Civil Engineering Reference
In-Depth Information
B.2 Numerical solutions
In most cases a numerical integration is the most effective solution, in which case Eq.
B.1 is to be replaced by:
1
NN
()
() ( )
()
¦¦
I
fx fx
ˆ
ˆ
exp
x
ˆ
β
=
⋅
⋅
− ⋅Δ
β
(B.3)
p
k
N
2
pk
==
11
and
N
is the number of integration points. It should be noted that
in general a finely meshed integration scheme is required, i.e. a large
N
. The reason for
this is of course that the exponential function is rapidly dropping at increasing values of
its argument. The solution to a good number of cases has been plotted in Figs. B.1 - B.3:
x xx
ˆ
ˆ
ˆ
where
Δ= −
p
k
Fig. B.1
Sinus type of typical mode shape functions
