1. Classical Laminated Plate Theory

2. Combining Constituents
Micro Mechanics
Experimental Data
CONSTITUENTS
STRUCTURE
COMPOSITE
STRUCTURAL
ELEMENT
ELEMENTARY
STRUCTURE Ey
Finite Element Analysis Ex G 

3. Material Forms

4. Plane-Stress Assumption
Fiber-reinforced materials are utilized in beams, plates, cylinders and other structures
Typically one characteristic geometric dimension is an order of magnitude less than the other two
Three of the six components of stress are generally much smaller than the other three

5. Plane Stress Inaccuracies
Errors in analysis near edges
The stresses s3, t23, t13 lead to delaminations
Bonded joints can not be modeled
Adhesive or cocured interface can not be evaluated
The stress components equated to zero are forgotten and no attempt is made to estimate their magnitude
erroneously assumed that e3 is zero

6. Stress Transformation

7. Compliance Transformation Equations

8. Reduced Stiffness Transformation

9. Classical Lamination Theory
The influence of fiber direction, stacking arrangements,
material properties, and more on structural response

10. Laminate Coordinate System
Laminate thickness H
Layer thickness h
not all layers same h
kth layer - hk
z-axis downward from geometric midplane
can be between layers
can be within a layer
Fiber angles identified relative to x axis

11. Laminate Nomenclature
Layer 1 is the most -z
Layer N is the most +z
To catergorize a laminate as symmetric a mirror about the geometric midplane
material properties
fiber orientation
thickness of layer

12. The Kirchhoff Hypothesis
Mid 1800’s, simplified analysis
Beams, plates, shells
metal, wood, concrete, and other materials

13. Initially Flat Laminated Plate Acted Upon by Various Loads
applied moments, M
distributed loads, q
inplane loads, N
point loads, P
Multiple layers of fiber reinforced material
Fibers parallel to the plane of the plate
Layers are perfectly bonded

14. Deformation of Lines Normal to Geometric Midplane
Before deformation are straight
Despite the deformations caused by the applied loads, line AA’ remains straight and normal to the deformed geometric midplane and does not change length

15. Consequences of Kirchhoff Hypothesis

16. Implications of the Kichhoff Hypothesis in X-Z Plane
No through-thickness strain
Small deformations
Two components of translation
uo in x direction, horizontal translation
wo in z direction, vertical translation
One component of rotation about y-axis

17. Resulting Displacement Field in XZ plane

18. Resulting Displacement Field in YZ plane

19. Strain-Displacement Relations from Theory of Elasticity

20. Laminate Strains

21. Laminate Strains Composed of Two Parts
Extensional Stain of the Reference Surface
Curvature of the Reference Surface
inverse of the radius of curvature
involves more than just second derivative
For small strains second derivative and curvature identical

22. Strain Notation

23. Laminate Strains using Revised Notation
The gyz, and gxz are zero because the Kirchhoff hypothesis
assumes that lines perpendicular to the reference surface
before deformation remain perpendicular after the
deformation; right angles in the thickness direction do not
change when the laminate deforms

24. Laminate Stresses

25. [0/90]s Laminate, Axial 1000me Laminate Stress & Strain

26. [0/90]s Laminate, Axial 1000me Material Stress & Strain

27. Aluminum, Axial 1000me

28. [0/90]s Laminate, kxo 3.33 m-1 Laminate Stress & Strain

29. [0/90]s Laminate, kxo 3.33 m-1 Material Stress & Strain

30. Aluminum, kxo 3.33 m-1

31. Definitions of Stress Resultants
Stress in each ply varies through the thickness
It is convenient to define stresses in terms of equivalent forces acting at the middle surface
Stresses at the edge can be broken into increments and summed
The resulting integral is defined as the stress resultant, Ni [force per length]

32. Stress Resultant in X direction

33. Stress and Moment Resultants
bend
bend
twist

34. Putting the Resultants in Matrix Form and Summing

35. Relating Stress to Strain

36. Performing the Integration

37. Defining Laminate Stiffness Terms

38. Constitutive Equations in Matrix Form

39. Symmetric Laminates
For every layer to one side of the laminate reference surface with a specific thickness, material properties, and fiber orientation, there is another layer an identical distance on the opposite side
All components of [B] are zero
6x6 set of equations decouples into two 3x3 sets of equations

40. Balanced Laminates
For every layer with a specified thickness, material properties, and fiber orientation, there is another layer with the identical thickness, material properties, but opposite fiber orientation somewhere in the laminate
If a laminate is balanced, A16 and A26 are always zero
Q16 & Q26 from opposite orientation have opposite signs

41. Effective Engineering Properties of a Laminate