Civil Engineering Reference
In-Depth Information
3.0
6.0
ξ=0.02
ξ=0.05
ξ=0.10
ξ=0.02
ξ=0.05
ξ=0.10
5.0
ξ=0.15
ξ=0.20
ξ=0.30
ξ=0.15
ξ=0.20
ξ=0.30
ξ=0.40
ξ=0.50
ξ=0.60
ξ=0.40
ξ=0 50
ξ=0.60
2.0
4.0
3.0
2.0
1.0
1.0
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Period (s)
Period (s)
Figure 3.7
Mean ratio of maximum relative velocity to pseudo-velocity (
left
) and maximum absolute acceleration
to pseudo - acceleration (
right
) as a function of the damping (
adapted from
Sadek
et al.
, 2000 )
1.00
2.0
The Kocaeli (Turkey) earthquake of August 17, 1999
Sakaria recording station
Critical damping ξ=5%
0.75
1.8
1.5
0.50
1.3
0 25
a
max
=
S
a
(
T
1
=0.5sec) = 0.83g
1.0
0.00
0 0
0
10.0
15.0
20.0
a
max
=
S
a
(
T
2
=1.5sec) = 0.44g
0.8
-0 25
PGA =0.63g @
t
= 5.92 sec
0.5
-0.50
PGA
0.3
-0.75
0.0
-1.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Time (seconds)
Natural Period of Vibration (seconds)
1.00
1.00
0.75
0.75
0.50
0.50
a
max
=
S
a
(
T
1
=0.5sec) = 0.83g
0 25
0.25
a
max
=
S
a
(
T
2
=1.5sec) = 0.44g
0.00
0.00
0 0
5 0
10 0
15.0
20.0
0 0
5.0
10.0
15.0
20.0
-0 25
-0 25
M
-0.50
-0.50
M
k
2
-0.75
-0.75
k
1
T
1
= 0.5 sec
ξ
T
2
= 1.5 sec
ξ
= 5%
= 5%
-1.00
-1.00
Time (seconds)
Time (seconds)
Figure 3.8
Derivation of elastic spectra
(i) Select the earthquake record from databanks, e.g. using those given in Section 3.7 ;
(ii) Select a
T
-
ξ
pair, i.e. the fundamental period of vibration and the damping ratio for the SDOF.
Values of interest for structural earthquake engineering applications range between 0.01 and 5
seconds for
T
, for very rigid and very fl exible structures, respectively, and 0% to 20% for
ξ
, for
lightly and highly damped systems, respectively as illustrated in Section 2.3.5;
(iii) Select a numerical method to integrate the equation of motion as expressed, for example, in
equation (3.14). Several reliable methods are available in the literature as also discussed in
Section 4.5.1.2: their numerical stability and accuracy are reviewed in several textbooks (e.g.
Hughes, 1987 ; Bathe, 1996 );
(iv) Compute the response history for the given earthquake record. The peak value as the spectral
displacement is
S
d
;
