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Two-Dimensional Elasticity 131

4
Note that ∑ =i N i = 1 at any point inside the element, as expected.
1
The displacement field is given by
4
4
u = ∑ N u , v = ∑ Nv (4.36)
ii
ii
i=1 i=1
which are bilinear functions over the element. The stress and strain fields are constant on
this type of elements.

4.4.5 Quadratic Quadrilateral Element (Q8)
This is the most widely used element for 2-D problems due to its high accuracy in analysis
and flexibility in modeling.
There are eight nodes for this element (Figure 4.14), four corners nodes and four midside
nodes. In the natural coordinate system (ξ, η), the eight shape functions are

1
N 1 = ( 1 − ξη − 1 ξ +η + 1)
)
)(
(
4
1
N 2 = ( 1 + ξη − )( 1)
1 η −ξ +
)(
4
1
N 3 = ( 1 + ξ)( 1 + η))(ξ+ η− 1 )
4
1
N 4 = (ξ − 1 )(η + 1 )(ξ −η + 1 )
4 (4.37)
1
)( − ξ
N 5 = ( − η 1 2 )
1
2
1
2
1
N 6 = ( + ξ )(1 −η )
1
2
1
2
N 7 = ( 1 + η)( 1 − ξ )
2
1
2
N 8 = ( 1 − ξ)( 1 − η )
2

3
= 1 7
4
6
8
5 2
1
y = –1 = –1 = 1
x
FIGURE 4.14
Quadratic quadrilateral element (Q8).