Civil Engineering Reference
In-Depth Information
However, this does not imply that the fluctuations of cross sectional forces are
insignificant. It only means that the resonant part of the force load effect is negligible
(see chapter 2.10). For such a structure the total value of a cross sectional force
component
F
at spanwise position
x
may be obtained from
()
()
()
F
x
Fx
k
x
≈
+
⋅
σ
(7.4)
max
r
r
p
F
r
B
The entire solution, including
, may then be obtained exclusively from static
σ
F
B
considerations, i.e. the determination of response spectra is obsolete. Since the solution
contains the combined mean and fluctuating load effects, it represents the maximum
value of the force load effect for a structure whose behaviour is defined as static.
The more general solution, covering static as well as dynamic structural behaviour is
given in Eq. 7.2. Having split the fluctuating part of the response into a background and
a resonant part, the maximum value of
F
at
x
may then be expressed by
()
()
2
()
2
()
F
x
=
Fx
+
k
⋅
σ
x
+
σ
x
(7.5)
max
r
r
p
F
r
F
r
B
R
where
F
and
σ
are obtained from static equilibrium conditions and
F
σ
is obtained
f
ro
m the resonant part of a modal frequency domain approach. For the determination of
F
the finite element type of approach that is shown below (chapter 7.2) is appropriate,
unless the structural system is so simple that a direct analytical establishment of the
equilibrium conditions is sufficient, in which case the solution is considered trivial.
Similarly, for the determination of the background quasi-static part
F
B
there are two
σ
F
B
alternatives. If the structural system is fairly complex a finite element approach is
appropriate, but if the system is fairly simple a direct approach based on influence
functions will suffice. Both methods are shown below (chapter 7.3).
For the determination of the resonant part
there is the possibility of establishing
σ
F
R
an equivalent load based on the inertia forces, i.e. the product of response acceleration
and the oscillating mass variation, but this option is only useful if the structural system is
very simple because the equivalent load pattern must reproduce the actual structural
displacements that are relevant for the mode shapes that have been excited. In chapter
7.4 a more general procedure is given, based on the linear relationship between cross
sectional stress resultants and the corresponding spanwise derivatives of the resonant
displacement response.
In a finite element formulation it is in the following assumed that the structural
system has been modelled by nodes with six degrees of freedom as shown in Fig. 7.3 and
