3.18 Response of SDF Systems Related to Earthquakes
115
which on integration gives
(DLF) a = 1 − cos pt +
sin pt
pt d
−
t
t d
for t ≤ t d
(3.151)
For t > t d ,
(DLF) a =
1
pt d
[sin pt d − sin p (t − t d )] − cos pt
(3.152)
For different values of t d /T, (DLF) a, max is plotted in Fig. 3.37c.
If ¨
x s0 = g and t d = 0.1, then the variation of the maximum relative displacement
with the natural frequency is plotted in Fig. 3.37d, and the curve of maximum absolute
acceleration versus natural frequency is plotted in Fig. 3.37e.
Figure 3.37d and e give some interesting reading. From Eq. (3.145), it can be
seen that as p approaches zero, (DLF) a , max also approaches zero. Therefore, z max
becomes indeterminate. However, it can be shown that as p approaches zero, z max
approaches maximum ground motion.
The results of the response spectra at two extreme values of natural frequencies
can be physically explained. A very high value of natural frequency indicates a
very stiff structure. The structure with a high value of p will move more or less
as a rigid body, and the mass will move more or less wholly with the ground, so
that the maximum relative displacement is nearly zero and the absolute maximum
acceleration of the mass approaches the maximum ground motion. Similarly, a very
low value of natural frequency indicates a very flexible structure. Therefore, due to
ground motion, the mass remains more or less stationary, while the ground moves
below it. The relative displacement, of the mass, therefore, becomes nearly equal
to the ground displacement, and the maximum acceleration of the mass becomes
negligible.
3.19 Techniques for Analysing Earthquake Response
Unfortunately, the records obtained from the earthquake are not as simple as the one
indicated in Fig. 3.37a. The accelerograph of the earthquake having amplitudes and
periods varying with time indicates a random movement. The acceleration associated
with ground movement has all three components. However, the horizontal component
of it is most important, as it has the most damaging effect for the majority of the
structures.
So far, very few records of earthquakes have been obtained. Only during the last
three decades, earthquake records are taken on a more systematic basis in some
countries. Still, their number is not sufficient to be used for statistical purposes. So,
prediction of future earthquake at a place becomes problematic. However, the past
115
which on integration gives
(DLF) a = 1 − cos pt +
sin pt
pt d
−
t
t d
for t ≤ t d
(3.151)
For t > t d ,
(DLF) a =
1
pt d
[sin pt d − sin p (t − t d )] − cos pt
(3.152)
For different values of t d /T, (DLF) a, max is plotted in Fig. 3.37c.
If ¨
x s0 = g and t d = 0.1, then the variation of the maximum relative displacement
with the natural frequency is plotted in Fig. 3.37d, and the curve of maximum absolute
acceleration versus natural frequency is plotted in Fig. 3.37e.
Figure 3.37d and e give some interesting reading. From Eq. (3.145), it can be
seen that as p approaches zero, (DLF) a , max also approaches zero. Therefore, z max
becomes indeterminate. However, it can be shown that as p approaches zero, z max
approaches maximum ground motion.
The results of the response spectra at two extreme values of natural frequencies
can be physically explained. A very high value of natural frequency indicates a
very stiff structure. The structure with a high value of p will move more or less
as a rigid body, and the mass will move more or less wholly with the ground, so
that the maximum relative displacement is nearly zero and the absolute maximum
acceleration of the mass approaches the maximum ground motion. Similarly, a very
low value of natural frequency indicates a very flexible structure. Therefore, due to
ground motion, the mass remains more or less stationary, while the ground moves
below it. The relative displacement, of the mass, therefore, becomes nearly equal
to the ground displacement, and the maximum acceleration of the mass becomes
negligible.
3.19 Techniques for Analysing Earthquake Response
Unfortunately, the records obtained from the earthquake are not as simple as the one
indicated in Fig. 3.37a. The accelerograph of the earthquake having amplitudes and
periods varying with time indicates a random movement. The acceleration associated
with ground movement has all three components. However, the horizontal component
of it is most important, as it has the most damaging effect for the majority of the
structures.
So far, very few records of earthquakes have been obtained. Only during the last
three decades, earthquake records are taken on a more systematic basis in some
countries. Still, their number is not sufficient to be used for statistical purposes. So,
prediction of future earthquake at a place becomes problematic. However, the past