Thermal Buckling of Angle-Ply Laminated Plates Using New Displacement Function

Critical buckling temperature of angle-ply laminated plate is developed using a higher-order displacement field. This displacement field used by Mantari et al based on a constant ‘‘m’’, which is determined to give results closest to the three dimensions elasticity (3-D) theory. Equations of motion based on higher-order theory angle ply plates are derived through Hamilton s principle, and solved using Navier-type solution to obtain critical buckling temperature for simply supported laminated plates. Changing (α2/ α1) ratios, number of layers, aspect ratios, E1/E2 ratios for thick and thin plates and their effect on thermal buckling of angle-ply laminates are studied in detail. It is concluded that, this displacement field produces numerical results close to 3-D elasticity theory with maximum discrepancy (7.4 %).


INTRODUCTION
Laminated composite plates are used widely in aeronautical, marine and mechanical industries as well as in other fields of modern engineering structures , those structure are often subjected to thermal load especially aircraft, launch vehicle and missiles structures, which may cause buckling of structure with certain boundary conditions, therefore there are many investigations about thermal buckling. Thangaratnam, 1989. used finite element method using semiloof elements to analyze critical buckling temperature for composite laminates under thermal load. The equation of motion for critical temperature is obtained by equating the second variation of total potential energy to zero. Different boundary condition for cross-ply and angle-ply symmetric and antisymmetic stacking are analyzed. Chang and Leu, 1991. studied thermal buckling of antisymmetric angle-ply laminated simply supported subjected to uniform thermal load using higher order deformation theory which account for transverse shear and transverse normal strain to obtain exact-closed form solution. Obtained results are compared with first-order shear deformation theory and Reddy's higher-order shear deformation theory and showed surprising discrepancies exist. Chen, et al, 1991, implemented finite element method to analyze thermal buckling temperature of composite plates under uniform or nonuniform thermal load. Thermal-elastic Mindlin plate theory is used by which the transverse shear deformation and rotatory inertia were taken into account. Meyers and Hyer, 1991, used Rayleigh-Ritz formulation to obtain thermal buckling and post buckling response of symmetrically laminated composite plates. Two different laminates with two types of simply supported edges, fixed and sliding are investigated. Uniform temperature change along these laminates thickness is considered. Noor and Burton, 1992, Presented three-dimensional analytical solution for thermal buckling multilayered angle-ply composite plates with temperaturedependent thermo elastic properties. The temperature is assumed to be independent of the surface coordinates, but has symmetric variation along plate thickness. Noor, et al , 1992, studied buckling of laminated plates under combined thermal and axial loadings. Multi parameter-reduction method based on a first-order shear deformation theory, in connection with mixed finite-element is developed to study the effect of different lamination and material parameters on stability of the plate. Noor, et al , 1992, developed three-dimensional elasticity solutions for the critical buckling temperature of composite plates. The pre buckling deformations are taken into account. Chen and Liu], 1993, used first-order plate theory to analyze thermal buckling of angle-ply composite plates subjected to a uniform temperature change with Levy-type boundary conditions. Prabhut and Dhanaraj, 1994, analyzed thermal buckling of symmetric cross-ply and antisymmetric angle-ply laminated composite plates subjected to uniform temperature distribution using finite element method which based on the first order shear deformation theory. Matsunaga, 2006, investigated thermal buckling of angle-ply laminated composite and sandwich plates based on two-dimensional global higher order shear deformation theory. Fundamental governing equations are derived by the principle of virtual work and solved using power series expansion of continuous displacement components for simply supported laminated composite and sandwich plates.[ Shiau et al , 2010, studied thermal buckling behavior of composite laminated plates by using finite element method.
The results indicate that the higher thermal buckling mode shapes are formed when the laminates produce higher bending rigidity along the fiber direction and higher in-plane compressive force in a direction perpendicular to the fiber direction. Shi et al], 2010, studied nonlinear thermal post buckling of antisymmetric angle-ply composite plates subjected on mechanical and thermal loads using finite element formulation. Bourada et al , 2011, used a new four-variable refined plate theory for thermal buckling analysis of functionally graded material (FGM) sandwich plates. The thermal loads are assumed as uniform, linear, and nonlinear temperature rises across the thickness direction. Abdul-Majeed , 2011, investigated thermal buckling of isotropic thermo elastic thin plates using governing differential equation and the Rayleigh-Ritz method. Three types of thermal distribution have been considered these are: uniform temperature, linear distribution and nonlinear thermal distribution across thickness. Naji , 2013, investigated critical buckling temperature of cross-ply and angle-ply composite laminated plate using classical laminated and higher order shear deformation plate theory. Equations of motion are solved using Navier and Levy methods for symmetric and anti-symmetric laminated plates. Singh, 2014, presented thermal buckling behavior of laminated composite curved panel embedded with shape memory alloy fiber based on higher order shear deformation plate theory. Variational principle with finite element modeling under uniform temperature loading is used to obtain the responses. Cetkovic and Gyorgy , 2016, analyzed thermal buckling of angle-ply laminates using Generalized Layer wise Plate Theory. Element stiffness matrix and geometric stiffness matrix are derived based on finite element formulation. Cetkovic , 2016, studied thermal buckling of composite plates using new version of Layer wise. From the strong form, analytical solution is derived based on Navier's type, while the weak form is analyzed using the isoperimetric finite element approximation. Chen, et al], 2016, investigated Vibration and buckling behavior of initially stressed and thermally stressed composite plate using variation method. The temperature is assumed uniform and linearly distributed through the plate thickness. Ounis and Belarbi , 2017, studied the thermal buckling behavior of laminated plates with rectangular cutouts using classical plate theory as a base for finite element method. Vescovini et al , 2017, used Ritz-based variable-kinematic formulation to study thermal buckling of composite plates and sandwich panels. They represented structure by means of sub laminates. Critical temperatures obtained were for, with and without accounting for the pre-buckling. Xing and Wang , 2017, concerned 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 separation-of-variable method. In present work, critical temperature of simply supported composite plate is obtained using high order shear deformation theory of plate based on displacement field used by Mantari et al , 2011 Effect of many thin and thick plate parameters, such as aspect ratio, E1/E2 ratio, α2/ α1 ratio for antisymmetric angle ply are investigated.
( ) is a shape functions to develop transverse shear strains and then stresses distribution along plate thickness.
For free boundary conditions at the top and bottom surfaces of the plate, the new proposed displacement field is, Mantari et al, 2011: The strain-displacement relations are, Reddy, 2004.

PRINCIPLES OF VIRTUAL WORK
The equations of motion will be derived depending on the new higher order theory using the Hamilton's principles ,Reddy, 2004.

EQUATIONS OF MOTION
The Euler-Lagrange equations are determined by substituting Eqs. (27 -28) into Eq. (24) to derive equations of motion as follows: δu: δv: The plane stress reduced stiffness is:

NAVIER'S SOLUTION
The generalized displacements are expanded in a double trigonometric series in terms of unknown parameters in Navier's method. To satisfy the boundary conditions of the problem, the restricted choice of the function in the series is selected.
Assuming the following displacements form to satisfied simply supported boundary conditions: Where: α = ℎ , = ℎ ( , , , 1 , 2 ), are arbitrary constants. And is the element of stiffness, from which the critical buckling temperature for the plate can be obtained.

RESULTS AND DISCUSSION
Using above analytical solutions of the HOSDT based on displacement field given by [Mantari et al], 2011, a computer program is built using MATLAB15 programming for thermal buckling of laminated angle ply composite plates. The parametric effect of side to-thickness ration (a/h), plate, aspect ratio a/b, modulus ratio E1/E2 and thermal expansion coefficient ratio (α2/ α1) on critical buckling temperature of laminated composite plates are analyzed. To verify the suggested above solution, obtained results are compared with three dimension elasticity theory proposed by ,Noor, 1992, which give good agreement with maximum discrepancy (7.4 %) for ten layers of anti symmetric angle ply with different thickness ratio (a/h) and different angles as shown in Table 1. Also, as compared with first-order thick plate theory proposed by ,Chen and Liu], 1993, for anti symmetric six layers angle ply [45/-45] plate, as listed in Table 2. with maximum discrepancy (9.4 %). Changing of aspect ratio effect on critical buckling temperature of ten layers laminated thick and thin antisymmetric plates, are listed in Table 3. Which show that critical temperature decreases as aspect ratio (a/b) increases, also it decrease with increasing (a/h) ratio which effected critical temperature larger than (a/b) ratio. Different critical thermal buckling modes for plates with different aspect ratio for [45 -45]2 angle-ply square plate are shown in Figs. 1-3. For these three figures the normalized critical temperature is (T * cr = T* α1*10*(b/h) 2 ) and material properties are, E1/E2=25, G12=G13=0.5 E2, G23=0.2 E2, (α2/ α1=3), ν12= ν13= ν23=0.25. Tables 4. and 5. show another comparison with , Chen and Liu , 1993, for antisymmetric laminated square thick and moderately thick plate (a/h =10 and a/h = 20) for different aspect ratio (b/a), angle orientation (30, 45 and 60) and number of layer (2, 4 and 8) which give closed results with maximum discrepancy (3.5 %). Table 6. show the effect of changing (E1/E2) on critical temperature for four and eight layers anti symmetric angle ply plates for different thickness ratio (a/h), since stiffness increase when increasing orthotropic ratio therefore normalized critical temperature increase. Effect of thermal expansion coefficient ratio (α2/ α1) on critical buckling temperature of four layer laminated thick and thin plates [with different (a/h) ratio], are listed in

CONCLUSIONS
Thermal buckling analysis of angle ply composite thick and thin plates is developed by using new displacement field developed by ,Mantari et al, 2011, but with changing a parameter ''m'', to 'm=.05' then obtained results are well agree with 3D elasticity theory solution and other plate solution methods. As expected critical temperature is decreased as thickness ratio and aspect ratio increased while the buckling temperature decreases with the increase of thermal expansion coefficient ratio α2/ α1 and is larger for thick, than thin laminates. thermal buckling mode of simply supported angle-ply plate does not change according to this displacement field.