1. Pendahuluan Material Komposit
BAB 4 Macromechanical Analysis of a Laminate
Classical Lamination Theory
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

2. Laminate Stacking Sequence
Gambar 4.1
Schematic of a lamina

3. Laminate Behavior Modulus Elastis The Stacking Position Thickness
Angles of Orientation
Coefficients of Thermal Expansion
Coefficients of Moisture Expansion

4. Strains in a
(4.1)
Gambar 4.2
A beam under (a) axial load, (b) bending moment, and (c) combined axial and bending moment.

5. Types of loads allowed in CLT analysis
Nx = normal force resultant in the x direction (per unit length)
Ny = normal force resultant in the y direction (per unit length)
Nxy = shear force resultant (per unit length)
Gambar 4.3
Resultant forces and moments on a laminate.

6. Gambar 4.3
Mx = bending moment resultant in the yz plane (per unit length)
My = bending moment resultant in the xz plane (per unit length)
Mxy = twisting moment resultant (per unit length)

7. Classical Lamination Theory

8. Gambar 4.4
Gambar 4.4
Gambar showing the relationship between displacements through the thickness of a plate to midplane displacements and curvatures.

9. Global Strains in a Laminate

10. Gambar 4.5
Gambar 4.5
Strain and stress variation through the thickness of the laminate.

11. END

12. Pendahuluan Material Komposit
BAB 4 Macromechanical Analysis of a Laminate
Relating Loads to Midplane Strains/Curvatures
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

13. Laminate Stacking Sequence
Gambar 4.1
Schematic of a lamina

14. Types of loads allowed in CLT analysis
Nx = normal force resultant in the x direction (per unit length)
Ny = normal force resultant in the y direction (per unit length)
Nxy = shear force resultant (per unit length)
Gambar 4.3
Resultant forces and moments on a laminate.

15. Types of loads allowed in CLT analysis
Mx = bending moment resultant in the yz plane (per unit length)
My = bending moment resultant in the xz plane (per unit length)
Mxy = twisting moment resultant (per unit length)

16. Stacking Sequence Gambar 4.6
Coordinate locations of plies in the laminate.

17. Stresses in a Lamina in a Laminate

18. Forces and Stresses

19. Forces & Midplane Strains/Curvatures

20. Forces & Midplane Strains/Curvatures

21. Integrating terms

22. Forces & Midplane Strains/Curvatures

23. Stiffness Matrices
[A] – Extensional stiffness matrix relating the resultant in-plane forces to the in-plane strains.
[B] – Coupling stiffness matrix coupling the force and moment terms to the midplane strains and midplane curvatures.

24. Stresses in a Lamina in a Laminate

25. Moments and Stresses

26. Moments & Midplane Strains/Curvatures

27. Moments & Midplane Strains/Curvatures

28. Moments & Midplane Strains/Curvatures

29. Stiffness Matrices
[A] – Extensional stiffness matrix relating the resultant in-plane forces to the in-plane strains.
[B] – Coupling stiffness matrix coupling the force and moment terms to the midplane strains and midplane curvatures.
[D] – Bending stiffness matrix relating the resultant bending moments to the plate curvatures.

30. Forces, Moments, Midplane Strains, Midplane Curvatures

31. END

32. Pendahuluan Material Komposit
BAB 4 Macromechanical Analysis of a Laminate
Laminate Analysis Steps
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

33. Laminate Stacking Sequence
Gambar 4.1
Schematic of a lamina

34. Forces, Moments, Strains, Curvatures

35. Steps

36. Steps

37. Step 1: Analysis Procedures for Laminate
Step 1: Find the reduced stiffness matrix [Q] for each ply

38. Step 2: Analysis Procedures for Laminate
Step 2: Find the transformed stiffness matrix [Q] using the reduced stiffness matrix [Q] and the angle of the ply.

39. Step 3: Analysis Procedures for Laminates
Step 3: Find the coordinate of the top and bottom surface of each ply.
Gambar 4.6
Coordinate locations of plies in the laminate.

40. Step 4: Analysis Procedures for Laminates
Step 4: Find three stiffness matrices [A], [B], and [D]

41. Step 5: Analysis Procedure for Laminates
Step 5: Substitute the three stiffness matrices [A], [B], and [D] and the applied forces and moments.

42. Step 6: Analysis Procedures for Laminates
Step 6: Solve the six simultaneous equations to find the midplane strains and curvatures.

43. Step 7: Analysis Procedures for Laminates
Step 7: Find the global strains in each ply.

44. Step 8: Analysis Procedure for Laminates
Step 8: Find the global stresses using the stress-strain equation.

45. Analysis Procedures for Laminated Composites
Step 9: Find the local strains using the transformation equation.

46. Step 10: Analysis Procedures for Laminates
Step 10: Find the local stresses using the transformation equation.

47. END

48. Pendahuluan Material Komposit
BAB 4 Macromechanical Analysis of a Laminate
Laminate Analysis: Example
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

49. Laminate Stacking Sequence
Gambar 4.1
Schematic of a lamina

50. Problem
A [0/30/-45] Graphite/Epoxy laminate is subjected to a load of Nx = Ny = 1000 N/m. Use the unidirectional properties from Table 2.1 of Graphite/Epoxy. Assume each lamina has a thickness of 5 mm. Find
the three stiffness matrices [A], [B] and [D] for a three ply [0/30/-45] Graphite/Epoxy laminate.
mid-plane strains and curvatures.
global and local stresses on top surface of 300 ply.
percentage of load Nx taken by each ply.
Gambar 4.7
Thickness and coordinate locations of the three-ply laminate.

51. Solution
A) The reduced stiffness matrix for the Oo Graphite/Epoxy ply is

52. Matrices Qbar Untuk Laminas

53. Coordinates of top & bottom of plies
The total thickness of the laminate is h = (0.005)(3) = m. h0= m h1= m h2= m h3= m
Gambar 4.7
Thickness and coordinate locations of the three-ply laminate.

54. Calculating [A] matrix

55. The [A] matrix

56. Calculating the [B] Matrix

57. The [B] Matrix

58. Calculating the [D] matrix

59. The [D] matrix

60. Setting up the 6x6 matrix
B) Since the applied load is Nx = Ny = 1000 N/m, the mid-plane strains and curvatures can be found by solving the following set of simultaneous linear equations

61. Mid-plane strains and curvatures

62. Global Strains/Stresses at top of 30o ply
C) The strains and stresses at the top surface of the 300 ply are found as follows. The top surface of the 300 ply is located at z = h1 = m.
Gambar 4.7 Thickness and coordinate locations of the three-ply laminate.

63. Global strains (m/m) εx εy Ply # Position 1 (00) Top Middle Bottom
1 (00)
Top
Middle
Bottom
8.944 (10-8)
1.637 (10-7)
2.380 (10-7)
5.955 (10-6)
5.134 (10-6)
4.313 (10-6)
(10-6)
(10-6)
(10-6)
2 (300)
3.123 (10-7)
3.866 (10-7)
3.492 (10-6)
2.670 (10-6)
(10-7)
2.655 (10-7)
3(-450)
4.609 (10-7)
5.352 (10-7)
1.849 (10-6)
1.028 (10-6)
1.291 (10-6)
2.316 (10-6)

64. Global stresses in 30o ply

65. Global stresses (Pa) Ply # Position σx σy τxy 1 (00) Top Middle Bottom
1 (00)
Top
Middle
Bottom
3.351 (104)
4.464 (104)
5.577 (104)
6.188 (104)
5.359 (104)
4.531 (104)
(104)
(104)
(104)
2 (300)
6.930 (104)
1.063 (105)
1.434 (105)
7.391 (104)
7.747 (104)
8.102 (104)
3.381 (104)
5.903 (104)
8.426 (104)
3 (-450)
1.235 (105)
4.903 (104)
(104)
1.563 (105)
6.894 (104)
(104)
(105)
(104)
4.091 (104)

66. Local Strains/Stresses at top of 30o ply
The local strains and local stress as in the 300 ply at the top surface are found using transformation equations as

67. Local strains (m/m) Ply # Position ε1 ε2 γ12 1 (00) Top Middle Bottom
1 (00)
Top
Middle
Bottom
8.944 (10-8)
1.637 (10-7)
2.380 (10-7)
5.955(10-6)
5.134(10-6)
4.313(10-6)
-3.836(10-6)
-2.811(10-6)
-1.785(10-6)
2 (300)
4.837(10-7)
7.781(10-7)
1.073(10-6)
4.067(10-6)
3.026(10-6)
1.985(10-6)
2.636(10-6)
2.374(10-6)
2.111(10-6)
3 (-450)
1.396(10-6)
5.096(10-7)
-3.766(10-7)
1.661(10-6)
1.800(10-6)
1.940(10-6)
-2.284(10-6)
-1.388(10-6)
-4.928(10-7)

68. Local stresses in 30o ply

69. Local stresses (Pa) Ply # Position σ1 σ2 τ12 1 (00) Top Middle Bottom
1 (00)
Top
Middle
Bottom
3.351 (104)
4.464 (104)
5.577 (104)
6.188 (104)
5.359(104)
4.531 (104)
(104)
(104)
(104)
2 (300)
9.973 (104)
1.502 (105)
2.007 (105)
4.348 (104)
3.356 (104)
2.364 (104)
1.890 (104)
1.702 (104)
1.513 (104)
3 (-450)
2.586 (105)
9.786 (104)
(104)
2.123 (104)
2.010 (104)
1.898 (104)
(104)
(103)
(103)

70. D) Portion of load taken by each ply
Portion of load Nx taken by 00 ply = 4.464(104)(5)(10-3) = N/m
Portion of load Nx taken by 300 ply = 1.063(105)(5)(10-3) = N/m
Portion of load Nx taken by -450 ply = 4.903(104)(5)(10-3) = N/m
The sum total of the loads shared by each ply is 1000 N/m, ( ) which is the applied load in the x-direction, Nx.
Gambar 4.7
Thickness and coordinate locations of the three-ply laminate.

71. Percentage of load Nx taken by 00 ply Percentage of load Nx taken by 300 ply Percentage of load Nx taken by -450 ply

72. END