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Classical Laminated Plate Theory : Assumptions and Approximations
1. In-plane displacements u and v vary linearly through the thickness.
b g
2. Transverse displacement w is constant through the thickness.  z  0
3.  z  0
F
G
H
4.  yz x , y , 
IJ F
K G
H
IJ
K
h
h
  xz x , y , 
0
2
2
5.  yz  0 but  yz  0
 xz  0 but  xz  0
6.
f x , f y  constant in z  direction
For first order shear deformation theory we would permit constant  yz and  xz through the
thickness.
Displacements (Kinematics)
First order Taylor series for u and v about midplane (z = 0) (Not neutral surface)
x-Displacement
Note:
 xz 
b g b g uz bx, y, 0gz
u x, y, z  u x, y, 0 
u w
 0
z x

u
w

z
x
b g
 u x , y, z uo z
y-Displacement
w
x
where
b g bg
u x , y,0 u0 x , y
b g b g vz bx, y, 0gz
v x, y, z  v x, y, 0 
v w
Note:  yz   0
z y

b g
v
w

z
y
 v x , y, z vo z
w
y
where
b g bg
v x , y,0 v0 x , y
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STRAINS
x 
u uo
2 w

z 2
x
x
x
y 
v vo
2 w

z 2
y
y
y
 xy 

 w
u v u0
 2 w v o


z

z 2
y x
y
xy x
xy
uo v o
2 w

 2z
y
x
xy
Define:
L
 w O

M
P
L
 OMw P
M
P
M  w P
M
 P
M
P
M
P
y P
M
M
 P
 wP
N
QM
2
M
N xy P
Q
L
 OL
 OL
 O
M
P
M
P
M
P
M
 P
M
 P
 zM
 P
M
P
M
P
M
P
M
P
M
P
M



N QN Q N P
Q
2
2
x
2
y
2
2
xy
x
o
x
x
y
o
y
y
xy
o
xy
xy