Civil Engineering Reference
In-Depth Information
3.2.5
Supplementary remarks
The procedures presented in the previous sections for the frequency analysis
concentrate on the dominant characteristics of the structure while items whose
influence on the dynamic response is normally small are neglected. Under certain
circumstances, however, some phenomena and characteristics of usually
secondary importance may increase their effect and therefore they may be no
longer negligible. Such cases are discussed briefly in this section.
(a)
Compressive forces
In analysing the lateral vibrations of simply supported beams subjected to
concentrated end-forces and cantilevers under their uniformly distributed weight,
Timoshenko [1928] took into account the effect of axial forces. His results showed
that the axial compressive forces reduce the value of the frequencies of natural
vibration by the factor of
(3.79)
where
F
is the magnitude of the axial load and
F
cr
is the corresponding critical load.
The reduction factor is considered exact for simply supported beams subjected to
concentrated end-forces and can be used as reasonable approximation for cantilevers
under uniformly distributed axial load.
Timoshenko's results can be generalized and the same approach can be applied
to the
coupled
vibrations of thin walled cantilevers of open cross-section. Taking
into account the effects of the uniformly distributed axial forces, the relevant
formulae can be modified. This leads to the following formulae for the first basic
frequencies
(3.80)
(3.81)
which now take into consideration the effect of the axial forces. Critical loads
N
cr,X
, N
cr,Y
and
N
cr,
ϕ
are the basic critical loads and are obtained from the stability analysis as
described in section 3.1.1.
