H. Krawinkler, July 2002
Matching of Equivalent Pulses to Near-Fault Ground Motions:
1. Select pulse type. In general, the pulse shown in Fig. 1 (square acceleration pulse,
P2) is adequate. This pulse is fully defined by a pulse period, Tp, and an effective
pulse acceleration ap,eff (ag,max in Fig. 1) or an effective pulse displacement up,eff (ug,max
= ag,max(Tp2/16) in Fig. 1).
2. Find best match for the pulse period, Tp. “Best” is defined here as the Tp that
minimizes the “error” in predicting the elastic drift response of a representative
structure to a near-fault ground motion from an equivalent pulse. A 20-story frame is
used as a representative structure, and the “error” is defined as the sum of the square
of the differences in story drifts obtained from the ground motion response and the
pulse response, see Fig. 2. The process of obtaining the pulse period is as follows:

For a pulse (defined by period Tp and effective pulse displacement up,eff) compute
the elastic story drift in each story i,  p,i , for the 20-story structure with a first
mode period T1 = Tp, for  = 0.5 to 2.0 with an increment in  of 0.1. The story
drifts for the 16  values are shown in Fig. 3.

For near-fault ground motion:
a. Select a safe upper and lower bound for Tp (e.g., 0.5 to 4.0 seconds)
b. For a test value of Tp (starting with lower bound (e.g., 0.5 sec.)):

Compute the elastic story drift in each story i,  r,i , for the 20-story
structure with a first mode period T1 = Tp, for  = 0.5 to 2.0 with an
increment in  of 0.1. This implies 16 T1 cases for each test value of Tp.

For each T1 () case:
i. For each story i compute the difference in story drift between the
ground motion response and the pulse response, i.e.,
diff   r,i   p ,i   r,i  u p ,test ( p ,i / u p ,test )   r,i  u p ,test d p ,i
(d p ,i   p ,i / u p ,test )
ii. Normalize this difference as follows:

ei 
 r,i  u p ,test d p ,i
 r,i
iii. The normalized difference is squared and summed up over the
height (20 values), i.e.,
2
20  d  
20  d  

d p ,i 
p ,i
2
   (ei )   1  u p ,test    20  2u p ,test      u p ,test   p,i 
 r ,i 
i 1
i 1 
i 1   r , i 
i 1   r , i 

20
 2
20
1
2
H. Krawinkler, July 2002

The so defined error is calculated for each T1 () and is summed over all
16  values, i.e.,
 d p ,i  2 320  d p ,i 
     320  2u p ,test      u p ,test    
i 1   r , i 
i 1   r , i 
320

2
c. The combined error is calculated for each test value of Tp and is plotted versus
Tp. If this plot shows a clear valley (low point), it is postulated that the period
associated with this valley is the pulse period for the near-fault record.
d. It further can be postulated that error minimization provides an estimate of the
effective pulse displacement up,eff, i.e.,
2
320  d  
320  d  

 2  p,i   2u p , eff   p,i   0
u p ,test
i 1   r , i 
i 1   r , i 
 d 
   p,i 
i 1   r , i 
320
u p , eff
 d p ,i 
  



i 1   r , i 
320
2
Assessment:
In many (most) cases this process results in an unambiguous identification of a pulse
period, and in some (but not many) cases the process is less successful. Typical
examples are shown in Fig. 4. The first graph shows a clear pulse period, whereas the
second graph shows two error valleys, in which case the choice of Tp becomes a matter
of judgment.
The process of estimating the effective pulse displacement (pulse intensity), as mentioned
in d, above, is not the best choice because it utilizes only elastic structural systems. A
better approach, which is based on the response of inelastic systems, is summarized in
Krawinkler, H., and Alavi, B., “Development of Improved Design Procedures for NearFault Ground Motions,” SMIP98 Seminar on Utilization of Strong-Motion Data,
Oakland, Sept. 15, 1998, pp. 21-41.
2
H. Krawinkler, July 2002
Ground Acceleration Time History
Pulse P2
1.5
1
ag / ag,max
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
3
3.5
3
3.5
3
3.5
t / Tp
Ground Velocity Time History
Pulse P2
vg / vg,max = vg / (ag,max.Tp / 4)
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
t / Tp
Ground Displacement Time History
Pulse P2
1
2
ug / ug,max = ug / (ag,max.Tp / 16)
1.5
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
t / Tp
Figure 1. Pulse P2 Ground Acceleration, Velocity, and Displacement Time Histories
3
H. Krawinkler, July 2002
pulse
record
Height
r,i – up(p,i/up)
= r,i – updp,i
Elastic Story Drift
Figure 2. Drift Matching for Individual Record and Pulse
Elastic Story Drift demand
Pulse P2, 20story, without P-D
1.2
T / Tp
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Relative Height
1
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
dp,i = max, i / ug, max
Figure 3. Pulse Response Story Drifts for Structures with Different T/Tp
4
H. Krawinkler, July 2002
Elastic story drift demand differences (square)
Pulse P2, Record NR94rrs, 20story
100
Minimized difference,  square
Square difference
80
60
40
20
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Tp
Elastic story drift demand differences (square)
Pulse P2, Record NR94sce, 20story
100
Minimized difference,  square
Square difference
80
60
40
20
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Tp
Figure 4. Examples of Pulse Period Identification from Error Analysis
5
4.0