CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.
-
Upload
zechariah-samuel -
Category
Documents
-
view
436 -
download
32
Transcript of CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.
![Page 1: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/1.jpg)
CHAPTER 4MACROMECHANICAL ANALYSIS
OF LAMINATES
Dr. Ahmet Erkliğ
![Page 2: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/2.jpg)
Laminate Code
A laminate is made of a group of single layers bonded to each other. Each layer can be identified by its location in the laminate, its material, and its angle of orientation with a reference axis.
![Page 3: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/3.jpg)
Laminate Code
![Page 4: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/4.jpg)
Laminate Code
[0/–45/90/60/30]or
[0/–45/90/60/30]T
[0/-45/902/60/0]
T stands for a total laminate.
subscript s outside the brackets represents that the three plies are repeated in the reverse order.
![Page 5: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/5.jpg)
Laminate Code
![Page 6: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/6.jpg)
Special Types of Laminates
• Symmetric laminate: for every ply above the laminate midplane, there is an identical ply (material and orientation) an equal distance below the midplane
• Balanced laminate: for every ply at a +θ orientation, there is another ply at the – θ orientation somewhere in the laminate
• Cross-ply laminate: composed of plies of either 0˚ or 90˚ (no other ply orientation)
• Quansi-isotropic laminate: produced using at least three different ply orientations, all with equal angles between them. Exhibits isotropic extensional stiffness properties
![Page 7: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/7.jpg)
Question
![Page 8: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/8.jpg)
![Page 9: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/9.jpg)
1D Isotropic Beam Stress-Strain Relation
![Page 10: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/10.jpg)
Strain-Displacement Equations
The classical lamination theory is used to develop these relationships. Assumptions:• Each lamina is orthotropic.• Each lamina is homogeneous.• A line straight and perpendicular to the
middle surface remains straight and perpendicular to the middle surface during deformation
![Page 11: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/11.jpg)
Strain-Displacement Equations
• The laminate is thin and is loaded only in its plane (plane stress)
• Displacements are continuous and small throughout the laminate
• Each lamina is elastic• No slip occurs between the lamina interfaces
![Page 12: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/12.jpg)
Strain-Displacement Equations
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)
![Page 13: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/13.jpg)
Strain-Displacement Equations
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)
![Page 14: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/14.jpg)
Strain-Displacement Equations
![Page 15: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/15.jpg)
Strain-Displacement Equations
![Page 16: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/16.jpg)
Strain-Displacement Equations
Midplane strains in the laminate
Curvatures in the laminate
Distance from the midplane in the thickness direction
![Page 17: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/17.jpg)
Strain-Displacement Equations
![Page 18: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/18.jpg)
Strain and Stress in a Laminate
![Page 19: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/19.jpg)
Strain and Stress in a Laminate
![Page 20: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/20.jpg)
Coordinate Locations of Plies in a Laminate
Consider a laminate made of n plies. Each ply has a thickness of tk . Then the thickness of the laminate h is
![Page 21: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/21.jpg)
The z-coordinate of each ply k surface (top and bottom) is given by
Ply 1:
Ply k: (k = 2, 3,…n – 2, n – 1):
Ply n:
Coordinate Locations of Plies in a Laminate
![Page 22: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/22.jpg)
Integrating the global stresses in each lamina gives the resultant forces per unit length in the x–y plane through the laminate thickness as
Similarly, integrating the global stresses in each lamina gives the resulting moments per unit length in the x–y plane through the laminate thickness as
![Page 23: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/23.jpg)
![Page 24: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/24.jpg)
![Page 25: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/25.jpg)
The midplane strains and plate curvatures are independent of the z-coordinate. Also, the transformed reduced stiffness matrix is constant for each ply.
![Page 26: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/26.jpg)
![Page 27: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/27.jpg)
Force and Moment Resultant
![Page 28: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/28.jpg)
Force and Moment Resultant
![Page 29: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/29.jpg)
Force and Moment Resultant
[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.
![Page 30: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/30.jpg)
Force and Moment Resultant
![Page 31: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/31.jpg)
Analysis Procedures for Laminated Composites
1. Find the value of the reduced stiffness matrix [Q] for each ply using its four elastic moduli, E1 , E2 , ν12 , and G12
2. Find the value of the transformed reduced stiffness matrix [] for each ply using the [Q] matrix calculated in step 1 and the angle of the ply
3. Knowing the thickness, tk , of each ply, find the coordinate of the top and bottom surface, hi , i = 1…, n, of each ply.
4. Use the [] matrices from step 2 and the location of each ply from step 3 to find the three stiffness matrices [A], [B], and [D]
![Page 32: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/32.jpg)
5. Substitute the stiffness matrix values found in step 4 and the applied forces and moments
6. Solve the six simultaneous equations to find the midplane strains and curvatures.
7. Now that the location of each ply is known, find the global strains in each ply
8. For finding the global stresses, use the stress–strain9. For finding the local strains, use the transformation10. For finding the local stresses, use the transformation
Analysis Procedures for Laminated Composites
![Page 33: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/33.jpg)
ExampleFind the three stiffness matrices [A], [B], and [D] for a three-ply [0/30/-45] graphite/epoxy laminate as shown in Figure. Assume that each lamina has a thickness of 5 mm.
![Page 34: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/34.jpg)
![Page 35: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/35.jpg)
SolutionStep 1: Find the reduced stiffness matrix [Q] for each ply
![Page 36: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/36.jpg)
Step 2: Find the transformed stiffness matrix [] using the reduced stiffness matrix [Q] and the angle of the ply
![Page 37: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/37.jpg)
![Page 38: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/38.jpg)
Step 3: Find the coordinate of the top and bottom surface of each ply using equation 4.20
The total thickness of the laminate is h = (0.005)(3) = 0.015 m.The midplane is 0.0075 m from the top and the bottom of the laminate.
h0 = –0.0075 mh1 = –0.0025 mh2 = 0.0025 mh3 = 0.0075 m
Ply n:
![Page 39: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/39.jpg)
Step 4: Find three stiffness matrices [A], [B], and [D]
![Page 40: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/40.jpg)
![Page 41: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/41.jpg)
![Page 42: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/42.jpg)
![Page 43: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/43.jpg)
![Page 44: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/44.jpg)
Example 2
A [0/30/–45] graphite/epoxy laminate is subjected to a load of Nx = Ny = 1000 N/m. Find,1. Midplane strains and curvatures2. Global and local stresses on top surface of
30° ply
![Page 45: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/45.jpg)
Solution
![Page 46: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/46.jpg)
![Page 47: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/47.jpg)
Find the global strains in each ply
![Page 48: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/48.jpg)
The strains and stresses at the top surface of the 30° ply are found as follows. First, the top surface of the 30° ply is located at z = h1 = –0.0025 m.
![Page 49: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/49.jpg)
Find the global stresses using the stress-strain equation
![Page 50: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/50.jpg)
Global stresses
![Page 51: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/51.jpg)
Find the local strains using the transformation equation
![Page 52: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/52.jpg)
Local strains
![Page 53: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/53.jpg)
Find the local stresses using the transformation equation
![Page 54: CHAPTER 4 MACROMECHANICAL ANALYSIS OF LAMINATES Dr. Ahmet Erkliğ.](https://reader035.fdocuments.in/reader035/viewer/2022081415/551a978d550346b52d8b612e/html5/thumbnails/54.jpg)
Local stresses