Navier-type bending analysis of general composite laminates under different types of thermomechanical loading

Due to the wide spread applications in engineering structures, composite members are receiving more attentions from the research communities, since they are stronger, stiffer, and more ductile than the sum of the individual elements. Analysis of deformation and stress fields in composite laminates is of fundamental importance in experimental determination of the lamina properties and exact solutions are useful in developing a numerical model. Zhang et al. [1] illustrated a new procedure for obtaining the static exact solution of composite laminates with piezothermo-elastic layers under cylindrical bending. Duc and Minh [3] presented a method to determine bending deflection of three-phase polymer composite plates consisting of reinforced glass fibers and titanium oxide particles. An investigation of the stochastic nonlinear bending response of a laminated composite plate resting on a two parameter Pastemak elastic foundation with Winkler cubic nonlinearity subjected to transverse distributed static load was given by Singh et al. [4]. Sturzenbecher and Hofstetter [5] analyzed bending response of cross-ply laminated composites based on models of Lekhnitskii and Ren. Cetkovic and Vuksanovic [6] used an original MATLAB computer code to perform finite element solution for bending, free vibrations and buckling of laminated composite and sandwich plates using a layerwise displacement model. Dash and Singh [7] used a higher order shear deformation theory to investigate a transverse bending of laminated composite plates in Green-Lagrange sense accounting for the transverse shear and large rotations. Shiyekar and Kant [9] applied higher order shear deformation theory to analyze piezoelectric fiber reinforced composites under bidirectional bending. Higher order shear and normal deformable plate theory and a meshless method was used by Xiao et al. [10] to analyze static infinitesimal deformations of thick laminated composite elastic plates under different boundary conditions. Kant and Swaminathan [13] also used higher order refined theory to present an analytical solution to static loading of simply supported composite and sandwich plates. Andrade et al. [8] presented geometrically nonlinear static and dynamic analysis of laminated composite plates and shells using the eight-node hexahedral element with one-point integration and compared outputs with results obtained by other authors using different element types. Xu and Wu [11] illustrated a twodimensional analytical solution for simply supported composite beams with interlayer slips by consideration of shear deformation effects. Kant et al. [12] presented a novel semi-analytical model for accurate estimation of stresses and displacements in composite and sandwich laminates. Their results were seen to compare well with the available three dimensional elasticity and analytical solutions. Shen [2] give the nonlinear analysis for bending of simply supported functionally graded nanocomposite plates subjected to a transverse uniform or sinusoidal loads in thermal environments. For this purpose, he used a higher order theory to derive governing equations. Also Mousavi and Tahani [14] presented an analytical solution for bending of moderately thick radially functionally graded sector plates with general boundary conditions using Kantorovich method. In spite of the abundant literature on the subject, the choice of mechanical response of composite plates under different types of loading in thermal environments with general configuration and boundary condition is critical issues. Classical lamination theory is used to derive governing equations and the Navier-type solution is applied to solve them for simply supported boundary conditions. Three types of mechanical loadings and two types of thermal distribution are considered to investigate the effect of loading, temperature service, geometry ratio and mechanical properties on static bending of laminated composites. A finite element code using ANSYS is also developed to evaluate accuracy of the presented solution.


Introduction
Due to the wide spread applications in engineering structures, composite members are receiving more attentions from the research communities, since they are stronger, stiffer, and more ductile than the sum of the individual elements.Analysis of deformation and stress fields in composite laminates is of fundamental importance in experimental determination of the lamina properties and exact solutions are useful in developing a numerical model.Zhang et al. [1] illustrated a new procedure for obtaining the static exact solution of composite laminates with piezothermo-elastic layers under cylindrical bending.Duc and Minh [3] presented a method to determine bending deflection of three-phase polymer composite plates consisting of reinforced glass fibers and titanium oxide particles.An investigation of the stochastic nonlinear bending response of a laminated composite plate resting on a two parameter Pastemak elastic foundation with Winkler cubic nonlinearity subjected to transverse distributed static load was given by Singh et al. [4].Sturzenbecher and Hofstetter [5] analyzed bending response of cross-ply laminated composites based on models of Lekhnitskii and Ren.Cetkovic and Vuksanovic [6] used an original MATLAB computer code to perform finite element solution for bending, free vibrations and buckling of laminated composite and sandwich plates using a layerwise displacement model.Dash and Singh [7] used a higher order shear deformation theory to investigate a transverse bending of laminated composite plates in Green-Lagrange sense accounting for the transverse shear and large rotations.Shiyekar and Kant [9] applied higher order shear deformation theory to analyze piezoelectric fiber reinforced composites under bidirectional bending.Higher order shear and normal deformable plate theory and a meshless method was used by Xiao et al. [10] to analyze static infinitesimal deformations of thick laminated composite elastic plates under different boundary conditions.Kant and Swaminathan [13] also used higher order refined theory to present an analytical solution to static loading of simply supported composite and sandwich plates.Andrade et al. [8] presented geometrically nonlinear static and dynamic analysis of laminated composite plates and shells using the eight-node hexahedral element with one-point integration and compared outputs with results obtained by other authors using different element types.Xu and Wu [11] illustrated a twodimensional analytical solution for simply supported composite beams with interlayer slips by consideration of shear deformation effects.Kant et al. [12] presented a novel semi-analytical model for accurate estimation of stresses and displacements in composite and sandwich laminates.
Their results were seen to compare well with the available three dimensional elasticity and analytical solutions.Shen [2] give the nonlinear analysis for bending of simply supported functionally graded nanocomposite plates subjected to a transverse uniform or sinusoidal loads in thermal environments.For this purpose, he used a higher order theory to derive governing equations.Also Mousavi and Tahani [14] presented an analytical solution for bending of moderately thick radially functionally graded sector plates with general boundary conditions using Kantorovich method.
In spite of the abundant literature on the subject, the choice of mechanical response of composite plates under different types of loading in thermal environments with general configuration and boundary condition is critical issues.Classical lamination theory is used to derive governing equations and the Navier-type solution is applied to solve them for simply supported boundary conditions.Three types of mechanical loadings and two types of thermal distribution are considered to investigate the effect of loading, temperature service, geometry ratio and mechanical properties on static bending of laminated composites.A finite element code using ANSYS is also developed to evaluate accuracy of the presented solution.

Displacement and strains
A rectangular plate of sides a and b with thickness h, shown in Fig. 1.Based on classical lamination plate theory, the following displacement field can be assumed: where u 0 , v 0 , w 0 are the displacements along the coordinate lines of a material point on xy-plane.
The von Karman strains associated with the displacement field in static loading can be computed using the strain-displacement relations for small strains: Note that the transverse strains are identically zero in classical plate theory.The first three strains have the form: If the temperature increment varies linearly, consistent with the mechanical strains, we can write: And the total strains are of the form: where A ij are extensional stiffnesses, D ij the bending stiffnesses and B ij the bending-extensional coupling stiffnesses, which are defined in terms of the transformed laminate stiffnesses ij Q as: Equations of equilibrium can be derived using variational principle which is not explained in details here (see [15]).Three equilibrium equations are as follows:

Boundary conditions
The Navier solutions can be developed for rectangular laminates with two sets of simply supported boundary conditions.Even for these boundary conditions, not all laminates permit the Navier solution.The two types of simply supported boundary conditions on the displacements and stress resultants used in classical lamination plate theory (CLPT) are given below [15].
Simply supported (SS-1): Simply supported (SS-2): The Navier solutions using SS-1 boundary conditions can be obtained only for laminates whose stiffnesses A 16 , A 26 , B 16 , B 26 , D l6 , D 26 , and A 45 are zero.Thus, the Navier solutions for the SS-1 boundary conditions can be developed for laminates with a single generally orthotropic layer, symmetrically laminated plates with multiple specially orthotropic layers, and antisymmetric cross-ply laminated plates.Similarly, the Navier solutions using SS-2 boundary conditions can be obtained only for laminates whose stiffnesses A 16 , A 26 , B 11 , B 12 , B 22 , B 66 , D 16 , D 26 , and A 45 are zero, i.e., for laminates with a single generally orthotropic layer, symmetrically laminated plates with multiple specially orthotropic layers, and antisymmetric angleply laminated plates.[15].In all cases of this study, nonlinear terms are omitted.

Navier solution methodology
The displacement boundary conditions of simply supported in (17-18) are satisfied by assuming the following form of the displacements: And the temperature increment is expanded as:

T z T x y z sin x sin y dxdy ab
Then thermal force and moment resultants can be assumed as: where By expansion of generalized displacements and loads in a double trigonometric series in terms of unknown parameters, Eqs. ( 13)-( 15) can be cast in differential operator form: where coefficients c ij are defined by: Eq. ( 30) are solved using the method of static condensation.This method allows the elimination of a selected set of variables and retains a desired set of variables.In order to this, the coefficients associated with the in-plane displacements are eliminated and those related with the transverse deflection are retained.Finally coefficients of unknown ,, mn mn mn U V W are determined. (for more details about this method, see Ref. [15]) Solution of Eq. (30) for each m, n = 1, 2,… gives ,, mn mn mn U V W , which can be used to compute the final solution.

Determination of stresses
The in-plane stresses in each layer of a laminate are calculated from constitutive relations.Accounting for mechanical and thermal effects: The transverse stresses in a laminate can be determined using the 3-D equilibrium equations for any C are functions to be determined using the boundary conditions and continuity of stresses at layer interfaces.

Mechanical loading
In all examples, material properties of the plate are assumed as: 25, 0 5, 0 2 , 0 3 In this study, three types of loads including uniform, center point and sinusoidal distributed loads in mechanical loading are considered.
Sinusoidal distributed load (SSL): The length-to-thickness ratio (i.e., a/h) is assumed to be 10 in all numerical examples.Furthermore, in the mechanical loading results the various nondimensionalized parameters used are: .s: All results presented in this section are compared with results of other numerical and analytical studies available in the literature as well as those obtained from the commercial finite element code ANSYS.Composite plates have been modeled in ANSYS by using three-dimensional 8-node layered elements which allows up to 250 different material layers in the thickness direction in each element without much increase of counting time.Mechanical properties of a unidirectional laminate at room temperature are used as initial values in finite element method.In this model, manual mesh is used and also the mesh is refined till no significant change in displacements and stresses are obtained.As mentioned before, the simply supported boundary conditions (SS-1) place a restriction on the laminate scheme of composite plates.First cross-ply lamination is selected for both mechanical and thermal loading.Table 1 contains nondimensionalized deflections and stresses for antisymmetric cross-ply laminates under different types of mechanical loads.Also the differences between analytical and finite element model are illustrated in percentage for each case.From these results, it can be concluded that, for the same laminate thickness, antisymmetric cross-ply laminates with four or more layers are more desirable than two layer laminates because of reduction in deflections and stresses.This behavior is due to the bending-stretching coupling coefficients (B ij ) which are dominant in case of two layers.

Fig. 2 Nondimensionalized center transverse deflection
versus plate aspect ratio of simply supported (SS-1) laminates Fig. 2-4 shows the effect of bending-stretching coupling and plate aspect ratio on the transverse deflection and stresses for a given z 0 under various types of mechanical loads.For comparison, results of symmetric laminates are also included.
The magnitude of deflections and stresses of symmetric laminates (0/90/90/0) are about two to three times that of antisymmetric (0/90/0/90) laminates for a/b > 1.For the uniformly distributed load there corresponds an aspect ratio, around a/b = 2.25 for (0/90) 2 and a/b = 3.5 for (0/90) s for which the deflection is the maximum of all aspect ratios.The effect of coupling coeffi-cients is to increase the stresses.These coupling coefficients decrease in magnitude with the increase in the number of layers in antisymmetric cross-ply laminates and the laminate essentially behaves like an especially orthotropic plate.The dependence of the coupling effect on the modulus ratio is illustrated in Fig. 5 for SSL and UDL.Distribution of the nondimensionalized maximum normal stress and transverse shear stress is shown in Fig. 6.These components are calculated through the thickness of antisymmetric cross-ply and orthotropic laminates under sinusoidal loadings.
The stress concentration is reduced in eight-layer cross-ply laminates than the two-layer one.So the latter plates experience larger stresses.Thus, the effect of the bending-stretching coupling present in two-layer plates on stresses is to increase the magnitude of stresses.As discussed before, one of the lamination scheme which admits Navier solution for simply supported boundary conditions (SS-2) is antisymmetric angle-ply which is selected in this study.The effect of bendingextension coupling and the dependence of the coupling on the modulus ratio can be seen from the deflections and stresses presented in Table 2 for antisymmetric angle ply laminates (-45/45) k for k = 1, 2, and 4, and subjected to center point load and uniformly distributed load.For each case, differences between analytical and finite element method is also calculated in percentage.Clearly, the bending-extension coupling is quite significant for two-layered plates, but the coupling decrease very rapidly as the number of layers is increased.Trend of chang-ing  is for a two-layer plate is different from four and eight layer ones.Two layer angle-ply laminates have increasing trend versus variation of lamination angle while this behavior is decreasing for four and eight layer plates.Maximum deflection takes place for lamination angle of about 27° for two-layer plates.Also the magnitude of deflections of orthotropic laminates was about two to three times that of antisymmetric angle-ply laminates for a/b > 3 under sinusoidal loading.Distribution of nondimensionalized transverse deflection versus plate aspect ratio for different types of loading are plotted in Fig. 9. From these results, it can be concluded that, for the same laminate thickness, antisymmetric angle-ply laminates with four or more layers are more desirable than two layer laminates because of reduction in deflections and stresses.This behavior is due to the bending-extension coupling coefficients which are dominant in case of two layers.
Lastly, nondimensionalized transverse deflections as a function of the modulus ratio for square laminates under sinusoidal transverse load are presented in Fig. 10.The effect of coupling is significant for all modulus ratios except for those close to unity.
Distribution of the nondimensionalized transverse shear stress is shown for multi layer antisymmetric angleply and orthotropic laminates under sinusoidally distributed transverse loads in Fig. 11.Unlike in antisymmetric cross-ply laminates, the stress xz  is not zero at (x, y) = (a / 2, 0), although small in magnitude compared to that at (x, y) = (0, b / 2).Note that through-thickness variations are significantly altered when the number of layers is increased (for the same total laminate thickness).

Thermal loading
In the case of thermal loading, thermal properties of composite plates are assumed as: where T 0 and T 1 are assumed to be 100 and 10°C respectively.Stress free temperature is assumed T = 0°C.
In numerical results, nondimensionalized parameters for thermal loading are: In thermal loading, because of the presented results were not found in the open literature, all predictions are compared with results of commercial finite element code ANSYS.Table 3 listed results of transverse deflections and stresses in square laminates subjected to uniformly and linear distributed temperature in thermal loads which are compared with finite element outputs.Also the difference between these two methods is reported in percentage for better evaluation of the results.12 and 13 show the distribution of transverse shear stress through the thickness of the composite plates under different types of distributed temperature in thermal loading.For comparison, results of orthotropic laminates are also included.
In the case of eight-layer laminates, minimum stress concentration was found for both linear and uniform distributed temperature.Only difference is that for linear distributed temperature, the normal transverse shear stresses are not zero in bottom of the plates which is due to different temperatures at top and bottom of the plates.
Distribution of normal stresses through the thickness of bending-stretching coupling for a given z 0 is compared for various types of thermal loadings (UDT and LDT) in Fig. 14.Also variation of transverse deflection versus plate aspect ratios for linear distributed temperatures is illustrated in Fig. 15.It can be concluded that, magnitude of maximum transverse deflection for two-layer cross ply is about three or four times greater than four-layer type.Thus, the effect of the bending-stretching coupling present in two-layer plates on transverse deflection is to increase the magnitude of deflection.

Conclusions
Navier solution was applied using classical laminated theory to analysis cross-ply and angle-ply composite laminated plates with simply supported boundary conditions.Laminates were considered under different types of mechanical and thermal loads.For mechanical analysis, sinusoidal, uniform distributed and center point loads and in the case of thermal loadings, uniform and linear distributed temperatures were used.Results which was not found in open literature, were compared with commercial finite element analysis ANSYS.Following comments can be highlighted: 1.It can be concluded that for the same laminate thickness, antisymmetric laminates with four or more layers were more desirable than two layer laminates because of reduction in deflections and stresses for all cases.This behavior was due to the bending-stretching coupling coefficients for cross-ply and bending-extension coupling coefficients for angle-ply laminates, which was dominant in case of two layers.
2. Also the magnitude of deflections and stresses of symmetric cross-ply laminates was about two to three times greater than that of antisymmetric cross-ply laminates for a/b > 1 under sinusoidal loading.These ratios in case of uniform distributed loads were 2.25 and 3.5 for symmetric and unsymmetrical cross-ply laminates respectively.
3. The effect of coupling coefficients is to increase the stresses.These coupling coefficients decrease in magnitude with the increase in the number of layers in antisymmetric cross-ply laminates and the laminate essentially behaves like an especially orthotropic plate.
4. Maximum transverse shear stress is reduced by increasing number of layer in cross-ply laminates.So that maximum shear stress in eight-layer laminates decreased about two times in compare with orthotropic plates.
5. Maximum deflection takes place for lamination angle of about 27° for two-layer plates.Also the magnitude of deflections of orthotropic laminates was about two to three times that of antisymmetric angle-ply laminates for a/b > 3 under sinusoidal loading.
6. Unlike in antisymmetric cross-ply laminates, the stress xz  is not zero at (x, y) = (a / 2, 0), although small in magnitude compared to that at(x, y) = (0, b / 2).Note that through-thickness variations are significantly altered when the number of layers is increased (for the same total laminate thickness).
7. In the case of eight-layer laminates, minimum stress concentration was found for both linear and uniform distributed temperature.Only difference is that for linear distributed temperature, the normal transverse shear stresses are not zero in bottom of the plates which is due to different temperatures at top and bottom of the plates.
8. Finally it was found that in all cases, finite element outputs were in good agreement with theoretical analysis results.An analytical and numerical solution for general laminated composite plates under different types of mechanical and thermal loading is presented based on classical lamination plate theory (CLPT).General lamination was evaluated by assumption of cross-ply and angle-ply laminated plates.In order to this, Navier-type method is applied for simply supported rectangular laminates and a finite element code using ANSYS is also developed to investigate the accuracy of the presented solution.Effects of lamination scheme, temperature service, geometry ratio and mechanical properties of composite laminates on static bending of laminated composite are also investigated and good agreement is found between evaluated results and those available in open literature.

Fig. 1
Fig. 1 Geometry of simply supported rectangular laminated plates used in the analytical solutions are coefficients to be determined.Considerations of Eqs.(19)-(21), shows that the mechanical transverse load q and thermal forces should also be expanded in double sine series.Thus:

Fig. 6
Fig. 6 Nondimensionalized maximum normal stress and transverse shear stress versus plate thickness under sinusoidal loading (SSL)

Fig. 10
Fig. 10 Nondimensionalized center transverse deflection versus modulus ratio for simply supported angleply laminates under sinusoidal loads of thermal loads including uniform and linear distributed temperature through the thickness are considered as follows:

Fig. 12 . 5 -Fig. 13
Figs. 12 and 13 show the distribution of transverse shear stress through the thickness of the composite plates under different types of distributed temperature in thermal loading.For comparison, results of orthotropic laminates are also included.In the case of eight-layer laminates, minimum stress concentration was found for both linear and uniform distributed temperature.Only difference is that for linear distributed temperature, the normal transverse shear stresses are not zero in bottom of the plates which is due to different temperatures at top and bottom of the plates.Distribution of normal stresses through the thickness of bending-stretching coupling for a given z 0 is compared for various types of thermal loadings (UDT and LDT) in Fig.14.Fig.12Nondimensionalizedmaximum transverse shear stress versus plate thickness under UDT -0.5

Fig. 15
Fig. 15 Distribution of transverse deflection versus plate aspect ratios for linear distributed temperatures

Table 1
Transverse deflections and stresses in square laminates subjected to uniformly distributed transverse load (UDL), sinusoidal load (SSL) or central point load (CPL)*

Table 2
Transverse deflections and stresses in square laminates subjected to uniformly distributed transverse load (UDL) and central point load (CPL)*

Table 3
Transverse deflections and stresses in square laminates subjected to UDT and LDT*