Basic Equations of the Classical Plate Theory
The Displacement Fields:
w
u ( x, y , z )  u 0 ( x, y )  z
x
v ( x, y , z )  v 0 ( x, y )  z
Strain-Displacement Relations:
u
u v
2w
2w
 xx  0  z 2
 xy  0  0  2 z
x
x
y x
xy
2
v
 w
 xz  0
 yy  0  z 2
y
y
 yz  0
 zz  0
w
y
w( x, y, z )  w( x, y )
Stress Resultants:
h/2
Mx 
 z
h/2
xx
My 
dz
h / 2
Nx 
Qy 

h / 2
h / 2
h/2
h/2
xx
M yx  M xy 
dz
 z
xy
dz
h / 2
h/2
  xz dz

yy
h / 2
h/2
Qx 
 z
h/2
Ny 
dz
h / 2

yz
dz
h/2
yy
N yx  N xy 
dz
Moment Curvature Relationships:
2w
2w
M x   D  2  2 
y 
 x

xy
dz
h / 2
h / 2
Force Displacement (Derivative) Relationships:
 u
v 
N x  K  0  0 
y 
 x
2w
2w
M y   D  2  2 
x 
 y
2
 w
M xy   D(1  v)
xy
 v
u 
N y  K  0  0 
x 
 y
1    u0  v0 
N xy  K
2  y
x 
Eh3
D
12(1  2 )
Equilibrium Equations: (transverse problem)
K
Eh
1  2
(in-plane problem)
M x M xy
N x N xy 
h
h
h 
h
h 

 Qx   xz ( )   xz ( )  0

  xz ( )   xz ( )  0
x
y
2
2
2 
x
y 
2
2 
M xy M y
N xy N y 
h
h
h 
h
h 

 Qy   yz ( )   yz ( )  0

  yz ( )   yz ( )  0
x
y
2
2
2 
x
y 
2
2 
Qx Qy

 p1 ( x, y)  p2 ( x, y)  0
x
y
Transverse Shear Forces in terms of displacement (no surface shear):
 2
 2
Qx   D
 w( x, y )
Qy   D
 w( x, y )
x
y
Governing Equations:
(in-plane loading)
4
D w( x, y)  p( x, y)
 4 ( x, y)  0


p( x, y)  p1 ( x, y)  p2 ( x, y)


Nx 
 2
 2
 2
N

,
,
N


y
xy
x 2
xy
y 2
Stress Distributions within a plate:
Flexural stresses (transverse problem)
Mxz
12 Dz   2 w
2w 

 3
  3  2  2 
h / 12
h  x
y 
Myz
12 Dz   2 w
2w 
 3
  3  2  2 
h / 12
h  y
x 
Membrane Stresses (in-plane problem)
N x K  u0
v 
 
 0 
h
h  x
y 
N
K  u v 
 yy  y   0  0 
 yy
h
h  x
y 
N
M z
K 1    u0 v0 
12 D1  z  2 w
 xy  xy 

 xy  3 xy  
3
h
2h  y
x 
xy
h / 12
h
Transverse shear stresses in flexural problem(from equilibrium-no surface shear):
2
2
3Qx   z  
3   z    2
 xz 
1 
   1  
 D  w( x, y )
2h   h / 2  
2h   h / 2   x
2
3Q y   z  2 
3   z    2


 yz 
1 
 w( x, y )
   1  
 D
2h   h / 2  
2h   h / 2   y
 xx
 xx 



