Thermal Buckling of Laminated Composite Plates Using a Simple Four Variable Plate Theory

I n this study, the thermal buckling behavior of composite laminate plates cross-ply and angle-ply all edged simply supported subjected to a uniform temperature field is investigated, using a simple trigonometric shear deformation theory. Four unknown variables are involved in the theory, and satisfied the zero traction boundary condition on the surface without using shear correction factors, Hamilton's principle is used to derive equations of motion depending on a Simple Four Variable Plate Theory for cross-ply and angle-ply, and then solved through Navier's double trigonometric sequence, to obtain critical buckling temperature for laminated composite plates. Effect of changing some design parameters such as, orthotropy ratio (E1/E2), aspect ratio (a/b), thickness ratio (a/h), thermal expansion coefficient ratio (α2/α1), are investigated, which have the same behavior and good agreement when compared with previously published results with maximum discrepancy (0.5%).


INTRODUCTION
Thermal buckling research is critical for structural components used in high-speed aircraft, rockets, and space vehicles, where thermal loads are caused by aerodynamic and solar radiation heating, as well as for nuclear reactors and chemical planets, which are typically subjected to an elevated temperature regime during their service lives, (Cetkovic, 2016), (Shen, 2013), concerned thermal buckling and post-buckling behavior which presented for fiberreinforced laminated plates subjected to in-plane temperature variation and resting on an elastic foundation, The governing equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the thermal effect (Mansouri and Shariyat, 2014). The related differential equations governing the system are solved using a novel differential quadrature process (DQM). Although the refined four parameters plate theory (RPT) needs less displacement parameters and is usually more accurate than the pth order generalization of theory of Reddy (GRT), both are less accurate than the third order five parameters of Reddy theory (TOST). (Shaterzadeh, Abolghasemi and Rezaei, 2014) studied thermal buckling analysis of symmetric and antisymmetric laminated composite plates with a cut-out, subjected to a uniform temperature rise for different boundary conditions, The stiffness matrices and thermal force vector are derived according to first-order shear deformation theory (FSDT). (Ounis, Tati and Benchabane, 2014) focused on the classical plate theory, and investigated the thermal buckling behavior of composite laminated plates under uniform temperature distribution. The present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. (Vosoughi and Nikoo, 2015) developed a hybrid method for maximizing fundamental natural frequency and thermal buckling temperature of laminated composite plates that is a new combination of the differential quadrature method (DQM) based on the first-order shear deformation theory (FSDT) of plates and are discretized using the (DQM). (Jin et al., 2015) used of the Digital Image Correlation (DIC) technique to investigate the thermal buckling of a circular laminated composite plate subjected to a uniform distribution of temperature load, the results of the buckling temperature from DIC were close to the theoretical buckling temperature of the circular plate found using a simply supported boundary condition (Cetkovic, 2016). Thermal buckling of laminated composite plates was investigated using Reddy's Layerwise theory and a new version of Reddy's Layer-wise Theory. The strong form is used to derive Navier's analytical solution, while the weak form is discretized using the isoparametric finite element approximation. (Hussein and Alasadi, 2018) used two methods to study the stress analysis of composite plates subjected to the uniform temperature at various factors. The first method is an experimental test by using a dial gauge, and the second method is based on a finite element solution using a computer program resulted in the thermal strain increases with increasing temperature difference (∆T) and decreased with increasing the fiber volume fraction ( ). (Xing and Wang, 2017) investigated the critical buckling temperature of functionally graded rectangular thin plates. closed form solutions for the critical thermal parameter are obtained for the plate with different boundary conditions under uniform, linear and nonlinear temperature fields using the separationof-variable method. (Vescovini et al., 2017) used Ritz-based variable kinematic formulation to research thermal buckling of composite plates and sandwich panels. They represented arbitrary groups of plies composing the panel. Critical temperatures obtained were for, with and without accounting for the pre-buckling. (Tran, Wahab, and Kim, 2017) developed a six-variable quasi-3D model with one additional variable in the transverse displacement of higher-order shear deformation theory (HSDT), resulting in a temperature rise in a plate structure that produces nonzero transverse normal strain. The governing equation is discretized by isogeometric analysis (IGA). (Manickam et al., 2018) used a finite element approach based on first-order shear deformation theory, investigated the thermal buckling behavior of variable stiffness laminated composite plates subjected to thermal loads. The developed governing equations are solved using an eigenvalue method, in accordance with the concept of minimizing total potential energy. (Sadiq and Majeed, 2019) using a higher-order displacement field, Mantari et al. determined the critical buckling temperature of an angle-ply laminated plate. This displacement field is based on a constant "m" chosen to generate results consistent with three-dimensional elasticity (3-D) theory. The equations of motion for simply assisted laminated plates based on higher-order theory were deduced and solved using Hamilton's principle. (Kiani, 2020) studied thermal buckling nature of composite laminated skew plates reinforced by graphene platelets. The formulation is based on the first-order shear deformation plate theory. It is presumed that each layer of the composite laminated plate can have different volume fraction of graphene platelets leading to a through-thethickness piecewise functionally graded medium. The thermal buckling behavior of various shapes of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates is investigated by (Torabi, Ansari and Hassani, 2019) using higher-order shear deformation plate theory. Using Hamilton's principle, discretized equations of motion are finally obtained. A wide range of numerical results is also presented to analyze the thermal buckling behavior of various shapes of FG-CNTRC plates. (Do and Lee, 2019) used a mesh-free approach to describe the buckling behavior of multilayered composite plates in thermal environments, including a functionally graded material (FGM) layer. Thermal buckling of a composite plate laminated with an FGM layer is modeled using an improved Moving Kriging (MK) meshless approach based on n th -order shear deformation theory. (Tocci Monaco et al., 2020) used second-order strain gradient theory, and investigated the vibrations and buckling of thin laminated composite nanoplates in a humid-thermal setting. Hamilton's theorem is used to solve equations of motion. To amass analytical data the Navier displacement area was considered for both cross-ply and angle-ply laminates, and the findings revealed a wide range of angle-ply cases that are not often encountered in the published literature. (Yang et al., 2020) examined the effect of geometrical nonlinearities associated with pressure loads on the thermal buckling and dynamic properties of composite plates. Thermal buckling and modal analysis was performed on a four-sided simply supported rectangular composite plate subjected to a variety of pressure fields. The numerical results indicate that as the pressure increases, both the mode frequencies and critical buckling temperature of the plate increase. The thermal buckling behavior of a composite plate structure with a number of nano fractions was investigated by (Al-Waily, Al-Shammari and Jweeg, 2020) using analytical and Journal of Engineering Volume 27 September 2021 Number 9 4 numerical methods. The general motion equation for thermal buckling load was derived and the results were compared to the numerical results. (Alabas and Majid, 2020) and based on classical laminated plate theory, used the improved Rayleigh-Ritz method and Fourier series to evaluate the thermal buckling behavior of laminated composite thin plates with a general elastic boundary condition applied to an in-plane uniform temperature distribution (CLPT).
In the present work, the efficiency of a four-variable refined trigonometric shear deformation theory for thermal buckling analysis of cross-ply and angle-ply laminated composite plates is investigated. The theory does not require a problem-dependent shear correction factor. Finally, the numerical results obtained using the present theory are compared to those obtained using other theories and show a high degree of agreement.

VIRTUAL WORK PRINCIPLE
Where ( ) is Virtual strain energy and ( ) is Virtual external work done are given, (Reddy, 2003), the Virtual strain energy ( ) is: Substituting Eqs (7) in Eqs (6): Now using integration by parts, the form for Virtual strain energy (δU) is found: Journal of Engineering Volume 27 September 2021 Number 9 While the Virtual external work done by thermal applied load ( ) is : When integrating by parts Eq (11) and using divergence theory: Now substituting for , from Eqs (9),(12) in Eq (5):

NAVIER SOLUTION
The Navier solution is used to analyze laminated composite plates for bending, buckling, and free vibration. cross and angle ply simply supported at all four edgings satisfactorily for the following boundary conditions (Sayyad, Shinde, and Ghugal, 2016)

NUMERICAL RESULTS AND DISCUSSION
Trigonometric displacement function is used in the present work to analyze the critical temperature of the simply supported laminate plate both cross-ply and angle-ply for the first time. In this portion, the effect of changing some design parameters such as orthotropy ratio (E1/E2), aspect ratio (a/b), thickness ratio (a/h), thermal expansion coefficient ratio (α2/α1), are investigated, which have the same behavior and good agreement when compared with previously published results.

Verification of Results
To verify the derived equations and program built using Matlab, present work results are verified by comparison with the numerical results obtained with different theories used by researchers and give good agreement as shown in the following tables, the discrepancy and the results are very close between the present work with Refined Plate Theory (RPT) that used a different displacement field.  Table 1 shows the normalized critical buckling temperature of the symmetric different cross-ply laminated composite square plate with the thickness ratio (a/h) ranging from (4 to 1000) and The dimensionless of critical temperatures was ( Tcr = T * a 2 * h/π 2 * D22 ) with material properties using (Material 1) (Mansouri and Shariyat, 2014), Observed that the critical buckling temperature increases (the result listed is inversely since it divided by D22) when a number of layers increases for all thickness ratio due to stiffness increases, The results of present work are compared with other theories and give good agreement. Table 2 and Fig. (1) (a) show a good agreement between present theory results and other theories that study the influence of the thermal expansion coefficients ratio (α2/ α1) on critical temperature for the antisymmetric angle-ply (45/−45)3 laminated plates with the dimensionless (Tcr =10 3 * Journal of Engineering Volume 27 September 2021 Number 9 11 T*α0), thickness ratio (a/h=10) and material properties using (Material 2) (Mansouri and Shariyat, 2014), from which the normalized critical temperature decreases when increasing thermal expansion coefficient ratio due to softening the plates. Also, in Table 3 and Fig.(1)(b) effects of different modulus ratio (E1/E2) ranging from (2 to 50) on critical buckling temperature for (6) layers of antisymmetric angle-ply (45/−45)3 plates with thickness ratio (a/h = 10), with the dimensionless (Tcr =10 3 * T*α0) and the material properties using (Material 2) (Mansouri and Shariyat, 2014) is investigated and noted that when increasing modulus ratio, the normalized critical temperature increase because stiffness increase. Table 4 displays another comparison with the results of previous theories, the effect of changing in thickness ratio (a/h ) ranging from (10/3 to 100) on critical buckling temperatures for antisymmetric (10 layer) angle-ply (±θ)5 laminated plates all edges simply-supported with material properties using (Material 3) (Mansouri and Shariyat, 2014) and dimensionless (Tcr =T*α0), observed that the maximum critical buckling temperature is obtained for the plate with 45 o angle, the results are close to results of previous theories.

Effect of Design Parameters
The behavior of critical temperature with changing some design parameters is investigated. The critical temperature is decreasing when the thickness ratio (a/h) and Coefficient of Thermal Expansion (CTE) ratio increased and it increases when the number of plies and orthotropy ratio is increased for angle ply laminated plates while maximum thermal buckling temperature is obtained for the plate with 45 o angle, as shown in the following tables below this behavior is caused by stiffness differences. 7.2.1 Antisymmetric and Symmetric Cross-ply of the simply supported various composite laminate thick and thin plate For this type, the dimensionless (Tcr = T*a 2 *h/π 2 * D22 ) and material properties using (Material1) (the results listed of (a/h) ratio is inversely since it divided by D22). Table 5 displays the influence of changing of thickness ratio (a/h) and aspect ratio (a/b) on critical buckling temperatures of laminate plate [(0/90)S,(0/90)3]. The results show that critical buckling temperatures increase when the aspect ratio (a/b) increases along with the thickness ratio (a/h). Table 6 shows the effect of thermal expansion coefficient ratio (α2/α1) on critical buckling temperature for different thickness ratio (a/h) of laminate plate [(0/90)S,(0/90)3] as expected, the critical buckling temperature decrease when (α2/ α1) increase along (a/h) ratio, there is inversely proportional between the critical buckling temperature parameter and (α2/α1) due to thermal buckling strength decrease of the plate when increasing α2. Effect of changing orthotropy ratio (E1/E2) on critical buckling temperature for antisymmetric [(0/90),(0/90)2, (0/90)4] plates are listed in Table 7 with varied thickness ratio (a/h), the results are clear that the nondimensionalized buckling load decreases for antisymmetric laminates as the modulus ratio (E1/E2) increases.

Journal of Engineering
Volume 27 September 2021 Number 9 12 7.2.2 Antisymmetric angle-ply of the simply supported various composite laminate plate For this type, the dimensionless (Tcr = T* α0) and material properties given by (Material 3).

CONCLUSIONS
The thermal buckling behaviors of laminate plates have been described and discussed in this work by using a simple trigonometric shear deformation theory. The most significant characteristic of this theory is that it contains only four unknowns, as opposed to five in firstorder shear deformation theory and other higher-order theories. Following the above discussions, the preliminary results are summarized as follows: 1-Design parameters as (modulus ratio, aspect ratio, thickness ratio, (α2/α1) ratio), in addition to the number of layers, have the same behavior that obtained by other different plate theories.
2-The angle ply gives higher critical temperature buckling than cross-ply for the same materials because of stiffness differences 3-Results obtained from present work agree well with other RPT that use different displacement fields for both thick and thin plates also with HSDPT for thin plate,s while discrepancy increased for thick plates.
4-The critical thermal buckling is increased for symmetric and antisymmetric laminates angleply when the modulus ratio increases while it decreases for antisymmetric laminates cross-ply.