116
3 Forced Vibration of Single Degree of Freedom System
earthquake recordings have indicated that they all follow a particular pattern. As
such, the same trend can be assumed for future earthquakes.
Due to the limited extent of data available on earthquake motions, a simplified approach is adopted in design. A standard earthquake may be assumed for
this purpose. If one is interested in determining the response on the basis of this
earthquake, one has to numerically integrate the record to compute DLF.
However, numerical integration of the earthquake records is time consuming. For
design purposes, some standard spectra are adopted. In IS 1893, the acceleration and
velocity spectra based on studies of four strongest earthquakes have been indicated.
To take into account the seismicity of various zones, the ordinates of the spectra are to
be multiplied by a coefficient specified in the table of the code. The magnitude of this
coefficient depends on a number of factors. They are the magnitudes, duration, form
of the expected earthquake, distance of the place from the expected earthquake, soil
conditions and the deformation characteristics of the structure. The average velocity
and acceleration curves of ground motions adopted by IS 1893 are given in Fig. 3.38.
If S v is the value of the maximum relative velocity as given by the velocity spectra,
then the following relationship is valid
˙
z max ∼ = S v
z max ∼ =
S v
p
¨
x max ∼ = pS v
⎫
⎪ ⎬
⎪ ⎭
(3.153)
The above relations are only approximate, but are fairly accurate for design
purposes. Equations (3.153) are derived as follows for the undamped cases. The
relative velocity is given by
˙
z = ˙
x − ˙
y =
d
dt
(x − y) =
d
dt
z
(3.154)
Substituting the value of z from Eq. (3.153) into Eq. (3.154), one obtains
˙
z =
d
dt
⎡
⎣ −
¨
x s0
p
t
0
f a (τ ) sin p (t − τ ) dτ
⎤
⎦
(3.155)
= −
1
p
t
0
¨
x s0 [ p f a (τ ) cos p (t − τ ) dτ
(3.156)
As S v indicates the maximum value of relative velocity
S v = ˙
z max =
⎡
⎣
t
0
¨
x s0 f a (τ ) cos p (t − τ ) dτ
⎤
⎦
max
(3.157)
3 Forced Vibration of Single Degree of Freedom System
earthquake recordings have indicated that they all follow a particular pattern. As
such, the same trend can be assumed for future earthquakes.
Due to the limited extent of data available on earthquake motions, a simplified approach is adopted in design. A standard earthquake may be assumed for
this purpose. If one is interested in determining the response on the basis of this
earthquake, one has to numerically integrate the record to compute DLF.
However, numerical integration of the earthquake records is time consuming. For
design purposes, some standard spectra are adopted. In IS 1893, the acceleration and
velocity spectra based on studies of four strongest earthquakes have been indicated.
To take into account the seismicity of various zones, the ordinates of the spectra are to
be multiplied by a coefficient specified in the table of the code. The magnitude of this
coefficient depends on a number of factors. They are the magnitudes, duration, form
of the expected earthquake, distance of the place from the expected earthquake, soil
conditions and the deformation characteristics of the structure. The average velocity
and acceleration curves of ground motions adopted by IS 1893 are given in Fig. 3.38.
If S v is the value of the maximum relative velocity as given by the velocity spectra,
then the following relationship is valid
˙
z max ∼ = S v
z max ∼ =
S v
p
¨
x max ∼ = pS v
⎫
⎪ ⎬
⎪ ⎭
(3.153)
The above relations are only approximate, but are fairly accurate for design
purposes. Equations (3.153) are derived as follows for the undamped cases. The
relative velocity is given by
˙
z = ˙
x − ˙
y =
d
dt
(x − y) =
d
dt
z
(3.154)
Substituting the value of z from Eq. (3.153) into Eq. (3.154), one obtains
˙
z =
d
dt
⎡
⎣ −
¨
x s0
p
t
0
f a (τ ) sin p (t − τ ) dτ
⎤
⎦
(3.155)
= −
1
p
t
0
¨
x s0 [ p f a (τ ) cos p (t − τ ) dτ
(3.156)
As S v indicates the maximum value of relative velocity
S v = ˙
z max =
⎡
⎣
t
0
¨
x s0 f a (τ ) cos p (t − τ ) dτ
⎤
⎦
max
(3.157)