1. Laminated plates

2. definitions
Unidirectional: orientation direction is identical for all laminates
Angle-ply:
Fiber orientation alternates between +theta to –theta from the stress normal
…/q/-q/q/-q/…
Cross ply:
Fibers in alternating plies are at right angles to each other
…/0/90/0/90/…
Symmetric: fiber orientation in the stack is symmetric about the center line of the laminate

3. Symmetric laminate examples
7 ply stack:
This stack is symmetrical about the centerline of the 4th ply
The stack order is a ‘mirror’ reflection about the centerline
We will need the average height of the ply above the centerline.
Ply #
orientation
Distance above ctrln 1 +3 2
+45 +2 3
-45 +1 4 90 5 -1 6 -2 7 -3

4. Symmetric laminate examples
8 ply stack:
This stack is symmetrical about the interface between ply 4 and ply 5
The stack order is a ‘mirror’ reflection about the centerline
We will need the average height of the ply above the centerline.
Ply #
orientation
Distance above ctrln 1
+ 3 ½ h 2
+45
+2 ½ h 3
-45
+ 1 ½ h 4 90
+ ½ h 5
- ½ h 6
- 1 ½ h 7
- 2 ½ h 8
-3 ½ h

5. Antisymmetric q(z)=-q(-z) q/-q/q/-q is antisymmetric
Ply orientation is antisymmetric about the centerline of the laminate
For each ply with fiber orientation angle, q, above the midplane, this is a ply of fiber orientation angle, –q, with identical material and thickness and equal distance below the midplane
q(z)=-q(-z)
q/-q/q/-q is antisymmetric
q/-q/-q/q is symmetric

6. Unsymmetric laminate Unsymmetric: q1/q2/q3; random placement of angles
Quasi-isotropic laminate: equal angles between adjacent lamina, increment is p/n
+60/0/-60
+45/0/-45/-90
0/+60/-60
0/+45/-45/90

7. Quasi-isotropic laminates
Quasi-isotropic laminate: equal angles between adjacent lamina, increment is p/n
+60/0/-60
+45/0/-45/-90
0/+60/-60
0/+45/-45/90
A common quasi-isotropic symmetrical stacking sequence is: [0/±45/90]s

8. Code examples
[0/45/90]s: 1st angle is for the outermost ply, last angle is for the innermost ply; s subscript means symmetrical
The bar over 90 indicates that the plan of symmetry passes midway through the thickness of the 90 degree laminate
±45 implies adjacent +45 and -45 degrees laminates
04: four adjacent zero degree plies
(±45)2: two grouped, adjacent ±45 degree plies

9. In-class examples Sketch the following: [0/45/90]s [0/-45/90]s

10. Young B35 classroom (30 student seats and 1 teacher station)
                Thursday 4/3 3:30pm-4:45pm
                Thursday 4/10 3:30pm-4:45pm
               
Peggy M. Akridge
Manager, Student Computing Services
UK Analytics and Technologies
University of Kentucky
Lamination theory

11. Step-by-step procedure
Calculate stiffness matrices for the laminate
Calculate the midplane strains and curatures for the laminate due to specific applied forces
Calculate in-plane stresses, exx, eyy, gxy, for each lamina
Calculate in-plane strains, sxx, syy, txy, for each lamina

12. Assumptions Laminate is thin, wide: w>>t
Perfect interlaminar bonds
Strain distribution in the thickness direction is linear
All laminas are macroscopically homogeneous and behave linearly elastically

13. Laminate strains Since strain distribution in z direction is linear,
Geometric midplane = xy axes, z axis defines thickness direction
Total laminate thickness = h
Thickness of various lamina are ti
Total lamina = N

14. Laminate strains Since strain distribution in z direction is linear,
e0ii = midplane normal strains
g0xy = midplane shear strain
Kii bending curvatures of laminate
Kxy = twisting curvature of laminate