Page 92 - Finite Element Analysis with ANSYS Workbench

5.1 Basic Equation 83

 2 w  2 w   2  w  2 w  
M   x D      ; M   y D  2   
 x  2 y  2   x  y  2 
 2 w Eh 3
M  xy D  (1  )  x y ; D  12(1  2 )


In these bending moment component equations, D
represents the plate flexural rigidity that depends on the Young’s
,
modulus E the Poisson’s ratio , and the plate thickness h.
Substituting these bending moment component equations into the
governing differential equation above yields the final form of the
plate differential equation. The final form is of fourth-order
differential equation containing only one unknown of the deflection
w .

5.1.2 Related Equations
From the relations of the in-plane displacements u and
v with the deflection w, the strain components become,
  u  x   z  2 x  w
x
2
  v  y   z  2 y  w
y
2
w
2
  xy u  y   v  x   2z   x y

Then, the stress components are,

  E (  )   E   2 w    2 w   z 
x


1  2 x y 1  2  x  2  y  2  
E
E
  y 1  2 ( x   y )  1  2    2  x  w  2   2 y  w   2   z 


E 
E
  2(1  )   1      x y  2 w     z 
xy
xy

