FINITE ELEMENT METHOD 333Since the interpolation functions for the coordinates (x,y) and displacements (u,v) are the same, the
element is sometimes called the isoparametric element. Because of the term ξη (or xy) in interpolation
(Eq. 11.11a or 11.11b), the element is also known as the bilinear element. The strain vector can be
computed in a similar manner as in Eq. (11.10h), that is
= =00 =00 = =ε
ε
γx
y
xyxyyxuxyyx⎛⎝⎜
⎜⎜⎞⎠⎟
⎟⎟∂ ∂ ∂ ∂ ∂ ∂∂
∂⎛⎝⎜
⎜
⎜
⎜
⎜⎞⎠⎟
⎟
⎟
⎟
⎟⎛
⎝
⎜⎞
⎠
⎟∂ ∂ ∂ ∂ ∂ ∂∂
∂⎛⎝⎜
⎜
⎜
⎜
⎜⎞⎠⎟
⎟
⎟
⎟
⎟vNu AGu Bu (11.11i)whereB = AGis the strain-displacement matrix with matrices AandG given as
A
JJJ
JJ
JJ J J=^1
det ( )–00
00–
––22 12
21 11
21 11 22 12⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥(11.11j)and
G =^1
4- (1 – ) 0 (1 – ) 0 (1 + ) 0 – (1 + ) 0
- (1 – ) 0 – (1 + ) 0 (1 + ) 0 (1 – ) 0
0 – (1 – ) 0 (1 – ) 0 (1 + ) 0 – (1 + )0 – (1 – ) 0 – (1 + ) 0 (1 + ) 0 (1 – )ηηηηξξξξηηη ηξξξξ⎡⎣⎢
⎢
⎢
⎢⎢
⎢
⎢
⎢⎤⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥(11.11k)whereJ is the Jacobian matrix given using Eqs. (11.11f) and (11.11g) as
J =(1 – ) ( – ) + (1 + ) ( – ) (1 – ) ( – ) + (1 + ) ( – )
(1 – ) ( – ) + (1 + ) ( – ) (1 – ) ( –1
4 211
4 341
4 211
4 34
1
4 411
4 321
4 4ηηηη
ξξ ξxx xx yy yy
xx xx y ) + (1 + ) (yyy 1 14 ξ 32 – )⎡⎣⎢
⎢⎤⎦⎥
⎥(11.11l)For a plane stress, the elasticity matrix Dis defined in Eq. (11.10l) using which we can compute
the stress vector σσσσσ. The element stiffness matrix ke is given similar to that of the triangular element
as
ke B DB J B DB
V= t TTdxdy = t det ( ) d d
–11–11
∫∫∫
ξη (11.11m)wheretis the out-of-plane thickness of the element. Note that the matrices B and Jare functions of
the local coordinates ξ and η. The expression within the integral is usually computed using the four-
point Gauss integration approach, that is
kJBDB JBDBe T
i iiiiiT
= tddtwdet ( ) = det ( ( , )) ( , ) ( , )ii
–11–11
=14
∫∫