Page 130 - Finite Element Analysis with ANSYS Workbench

7.1 Basic Equations 121

2D Plate
 2 w Eh 3  4 w  4 w   4 w 
 h    2  

t  2 12(1  2 )  x 4  x y   2 2 y  4 
The deflection w  w (, , ) varies with the coordinates x, y and
y
x
t
time ,t  is the material density, h is the plate thickness, E is the
material Young’s modulus and  is the Poisson’s ratio.
3D Solid
 2 u      
  x  xy  xz
t  2 x  y  z 
 2 v   xy   y   yz
   
t  2 x  y  z 
 2 w     yz  


  xz z
t  2 x  y  z 
The displacement components u  u (, , , ), v  v (, , , ), w 
y
x
t
z
t
x
y
z
(, , , ) vary with the coordinates ,x y z and time t . The
wx y z t ,
quantities    are the normal stress components while  xy ,
,
,
z
x
y
 xz ,  are the shearing stress components.
yz

7.1.2 Related Equations
After the displacement unknowns are solved from the
above differential equations, the stresses can be determined by
using the related equations as follows.
1D Truss
The stress  is determined from the computed dis-
placement u as,
 u
  E
x 
1D Beam
The stress  of the beam is determined at any z
coordinate from the computed deflection w as,