Page 320 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.2 Stiffness and Mass for the Beam Element 307

The preceding displacements can be considered to be the superposition of
the four shapes, labeled (Pi(x), (P2(x), cp^ix), and (pj^x), shown in Fig. 10.2-2. The
forces and moments required at the two ends were found in Chapter 6 and are
shown in Fig. 10.2-3 with the factor EI/P omitted. The diagram immediately leads
to the force-stiffness equation:
12 6/ 1 -1 2 61 '
EL 61 4/2 1 -61 2/2 (10.2-1)
-12 - 6 / i 12 -61 Vi
6/ 2/2 1 -61 4 l \ 1^2 j
2 )
Equation (10.2-1) for the stiffness was obtained from the given forces and
moments shown in Fig. 10.2-3. The stiffness matrix as well as the mass matrix can
also be determined from the potential and kinetic energy, provided the shape
functions (p^ix) of the beam are known.
For the development of the general equation of the beam, which is a cubic
polynomial, the deflection is expressed in the form
v { x ) = Pi + P 2 i + (10.2-2)
where
X
^ = j and p¿ = constants
Differentiating yields the slope equation

i e { x ) = P 2 + 2P2^ + (10.2-3)
If we apply the boundary conditions ^ = 0 and ^ = 1, the boundary equations can

Figure 10.2-3.