Page 193 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

180 Properties of Vibrating Systems Chap. 6

Figure 6.4-1. Beam with arbitrary
end displacements.
terms of the superposition of four displaeements taken separately, as shown in Fig.
6.4-2. Shown also are the end forces and moments required to maintain the
equilibrium of the separate displacements, which can be simply determined by the
area-moment method. They relate to the following stiffness matrix:
FA fcii ^12 k 13 kx4~ i F
M, ^21 ^22 F , A 24
^31 k^2 <
F2 ^33 ^34 II2
Mj ^ J _^41 A: 42 ^43 ^44 _i^2,

where each column represents the force and moment required for each of the
displacements taken separately. The positive sense of these coordinates is arbi­
trary; however, the configuration shown in Fig. 6.4-1 conforms to that generally
used in the finite element method.
^ ]Z E I
cl 6 E/
„2 ^1
V 12E7
E=— 5- 1/1
i ^

M . 5 S e , ( _ U ^
0
^2 1
6 EJ
12EJ /W= ,2 ^2

12EJ
.3 ''2

M -— e^
1 0^ ^
_ Z E I
M
6£

Figure 6.4-2. Stiffness of beam element.