Document Type : Research Paper
Authors
Research Scholar, Department of Civil Engineering, SRES’s Sanjiv ani College of Engineering, Savitribai Phule Pune University, Kopargaon423603, Maharashtra, India
Abstract
Keywords
Higherorder Displacement Model for Cylindrical Bending of Laminated and Sandwich Plates Subjected to Environmental Loads
N.S. Naik^{ *}, A.S. Sayyad
Research Scholar, Department of Civil Engineering, SRES’s Sanjiv ani College of Engineering, Savitribai Phule Pune University, Kopargaon423603, Maharashtra, India
KEYWORDS 

ABSTRACT 
Cylindrical bending Laminated Sandwich Shear deformation Normal deformation 
In this research article, thermal and hygrothermal stress analysis of composite layered and sandwich plate having one dimension infinitely long and simply supported on the edges is presented using a new fifthorder theory. The proposed theory considers, the effect of thickness stretching. The present theory uses a polynomial shape function to account for transverse shear deformation using the expansion of thickness up to the fifthorder while to consider the effect of thickness stretching the derivative of shape function is used in the transverse displacement. In this theory, the shear strain variation is assumed to be parabolic across the thickness. The present displacement field satisfies zero shear stress condition both at the top and bottom surfaces and avoids the use of a shear correction factor. The governing equations are derived using the virtual work principle. For solution of problem, Navier’s solution technique is used. The results generated using the present theory are compared with the existing elasticity solution wherever it is available. However, many results for the cylindrical flexural analysis of laminated and sandwich plates subjected to environmental loading are presented for the first time in this paper. 
Composite material is characterized by its low density, highmodulus, high strength and low weight and its flexibility to tailor it as per the structural requirements. Because of these important properties it is widely used in many branches of engineering viz. civil engineering, aerospace engineering, mechanical engineering etc. Composite material during its life span has to carry different types of loadings, like mechanical load and environmental loads (Thermal/hygrothermal). Environmental loads like temperature and moisture result in degradation of the properties and reduction in the strength. Hence, considering the use of composite material in important applications and effect of environmental loads on it, it is necessary to analyze composite structures for mechanical as well as environmental loads.
The literature available on cylindrical bending analysis of composite laminates subjected to environmental loading is scare. The review of available literature on the development of theories used to analyze beams, plates and shell is documented by Timoshenko and WoinowskyKrieger [1], Todhunter and Pearson [2] and Carrera et al. [3]. Pagano [46] developed bench mark exact solution for 1D and 2D bending analysis of laminated composite plates and sandwiches. Kirchhoff [7] developed the simplest theory (Classical Plate Theory i.e. CPT) ignoring the shear deformation effect. CPT cannot be applied to plates having a considerable thickness and in which shear deformation effect is significant. To overcome the drawback of CPT, first time Mindlin [8] developed first order shear deformation theory (FSDT) assuming constant shear strain across the thickness. Hence FSDT does not satisfy zero shear stress conditions at the top and bottom surfaces of the plate and needs a shear correction factor. The limitations of CPT and FSDT initiated the need of shear deformation theories with a higher order. Sayyad and Ghugal [911] used exponential theory for the study of flexural and vibrational analysis of thick plates. Ghugal and Dahake [12] developed a trigonometric shear deformation theory using the sinusoidal function for the analysis of deep beams carrying parabolic load. Sayyad and Ghugal [13] studied the flexural behavior of softcore sandwich beams using trigonometric shear deformation theory. Sayyad and Ghugal [14] presented an n^{th} order theory with consideration of the shear deformation effect for the cylindrical bending analysis of composites. Sayyad and Ghugal [15] also studied bending, buckling and free vibrations of homogeneous beams using single variable refined beam theories. Shinde and Sayyad [16] analyzed isotropic, functionally graded, laminated and sandwich beams using a quasi3D polynomial shear deformation theory. Sayyad and Ghugal [17] investigated bending, buckling, and free vibration responses of functionally graded material beams using hyperbolic shear deformation theory. Recently Sayyad and Naik [18] developed a new quasi 3D model for the accurate prediction of transverse shear stresses in the laminated composites and sandwiched plates. Plucinski and Jaskowiech [19] presented threedimensional analysis of laminated plate subjected to mechanical load using twodimensional numerical model. A detailed review of such higherorder theories and solutions is presented by Sayyad and Ghugal [2022].
Because of the various applications of composite materials in the field of aerospace engineering where the material is subjected to thermal and hygrothermal stresses; many researchers have presented various theories for the thermal and hygrothermal stress analysis of the composite laminates. This section of the paper deals with literature related to thermal, thermomechanical and hygrothermal stress analysis of composite plates. Cho et al. [23] presented layerwise theory for analysis of laminates under thermal loading. Bhaskar et al. [24] presented thermoelastic solutions for composite laminates within the framework of linear uncoupled thermoelasticity. Carrera [25] compared different theories formulated on the basis of the principle of virtual work and the Reissner mixed variational theorem (RMVT). The higherorder theory developed by Rohwer et al. [26] for analysis of laminated plates in thermal environment predicts inplane stresses accurately when applied to thick plates carrying a mechanical load but gives less accurate predictions for sinusoidal thermal loading. A finite element model was proposed by Robaldo and Carrera [27] for the thermoelastic analysis of anisotropic plates. Kant and Shiyekar [28] developed a complete analytical model for the thermal stress analysis of composite laminates under gradient thermal load. A fourvariable plate theory was developed by Sayyad et al. [29] for the thermoelastic flexural analysis of laminated composite plates. Sayyad et al. [30] presented thermal stress analysis of layered composites using exponential shear deformation theory. Zenkour and Radwan [31] presented a hyperbolic model for the analysis of layered plates under thermal load and resting on elastic foundations. Shahravi et al. [32] presented an analytical approach to study the thermal deflections of simply supported composite plates under sinusoidal thermal load. Evran [33] presented finite element analysis of laminated composite plates under constant temperature load using Taguchi method.
Carrera and Nali [34] presented an advanced finite element formulation for the layered plates carrying mechanical, thermal, electrical and magnetic fields. Ali et al [35] proposed a theory for the thermal and mechanical stress analysis of laminated plates, the proposed theory can be used for thick plates and for any combination of thermal and mechanical loading. Zenkour [36] used unified shear deformation theory for flexural analysis of laminated plates under combined thermal and mechanical load. Kant et al. [37] investigated the thermomechanical response of laminated composites using semianalytical model. Nali and Carrera [38] studied buckling of composite plates under combined thermomechanical load. Wu et al. [39] presented a refined higherorder theory for angleply composite laminate subjected to thermomechanical loads. Ghugal and Kulkarni [40, 41] presented a refined sinusoidal theory for thermomechanical stress analysis of crossply laminates. Zenkour et al. [42] used unified theory for investigation of bending of crossply laminated plates under thermomechanical loads. Wu and Xiaohui [43] presented thermomechanical analysis of multilayered plates using Reddytype plate theory considering the effect of transverse normal strain.
Patel et al. [44] studied characteristics of thick composite laminated plates under hygrothermal load using a higherorder theory. Zenkour [45] has investigated the static response of angleply laminated plates for variation in temperature and moisture concentrations. A higherorder globallocal model is proposed by Wu and Lo [46] for the hygrothermomechanical analysis of laminated composite plates. Najafi et al. [47, 48] studied the environmental effects on mechanical properties of glass/epoxy and fiber metal laminates through experimental investigations because of hygrothermal and isothermal aging. Akbas [49] investigated nonlinear static analysis of composite beams under hygrothermal effects using finite element method. Sayyad and Ghugal [50] presented a simple four variable shear deformation theory for the bending of functionally graded plates subjected to nonlinear hygrothermomechanical loading. A refined quasi3D model considering the effect of transverse normal strain and shear deformation for the bucking response of functionally graded plates on elastic foundations under hygrothermomechanical loading is proposed by Zenkour and Radwan [51]. Garg and Chalak [52] presented a critical review of literature related to the behavior of laminated composites and sandwich structures subjected to hygrothermal loading. Moleiro et al. [53] developed an exact 3D hygrothermal elasticity solution for simply supported rectangular composite plates. Das and Niyogi [54] studied free vibrations of epoxybased cross ply laminated plates subjected to hygrothermal loading. Recently Naik and Sayyad [55] presented an analysis of laminated plates subjected to mechanical and hygrothermal loads using fifthorder shear and normal deformation theory.
In this paper the fifth order shear and normal deformation theory is applied for the cylindrical bending of laminated composite and sandwich plates under thermal and hygrothermal loads. The present theory is developed by Naik and Sayyad [5658] for laminated and sandwich plates and by Ghumare and Sayyad [5961] for the analysis of functionally graded plates subjected to mechanical and thermal loads.
Following points summarizes the features of the present study.
2. Sufficient literature is available on the bidirectional bending of layered plates under thermal loading, while limited literature is available on the hygrothermal stress analysis of composite plates. Based on the above fact, in the present investigation cylindrical bending of laminated composite plates under thermal and hygrothermal loading is presented.
3. In the present study hygrothermal cylindrical flexural analysis of laminated and sandwich plates is presented for the first time considering the effects of thickness stretching.
4. In the present study, the detailed numerical results and through the thickness distributions of stresses are presented for cylindrical bending of layered and sandwiched plates which will help the researchers to compare and validate their studies.
Following are the some of the advantages of the present theory over the other higher order theories; which can be summarized as below
• The present theory is computationally simple as compared to other nonpolynomial theories as it uses polynomial shape function.
• In the wellknown theory of Reddy [63], the thickness coordinate is expanded up to thirdorder and it ignores the effect of thickness stretching, while the present theory the thickness coordinates are expanded up to fifth order and hence the present theory improves the accuracy.
In the present study, a rectangular plate of orthotropic fibrous composite is considered. The plate is having length ‘a’ along xdirection and ‘b’ along the ydirection. The thickness of the plate ‘h’ is measured in zdirection. The dimension b>>a, and hence the plate is subjected to cylindrical bending. For the plate under consideration, since the dimension of the plate along the ydirection is assumed to be very long as compared to other dimensions in x and zdirections, the strain in the ydirection is neglected. The plate is subjected to an out of plane mechanical load q(x), which is acting on the top surface of the plate located at z = h/2 and sinusoidal thermal and hygroscopic load on the top surface. Fig. 1 shows the plate under consideration and the geometry of the plate.
Fig. 1. Plate under consideration and coordinate system
Following are the assumptions made in the development of the present theory.
1) The present theory is displacementbased shear deformation theory
2) The inplane displacements (u) includes three components viz. extension, bending and shear.
3) The transverse displacement (w) considers the effect of shear and thickness stretching.
4) Threedimensional Hooke’s law is used to determine stresses.
In the present work, the inplane displacement ‘u’ and the transverse displacement ‘w’, are considered in polynomial form to accommodate the effect of transverse shear and thickness stretch. The effect of transverse deformation is considered through polynomial shape function expanded up to fifth order in terms of the thickness coordinate. While the derivative of shape function is used in the transverse displacement to accommodate the effect of thickness stretching. There are six variables and the displacement field satisfy the traction free boundaries at the top and bottom surfaces of the plate. The assumed displacement field of the present theory is written as
(1)
where ‘u’ in inplane displacement at any point on the plate in xdirection and ‘w’ is the displacements in zdirection. ‘u_{0}’ and ‘w_{0}’ are the inplane displacements of midplane in x and zdirections respectively. are the rotations about yaxis to account the effect of transverse shear deformation. represent higherorder transverse crosssectional deformation modes which account the effect of thickness stretching. Eq. (2) shows the nonzero strain components in the present displacement field.
(2)
The coordinate system (xyz) is used to express the stressstrain relationship. For the k^{th} lamina the stresses and the strain are related through the relationship given in Eq. (3).
(3)
where Q_{11}, Q_{13}, Q_{33} and Q_{55} are the reduced elastic constants in xz plane and is the inplane stress acting along xdirection, is the stress acting along zdirection and is the transverse shear stress acting along the zdirection. are the inplane and normal strains along x and zdirections respectively, are the coefficients of linear thermal and moisture expansion in x and zdirections respectively. The below mentioned Eq. (4) states the relationship between elastic constants and engineering constants.
(4)
here are the moduli of elasticity, is the modulus of shear and are Poisson’s ratios; the subscripts 1, 2, 3 correspond to the coordinate system of fibers, while x, y, z directions represent the coordinate systems for the plate. In the present study, the laminated and sandwich plates are analysed for thermal, mechanical and hygrothermal loading. The variations of thermal and moisture load are assumed along the thickness of the plate and are given in Eq. (5).
(5)
where, T_{0}, T_{1}, T_{2} and T_{3} are thermal loads, C_{0}, C_{1}, C_{2} and C_{3} are the moisture concentrations.
The variationally consistent governing equations and the boundary conditions corresponding to them are derived using the principle of virtual work given in Eq. (6)
(6)
Integrating Eq. (6) by parts and equating the coefficients of to zero, six governing equations can be obtained. The Eq. (7) below gives governing equations in terms of stress resultants.
(7)
The boundary conditions along edges (x=0, x=a) are stated in Eq. (8).
(8)
In the above governing equations is the inplane force resultant; is the moment resultant; are the shear moment resultant; are the transverse shear and transverse normal stress resultants. Eq. (9) below gives the expressions for all above stress resultants used in the governing equation.
(9)
Governing equations in terms of unknown variables can be developed using the expressions of stress resultants. These governing equations along with the mechanical, thermal and moisture coefficients are mentioned in Appendix A.
To obtain the analytical solutions of the governing equations for the plates under considerations Navier’s solution is used. It is well known that this solution is applicable for simply supported boundary conditions only. For other boundary conditions, numerical methods such as FEM, FDM, GDQ, Meshfree method and other methods can be used.
In the present study, the plates under consideration are carrying a sinusoidally distributed mechanical and environmental loads on the top surface, whereas the thermal and moisture load are varying linearly across the thickness of the plates. Following are the kinematic boundary conditions.
(10)
For the unknown displacements to be determined and to satisfy the abovementioned boundary conditions following form of closed form solution is used.
(11)
In the above equation are the unknowns to be determined. The mechanical, thermal and moisture loadings are also expressed using double trigonometric Fourier series as stated in Eq. (12).
(12)
In the Eq. (12), T_{0} and T_{1} represent constant and linear temperature profiles respectively, while T_{2 }and T_{3 }represent the nonlinear temperature profiles. Similarly, C_{0} represents constant moisture load, C_{1} represents the linear moisture profile, C_{2} and C_{3} represent the nonlinear moisture profile. For the sinusoidally distributed loads, positive integers m and n are taken as unity. Substitution of Eqs. (11) and (12) in governing equation gives a set of equations which are expressed in matrix form as given in Eq. (13) below.
(13)
The stiffness coefficients [K_{ij}] and the force vectors used in the Eq. (13) are given in the appendix A.
Values of the unknowns i.e. obtained from the solution of Eq. (13) are further used to determine the unknown displacements from the Eq. (11). After knowing the values of all the unknown variables, one can determine all the displacements and stresses for the plate under consideration using Eqs. (1)  (4).
Transverse shear stress is calculated using equilibrium equation. If the constitutive equation is used to calculate the transverse shear stress, it results in discontinuity at the layer interface, hence to have a single value of transverse shear stress at the layer interface and to avoid discontinuity at the layer interface, transverse shear stress is calculated using equilibrium equation of theory of elasticity. Because at the layer interface, there must be the same stress in the upper and the lower layer, and this condition is satisfied using equilibrium equation. Eq. (14) states the equilibrium equation used to calculate the transverse shear stress.
(14)
In addition to continuity of the transverse shear stress, this theory also satisfies the continuity condition for inplane displacement and transverse displacement as stated in Eq. (15).
(15)
This section deals with the numerical results corresponding to the thermal, hygrothermal and mechanical stress analysis of laminated composite and sandwich plates. The results are presented in Tables 16 and graphically plotted in Figs. 218. Solutions of the following problems are presented in the present study.
Problem 1: Thermal stress analysis of threelayered (0^{0}/90^{0}/0^{0}) laminated plate.
Problem 2: Stress analysis of threelayered (0^{0}/core/0^{0}) sandwiched sandwich plate under mechanical loading.
Problem 3: Thermal stress analysis of threelayered (0^{0}/core/0^{0}) sandwiched plate.
Problem 4: Hygrothermal stress analysis of twolayered (0^{0}/90^{0}) laminated plate.
Problem 5: Hygrothermal stress analysis of threelayered (0^{0}/90^{0}/0^{0}) laminated plate.
Problem 6: Hygrothermal stress analysis of threelayered (0^{0}/core/0^{0}) sandwiched plate.
Following material properties and nondimensional forms are used in the present study and the effect of temperature and moisture is considered through the strain and inplane forces due to temperature and moisture.
Problem 1:
Material properties [24]
(16)
Nondimensional forms
(17)
Problem 2:
Material properties
Skin material [64]
(18)
Nondimensional forms
(19)
Problem 3:
Material properties
(20)
Nondimensional forms
(21)
Problem 4 and 5:
Material properties
(22)
Nondimensional forms
(23)
In the material properties mentioned above, 1, 2 and 3 refer to directions parallel and perpendicular to the fibers respectively.
Problem 6
Carbon fiber reinforced polymer skin material
(24)
PVC foam core material
(25)
Nondimensional forms
(26)
Problem 1: In this problem bending of a (0^{0}/90^{0}/0^{0}) laminate plate subjected to sinusoidally distributed thermal load over the surface of the plate and linearly distributed across the thickness of the plate is discussed. The plate is having thickness of each layer as h/3. The material properties used for this problem are given in Eq. (16), while Eq. (17) gives nondimensional forms for the calculations of displacements and stresses. Numerical results obtained are summarized in Table 1 for this problem. The results obtained by the present FOSNDT are compared with the elasticity solution presented by Bhaskar et al. [24]. The comparison of results is done for different aspect ratio (a/h = 4, 10, 20, 50, 100). The comparison reveals that the present theory gives results which are close to the elasticity solution.
Variation in transverse displacement with aspect ratio is plotted in Fig. 2, while throughthethickness variations of displacements and stresses are presented in Figs. 35 for a/h = 4.
Problem 2: In this problem the present theory before it is application to thermal stress analysis of sandwich plate, it is applied to a sandwich plate subjected to sinusoidal mechanical loading. The top and bottom layers are having thickness 0.1h, while the thickness of the middle core is 0.8h. The material properties and the nondimensional forms used are given in Eqs. (18) and (19). The results obtained using present theory are compared with those predicted by sinusoidal shear and normal plate theory (SSNPT) of Sayyad and Ghugal [66], higher order shear deformation theory (HSDT) of Reddy [63], first order shear deformation theory (FSDT) of Mindlin [8] and classical plate theory (CPT) of Kirchhoff [7]. The comparison is shown in Table 2 revels that the present theory gives results in good agreement with SSNPT and HSDT.
Table 1. Comparison of deflections and stresses for (0^{0}/90^{0}/0^{0})laminated plate under cylindrical bending subjected to the linear temperature field
S 
Model 
% Error 
% Error 
% Error 
% Error 

4 
Present 
7.206 
3.5341 
18.55 
1.255 
375.6 
0.886 
3.033 
7.173 
Bhaskar et al. [24] 
7.470 
 
18.32 
 
372.3 
 
2.830 
 

10 
Present 
4.976 
0.6588 
5.441 
0.610 
375.1 
0.914 
2.441 
5.387 
Bhaskar et al. [24] 
5.009 
 
5.408 
 
371.7 
 
2.580 
 

20 
Present 
4.564 
0.5447 
3.476 
0.086 
375.0 
0.942 
1.376 
4.510 
Bhaskar et al. [24] 
4.589 
 
3.479 
 
371.5 
 
1.441 
 

50 
Present 
4.444 
0.5148 
2.920 
0.443 
374.9 
0.942 
0.569 
4.208 
Bhaskar et al. [24] 
4.467 
 
2.933 
 
371.4 
 
0.594 
 

100 
Present 
4.426 
0.5169 
2.840 
0.525 
374.9 
0.942 
0.286 
4.026 
Bhaskar et al. [24] 
4.449 
 
2.855 
 
371.4 
 
0.298 
 

CPT [7] 
4.444 
 
2.829 
 
371.4 
 
 
 
For a thin plate having an aspect ratio equal to 100, present theory gives results almost equal to those predicted by the other theories. It shows that the present theory can be applied efficiently to analyse sandwich plates as well.
Problem 3: In this problem bending response of (0^{0}/core/0^{0}) sandwich plate subjected to sinusoidal thermal load is studied. For the plate under consideration; the face sheets are having a thickness equal to 0.1h and the core is of thickness 0.8h. The material properties stated in Eq. (20) are used in the present example and the nondimensional forms stated in Eq. (21) are used for the calculations of unknown displacements and stresses.
Fig. 2. Variation of nondimensional transverse displacement ( ) with respect to the aspect ratio (S) for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear thermal load
Fig. 3. Variation of nondimensional inplane displacement ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear thermal load
Fig. 4. Variation of nondimensional inplane normal stress ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear thermal load
Table 2. Comparison of deflections and stresses for (0^{0}/core/0^{0}) sandwich plate under cylindrical bending subjected to sinusoidal mechanical load
a/h 
Model 

4 
FOSNDT 
1.9096 
8.6150 
28.8295 
1.4005 


SSNPT [66] 
1.8901 
8.4532 
28.9670 
1.3841 


0 
HSDT [63] 
1.9081 
8.5369 
28.6061 
1.3855 

0 
FSDT [8] 
1.3295 
5.4694 
19.9320 
1.4089 

0 
CPT [7] 
1.3295 
1.3225 
19.9320 
1.4089 
10 
FOSNDT 
22.203 
2.4914 
133.100 
3.5242 


SSNPT [66] 
22.092 
2.4739 
133.754 
3.5122 


0 
HSDT [63] 
22.235 
2.4889 
133.340 
3.5128 

0 
FSDT [8] 
20.773 
1.9860 
124.575 
3.5223 

0 
CPT [7] 
20.773 
1.3225 
124.575 
3.5223 
100 
FOSNDT 
20781.0 
1.3337 
12433.0 
35.285 


SSNPT [66] 
20680.2 
1.3272 
12477.5 
35.220 


0 
HSDT [63] 
20.788.4 
1.3342 
12466.4 
35.222 

0 
FSDT [8] 
20773.2 
1.3291 
12457.3 
35.221 

0 
CPT [7] 
20773.2 
1.3225 
12457.3 
35.221 
This problem is presented for the first time in this paper and no results are reported in the literature, hence only present results are tabulated in Table 3. Variation in transverse displacement with respect to aspect ratio is plotted in Fig. 6 and variations of normalized displacements and stresses along the thickness are plotted in Figs. 79 for a/h = 4. Since in the literature no results are available for this problem, the results presented in this paper will serve as a benchmark solution for the future research.
Fig. 5. Variation of nondimensional transverse shear stress ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear thermal load
Table 3. Deflections and stresses in (0^{0}/core/0^{0}) sandwich plate subjected to the linear temperature field using present FOSNDT
S 

4 
0.0990 
20.589 
0.0526 
0.0075 
10 
0.0387 
2.5890 
0.0058 
0.0012 
20 
0.0193 
0.6220 
0.0017 
0.0000 
50 
0.0077 
0.0980 
0.0002 
0.0000 
100 
0.0039 
0.0250 
0.0000 
0.0000 
Fig. 6. Variation of nondimensional transverse displacement ( ) with respect to aspect ratio (S) for (0^{0}/core/0^{0}) sandwich plate subjected to linear thermal load
Fig. 7. Variation of nondimensional inplane displacement ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear thermal load
Fig. 8. Variation of nondimensional inplane normal stress ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear thermal load
Problem 4: In this problem bending of (0^{0}/90^{0}) laminated plate subjected to linearly varying hygrothermal load, and having material properties given in Eq. (22) is presented. The nondimensional forms stated in Eq. (23) are used to calculate the displacement and stresses.
Each layer of the plate is having a thickness equal to h/2. Only results obtained using the present theory are summarized in this paper in Table 4 and the same are plotted for a/h = 4 in Figs. 10 12.
Fig. 9. Variation of nondimensional transverse shear stress ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear thermal load
Table 4. Deflections and stresses in (0^{0}/90^{0}) laminated plate subjected to the linear hygrothermal loading for present FOSNDT (T_{0}=100.0; T_{0M}=0; T_{1M}=T_{0}; T_{2M}=0; T_{3M}=0; C_{0}=3x10^{4}; C_{0M}=0; C_{1M}=C_{0}; C_{2M}=0; C_{3M}=0)
S 

4 
0.0482 
1.6432 
2.2784 
0.1326 
10 
0.0191 
1.4783 
2.1355 
0.0529 
20 
0.0096 
1.4539 
2.1433 
0.0265 
50 
0.0038 
1.4470 
2.1480 
0.0106 
100 
0.0019 
1.4460 
2.1488 
0.0053 
Fig. 10. Variation of nondimensional inplane displacement ( ) along the thickness for (0^{0}/90^{0}) laminated plate subjected to linear hygrothermal load
Fig.11. Variation of nondimensional inplane normal stress ( ) along the thickness for (0^{0}/90^{0}) laminated plate subjected to linear hygrothermal load
Problem 5: In this problem bending of (0^{0}/90^{0}/0^{0}) laminated plate subjected to linearly varying hygrothermal load is presented. All the layers are having equal thickness i.e., h/3 and having the material properties stated in Eq. (22). The nondimensional forms mentioned in the Eq. (23) are used to obtain results. The results obtained using the present theory are tabulated in Table 5 and plotted in Figs. 1315.
Problem 6: A sandwich plate (0^{0}/core/0^{0}) subjected to linearly varying hygrothermal load is analyzed in this problem. Thickness of each face sheet is assumed as 0.1h and core is having a thickness of 0.8h, where h is the thickness of the plate under consideration.
Fig. 12. Variation of nondimensional transverse shear stress ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear thermal load
Fig. 13. Variation of nondimensional inplane displacement ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear hygrothermal load
Table 5. Deflections and stresses in (0^{0}/90^{0}/0^{0}) laminated plate subjected to the linear hygrothermal loading for present FOSNDT (T_{0}=100.0; T_{0M}=0; T_{1M}=T_{0}; T_{2M}=0; T_{3M}=0; C_{0}=3x10^{4}; C_{0M}=0; C_{1M}=C_{0}; C_{2M}=0; C_{3M}=0)
S 

4 
0.0421 
1.2590 
0.3504 
0.0237 
10 
0.0165 
1.0824 
0.1013 
0.0075 
20 
0.0082 
1.0558 
0.0556 
0.0035 
50 
0.0033 
1.0483 
0.0423 
0.0014 
100 
0.0016 
1.0473 
0.0403 
0.0000 
The plate is having a skin of orthotropic carbon fiber reinforced polymer (CFRP) and the core is made of polyvinyl chloride (PVC). The face sheet is having material properties as given in Eq. (24) while those used for the core are given in Eq. (25). The nondimensional displacements and stresses are calculated using the nondimensional forms stated in Eq. (26). The results are presented in Table 6 and are plotted in Figs. 1618. Only the results obtained using present FOSNDT are given in the table as no results are available in the literature for the cylindrical bending of sandwich plates subjected to hygrothermal loading.
Fig. 14. Variation of nondimensional inplane normal stress ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear hygrothermal load
Fig. 15. Variation of nondimensional transverse shear stress ( ) along the thickness for (0^{0}/90^{0}/0^{0}) laminated plate subjected to linear hygrothermal load
Table 6 Deflections and stresses in (0^{0}/core/0^{0}) sandwich plate subjected to the linear hygrothermal loading for present FOSNDT (T_{0}=100.0; T_{0M}=0; T_{1M}=T_{0}; T_{2M}=0; T_{3M}=0; C_{0}=3x10^{4}; C_{0M}=0; C_{1M}=C_{0}; C_{2M}=0; C_{3M}=0)
S 

4 
0.0878 
3.9846 
6.4290 
0.0269 
10 
0.0307 
0.2617 
2.9577 
0.0083 
20 
0.0151 
0.0520 
2.4656 
0.0040 
50 
0.0060 
0.0077 
2.3280 
0.0016 
100 
0.0030 
0.0019 
2.3084 
0.0000 
Fig.16. Variation of nondimensional inplane displacement ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear hygrothermal load
Fig. 17. Variation of nondimensional inplane normal stress ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear hygrothermal load
Fig. 18. Variation of nondimensional transverse shear stress ( ) along the thickness for (0^{0}/core/0^{0}) sandwich plate subjected to linear hygrothermal load
This paper presents a new displacementbased fifthorder shear and normal deformation theory (FOSNDT) for the thermal and hygrothermal stress analysis of layered composite and sandwiches having one dimension infinitely long. To assess the performance of the present theory, the results are generated for cylindrical bending of 0^{0}/90^{0}/0^{0} plate carrying linear thermal load. The results obtained are compared with the available elasticity solution. The comparison shows that the present theory predicts the inplane displacement, transverse displacement in close agreement with the elasticity solution. The percentage error in the prediction of inplane displacement is 3.5341% for aspect ratio 4 and reduces to 0.51% for a plate having aspect ratio 100. For the transverse displacement and inplane stress the percentage error in prediction decreases with increase in aspect ratio. Thus, it can be concluded that the present theory under predicts the inplane displacement and over predicts the transverse displacement and inplane stresses. The transverse shear stress prediction using the present theory is on the higher side as compared to the elasticity solution. Similarly for the sandwich plate under mechanical loading the present theory predicts the behaviour in good agreement with other well established theories.
For cylindrical bending of layered composites and sandwiches under hygrothermal load results are not reported in the literature for comparison. Hence, in the present paper the results generated for cylindrical bending of laminated and sandwich plates for hygrothermal loading are presented without comparison, but after validating the theory for thermal loading, and mechanical loading thus these results will serve as a benchmark for the future research.
Appendix A
Following are the governing equations derived by applying principle of virtual work in Eq. (6),
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
Mechanical coefficients
(A.7)
Thermal Coefficients
(A.8)
Moisture Coefficients
(A.9)
Following are stiffness matrix coefficients [K] used in Eq. (13)
(A.10)
Following are force vectors used in Eq. (13)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
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