l}\right\}\mathrm{d}A$

(12)

where $\{{\stackrel{\mathrm{ˉ}}{\mathit{ϵ}}}_l\}$ is the linear strain vector at the reference plane (Z = 0) and [D] is the laminate stiffness matrix.

and

$\{{\stackrel{\mathrm{ˉ}}{\mathit{ϵ}}}_l\}={\left(\begin{array}{ccccccccccccc}{\mathit{ϵ}}_1^0& {\mathit{ϵ}}_2^0& {\mathit{ϵ}}_6^0& k_1^0& k_2^0& k_6^0& k_1^2& k_2^2& k_6^2& {\mathit{ϵ}}_4^0& {\mathit{ϵ}}_5^0& k_4^2& k_5^2\end{array}\right)}^T,$

Strain energy due to elastic foundation

The potential energy (π₂) for linear elastic foundation having shear deformable layers can be written as

$\begin{array}{cc}& {\mathrm{\Pi }}_2=\frac{1}{2}{\int }_A\left\{K_1{w}^2+K_2\left[{\left(w_{,x}\right)}^2+{\left(w_{,y}\right)}^2\right]\right\}\mathrm{d}A\\ & {\mathrm{\Pi }}_2=\frac{1}{2}{\int }_A{\left\{\begin{array}{c}w\\ w_{,x}\\ w_{,y}\end{array}\right\}}^T\left[\begin{array}{ccc}{K}_1& 0& 0\\ 0& K_2& 0\\ 0& 0& K_2\end{array}\right]\left\{\begin{array}{c}w\\ w_{,x}\\ w_{,y}\end{array}\right\}\mathrm{d}A\\ & \end{array}$

(13)

External work done

Due to uniform change in temperature, prebuckling stresses in the plate are generated. The in-plane prebuckling stress resultants are the reason for buckling. The work done by the in-plane thermal stress resultants in producing out-of-plane displacement ‘w’ is expressed as

$\begin{array}{cc}& \begin{array}{c}W=\frac{1}{2}{\int }_A\left[N_x{\left(w{,}_x\right)}^2+N_y{\left(w{,}_y\right)}^2+2N_{xy}\left(w{,}_x\right)\left(w{,}_y\right)\right]\mathrm{d}A\\ \end{array}\\ & =\frac{1}{2}{\int }_A{\left\{\begin{array}{c}w{,}_x\\ w{,}_y\end{array}\right\}}^T{\left[\begin{array}{cc}{N}_x& N_{xy}\\ N_{xy}& N_y\end{array}\right]}^{(e)}\left\{\begin{array}{c}w{,}_x\\ w{,}_y\end{array}\right\}\mathrm{d}A,\end{array}$

(14)

where, N_x , N_y and N_xy are in-plane applied thermal compressive stress resultants per unit length.

Kinetic energy of the laminate

The kinetic energy (T) of the vibrating laminated plate can be expressed as

$T=\frac{1}{2}{\displaystyle {\int }_v{\text{ρ}}^{(k)}}{\left\{\dot{\widehat{u}}\right\}}^T\left\{\dot{\widehat{u}}\right\}\text{d}V$

(15)

where ρ and $\left\{\dot{\widehat{u}}\right\}={\left\{\begin{array}{ccc}\dot{\overline{u}}& \dot{\overline{v}}& \dot{\overline{w}}\end{array}\right\}}^T$ are the density and velocity vector of the plate, respectively.

Finite element model

In the FEM, the domain is discretized in to a set of finite elements. Over each of the elements, the displacement vector and element geometry are represented as

$\begin{array}{cc}& \left\{\mathrm{\Lambda }\right\}=\sum _{i=1}^{NN}{\mathit{\varphi }}_i\{\mathrm{\Lambda }{\}}_i;x=\sum _{i=1}^{NN}{\mathit{\varphi }}_i{x}_i;\\ & \mathrm{a}\mathrm{n}\mathrm{d}y=\sum _{i=1}^{NN}{\mathit{\varphi }}_i{y}_i\end{array}$

(16)

where φ _i is the interpolation (shape function) function for the ith node, {Λ} _i is the vector of unknown displacements for the ith node, NN is the number of nodes per element and x_i and y_i are Cartesian coordinates of the ith node.

Strain energy of the laminated plate

The linear functional are computed for each element and then summed over all the elements in the domain to get the total function. Following this, Eq. (16) can be written as

$\mathrm{\Pi }=\sum _{e=1}^{NE}\left({\mathrm{\Pi }}_l^{\left(e\right)}\right)$

(17a)

where, NE is the number of elements

${\mathrm{\Pi }}_l^{(e)}={\left\{\mathrm{\Lambda }\right\}}^{T(e)}{\left[K_l\right]}^{\left(e\right)}\left\{{\mathrm{\Lambda }}^{\left(e\right)}\right\}$

(17b)

Here, [K_l ]^(e) is the elemental linear stiffness matrix and {L^(e)} is the elemental nodal displacement vector

With $[K_l]=[K_b]+[K_s]<$

where global bending stiffness matrix [K_b ], shear stiffness matrix [K_s ], global displacement vector {q} and hygrothermal load vector [F] are defined in Appendix.

Strain energy due to elastic foundation

Using finite element notation, Eq. (16) after summing the entire element can be written

$\begin{array}{cc}& {\mathrm{\Pi }}_2=\left(\sum _{e=1}^{NE}{\mathrm{\Pi }}_{{}^{fl}}^{\left(e\right)}\right)\\ & {\mathrm{\Pi }}_{{}^{fl}}^{\left(e\right)}=\frac{1}{2}\underset{A}{\int }{\left\{\mathrm{\Lambda }\right\}}^{T(e)}{\left[K_{fl}\right]}^{\left(e\right)}\left\{{\mathrm{\Lambda }}^{(e)}\right\}\end{array}$

(18)

here [K_fl ]^(e) is the elemental linear foundation stiffness matrix.

Geometric stiffness matrix due to in-plane hygrothermal stress

Using the finite element model (Equation (16)), Equation (14) can also be written as

$\begin{array}{cc}& W=\left(\sum _{e=1}^{NE}{W}_{}^{\left(e\right)}\right)\\ & W^{(e)}=\frac{1}{2}\underset{A}{\int }{\left\{\mathrm{\Lambda }\right\}}^{T(e)}{\mathit{\lambda }}_{HT}{\left[Kg\right]}^{\left(e\right)}\left\{{\mathrm{\Lambda }}^{(e)}\right\}\mathrm{d}A.\end{array}$

(19)

where λ _HT and [Kg]^(e) are defined as the critical hygrothermal buckling load parameters and the elemental geometric stiffness matrix for the eth element, respectively.

Kinetic energy of laminated plate

Using Eq. (16), Eq. (14) can also be written as

$T=\sum _{e=1}^{NE}{T}^{(e)},$

(20a)

where

$T^{(e)}=\frac{1}{2}\sum _{e=1}^{NE}{{\left\{\dot{\mathrm{\Lambda }}\right\}}^{T^{(e)}}\left[M\right]}^{(e)}{\left\{\dot{\mathrm{\Lambda }}\right\}}^{(e)},$

(20b)

where [M]^(e) is consistent inertia matrix of eth element.

Adopting Gauss quadrature integration numerical rule, the elemental linear, foundation stiffness matrices and geometric stiffness matrix, respectively, can be obtained by transforming expression in Cartesian x and y coordinate systems to natural coordinate systems ξ and η.

Governing equation of motion

The governing equation for hygrothermal free vibration analysis of the laminated plate can be derived using the Lagrange’s equation of motion ³⁰ in terms of global matrices. This gives

$\mathit{\delta }\underset{t1}{\overset{t2}{\int }}({\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2-W-T)\mathrm{d}t$

(21a)

Substituting Eqs. (17b), (18), (19) and (20a) in Eq. (21a) will result in:

$\begin{array}{cc}& [M]\left\{q\right\}+\{K\}\{q\}=0\\ & [K]=\{[K_l]+[K_{fl}]-{\mathit{\lambda }}_{HT}[Kg]\}\end{array}$

(21b)

where

{q}, [K_l ], [K_fl ], [M] and λ _HT are defined as the global displacement vector, the global linear stiffness matrix, the global foundation stiffness matrix, the global mass matrix and the critical hygrothermal buckling parameter, respectively.

The above equation can be expressed in the form of linear generalized eigenvalue for hygrothermal linear free vibration as

$[K]\left\{q\right\}=\mathit{\lambda }[M]\{q\}$

(22)

Solutions-perturbation technique

• Step 1. By setting amplitude to zero, the random linear eigenvalue problem $\left[[K]\{q\}=\mathit{\lambda }\mathrm{[}M\mathrm{]}\{q\}\mathrm{,}\mathit{\lambda }={\mathit{\omega }}^2\right]$ is obtained from Eq. (21) by assuming that the system vibrates in its principal mode. Then the random linear eigenvalue problem is broken up into zeroth- and first-order equations using perturbation technique by neglecting higher-order equations. The zeroth-order linear eigenvalue problem is solved by normal eigen solution procedure and the first-order equation is solved for SD of natural frequency using the Taylor series expansion. The detail of the technique is presented in Figure 2.

• Step 2. The mean mode shape of the desired linear mode is normalized with respect to the given amplitude at the point of maximum deflection.

• Step 3. Using the normalized mode shape, the linear random stiffness matrix is computed. Again, the zeroth- and the first-order eigen equations are obtained using perturbation technique.

• Step 4. The zeroth- and the first-order equations are then solved to obtain new eigenvalue and corresponding eigenvectors and SD using updated zeroth- and first-order equations, respectively.

• Step 5. Ste7ps (2)–(4) are repeated until convergence is attained for {q}_max, ω² and the SD; and corresponding to the converged mode shape, the mean and SD are computed.

Figure 2.

(a) Overview of stochastic analysis procedure. (b) Flow chart of solution procedure of stochastic linear vibration problem.

Due to the result of randomness in basic input variables, all the quantities in Eq. (20), that is [K], {q} and λ are random as well. However, M is a constant mass matrix because it relates only the mass densities ρ that is taken as the deterministic in the present analysis. Further, it is quite logical to assume that the random input variables are independent of each other and random part of each input variable is small with respect to its mean value; fortunately, most of the engineering problems including composite materials fall in this category. Based on this, the mean-centered first-order perturbation techniques in which all the system parameters are expanded in Taylor series is used to determine the stochastic characteristics of the linear frequency of laminated composite plates.

Solution approach—SFEM for hygrothermal free vibration problem

Stochastic Finite Element Method (SFEM) approach has been adopted for obtaining the second-order statistics of dimensionless hygrothermal fundamental frequency response of laminated composite plate resting on elastic foundation with randomness in material properties, coefficients of hygroscopic expansion, coefficient of thermal expansion and foundation parameters. The material properties coefficients of hygroscopic expansion, coefficient of thermal expansion and foundation parameters are assumed to be the basic input RVs. Without any loss of generality, the RV can be spite up as the sum of a mean and a zero random part. ¹⁰ In general, a RV can be represented as the sum of its mean and zero mean RV, denoted by the superscripts ‘d’ and ‘r,’ respectively.

$K=K^d+K^r,{\mathit{\lambda }}_i={\mathit{\lambda }}_i^d+{\mathit{\lambda }}_i^r,q_i=q_i^d+q_i^r$

(23)

where ${\mathit{\lambda }}_i^d={\mathit{\omega }}_i^{d^2}$ , i = 1, 2…., p. The parameter p indicates the size of eigen problem.

Consider a class of problems where the random variation is very small as compared to the mean part of material properties. Further, it is quite logical to assume that the coefficient of variation in the derived quantities like λ^r, ω^r, q ^r and K ^r are also small as compared to mean values.

By substituting Eq. (23) in Eq. (22) and expanding the random parts in Taylor’s series (keeping the first-order terms, neglecting the second- and higher-order) one obtains as ³¹

For hygrothermal linear free vibration analysis

Zeroth-order perturbation equation:

$[K^d]\left\{q_i^d\right\}=\mathit{\lambda }[M]\{q_i^d\}$

(24)

First-order perturbation equation:

$[K^d]\left\{q_i^r\right\}+[K^r]\left\{q_i^d\right\}={\mathit{\lambda }}_i^r[M]\{q_i^d\}+{\mathit{\lambda }}_i^d[M]\left\{q_i^r\right\}$

(25)

Eq. (24) is the deterministic equations relating to the mean eigenvalues and corresponding mean eigenvectors, which can be determined by conventional eigen solution procedures. Eq. (25) the first-order perturbation approach is employed in the present study. ³²

Using this, Eq. (25) can be decoupled and the expression for hygrothermal free vibration is obtained. The FEM in conjunction with first-order perturbation has been found to be accurate and efficient. ^{31 ,33,34} According to this method, the RVs are expressed by Taylor’s series. Keeping the first-order terms, neglecting the second and higher order terms. Eq. (24) can be expanded up to first order, as the first order is sufficient to yield results.

${\mathit{\lambda }}_i^r=\sum _{j=1}^q{\mathit{\lambda }}_{i,j}^d{b}_j^r;\left\{q_i^r\right\}=\sum _{j=1}^p{q}_{i,j}^d{b}_j^r;\left[K^r\right]=\sum _{j=1}^q[K{,}_j^d]b_i^r;$

(26)

Using the above equation, the expression ${\mathrm{\lambda }}_{1i,j}^d$ is obtained.

Using Eq. (26), the variances of the eigenvalues can now be expressed as:

$\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathit{\lambda }}_{1i}\right)=\sum _{j=1}^p\sum _{k=1}^p{\mathit{\lambda }}_{1i,j}^d{\mathit{\lambda }}_{1i,k}^{d}Cov\left(b_j^r,b_k^r\right)$

(27)

where $Cov\left(b_j^r,b_k^r\right)$ is the cross-variance between $b_j^r$ and $b_k^r$ . The SD is obtained by the square root of the variance. ³¹

Results and discussion

The stochastic finite element analysis has been applied to obtain the mean and Coefficient of Variation (cov), considering different random parameters of the hygrothermal free vibration of laminated composite plate of graphite epoxy material resting on elastic foundations. The lamina coefficients of hygroscopic and thermal expansion including foundation stiffness parameters and material properties are modeled as basic RVs. The mean and SD of the hygrothermal free vibration are obtained considering all the random material input variables and thermal expansion coefficients, moisture expansion coefficients as well as foundation parameters taking separately as basic RVs as stated earlier. However, the results are only presented taking coefficient of variation ( ${\mathit{\omega }}_1^2$ ) of the system property equal to 0.10. However, the obtained results reveal that the stochastic approach would be valid up to COV = 0.20 ³⁴ as the nature of the SD variation is linear and passing through the origin. Hence, the presented results would be sufficient to extrapolate the results for other COV value, keeping in mind the limitation of FOPT. The thickness of all the lamina is assumed to be constant and of same material without varying individual property of materials used. The results obtained have been compared with MCS and those available in the literature. For the present study, a nine nodded serendipity element, which results in 63 DOF per element modified from HSDT-based C ⁰ finite element model has been used for discretizing the laminate. Based on convergence, a 4 × 4 mesh has been used throughout for evaluation of the results. Results have been computed by employing the full (3 × 3) integration rule for bending stiffness matrices, hygrothermal load vector and the geometric stiffness matrices and the reduced (2 × 2) integration rule for computing the shear stiffness matrices to avoid shear locking in the thin plates. The boundary conditions for the plate shown in Figure 3 are:

Figure 3.

Schematic representation of various boundary conditions for the plate.

All edges simply supported (SSSS [S1]):

$u=w={\mathit{\theta }}_y={\mathit{\psi }}_y=0,\mathrm{a}\mathrm{t}x=0,a;v=w={\mathit{\theta }}_x={\mathit{\psi }}_x=0\mathrm{a}\mathrm{t}y=0,b;$

All edges simply supported (SSSS [S2]):

$v=w={\mathit{\theta }}_y={\mathit{\psi }}_y=0,\mathrm{a}\mathrm{t}x=0,a;u=w={\mathit{\theta }}_x={\mathit{\psi }}_x=0\mathrm{a}\mathrm{t}y=0,b;$

All edges clamped (CCCC):

$u=v=w={\mathit{\psi }}_x={\mathit{\psi }}_y={\mathit{\theta }}_x={\mathit{\theta }}_y=0,\mathrm{a}\mathrm{t}x=0,a\mathrm{a}\mathrm{n}\mathrm{d}y=0,b;$

The plate geometry used is characterized by aspect ratios, a/b = 0.5, 1.0, 1.5 and 2, and side-to-thickness ratios, a/h = 5, 10, 20, 30, 40, 50, 60 and 100. The mean values of coefficients of hygroscopic expansion, coefficients of thermal expansion and material constants are used for computation. We now consider a second step as the elastic constants; thermal expansion coefficients and coefficients of hygroscopic expansion of each layers are assumed to be linear functions of temperature and moisture. The only exception is the Poisson’s ratio, which can reasonably be assumed as constant due to the weak dependency on temperature change. For the ratio of the SD to the mean of material and geometric properties varying from 0% to 20%, ³⁴ plate thickness ratios of a/h = 5, 10, 20, 30, 40, 50, 60 and 100 have been considered. The thermoelastic material properties such as E ₁₁, E ₂₂, G ₁₂, G ₁₃, G ₂₃, V ₁₂, α₁, α₂, β₂ and k ₁ and k ₂ are modeled as basic RVs. The input random variables b _i is related as b ₁ = E ₁₁, b ₂ = E ₂₂, b ₃ = G ₁₂, b ₄ = G ₁₃, b ₅ = G ₂₃, b ₆ = V ₁₂, b ₇ = α ₁, b ₈ = α ₂, b ₉ = β₂, b ₁₀ = k ₁ and b ₁₁ = k ₂. The following dimensionless quantities have been used in this study: the dimensionless expected mean fundamental frequency (ϖ_l) = ωa ² √(ρ₀/E ₂₂/h ²), nondimensionalized elastic foundation stiffness parameters $k_1=K_1{\mathit{\alpha }}^4/E_{22}^d{h}^3$ and $k_2=K_2{\mathit{\alpha }}^2/E_{22}^d{h}^3$ . Volume fractions (V_f ), initial temperature (T ₀) = 25°C, initial moisture C ₀ = 0%. T = (T ₀ + ΔT), C = (C ₀ + ΔC) where ΔT and ΔC are change in temperature and moisture in percentage, respectively. The following material properties are used for computation ²¹ :

$\begin{array}{l}{T}_0 = 25;C_0 = 0;\text{Δ}T = 00;\text{Δ}C = 0.0;V_f = 0.5;E_f = \text{23}0.0 \times \text{1e-9};\\ G_f = \text{9}.0 \times \text{1e-9};c_{fm} = 0;{\text{ρ}}_c = \text{1}.\text{5};V_f = 0.\text{2}0\text{3};{\text{α}}_f = -0.\text{54} \times \text{1e-6};\\ {\text{ρ}}_f = \text{175}0;c_{fm} = 0;v_m = 0.\text{34};{\text{α}}_m = \text{45} \times \text{1e-6};{\text{ρ}}_m = \text{12}00;\\ {\text{β}}_m = \text{2}.\text{68} \times \text{1e-3};{\text{β}}_f = 0;E_m = (\text{3}.\text{51} - 0.00\text{3} \times T - 0.\text{142} \times C)\times \text{1e-9};\\ G_m = E_m\left(\frac{2}{1+v_m}\right),E_{\text{1}0}=\left(V_f{E}_f + V_m{E}_m\right)\end{array}$

$\begin{array}{l}{\text{α}}_{11}=\frac{V_f{E}_f{\text{α}}_f+V_m{E}_m{\text{α}}_m}{V_{f}Ef+V_m{E}_m}\\ {\text{α}}_{22}=(1+{\upsilon }_f)V_f{\text{α}}_f+(1+{\upsilon }_m)V_m{\text{α}}_m-{\upsilon }_{12}{\text{α}}_{11}\\ {\text{β}}_{11}=\frac{V_f{E}_f{c}_{fm}{\text{β}}_f+V_m{E}_m{\text{β}}_m}{E_{11}(V_f{\text{ρ}}_f{c}_{fm}+V_m{\text{ρ}}_m)}\text{ρ}\end{array}$

$\begin{array}{cc}& {\mathit{\beta }}_{22}=\frac{V_f(1+{\mathit{\upsilon }}_f)c_{fm}{\mathit{\beta }}_f+V_m(1+{\mathit{\upsilon }}_m){\mathit{\beta }}_m}{(V_f{\mathit{\rho }}_f{c}_{fm}+V_m{\mathit{\rho }}_m)}\mathit{\rho }-{\mathit{\upsilon }}_{12}{\mathit{\beta }}_{11}\\ & \mathit{\rho }=V_f{\mathit{\rho }}_f+V_m{\mathit{\rho }}_m\\ & V_m\hspace{0.17em}\mathrm{+}\hspace{0.17em}{V}_f\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{1}\\ & E_{11}=V_f{E}_f+V_m{E}_m\\ & \frac{1}{E_{22}}=\frac{V_f}{E_f}+\frac{V_m}{E_m}-V_f{V}_m\frac{V_f^2\frac{E_m}{E_f}+\frac{E_f}{E_m}-2{\mathit{\upsilon }}_f{\mathit{\upsilon }}_m}{V_f{E}_f+V_m{E}_m}\\ & \frac{1}{G_{12}}=\frac{V_f}{E_f}+\frac{V_m}{G_m}\\ & {\mathit{\upsilon }}_{12}=V_f{\mathit{\upsilon }}_f+V_m{\mathit{\upsilon }}_m\end{array}$

$\begin{array}{cc}& E_{111}=-0.5\times 1\mathrm{e}\mathrm{-}3;E_{21}=-0.2\times 1\mathrm{e}\mathrm{-}3;\\ & G_{121}=-0.2\times 1\mathrm{e}\mathrm{-}3;G_{131}=-0.2\times 1\mathrm{e}\mathrm{-}3;\\ & G_{231}=-0.2\times 1\mathrm{e}\mathrm{-}3;{\mathit{\alpha }}_{11}=0.5\times 1\mathrm{e}\mathrm{-}3;\\ & {\mathit{\alpha }}_{21}=0.5\times 1\mathrm{e}\mathrm{-}3;{\mathit{\beta }}_{11}=0.5\times 1\mathrm{e}\mathrm{-}3;{\mathit{\beta }}_{21}=0.5\times 1\mathrm{e}\mathrm{-}3;\\ & E_1\left(TC\right)=E_{10}(1+E_{111}\left(T+C\right);E_2\left(TC\right)=E_{20}(1+E_{21}\left(T+C\right);\\ & G_{12}\left(TC\right)=G_{120}(1+G_{121}\left(T+C\right);G_{13}\left(TC\right)=G_{130}(1+G_{131}\left(T+C\right);\\ & G_{23}\left(TC\right)=G_{230}(1+G_{231}\left(T+C\right);{\mathit{\alpha }}_1\left(T\right)={\mathit{\alpha }}_{110}(1+{\mathit{\alpha }}_{11}T);\\ & {\mathit{\alpha }}_2\left(T\right)={\mathit{\alpha }}_{210}(1+{\mathit{\alpha }}_{21}T);{\mathit{\beta }}_1\left(C\right)={\mathit{\beta }}_1(1+{\mathit{\beta }}_{11}C);\\ & {\mathit{\beta }}_2\left(C\right)={\mathit{\beta }}_2(1+{\mathit{\beta }}_{21}C);\end{array}$

Temperature- and moisture-independent (TID) material property constants used for present study are

$E_{111}=0;E_{21}=0;G_{121}=0;G_{131}=0;G_{231}=0;{\text{α}}_{11}=0;{\text{α}}_{21}=0;{\text{β}}_{11}=0;{\text{β}}_{21}=0;$

Validation study for expected mean values

Table 1 shows the comparison of dimensionless linear fundamental frequency (ϖ _l ) for perfect (±45°)_2T laminated square plates, biaxial compression volume fractions (V _f  = 0.5, 0.6 and 0.7), plate thickness ratio (a/h = 20), aspect ratio (a/b = 1.0) where dimensionless length of plate (a) = 0.1, width of plate (b) = 0.1 with simple SSSS(S2) boundary support under environmental conditions. It is noticed that the present (HSDT) expected mean fundamental frequency of deterministic results are in good agreement and validated by the results of Huang and Zheng. ²⁰

Table 1.

Comparison of dimensionless linear fundamental frequency (ϖ_l) for perfect (±45°)_2T laminated square plates, plate thickness ratio (a/h = 20), aspect ratio (a/b = 1.0) where a = 0.1, b = 0.1.^a

Lay-up	Environmental conditions	Dimensionless linear fundamental frequency (ϖl) = ωa 2√(ρ0/E 22/h 2)
		Huang and Zheng 20			Present
		Vf  = 0.5	Vf  = 0.6	Vf  = 0.7	Vf  = 0.5	Vf  = 0.6	Vf  = 0.7
(±45°)2T	ΔT = 0°C, ΔC = 0%	12.786	13.727	14.678	12.534	13.715	14.955
	ΔT = 200°C, ΔC = 3%	9.173	10.075	10.729	9.163	10.080	10.684

^aInitial temperature T ₀ = 25°C, volume fractions (V_f ), initial moisture C ₀ = 0%. T = (T ₀ + ΔT), C = (C ₀ + ΔC) where ΔT and ΔC rise in temperature and moisture, respectively, simple support (S2) under environmental conditions and biaxial compression.

Validation study for random hygrothermal free vibration

The present results of random hygrothermal free vibration of laminated composite plate without foundation (k ₁ = 00, k ₂ = 00) and resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations obtained from present perturbation approach have been compared and validated with an independent MCS approach. Figure 4 plots the normalized SD, $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ ) of the hygrothermal free vibration versus the $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the random material property (E ₁₁) for all edges simply supported SSSS (S2), a/h = 20, V_f  = 0.6, angle-ply antisymmetric (±45°)_2T square laminated composite plate, ΔT = 100°C, ΔC = 0.010, with hygrothermomechanical properties varying from 0% to 20%. ³⁴ It is assumed that one of the material property (i.e. E ₁₁) changes at a time, keeping other as a deterministic with the mean value of the material properties. The present FOPT result obtained using first-order perturbation approach based on HSDT compared with independent MCS approach. For the MCS approach, the samples are generated using Mat Lab to fit the desired mean and SD. These samples used in response are solved repeatedly, adopting conventional eigenvalue procedure, to generate a sample of the hygrothermal free vibration. The number of samples used for MCS approach is 10,000 based on satisfactory convergence of the results. The normal distribution has been assumed for random number generations in MCS. However, the present perturbation approach used in the study does not have any limitation as regard to probability distribution of the system property. This is an advantage over the MCS. From the figure it is clear that close correlation is achieved between two results.

Figure 4.

Validation of present First Order Perturbation Technique (FOPT) results for hygrothermal free vibration with independent Monte Carlo simulation (MCS) results for only one material property E ₁₁ varying for all edges simply supported SSSS (S2), a/h = 20, volume fraction V _f = 0.6, angle-ply antisymmetric (±45°)_2T square laminated composite plate, ΔT = 100°C, ΔC = 0.010 with hygrothermomechanical properties varying from 0% to 20%.

Table 2 shows the effects of individual random variables b_i , {(i = 1–11) = 0.10} on the dimensionless mean (ϖ _l ) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plate without foundations (k ₁ = 00, k ₂ = 00), resting on Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 10), volume fraction (V_f  = 0.6), Simple support (S2) under environmental conditions. The dimensionless mean fundamental frequency (ϖ _l ) = ωa ² √(ρ₀/E ₂₂/h ²) is given within parentheses. It is observed that mean hygrothermal fundamental frequency increases drastically when plate is supported on Pasternak elastic foundation, the coefficient of variation for material property E ₁₁ is of significance compared to other random variables for with and without rise in temperature and moisture.

Table 2.

Effects of individual random variables b_i {(i = 1–11) = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundation (k ₁ = 00, k ₂ = 00), resting on Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 10), volume fraction (V_f  = 0.6).^a

(bi )	(k 1 = 00, k 2 = 00)		(k 1 = 100, k 2 = 00)		(k 1 = 100, k 2 = 10)
	COV, ${\mathit{\omega }}_1^2$		COV, ${\mathit{\omega }}_1^2$		COV, ${\mathit{\omega }}_1^2$
	ΔT = 0°C, ΔC = 0%	ΔT = 100°C, ΔC = 1%	ΔT = 0°C, ΔC = 0%	ΔT = 100°C, ΔC = 1%	ΔT = 0°C, ΔC = 0%	ΔT = 100°C, ΔC = 1%
E 11 (i = 1)	(10.9481) 4.73e-04	(10.9292) 4.79e-04	(14.8111) 2.60e-04	(14.5066) 2.73e-04	(20.3265) 8.52e-06	(19.7731) 1.49e-04
E 22 (i = 2)	0.0238	0.0014	0.0129	8.13e-04	2.18e-05	4.37e-04
G 12 (i = 3)	1.88e-04	2.05e-04	1.04e-04	1.17e-04	0.00099	6.41e-05
G 13 (i = 4)	3.13e-05	3.16e-05	1.69e-05	1.78e-05	2.51e-09	9.49e-06
G 23 (i = 5)	0.0157	0.0158	0.0085	0.0089	1.25e-06	0.0047
V 12 (i = 6)	0.0046	3.31e-04	0.0025	1.89e-04	5.63e-05	1.02e-04
α11 (i = 7)	2.81e-05	2.02e-04	1.53e-05	1.14e-04	1.12e-12	6.18e-05
α22 (i = 8)	3.07e-04	0.0014	1.68e-04	8.04e-04	1.35e-011	4.33e-04
β22 (i = 9)	1.18e-04	5.44e-04	6.44e-05	3.08e-04	5.19e-012	1.66e-04
k 1 (i = 10)	0	0	4.53e-02	4.32e-02	3.83e-010	2.32e-02
k 2 (i = 11)	0	0	0	0	6.55e-09	4.61e-02

^aSimple support (S2) under environmental conditions.

Effect of boundary conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ_l) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 20), volume fraction (V_f  = 0.6), under environmental conditions is shown in Tables 3–5. It is seen that the mean hygrothermal fundamental frequency is higher for clamp support compared to other support for with and without rise in temperature and moisture. The coefficient of variation of hygrothermal fundamental frequency is also varied for different combinations of random variables without foundations, Winkler and Pasternak elastic foundations. The mean hygrothermal fundamental frequency for Pasternak elastic foundations further increases and it is higher for clamp support compared to other support for with and without rise in temperature and moisture. The coefficient of variation in hygrothermal fundamental frequency is also varied for different combinations of random variables for Pasternak elastic foundations.

Table 3.

Effect of boundary conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundation (k ₁ = 00, k ₂ = 00), plate thickness ratio (a/h = 20) and volume fraction (V_f  = 0.6).

BCs	(k 1 = 00, k 2 = 00)
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7–8	i = 9	i = 10–11		i = 1…9	i = 7–8	i = 9	i = 10–11
SSSS S1	13.7737	0.0147	7.824e-04	2.98e-04	0	13.3740	0.0084	0.0038	0.0015	0
SSSS S2	13.7155	0.0148	7.890e-04	3.01e-04	0	13.3082	0.0084	0.0039	0.0015	0
CCCC	18.5889	0.0249	4.769e-04	1.82e-04	0	18.1450	0.0135	0.0023	8.80e-04	0
CSCS	15.9343	0.0203	6.226e-04	2.37e-04	0	15.6121	0.0113	0.0030	0.0011	0

Table 4.

Effect of boundary conditions and input random variables b_i {i = 1…9, 7−8, 9 and 10 = 0.10} on the dimensionless mean (ϖ_l) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 20) and volume fraction (V_f  = 0.6).

BCs	k 1 = 100, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7−8	i = 9	i = 10–11		i = 1…9	i = 7–8	i = 9	i = 10–11
SSSS S1	17.0126	0.0096	5.12e-04	1.95e-04	3.44e-02	16.4331	0.0056	0.0025	9.65e-04	3.37e-02
SSSS S2	16.9656	0.0097	5.15e-04	1.97e-04	3.46e-02	16.3797	0.0055	0.0026	9.70e-04	3.39e-02
CCCC	21.1016	0.0193	3.70e-04	1.41e-04	2.23e-02	20.5047	0.0106	0.0018	6.89e-04	2.16e-02
CSCS	18.8044	0.0145	4.47e-04	1.70e-04	2.81e-02	18.3007	0.0082	0.0022	8.29e-04	2.72e-02

Table 5.

Effect of boundary conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ_l) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 20) and volume fraction (V_f  = 0.6).

BCs	k 1 = 100, k 2 = 10
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7–8	i = 9	i = 10−11		i = 1…9	i = 7−8	i = 9	i = 10–11
SSSS S1	22.0548	0.0057	3.051e-04	1.16e-04	4.53e-02	21.2387	0.0033	0.0015	5.77e-04	4.49e-02
SSSS S2	22.0183	0.0057	3.061e-04	1.16e-04	4.55e-02	21.1932	0.0033	0.0015	5.79e-04	4.50e-02
CCCC	25.6879	0.0140	2.412e-04	9.21e-05	3.54e-02	24.8528	0.0078	0.0012	4.53e-04	3.47e-02
CSCS	23.6859	0.0098	2.757e-04	1.05e-04	4.06e-02	22.9260	0.0056	0.0014	5.17e-04	3.99e-02

Tables 6–8 show the effect of plate thickness ratios (a/h) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ_l) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, volume fraction (V_f  = 0.6), simple support (S2) under environmental conditions. It is observed that the mean hygrothermal fundamental frequency is higher for thin plates and decreases drastically for moderately thick plates resting on Winkler elastic foundation; however the coefficient of variations for different combinations of random variables is also varied with and without rise in temperature and moisture. It is interesting that mean hygrothermal fundamental frequency rises rapidly for thin plate and decreases drastically for moderately thick plates resting on Pasternak elastic foundation in similar environmental conditions. Table 7

Table 6.

Effect of plate thickness ratios (a/h) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without (k ₁ = 00, k ₂ = 00) elastic foundations, volume fraction (V_f  = 0.6).^a

a/h	k 1 = 00, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1… 9	i = 7–8	i = 9	i = 10–11		i = 1…9	i = 7–8	i = 9	i = 10–11
5	7.0981	0.0441	1.83e-04	7.02e-05	0	7.1150	0.0253	8.50e-04	3.23e-04	0
10	10.9481	0.0289	3.08e-04	1.18e-04	0	10.9292	0.0160	0.0014	5.44e-04	0
50	15.0880	0.0102	0.0041	0.0016	0	19.4276	0.0199	0.0131	0.0048	0
100	15.2812	0.0191	0.0159	0.0061	0	64.4922	0.1644	0.0370	0.0140	0

^aSimple support (S2) under environmental conditions.

Table 7.

Effect of plate thickness ratios (a/h) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations, volume fraction (V_f  = 0.6).^a

a/h	k 1 = 100, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7−8	i = 9	i = 10–11		i = 1…9	i = 7−8	i = 9	i = 10–11
5	10.1630	0.0991	3.20e-011	1.22e-011	3.58e-09	10.1627	0.0991	1.45e-010	5.53e-011	3.31e-09
10	14.8111	0.0157	1.68e-04	6.44e-05	4.53e-02	14.5066	0.0090	8.12e-04	3.08e-04	4.32e-02
50	18.0993	0.0071	0.0028	0.0011	3.05e-02	21.6488	0.0161	0.0105	0.0038	1.94e-02
100	18.2619	0.0134	0.0112	0.0043	2.99e-02	58.3662	0.1988	0.0443	0.0168	2.67e-03

^aSimple support (S2) under environmental conditions.

Table 8.

Effect of plate thickness ratios (a/h) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10}on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, volume fraction (V_f  = 0.6).^a

a/h	k 1 = 100, k 2 = 10
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7−8	i = 9	i = 10−11		i = 1…9	i = 7−8	i = 9	i = 10−11
5	10.1630	0.0991	2.07e-012	7.93e-013	3.73e-09	10.1627	0.0991	1.36e-011	5.20e-012	4.77e-09
10	20.3265	0.0991	1.35e-011	5.19e-012	6.56e-09	9.7731	0.0048	4.37e-04	1.66e-04	5.17e-02
50	22.9130	0.0044	0.0018	6.76e-04	4.21e-04	2.9737	0.8729	0.7960	0.3034	0.2910
100	23.0453	0.0084	0.0070	0.0027	4.16e-04	35.8314	0.4939	0.1065	0.0402	0.1400

^aSimple support (S2) under environmental conditions.

Effect of aspect ratio (a/b) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 30), volume fraction (V _f = 0.6), simple support (S2) under environmental conditions are shown in Tables 9–11. It is seen that under given environmental conditions and plate resting on Winkler elastic foundations, the mean hydrothermal fundamental frequency drastically increases when the aspect ratio is increased; whereas the coefficient of variation for different combinations of random variables is decreased. However, when the plate is resting on Pasternak elastic foundation, the mean and coefficient of variations of fundamental frequency further vary in similar environmental conditions.

Table 9.

Effect of aspect ratio (a/b) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10}on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without (k ₁ = 00, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 30), volume fraction (V_f  = 0.6).^a

(a/b)	(k 1 = 00, k 2 = 00)
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7–8	i = 9	i = 10–11		i = 1…9	i = 7–8	i = 9	i = 10–11
0.5	8.5313	0.0111	0.0046	0.0018	0	7.9384	0.0307	0.0248	0.0094	0
1.0	14.5520	0.0111	0.0016	6.03e-04	0	12.2977	0.0123	0.0102	0.0039	0
1.5	22.2741	0.0131	6.78e-04	2.57e-04	0	21.5942	0.0067	0.0034	0.0013	0
2.0	31.6174	0.0161	3.41e-04	1.27e-04	0	31.4234	0.0080	0.0016	5.95e-04	0

^aSimple support (S2) under environmental conditions.

Table 10.

Effect of aspect ratio (a/b) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 30), volume fraction (V_f  = 0.6).^a

a/b	k 1 = 100, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7–8	i = 9	i = 10–11		i = 1…9	i = 7–8	i = 9	i = 10–11
0.5	8.8896	0.0102	0.0042	4.91e-05	7.90e-03	8.2400	0.0285	0.0230	0.0087	8.39e-03
1.0	17.6523	0.0075	0.0011	4.10e-04	3.20e-02	15.5737	0.0076	0.0064	0.0024	3.76e-02
1.5	31.6417	0.0065	3.36e-04	1.27e-04	5.04e-02	30.4660	0.0034	0.0017	6.3e-04	4.97e-02
2.0	50.9353	0.0062	1.31e-04	4.91e-05	6.14e-02	49.4548	0.0032	6.65e-04	2.40e-04	5.96e-02

^aSimple support (S2) under environmental conditions.

Table 11.

Effect of aspect ratio (a/b) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 30) and volume fraction (V_f  = 0.6).^a

a/b	k 1 = 100, k 2 = 10
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1… 9	i = 7–8	i = 9	i = 10−11		i = 1… 9	i = 7−8	i = 9	i = 10−11
0.5	10.4812	0.0073	0.0030	0.0012	2.86e-02	10.1068	0.0190	0.0153	0.0058	4.51e-02
1.0	22.5574	0.0046	6.57e-04	2.51e-04	4.34e-02	20.5831	0.0044	0.0036	0.0014	4.78e-02
1.5	41.4938	0.0038	1.95e-04	7.41e-05	5.11e-02	39.8726	0.0020	9.85e-04	3.68e-04	5.07e-02
2.0	67.5518	0.0035	7.47e-05	2.79e-05	5.55e-02	64.9705	0.0028	4.28e-04	1.56e-04	5.63e-02

^aSimple support (S2) under environmental conditions.

Tables 12–14 show the effect of lay-up and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect laminated composite square plates without foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 40), volume fraction V_f  = 0.6), simple support (S2) under environmental conditions. It is observed that mean hygrothermal fundamental frequency drastically increases for cross-ply antisymmetric plate, whereas there is a sharp decrease of other lamina lay-up when the plate is resting on Winkler elastic foundations under given environmental conditions; however the coefficient of variation of hygrothermal fundamental frequency also vary for different combinations of random variables. The mean and coefficient of variation of hygrothermal fundamental frequency further vary for different combinations of random variables and the plate is resting on Pasternak elastic foundation.

Table 12.

Effect lay-up and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect laminated composite square plates without (k ₁ = 00, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 40) and volume fraction (V_f = 0.6).^a

Lay-up	k 1 = 00, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1… 9	i = 7−8	i = 9	i = 10−11		i = 1… 9	i = 7−8	i = 9	i = 10−11
(0°/90°)s	11.7725	0.0116	0.0043	8.19e-04	0	22.4175	0.0656	0.0099	0.0017	0
(0°/90°)2T	10.8800	0.0145	0.0050	0.0016	0	4.6524	0.1372	0.1269	0.0410	0
(±45°)s	13.9617	0.0150	0.0032	0.0012	0	1.6997	0. 1119	0.1014	0.0385	0
(±45°)2T	14.9053	0.0101	0.0027	0.0010	0	5.0206	0.1194	0.1091	0.0414	0

^aSimple support (S2) under environmental conditions.

Table 13.

Effect lay-up and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 40), volume fraction (V_f  = 0.6) and simple support (S2) under environmental conditions.

Lay-up	k 1 = 100, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1… 9	i = 7−8	i = 9	i = 10–11		i = 1… 9	i = 7−8	i = 9	i = 10 −11
(0°/90°)s	15.4433	0.0067	0.0025	4.76e-04	4.18e-02	22.6010	0.0639	0.0097	0.0017	1.78e-02
(0°/90°) T	14.7737	0.0079	0.0027	8.84e-04	4.57e-02	10.4556	0.0272	0.0251	0.0081	8.35e-02
(±45°)s	17.1707	0.0099	0.0021	7.96e-04	3.38e-02	9.7082	0.0343	0.0311	0.0118	9.69e-02
(±45°)2T	17.9464	0.0070	0.0018	7.05e-04	3.10e-02	10.7967	0.0258	0.0236	0.0090	7.83e-02

Table 14.

Effect Lay-up and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect laminated composite square plates resting on Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 40), volume fraction (V_f  = 0.6) and simple support (S2) under environmental conditions.

Lay-up	k 1 = 100, k 2 = 10
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7−8	i = 9	i = 10–11		i = 1… 9	i = 7−8	i = 9	i = 10−11
(0°/90°)s	20.8758	0.0037	0.0014	2.60e-04	5.07e-02	16.2538	0.8182	0.6627	0.1265	0.3009
(0°/90°)2T	20.3847	0.0041	0.0014	4.64e-04	5.32e-02	16.5099	0.0109	0.0101	0.0033	7.44e-02
(±45°)s	22.3055	0.0056	0.0012	4.66e-04	4.52e-02	16.8326	0.0110	0.0100	0.0038	7.34e-02
(±45°)2T	22.7910	0.0043	0.0011	4.37e-04	4.25e-02	17.2585	0.0101	0.0092	0.0035	6.81e-02

Effect of volume fraction (V_f ) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10}on the dimensionless mean (ϖ _l ) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 40), volume fraction (V_f  = 0.6), simple support (S2) under environmental conditions is shown in Tables 15–17. It is noticed that on varying the V_f , the mean and coefficient of variation of hygrothermal fundamental frequency change when the plate is resting on Winkler elastic foundation under given environmental conditions with different combinations of input random variables; however these values further increase when the plate is resting on Pasternak elastic foundation under similar environmental conditions and different combinations of input random variables.

Table 15.

Effect of volume fraction (V _f) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without (k ₁ = 00, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 40), volume fraction (V_f  = 0.6).^a

V f	k 1 = 00, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9)	i = 7−8	i = 9	i = 10−11		i = 1…9	i = 7−8	i = 9	i = 10−11
0.50	13.6197	0.0041	0.0033	9.61e-04	0	3.3628	0.2609	0.2464	0.0724	0
0.55	14.2662	0.0038	0.0029	9.83e-04	0	4.4100	0.1519	0.1417	0.0473	0
0.60	14.9053	0.0069	0.0027	0.0010	0	5.0206	0.1190	0.1091	0.0414	0
0.65	15.5449	0.0034	0.0025	0.0011	0	5.2438	0.1133	0.1007	0.0437	0
0.70	16.1954	0.0033	0.0023	0.0012	0	4.9523	0.1314	0.1146	0.0575	0

^aSimple support (S2) under environmental conditions.

Table 16.

Effect of volume fraction (V_f ) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 40), V_f  = 0.6.^a

V f	k 1 = 100, k 2 = 00
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9	i = 7−8	i = 9	i = 10−11		i = 1… 9	i = 7−8	i = 9	i = 10−11
0.50	16.3105	0.0028	0.0023	6.70e-04	3.02e-02	9.2147	0.0347	0.0328	0.0096	8.66e-02
0.55	17.1088	0.0027	0.0020	6.84e-04	3.04e-02	10.0488	0.0293	0.0273	0.0091	8.07e-02
0.60	17.9464	0.0047	0.0018	7.05e-04	3.10e-02	10.7967	0.0257	0.0236	0.0090	7.83e-02
0.65	18.8463	0.0023	0.0017	7.36e-04	3.19e-02	11.4620	0.0237	0.0211	0.0092	7.90e-02
0.70	19.8433	0.0022	0.0015	7.79e-04	3.33e-02	12.0367	0.0222	0.0194	0.0097	8.30e-02

^aSimple support (S2) under environmental conditions.

Table 17.

Effect of volume fraction (V_f ) and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 40), V_f  = 0.6.^a

Vf	k 1 = 100, k 2 = 10
	ΔT = 0°C, ΔC = 0%					ΔT = 100°C, ΔC = 1%
	Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$				Mean (ϖ l )	COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1…9)	i = 7−8	i = 9	i = 10–11		i = 1… 9	i = 7−8	i = 9	i = 10−11
0.50	20.6186	0.0018	0.0014	4.19e-04	4.19e-02	15.1971	0.0128	0.0121	0.0035	7.08e-02
0.55	21.6540	0.0017	0.0013	4.27e-04	4.21e-02	16.2097	0.0112	0.0105	0.0035	6.89e-02
0.60	22.7910	0.0043	0.0011	4.37e-04	4.25e-02	17.2585	0.0101	0.0092	0.0035	6.81e-02
0.65	24.0721	0.0015	0.0010	4.51e-04	4.33e-02	18.3712	0.0092	0.0082	0.0036	6.83e-02
0.70	25.5627	0.0013	9.18e-04	4.69e-04	4.45e-02	19.5887	0.0084	0.0073	0.0037	6.96e-02

^aSimple support (S2) under environmental conditions.

Tables 18 and 19 show the effects of environmental conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and $\mathrm{C}\mathrm{O}\mathrm{V}({\mathit{\omega }}_1^2)$ of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without elastic foundations (k ₁ = 00, k ₂ = 00), with Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 60), volume fraction (V_f  = 0.6) and simple support (S2). It is noticed that mean and coefficient of variations in hygrothermal fundamental frequency vary on changing environmental conditions for plates resting on Winkler elastic foundation with input random variables. The mean hygrothermal fundamental frequency and coefficient of variations value further increase for plates resting on Pasternak elastic foundation under similar environmental conditions with input random variables.

Table 18.

Effects of environmental conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates without (k ₁ = 00, k ₂ = 00) elastic foundations, plate thickness ratio (a/h = 60) and volume fraction (V_f  = 0.6).^a

Environmental conditions	Mean (ϖ l )	(k 1 = 00, k 2 = 00)
		COV, ${\mathit{\omega }}_1^2$
		bi
		i = 1...9)	i = 7−8	i = 9	i = 10−11
ΔT = 0°C, ΔC = 0%	15.1930	0.0109	0.0058	0.0022	0
ΔT = 100°C, ΔC = 1%	26.5638	0.1356	0.0899	0.0343	0
ΔT = 200°C, ΔC = 3%	73.3668	1.5249	0.0789	0.0297	0
ΔT = 300°C, ΔC = 5%	55.8570	1.4326	0.0371	0.0078	0

^aSimple support (S2).

Table 19.

Effects of environmental conditions and input random variables b_i {i = 1…9, 7–8, 9 and 10 = 0.10} on the dimensionless mean (ϖ _l ) and coefficient of variation (ω₁ ²) of the fundamental frequency of perfect angle-ply (±45°)_2T laminated composite square plates resting on Winkler (k ₁ = 100, k ₂ = 00) and Pasternak (k ₁ = 100, k ₂ = 10) elastic foundations, plate thickness ratio (a/h = 60) and volume fraction (V_f  = 0.6).

Environmental conditions.	Mean (ϖ l )	(k 1 = 100, k 2 = 00)				Mean (ϖ l )	(k 1 = 100, k 2 = 10)
		COV, ${\mathit{\omega }}_1^2$					COV, ${\mathit{\omega }}_1^2$
		bi					bi
		i = 1… 9	i = 7−8	i = 9	i = 10−11		i = 1…9	i = 7–8	i = 9	i = 10–11
ΔT = 0°C, ΔC = 0%	18.1874	0.0076	0.0040	0.0015	3.02e-02	22.9836	0.0047	0.0025	9.68e-04	4.18e-02
ΔT = 100°C, ΔC = 1%	28.2262	0.1201	0.0796	0.0304	1.14e-02	11.3095	0.0862	0.0654	0.0241	0.3900
ΔT = 200°C, ΔC = 3%	73.9259	1.5019	0.0777	0.0292	1.50e-03	104.7245	0.2432	0.0122	0.0044	3.39e-02
ΔT = 300°C, ΔC = 5%	56.5132	1.3996	0.0362	0.0076	2.30e-03	84.3942	0.6300	0.0163	0.0034	5.53e-02

Conclusions

A C ⁰ FEM in conjunction with FOPT is employed to compute the mean and COV of the hygrothermal fundamental frequency of the laminated composite plate with random change in all input variables, aspect ratios and Winkler and Pasternak elastic foundations. Assumptions are made for uniform distribution of temperature and moisture over entire surface and through the thickness of laminated composite plate. The following main things are concluded from the limited present investigation.

The first-order perturbation technique gives acceptable results for the range of COV taken in the study. It is observed that COV for E ₁₁ and E ₂₂ is of significance compared to all other random input variables in case plates resting on Winkler and Pasternak foundations.

The characteristics of hygrothermal fundamental frequency of laminated composite plate is significantly influenced by various support conditions, plate thickness ratios, aspect ratios, volume fraction, elastic foundations, temperature and moisture changes. The mean and COV of fundamental frequency plate is significant when the plate is subjected to hygrothermoelastic material properties.

The clamp-supported plate vibrates at slightly higher temperature and moisture compared to other supports when resting on Pasternak elastic foundation. Vibration is less dominant in moderately thick plate compared to thin plate. It is also noticed that with Pasternak foundation support the mean fundamental frequency is higher and COV of fundamental frequency is lower compared to Winkler foundation.

The sensitivity of hygrothermal fundamental frequency, COV due to variation in material properties, is dependent on thickness ratio and boundary conditions of the laminate. The study is meaningful for the aerospace applications where such environmental conditions may occur besides other factors. The importance of elastic foundations on which laminated composite plate is resting is quite important from the design point of view.