Civil Engineering Reference
In-Depth Information
7.3 FURTHER APPLICATIONS
As the magnitude of the vertical load on the building increases in relation to the critical
load, the value of the stresses and deformations which are caused by the horizontal
load increases and the value of the natural frequencies decreases. The critical load
ratio can be used to allow for these second-order effects.
In making use of the magnification factor
(7.6)
the accuracy of the formulae given in
Chapters 4
and
5
for the stresses and deformations
can be improved. Factor
v
in formula (7.6) is the global critical load ratio. Values of the
stresses and deformations multiplied by magnification factor (7.6) take into account
the fact that the stresses and deformations of a building under horizontal loads increase
due to the vertical load.
The accuracy of the formulae for the natural frequencies can also be improved in a
similar way. In the dynamic analysis of beam-columns it is shown that the axial
compressive forces reduce the value of the natural frequencies of vibration by a factor
of (1—
v
)
1/2
[Timoshenko and Young, 1955]. In allowing for this effect, the accuracy of
the formulae given in section 3.2 can be improved by incorporating the factor into the
formulae:
(7.7)
In formula (7.7),
f
c
is the frequency which includes the effect of the vertical load,
f
is the
frequency which is calculated by ignoring this effect (using the formulae given in
section 3.2) and
v
is the relevant critical load ratio. The individual formulae for the basic
natural frequencies are given in section 3.2.5.a.
