# EFFECT OF LAMINATION ANGLE AND THICKNESS ON ANALYSIS OF COMPOSITE PLATE UNDER THERMO MECHANICAL LOADING

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3 6. Straight lines normal to the middle surface remain straight and normal to that surface after deformation. 7. Transverse normal strain ε z is negligible compared to the in-plane strains ε x and ε y. 8. Strain-displacement and stress-strain relations are linear. 2.2 Stress analysis of lamina A lamina is a thin layer of a composite material that is generally of a thickness on the order of.5 in. (.125 mm). A laminate is constructed by stacking a number of such lamina in the direction of the lamina thickness. The constitutive equations for a general linear elastic solid relate the stress and strain tensors through the expression: σ ij = C ijkl ε kl, ε ij = S ijkl σ kl (i, j, k, l = 1,2,3) (1) Where C ijkl = stiffness components, S ijkl = compliance components. Fiber reinforced composite materials are considered to have orthotropic elasticity because these materials possess three mutually perpendicular planes of the elastic symmetry. For a unidirectional reinforced lamina in the 1-2 plane as shown in Figure, a plane stress state is defined by: σ 3 =, τ 23 =, τ 31 = σ 1, σ 2, τ 12 Moreover the stress- strains relations in 1-2 planes: where Q 11 = E 1 1 ν 12 ν 21 σ 1 Q 11 Q 12 Q 11 ε 1 [ σ 2 ] = [ Q 12 Q 22 ] [ ε 2 ] (2) σ 3 Q 66 γ 12,Q 22 = E 2 1 ν 12 ν 21 Strain- stress relation is obtained as:, Q 12 = Stress strain relation in x y coordinate system: ν 12 E 2 1 ν 12 ν 21, Q 66 = G 12, ν 12 E 1 = ν 21 E 2 [ε] = [Q] 1 [σ] (3) σ x Q 11 Q 12 Q 16 ε x [ σ y ] = [ Q 12 Q 22 Q 26 ] [ ε y ] where [Q ] = [T] 1 [Q][R][T][R] 1 (4) τ xy γ xy Q 16 Q 26 Q Stress analysis in laminate C 2 S 2 2CS 1 T = [ S 2 C 2 2CS ] and R = [ 1 ], SC SC C 2 S 2 2 where C = cos θ, S = sin θ Determination of laminate stiffness [A B D] matrix Consider a laminate made of n plies shown in Figure 4.6. Each ply has a thickness of tk. Then n the thickness of the laminate h is h = k=1 t k Then, the location of the mid plane is h/2 from the top or the bottom surface of the laminate. The z-coordinate of each ply k surface (top and bottom) is given by: k 1 h k 1 = h 2 + t 1 (top surface), h k 1 = h 2 + t Volume 67, No. 1, (217) 217 SjF STU Bratislava 7 k 1 (bottom surface) (5)

4 (a) (b) Fig. 1 (a) Rotation of Principle material axis from reference x-y axis, (b) location of plies in laminate [15] (a) (b) Fig. 2 Composite plate subjected to (a) Normal force/ length, (b) Bending moment / length [15] (a) (b) Fig. 3 Composite plate subjected to (a) combined mechanical load (normal force and bending moment), (b) Thermo mechanical load Force per unit length (N) and the bending moment per unit length(m) of a laminate are given as: N x A 11 A 12 A 16 [ N y ] = [ A 12 A 22 A 26 ] [ N xy A 16 A 26 A 66 ε x ε y γ xy B 11 B 12 B 16 ] + [ B 12 B 22 B 26 ] [ B 16 B 26 B 66 k x k y ] (6) k xy SjF STU Bratislava Volume 67, No. 1 (217)

5 M x B 11 B 12 B 16 [ M y ] = [ B 12 B 22 B 26 ] [ M xy B 16 B 26 B 66 ε x ε y γ xy D 11 D 12 D 16 ] + [ D 12 D 16 D 22 D 26 D 26 ] [ D 66 N x, N y, = normal force per unit length, N xy = shear force per unit length, k x k y ] (7) k xy M x, M y, = Bending moment per unit length, M xy = Twisting moments per unit length ε = midplane strain of laminate in x-y coordinate and k =laminate curvature. Combining the above two matrix as: { N B } = [A M B D ] [ε k ] (8) n A ij (extensional stiffness matrix) = k=1 [Q ] ij k (h k h k 1 ), i = 1,2,6, j= 1,2,6 B ij (extension bending coupling matrix) = 1 n [Q ] 2 ij k (h 2 2 k=1 k h k 1 ), i = 1,2,6, j= 1,2,6 D ij (bending stiffness matrix) = 1 n [Q ] 3 ij k (h 3 3 k=1 k h k 1 ), = 1,2,6, j= 1,2, Determination of stress of laminate The mid plane strain and curvature can be determined from ABD matrix as: ε k = A B N B D 1 [ M ] (9) Stresses of laminate in the k th lamina as: Strain of laminate in the k th lamina as: 2.4 Thermo mechanical stresses [σ] k = [Q ] k [ε ] + h[q ] k [k] (1) [ε] k = [ε ] + h[k] (11) Thermal stresses are developed due to cooling down or heating from processing temperatures to operating temperatures different from processing temperatures. Each ply in a laminate gets stressed by the deformation differences of adjacent lamina. The mechanical strains developed by thermal loads are: [ε M ] = [ε] [ε T ] where [ε] = [ε ] + z[k] (12) The thermal stresses are given by: Mid plane strain and curvature is calculated by: where [σ T ] = [Q ][ε M ] (13) [ NT M T] = [A B B [N T n ] = T [Q ] ij [α] k k (h k h k 1 ) D ] [ε k ] (14) k=1, n [M T ] = 1 2 T [Q ] ij [α] k k (h 2 2 k h k 1 ) k=1 Volume 67, No. 1, (217) 217 SjF STU Bratislava 9 (15)

8 SHEAR STRESS XY () STRESS X() STRESS Y() Table 3 Comparison of stress ratio () obtained from Classical Lamination Theory with FEA (ANSYS) for thermo mechanical load. THERMO MECHANICAL LOAD STRESS RATIO CLT ANSYS LAMINA a/h STACKING SEQUENCE STACKING SEQUENCE []s [3]s [45]s [6]s [9]s []s [3]s [45]s [6]s [9]s T1 B1 T2 B2 T3 B3 T4 B T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE(DEG) LAMINATION ANGLE(DEG) List of symbols used σ Stress G Shear modulus ε Strain θ Poisson s ratio θ Lamination angle(deg) T1 Top of layer 1 γ Shear strain at plane B1 Bottom of layer1 τ Shear stress at plane T2 Top of layer 2 E Young s Modulus B2 Bottom of layer 2 h Total laminate thickness T3 Top of layer 3 z thickness B3 Bottom of layer3 a Plate length T4 Top of layer T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE (DEG) Fig. 4 Composite plate subjected to combined mechanical load (N x = 1 N m, M x = 4 N m and a h = 5) a) stress in x direction b) stress in y direction c) Shear stress XY for different lamination angle SjF STU Bratislava Volume 67, No. 1 (217)

9 SHEAR STRESS XY, STRESS X, STRESS Y, LAMINATION ANGLE (DEGREE) T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE(DEGREE) T1 B1 T2 B2 T3 B3 T4 B T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE(DEGREE) Fig. 5 Composite plate subjected to thermal load ( T = 75 ) a) stress in x direction b) stress in y direction c) Shear stress XY for different lamination angle. Volume 67, No. 1, (217) 217 SjF STU Bratislava 13

10 SHEAR STRESS XY, STRESS X, STRESS Y, T1 B1 T2 B2 T3 B3 T4 B T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE (DEGREE) LAMINATION ANGLE(DEGREE) T1 B1 T2 B2 T3 B3 T4 B LAMINATION ANGLE(DEGREE) Fig. 6 Composite plate subjected to Thermo mechanical load (N x = 1 N m, M x = 4 N m, T = 75 and a h = 5) a) stress in x direction b) stress in y direction c) Shear stress XY for different lamination angle SjF STU Bratislava Volume 67, No. 1 (217)

11 z/h z/h z/h z/h Stresses(Mechanical load, Nx & Mx) on Laminate (a/h=5, [+-]as) TaoXY XY Stresses(Mechanical load, Nx & Mx) on Laminate (a/h=5, [+-3]as) TaoXY XY Stresses(Mechanical load, Nx & Mx) on Laminate (a/h=5, [+-45]as) Tao XY Stresses(Mechanical load, Nx & Mx) on Laminate (a/h=5, [+-6]as) Tao XY Fig. 7 Variation of stress with laminate thickness ( z h ) for lamination angle (a) [+- ] as & [+-9] as (b) [+-3] as (c) [+-45] as and (d) [+- 6] as for combined mechanical load (N x = 1 N m, M x = 4 N m and a h = 5) Volume 67, No. 1, (217) 217 SjF STU Bratislava 15

12 z/h z/h z/h z/h Stresses(Thermal load,deltat) on Laminate (a/h=5 & [+-]as & [+-9]as) Tao XY Stresses(Thermal load,deltat) on Laminate (a/h=5 & [+-3]as) Tao XY x Stresses(Thermal load,deltat) on Laminate (a/h=5 & [+-45]as) Tao XY Stresses(Thermal load,deltat) on Laminate (a/h=5 & [+-6]as) Tao XY Fig. 8 Variation of stress with laminate thickness ( z h ) for lamination angle (a) [+- ] as & [+-9] as (b) [+-3] as (c) [+-45] as and (d) [+- 6] as for thermal load ( T = 75 and a h = 5) SjF STU Bratislava Volume 67, No. 1 (217)

13 z/h z/h z/h z/h Stresses(Thermo Mechanical load, Nx,Mx & deltat) on Laminate (a/h=5 & [+-] & [+-9]) TaoXY XY Stresses(ThermoMechanical load, Nx,Mx, deltat) on Laminate (a/h=5 & [+-3]) TaoXY XY Stresses(Thermo Mechanical load, Nx,Mx & deltat) on Laminate (a/h=5 & [+-45]) TaoXY XY Stresses(Thermo Mechanical load, Nx,Mx & deltat) on Laminate (a/h=5 & [+-6]) TaoXY XY Fig. 9 Variation of stress with laminate thickness ( z h ) for lamination angle (a) [+- ] as & [+-9] as (b) [+-3] as (c) [+-45] as and (d) [+- 6] as for thermo mechanical load (N x = 1 N m, M x = 4 N m, T = 75 and a h = 5) Volume 67, No. 1, (217) 217 SjF STU Bratislava 17

14 STRESS RATIO STRESS RATIO MECH THERM THERMOMECH [+-3] MECH THERM THERMOMECH a/h a/h [+-45] MECH THERM THERMOMECH [+-6] MECH THERM THERMOMECH a/h a/h Fig. 1 Comparison of the variation of stress ratio () with thickness ratio (a/h) for mechanical, thermal and thermo mechanical load for different lamination angle a/h = 5 a/h = 1 a/h = 2 THERMO MECHANICAL LOAD 1.8 a/h = 5 MECHANICAL LOAD a/h = a/h = LAMINATION ANGLE (DEG) LAMINATION ANGLE(DEG) Fig. 11: variation of stress ratio with lamination angle for different a/h ratio for mechanical and thermo mechanical load SjF STU Bratislava Volume 67, No. 1 (217)

15 STRESS RATIO STRESS RATIO 1.7 MECHANICAL LOAD 1.15 THERMO MECHANICAL LOAD DEG 3 DEG 45 DEG 6 DEG 9 DEG DEG 3 DEG 45 DEG 6 DEG 9 DEG a/h a/h Fig. 12: variation of stress ratio with a/h ratio for different lamination angle for mechanical and thermo mechanical load Mechanical load [+-] & [+-9] 1.25 Mechanical load [+-3] 1.2 a/h = 5 a/h = 1 a/h = a/h = 5 a/h = 1 a/h = NO. OF LAMINA NO. OF LAMINA 1.25 Mechanical load [+-6] 1.25 Mechanical load [+-45] 1.2 a/h = 5 a/h = 1 a/h = a/h = 5 a/h = 1 a/h = NO. OF LAMINA NO. OF LAMINA Volume 67, No. 1, (217) 217 SjF STU Bratislava 19

16 1.25 Thermo Mechanical load [+-] & [+-9] [+-3] Thermo mechanical load 1.2 a/h = 5 a/h = 1 a/h = a/h = 5 a/h = 1 a/h = NO. OF LAMINA NO. OF LAMINA 1. [+-45] Thermo mechanical load 1. [+-6] Thermo Mechanical load a/h = 5 a/h = 1 a/h = a/h = 5 a/h = 1 a/h = NO. OF LAMINA NO. OF LAMINA Fig. 13 variation of stress ratio with no. of lamina for different lamination angle for mechanical and thermo mechanical load. 5. Conclusion Composite plate subjected to thermo mechanical load find wide application in aerospace, military, medical, automobile sector etc. This paper present a model to analyze a composite plate subjected to different loading conditions. Classical laminate theory is used to solve the model. It is concluded from the analysis that the optimum lamination angle for combined mechanical load is [+-3] and for thermo mechanical load is [+-6] at which the stress ratio is minimum. The stress ratio () decreases with the increase in a h ratio and number of lamina. The problem is solved by using both CLT method and FEA.The result of both the method shows good agreement with each other with minor percentage error (3% -5%). Finite element tool (ANSYS) may be a more appropriate method to find the approximate solution to the stress and strain analysis of complex composite plate problem where it is not possible to solve the mathematical model. REFERENCES [1] L.W. Chen, L.Y. Chen. Thermal deformation and stress analysis of composite laminated plates by finite element method, Computers and Structures, Vol. 35, 199 (6), SjF STU Bratislava Volume 67, No. 1 (217)

17 [2] O. Sayman. An elastic plastic thermal stress analysis of aluminium metal matrix composite beams, Composite Structures, 21 (53), [3] O. Ozcan, N. B. Bektas. Elasto plastic stress analysis in simply supported thermo plastic laminated plates under thermal loads, Composite Science and Technology, 21 (61), [4] F. R. Jones, F. Zhao. Thermal loading of short fibre composites and the induction of residual shear stresses, Composites: Part A, 27, [5] C. H. Wu, T. R. Tauchert. Thermoelastic analysis of laminated plates, 1. Symmetric specially orthotropic laminates, Thermal Stresses, 198 (3), [6] A.K. Noor, W.S. Burton, Assessment of shear deformation theories for multilayered composite plates. Applied Mechanics Reviews, 1989, (42), [7] S. Savithri, T. K. Varadan. A Simple Higher Order Theory for Homogeneous Plates. Mechanics: Research Communications, 1992 (19), No. 1, [8] P. Vidal, O. Polit. A thermomechanical finite element for the analysis of rectangular laminated beams. Finite. Elem. Anal. Des., 26 (42), [9] T. B. Zineb et al. Analysis of High Stress Gradients in Composite Plates with Rapidly Varying Thickness. Composites Science and Technology, 1998 (58), [1] M. G. Joshi, S. B. Biggers, Jr. Thickness Optimization for Maximum buckling loads in Composite Laminated plates, Composites: Part B, 1996 (27B), [11] H. Javadi, I. Rajabi, V. Yavari, M. H. Kadivar. Three-dimensional solutions for contact area in laminated composite pinned joints with symmetric and non-symmetric stacking sequences, Journal of Mechanical Engineering Strojnícky časopis, 27 (58), No. 6, [12] E. Kormaníková. Optimal design of laminate circular cylindrical shell 34. Journal of Mechanical Engineering Strojnícky časopis, 27 (58), No. 6, [13] S. S. Choudhary, V.B. Tungikar. A simple higher-order theory for dynamic analysis of composite plate. Journal of Mechanical Engineering Strojnícky časopis, 21 (61), No. 3, [14] Autar K. Kaw, Mechanics of Composite material, second edition, Taylor & Francis, 26. [15] Khoo Ban Han, Fiber reinforced composite laminate plate for varying thickness, Thesis, University of Technology, Malysia Volume 67, No. 1, (217) 217 SjF STU Bratislava 21

18 SjF STU Bratislava Volume 67, No. 1 (217)

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