The Finite Element Method for the Analysis of Non-Linear and ...

Example
In order to define the required stresses and strains we need to determine the deformation
gradient t 0S, which in 2D is written as:
⎡
∂ t x 1 ∂ t ⎤
x 1
t
0 X = ⎢ ∂ 0 x 1 ∂ 0 x 2
⎣ ∂ t x 2 ∂ t ⎥
x 2 ⎦ (2)
∂ 0 x 1 ∂ 0 x 2
The coordinates of a random point within the element are given as the weighted sum of
the nodal coordinates, where the weights are the shape functions. However, the shape
functions are written with respect to the r, s system (isoparametric representation) as:
N 1 = 1 4 (1 + r)(1 + s) N 2 = 1 (1 − r)(1 + s)
4
N 3 = 1 4 (1 − r)(1 − s) N 4 = 1 (3)
4 (1 + r)(1 − s)
From the given figure however we see that the 0 x 1 , 0 x 2 system is related to the r, s
system through:
0 x 1 = r + 1
0 x 2 = s + 1 ⇒
N 1 = 1 4 0 x 1 0 x 2 N 2 = 1 4 (2 − 0 x 1 )( 0 x 2 )
(4)
N 3 = 1 4 (2 − 0 x 1 )(2 − 0 x 2 ) N 4 = 1 4 0 x 1 (2 − 0 x 2 )
Institute of Structural Engineering Method of Finite Elements II 12