Page 329 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

316 Introduction to the Finite Element Method Chap. 10

Beam element. The stiffness and mass matrices for the beam element are
of order 4 x 4 , whereas the transformation matrix is 6 X 6. Thus, to transform
these matrices for the global coordinates, we need to modify them by adding the
axial components rearranged as follows:
1 0 0 1 - 1 0 0 ■
0 0 0 1 0 0 0
1 i’l
EA 0 0 0 1 0 0 0
1
I
- 1 0 0! 1 0 0 111
0 0 0! 1 0 0 0 f'2
0 0 0! 0 0 0 ^2

"2 0 0 1 1 0 o '
0 0 0 1 0 0 0
ml 0 0 0 1 ^ 0 0
T 1 0 0 ! 2 0 0
0 0 0 1 0 0 0
0 0 0 1 0 0 0

The element matrices to be used in the transformation then become
R 0 0 ! - R 0 0
0 12 61 \ 0 - 12 61
El 0 6/ 4/2 ! 0 -61 2/2
(10.4-5)
- R 0 0 i R 0 0
0 -12 -61 ! 0 12 -61
1
0 6/ 2/2 ! 0 -61 4/2
EA Al^
where R =
I El I

N 0 0 1 iN 0 0
0 156 22/ 1 0 54 -13/
ml 0 22/ 4/2 0 13/ -3/2
m = (10.4-6)
\N 0 0 1 N 0 0
0 54 13/ 0 156 -22/
0 -13/ -3/2 0 -22/ 4/2

where N