Page 78 - Dynamic Loading and Design of Structures
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Table 2.4 Maximum values of the DLF for load F(t).
Eigenvector tr/T (DLF)max
1 0.26 1.89
2 0.77 1.28
3 1.12 1.11
Figure 2.23 The flexural beam as a continuous dynamic system.
2.4.1 Equations of motion for continuous beams
As example, we will examine the flexural beam, which is one of the basic unidimen-sional
structural elements. Referring to Figure 2.23, the equation of dynamic equilibrium of a
continuous beam element is
(2.72)
where EI is the flexural rigidity, m is the mass per unit length, p is the distributed load and y(t,
x) is the transverse displacement. For free vibrations, we have that p(t, x)=0 and
(2.73)
where Φ n(x) is the nth eigenvector. The original equation of motion can be split into two,
which respectively govern the temporal and spatial variation of the displacement y(t, x) as
(2.74)
The solution for the time function f (t) and the eigenvector Φ(x) are given below as
n
n
(2.75)
