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Two-Dimensional Elasticity 129

4.4.3 Quadratic Triangular Element (LST or T6)
This type of element (Figure 4.11) is also called quadratic triangular element. There are six
nodes on this element: three corner nodes and three midside nodes. Each node has two
DOFs as before. The displacements (u, v) are assumed to be quadratic functions of (x, y),

2
u = b + bx + by + bx + bxy + b y 2 (4.31)
1
6
5
4
3
2
v = b + bx + by + bx + bxy + b y 2
2
7 8 9 10 11 12
where b (i = 1, 2, …, 12) are constants. From these, the strains are found to be
i
+ b x + b y
2
ε= b 2 4 5
x
+ b x + b y (4.32)
2
ε= b 9 11 12
y
(
2
(
(
γ xy = b 3 + b 8 ) + b 5 + b 10 ) x + 2 b 6 + b b 11 ) y
which are linear functions. Thus, we have the “linear strain triangle” (LST), which pro-
vides better results than the CST.
In the natural coordinate system, the six shape functions for the LST element are
N 1 =ξ( 2ξ− 1)
N 2 =η( 2η− 1)
N 3 =ζ( 2ζ− 1)
(4.33)
N 4 =ξη
4
N 5 = 4ηζ
N 6 =ζξ
4

in which ζ = 1 − ξ − η. Each of these six shape functions represents a quadratic form on the
element as shown in Figure 4.12.

v 3

3 u 3
v
v 6 5
5
u 6 6 u 5
v 2
v 1
y u
1 4 u 4 2 2
u 1 v 4
x

FIGURE 4.11
Quadratic triangular element (T6).