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

2D Axisymmetric, Plane Strain and Plane Stress Elements
Shape Functions
n = 4 nodes
N1 a = 1 4 (1 + r)(1 + s), N 2 a = 1 4 (1 − r)(1 + s), N 3 a = 1 (1 − r)(1 − s),
4
N4 a = 1 (1 + r)(1 − s)
4
n = 8 nodes
N b 5 = 1 2 (1 − r2 )(1 + s), N b 6 = 1 2 (1 − s2 )(1 − r), N b 7 = 1 2 (1 − r2 )(1 − s),
N b 8 = 1 2 (1 − s2 )(1 + r)
N b 1 = N a 1 − 1 2 N b 5 − 1 2 N b 8 , N b 2 = N a 2 − 1 2 N b 6 − 1 2 N b 7 , N b 3 = N a 3 − 1 2 N b 6 − 1 2 N b 7 ,
N4 b = N 4 a − 1 2 N 7 b − 1 2 N 8
b
n = 9 nodes
N9 c = (1 − r2 )(1 − s 2 ),
Ni c = N i b − 1 4 N 9 c, i=1,2,3,4
Nj c = N j b − 1 2 N 9 c, i=5,6,7,8
Institute of Structural Engineering Method of Finite Elements II 3