Page 325 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

312 Introduction to the Finite Element Method Chap. 10

10.4 ELEMENT STIFFNESS AND ELEMENT MASS
IN GLOBAL COORDINATES

Axial element. For axial elements, the end moments are zero and the end
forces and displacements are colinear with the element length. Thus, for systems
involving only axial elements, the 6 x 6 transformation matrix reduces to the
following 4 x 4 matrix:
C S 1 0
— 5 c
T = (10.4-1)
0 1 c s
1 — s c
1
We note here that the stiffness and mass matrices for the axial element are of
order 2 x 2 and, therefore, must be rewritten as a 4 X 4 matrix as follows:

1 0 ! - 1 0 “
EA 1 - 1 EA 0 0 ! 0 0
-1 I - 1 ■q’ T 1 0 Ui
0 0 1 0 0 \^'2
1
(10.4-2)
/ ^ \
2 0 ! 1 O’
ml 2 1 W i ml 0 0 1 0 0
~6 1 2 < ¿¿2 1 0 1 2 0
K J 0 0 I I 0 0 2j

These 4 x 4 matrices can then be substituted into Eq. (10.3-8) in order to convert
them to the global coordinates:
1
— c -CS
CS 1
EA CS — CS -s^
k = T^kT = (10.4-3)
- — CS 1 c 2 CS
— CS i CS s^ _

~2c" 2cs 1 — CS
1
ml 2cs 2s^ 1 CS s^

m = T niT = (10.4-4)
~6 C^ CS 1 2 c 2 2cs

CS s^ 2cs 2s^ _
Example 10.4-1
Determine the stiffness matrix for the 3, 4, 5 oriented pinned truss of Fig. 10.4-1.
Solution: The strueture is eomposed of three pinned members a , h , c with joints 1, 2, 3.
Each joint has 2 DOF in the global system, and the six forces and displacements are