Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading

Composite Structures 93 (2011) 1187–1200

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Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading Achchhe Lal a,⇑, B.N. Singh b, Sushil Kale a a b

Department of Mechanical Engineering, S.V. National Institute of Technology, Surat 395 007, India Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721 302, India

a r t i c l e

i n f o

Article history: Available online 3 December 2010 Keywords: Laminated composite cylindrical shell panel Random system properties Second order statistics Post buckling

a b s t r a c t In this paper, the effect of random system properties on the post buckling load of geometrically nonlinear laminated composite cylindrical shell panel subjected to hygrothermomechanical loading is investigated. System parameters are assumed as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration using micromechanical approach. A direct iterative based C0 nonlinear finite element method in conjunction with first-order perturbation technique proposed by present author for the plate is extended for shell panel subjected to hygrothermomechanical loading to compute the second-order statistics (mean and variances) of laminated composite cylindrical shell panel. The effect of random system properties, plate geometry, stacking sequences, support conditions, fiber volume fractions and temperature and moisture distributions on hygrothermomechanical post-buckling load of the laminated cylindrical shell panel are presented. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforced plastic composites are being increasingly used in the aerospace and other allied industry because of their properties, namely high specific strength, high specific stiffness and low specific density. They are subjected to environmental conditions during their service life. They operate in a variety of thermal and moisture environments that may have a pronounced impact on their performance during their service life. These hygrothermal effects are a result of the temperature and moisture content variations and are related to the difference in the thermal and hygro properties of the constituents. Accurate analysis of the hygrothermal effects to find the nature and extent of their deleterious effect upon the performance is an important component of the overall design and analysis. In certain cases, the hygrothermal loads turn out to be the primary one and control the design. Hygrothermal gradients are built across wall thickness. Due to the boundary constraints, compressive stresses acting on the edges of the component are induced, which may cause buckling, especially in thin walled members. For this reason, accurate prediction of the post buckling response of the fiber composite laminated structures un⇑ Corresponding author. Tel.: +91 0261 2201572; fax: +91 0261 2228394. E-mail addresses: [email protected] (A. Lal), [email protected]. ernet.in (B.N. Singh). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.11.005

der different environmental conditions is required for efficient and optimal use of the materials. The response of a structure under different hygrothermomechanical loading conditions taking separately or combined depends on its system properties. Uncertainty in the system properties, which is inherent leads to uncertainties in the response behaviour of the structure. For accurate analysis of structural behaviour, the random variation in the system properties are incorporated in the analysis so that the predicted response may not differ significantly from the observed values leaving the structures unsafe. In available literature much of the published works on hygrothermal initial and/or post buckling analysis with temperature independent and dependent thermo-elastic material properties are based on deterministic analysis. Notably among them are Whitney and Ashton [1], Flaggs and Vinson [2], Lee et al. [3], SaiRam and Sinha [4], Shen [5] and Patel et al. [6], Chen and Chen [7], Shen [8–11], Srikanth and Kumar [12], Shariyat [13], and Pandey et al. [14]. All the above mentioned literature are based on assumption of complete determinacy of structural parameters, which give only mean response and misses the deviation caused by inherent random system properties. Due to dependency of large number of parameters in manufacturing and fabrication process of composite components, the system properties are generally random in nature. In the presence of inherent random system properties, they may have affected the fundamental characteristics of the

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Nomenclature a, b bi E11, E22 G12, G13, h KL, KNL K(G) NE, N Nn1 Nn2 NN /i

shell length and breadth basic random system properties longitudinal and Transverse elastic moduli G23 shear moduli thickness of the shell linear and nonlinear bending stiffness matrix thermal geometric stiffness matrix number of elements, number of layers in the laminated plate N n1 n2 in-plane thermal post-buckling loads number of nodes per element shape function of ith node

structural response like stability and stress analysis. The allowable stresses for conventional material can be expected to very closed to their mean values as a fewer parameters are involve in manufacturing/fabrication/processing as compared to composites. However in the case of composite, in spite of effective quality control in manufacturing/fabrication/processing process as we normally do in sensitive applications like aerospace, etc., the allowable stress will normally have to be much less than respective mean values due to wide scattered data. A similar behaviour is expected in the case of basic characteristics of material like elastic moduli, etc. Finally all these would negate the basic idea of weight optimisation, which is one of the important parts of design consideration. Therefore for sensitive applications and higher reliability of structures for factor of safety and weight optimization point of view, the modelling and analysis of structures can be done only by assuming the system properties as random. This can be done only using probabilistic/stochastic analysis. Probabilistic analysis provides a tool for incorporating structural modelling uncertainties in the analysis of structural response like buckling response. Even though the risk and failures can be evaluated accurately by adopting stochastic approach, more over this analysis is more pertinent to composite as compared to conventional materials. A limited number literature is available on the analysis of the structures with random system properties. Nakagiri et al [15] studied all edges simply supported of a laminate with stochastic finite element method taking fiber orientation, layer thickness and number of layers as random variables. Englested and Reddy [16] contributed metal matrix composites based on probabilistic micro mechanics nonlinear analysis. Singh et al. [17] studied the buckling of composite cylindrical panels with uncertain material properties using probabilistic approach. Yang and Liew [18] investigated the effect of random system properties on the elastic buckling of functionally graded material rectangular plates resting on elastic foundation subjected to uniform in-plane edge compression. They have used first-order shear deformation plate theory and a mean-centered first-order perturbation procedure is used to examine the stochastic characteristics of the buckling load. Effect of random system properties on initial buckling of composite plates resting on elastic foundation using stochastic finite element method studied using higher order shear deformation theory (HSDT) by Lal et al. [19]. Post buckling of laminated composite plate on elastic foundation with random system properties based on HSDT in conjunction with first order perturbation technique (FOPT) is studied by Singh et al. [20]. Chen et al. [21] outlined the probabilistic method to evaluate the effects of uncertainties in geometric and material properties. Hygrothermal effects on the buckling of laminated composite plates with random geometric and temperature independent material properties by Singh and Verma [22]. Zhang and Ellinwood [23] have

q

vector of unknown displacements, displacement vector of eth element u, v, w displacements of a point on the mid plane of plate  v wÞ  ðu displacement of a point along (n1 n2 f) {}, {} stress vector, strain vector /1 and /2 rotations of normal to mid plane about the n1 and n2 and axis respectively (n1 n2 f) Cartesian coordinates RVs random variables DT and DC difference in temperatures and moisture concentration

analyzed the effect of material properties on the elastic stability of structural members and frames using stochastic finite element method. They used mean cantered FOPT using FEM to evaluate the buckling load. Onkar et al. [24] used a generalized layer wise stochastic finite element formulation for the buckling analysis of homogeneous and laminated plates with random material properties using FOPT in conjunction with HSDT. Lal et al. [25] evaluated the second order statistics of fundamental frequency of laminated composite plates in thermal environments. They used C0 finite element method in conjunction with first deformation perturbation technique using HSDT to examine the effect of temperature loading on the free vibration characteristics of plate. In the same direction, Pandit et al. [26] modelled the stochastic approach to examine the fundamental frequency of soft core sandwich plates with random material properties using the C0 FEM combined with FOPT and layer wise theory. However, no work dealing with hygrothermomechanical postbuckling analysis of the laminated composite shell panel with random system properties having temperature independent and dependent thermo-elastic material properties is reported in the literature to the best of the authors’ knowledge. Contribution of this paper is to compute the second-order statics of post-buckling load of laminated composite shell panel subjected to hygrothermomechanical and thermomechanical loading in the presence of small random variation in the system properties using HSDT with von-Karman nonlinearity. A direct iterative based stochastic finite element method (DISFEM) is used to solve the random eigen solution. Typical numerical results are presented for hygrothermomechanical and thermomechanical post-buckling load with different geometrical parameters, support condition, moisture and temperature variation, etc. It is found that small variation of random system properties variation of laminate composite shell panel significantly affect the post-buckling load. The present probabilistic approach would be validating only small random variation compared to their mean value, which is usually satisfied by most of the engineering applications. 2. Formulation 2.1. Displacement field model A typical composite laminated cylindrical shell panel of radius R, thickness h and subjected to in-plane loads is shown in Fig. 1 Coordinates used for the study are n1 axial, n2 circumferential and f radial surface metrics. It is assumed that the panels are composed of thin orthotropic layers and are perfectly bonded together. The displacement field model based on the HSDT after satisfying transverse shear conditions on the top and bottom of the plate can be expressed as [27]

A. Lal et al. / Composite Structures 93 (2011) 1187–1200

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Fig. 1. Geometry of laminated composite cylindrical shell.

 ¼ u þ f1 ðfÞ/1 þ f2 ðfÞh1 u   f v ¼ 1 þ v þ f2 ðfÞ/2 þ f2 ðfÞh2 R  ¼w w

ð1Þ

where h1 ¼ w;n1 ; h2 ¼ w;n2 ð; Þ denotes partial differential, f1(f) = C1f  C2f3 and f2(f) = C4f3 with C1, and C4 are the constants and defined as C1 = 1; C2 = C4 = 4/3h2. Based on the displacement field as given in Eq. (1), a C1 continuity is minimum requirement. A choice of C1 continuity may be reduced to C0 by assuming derivatives as independent field variables, which would definitely help in achieving the computation efficiency. But, in this process artificial constraints are imposed, which should be enforced variationally through a penalty approach apart from increase in the field variables. However, a comparison presented in the literature [28] indicates that without enforcing these constraints, the accurate results could be obtained. The displacement vector for the modified model can be written as

fqg ¼ ðu

v

w h2 h1 /2 /1 ÞT

ð2Þ

where feLij g is the linear strain vector [14], feNL ij g is the von-Karman based geometric nonlinear strain vector and the hygrothermal strain vector feTH ij g is defined as

8 9 an1 DT þ bn1 DC > > > > > > > > > > > < an2 DT þ bn2 DC > = TH feij g ¼ an1 n2 DT þ bn1 n2 DC > > > > > > 0 > > > > > > : ; 0

ð4Þ

Note that suffixes 1, 2 and 3 denote components along n1, n2 and f directions, respectively. Here, an1 ; an2 an1 n2 and bn1, bn2, bn1n2 are coefficients of thermal expansion along the n1, n2 and n1n2 directions respectively, which can be obtained from the thermal expansion and contraction coefficients due to temperature and moisture in the longitudinal (a1) and transverse (a2) directions of the fibers using transformation matrix and DT and DC are the change in temperature and moisture in the shell panel subjected with uniform temperature rise and moisture rise can be defined as DT = T  T0 and DC = C  C0 [22]

2.2. Strain displacement relations 2.3. Stress–strain relations For a structure considered here, the relevant strain vectors consisting of mid plane deformation, rotation of normal and higher order terms and hygrothermal strains with the displacement of kth layer are expressed as TH feij g ¼ feLij g þ feNL ij g  feij g

ð3Þ

The constitutive law of thermo-elasticity for material under consideration relates the stresses with strains in a in-plane stress state for the kth orthotropic lamina of a laminate consisting of N layers, having fibres oriented in any arbitrary orientation with respect to the reference axes

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rij ¼ C ijkl eij

ð5Þ

where Cijkl is reduced elastic stiffness matrices. 2.4. Micromechanical approach The material properties of the fiber composite at different moisture concentration and temperature are evaluated using micromechanical model. The degradation of the fiber composite material properties is estimated by degrading the matrix property only. The matrix mechanical property retention ratio is expressed as [29]

 Fm ¼

T gw  T T g0  T 0

12 ð6aÞ

2.5. Total potential energy of the nonlinear state of the system Structural thermal post buckling is essentially a nonlinear phenomenon and this can be modelled by a nonlinear strain displacement relation in von-Karman sense by assuming that pre buckling deformations are large and applied loads are proportional to a single load parameter. Let the shell panel system properties be subjected to some reference in-plane thermal loading qref i . The shell panel is assumed to be nonlinear elastic with a stochastic elasticity tensor field Cijkl. The total potential energy corresponding to the nonlinear state of the system for uncertain stiffness can be written as [33]

Z Z Y ref 1 ref ref C ijkl eref qref ðui Þ ¼ ij ekl dX  i ui dC1 2 X C1

ði; j; k; l ¼ 1; 2; 3Þ ð8Þ

where T = T0 + DT and T is the temperature at which material property is to be predicted; T0 is the reference temperature, Tgw and Tgo are glass transition temperature for wet and reference dry conditions, respectively. The glass transition temperature for wet material is determined as [30]

T gw ¼ ð0:005C 2  0:10C þ 1:0ÞT g0

ð6bÞ

where C = C0 is the weight percent of moisture in the matrix material. C0 = 0 weight percent and C is the increase in moisture content. The elastic constants are obtained from the following equations [14,31]:

d

E11 ¼ Ef 1 V f þ F m Em V m ;  pffiffiffiffiffi E22 ¼ 1  V f F m Em þ

G12

pffiffiffiffiffi F m Em V f  pffiffiffiffiffi 1  V f 1  F mE Em f2 pffiffiffiffiffi  pffiffiffiffiffi F m Gm V f  ; ¼ 1  V f F m Gm þ pffiffiffiffiffi 1  V f 1  FGm Gm

ð6cÞ

f 12

ð6dÞ

where ‘‘V’’ is the volume fraction and subscripts ‘‘f’’ and ‘‘m’’ are used for fiber and matrix, respectively. The effect of increased temperature and moisture concentration on the coefficients of thermal expansion (a) and hygroscopic expansion (b) is opposite from the corresponding effect on strength and stiffness. The matrix hygrothermal property retention ratio is approximated as

1 Fh ¼ Fm

ð6eÞ

Coefficients of thermal expansion and hygroscopic expansion are expressed as [31,32]

b11

Ef 1 V f af 1 þ F m Em V m F h am ; Ef 1 V f þ F m Em V m

a22 ¼ ð1 þ mf 12 ÞV f af 2

þ ð1 þ mm ÞV m F h am  m12 a11 Ef 1 V f bf 1 þ F m Em V m F h bm ¼ ; b22 ¼ ð1 þ mf 12 ÞV f bf 2 Ef 1 V f þ F m Em V m þ ð1 þ mm ÞV m F h bm  m12 b11

ð6fÞ

ref qref i dui dC1 ¼ 0

ð9Þ

d

Z Y p ðui Þ ¼

X

rcrij epij dX þ

1 2

Z X

rpij epij dX 

Z C1

p qcr i ui dC1

ði; j ¼ 1; 2; 3Þ ð10Þ

where r is the current thermal linear state of stress (pre-buckled) ref due to the critical load qcr and is given by rcr ij ¼ kcr rij . cr ij

2.6. Discretization In the present study a C0 nine-nodes isoparametric finite element with 7 degree of freedoms (DOFs) per node is employed. For this type of element, the displacement vector and the element geometry are expressed as NN X i¼1

ui qi ; n1 ¼

NN X i¼1

ui n1i ; and n2 ¼

NN X

ui n2i

ð11Þ

i¼1

ð6gÞ

eij ¼ Hmn eKl

ð7Þ 

C1

Having solved Eq. (9), for linear displacements uref due to the referi ence thermal load, we look for load parameters k such that the system becomes unstable. Among all k the critical or minimum value of the load parameter is denoted by kcr (with total load qcr = kcrqref). From this equilibrium position, we perturb the system by an amount upi (with strain epij ) such that the system goes to new equilibrium position. The total potential energy due to perturbation of the system can be written as

q¼

The strain tensor at any point ei , may be expressed in terms of strain vector ekl defined in terms of unknowns at the reference plane as [26]

where

Z Z Y ref ref ðui Þ ¼ C ijkl eref d e d X  ij kl X

m12 ¼ mf 12 V f þ mm V m

a11 ¼

where X denotes the unreformed configuration of the shell panel and its boundary is denoted by C = C0 [ C1, C0 denotes the Dirichlet part and C1 denotes the Neumann part of the lateral boundary of the plate. eref ij denotes the nonlinear strain tensor defined by Eq. (3) and qref is the boundary traction. In the present analysis, the referi ref ref ence loads qref and qref 1 ¼ q 2 ¼ q3 ¼ 0 are taken. The exact solution minimizes P on the set of all kinematically admissible functions denoted by V, i.e. u 2 V such that, V = {u 2 H1(X): M(u)} = 0 on where M( ) is an operator defined by the edge condition. This yield



eKl ¼ e01 e02 e06 k01 k02 k06 k21 k22 k26 e04 e05 k04 k05 ; ðm ¼ 1; 2; 3; . . . ;

6 and n ¼ 1; 2; . . . ; 33Þ and Hmn is the function of Z and unit step functions and eKl is reference plane strain tensor.

where ui is the interpolation function for the ith node, q is the vector of unknown displacements for the ith node, NN is the number of nodes per element and n1i and n2i are Cartesian Coordinate of the ith node. The total strain energy (P) of laminates for whole domain can be expressed using Eq. (10) and the strain–displacement relation [20] as

Y p ðGÞ NL2 NL3 ðui Þ ¼ qi K Lij qj þ qi K NL1 ij qj þ qi K ij qj þ qi K ij qj  kcr qi K ij qj

ð12Þ

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2.7. Governing equation Now find kcr and u 2 V, u R 0 such that yields [20]

Q

is minimum. This

  ðGÞ K Lij þ K NL qkj ¼ 0 ij  kcr K ij

ð13Þ

After simplification of Eq. (13) we get the generalised eigen value problem

  ðGÞ K ij  kcr K ij qkj ¼ 0

ð14Þ

1 NL1 where, K ij ¼ K Lij þ K NL with K NL þ K NL2 þ 12 K NL3 here ij ij ¼ 2 K ij ij ij ðGÞ NL2 NL3 k K Lij ; K NL1 ; K ; K ; K ; q are linear, nonlinear, geometric stiffij ij ij j ij ness matrices and displacement vector, respectively. The shell panel stiffness matrix Kij and geometric stiffness maðGÞ trix K ij are random in nature, being dependent on the system geometric and thermo-elastic properties. Therefore the eigenvalue and eigenvectors also become random. Eq. (14) is the governing equation for the thermal post buckling analysis of laminated composite shell panel. In deterministic environment the solution of this equation can be obtained using standard solution procedure i.e., direct iterative method or reputation method subspace iteration method, ðGÞ etc. However in random environment, the matrices [Kij] and ½K ij  are random in nature and therefore eigen value and eigen vector are also random. It is therefore not possible to obtain post-buckling load statistics from Eq. (14). To achieve this, a probabilistic approach, i.e. direct iterative based C0 nonlinear finite element in conjunction with FOPT proposed earlier by the first two authors [20] is extended for this problem.

solutions using the first order perturbation technique as presented in the next section. (iv) Steps (ii) and (iii) are repeated by replacing fqLj g by fqNL j g in the step (ii) to obtain the converged mean and standard deviation of the nonlinear critical buckling load knl to a prescribed accuracy (103) (v) Steps (i)–(iv) are repeated for various value of C. 3.2. Solution: perturbation technique In the present analysis, the lamina material properties, thermal expansion coefficients and the geometric properties are treated as independent random variables (RVs). The overview of detailed stochastic analysis procedure is given in Fig. 2. In general, without any loss of generality any arbitrary random variable can be represented as the sum of its mean and zero mean random part, denoted by superscripts ‘d’ and ‘r’, respectively [34]

K ij ¼ K dij þ K ij;r

ðGÞ

ðGÞd

K ij ¼ K ij

ðGÞ

þ K ij ;r

ki ¼ kdi þ ki;r

qj ¼ qdj þ qrj ð15Þ

K dij

ðGÞd K ij

and are the mean elastic stiffness and the geometric matriðGÞ ces of the structure respectively. Kij,r and K ij ;r are the first order partial derivative of elastic stiffness matrix and geometric stiffness matrix of the structure respectively. By substituting Eq. (15) in Eq. (14) and expanding the random parts in Taylor’s series keeping the first order terms and neglecting the second and higher order terms, the following system of equations are obtained: Zeroth-order

3. Solution approach



3.1. A DISFEM for hygro-thermal post buckling problems The nonlinear eigenvalue problem as given in Eq. (14) is solved by employing a direct iterative method in conjunction with the mean cantered first order perturbation technique assuming that the random changes in eigenvector during iterations does not affect the nonlinear stiffness matrices with the following steps. (i) By setting amplitude to zero, the random linear eigenvalue ðGÞ problem ½K Lij qj ¼ kK j  is obtained from the Eq. (14) by assuming that the system vibrates in its principal mode. Then the random linear eigen value 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 to obtain the linear critical load parameters kl and the linear eigen vector {ql}. The first order perturbation equation is used to obtain the standard deviation of the thermal post buckling which is presented in next subsection of perturbation technique. (ii) For a specified maximum deflection C at a centre of the shell panel, the linear normalized eigen vector is scaled up by C times, so that resultant vector will have a displacement C at the maximum deflection point. (iii) Using the scale-up eigenvector, the nonlinear terms in the stiffness matrix ½K NL ij  can be obtained. The problem may now be treated as a linear eigen value problems with a new updated stiffness matrix. The random eigen value problem can again be broken up into zeroth and first order equation using perturbation technique. The deterministic zeroth order can be used to obtain nonlinear critical load kNL and eigenvector {qNL} and the random first order equations can be used to obtain the standard deviation (SD) of the eigen

ðGÞd

K dij þ kdk K ij

 qdj ¼ 0

ði; j; k ¼ 1; 2; 3; ::; n : no sum ov er kÞ

ð16Þ

First order



ðGÞd

K dij þ kdk K ij

   ðGÞ ðGÞd qj;r þ K ij;r þ kdk K ij ;r þ kk;r K ij qj

¼ 0 ði; j; k ¼ 1; 2; 3; . . . ; n : no sum ov er k r ¼ 1; 2; . . . :RÞ

ð17Þ

d the kth mean eigen vector comprised of  qj represents  U dlm V dlm W dlm components. Similarly qdj;r denotes the first order partial derivative of the kth eigen vector with ðU lm;r V lm;r W lm;r Þ components. It may be noted again that kdcr is the minimum among all kdk .

3.3. Mean buckling analysis The zeroth order equation is used to obtain the mean buckling load. The computation of buckling load and mode shapes under initial stress condition requires the following steps: A linear elastostatic problem is solved first for the reference loading qref with given constraints for the shell panel. The linear solution is used for computing the initial stress tensor rref ij . The stress tensor rref ij at each integration point is used to compute the geometric matrix. After the geometric matrix is available, the system stiffness matrix is modified by geometric matrix to solve the eigen value problem posed by Eq. (16). This is generalizes eigen value problem ðGÞd where K dij and K ij are symmetric matrices and are generally found ðGÞd to be positive definite. In some cases K ij can be positive semi-definite which can overcome by shift invert transformation. In such cases the right most eigen value gives the minimum value of the mean buckling load parameter. The critical or minimum mean

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SDT

Stress Strain Analysis

Potential Energy of Shell

Work done due to hygrothermomechanical in plane loading

Physical problem with uncertainties

Finite element discritization and assembly Apply boundary condition and solve discritize equation using minimum potential energy theory

Deterministic analysis Obtain mean structural response

Stochastic analysis

First order peterubation (solution of sensitivity equation)

Monte Carlo Simulation (assign a random sample of material and geometric properties with repeated over sampling point

Obtain structural response for mean and standard deviation Fig. 2. Overview of stochastic analysis procedure.

buckling load of the structure is obtained by multiplying the load parameter k0cr with reference load. 3.4. Variance of buckling analysis The first order equation as given in Eq. (17) is used to obtain the first order partial derivatives of eigenvalue with respect to the basic random variables which are then used to the buckling load covariance. Having solved Eq. (17) for minimum mean eigenvalue kdcr and corresponding mean eigen vector qdi . This gives

    ðGÞd ðGÞd qdi K dij þ kdcr K ij qi;r ¼ kcr;r qdi K ij qdj   ðGÞ  qdi K ij;r þ kdcr K ij;r qdj ðGÞd

ð18Þ

Since both K dij and K ij are symmetric, the left hand side equals zero by definition of the zeroth order equation. By employing ðGÞd K ij orthonormality conditions, the first term on the right hand side equation reduces to kcr,r. The expression for the first order derivative of the eigenvalue using outlined procedure in Ref. [34] is the written as

  ðGÞ kcr ;r ¼ qdi K ij ;r þ K ij ;r qdj

ði; j ¼ 1; 2; . . . ; n; r

¼ 1; 2; . . . ; RÞ

ð19Þ

It may be noted again that the eigen vector in the above expression is d

d

kcr ¼ kdcr ðbl Þ þ

@kcr ðbl Þ d ðbl  bl Þ ðl ¼ 1; 2; . . . ; RÞ @bl

ð20Þ

The second-order statics of critical load parameter can be evaluated by first squaring and then expectation of the above equation. The statistics of post-buckling load can then be obtained by multiplying the statistics of critical load parameter with the mean reference load. 4. Results and discussion In the present study, a computer program in MATLAB software has been developed to find out the second-order statistics of the post-buckling load for laminated composite cylindrical shell panel with hygrothermomechanical loading having random system

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properties. The influence of randomness in the system properties on thermal post buckling behaviour of laminated composite shell panel with varying amplitude ratios, boundary conditions, plate geometry are investigated in detailed. A nine noded Lagrange isoparametric element with 63 DOFs per element for the present HSDT model has been used for discretizing the laminate and (8  8) mesh has been used throughout the study based on the convergence of the results. The mean and standard deviation of the post-buckling load are obtained considering the random material input variables, thermal expansion coefficients and lamina plate thickness taking combined as well as separately as basic random variables (RVs) as stated earlier. However, the results are only presented taking COV of the system property equal to 0.10 [35] as the nature of the SD (Standard deviation) 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 [35,36]. The basic random variables such as E1, E2, G12, G13, G23, t12, a1, a2 and h are sequenced and defined as

b1 ¼ E11 ; ¼ a11 ;

b2 ¼ E22 ;

b3 ¼ G12 ;

b4 ¼ G13 ;

b5 ¼ G23 ;

b6

b7 ¼ a22 ; b8 ¼ h

The dimensionless post-buckling load for cylindrical shell panel is expressed as [14]

kcrnl ¼

Ncrnl b

2

d Ed22 ðh Þ3

where, Ncrnl is the dimensionless hygrothermal post-buckling load (E22 is taken at DC = 0%, DT = 0 °C, Vf = 0.6). The mean values of E22 and h have been used for obtaining dimensionless post-buckling load and its scattering and amplitude ratio (Wmax/h). In the present study various combinations of edge support conditions namely clamped (CCCC) and simply supported (SSSS) have been used for the investigation. For example, CSCS means clamped edges at n1 = 0, a while simply supported at edges at n2 = 0 and b. The boundary conditions for the shell panel are

and 100, and curvature to side ratio (R/b) = 20, 35, 40, 50, 55, 70 and 75 respectively and volume fraction (Vf) = 0.5, 0.6 and 0.8. 4.1. Validation results for mean dimensionless hygrothermomechanical post-buckling and buckling load In order to validate the accuracy of present deterministic FEM results, an example is solved and compared with those available in the literatures. 4.2. Numerical examples: mean and coefficient of variation of initial buckling load and hygrothermomechanical post-buckling load Table 1 shows the validation of the present solution methodology. The obtained results for non dimensional hygrothermomechanical post-buckling load are compared with the results obtained by Pandey et al. [37] for cross ply, clamped, square plate (a/h = 10) for volume fraction Vf = 0.4, 0.5 and 0.6. It is seen that the present results using C0 nonlinear finite element method are in good agreement with Chebyshey polynomial based analytical solution [37]. It is also seen that the buckling load decreases with increase in moisture concentration due to degradation in material properties at higher moisture concentration. Table 2 shows the results of dimensionless mean initial buckling load of laminated cylindrical shell panels for two stacking sequences of [0°/90°] and [0°/90°/90°/0°] having all edges simply supported and compared to those published results of Singh et al. [17]. Clearly, it is seen that the present results obtained by HSDT based C0 finite element method are in good agreement with HSDT based exact state-space technique. In another comparison, the results of dimensionless mean hygrothermomechanical post-buckling load for anti-symmetric 4layers cross ply [0/90]2T laminated square cylindrical panels with all edges simply supported under four sets of environmental conditions having b/h = 100 and R/a = 10 are shown in Table 3. From the table the results confirm that the buckling load of the laminated composite cylindrical shell panels decreases with increasing the moisture and temperatures. From the table, it can be seen that

All edges simply supported (SSSS):

v ¼ w ¼ h2 ¼ /2 ¼ 0;

at n1 ¼ 0; a;

u ¼ w ¼ h1 ¼ /1 Table 1 Compression of non-dimensionalized critical hygrothermomechanical post-buckling load of clamped support four layers cross ply [0°/90°]2T laminated composite square plate with uniaxial loading (Nx) and volume fractions Vf (=0.4, 0.5, 0.6), a/h = 10, DT = 0 °C.

¼ 0 at n2 ¼ 0; b All edges clamped (CCCC):

u ¼ v ¼ w ¼ /1 ¼ /2 ¼ h1 ¼ h2 ¼ 0;

at n1

¼ 0; a and n2 ¼ 0; b Two opposite edges clamped and other two simply supported (CSCS):

u ¼ v ¼ w ¼ /1 ¼ /2 ¼ h1 ¼ h2 ¼ 0; at n1 ¼ 0 and n2 ¼ 0

v ¼ w ¼ h2 ¼ /2 ¼ 0;

Vf

DC %

Pandey et al. [37]

Present

0.4

0 1 3

14.178 13.253 11.712

16.5458 14.7473 12.1112

0.5

0 1 3

17.405 16.183 14.703

19.8233 17.8269 14.8350

0.6

0 1 3

20.913 19.742 18.304

23.6694 21.5248 18.2190

at n1 ¼ a; u ¼ w ¼ h1 ¼ /1 ¼ 0; at n2 ¼ b

For the computational analysis the following properties of composite material are considered to be dependent on temperature and moisture. The material properties taken in analysis are at reference temperature 21 °C and moisture concentration at 0% are given as [37]

Ef1 ¼ 220 GPa; ¼ 8:79 Gpa;

Ef2 ¼ 13:79 GPa;

Em ¼ 3:45 GPa;

Gf1

mf12 ¼ 0:2; mm ¼ 0:35 af1

¼ 0:99  10e6  =C; af2 ¼ 10:08  10e6  =C;

bm

Table 2 Comparison of dimensionless mean buckling loads, under uniaxial compassion with all edges simply supported square composite cylindrical panels with R/b = 5 and (E11 = 40E22, G12 = G13 = 0.6E22, G13 = 0.5E22, t12 = 0.25). Lamination layers



¼ 0:33; T g0 ¼ 216 C The shell panel geometry used is characterized by aspect ratios (a/ b) = 1 and 2, side to thickness ratios (b/h) = 10, 25, 35, 40, 50, 60, 80

[0°/90°] [0°/90°/90°/0°]

Present

Singh et al. [17]

b/h = 10

b/h = 100

b/h = 10

b/h = 100

11.7528 18.02897

26.3227 30.0491

11.7894 18.0414

26.35685 30.1541

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Table 3 Comparison of dimensionless buckling load (KN) for [0/90]2T square laminated cylindrical panels under four sets of environmental conditions (b/h = 100, a/R = 1.0, b = 0.1 m). Environmental conditions

Vf = 0.5

Vf = 0.6

Present

Shen [11]

Present

Shen [11]

Present

Shen [11]

DT = 0 °C, DC = 0% DT = 100 °C, DC = 1% DT = 200 °C, DC = 2% DT = 300 °C, DC = 3%

9.2623 8.8731 8.2563 8.1680

9.1853 8.7211 8.2439 7.7530

11.053 10.632 10.231 9.9248

10.8900 10.3696 9.8304 9.2710

13.275 12.886 12.0324 11.7361

13.0708 12.4859 11.8729 11.2295

the present results are in good agreement with the available analytical published results [11]. The difference in the results may be due to application of boundary conditions. 4.3. Validation results for probabilistic hygrothermomechanical postbuckling load The results have been validated by comparing present C0 nonlinear EEM with direct iterative method in conjunction with first order perturbation technique (DISFEM) with an independent MCS approach subjected to amplitude ratio Wmax/h (=0.2 and 0.4). As mentioned earlier, no results are available in the reported literature for normalized standard deviation for the nonlinear problems. Hence, the standard results for comparison have been computed by using independent MCS. A good agreement is observed among them. Table 4 shows the validation study of present DISFEM with independent MCS approach for normalized standard deviation, SD (i.e. the ratio of the standard deviation (SD) to the mean value), of hygrothermal post-buckling load versus the SD to the mean value of the random material constants (bi = E11) for an all simply supported square [0°/90°] laminated composite cylindrical shell panel, b/h = 100, and R/b = 10, DT = 100 °C and DC = 2% with different amplitude ratios subjected to uniform temperature and moisture concentration distribution changing from 0% to 20%. It is assumed that all of the material property (i.e., bi, i = 1) change at a time keeping other as deterministic. The present DISFEM results obtained by using first-order perturbation approach yield very close results with independent MCS approach. For the MCS approach, the samples are generated using MAT LAB software to fit the desired mean and SD. These samples are used in response equation which is solved repeatedly, adopting conventional eigenvalue procedure, to generate a sample of the hygrothermal post bucking load. The number of samples used for MCS approach is 12,000 based on satisfactory convergence of the results. The nor-

Table 4 Validation study of present DISFEM method and MCS method for amplitude ratios (Wmax/h = 0.2, 0.4) on the dimensionalized mean and dispersion of hygrothermomechanical post-buckling load of cross ply [0°/90°] laminated composite cylindrical shell with (SSSS) support boundary conditions and uniaxial loading (Nx) having only E11 as random keeping others deterministic Vf = 0.6, b/h = 100, R/b = 10, for moisture and temperature having COV, bi, i = 1, DT = 100 °C, DC = 2%. Validation study

DISFEM

MCS

Coefficient of co-relation (COC)

Vf = 0.7

mal distribution has been assumed for random number generations in MCS. However, the present DISFEM used in the study does not put any limitation as regard to probability distribution of the system property. This is an advantage over the MCS [38]. It can also be observed that the DISFEM for present analysis is sufficient to give accurate results for the level of variations considered in the basic random variables. The mean response values in the two methods are almost same. Table 5 shows the effects of individual random system property keeping others as deterministic with mean amplitude ratios (Wmax/ h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply [0°/90°/90°/0°] laminated composite cylindrical shell panel with b/h = 10, R/b = 30, Vf = 0.6, simple support (SSSS) support boundary condition and uniaxial loading for moisture and temperature DT = 0 °C, DC = 0% and DT = 150 °C, DC = 1.5% having COV, bi, {i = (1–8) = 0.10. It is observed that the hygrothermal effects with individual random system property a11 and a22 have least value of COV where as E11 shows the highest COV values for both temperature and moisture condition. Table 6 shows the effects of side to mean thickness ratios (b/h) with mean amplitude ratios (Wmax/h) on the dimensionalized

Table 5 Effects of individual random system property keeping others as deterministic with amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post buckling load of cross ply [0°/90°/90°/0°] laminated composite cylindrical shell with b/h = 10, R/b = 30, Vf = 0.6, simple support (SSSS) support boundary condition and uniaxial loading (Nx) for moisture and temperature having COV, bi, {i = (1–8) = 0.10}. (The values in the bracket indicate dimensionalized mean hygrothermomechanical post buckling load.) bi

Wmax/h

DT = 0 °C, DC = 0%, kcr l = 12.3321

DT = 150 °C, DC = 1.5%, kcr l = 6.3369

E11 (i = 1)

0.2 0.4 0.6

0.0630 (12.4430) 0.0567 (12.7476) 0.0575 (13.1768)

0.0503 (6.4055) 0.0513 (6.5821) 0.0523 (6.5821)

E22 (i = 2)

0.2 0.4 0.6

0.0243 0.0240 0.0238

0.0012 0.0012 0.0012

G12 (i = 3)

0.2 0.4 0.6

0.0078 0.0077 0.0076

0.0048 0.0047 0.0046

G13 (i = 4)

0.2 0.4 0.6

0.0375 0.0366 0.0357

0.0500 0.0488 0.0476

Wmax/h = 0.2

Wmax/h = 0.4

G23 (i = 5)

COV, Ncr nl kcr nl = 5.7012 (5.6115)

COV, Ncr nl kcr nl = 5.8967 (5.6115)

0.2 0.4 0.6

0.0065 0.0064 0.0063

0.0087 0.0085 0.0084

a11 (i = 6)

0.05 0.1 0.15 0.20

0.0343 0.0685 0.1029 0.1372

0.0319 0.0637 0.0957 0.1276

0.2 0.4 0.6

7.2264e005 7.0546e005 6.8244e005

7.4697e004 7.2678e004 7.0194e004

a22 (i = 7)

0.2 0.4 0.6

4.7374e004 4.6248e004 4.4742e004

0.0058 0.0056 0.0054

0.05 0.1 0.15 0.20

0.0345 0.0690 0.1035 0.1380.

0.0321 0.0642 0.0963 0.1284

h (i = 8)

0.2 0.4 0.6

0.0128 0.0155 0.0183

0.0093 0.0133 0.0175

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Table 6 Effects of thickness ratios (b/h) with amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post-buckling load of cross ply [0°/90°/ 90°/0°] laminated composite cylindrical shell with simple support (SSSS) and CCCC support boundary condition and uniaxial loading (Nx) for moisture and temperature R/b = 50, Vf = 0.6 having COV, bi, {i = (1, . . ., 8), (6, 7), (8) and (8) = 0.10}. Lamina layup

b/h

Wmax/h

Mean kcr

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

SSSS

40

0.2 0.4 0.6 kcrl

18.1508 18.4976 19.0501 (18.0276)

0.0816 0.0818 0.0820

0.0054 0.0053 0.0052

0.0274 0.0286 0.0295

14.5409 14.8279 15.2836 (14.4404)

0.0947 0.0943 0.0937

0.0569 0.0558 0.0541

0.0266 0.0281 0.0292

60

0.2 0.4 0.6 kcrl

18.6844 19.0380 19.6014 (18.5568)

0.0833 0.0834 0.0835

0.0119 0.0117 0.0113

0.0285 0.0296 0.0304

15.7006 16.0042 16.4886 (15.5934)

0.1409 0.1390 0.1363

0.1237 0.1214 0.1178

0.0281 0.0294 0.0304

80

0.2 0.4 0.6 kcrl

18.9884 19.3472 19.9178 (18.8566)

0.0853 0.0853 0.0854

0.0208 0.0204 0.0198

0.0291 0.0302 0.0309

16.3314 16.6447 17.1450 (16.2197)

0.2231 0.2193 0.2136

0.2163 0.2122 0.2060

0.0287 0.0300 0.0309

40

0.2 0.4 0.6 kcrl

65.3948 66.0721 67.1191 (65.1072)

0.0807 0.0807 0.0808

0.0015 0.0015 0.0015

0.0168 0.0170 0.0173

50.9187 51.4714 52.3156 (50.6820)

0.0788 0.0789 0.0790

0.0162 0.0161 0.0158

0.0132 0.0136 0.0142

60

0.2 0.4 0.6 kcrl

72.2668 73.0324 74.2034 (71.9184)

0.0838 0.0839 0.0840

0.0031 0.0030 0.0030

0.0212 0.0212 0.0211

60.3272 60.9820 61.9763 (60.0268)

0.0853 0.0853 0.0853

0.0322 0.0319 0.0313

0.0187 0.0188 0.0190

80

0.2 0.4 0.6 kcrl

76.2379 77.0744 78.3357 (75.8344)

0.0858 0.0858 0.0859

0.0052 0.0051 0.0050

0.0234 0.0232 0.0231

65.8853 66.6204 67.7228 (65.5280)

0.0946 0.0944 0.0941

0.0536 0.0530 0.0521

0.0214 0.0214 0.0214

CCCC

nl

(DT = 0 K, DC = 0%) COV, Ncr nl bi

mean and dispersion of hygrothermal post-buckling load of cross ply [0°/90°/90°/0°] laminated composite shell panel with simple support (SSSS) and CCCC support boundary conditions and uniaxial laoding for different moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. It is seen that for simple support S2 the mean hygrothermal load increases as the side to thickness

Mean kcr

nl

(DT = 150 K, DC = 1.5%) COV, Ncr nl bi

ratio and amplitude ratio increases. As side to thickness ratio increases the dispersion of the shell panel increases for all input random change material properties bi, (i = 1, . . ., 7) except bi, (i = 7, 8). Also it can be noted that for simple support SSSS and clamped support CCCC condition, if we increase the temperature and moisture percentage the mean hygrothermal post-buckling load decreases.

Table 7 Effects of aspect ratios (a/b), support conditions (SSSS, CCCC) with amplitude ratios (Wmax/h) and environmental conditions on the dimensionalized mean and dispersion of hygrothermomechanical post buckling load of cross ply [0°/90°/90°/0°] laminated composite shells with simple support (SSSS) and CCCC support boundary conditions and uniaxial loading (Nx), b/h = 50, R/b = 50, Vf = 0.6, for moisture and temperature (having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Support condition

a/b

Wmax/h

DT = 0 °C, DC = 0% Mean kcr nl

SSSS support

CCCC support

COV, Ncr bi

DT = 150 °C, DC = 1.5% Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i=8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

1

0.2 0.4 0.6 kcr l

18.4657 18.8162 19.3750 (18.3403)

0.0825 0.0826 0.0828

0.0083 0.0082 0.0079

0.0385 0.0292 0.0301

15.2264 15.5230 15.9957 (15.1222)

0.1131 0.1121 0.1106

0.0871 0.0854 0.0829

0.0275 0.0289 0.0299

2

0.2 0.6 0.4 kcr l

4.8024 4.8932 5.0368 (4.7684)

0.0889 0.0888 0.0886

0.0322 0.0316 0.0307

0.0294 0.0305 0.0312

4.1870 4.2671 4.3942 (4.1582)

0.3367 0.3305 0.3213

0.3341 0.3278 0.3183

0.0292 0.0304 0.0313

1

0.2 0.4 0.6 kcr l

69.4019 70.1263 71.2412 (69.0830)

0.0825 0.0826 0.0827

0.0022 0.0022 0.0022

0.0194 0.0195 0.0196

56.3635 56.9710 57.8977 (56.0939)

0.0820 0.0820 0.0821

0.0235 0.0233 0.0229

0.0165 0.0168 0.0171

2

0.2 0.4 0.6 kcr l

19.7524 19.9769 20.3106 (19.6386)

0.0873 0.0873 0.0874

0.0078 0.0077 0.0076

0.0246 0.0244 0.0242

17.4448 17.6457 17.9428 (17.3423)

0.1101 0.1096 0.1088

0.0802 0.0793 0.0779

0.0230 0.0229 0.0227

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Table 8 Effects of lamina layup with amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post buckling load of cross ply and angle laminated composite cylindrical shell with simple support SSSS and CCCC support boundary condition and uniaxial loading (Nx), b/h = 100, R/b = 20, Vf = 0.6, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Support condition

Lamina layup

Wmax/h

DT = 0 °C, DC = 0% Mean kcr nl

SSSS support

CCCC support

COV, Ncr bi

DT = 150 °C, DC = 1.5% Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

[0°/90°/0°/90°]

0.2 0.4 0.6 kcrl

17.6336 18.1914 18.8004 (17.7980)

0.0840 0.0839 0.0837

0.0347 0.0340 0.0329

0.0377 0.0387 0.0391

14.9600 15.2964 15.8267 (14.8304)

0.3757 0.3676 0.3555

0.3740 0.3658 0.3535

0.0397 0.0409 0.0414

[0°/45°/0°/45°]

0.2 0.4 0.6 kcrl

22.4816 22.8150 23.3023 (22.3081)

0.0820 0.0819 0.0818

0.0196 0.0193 0.0189

0.0401 0.0406 0.0408

19.3991 19.6815 20.0988 (19.2579)

0.2129 0.2101 0.2061

0.2082 0.2052 0.2009

0.0414 0.0421 0.0424

0°/90°/0°/90°]

0.2 0.4 0.6 kcrl

79.0937 80.3062 81.9466 (78.3155)

0.0865 0.0866 0.0866

0.0078 0.0077 0.0075

0.0305 0.0302 0.0298

69.5754 70.6491 72.0940 (68.8839)

0.1100 0.1093 0.1084

0.0804 0.0792 0.0776

0.0297 0.0294 0.0291

[0°/45°/0°/45°]

0.2 0.4 0.6 kcrl

69.0303 69.8496 71.0239 (68.5689)

0.0814 0.0814 0.0814

0.0064 0.0063 0.0062

0.0310 0.0307 0.0302

59.9735 60.6809 61.6967 (59.5789)

0.0962 0.0958 0.0952

0.0673 0.0665 0.0655

0.0306 0.0303 0.0398

In the case of shell panel with clamped support CCCC conditions the mean hygrothermal post-buckling load increases much more compared to shell panel with simple support S2 conditions. Simply supported shell panel is more sensitive as compared to clamped supported shell panel. Table 7 shows the effects of aspect ratios (a/b) with mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply [0°/90°/ 90°/0°] laminated composite shell panel with simple support (SSSS) and CCCC support boundary condition and uniaxial loading, with b/h = 50, R/b = 50, Vf = 0.6, for two different temperatures and moisture percentage i.e. DT = 0 °C, DC = 0% and DT = 150 °C, DC = 1.5% (having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. It is seen as that the aspect ratio increases i.e. for the square cylindrical shell panel becomes rectangular cylindrical shell panel, the dimensionalized mean hygrothermal load decreases considerably for both support condition SSSS and CCCC. However the dispersion of rectangular shell panel decreases with random change in shell panel thickness.

Table 8 shows the effects of lamina layup with mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply and angle ply laminated composite cylindrical shell panel with simple support (SSSS) and CCCC support boundary condition and uniaxial loading, b/ h = 100, R/b = 20, Vf = 0.6, for two different temperatures and moisture percentage, i.e. DT = 0 °C, DC = 0% and DT = 150 °C, DC = 1.5% for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Here we can see that when the lamina layup changes from cross ply to angle-ply layup the mean dimensionalized mean hygrothermal post-buckling load increases for support condition S2 where as for clamped support condition it decreases., also as the amplitude ratio increases COV increases for bi, {i = (1, . . ., 7), and (8) = 0.10}. Table 9 effect of volume fraction with mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell panel with simple support (SSSS) support boundary condition and uniaxial loading, b/h = 25, R/b = 40,

Table 9 Effect of volume fraction with amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell with simple support SSSS support boundary condition and uniaxial loading (Nx), b/h = 25, R/b = 40, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Vf

Wmax/h

DT = 0 °C, DC = 0% Mean kcr nl

DT = 150 °C, DC = 1.5% COV, Ncr bi

Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i, . . ., 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

0.4

0.2 0.4 0.6 kcr l

5.2932 5.4092 5.5935 (5.2521)

0.0748 0.0751 0.0755

0.0039 0.0038 0.0037

0.0755 0.0375 0.0381

3.7778 3.8663 4.0057 (3.7468)

0.0817 0.0817 0.0816

0.0403 0.0394 0.0380

0.0365 0.0379 0.0388

0.5

0.2 0.4 0.6 kcr l

6.4570 6.5999 6.826 (6.4065)

0.0760 0.0763 0.0767

0.0030 0.0030 0.0029

0.0365 0.0376 0.0383

4.7412 4.8526 5.0279 (4.7021)

0.0810 0.0811 0.0813

0.0320 0.0312 0.0302

0.0365 0.0379 0.0388

0.6

0.2 0.4 0.6 kcr l

7.6810 7.8515 8.1221 (7.6208)

0.0780 0.0783 0.0786

0.0025 0.0024 0.0023

0.0365 0.0377 0.0384

5.8037 5.9397 6.1538 (5.7560)

0.0810 0.0811 0.0814

0.0267 0.0261 0.0252

0.0366 0.0380 0.0388

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Table 10 Effect of curvature to side ratio with amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post-buckling load of cross ply [0°/45°/ 45°/0°] laminated composite cylindrical shell with simple support SSSS support boundary condition with uniaixial (Nx) loading, b/h = 35, Vf = 0.6, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. R/b

Wmax/h

DT = 0 °C, DC = 0% Mean kcr nl

DT = 150 °C, DC = 1.5% COV, Ncr bi

Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

35

0.2 0.4 0.6 kcrl

9.7077 9.8391 10.0464 (9.6568)

0.0819 0.0820 0.0820

0.0028 0.0027 0.0027

0.0351 0.0360 0.0366

7.6453 7.6897 10.0186 (7.5843)

0.0597 0.0598 0.0643

0.0123 0.0122 0.0142

0.0633 0.0625 0.0613

55

0.2 0.4 0.6 kcrl

9.6783 9.8062 10.0101 (9.6309)

0.0820 0.0821 0.0821

0.0028 0.0027 0.0027

0.0351 0.0361 0.0367

7.6526 8.6203 9.8628 (7.5819)

0.0597 0.0703 0.0648

0.0123 0.0122 0.0142

0.0633 0.0625 0.0613

75

0.2 0.4 0.6 kcr l

9.6685 9.7948 9.9971 (9.6228)

0.0820 0.0821 0.0821

0.0028 0.0027 0.0027

0.0352 0.0361 0.0367

7.6442 7.6847 7.8120 (7.5811)

0.0597 0.0598 0.0685

0.0123 0.0122 0.0142

0.0633 0.0625 0.0613

Flat plate

0.2 0.4 0.6 kcr l

19.3977 19.6432 20.0440 (19.3150)

0.0821 0.0822 0.0822

0.0028 0.0027 0.0027

0.0352 0.0363 0.0370

15.9052 16.1080 16.4375 (15.8366)

0.0833 0.0833 0.0833

0.0290 0.0286 0.0281

0.0338 0.0350 0.0360

for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. The table indicates that s the amplitude ratio and volume fraction increases the non-dimensionalized mean increases for both temperature and moisture percentage condition. However when volume fraction remains same and the temperature and moisture percentage increases from DT = 0 °C, DC = 0% to DT = 150 °C, DC = 1.5%, the dimensionalized mean decreases. The dispersion value increases for random change in bi, {i = (1, . . ., 7), and (8) = 0.10} and decreases for bi, {i = (6, 7). Table 10 shows effect of curvature to side ratio (R/b) with mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply [0°/45°/ 45°/0°] laminated composite cylindrical shell panel with simple support (SSSS) support boundary condition and uniaxial loading, b/h = 35, Vf = 0.6, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. It can be seen that thought the curvature to side ratio increases significantly the dimensionalized mean hygrothermal post-buckling load decreases and slight change in hygrothermal post-buckling load dispersion for bi,

{i = (1, . . ., 7), (6, 7) and (8) = 0.10}. However the mean hygrothermal post-buckling load and its dispersion is highest for flat plate. Table 11 shows effect of constant moisture concentration with change in uniform temperature and mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal post-buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell panel with (CSCS) support boundary conditions, b/h = 80, R/b = 70, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. It can be seen from the table that as we increase the temperature keeping the moisture content same, the dimensionalized mean hygrothermal post-buckling load decreases. Also the hygrothermal post buckling dispersion for bi, {i = (1, . . ., 7), (6, 7)} increases with least change for bi, {i = (8) = 0.10} Table 12 shows effect of constant temperature with change in moisture concentration and mean amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermal postbuckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell panel with (CSCS) support boundary conditions,

Table 11 Effect of constant moisture concentration with change in temperature and amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post-buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell with (CSCS) support boundary condition and biaxial loading (Nx = Ny), b/h = 80, R/b = 70, Vf = 0.5, 0.6 and 0.8 for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Vf

Wmax/h

DT = 0 °C, DC = 0% Mean kcr nl

DT = 100 °C, DC = 0% COV, Ncr bi

Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

0.5

0.5 1.0 1.5 kcr l

14.5543 16.2093 18.5894 (13.9292)

0.0827 0.0834 0.0840

0.0139 0.0125 0.0109

0.0369 0.0349 0.0314

13.4188 14.9687 17.1860 (12.8331)

0.1233 0.1172 0.1108

0.0905 0.0811 0.0706

0.0364 0.0345 0.0312

0.6

0.5 1.0 1.5 kcr l

17.3001 19.2715 22.1059 (16.5553)

0.0839 0.0845 0.0850

0.0114 0.0102 0.0089

0.0370 0.0349 0.0315

16.1205 20.6477 20.6477 (15.4167)

0.1123 0.1035 0.1035

0.0747 0.0670 0.0583

0.0365 0.0313 0.0313

0.8

0.5 1.0 1.5 kcr l

23.1948 25.8311 29.6273 (22.1983)

0.6898 0.6280 0.6280

0.0087 0.0079 0.0068

0.0373 0.0316 0.0316

22.1570 24.698828 .3482 (21.1964)

0.1406 0.1307 0.1307

0.0560 0.0503 0.0438

0.0370 0.0315 0.0315

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Table 12 Effect of constant temperature with change in moisture concentration and amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of hygrothermomechanical post buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell with (CSCS) support boundary condition and biaxial loading (Nx = Ny), b/h = 80, R/b = 70, Vf = 0.5, 0.6 and 0.8 for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. Vf

Wmax/h

DT = 100 °C, DC = 1% Mean kcr nl

COV, Ncr bi

DT = 100 °C, DC = 2% Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i, . . ., 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

0.5

0.5 1 1.5 kcr l

13.3151 14.8532 17.0534 (12.7339)

0.1120 0.1062 0.1004

0.0905 0.0811 0.0707

0.0363 0.0345 0.0312

13.2130 14.7393 16.9227 (12.6361)

0.1030 0.0970 0.0913

0.0906 0.0812 0.0707

0.0363 0.0345 0.0312

0.6

0.5 1 1.5 kcr l

16.0067 17.8560 20.5021 (15.3078)

0.1037 0.0997 0.0958

0.0748 0.0670 0.0584

0.0364 0.0346 0.0313

15.8944 17.7308 20.3584 (15.2004)

0.0960 0.0922 0.0887

0.0748 0.0671 0.0584

0.0364 0.0346 0.0312

0.8

0.5 1 1.5 kcr l

22.0491 24.5786 28.2103 (21.0931)

0.1354 0.1309 0.1264

0.0561 0.0503 0.0438

0.0370 0.0350 0.0315

21.9421 24.4594 28.0735 (20.9907)

0.1303 0.1262 0.1221

0.0561 0.0503 0.0438

0.0370 0.0349 0.0315

b/h = 80, R/b = 70, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. The table indicates that keeping the temperature constant and increasing the moisture content the dimensionalized mean hygrothermal post-buckling load decreases. However for the hygrothermal post buckling dispersion results, it is observed that no definite trend is observed for various fibre volume fractions. The effect of different in-plane loadings with different amplitude ratios and two sets of constant moisture loadings on dimensionless mean and dispersion for COV, bi, {(i = 1, . . ., 7), (7, 8) and (9)} of thermomechanical post-buckling load of anti-symmetric, cross ply [0°/90°/0°/90°] square laminated cylindrical shell panels with CSCS support conditions having b/h = 80, R/b = 70 and Vf = 0.6 is shown in Table 13. It is observed that the thermomechanical post-buckling load is lowest when edge has in-plane biaxial loading combined with shear loading and highest for in-plane uniaxial loading. It can also be seen that the dispersion in the thermomechanical post-buckling load is highest for in-plane uniaxial loading and lowest for in-plane uniaxial loading combined with

shear loading. This is because of the axial loading help to prone buckling. As the moisture loading increases the thermomechanical post buckling load and its dispersion decreases. This is because of increase in moisture concentration decreases the stiffness, thus lowering the strength of the structure and its stability. Table 14 shows the dimensionless mean and dispersion for COV, bi, {(i = 1, . . ., 7), (7, 8) and (9)} of post-buckling load of anti-symmetric, cross ply [0°/90°/0°/90°] square laminated cylindrical shell panels with amplitude ratios for two set of thermal loading conditions and subjected to different in-plane loading conditions having b/h = 80, R/b = 70 and Vf = 0.6. It is observed as expected that the post-buckling load is lowest when edge has in-plane biaxial loading combined with shear loading and highest for in-plane uniaxial loading. It can also be seen that the dispersion in the post-buckling load is strongest for in-plane uniaxial loading and lowest for in-plane uniaxial loading combined with shear loading. This is because of the axial loading help to prone buckling load. As the temperature loading increases the post-buckling load decreases while dispersion increases. This is because of increase in tempera-

Table 13 Effect of in-plane loading with constant temperature, change in moisture concentration and amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of thermomechanical post-buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell with (CSCS) support boundary condition and biaxial and shear loading, b/h = 80, R/b = 70, Vf = 0.6 for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. In plane loading

Wmax/h

DT = 100 °C, DC = 1% Mean kcr nl

COV, Ncr bi

DT = 100 °C, DC = 2% Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6,7)

(i = 8)

Biaxial

0.5 1 1.5 kcr l

16.1182 17.9803 20.6449 (15.4144)

0.1036 0.0997 0.0958

0.0748 0.0670 0.0584

0.0365 0.0346 0.0313

16.1158 17.9777 20.6420 (15.4122)

0.0959 0.0922 0.0887

0.0748 0.0670 0.0584

0.0365 0.0346 0.0313

Biaxial with shear

0.5 1 1.5 kcr l

15.1236 16.8425 19.3328 (14.4807)

0.1007 0.0968 0.0930

0.0705 0.0626 0.0538

0.0356 0.0337 0.0305

15.1214 16.8400 19.3300 (14.4785)

0.0932 0.0895 0.0862

0.0705 0.0626 0.0538

0.0356 0.0337 0.0305

Uniaxial

0.5 1 1.5 kcr l

31.5174 35.3150 40.5795 (30.0496)

1.0488 0.1005 0.0963

0.0764 0.0682 0.0594

0.0336 0.0319 0.0288

31.5128 35.3099 40.5738 (30.0451)

0.0974 0.0932 0.0894

0.0765 0.0682 0.0594

0.0336 0.0319 0.0288

Uniaxial with shear

0.5 1 1.5 kcr l

26.0650 28.9526 33.425 (24.9754)

0.0973 0.0934 0.0897

0.0655 0.0574 0.0437

0.0326 0.0309 0.0277

26.0610 28.9481 32.812 (24.9715)

0.0902 0.0867 0.0862

0.0655 0.0574 0.0437

0.0326 0.0309 0.0277

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Table 14 Effect of in-plane loading with constant temperature and amplitude ratios (Wmax/h) on the dimensionalized mean and dispersion of thermomechanical post-buckling load of cross ply [0°/90°/0°/90°] laminated composite cylindrical shell with (CSCS) support boundary condition and biaxial and shear loading, b/h = 80, R/b = 70, Vf = 0.6, for moisture and temperature having COV, bi, {i = (1, . . ., 7), (6, 7) and (8) = 0.10}. In plane loading

Wmax/h

DT = 50 °C, DC = 0 Mean kcr nl

COV, Ncr bi

DT = 150 °C, DC = 0 Mean kcr nl

nl

COV, Ncr bi

nl

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

(i = 1, . . ., 7)

(i = 6, 7)

(i = 8)

Biaxial

0.5 1 1.5 kcr l

16.7715 18.6937 21.4522 (16.0453)

0.0923 0.0913 0.0902

0.0407 0.0365 0.0318

0.0368 0.0348 0.0314

15.1831 16.9601 19.4880 (14.5106)

0.1467 0.1372 0.1273

0.1188 0.1064 0.0926

0.0358 0.0341 0.0309

Biaxial with shear

0.5 1 1.5 kcr l

15.7231 17.4971 20.0749 (15.0596)

0.0909 0.0898 0.0887

0.0384 0.0341 0.0293

0.0359 0.0340 0.0306

14.2506 15.8885 18.2476 (13.6373)

0.1408 0.1314 0.1214

0.1120 0.0993 0.0852

0.0349 0.0332 0.0301

Uniaxial

0.5 1 1.5 kcr l

32.8129 36.7357 42.1934 (31.2977)

0.0924 0.0912 0.0900

0.0416 0.0372 0.0324

0.0341 0.0323 0.0291

29.6531 33.2692 38.2583 (28.2509)

0.1487 0.1385 0.1281

0.1216 0.1084 0.0943

0.0325 0.0310 0.0281

Uniaxial with shear

0.5 1 1.5 kcr l

27.0752 30.0547 34.2518 (25.9514)

0.0889 0.0877 0.0865

0.0356 0.0312 0.0265

0.0331 0.0303 0.0282

24.5482 27.2584 31.3790 (23.5108)

0.1339 0.0597 0.0862

0.1042 0.1003 0.0986

0.0316 0.0294 0.0273

ture decreases the stiffness, thus lowering the strength of the structure and its stability. Although for the same parameters the decrease in thermomechanical post buckling subjected to thermal loading is more than moisture loading because the thermal loadings are more sensitive to buckling. 5. Conclusions A DISFEM probabilistic procedure is presented to study the second-order statistics (mean and coefficients of variation) of hygrothermo-mechanically post buckling of laminated composite cylindrical shell panel with randomness in material properties, expansion of thermal coefficients, shell panel thickness. The effects of the mean value of combination of multiple random variables varying simultaneously or individually, shell panel geometric parameters, boundary condition and various modes of moisture and temperature change have been examined in analysis. The following conclusion can be drawn from this study: 1. The mean dimensionless hygrothermal post-buckling load of the shell panel is the most affected with random change in E11, E22, G13 and a11. The strict control of these random parameters is therefore required for reliable design of laminated composite shell panel. 2. From reliability point of view, CCCC support shell panel is the most desirable as compared to other support shell panel. 3. In general, the square shell panel is more sensitive as compared to rectangular shell panel. 4. It is also observed for anti-symmetric lamina layup keeping the same support condition; the dimensionless hygrothermal postbuckling load increases for cross-ply layup as compared to angle-ply layup. The different curvatures to side ratio have least impact on the non dimensionalized hygrothermal post-buckling load for cross ply lamination scheme. 5. The dimensionless hygrothermal post-buckling load is more sensitive to change in moisture concentration as compared to change in temperature condition. 6. The effect of randomness in the thermal expansion coefficients, the lamina shell panel thickness on the coefficients of nonlinear hygrothermal post-buckling load subjected to hygrothermome-

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