Page 205 - Finite Element Modeling and Simulations with ANSYS Workbench

190 Finite Element Modeling and Simulation with ANSYS Workbench

The displacement, strains, and stresses can be written in terms of the main variable—
deflection w = w(x, y), as follows:

Displacement:
w = w xy), ( deflection)
(,
∂ w
u =− z
∂ x (6.7)
∂ w
v =− z
∂ y

Strains:
2
∂ w
ε= − z
x
∂x 2
2
∂ w
ε= − z (6.8)
y
∂y 2
2
∂ w
γ xy = − z 2
∂∂xy
Note that there is no stretch of the mid-surface caused by the deflection of the plate.
Stresses (plane stress state):


 σ  1 ν 0  ε x 
x
  E   
 σ  = 2  ν 1 0   ε y  (6.9)
y
  1 −ν 0 0 1 ( −ν 2)/  

 γ xy
 τ xy    
or,
 ∂ w 
2
 2 
 σ  1 ν 0   ∂x 


x

2
  E   ∂ w 
 σ  =−z 2  ν 1 0   2  (6.10)
y
y
  1 − ν 0 0 1 ( −ν  )  ∂y 
 τ xy     ∂ 2 w 
 
xy 
  ∂∂ 
The following equation governs the equilibrium of the plate in the z-direction:
D∇ 4 w = qx y) (6.11)
(,

where
 ∂ 4 ∂ 4 ∂ 
4
4
∇≡  4 + 2 2 2 + 4 
 ∂x ∂x ∂y ∂y 