Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading

Accepted Manuscript Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading Nikhil Garg, Karkhanis Rahul Sanjay, Rosalin Sahoo, P.R. Maiti, B.N. Singh PII: DOI: Reference:

S0263-8223(18)31588-5 https://doi.org/10.1016/j.compstruct.2018.10.064 COST 10315

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

1 May 2018 1 October 2018 22 October 2018

Please cite this article as: Garg, N., Rahul Sanjay, K., Sahoo, R., Maiti, P.R., Singh, B.N., Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.10.064

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Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading Nikhil Garga, Karkhanis Rahul Sanjaya, Rosalin Sahoob,*, P. R. Maitic, B. N. Singhd a

Post Graduate Student, b Assistant Professor, c Associate Professor, d Professor

abc d

Department of Civil Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India.

Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India.

*

Corresponding Author, Ph No. +91-9930425013; E-mail address: [email protected]

Abstract In the present work, the static response of symmetric and anti-symmetric laminated composite and sandwich plates is investigated for elevated temperature and moisture conditions. Analysis is carried out using recently developed zigzag model which is based on a trigonometric (secant) shear deformation in the plate. The theory incorporates inter-laminar shear stress continuity along with the traction free boundary conditions at the top and bottom surfaces. The finite element method is employed for analysis, using the eight noded isoparametric serendipity element, accounting for the C0 continuity. Numerous examples covering the various features such as effect of thermal and moisture coefficients, material anisotropy, boundary and loading conditions and span to thickness ratio are solved for symmetric as well as anti-symmetric plates. Results in the form of deflection and stresses are presented and validated with the available results in the existing literature. Further, few new results are also provided in this work, to set a benchmark study for the future research. Keywords: Zigzag theory; composite laminate; sandwich plates; hygrothermal analysis; static response. 1. Introduction Recently, the use of fibre reinforced composites has highly increased in various industries like aerospace, civil, automobile, sports etc. Composites have the advantage of the unique properties of the constituent materials and their interaction with each other to obtain a tailored behaviour as a final outcome. The increased use of composites is due to the fact that these possess various enhanced mechanical properties, like high strength to weight ratio, high stiffness to weight ratio, high damping, high toughness etc. Furthermore, use of hybrid reinforced composites also enhances thermal and damping properties.

Due to such a wide range of applications of composites, a lot of research has been done in an attempt to study the behaviour of the material. The exact solution has been obtained by Pagano [1] and Pagano and Hatfield [2], considering the plate as a three dimensional (3D) elastic problem. But, that 3D analysis is restricted to simple boundary conditions, geometry and loadings. Also, it becomes very tedious to perform analysis for practical requirements which involve a large number of plates stacked over each other. Hence, there was a requirement to develop a model which can be applied to practical problems and is computationally efficient. At first, an extension to the Kirchoff’s plate theory gave rise to Classical Laminated Plate Theory (CLPT). However, it does not consider the shear deformation effects, as has been explored by Yang et al. [3] and Whitney and Leissa [4]. Hence, Reissner [5,6] and Mindlin [7] proposed the First Order Shear Deformation Theory (FSDT) which accounts for a constant shear deformation through the thickness. However, FSDT gives erroneous results especially for thick plates. Hence, researchers moved towards the higher order shear deformation theories (HSDT). Some of these theories are based on Taylor series expansion as analysed by Lo et al [8, 9], Reddy [10, 11] while many other researchers worked with non-polynomial terms in shear strain function [12–14]. All these theories are collectively termed as Equivalent Single Layer (ESL) theories. ESL theories predict global response of laminated plates like critical buckling loads, fundamental frequencies etc. very accurately, especially for the thin plates. However, when laminated components are used as the primary structural components, there arose a need for a theory which can accurately assess the localised regions and probable regions of damage initiation. ESL theories show an erroneous variation in these parameters. To overcome these drawbacks of ESL theories, Layer Wise (LW) models were presented [15, 16]. In these models, a unique displacement field is considered for each layer and conditions are imposed to incorporate the continuity in shear stresses in adjacent layers. One of the major shortcomings of LW models is that the number of variables depends upon the number of layers in the plate and hence, for plates of practical aspects, computational cost becomes too high. The zigzag model based theories eliminated this drawback. Zig-zag theories include all the concepts of LW models, but with a constant number of variables which are independent of number of layers. Various authors like Cho and Parmerter [17], Kapuria and Kulkarni [18], Sahoo and Singh [19, 20], Chalak et al. [21], Pandit et al. [22] etc. have worked in order to improve the presented zigzag model.

The use of composites has also increased in environments with highly elevated temperatures like nuclear power plants as well as in off-shore structures where humidity is very high. Temperature and moisture have adverse effects on the performance of the composite materials. Generally, moisture and high temperature reduce the strength and stiffness of the material and hence, the observed transverse deflection is more. Also, extreme environmental conditions can contribute to the ultimate failure of structures too. So, it generates a need for assessing the performance of composites under thermal and hygrothermal loading conditions so that a safe and superior designing can be performed. Hence, the analysis of plates under enhanced thermal and moisture conditions has become a topic of research interest for researchers. Wu and Tauchert [23] used CLPT for the analysis for different temperature fields. Reddy and Hsu [24] used FSDT and presented analytical solutions for the thermomechanical response of laminated plate. Sai Ram and Sinha [25] presented finite element solutions for the bending characteristic of laminated plates under hygrothermal environment. Zenkour [26, 27] and Zenkour and Alghanmi [28] studied the thermal behaviour of the plate using Sinusoidal shear deformation Plate Theory (SPT) and compared results with FSDT and HSDT. Joshan et al. [29, 30] assessed Inverse Hyperbolic Shear Deformation Theories (IHSDT) and presented analytical results for the hygro-thermo-mechanical response of plate. Also, Mechab et al. [31] presented results for thermo-mechanical analysis of plate working on higher order theory. Apart from these numerical approaches, experimental research work has also been carried out in this area to determine the actual response of laminated composites subjected to different conditions. Jin et al. [32] performed the experimental studies to investigate the thermal buckling behaviour of the circular laminated plates using the digital correlation method. It was found that the results were in agreement with the finite element analysis results of the same plate. Also, Lau et al. [33] studied the quantification of fracture energy in laminated plates at the nanoscale using experimental approach. It may be observed from the literature review that mostly ESL theories are exploited for the analysis of laminated composite and sandwich plates for the elevated thermal environment. The static analysis of laminated composite and sandwich plates under hygrothermal environment in the framework of zigzag model is scarcely available. According to the author’s knowledge, no results have been published so far for the static response of laminated composite and sandwich plates under enhanced moisture and temperature conditions, which incorporates the zigzag concept along with the use of non-polynomial secant trigonometric function to represent the shear deformation in the plate.

This paper deals with the assessment of the recently developed trigonometric zigzag theory [34] for the static analysis of laminated composite and sandwich plates under hygro-thermomechanical loading conditions. In this paper, an attempt is made to further improve the model by modifying the parameters involved in the displacement field. The effect of the span to thickness ratio, material anisotropy, ratio of coefficient of thermal expansion, stacking pattern, boundary conditions and loading conditions is studied. Results of present theory are compared with the results available in the literature to show the applicability of the theory. 2. Mathematical Formulation Fig 1 shows the position of Cartesian co-ordinate system adopted for the plate. Origin is chosen at a corner at mid height. Plate has the dimensions ‘a’, ‘b’, and ‘h’ which represents length, breadth and height respectively. Plate is composed of layers of orthotropic material stacked over each other. 2.1 Assumptions The mathematical formulation is based on the following assumptions: • The middle plane of the plate is taken as the reference plane. • The laminated plate consists of arbitrary number of linearly, homogenous, elastic orthotropic layers perfectly bonded to each other. • The flat plates considered here are free from any kind of initial imperfections. • The analysis follows linear constitutive relations i.e. obey generalized Hooke’s law for the material. • The lateral displacements are small compared to plate thickness. • Normal strain in z-direction is neglected. 2.2 Displacement Field A generalised zigzag field is presented in equation (1) using the Heaviside step function to incorporate the zigzag effect. The in plane displacement pattern assumed by zigzag field is represented in the Fig 2.

n 1

u w U ( x, y, z )  u0 ( x, y )  z 0   ( z  ziu )  H ( z  ziu )   ixu x i 1

nl 1

  ( z  z lj )  H ( z  z lj )   xlj  [ g ( z )  x z ] βx j 1

n 1

V ( x, y, z )  v0 ( x, y )  z

w0 u   ( z  ziu )  H ( z  ziu )   iyu y i 1

(1)

nl 1

  ( z  z lj )  H ( z  z lj )   ylj  [ g ( z )  y z ] βy j 1

W ( x, y, z )  w0 ( x, y )

Here,

u0 , v0 and w0 represents the mid plane displacements in x, y and

respectively.

z

direction

βx and βy are higher order degrees of freedom.  xu ,  xl ,  yu and  yl depend

upon the layer considered and indicates the slope of the concerned layer.

H ( z  ziu ) and

H ( z  z lj ) are the Heaviside step functions. nu and nl represents the number of layers above and below the reference plane respectively. x and y are constant values for a particular stacking pattern, which are determined by applying the traction free conditions at top and bottom surface of the plate. The function g ( z ) describes the non-linear variation of shear deformation. In this formulation, g ( z ) is considered to be a trigonometric secant function [34]. The expression used for representing function g ( z ) is z  sec(rz / h) . Value of shear stress parameter ‘ r ’ is chosen as 0.1, by comparing the results with the exact solutions, along the same lines as reference [14]. The model considered in [34] incorporates different values of x and y for each of the layers in the plate. However, as mentioned, the present model considers a constant or layer independent value for the same. It increases the computational efficiency of the present model. Two essential conditions which are imposed on the presented field are: Traction free boundary condition As mentioned earlier, traction free boundary conditions are imposed on the plate to ensure that the out of plane shear stresses vanish at top and bottom surfaces. Hence

τ xz = 0; τ

yz

 0 at  h / 2 is ensured. Values of constant x and y are determined by using

these conditions. The obtained values of these parameters are presented in Appendix. Inter-laminar shear stress continuity Conditions are imposed to maintain τ

k 1 xz

=τ

k xz

;τ

conditions are used to obtain the value of  ,  ,  xu

the variables  ,  ,  xu

xl

yu

and 

yl

xl

k 1 yz

yu

τ

k yz

at each layer interface. These

and 

yl

in terms of βx or βy . Hence,

do not contribute towards the number of independent

degrees of freedom. 2.3 Constitutive equations As shown by the displacement field in equation (1), transverse displacement is not considered to be dependent on z co-ordinate, which makes it possible to neglect the normal strain in this particular direction. The stress-strain relationship for a particular lamina in global axes can be represented as:  xx   Q11   Q  yy   12 τ xy   Q16    τ xz   0 τ yz      0

Q12

Q16

0

Q 22

Q 26

0

Q 26

Q 66

0

0

0

Q 44

0

0

Q 45

0   xx      β1      1   β  0   yy 2     2              T   C  β 0  xy 3    3     0  Q 45     xz   0          0    Q 55    yz   0 

(2)

It can be represented as

   Qij     T  β C 

(3)

Here Qij  represents the transformed rigidity matrix in global axes.    and β are the matrices representing transformed thermal coefficient and moisture coefficient respectively. The equation (2) and equation (3) is valid only in the temperature range in which the composite behaves linearly. At very high temperatures, the material begins to show nonlinearity. For the analysis at those temperatures, nonlinear stress-strain relationship must be considered. 2.4 Loading conditions Mechanical load

Two different type of mechanical loading are considered in the analysis, i.e. sinusoidal loading (SSL) and uniformly distributed loading (UDL). Expression for these are given as: x   y  q  q0 sin    sin    a   b 

q  q0

(For SSL)

(4)

(For UDL)

(5)

Temperature field Temperature distribution in plate is given as: ∆ T ( x, y, z )  z T2 sin   x   sin   y  h

 a 

 b 

(6)

Moisture field In the similar fashion, moisture distribution is given as: C ( x, y, z ) 

z x   y  C2 sin    sin   h  a   b 

(7)

These temperature and moisture fields vary linearly through thickness, which creates a thermal and moisture load vector in the plate which is formulated by finite element modelling as shown in equation (22) in the next section.

3. Finite Element formulation 3.1 Transforming to C0 continuity The presented displacement field carries first order derivatives of transverse displacements with respect to x and y direction. Hence, it requires C1 continuity model at the element interfaces during its finite element implementation. Hence, to avoid the complexities associated with the C1 continuity requirements, finite element model with C0 continuity is used instead of C1 and two more independent constants are introduced in the field as follows

w0 w   x and 0   y x y Hence the modified displacement field becomes

(8)

nu 1

U ( x, y, z )  u0 ( x, y )  z x   ( z  ziu )  H ( z  ziu )   ixu i 1

nl 1

  ( z  z lj )  H ( z  z lj )   xlj  [ g ( z )  x z ] βx j 1

(9)

nu 1

V ( x, y, z )  v0 ( x, y )  z y   ( z  ziu )  H ( z  ziu )   iyu i 1

nl 1

  ( z  z lj )  H ( z  z lj )   ylj  [ g ( z )  y z ] βy j 1

W ( x, y, z )  w0 ( x, y )

As  ,  ,  xu

xl

yu

and  can be expressed in terms of βx and βy , FEM model is formulated yl

with seven degrees of freedom, i.e. u0 , v0 , w0 , x , y ,βx and βy . An eight noded serendipity plate element is considered for the finite element formulations. Lagrangian shape functions are used for this element which are taken from Cook et al. [35]. The field variables and element geometry can be expressed in terms of shape functions as: n

   Ni i ; i 1

n

n

x   Ni xi ;

y   Ni yi

i 1

(10)

i 1

3.2 Strain-Displacement relations Linear strain-displacement relations are used in the analysis, which are expressed as follows: xx yy  xy  xz

 U  yz     x T

V y

U V  y x

W U  x z

W V   y z 

T

(11)

This can be represented as    H  Where   10

02

06

k11

k21

k61

(12) k12

k23

k64

k65

04

50

k46

k57 

T

(13)

Matrix  H  and terms in matrix    are presented in Appendix. Further,

   can be expressed in terms of differentials of shape functions and degrees of

freedom of all eight nodes of element. The equation generated can be expressed as

   B

(14)

Where,  B  is a matrix of partial derivatives of interpolation functions with respect to x and

y and    u01 v01 w01  x1  y1 βx1 βy1...

T

(15)

Using equation (13) and equation (15), strain matrix can be written as follows

   H  B

(16)

3.3 Energy Calculation Potential energy in the hygrothermal conditions is given as

  Us W

(17)

Here

U s is strain energy and W is work done by hygrothermal load.

Us 

T 1 Qij   dxdydz     k 2 

(18)

Substituting value of  from equation (17)

Us 

T 1 T     B  D B dxdydz  2

Where, D    H T Qij   H  dz

(19) (20)

k

Now considering work done due to hygrothermal load, W     P T

Where  P 

(21)

  B  H  T

T

Qij  hg  dxdydz  k

(22)

hg  represents the initial strain in plate due to temperature or moisture. hg    T  β C

(23)

By using the minimization of potential energy, equilibrium equation is obtained as:

 Ke    Pe 

(24)

Where  Ke     B   D B  dxdy T

(25)

Equation (25) denotes the elemental equilibrium equation. For predicting global response of plate, elemental stiffness matricess and elemental load vectors are assembled suitably to get the global stiffness matrix and global load vector. The equilibrium equation for plate can be written as:

 K     P 

(26)

Where,  K  is global stiffness matrix, and  P  is global load vector. 4. Numerical results and discussion 4.1 Material Properties In the complete analysis presented in this paper, two different types of material models are considered, one for laminated composite plates and other for sandwich plates. Elastic material model for sandwich plates (MM1) [1] For face sheets: (Graphite-Epoxy) E1 f E2 f

 25,

G G12 G13   0.5, 23  0.2,  12   23   13  0.25 E2 f E2 f E2 f

For core: E1c E G G G  2c  0.04, 12  0.016, 13  23  0.06,  12   23   13  0.25 E2 f E2 f E2 f E2 f E2 f

Here, subscript f represent the face sheets and c represents core.

Elastic material model for laminated composite plates(MM2) (Graphite-Epoxy composite) [26] G G E1 G  25, 12  13  0.5, 23  0.2,  12   23   13  0.25 E2 E2 E2 E2

 xx = 10-6,  yy = 3 x 10-6,  xx = 0,  yy = 0.44 (wt. % H2O)-1 Here, subscript 1 and 2 represents the directions parallel and perpendicular to the fibres respectively. 4.2 Boundary conditions

Two types of boundary conditions are used in the present analysis i.e. either simply supported or clamped, on all four edges. The degrees of freedom which will be zero under these conditions are: Simply supported boundary condition

v  w   y βy  0 at x  0 & x  a and

u  w   x βx  0 at

y0 &yb

Clamped boundary condition

u  v  w   x   y βx βy  0 at

x  0, x  a, y  0 & y  b

4.3 Non-dimensional parameters Non-dimensional parameters used throughout this paper are presented in this section. Depending upon the loading case, four types of results are presented in this paper, i.e. results with mechanical loading, with thermal loading, with thermo-mechanical loading and with hygrothermal loading. Non-dimensional parameters (P1) used for mechanical loading are:

w  w *(100E2 h3 / b4q0 )  xx 

 yy τ

xy

τ

yz

τ

xz

(27)  h     bq0

h    xx  b

h  yy b

h  xy b



 yz  xz  

(28)

Non-dimensional parameter (P2) used for thermal and hygrothermal loading is:

w  w *(10h /  xx  a 2  T 2 )

(29)

Non-dimensional parameter (P3) used for thermo-mechanical loading is:   q a4    T a2   w  w *   03    xx 2    h     10h   

1

 E1  1 12  E2     4  where      4G12   1  12 21  12   

(30)

4.4 Calculation of average error Average error is calculated in the presented results to assess the accuracy of the present theory. It is calculated by taking arithmetic mean of percentage errors in each of the quantity.

Percentage error for a quantity is calculated as: result under consideration and

w  % Error   p  1 100 w  e 

where, wp represents the

we denotes the corresponding result given by the exact

elasticity solution. 4.5 Numerical examples Examples presented here are categorised into four parts. Mechanical response of symmetric plates, hygro-thermo-mechanical response of single layered plate, symmetric laminates and antisymmetric laminate is shown. 4.5.1 Symmetric laminated composite and sandwich plates under mechanical loading (a) Three layered sandwich plate (0/C/0) subjected to mechanical load A three layered symmetric sandwich plate is considered having core thickness 0.8h and thickness of face sheets 0.1h each, where h is the total thickness. Plate is made up of the material which have properties described as MM1. Non-dimensional results are presented using parameter P1. Table 1 indicates the non-dimensional deflection and stresses of aforementioned plate for various span to thickness ratio, when plate is subjected to sinusoidal mechanical load ( q0 =1). Plate is considered to be simply supported on all four edges. At first, convergence study is presented for the adopted finite element model. Results for a / h  100 , are provided for mesh size varying from 6x6 to 16x16. It is observed from the table that displacements show an excellent convergence at 8x8 mesh size, however, stresses show convergence at a finer mesh (16x16) as they are the derived quantities. Hence, a mesh size of 16x16 is adopted for all the subsequently presented results. Results are compared with various results available in the literature like Sahoo and Singh [34], Chalak et al. [21], Pandit et al. [22], and Singh et al. [36]. Also, all of these results are compared with the exact 3D analysis performed by Pagano [1] and their degree of accuracy is measured in terms of average error. The average error involved in the present theory is significantly lower for thicker plates, than other theories. It can be observed that the present theory is predicting more accurate results compared to other existing theories. Fig 3 indicates the results of non-dimensional transverse deflection when plate is subjected to SSL and UDL mechanical loads ( q0 =1) separately, under clamped boundary condition. It can

be observed that non dimensional deflection of plate subjected to UDL is higher than that of plate subjected to SSL for all span to thickness ratios. Also, the difference in non-dimensional deflections under both the loadings is high in case of thick plates but it rapidly decreases and becomes negligible as plate becomes thinner. (b) Three layered laminated plate (0/90/0) subjected to mechanical load A three layered symmetric cross ply laminate (0/90/0) is considered, in which all the layers have equal thickness. MM2 material model is used for the analysis. The plate is analysed for simply supported condition on all the four edges. P1 parameter is used to present nondimensional results. In Table 2, non-dimensional transverse deflection is presented for various span to thickness ratio when the plate is subjected to UDL. Results are compared with the exact results given by Pagano [1] to estimate the error. Along with this, results are also matched with theories presented by Sahoo and Singh [34], Singh et al. [36], Sheikh and Chakrabarti [37], and Reddy [10] to prove the superiority of present theory over those. Average error for results of all the theories is calculated for assessing the accuracy of the given results. It can be noticed that present theory predicts the results with an error of 0.3% only whereas all other presented theories have errors more than 3%. Also, it is observed that the other theories only predicts accurate results in case of thin plates, however, the present theory gives accurate results for thin as well as thick plates. (c) Four layered laminated plate (0/90/90/0) subjected to mechanical load In this example, a (0/90/90/0) plate is considered to be simply supported on all the four edges. Plate is assumed to be made up of material having properties MM2. Thickness of each layer in the plate is equal. P1 parameter is used for calculating the non-dimensional deflections and stresses. Table 3 indicates the results for non-dimensional deflection and stresses for the plate subjected to sinusoidal mechanical loading. Results are compared with the various results available in the literature including the exact solutions given by Pagano and Hatfield [2], trigonometric zigzag formulation of Sahoo and Singh [34], HSDT using analytical approach by Reddy [10] and using finite strip approach by Akhras et al. [38], non-polynomial shear deformation theories of Grover et al. [14] and Mantari et al. [39] and the layerwise formulation of Ferreira [40] and Roque et al. [41]. It can be observed that present theory predicts results that are in good agreement with the other theories. Also, the non-dimensional deflections predicted by present theory are the most accurate among all the other cited

theories. Along with that, stresses are also predicted with sufficient accuracy by the present formulation. 4.5.2 Single layered orthotropic plate (0) for enhanced thermal and moisture conditions An orthotropic plate (single layer laminate) is considered to have orientation of fibres at 0o. Plate is simply supported on all four edges and subjected to three different load cases, i.e. thermal load, thermo-mechanical load and hygrothermal load. Suitable non-dimensional parameter is used for the corresponding loading case as described in section 4.3. Material properties given by MM2 are used for the analysis Table 4 shows the results of maximum transverse non-dimensional deflections of the plate for various span to thickness ratio. First, for the thermal loading case, only temperature variation in the plate is considered and hence,

q0 = C2 = 0 and the value of T2 is taken as 100 oC.

Presented results are compared with the IHSDT results [29], FSDT formulation of Reddy and Hsu [24], and results of HSDT and SPT given by Zenkour [26]. It can be observed that the results predicted by present theory are in good agreement with previously presented theories for thin as well as thick plates. Further, for thermo-mechanical loading, SSL in transverse direction is also considered along with the temperature variation in the plate. Values used in the analysis are C2 = 0,

q0 =100

and T2 =100 oC. Results are compared with the aforementioned literature and are found to be in a range with those. Moreover, in case of hygrothermal conditions, both the temperature and moisture are considered to vary linearly through the thickness in a sinusoidal manner in the xy plane. Values of parameters are taken as

q0 = 0, T2 = 300 oC and C2 = 0.01%. Results of present

theory are compared with FSDT, HSDT, and SPT analysed by Zenkour [28]. It can be noted that for all three cases, the non-dimensional deflection increases with decrease in span to thickness ratio. Also, in this particular example, due to presence of only one layer, results predicted by all the theories are very close to each other, even for the thick plates. However, considering small difference that is present in the results, it can be observed that FSDT overpredicts the deflections whereas IHSDT under-predicts it as compared to other higher order theories and the present zigzag theory.

4.5.3 Symmetric cross ply laminates under enhanced temperature and moisture conditions. (a) Three layered laminated composite plate (0/90/0) A symmetric cross ply laminate (0/90/0) is considered for analysis. All the layers are of equal thickness and built up with the orthotropic material having properties described as MM2. Plate is in simply supported conditions at all the four edges. Table 5 provides the maximum non-dimensional transverse deflection of considered square laminated plate for thermal, thermo-mechanical and hygrothermal loading cases. P2 and P3 are used as non-dimensional parameters for the corresponding loads. Firstly, the results are given for sinusoidal thermal loading ( q0 = C2 = 0, T2 = 100oC) and are compared with the different theories including FSDT [24], HSDT [26], SPT [26] and IHSDT [29]. The results provided by the present theory are matching with the other theories. For thin plates (a/h=100), difference in the results predicted by different theories is negligible. However, the difference is substantial in the case of thick plates (a/h=5). Also, non-dimensional deflections predicted by HSDT are more than that of predicted by FSDT. Further, the present zigzag theory predicts results which are higher than the HSDT. Further in the Table 5, results are presented for thermo-mechanical ( q0 =100, C2 = 0, T2 = 100 o

C) as well as hygrothermal ( C2 = 0.01%, T2 = 300oC) loading. Results of thermo-mechanical

loading are compared with above mentioned literature and of hygrothermal loading are compared with Zenkour and Alghanmi [28]. Again, for both the loading cases, an increasing trend of deflections is observed while moving from the results of FSDT to HSDT and then HSDT to present zigzag theory. Moreover, maximum transverse non-dimensional deflection of rectangular plate under thermal and hygrothermal loading conditions is presented in Table 6 and Table 7 respectively. Results are provided for various aspect ratios (1/3, 0.5, 1.5, and 2) and span to thickness ratios (5, 10, 20, 50, and 100). Results given by present theory are compared with aforesaid respective literature [24,26,28,29]. For this case also, results of the present theory are on higher side than those of predicted HSDT and FSDT. This is because in case of thick plates, shear deformation plays a key role in determining the response of the plate. Every theory assumes a different variation of shear deformation across

thickness. It is well established that HSDT predicts the better results than FSDT due to consideration of non-linear variation in shear deformation. Zigzag theory considers interlaminar stress continuity and represents a more accurate picture of real scenario than HSDT and hence it provides more precise results. (b) Four layered laminated plate (0/90/90/0). The next example is considered with the four layered symmetric laminate (0/90/90/0). MM2 material model is used for the analysis. Thickness of each layer in the plate is equal and all the edges of plate are simply supported. Suitable non-dimensional parameters are used according to the loading case considered. In Fig 4, the effect of enhanced thermal conditions on mechanical response of the plate is studied. Response is measured in terms of maximum non-dimensional transverse deflection. The simply supported plate is subjected to a sinusoidal mechanical load ( q0 = 100) and its response is compared with the plate which is subjected to same load along with an additional thermal gradient ( q0 = 100, T2 =100oC). The results obtained for various span to thickness ratio and show that non-dimensional deflection increases due to presence of temperature gradient. Also, it can be observed that for thin and moderately thick plates, non-dimensional deflection increases by a constant value for any span to thickness ratio. However, in case of thick plates, difference in the non-dimensional deflections of both the loads increases as plate becomes thicker. Hence, it can be concluded that effect of thermal gradient shows a more pronounced effect on thick plates. Further, the plate is subjected to thermal loading ( T2 =100oC). Fig 5 shows the variation of non-dimensional deflection of this plate with ratio of thermal coefficients (  yy /  xx ) for various span to thickness ratio. It can be observed from the plot that non-dimensional deflection varies linearly with the ratio of thermal coefficient, irrespective of the a / h ratio. However, the rate of increase in the deflection is higher in case of thick plates compared to thin plates. Variation of non-dimensional deflection with span to thickness ratio is presented in Fig 6 when the plate subjected to hygrothermal ( C2 = 0.01%, T2 = 300 oC) and thermal load ( T2 =100oC) separately. The presented results are compared with the results developed by Zenkour and Alghanmi [28] using SPT. It can be observed from the plot that present theory

predicts the results that are similar to those of SPT. Non-dimensional deflection decreases with an increase in the a / h ratio, under both the loading conditions. However, the rate at which non-dimensional deflection decreases, is much higher for thick plates under hygrothermal load as compared to plates subjected to thermal load. To study the effect of material anisotropy on static response of thick plate ( a / h  10 ) under hygrothermal and thermal environment, a curve of non-dimensional deflection is plotted with respect to the modular ratio ( E1 / E2 ) in Fig 7. For thermal loading, non-dimensional deflection increases as modular ratio increases. However, for hygrothermal loading case, initially, a decrement in the non-dimensional deflection is observed with an increase in modular ratio for lower degree of anisotropy. But afterwards, it also follows an increasing trend. Also, it can be observed that rate of change in non-dimensional deflection is higher under the hygrothermal loading conditions. 4.5.4 Anti-symmetric cross ply (0/90) laminate under thermal, thermo-mechanical and hygrothermal loading conditions. An anti-symmetric two layered laminated composite plate (0/90) is considered. Both the layers of the laminate are of equal thickness and made up of orthotropic material having properties MM2. P2 and P3 are used as non-dimensional parameters corresponding to their load cases. Results for maximum transverse non-dimensional deflections of simply supported plate under different loading conditions are presented in Table 8. In case of thermal loading ( T2 =100oC), results of present theory are compared with Joshan et al. who worked with sine inverse hyperbolic [29] and tangent inverse hyperbolic [30] shear strain functions, FSDT formulation of Reddy and Hsu [24], HSDT and SPT formulation of Zenkour [26], and the higher order formulation of Mechab et al. [31]. It can be seen that the results predicted by the present theory are in good agreement with the other theories. Also, Table 8 provides results for the plate subjected to thermo-mechanical ( q0 = 100, T2 =100oC) and hygrothermal ( C2 = 0.01%,

T2 = 300 oC) loading. The static response of hygrothermal loading is compared with the results presented in the literature [28, 30]. It can be noticed that results of present theory lie in between the results of FSDT and various compared higher order theories. Also, the difference in the results presented by various theories almost vanishes for the thin plates. The presented theory considers the zigzag variation of displacements which is a more accurate

representation than that of assumed by HSDT. Hence, it can be concluded that results obtained by present theory are more precise. Fig 8 shows the variation of non-dimensional deflections of the plate described above, with material anisotropy. Results are plotted for thick plate (a/h=5) for thermal and hygrothermal loading for simply supported and clamped boundary conditions. It can be observed from the figure that the non-dimensional deflection is more in case of simply supported condition for a particular loading case. Also, the figure indicates that the effect of material anisotropy on the non-dimensional deflection is distinct only in the case of simply supported plate. For the clamped plate, the rate of change of non-dimensional deflection with material anisotropy is very close to zero. Further, it is observed that for the case of simply supported plate, nondimensional deflections increase more rapidly for hygrothermal conditions as compared to thermal conditions. 5. Conclusion In this paper, recently presented trigonometric zigzag theory is improved and assessed for hygro-thermo-mechanical analysis of laminated composite and sandwich plates. The presented displacement field in the theory inherently satisfies the traction free boundary condition, inter-laminar shear stress continuity, and in-plane displacement continuity. Zig-zag variation is incorporated in the field with the help of Heaviside step function and number of variables involved are only seven. Accounting for zigzag concept not only decreases the computational cost as compared to layer-wise models, but also increases the accuracy of the results because it represents the very realistic variation of stresses and displacement. An efficient finite element method is employed using the C0 continuity to establish results for deflection and generated stresses under different conditions such as aspect ratio of plate, thickness to width ratio of plate, loading condition etc. Results are shown for an optimum mesh size suitable for required accuracy (as shown in convergence study). It is observed that results for the static response of laminated composite and sandwich plates are in good agreement with the results presented by other theories. From the results presented, the following conclusions can be drawn: 1. Non-dimensional deflection decreases with increase in the span to thickness ratio of the plate. 2. Non-dimensional deflection is higher for uniformly distributed mechanical loading as compared to sinusoidal mechanical loading.

3. Higher value of non-dimensional deflection is observed for the hygrothermal environment than thermal conditions due to the presence of moisture concentration. 4. Difference between the non-dimensional deflection for thermal and hygrothermal loading conditions increases substantially when plate becomes thicker. Similar pattern is observed for difference in non-dimensional deflection for mechanical and thermo-mechanical loading. 5. Rate of change of non-dimensional deflection with respect to span to thickness ratio and modular ratio for hygrothermal conditions is significantly higher compared to thermal conditions. Hence, the present theory is recommended for the study of static characteristics of laminated and sandwich flat panels under hygro-thermo-mechanical environment. Acknowledgement The corresponding author acknowledges the support due to Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India under grant number ECR/2016/001926. Appendix For (0/90/0) and (0/90/90/0) plates: x 

 g (h / 2)  g ( z1 )  Rx 1  Rx ( 1)

Where, Rx  Q55( 2)  1 Q55

y 

 g (h / 2)  g ( z1 )  Ry

Ry 

1  Ry

Q45( 1) 1 Q45( 2)

Here, the ‘+’ sign denotes the layers above the reference axis and the ‘–‘ sign denotes the layers below it.

1 0  [ H ]  0  0 0 Where

0 1 0 0 0

0 0 1 0 0

z 0 0 0 0

0 z 0 0 0

0 0 z 0 0

p1 0 0 0 0

0 p2 0 0 0

0 0 p1 0 0

0 0 p2 0 0

0 0 0 0 1

0 0 0 0 0 0  0 0 0  1 0 q2  0 q1 0 

nl 1

nu 1

p1  g ( z )   ( z  ziu )  H ( z  ziu )  Rix  [g ( ziu )  x ]   ( z  z lj )  H ( z  z lj )  R jx  [ g ( z lj )  x ] i 1

j 1

nu 1

nl 1

p1  g ( z )   ( z  ziu )  H ( z  ziu )  Riy  [g ( ziu )  y ]   ( z  z lj )  H ( z  z lj )  R jy  [ g ( z lj )  y ] i 1

j 1

nl 1

nu 1

q2  g ( z )   H ( z  ziu )  Rix  [g ( ziu )  x ]   H ( z  z lj )  R jx  [ g ( z lj )  x ] i 1

j 1

nu 1

nl 1

i 1

j 1

q1  g ( z )   H ( z  ziu )  Riy  [g ( ziu )  y ]   H ( z  z lj )  R jy  [ g ( z lj )  y ]

The individual terms in equation (13): 10 

v  u v  βx u0   ; 02  0 ; 06   0  0  ; k11    x  x y x x  x  y x 

βy   y  1  y  ; k2    y   y

 ; 

βy βy      βy  βx βx βx 3 2 ; k2  ; k64  ; k65  ; k61     x  y   x  y  ; k1  y y x x x  y x    y  w  04   0  yβy   y  ;  y 

 w  50   0  xβx   x  ;  x 

k46 βy ;

k57 βx

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List of Figures Fig 1. Co-ordinate system adopted for the analysis of the plate. Fig 2. Zigzag variation of in-plane displacements along the thickness of the plate. Fig 3. Variation of non-dimensional deflection of clamped sandwich plate (0/C/0) subjected to SSL and UDL with span to thickness ratio. Fig 4. Variation of non-dimensional deflection of simply supported (0/90/90/0) plate subjected to mechanical and thermo-mechanical loading with span to thickness ratio. Fig 5. Variation of non-dimensional deflection for (0/90/90/0) plate with ratio of thermal coefficients under thermal loading for various span to thickness ratio. Fig 6. Variation of non-dimensional deflection of (0/90/90/0) plate with span to thickness ratio under thermal and hygrothermal loading. Fig 7. Variation of non-dimensional deflection of (0/90/90/0) plate with modular ratio under thermal and hygrothermal loading.

Fig 8. Variation of non-dimensional deflection of two layered anti-symmetric cross ply (0/90) with modular ratio under thermal and hygrothermal load for different boundary conditions.

List of Tables a/h

References

a,b  2 2

w

a,b,h  2 2 2

 xx 

a b h 2 2 2

 yy  , , 

τ

xy

 0, 0, h    2 

τ

xz

 0, b , 0     2 

τ

yz

 a , 0, 0    2 

Avg. Error

100 Present (6x6)

0.8920

1.1463

0.0574

0.0455

0.3569

0.0345

-

Present (8x8)

0.8923

1.1253

0.0563

0.0447

0.3508

0.0333

-

Present (10x10)

0.8923

1.1154

0.0558

0.0444

0.3479

0.0329

-

Present (12x12)

0.8924

1.1100

0.0556

0.0441

0.3463

0.0328

-

Present (14x14)

0.8924

1.1067

0.0554

0.0440

0.3453

0.0327

-

Present (16x16)

0.8924

1.1045

0.0553

0.0439

0.3446

0.0326

4.24

Pagano [1]

0.8917

1.0980

0.0552

0.0433

0.3040

0.0297

-

Sahoo and Singh [34]

0.8921

1.1119

0.0557

0.0442

0.3048

0.0263

2.67

Chalak et al. [21]

0.8814

1.0982

0.0592

0.0433

0.3426

0.0332

5.48

Pandit et al. [22]

0.8917

1.1093

0.0547

0.0434

0.3412

0.0324

3.92

Singh et al. [36]

0.9017

1.1020

-

0.0453

0.4079

-

10.07

Present

0.9348

1.1060

0.0572

0.0449

0.3437

0.0335

2.96

Pagano [1]

0.9348

1.0990

0.0569

0.0446

0.3230

0.0306

-

Sahoo and Singh [34]

0.9338

1.1134

0.0576

0.0451

0.3040

0.0270

3.57

Chalak et al. [21]

0.9234

1.0997

0.0611

0.0443

0.3300

0.0341

3.82

Pandit et al. [22]

0.9341

1.0948

0.0566

0.0445

0.3403

0.0333

2.56

Singh et al. [36]

0.9458

1.1050

-

0.0465

0.3617

-

4.49

Present

1.2262

1.1170

0.0702

0.0515

0.3376

0.0394

2.77

Pagano [1]

1.2264

1.1161

0.0700

0.0511

0.3174

0.0361

-

Sahoo and Singh [34]

1.2257

1.1244

0.0706

0.0515

0.2986

0.0315

3.52

Chalak et al. [21]

1.2121

1.1103

0.0742

0.0508

0.3272

0.0399

3.65

Pandit et al. [22]

1.2252

1.1055

0.0694

0.0509

0.3342

0.0392

2.70

Singh et al. [36]

1.2424

1.1100

-

0.0536

0.3429

-

3.69

Present

2.2010

1.1603

0.1098

0.0717

0.3190

0.0574

1.54

Pagano [1]

2.2004

1.1530

0.1104

0.0707

0.3000

0.0575

-

Sahoo and Singh [34]

2.1999

1.1680

0.1105

0.0712

0.2821

0.0454

4.86

50

20

10

4

Chalak et al. [21]

2.1775

1.1528

0.1143

0.0705

0.3058

0.0360

7.37

Pandit et al. [22]

2.0020

1.1483

0.1086

0.0709

0.3158

0.0527

4.16

Singh et al. [36]

2.2389

1.1594

-

0.0707

0.3287

-

2.97

Present

7.6392

1.5403

0.2532

0.1484

0.2540

0.1164

3.70

Pagano [1]

7.5962

1.5560

0.2595

0.1437

0.2390

0.1072

-

Sahoo and Singh [34]

7.6351

1.5503

0.2549

0.1469

0.2225

0.0913

4.44

Chalak et al. [21]

7.5822

1.5306

0.2581

0.1445

0.2436

0.1147

1.97

Pandit et al. [22]

7.6552

1.5218

0.2506

0.1468

0.2520

0.1156

3.64

Singh et al. [36]

7.8556

1.5488

-

0.1671

0.2611

-

7.35

Table 1 Non-dimensional transverse deflections and stresses for (0/C/0) simply supported sandwich plate subjected to SSL ( q0

 1 ) (MM1)

Non-dimensional deflection ( w) References a/h=100

a/h=50

a/h=20

a/h=10

a/h=4

a/h=2

Avg. Error

Present

0.6713

0.6872

0.7960

1.1581

3.0611

7.7991

0.36

Pagano [1]

0.6712

-

-

1.1533

3.0416

-

-

Sahoo and Singh [34]

0.6706

0.6843

0.7785

1.0981

2.9096

7.7634

3.07

Singh et al. [36]

0.6706

0.6843

0.7785

1.0981

2.9096

7.7632

3.07

Sheikh and Chakrabarti (HSDT) [37]

0.6708

0.6841

0.7763

1.0910

2.9093

7.7670

3.27

Sheikh and Chakrabarti (FSDT) [37]

0.6707

0.6813

0.7588

1.0235

2.6608

7.7068

7.95

Reddy (HSDT) [10]

0.6705

0.6838

0.7760

1.0900

2.9091

7.7671

3.32

Reddy (FSDT) [10]

0.6697

0.6807

0.7573

1.0219

2.6596

7.7062

8.06

Table 2 Non-dimensional transverse deflection of three layered simply supported symmetric laminated (0/90/0) plate subjected to UDL (MM2) ( q0

a/h 4

References

a,b  2 2

w

a,b,h  a,b,h τ yy    2 2 2 2 2 4

 xx 

xy

 1)

 0, 0, h  τ  0, b , 0  τ   xz   2   2 

yz

 a , 0, 0    2 

Present

1.9221

0.6868

0.7131

0.0430

0.2237

0.2335

Pagano and Hatfield [2]

1.9540

0.7200

0.6630

0.0470

0.2190

0.2910

Sahoo and Singh [34]

1.9016

0.7149

0.6391

0.0467

0.2366

0.2913

Grover et al. [14]

1.8935

0.6650

0.6322

0.0441

0.2063

0.2389

10

Mantari et al. [39]

1.9210

0.7400

0.6350

0.0480

0.2540

0.2690

Reddy [10]

1.8937

0.6650

0.6320

0.0440

0.2060

0.2390

Akhras et al. [38]

1.8939

0.6806

0.6463

0.0450

0.2109

0.2444

Ferreira [40]

1.9075

0.6420

0.6228

0.0441

0.2166

-

Roque et al. [41]

1.8842

0.7560

0.6777

0.0430

0.1885

-

Present

0.7362

0.5611

0.4112

0.0275

0.3121

0.1478

Pagano and Hatfield [2]

0.7430

0.5590

0.4010

0.0280

0.3010

0.1960

Sahoo and Singh [34]

0.7210

0.5682

0.3962

0.0276

0.3385

0.1963

Grover et al. [14]

0.7147

0.5456

0.3888

0.0268

0.2639

0.1531

Mantari et al. [39]

0.7300

0.5610

0.3950

0.0280

0.3350

0.1760

Reddy [10]

0.7147

0.5460

0.3890

0.0270

0.2640

0.1530

Akhras et al. [38]

0.7149

0.5589

0.3974

0.0273

0.2697

0.1568

Ferreira [40]

0.7309

0.5496

0.3956

0.0273

0.2888

-

Roque et al. [41]

0.7350

0.5637

0.4055

0.0272

0.2908

-

Table 3 Non-dimensional deflection and stresses of simply supported symmetric cross ply laminate (0/90/90/0) subjected to SSL ( q0  1 ) (MM2) Non-dimensional deflection ( w) Loading

References a/h=5

a/h=6.25 a/h=10 a/h=12.5 a/h=20 a/h=25 a/h=50 a/h=100

Present

1.0711

1.0593

1.0438

1.0395

1.0345

1.0333 1.0316

1.0312

Sinusoidal

Joshan et al. [29]

1.0693

1.0586

1.0435

1.0394

1.0345

1.0333 1.0317

1.0313

Thermal Load

Reddy and Hsu [24]

1.0721

1.0602

1.044

1.0396

1.0346

1.0334 1.0317

1.0313

( T2 = 100 oC)

Zenkour (HSDT) [26]

1.0711

1.0597

1.0439

1.0396

1.0346

1.0334 1.0317

1.0313

Zenkour (SPT) [26]

1.0708

1.0595

1.0438

1.0396

1.0346

1.0334 1.0317

1.0313

Sinusoidal

Present

2.7967

2.1706

1.4643

1.2961

1.1112

1.0681 1.0104

0.996

thermo-

Joshan et al. [29]

2.7102

2.1244

1.4502

1.2877

1.1082

1.0663 1.0101

0.9959

mechanical Load

Reddy and Hsu [24]

2.8332

2.1868

1.4671

1.2973

1.115

1.0683 1.0105

0.9962

( T2 = 100 oC,

Zenkour (HSDT) [26]

2.7769

2.1631

1.4634

1.2958

1.1113

1.0682 1.0105

0.9961

q0 = 100)

Zenkour (SPT) [26]

2.7654

2.1575

1.4621

1.2951

1.1111

1.0681 1.0105

0.996

Sinusoidal

Present

2.2725

2.1035

1.8703

1.8069

1.7332

1.7154 1.6911

1.685

hygrothermal

Zenkour and

Load

Alghanmi (FSDT) [28] 2.2859

2.11

1.8717

1.8075

1.7334

1.7155 1.6913

1.6851

( T2 = 300 oC,

Zenkour and Alghanmi (HSDT)

C2 = 0.01%)

[28]

2.2709

2.103

1.8704

1.8069

1.7333

1.7155 1.6913

1.6913

2.2669

2.1009

1.8699

1.8067

1.7332

1.7154 1.6912

1.6851

Zenkour and Alghanmi (SPT) [28]

Table 4 Non-dimensional transverse deflection of simply supported single layered orthotropic square plate (0) (MM2)

( yy /  xx  3)

Non-dimensional deflection ( w) Loading

References a/h=5

a/h=6.3 a/h=10 a/h=12.5 a/h=20 a/h=25 a/h=50 a/h=100

Present

1.2732

1.2261

1.1576

1.1369

1.1119 1.1057

1.0972

1.0950

Sinusoidal

Joshan et al. [29]

1.2517

1.2125

1.1507

1.1323

1.1101 1.1045

1.0970

1.0950

thermal load

Reddy and Hsu [24]

1.2224

1.1870

1.1365

1.1224

1.0158 1.1018

1.0963

1.0949

( T2 = 100 oC)

Zenkour (HSDT) [26]

1.2452

1.2057

1.1463

1.1292

1.1087 1.1036

1.0967

1.0950

Zenkour (SPT) [26]

1.2472

1.2077

1.1475

1.1300

1.1090 1.1039

1.0968

1.0950

Sinusoidal

Present

3.5205

2.6996

1.7345

1.4793

1.1877 1.1178

1.0231

0.9991

thermo-

Joshan et al. [29]

3.4075

2.6317

1.6845

1.4443

1.1726 1.1080

1.0206

0.9966

mechanical load

Reddy and Hsu [24]

3.0377

2.9983

1.5384

1.3451

1.1312 1.0811

1.0138

0.9973

( T2 = 100 oC,

Zenkour (HSDT) [26]

3.2948

2.5394

1.6366

1.4115

1.1587 1.0989

1.0183

0.9980

Zenkour (SPT) [26]

3.3238

2.5637

1.6493

1.4202

1.1624 1.1013

1.0189

0.9982

Present

4.0748

3.6022

2.8721

2.6559

2.3945 2.3296

2.2403

2.2174

Sinusoidal

Zenkour and

hygrothermal

Alghanmi (FSDT) [28] 3.5659

3.1907

2.6560

2.5067

2.3309 2.2882

2.2297

2.2148

load

Zenkour and

( T2 = 300 oC,

Alghanmi (HSDT) 3.8261

3.3989

2.7623

2.5794

2.3616 2.3081

2.2348

2.2161

3.8499

3.4207

2.7749

2.5882

2.3654 2.3106

2.2355

2.2163

q0 = 100)

[28]

C2 = 0.01%)

Zenkour and Alghanmi (SPT) [28]

Table 5 Non-dimensional transverse deflection of simply supported three layered symmetric square plate (0/90/0) subjected to thermal, thermo-mechanical and hygrothermal loading. (MM2) ( yy /  xx  3) Aspect ratio 1/3

Non-dimensional deflection ( w) References Present

a/h=5

a/h=10

a/h=20

a/h=50

a/h=100

1.1129

1.0744

1.063

1.0597

1.0592

1/2

1.5

2

Joshan et al. [29]

1.1097

1.0733

1.0628

1.0597

1.0593

Reddy and Hsu [24]

1.0998

1.0701

1.0619

1.0596

1.0593

Zenkour (HSDT) [26]

1.1073

1.0724

1.0625

10567

1.0593

Zenkour (SPT) [26]

1.1081

1.0726

1.0626

1.0597

1.0593

Present

1.1817

1.1054

1.0821

1.0751

1.0741

Joshan et al. [29]

1.174

1.1029

1.0814

1.075

1.0741

Reddy and Hsu [24]

1.1535

1.0959

1.0795

1.0748

1.0741

Zenkour (HSDT) [26]

1.1689

1.1008

1.0808

1.075

1.0741

Zenkour (SPT) [26]

1.1704

1.1014

1.081

1.075

1.0741

Present

1.0325

1.0054

0.9909

0.9855

0.9847

Joshan et al. [29]

1.0156

1.0005

0.9896

0.9853

0.9847

Reddy and Hsu [24]

1.0157

0.9973

0.9883

0.9851

0.9847

Zenkour (HSDT) [26]

1.0169

0.9997

0.9892

0.9853

0.9847

Zenkour (SPT) [26]

1.0167

0.9999

0.9893

0.9853

0.9847

Present

0.7274

0.7434

0.757

0.7631

0.7641

Joshan et al. [29]

0.7187

0.7432

0.7574

0.7632

0.7642

Reddy and Hsu [24]

0.7355

0.7508

0.7601

0.7638

0.7643

Zenkour (HSDT) [26]

0.7237

0.7455

0.7583

0.7634

0.7642

Zenkour (SPT) [26]

0.7225

0.7449

0.7581

0.7634

0.7642

Table 6 Non-dimensional transverse deflection of simply supported three layered symmetric square plate (0/90/0) for different aspect ratios under thermal loading conditions. (MM2). ( T2 = 100 oC)

( yy /  xx  3) Aspect Ratio

1/3

1/2

Non-dimensional deflection ( w) References a/h=5

a/h=10

a/h=20

a/h=50

a/h=100

Present

1.7264

1.4493

1.3676

1.3437

1.3402

FSDT [28]

1.6324

1.4178

1.3591

1.3423

1.3399

HSDT [28]

1.6879

1.4345

1.3635

1.343

1.3401

SPT [28]

1.6934

1.4366

1.3641

1.3431

1.3401

Present

2.3205

1.7451

1.5683

1.5158

1.5081

FSDT [28]

2.1099

1.6728

1.5486

1.5126

1.5074

HSDT [28]

2.229

1.7101

1.5585

1.5142

1.5078

SPT [28]

2.2409

1.7147

1.5598

1.5144

1.5078

1.5

2

Present

4.1237

3.3373

2.8895

2.7214

2.6953

FSDT [28]

3.7212

3.109

2.8121

2.7078

2.6919

HSDT [28]

3.9059

3.2155

2.8477

2.7141

2.6935

SPT [28]

3.9206

3.2273

2.8521

2.7148

2.6937

Present

3.2301

2.9226

2.6909

2.59

2.5735

FSDT [28]

3.0204

2.7826

2.6372

2.5801

2.5711

HSDT [28]

3.1037

2.8434

2.6605

2.5844

2.5722

SPT [28]

3.1094

2.8496

2.6631

2.5849

2.5724

Table 7 Non-dimensional transverse deflection of simply supported three layered symmetric square plate (0/90/0) for different aspect ratio under hygrothermal loading. (MM2). ( T2 = 300oC and C2 = 0.01%) ( yy /  xx  3)

Non-dimensional deflection Loading

References

a/h=5

a/h=6.25 a/h=10 a/h=12.5 a/h=20 a/h=25 a/h=50 a/h=100

Present

1.6845

1.6818

1.6786

1.6778

1.677

1.6768

1.6765

1.6764

Joshan et al. [29]

1.6955

1.6888

1.6814

1.6796

1.6777 1.6773

1.6767

1.6766

Joshan et al. [30]

1.6894

1.6848

1.6798

1.6786

1.6773

1.677

1.6767

1.6766

Reddy and Hsu [24]

1.6765

1.6765

1.6765

1.6765

1.6765 1.6765

1.6765

1.6765

Zenkour (HSDT) [26]

1.6894

1.6848

1.6798

1.6786

1.6773

1.677

1.6767

1.6766

Zenkour (SPT) [26]

1.691

1.6858

1.6802

1.6789

1.6774 1.6771

1.6767

1.6766

Mechab et al. [31]

1.6896

1.6849

1.6798

1.6786

1.6773

1.677

1.6767

1.6766

Sinusoidal

Present

3.8732

3.3608

2.8033

2.6742

2.5343 2.5019

2.4588

2.4481

thermo-

Joshan et al. [29]

3.6797

3.2433

2.7602

2.6471

2.524

2.4955

2.4574

2.4478

mechanical load

Joshan et al. [30]

3.8302

3.3362

2.795

2.6692

2.5325 2.5009

2.4587

2.4482

( T2 = 100 oC,

Reddy and Hsu [24]

4.0415

3.4666

2.8438

2.7001

2.5443 2.5083

2.4597

2.4541

Zenkour (HSDT) [26]

3.812

3.3273

2.7927

2.6679

2.5321 2.5006

2.4586

2.4481

Zenkour (SPT) [26]

3.7821

3.309

2.7859

2.6636

2.5304 2.4996

2.4584

2.4481

Present

7.8613

7.8484

7.8335

7.8299

7.826

7.8251

7.8238

7.8235

Joshan et al. [30]

7.884

7.8625

7.839

7.8335

7.8276 7.8262

7.8244

7.8239

Zenkour and Alghanmi

7.884

7.8625

7.839

7.835

7.8276 7.8262

7.8234

7.8238

Sinusoidal thermal load o

( T2 = 100 C)

q0 = 100)

Sinusoidal hygrothermal load

( T2 = 300 oC,

(HSDT) [28] Zenkour and Alghanmi

C2 = 0.01%)

(FSDT) [28]

7.8238

7.8238

7.8238

7.8238

7.8238 7.8238

7.8238

7.8238

7.8913

7.8672

7.8408

7.8247

7.828

7.8245

7.8239

Zenkour and Alghanmi (SPT) [28]

7.8265

Table 8 Non-dimensional transverse deflection of simply supported two layered anti-symmetric square plate (0/90) (MM2)

( yy /  xx  3)

z

y

b x a Fig 1. Co-ordinate system adopted for the analysis of the plate.

Znu Zn-1

αxunu-1

nu nu-1

u 1

3 Z2

u

Z1

u

2

1

1

-2

αxl1

l

x

-1

x

Z1l Z2

αxu1

-3

nl-1 Zn-1 Zn

l

αxlnl-1

l

nl

1

Fig 2. Zigzag variation of in-plane displacements along the thickness of the plate.

Non-dimensional deflection

9

8 7

Uniformly distributed mechanical load

6

Sinusoidal mechanical load

5 4 3 2 1 0 4

6.25

10

12.5

20

25

50

100

Span to thickness ratio

Fig 3. Variation of non-dimensional deflection of clamped sandwich plate (0/C/0) subjected to SSL and UDL with span to thickness ratio.

Non-dimensional deflection

5 4.5 4 3.5

Mechanical (SSL) load

3

Thermo-mechanical load

2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

80

90

100

110

Span to thickness ratio

Fig 4. Variation of non-dimensional deflection of simply supported (0/90/90/0) plate subjected to mechanical and thermo-mechanical loading with span to thickness ratio.

Non-dimensional deflection

4.5 4

a/h=5 a/h=10 a/h=20 a/h=50 a/h=100

3.5 3 2.5 2 1.5 1 2

3

4

5

6

7

8

9

10

Ratio of thermal coefficients

Fig 5. Variation of non-dimensional deflection for (0/90/90/0) plate with ratio of thermal coefficients under thermal loading for various span to thickness ratio.

Non-dimensional deflection

9

Thermal (SPT) [28] Hygrothermal (SPT) [28] Thermal (Present) Hygrothermal (Present)

8 7 6 5 4 3 2 1 0 2

4

6

8

10

12

14

16

18

20

Span to thickness ratio

Fig 6. Variation of non-dimensional deflection of (0/90/90/0) plate with span to thickness ratio under thermal and hygrothermal loading.

Non-dimensional deflection

5.2 4.7 4.2 3.7 Thermal loading

3.2

Hygrothermal Loading

2.7 2.2 1.7 1.2 0.7 5

10

15

20

25

30

35

40

Modular ratio Fig 7. Variation of non-dimensional deflection of (0/90/90/0) plate with modular ratio under thermal and hygrothermal loading.

Non-dimensional deflection

9 8 7

Simply Supported Simply Supported Clamped Clamped

6 5 4 3 2 1 0 5

10

15

20

25

30

35

40

Modular ratio Fig 8. Variation of non-dimensional deflection of two layered anti-symmetric cross ply (0/90) with modular ratio under thermal and hygrothermal load for different boundary conditions.