Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 65

2 1 2 5 1

η = +1 η = +1

η η

ξ 6 ξ 8

o o

η = −1 η = −1

3
(a)

4 3 7 4

4-node rectangle element (b) 8-node rectangle element

2 6 5 1

η = +1

η

7 12

ξ

o

8 11

η = −1

3 9 10 4

(c) 12-node rectangle element

Figure 3.2 (a-c) Three different types of rectangle elements.

The displacement functions of an eight-node element are

1
4 1

( ) ( ) ( )

N = + ξ 1 + η ξ η + − 1

1

1
4 1

( ) ( ) − + − ( )

N = − ξ 1 + η ξ η 1

2

1
4 1

( ) ( ) − − − ( )

N = − ξ 1 − η ξ η 1

3

(3.20)

1
4 1

( ) ( ) ( )

N = + ξ 1 − η ξ η − − 1

4

1
2

1
2

( ) ( )

( ) ( )

2 2

N = 1 − ξ 1 + η , N = 1 − η 1 − ξ

5 6

1
2

1
2

( ) ( ) ( ) ( )

2 2

N = 1 − ξ 1 − η , N = 1 − η 1 + ξ

7 8