Page 338 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.6 Spring Constraints on Structure 325

P
©

(Q) (b)
Figure 10.6-1.

Solution: Wc first establish the stiffness matrix of the beam without the springs [Fig.
10.6-l(b)], but with loads P and M acting at station 2. The stiffness matrix for each
section 1-2 and 2-3 can be assembled from the beam element matrix, Eq. (10.2-1).
Noting that = 0^ = 0, wc need only evaluate the portion of the matrix
associated with the coordinates ¿s and 6^, which becomes

El
4 T - + ^

With the springs acting at station 2, the force vector is replaced by

Pi - kT'i
M, - KO,
Shifting the spring forces to the right side of the equation, we obtain

12 TT +
El 4 1; r,
M-,
^1/7 + 7t ) +

Because the force F2 in the global system is positive in the upward direction, and
is positive counterclockwise, the previous equation can be rearranged to

;> /■ ki^
1 2 ^ El -61
-P
M / /“
l-\
— 6/ — —— © El
Wr ¡¡1
which defines the stiffness matrix for the beam with the spring constraints.