Page 330 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.5 Vibrations Involving Beam Elements 317

These 6 x 6 element matrices are transformed to global coordinates (with
bars over the letters) by the equations ~k = T^kT and m = T^mT:

(Rc^ + U s ^ ) {R - \2)cs - 6 / 5 1 { - R c - - 12 5 2 ) ( - R + \2)cs —()ls u

(R - \2)cs (Rs~ + 1 2 c 2) 6 /r i ~ R + 12 ) c’5 i -R s^ - 1 2 C-2 ) 6lc r
— 6 Is 6lc 4 /2 6/5 — 6 /c 2 /2 Ô
r
( - Rc^ - Ì2s^) ( - R + \2)cs 6/5 1 ( Rc “ -r 125*') (R - 1 2 )c5 6 Is

( — R -f- 12 ) c\s’ ( - R s ^ - \2 c") — 6/c’ 1 (R - 1 2 )c5 (Rs^ + 12 c-2) —61c
— 6 Is 6lc 2 /2 1 6/5 — 6 /c 4 /2 _
( 1 0 . 4 - 7 )
(Nc^ + 15 6 5 2 ) { N — 1 5 6 ) c5 - 22/5 (^A^c 2 + 54 52) { \ N - 5 4 ) c5 13/5

(N — 1 5 6 ) c5 (Ns^ + 1 5 6 c 2) 22 Ic ( { N - 54)cs C N 52 + 5 4 c 2 ) - 13/c
ml - 2 2 / 5 22 Ic 4 /2 1 - 13/5 13/c - 3 / 2
4 2 0 ({N c^ + 54s^) { \ N - 5 4 ) c5 - 13/5 1 (Nc^ + 1 5 6 5 2 ) (N - 1 5 6 ) c5 22 Is
( \ N - 5 4 ) c5 ( { N s ^ + 5 4 c 2 ) 13/c 1 (iV - 1 5 6 ) c5 (Ns^ + 1 5 6 c 2) -221c
13/5 - 1 3 / c - 3 / 2 1 22/5 - 2 2 k 4 /2
(10.4-8)

10.5 VIBRATIONS INVOLVING BEAM ELEMENTS

To illustrate the finite element method for beams, we consider some problems
solved in Chapters 6 and 7. The object here is, first, to show how to assemble the
system equation using two elements and, second, to reduce the degrees of freedom
of the equation by elimination of rotational coordinates.
Example 10.5-1
The beam in Fig. 10.5-1 is considered as two equal elements of length 1/2, whose
stiffness and mass matrices are given by Eqs. (10.2-1) and (10.2-10). With 1/2
substituted for /, the element matrices are as follows:
Element a:

12 3/ 1 -12 31
Stiffness 3 3/ 1 -31 0.5/2 Displacement vector < r '
\ i'2
] -12 -3 / 1 12 - 3/
3/ 0.5/2 1 -31 /2 / 2J
156 11/ ' 54 -6.5/
j' m/ 11/ /2 1 6.5/ -0.751^
Mass
840 ; 54 6.5/ ; 156 -11/
-6.5/ -0.75/2 1 -11/ / 2