Thermal Buckling Analysis of Laminated Composite plates with General Elastic Boundary Supports

In this study, the modified Rayleigh-Ritz method and Fourier series are used to determine the thermal buckling behavior of laminated composite thin plates with a general elastic boundary condition applied to in-plane uniform temperature distribution depending upon classical laminated plate theory(CLPT). A generalized procedure solution is developed for the RayleighRitz method combined with the synthetic spring technique. The transverse displacement of the orthotropic rectangular plates is not a different term as a new shape expansion of trigonometric series. In this solution approach, the plate transverse deflection and rotation due to bending are developed into principle Fourier series with a sufficient smoothness auxiliary polynomial function, the variable of boundary condition can be easily done by only change the boundary spring stiffness of at the all boundaries of laminated composite plate without achieving any replacement to the solution. The accuracy of the current outcome is verified by comparing with the result obtained from other analytical methods in addition to the finite element method (FEM), so the excellent of this technique is proving during numerical examples.

constant thermal load for symmetrically balanced laminates is scrutinized thermal responses for multiple material models. The coefficients of thermal expansion are differentiated to research the influence on the optical fiber path and the critical temperature of buckling. This research obtains in comparison to straight fiber configurations, curved fiber path configurations increase of resistance to thermal buckling that provide a 36:9%. (Jinqiang Li, et al., 2016) studied the influence of random system properties of critical thermal buckling temperature of composite laminated plates through temperature-dependent properties by applying a micromechanical approach, depends on the CLPT in coupling by Hamilton's basic. To find the second-order statistics (mean and standard deviation) of the critical thermal buckling temperature they are applying a mean-centered first-order perturbation technique. Of random volume fractions, material properties, coefficients of thermal expansion, angels of layer, aspect ratios, and B. Cs, their effect on the critical temperature of thermal buckling is studied. (M. Cetkovic, 2016) by applying the Layerwise Theory of Reddy were subdued employing the (PVD) principle of virtual displacements the thermal of buckling laminated plates of the composite are studied. The isoparametric finite element approximation is found from the weak term, while the strong term is discretized utilizing the solution of Navier's analytical. The influencing on thermal buckling of isotropic, orthotropic, and laminated composite plates of ratio elastically E1/E2, side to thickness ratio, mesh refinement, ratio coefficient of thermal expansion, temperature distribution, aspect ratio, and B.Cs. (S. Kamarian, et al., 2016) applied a powerful meta-heuristic algorithm called a firefly algorithm (FA), which is based on the flashing behavior of fireflies to transact with stacking sequence optimization of laminated composite plates for maximizing the critical buckling temperature. The main objective of their study was to offer the capability of FA in the optimization of composite structures. (Loc V. Tran, et al., 2017) studied the two types of a thermal plate are take into consideration thermal bending and thermal buckling, a rise of temperature in structure plate produces normal non-zero transverse strain, a 6-variable model of quasi-3D, and one variable in displacement transverse of(HSDT) higher-order shear deformation theory to take into account. Thus, the effects of transverse shears and normal strain in a laminated composite plate are advanced. By (IGA) isogeometric analysis the governing equation is obtained. Because of the existence of bending coupling extension. (Abdouna, et al., 2017) a methodological approach based on the perturbation and homotopic methods is applied to study thermal buckling and post-buckling analyses with dependent properties on the temperature of the plates anisotropic laminated. A power law apportionment in the form of temperature is utilized, and to a uniform temperature difference, the structure is submitted. The method of finite element is applied for solutions numerically, the temperature and the displacement of power series expansions are progressing. The influence of structure geometry, boundary conditions and temperature-dependent properties on the thermal post-buckling and buckling behaviors are estimated. (Rajesh Kumar, et al., 2017) by distribution of uniform temperature by its thickness submitted to center on thermal heating on the plate area reported a solution of semi-analytical of thermal post-buckling and buckling of plate's rectangular composite. By fixing the thermos elasticity problem employs Airy's stress function the analytical terms for these in-plane allocation pre buckling stress during the composite plate are advanced. Applying a vibrational principle, the plate nonlinear stability equations are subedited, and applying First-order shear deformation theory incorporating von-Kármán geometric nonlinearity to model the plate. (Heated region area, aspect ratio, and the B.Cs) are examined the effect on the critical temperature of buckling and equilibrium paths. This study investigates the critical temperature of buckling laminated composite thin plate with general boundary condition, based on classical laminated plate theory and using Rayleigh-Ritz method, the displacement function as Fourier cosine series plus a continuous function so the Journal of Engineering Volume 26 March 2020 Number 3 4 effect of boundary conditions, aspect ratio, length to thickness ratio, angle-ply orientation, and different composite materials properties are examined.

2.THEORETICAL ANALYSIS 2.1 Classical Laminated Plate Theory
As a statically equivalent single layer owning complex constituent disposal are the equivalent single layer ESL laminated plate theories, so a variable properties laminated plate is processed, lessening the 3-D continuum problems to 2-D problems. By presuming the shape of the stress or displacement fields like a linear integration of unknown coordinate of thickness and functions, the equivalent single layer theories are amelioration: (REDDY, 2004).
Where , z, ( , ), ,t, are the stress or displacement component, the coordinate of thickness, the plane coordinates, functions and time to be determined respectively. The governing equations are calculated with the formulation of factual displacement, when are displacements. The ESL is (CLPT)the classical plate laminated theory, which is a widening of the (classical) plate Kirchhoff theory to laminated plates of composite be based on the displacements field.
Where the coordinate direction (x, y, z) the ( , , ) are the displacement component, respectively, at the mid-plane (i.e., z = zero). The differential equation of a thin laminated plate under thermal buckling load is written as: (Henry Khov, 2009).  Extension stiffness matrix In this study, the uniform temperature distribution is taken into consideration. The plate temperature is initially supposed to be where final temperature is uniformly raised to critical value in which the plate buckles. The temperature variation is then

Boundary Conditions:
The twisting and bending, shear forces can express of the displacement function as ( Where 0 , 1 0 , 1 are the stiffness of spring in rotations, 0 , 1 0 , 1 are the stiffness of spring in transitional, = 0 ( = 0 ), respectively. Eq. (11) -(18) explain a set of different B.Cs from which, the free boundary condition can be found by putting the spring constant to zero while by setting the spring constant into infinity the clamped condition can be got (in actual calculation as a large number) and the simply supported can be obtained by setting the spring constant in between larger than 0 and small than ∞ (0 < K < ∞). Fig. 2. Form Eq. (6)-(18), the boundary conditions can be finally expressed as:

Admissible Functions:
The permissible functions take a significant rule in the Rayleigh-Ritz technique. The beam functions products are the orderly selection as the permissible functions, and the displacement function can be written as (W.L. Li, 2004).
Where ( ) or ( ) are the characteristic beams functions containing the same B.Cs in the xdirection and y-direction, sequentially. As a linear collection of trigonometric and hyperbolic functions, the functions of beam can be broadly acquired. They contain a few unbeknown parameters, and from the boundary conditions are obtained. Accordingly, then, all boundary conditions essentially drive to various forms of beam functions. In veritable employments, this is plainly inappropriate, for a various boundary beam not to remember the boredom of obtaining the characteristic functions. An advanced method of Fourier series has been suggested for beams with qualitative boundary at ends to eschew this obstacle, in which the characteristic functions are express in the term of (Li, 2000).
Where p (x) can be a counted a qualitative continuous function that, whatever of B.Cs, is permanently elected to accept the subsequent equations: ( 28) Still, as a continuous function that accepts Eq. (25)-(28), p(x) has only been understood, its formula isn't worried by consideration to the convergence of the series of the solution. Thus, the function p (x) can be option in different wanted shapes. As shown, postulate that p (x) is a polynomial function Where ( )is the Legendre function is the expansion constant of order n. The above expression for the function P(x) can be expressed as: (15 4 − 30 2 2 + 7 4 )/360 (6 − 2 2 − 3 2 )/6 (3 2 − 2 )/6 } The results in Eq. (30)-(32) are obtained from much directly, but little general approaches (Li, 2004).
To find the unknown the coefficient of boundary, 0 , 1 , 0 , and 1 , substitution of Eq. (24) and ( For a totally free beam, the matrix Ha will be one matrix. Via, this case can be controlled to a few stretches by fictitious deliver springs to the edges of a beam with littlest stiffness. It has been exhibited in (Li, 2002). In such remediation via the matrix might be unconditional, although the functions are much appropriate for specific conditions and can be instantly applied in the method of Rayleigh-Ritz as the permissible functions. Where The terms for ( ), can be, correspondingly, found from Eq. (32), (34) and (35) by little shifting the x-concerning parameters by the y-concerning.

Determination of Critical Buckling Temperature:
The plate is applied to thermal buckling load , respectively. The total mechanical energy can be formed in the following terms: Where are the coefficient of the nonzero impersonality Amn. Obtained the first term determinant of Eq. (43), and equating it to zero to find critical buckling temperature. The ∆ is obtained by fixing the eigenvalue problem. For M and N more than 1and various edge conditions, the analysis's solution will be complicated, so it must use computer programming to find ∆ . In this investigation, MATLAB R2017b is applied to fix the eigenvalue problem to obtain the critical buckling temperature.

Numerical Results
With elastic boundary conditions supported the laminated plate is assumed to be embedded in thermal environments by a uniform temperature rise in the depth of the plate, by eigenvalue problem results from Ritz solution, its critical temperature of the buckling is determined. To verify the buckling temperature of thin plates calculated based on the admissible function used and activity of the computer program using Matlab 17 presented results are compared with those obtained by other researchers. Antisymmetric angle laminated plates are in good agreement when compared with those obtained by (Houdayfa, et al., 2014) and (Sarath, 2000). Although they used different solution methods as shown in Table 1, the plate contains 10 layers with various orientations angles and simply supported in all edges are possessed to investigate the influence of the fiber angle on the critical temperature of buckling, higher lamina angle leads to higher critical temperature of an antisymmetric angle plies for angles (0 o -45 o ). This is due to the increase of stiffness and thus increase the critical temperature of buckling. For numerical results, the material properties are E1 = 15Gpa, E2 = 1Gpa, G12 = 0.5Gpa, v12 = 0.3, α1 = 0.015*10 -6 1/ o C, α2 = 1*10 -6 1/ o C, Table 1. The critical temperature of buckling at angle-ply simply supported square composite plates, ΔTcr = Tcr * α2, N=10, thickness ratio =100.
Two types, symmetric cross and angle ply-simply supported plates (eight layered ) under uniform temperature are investigated in Table 2, compared by the results obtained by (Houdayfa, et al., 2014) using a finite element based upon the classical plate theory. It is well seen that critical temperature of angle ply greater than cross-ply due to the increase of the stiffness and thus increase the critical temperature of buckling, it is shown that numerical results are in good agreement, the material properties are E1 = 22.5Mpa, E2 = 1.17Mpa, G12 = 0.66Mpa, v12 = 0.22, α1 = -0.04*10 -6 1/ o C, α2 = 16.7*10 -6 1/ o C In Table 3 the critical buckling temperature of laminated plate with different thickness ratio are compared with published results of (Achchhe, et al., 2009). The example of plate have stacking sequence of [45/-45]3T with simply supported in all edges applied to uniform temperature distributions, as noted the decrease in thickness ratio increases the buckling temperature due to stiffness increase of plate so it needs high temperature to buckle, the results of present study are very close to those obtained by Achchhe Lal et al. using HSDT, the material properties are E1=0.081Gpa, E2=10.3Gpa, G12=0.031Gpa, G23=0.0304Gpa v12=0.21, α1= -0.21*10 -6 1/ o C, α2=16*10 -6 1/ o C, α0 = 1*10 -6 1/ o C. Table 3. Comparison of buckling loads ΔTcr ( o C) for perfect, angle-ply (±45)3T square laminated composite plates having simply supported boundary conditions ΔTcr=Tcr* α0 *10 3 The critical temperature of four-layered cross-ply symmetric laminated thin plates with various boundary conditions are compared in Table 4 with results obtained by (Rajesh Kumar, et al.,  2017). They used, first-order shear deformation theory to model the plate and also compared by (Kamariana, et al., 2016) who showed that the temperature buckling is highest in CCCC boundary condition and lowest in SSSS boundary condition because the stiffness of clamped is the larger than simply supported plate. Also the numerical results show good agreement, where the material properties are E1=15Gpa, E2=1Gpa, G12=0.5Gpa, G23=0.3356Gpa, v12=0.3, α1=0.015*10 -6 1/ o C, α2=1*10 -6 1/ o C, α0 = 1*10 -6 1/ o C. Table 4. Critical Buckling Temperature ΔTcr = Tcr * α0 *10 2 , a/b=1, a/h=100, (0/90/90/0) In Table 5 another comparison is shown, with the closed-form solution based on the FSDT obtained by (Mohammad-Zaman Kabir, et al., 2016); plates with CCCC and SSSS boundary conditions with good agreement. Two materials are compared in this case, material 1 (E1=38.16Gpa, E2=9.65Gpa, G12=6.21Gpa, v12=0.26, α1=4*10 -6 1/ o C, α2=16*10 -6 1/ o C) and  11 material 2(E1=155Gpa, E2=8.07Gpa, G12=4.55Gpa, v12=0.22, α1= -0.07*10 -6 1/ o C, α2=16*10 -6 1/ o C). Table 5. Critical Buckling Temperature ΔTcr ( o C) a=b=200mm, h=2mm, (0/90/0/90) Effect of aspect ratio on critical temperature of laminated angle plate is obtained in Table 6; results are compared with (Wen-Chi Chen, et al., 1993), who used first-order transverse shear deformation theory. It is noted that increasing the aspect ratio increases the buckling temperature due to the increase in the stiffness of the plate. The accuracy of the present approach is validated for evaluating the influence of aspect ratio on the critical buckling temperature; the material properties are E1=40Gpa, E2=1Gpa, G12=0.5Gpa, G23=1Gpa v12=0.25, α1=1*10 -6 1/ o C, α2=1*10 -6 1/ o C. Table 6. Critical Buckling Temperature ΔTcr ( o C), a /h=100, (45/-45) Table 7 shows the influence of rotational stiffness spring on the critical buckling temperature the value of rotational stiffness spring coefficient in simply support is Zero while for clamped edges coefficient of rotational spring is large number; critical temperature increases when the rotational stiffness spring coefficient is increased as mentioned before, the material properties E1=40Gpa, E2=1Gpa, G12=0.5Gpa, v12=0.25, α1=1*10 -6 1/ o C, α2=2*10 -6 1/ o C. The influence of other boundary conditions on critical buckling temperature for the plate with a symmetric and antisymmetric cross and angle ply is shown in Table 8. Some of the results are compared with (Wen-Chi Chen, et al., 1993). The publisher used first-order transverse shear deformation theory for antisymmetric angle ply, properties of material are E1=25Gpa, E2=1Gpa, G12=0.5Gpa,G23=0.2Gpa, v12=0.25, α1=1*10 -6 1/ o C, α2=3*10 -6 1/ o C. Table 8. Critical Buckling Temperature ΔTcr = Tcr *((a/h) 2 )* α1*10, a /h=50, a/b=1 The effect of different (ratio of modulus elastically (E1/E2) and the ratio of thermal expansion coefficient (α2/α1) on critical temperature of a symmetric and antisymmetric angle-ply and cross-ply 4-layer simply supported and clamped edges plate applied to uniform temperature distribution is tested. Material properties are: E2=1Gpa, G12=0.5Gpa, v12=0.25, α1=1*10 -6 1/ o C, α2=2*10 -6 1/ o C, a/h=100, a/b =1. In Fig. 3 and 4 for cross-ply and angle symmetric and antisymmetric plate the simply and clamped supported plate, there is small change (inversely) in stiffness of laminates with increase in E1. It should be observed that coupling between extensions and bending is not taken in this investigation. The influence of coupling is clearer in antisymmetric angle -ply laminates. From Fig. 5 and 6, it is obvious that the critical temperature of buckling lowering as the thermal expansion ratio raises, exhibit the effect of thermal expansion coefficient ratio (α2/α1) on critical temperatures of plates when E1=40Gpa, as shown in Fig. 5  and 6 for cross and angle ply plate the simply and clamped supported plate for symmetric and antisymmetric, there is an inverse proportionality between the critical buckling temperature parameter and (α2/α1) due to the stiffness of plate decreasing as the thermal expansion coefficient ratio (α2/α1) increase. In this study, analysis E1=40Gpa and α2 are diverse when α1 is staying constant.

CONCLUSIONS
In the present work, the critical buckling temperature of a composite laminated cross and angle plates with general boundary conditions is developed using the Ritz method based on a function proposed by (Li, 2002) for the first time. The obtained results are compared with those found by many researchers who studied thermal buckling of a composite laminated by different solution methods. The main conclusions from obtained results as excepted, are: 1-The influence of boundary conditions on the critical temperature buckling is very clear.
Clamped edges conditions result in high critical temperature buckling due to offering high stiffness compared by the simply and free, supported conditions. 2-The critical temperature buckling is inversely proportional to design parameters (aspect ratio, thickness ratio, lamination angle, modulus elastically ratio (E1/E2) and thermal expansion ratio (α1/α2)). 3-The angle ply gives higher critical temperature buckling than cross-ply for the same materials.
h the thickness of the laminate mm ℎ , ℎ thickness at the top and bottom of the laminate mm hk, hk-1 distances from the reference plane of the laminate to the two surfaces of the kth ply