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62 Chapter 4 Plan Stress Analysis

   
x  xy  0
x  y 
   
and xy  y  0
x  y 
where  and  is the normal stress in the x and y direction,
x
y
respectively, while  is the shearing stress. The basic unknowns
xy
of the two equations above are the u and v displacement in the x-
and y-direction, respectively.

4.1.2 Related Equations
The normal stresses  and  together with the shear-
y
x
ing stress  can be written in forms of the strain components
xy
according to the Hook’s law as,
  x  1  0     x 
     y    E  1   0    y       C



(3 1)   1    2 0 0 ( 1 )  2     (3 3) (3 1)
  xy      xy 
where  and  is the normal strain in the x- and y-direction,
y
x
respectively, while  is the shearing strain. The elasticity matrix
xy
  depends on the material Young’s modulus E and the
C
Poisson’s ratio  . For small deformation theory, these strain
components varies with the displacement u and v in the x- and y-
direction as,
u  v   u v 
  x  ;  y  y  ;  xy  y   x 
x
The stress-strain relations and strain-displacement
relations above lead to the two partial differential equations in the
forms,

 E   u   v    E   u   v  
             0
x   1 2   y     y    2  (   1    ) x    y  x     