Higher-order 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, Kopargaon-423603, Maharashtra, India

1.     Introduction

Composite material is characterized by its low density, high-modulus, 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 Woinowsky-Krieger [1], Todhunter and Pearson [2] and Carrera et al. [3].  Pagano [4-6] 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 [9-11] 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 soft-core 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 quasi-3D 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 3-D model for the accurate prediction of transverse shear stresses in the laminated composites and sandwiched plates. Plucinski and Jaskowiech [19] presented three-dimensional analysis of laminated plate subjected to mechanical load using two-dimensional numerical model. A detailed review of such higher-order theories and solutions is presented by Sayyad and Ghugal [20-22].

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 layer-wise 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 higher-order theory developed by Rohwer et al. [26] for    analysis of laminated plates in thermal environment predicts in-plane 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 four-variable 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 semi-analytical model. Nali and Carrera [38] studied buckling of composite plates under combined thermomechanical load. Wu et al. [39] presented a refined higher-order theory for angle-ply composite laminate subjected to thermomechanical loads. Ghugal and Kulkarni [40, 41] presented a refined sinusoidal theory for thermomechanical stress analysis of cross-ply laminates. Zenkour et al. [42] used unified theory for investigation of bending of cross-ply laminated plates under thermomechanical loads. Wu and Xiaohui [43] presented thermomechanical analysis of multi-layered plates using Reddy-type 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 higher-order theory. Zenkour [45] has investigated the static response of angle-ply laminated plates for variation in temperature and moisture concentrations. A higher-order global-local 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 quasi-3D 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 epoxy-based 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 fifth-order 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 [56-58] for laminated and sandwich plates and by Ghumare and   Sayyad [59-61] for the analysis of functionally graded plates subjected to mechanical and thermal loads.

Following points summarizes the features of the present study.

1. The main motivation behind the present theory is the contribution and recommendations given by Ali et al. [35], Bhaskar et al. [24] and Erasmo Carrera [62] regarding the stretching of the thickness of the laminated composite plates.  Carrera [62] studied the effect of the normal strain on the analysis of homogeneous and layered plates under thermal load and recommended the consideration of the thickness stretching effect for thermal stress analysis of plates particularly when the temperature and moisture concentration varies through the thickness of the plate.   Carrera has also recommended to expand the thickness coordinate up to fifth or seventh order for the accurate prediction of the behavior of laminated and sandwich plates under hygrothermomechanical load.  Therefore, in the present investigation fifth order shear and normal deformation theory (FOSNDT) is developed for the cylindrical bending of laminated composite and sandwich plates subjected to thermal and hygrothermal loading.

2.      Sufficient literature is available on the bi-directional 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 non-polynomial    theories as it uses polynomial shape function.

•        In the well-known theory of Reddy [63], the thickness co-ordinate is expanded up to third-order and it ignores the effect of thickness stretching, while the present theory the thickness co-ordinates are expanded up to fifth order and hence the present theory improves the accuracy.

2.     Plate Geometry

In the present study, a rectangular plate of orthotropic fibrous composite is considered.  The plate is having length ‘a’ along x-direction and ‘b’ along the y-direction. The thickness of the plate ‘h’ is measured in z-direction. 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 y-direction is assumed to be very long as compared to other dimensions in x- and z-directions, the strain in the y-direction 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

3.     Mathematical Formulation

3.1.  Displacement field

Following are the assumptions made in the development of the present theory.

1)      The present theory is displacement-based shear deformation theory

2)      The in-plane displacements (u) includes three components viz. extension, bending and shear.

3)      The transverse displacement (w) considers the effect of shear and thickness stretching.

4)      Three-dimensional Hooke’s law is used to determine stresses.

     In the present work, the in-plane 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 in-plane displacement at any point on the plate in x-direction and ‘w’ is the displacements in z-direction. ‘u₀’ and ‘w₀’ are the in-plane displacements of mid-plane in x and z-directions respectively.  are the rotations about y-axis to account the effect of transverse shear deformation.  represent higher-order transverse cross-sectional deformation modes which account the effect of thickness stretching.  Eq. (2) shows the non-zero strain components in the present displacement field.

(2)

3.2. Constitutive equations

The co-ordinate system (x-y-z) is used to express the stress-strain relationship. For the k^th  lamina the stresses and the strain are related through the relationship given in Eq. (3).

        (3)

where Q₁₁, Q₁₃, Q₃₃ and Q₅₅ are the reduced elastic constants in x-z plane and   is the in-plane stress acting along x-direction,  is the stress acting along z-direction and   is the transverse shear stress acting along the z-direction.    are the in-plane and normal strains along x and z-directions respectively,   are the coefficients of linear thermal and moisture expansion in x and z-directions 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, T₀, T₁, T₂ and T₃ are thermal loads, C₀, C₁, C₂ and C₃ are the moisture concentrations.

3.3.  Governing equations and boundary conditions   

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 in-plane 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. 

4.     Analytical Solutions

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 above-mentioned 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), T₀ and T₁ represent constant and linear temperature profiles respectively, while T₂ and T₃ represent the non-linear temperature profiles.  Similarly, C₀ represents constant moisture load, C₁ represents the linear moisture profile, C₂ and C₃ represent the non-linear 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 in-plane displacement and transverse displacement as stated in Eq. (15).

          (15)

5.     Solved Numerical Problems

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 1-6 and graphically plotted in Figs. 2-18.  Solutions of the following problems are presented in the present study.

Problem 1: Thermal stress analysis of three-layered (0⁰/90⁰/0⁰) laminated plate.

Problem 2: Stress analysis of three-layered (0⁰/core/0⁰) sandwiched sandwich plate under mechanical loading.

Problem 3: Thermal stress analysis of three-layered (0⁰/core/0⁰) sandwiched plate.

Problem 4: Hygrothermal stress analysis of two-layered (0⁰/90⁰) laminated plate.

Problem 5: Hygrothermal stress analysis of three-layered (0⁰/90⁰/0⁰) laminated plate.

Problem 6: Hygrothermal stress analysis of three-layered (0⁰/core/0⁰) sandwiched plate.

Following material properties and non-dimensional forms are used in the present study and the effect of temperature and moisture is considered through the strain and in-plane forces due to temperature and moisture.

Problem 1:

Material properties [24]

         (16)

Non-dimensional forms

                 (17)         

Problem 2:

Material properties

Skin material [64]

(18)

Non-dimensional forms

   (19)

Problem 3:

Material properties

        (20)

Non-dimensional forms

                         (21)

Problem 4 and 5:

Material properties

            (22)

Non-dimensional 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)

Non-dimensional forms

          (26)

6.   Results and Discussion

Problem 1:  In this problem bending of a (0⁰/90⁰/0⁰) 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   non-dimensional 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 through-the-thickness variations of displacements and stresses are presented in Figs. 3-5 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 non-dimensional 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 (0⁰/90⁰/0⁰)laminated plate under cylindrical bending subjected to the linear temperature field

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 (0⁰/core/0⁰) 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 non-dimensional forms stated in Eq. (21) are used for the calculations of unknown displacements and stresses. 

Fig. 2. Variation of non-dimensional transverse displacement (  ) with respect to the aspect ratio (S) for (0⁰/90⁰/0⁰) laminated plate subjected to linear thermal load

Fig. 3. Variation of non-dimensional in-plane displacement ( ) along the thickness for (0⁰/90⁰/0⁰) laminated plate subjected to linear thermal load

Fig. 4. Variation of non-dimensional in-plane normal stress ( ) along the thickness for (0⁰/90⁰/0⁰) laminated plate subjected to linear thermal load

Table 2. Comparison of deflections and stresses for (0⁰/core/0⁰) sandwich plate under cylindrical bending subjected to sinusoidal mechanical load