Transcript EML 4230 Introduction to Composite Materials

Chapter 4 Macromechanical Analysis of a Laminate
Classical Lamination Theory
Dr. Autar Kaw
Department of Mechanical Engineering
University of South Florida, Tampa, FL 33620
Courtesy of the Textbook
Mechanics of Composite Materials by Kaw
Fiber Direction

x
y
z
• elastic moduli
• the stacking position
• thickness
• angles of orientation
• coefficients of thermal expansion
• coefficients of moisture expansion
 xx =
P
z
(a)
M
M
x
 xx =
Mz
I
 xx =
z

 1 
 z 
=
P
+

 M
 xx 
 AE 
 EI 
z
(b)
M
M
x
P
x
P
z
(c)
(4.1)
P
 xx =
AE
x
P
P
A
1
=  0 + z  

=  0 + z
x
x
z
z
Mxy
Myx
Nx
y
Nyx
Nxy
My
y
Ny
(a)
(b)
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)
Mx
x
x
z
z
Mxy
Myx
Nx
y
Nyx
Nxy
My
y
Ny
(a)
(b)
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)
Mx
 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. ( γxz = γyz = 0 ) .
The laminate is thin and is loaded only in its plane (plane stress)
(σz = τxz = τyz = 0 ) .
Displacements are continuous and small throughout the laminate
(| u |, | v |, | w |  | h |) , where h is the laminate thickness.
Each lamina is elastic.
No slip occurs between the lamina interfaces.
u0
h/2
Mid-Plane
h/2
x
z

A
z
A 
z
Cross-Section
Before Loading
Cross-Section
after Loading
wo
2




w0 
 u0
 


2 
x 
x 


 ε x 



  
2


 v0 
 
 w0 
 ε y = 
 + z 
2 

y
y 
  


γ xy  



  u0


2
 v0 
+


 2  w0 
x 
 xy 
 y
 0
 κx
ε
x
 


 0
 
  ε y  + z  κ y .
 
 
κ xy 
γ0xy 
 
Mid-Plane
z
Laminate
Strain Variation
Stress Variation