# Seismic-Parameter-Based Statistical Procedures for the Approximate Assessment of Structural Damage

Authors: A. Elenas
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This study describes two statistical methodologies to estimate the postseismic damage status of structures based on seismic parameters as novel combined procedures in earthquake engineering. Thus, a multilinear regression analysis and discriminant analysis are utilized considering twenty seismic parameters. Overall damage indices describe the postseismic damage status. Nonlinear dynamic analyses furnish the damage indices, which are considered as exact indices and references for the subsequent study. The aim is to approximate the postseismic damage indices or the damage grade of buildings using statistical methods, thus avoiding complex nonlinear dynamic analyses. The multilinear regression procedure evaluates the damage indices explicitly, and the discriminant analysis furnishes the damage grade of the structures. The proposed methods are applied to a frame structure. A set of 400 natural accelerograms is used for the training phase of the models. The quality of the models is tested initially by the same set of natural accelerograms and then by a blind prediction using a second set of synthetic accelerograms. The results of both proposed methods have shown a correct classification percentage ranging from 87.75% to 97.50% and from 70% to 90% for the sets of the natural and synthetic accelerograms, respectively.


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## Body

## 1. Introduction

Several seismic parameters have been presented in the literature during the last several decades. These parameters can be used to express the intensity of the seismic excitations and to simplify their description. Postseismic field observations and numerical investigations have indicated the interdependency between the seismic parameters and damage status of buildings after earthquakes [1, 2]. The latter can be expressed by proper damage indices, whereas the interdependency between the considered quantities can be quantified numerically by appropriate correlation coefficients.This paper expands on the above, providing alternative statistical procedures for the approximate assessment of the structural Postseismic damage grade. Thus, the first step in the proposed new methodology is to choose a set of seismic parameters that represents the properties of the accelerograms. Next, the intensity parameters chosen in the first step are evaluated for a set of natural or artificial accelerograms. The damage indices for all considered accelerograms are subsequently calculated for the building of interest. Statistical analyses are applied to the resulting data from the previous steps to extract a statistical model for the prediction of the Postseismic damage status of structures. Finally, the new methodology is verified by a blind prediction.As mentioned previously, the proposed statistical methods are used to estimate the Postseismic structural damage status. Approximation methods are well known and broadly used in earthquake engineering. The evaluation of the fundamental period of a building from its geometric dimensions is such an approximation provided by several antiseismic codes. Thus, the fundamental period can be approximated using a simple formula depending on the type of structure (e.g., steel, reinforced concrete, masonry, frame, or wall) and the geometric dimensions of the examined building instead of needing to solve a complicated eigenvalue problem that depends on mass and stiffness matrices.The proposed methodologies can be used, for instance, by public authorities for a quick and accurate estimation of the Postseismic damage status or for estimating damage scenarios for specific buildings. The proposed techniques are simpler to apply than the more complicated and time-consuming nonlinear dynamic analyses. The novel application of the proposed statistical procedures in the specific field of earthquake engineering is also feasible for nonspecialists, in contrast to nonlinear dynamic analyses, which can only be performed by expert engineers.


## 2. Seismic Intensity Parameters

In general, the intensity parameters can be classified into peak, spectral, and energy parameters. Twenty parameters have been selected to represent the seismic excitation. They have been chosen from all three seismic parameter categories. The following seismic parameters are considered: peak ground acceleration (PGA); peak ground velocity (PGV); the ratio PGA/PGV; Arias’ intensity (I
0); root mean square acceleration (RMSa); the strong motion duration of Trifunac/Brady (T
0.90); seismic power (P
0.90); the spectral intensities of Housner (SIH), Kappos (SIK), and Martinez-Rueda (SIMR); effective peak acceleration (EPA); maximum effective peak acceleration (EPA
max
⁡); seismic energy input (E
inp); cumulative absolute velocity (CAV); the seismic damage potential of Araya/Saragoni (DP
A
/
S); central period (CP); spectral acceleration (SA); spectral velocity (SV); spectral displacement (SD); and the intensity of Fajfar/Vidic/Fischinger (I
F
/
V
/
F). Table 1 provides an overview of the used parameters and the literature references.Table 1
Seismic parameters.
Number
Seismic parameters
Reference
1
PGA
[4]
2
PGV
[4]
3
PGA/PGV
[4]
4
I
0
[3]
5
RMSa
[4]
6
T
0.90
[6]
7
P
0.90
[7]
8
SIH
[8]
9
SIK
[9]
10
SIMR
[10]
11
EPA
[11]
12
EPAmax
[11]
13
E
inp
[13]
14
CAV
[1]
15
DP
A
/
S
[14]
16
CP
[15]
17
SD
[17]
18
SV
[17]
19
SA
[17]
20
I
F
/
V
/
F
[18]The Arias intensity [3] is a measure of the total energy content of seismic excitation and is defined by the following relation:
(1)
I
0
=
π
2
g
∫
0
t
e
(
x
¨
)
2
d
t
.Here,I
0 is the Arias intensity, t
e is the total seismic duration, and x
¨ is the seismic ground acceleration.The root mean square acceleration RMSa [4] is a measure of the mean energy content calculated by the following relation:
(2)
RMS
a
=
1
t
e
∫
0
t
e
(
x
¨
)
2
d
t
.The Husid diagram [5] is the time history of the seismic energy content scaled to the total energy content. It is defined by the following relation:
(3)
H
(
t
)
=
∫
0
t
(
x
¨
)
2
d
t
∫
0
t
e
(
x
¨
)
2
d
t
,
where H
(
t
) is the Husid diagram as a function of time t.The strong motion duration of Trifunac/Brady is defined as the time elapsed between 5% and 95% of the Husid diagram [6] and is defined by the following relation:
(4)
T
0.90
=
T
0.95
-
T
0.05
,
where T
0.90 is the strong motion duration and T
0.95 and T
0.05 are the time elapsed at 95% and 5% of the Husid diagram, respectively.The powerP
0.90 [7] is a measure of the energy content per time unit of the seismic excitation and is defined by the following relation:
(5)
P
0.90
=
H
0.95
-
H
0.05
T
0.90
.Here,P
0.90 is the power of the seismic excitation, H
0.95 and H
0.05 the nonnormalized energy level, at 95% and 5% of the Husid diagram, respectively, and T
0.90 is the strong motion duration of Trifunac/Brady.According to Housner [8], the spectrum intensity is given by the relation
(6)
SI
H
=
∫
0.1
2.5
PSV
(
T
,
ξ
=
2
%
)
d
T
,
where PSV is the pseudovelocity curve, T is the natural period of an single-degree-of-freedom (SDOF) system, and ξ is the damping coefficient. In the present work, the integral in (6) is evaluated for a damping ratio equal to 2% of the critical damping, as recommended by Housner when he introduced this quantity [8]. However, it is now generally accepted that the Housner spectral intensity is valid for any level of damping (e.g., 5% of critical damping) [7].In the definition of the spectrum intensity after Kappos [9]:
(7)
SI
K
=
∫
T
n
-
t
1
T
n
+
t
2
PSV
(
T
,
ξ
)
d
T
,
the integration limits are dependent on the fundamental period of the structure T
n and the periodintervals t
1 and t
2. Kappos suggested t
1
=
t
2
=
0.2
T
n.Martinez-Rueda [10] introduced a spectral intensity definition accounting for the dynamic properties of the structure which is defined as
(8)
SI
MR
=
1
T
h
-
T
y
∫
T
y
T
h
PSV
(
T
,
ξ
)
d
T
,
where T
h the hardening period and T
y is the yield period, which are computed using the tangent stiffness of the hardening branch and using the tangent stiffness after the appearance of the first plastic hinge of the idealized lateral response of the structure, respectively. The required idealized lateral response of the structure for the evaluation of T
h and T
y has been performed by a nonlinear pushover analysis as suggested by Martinez-Rueda [10].The effective peak acceleration (EPA) [11, 12] is the average of the spectral ordinates of the elastic acceleration response spectrum (for 5% critical damping) in the period interval [0.1 s, 0.5 s], divided by a standard value 2.5. It is defined by the following relation:
(9)
EPA
=
SA
¯
0.1
s
÷
0.5
s
(
ξ
=
0.05
)
2.5
.The maximum EPA (EPA
max
⁡) is a variation of the EPA definition. Here, the period interval is not constant (as in the previous case from 0.1 to 0.5 s) but is a sliding window of 0.4 s, which sweeps the entire elastic acceleration response spectrum for 5% critical damping and provides the maximum of all possible EPA values calculated in this manner.The input seismic energy [13] is defined by the following relation:
(10)
E
inp
=
m
∫
0
t
e
(
u
¨
+
x
¨
)
x
˙
d
t
.Here,E
inp is the input seismic energy of a SDOF system, t
e is the total seismic duration, m is the mass, u is the displacement of the system relative to the ground, and x is the ground displacement.The cumulative absolute velocity (CAV) [1] is defined as the area under the absolute accelerogram:
(11)
CAV
=
∫
0
t
e
|
x
¨
|
d
t
.Araya and Saragoni [14] introduced the destructiveness potential (DP
A
/
S) of a seismic excitation, defined as
(12)
DP
A
/
S
=
I
0
ν
0
2
,
where I
0 is the Arias intensity and ν
0 is the intensity of the seismic acceleration zero-crossings. ν
0 is defined as the number of intersections of the acceleration ordinates with the time axis (points with zero acceleration) per time unit in a seismic acceleration time-history (accelerogram).The central period (CP) is the reciprocal value of the number of positive zero-crossings (points where the sign of the considered function changes from negative to positive) of the seismic acceleration per time unit [15]. The CP is a frequency content indicator and not an authentic seismic intensity parameter. However, this parameter has been used to characterize the seismic damage potential of seismic excitations (together with the term PGA/PGV and the strong motion duration of Trifunac/Brady T
0.90) [16]. For this reason, the CP has been included in the set of the seismic intensity parameters used in the present study.The response spectra describe the maximum response of a SDOF system to a particular input motion as a function of the natural period (or natural frequency) and the damping ratio of the SDOF system [17]. For each ground motion measurement there are displacement (SD), velocity (SV), and acceleration (SA) response spectra, respectively.Finally, the seismic intensity of Fajfar/Vidic/Fischinger [18] is defined as
(13)
I
F
/
V
/
F
=
v
max
⁡
(
T
0.90
)
0.25
,
where v
max
⁡ is the peak ground velocity (PGV) of the ground motion and T
0.90 is the strong motion duration of Trifunac/Brady.


## 3. Seismic Acceleration Time Histories

The expected damage potential of a seismic excitation on a particular structure is the prime consideration for the selection of the accelerograms for the presented methodology. Seismic excitations that generate a wide spectrum of damage, from negligible to severe, are considered for statistical reasons. Thus, the present investigation utilized 400 worldwide natural acceleration records with strong seismic activity. Table2 provides data for all the utilized seismic events (event name, country, date, record station, component, and PGA). Figure 1 is a scatter plot of standard (Richter) magnitude versus epicentral distance of all utilised seismic excitations. Furthermore, Tables 3 and 4 provide the number of accelerograms used per country and per PGA range. All of the above-mentioned seismic acceleration time histories have been investigated by a computer-supported evaluation of their seismic parameters, as presented in the previous section and in Table 1.Table 2
Data for the utilized seismic events.
Number
Seismic event
Country
Date
Station
Component
PGA
Magnitude
Epicentral distance
(g)
(km)
1
Long Beach
USA
10/3/1933
No. 288
S08W
0.131
6.4
38.1
2
Long Beach
USA
10/3/1933
No. 288
N82W
0.152
6.4
38.1
3
Long Beach
USA
10/3/1933
No. 131
NS
0.193
6.4
19.2
4
Long Beach
USA
10/3/1933
No. 331
EW
0.156
6.4
19.2
5
Long Beach
USA
10/3/1933
No. 136
N39E
0.100
6.4
45.2
6
Long Beach
USA
10/3/1933
No. 136
N51W
0.064
6.4
45.2
7
Lower California
USA
30/12/1934
No. 117
S00W
0.157
6.5
65.6
8
Lower California
USA
30/12/1934
No. 117
S90W
0.179
6.5
65.6
9
Helena Montana
USA
3/10/1935
No. 323
S00W
0.144
6.2
6.3
10
Helena Montana
USA
3/10/1935
No. 323
S90W
0.143
6.2
6.3
11
Imperial Valley
USA
18/5/1940
No. 117a
S00E
0.341
6.9
16.9
12
Imperial Valley
USA
18/5/1940
No. 117b
S00E
0.342
6.9
16.9
13
Imperial Valley
USA
18/5/1940
No. 117a
S90W
0.210
6.9
16.9
14
Imperial Valley
USA
18/5/1940
No. 117b
S90W
0.210
6.9
16.9
15
Santa Barbara
USA
30/6/1941
No. 283
N45E
0.234
5.5
14.9
16
Santa Barbara
USA
30/6/1941
No. 283
S45E
0.172
5.5
14.9
17
Northern California
USA
9/3/1949
No. 28
N89W
0.194
5.2
19.9
18
Puget Sound
USA
13/4/1949
Olympia
N176E
0.161
7.1
26.4
19
Puget Sound
USA
13/4/1949
Olympia
N266E
0.274
7.1
26.4
20
Western Washington
USA
13/4/1949
No. 325
N04W
0.162
7.1
17.0
21
Western Washington
USA
13/4/1949
No. 325
N86E
0.275
7.1
17.0
22
Kern County
USA
21/7/1952
No.283
S48E
0.129
7.5
87.7
23
Kern County
USA
21/8/1952
No. 1095
N21E
0.152
7.5
46.4
24
Kern County
USA
21/8/1952
No. 1095
S69E
0.176
7.5
46.4
25
Eureka
USA
21/12/1954
89509
349
0.165
6.6
23.5
26
Eureka
USA
21/12/1954
89509
79
0.253
6.6
23.5
27
Eureka
USA
21/12/1954
No. 1022
79
0.149
6.6
35.0
28
Eureka
USA
21/12/1954
No. 1022
N46W
0.189
6.6
35.0
29
Eureka
USA
21/12/1954
No. 1023
N44E
0.156
6.6
39.7
30
Eureka
USA
21/12/1954
No. 1023
N46W
0.197
6.6
39.7
31
Southern California
USA
18/3/1957
No. 272
NS
0.163
4.7
6.3
32
Southern California
USA
18/3/1957
No. 272
EW
0.086
4.7
6.3
33
Hollister
USA
8/4/1961
No. 028
N89W
0.176
5.6
22.2
34
Hollister
USA
8/4/1961
No. 028
N89W
0.169
5.6
22.2
35
Hokkaido
Japan
23/4/1962
Kushiro Met.
EW
0.281
7.0
75.0
36
Hokkaido
Japan
23/4/1962
Kushiro Met.
NS
0.517
7.0
75.0
37
Niigata
Japan
16/6/1964
Niigata
NS
0.129
7.5
59.0
38
Niigata
Japan
16/6/1964
Niigata
EW
0.171
7.5
59.0
39
Ibaraki
Japan
14/11/1964
Atom. En. Res
NS
0.260
5.1
8.0
40
Suruga Bay
Japan
20/4/1965
Shimizu
EW
0.153
6.1
30.0
41
Puget Sound
USA
29/4/1965
Olympia
N176E
0.134
6.5
63.0
42
Puget Sound
USA
29/4/1965
Olympia
N266E
0.195
6.5
63.0
43
Puget Sound
USA
29/4/1965
No. 325
S04E
0.134
6.5
61.0
44
Puget Sound
USA
29/4/1965
No. 325
S86E
0.194
6.5
61.0
45
Etorofu
Japan
5/4/1966
Hoshina-A
EW
0.500
5.4
4.0
46
Parkfield California
USA
27/6/1966
No. 013
N65E
0.480
5.6
3.5
47
Parkfield California
USA
27/6/1966
No. 014
N05W
0.348
5.6
3.5
48
Parkfield California
USA
27/6/1966
No. 014
N85E
0.426
5.6
7.1
49
Parkfield California
USA
27/6/1966
No. 015
N50E
0.233
5.6
7.1
50
Parkfield California
USA
27/6/1966
No. 015
N40W
0.270
5.6
11.2
51
Parkfield California
USA
27/6/1966
No. 097
N65W
0.264
5.6
11.2
52
Parkfield California
USA
27/6/1966
No. 097
S25W
0.341
5.6
12.0
53
Parkfield California
USA
27/6/1966
No. 014
N05W
0.348
5.6
12.0
54
Parkfield California
USA
27/6/1966
No. 014
N85E
0.426
5.6
14.0
55
Ariake
Japan
12/11/1966
Ariake
N40E
0.129
5.5
19.0
56
Ibaraki
Japan
19/11/1967
Atom. En. Res
NS
0.464
6.0
50.0
57
Ibaraki
Japan
19/11/1967
Atom. En. Res
EW
0.350
6.0
50.0
58
Wakayama
Japan
30/3/1968
Wakayama Harbor
NS
0.293
5.0
5.0
59
Wakayama
Japan
30/3/1968
Wakayama Harbor
EW
0.442
5.0
5.0
60
Hyuganada
Japan
1/4/1968
New Ind. City
NS
0.288
7.5
60.0
61
Hyuganada
Japan
1/4/1968
New Ind. City
EW
0.351
7.5
60.0
62
Honshu
Japan
16/5/1968
Muroran Harbor
EW
0.155
8.2
290.0
63
Honshu
Japan
16/5/1968
Shin-Ishikari B.
L
0.182
8.2
319.0
64
Honshu
Japan
16/5/1968
Shin-Ishikari B.
T
0.194
8.2
319.0
65
Honshu
Japan
16/5/1968
Horoman Bridge
L
0.126
8.2
159.0
66
Honshu
Japan
16/5/1968
Miyako Harbor
NS
0.165
8.2
187.0
67
Honshu
Japan
16/5/1968
Miyako Harbor
EW
0.158
8.2
187.0
68
Honshu
Japan
16/5/1968
Aomori Harbor
NS
0.227
8.2
243.0
69
Honshu
Japan
16/5/1968
Aomori Harbor
EW
0.197
8.2
243.0
70
Honshu
Japan
16/5/1968
Hachinohe Harbor
NS
0.312
8.2
187.0
71
Honshu
Japan
16/5/1968
Hachinohe Harbor
EW
0.206
8.2
187.0
72
Honshu
Japan
16/5/1968
Miyako Harbor
NS
0.155
8.2
226.0
73
Honshu
Japan
16/5/1968
Miyako Harbor
EW
0.124
8.2
226.0
74
Aomori Prefecture
Japan
23/5/1968
Miyako Harbor
NS
0.133
6.3
78.0
75
Aomori Prefecture
Japan
12/6/1968
Miyako Harbor
NS
0.190
7.2
81.0
76
Aomori Prefecture
Japan
12/6/1968
Miyako Harbor
EW
0.164
7.2
81.0
77
Saitama Prefecture
Japan
1/7/1968
Shinagawa
EW
0.127
6.1
58.0
78
Hokkaido
Japan
21/1/1970
Horoman Bridge
L
0.149
6.7
46.0
79
Hokkaido
Japan
21/1/1970
Horoman Bridge
T
0.215
6.7
46.0
80
Iwate Prefecture
Japan
1/4/1970
Miyako Harbor
NS
0.190
5.8
17.0
81
Iwate Prefecture
Japan
1/4/1970
Miyako Harbor
EW
0.162
5.8
17.0
82
Miyazaki Prefecture
Japan
26/7/1970
New Ind. City
EW
0.140
6.7
21.0
83
San Fernando
USA
9/2/1971
No. 279
S16E
1.148
6.6
6.7
84
San Fernando
USA
9/2/1971
No. 279
S74W
1.055
6.6
6.7
85
San Fernando
USA
9/2/1971
No. 151
N54W
0.123
6.6
41.5
86
San Fernando
USA
9/2/1971
No. 110
N21E
0.309
6.6
29.8
87
San Fernando
USA
9/2/1971
No. 110
N69W
0.265
6.6
29.8
88
San Fernando
USA
9/2/1971
No. 111
S00W
0.153
6.6
29.8
89
San Fernando
USA
9/2/1971
No. 135
S00W
0.167
6.6
34.8
90
San Fernando
USA
9/2/1971
No. 135
N90E
0.207
6.6
34.8
91
San Fernando
USA
9/2/1971
No. 217
S00W
0.147
6.6
38.6
92
San Fernando
USA
9/2/1971
No. 137
N50W
0.126
6.6
41.2
93
San Fernando
USA
9/2/1971
No. 137
S40W
0.169
6.6
41.2
94
San Fernando
USA
9/2/1971
No. 285
S51E
0.203
6.6
32.9
95
San Fernando
USA
9/2/1971
No. 285
S75W
0.174
6.6
32.9
96
San Fernando
USA
9/2/1971
No. 199
S00E
0.158
6.6
38.8
97
San Fernando
USA
9/2/1971
No. 199
N90E
0.162
6.6
38.8
98
San Fernando
USA
9/2/1971
No. 122
S70E
0.266
6.6
31.0
99
San Fernando
USA
9/2/1971
No. 122
S20W
0.209
6.6
31.0
100
San Fernando
USA
9/2/1971
No. 166
S53E
0.236
6.6
41.2
101
San Fernando
USA
9/2/1971
No. 166
S37W
0.192
6.6
41.2
102
San Fernando
USA
9/2/1971
No. 416
EW
0.162
6.6
36.6
103
San Fernando
USA
9/2/1971
No. 148
N37E
0.196
6.6
39.9
104
San Fernando
USA
9/2/1971
No. 148
S53E
0.188
6.6
39.9
105
San Fernando
USA
9/2/1971
No. 160
N30W
0.242
6.6
40.9
106
San Fernando
USA
9/2/1971
No. 160
S60W
0.221
6.6
40.9
107
San Fernando
USA
9/2/1971
No. 211
NS
0.154
6.6
38.8
108
San Fernando
USA
9/2/1971
No. 211
EW
0.130
6.6
38.8
109
San Fernando
USA
9/2/1971
No. 266
S90W
0.189
6.6
34.9
110
San Fernando
USA
9/2/1971
No. 264
NOOE
0.198
6.6
38.1
111
San Fernando
USA
9/2/1971
No. 264
N90E
0.182
6.6
38.1
112
San Fernando
USA
9/2/1971
No. 267
S82E
0.208
6.6
30.7
113
San Fernando
USA
9/2/1971
No. 267
S08W
0.139
6.6
30.7
114
San Fernando
USA
9/2/1971
No.  466
N11E
0.220
6.6
28.3
115
San Fernando
USA
9/2/1971
No.  466
N79W
0.146
6.6
28.3
116
San Fernando
USA
9/2/1971
No. 461
S81E
0.140
6.6
27.6
117
San Fernando
USA
9/2/1971
No. 461
S09W
0.129
6.6
27.6
118
San Fernando
USA
9/2/1971
No. 126
S69E
0.168
6.6
27.6
119
San Fernando
USA
9/2/1971
No. 126
S21W
0.144
6.6
27.6
120
San Fernando
USA
9/2/1971
No. 128
N21E
0.346
6.6
23.9
121
San Fernando
USA
9/2/1971
No. 128
N69W
0.277
6.6
23.9
122
San Fernando
USA
9/2/1971
No. 220
N00E
0.164
6.6
29.1
123
San Fernando
USA
9/2/1971
No. 220
S90W
0.148
6.6
29.1
124
San Fernando
USA
9/2/1971
No. 226
S89W
0.154
6.6
34.8
125
San Fernando
USA
9/2/1971
No. 226
S01E
0.156
6.6
34.8
126
San Fernando
USA
9/2/1971
No. 279
S74W
1.055
6.6
6.7
127
San Fernando
USA
9/2/1971
No. 128
N69W
0.278
6.6
23.9
128
Adak
USA
2/5/1971
Adak
N090E
0.204
7.1
78.9
129
Adak
USA
2/5/1971
Adak
N090E
0.205
7.1
78.9
130
Santiago
Chile
9/7/1971
Santiago Univ.
N10W
0.156
7.5
117.0
131
Chiba
Japan
11/10/1971
Kashima Harbor
EW
0.169
5.2
12.0
132
Kushiro
Japan
11/10/1971
Kushiro
NS
0.143
5.8
33.0
133
Central California
USA
4/9/1972
USGS 1211
N61E
0.471
4.8
9.8
134
Central California
USA
4/9/1972
USGS 1211
N29W
0.516
4.8
9.8
135
Central North Island
New Zealand
5/1/1973
ATENE C
N40E
0.143
6.7
67.0
136
Central North Island
New Zealand
5/1/1973
ATENE C
N50W
0.153
6.7
67.0
137
Nemuro
Japan
17/6/1973
Kushiro
NS
0.201
7.4
112.0
138
Nemuro
Japan
17/6/1973
Kushiro
EW
0.127
7.4
112.0
139
Lefkada
Greece
4/11/1973
Lefkada
L
0.041
5.0
19.0
140
Lefkada
Greece
4/11/1973
Lefkada
T
0.074
5.0
19.0
141
Lefkada
Greece
4/11/1973
Lefkada
L
0.515
5.8
20.0
142
Lefkada
Greece
4/11/1973
Lefkada
T
0.258
5.8
20.0
143
Hollister
USA
28/11/1974
Gilroy Array
S67W
0.135
5.2
12.0
144
Hollister
USA
28/11/1974
Hollister
N87W
0.163
5.2
9.8
145
Cape Mendocino
USA
12/1/1975
Petrolia
N75E
0.180
5.2
19.0
146
Oita Prefecture
Japan
21/4/1975
Beppu Harbor
EW
0.138
6.4
41.0
147
Cape Mendocino
USA
6/5/1975
Shelter Cove
N70E
0.168
5.3
63.0
148
Humboldt County
USA
7/6/1975
Petrolia
S60E
0.199
5.2
31.0
149
Humboldt County
USA
7/6/1975
Petrolia
N75E
0.158
5.2
31.0
150
Whittier Narrows
USA
7/6/1975
Petrolia
N15W
0.128
5.2
31.0
151
Oroville
USA
6/8/1975
USGS 0001
N90E
0.292
4.7
7.6
152
Oroville
USA
6/8/1975
USGS 0001
N00E
0.620
4.7
7.6
153
Oroville
USA
11/8/1975
USGS 0001
S55E
0.389
4.4
1.0
154
Oroville
USA
11/8/1975
USGS 0001
N35E
0.284
4.4
1.0
155
Denizli
Turkey
19/8/1976
Denizli Met.
NS
0.344
4.4
1.0
156
Vrancea
Romania
4/3/1977
Bucharest
EW
0.175
7.4
170.0
157
Vrancea
Romania
4/3/1977
Bucharest
SN
0.202
7.4
170.0
158
Vrancea
Romania
4/3/1977
Bucharest
NS
0.194
7.4
165.0
159
Vrancea
Romania
4/3/1977
Bucharest
EW
0.163
7.4
165.0
160
Izmir
Turkey
16/12/1977
Izmir
NS
0.409
5.3
6.0
161
Izmir
Turkey
16/12/1977
Izmir
EW
0.136
5.3
6.0
162
Miyagi Prefecture
Japan
12/6/1978
Ofunato Harbor
N41E
0.207
7.4
116.0
163
Miyagi Prefecture
Japan
12/6/1978
Ofunato Harbor
E41S
0.222
7.4
116.0
164
Miyagi Prefecture
Japan
12/6/1978
Siogama Harbor
NS
0.314
7.4
120.0
165
Miyagi Prefecture
Japan
12/6/1978
Siogama Harbor
EW
0.288
7.4
120.0
166
Dursunbey
Turkey
18/7/1979
Dursunbey
NS
0.234
5.2
10.0
167
Dursunbey
Turkey
18/7/1979
Dursunbey
EW
0.309
5.2
10.0
168
Coyote Lake
USA
6/8/1979
USGS 1409
S40E
0.251
5.9
14.4
169
Coyote Lake
USA
6/8/1979
USGS 1409
N50E
0.188
5.9
14.4
170
Coyote Lake
USA
6/8/1979
USGS 1413
N40W
0.299
5.9
10.3
171
Coyote Lake
USA
6/8/1979
USGS 1413
S50W
0.408
5.9
10.3
172
Imperial Valley
USA
15/10/1979
El Centro
230
0.367
6.6
27.6
173
Imperial Valley
USA
15/10/1979
El Centro
140
0.517
6.6
27.6
174
Imperial Valley
USA
15/10/1979
El Centro
230
0.349
6.6
18.5
175
Imperial Valley
USA
15/10/1979
El Centro
140
0.483
6.6
18.5
176
Imperial Valley
USA
15/10/1979
El Centro
230
0.405
6.6
33.9
177
Imperial Valley
USA
15/10/1979
El Centro
140
0.309
6.6
33.9
178
Imperial Valley
USA
15/10/1979
El Centro
140
0.138
6.6
27.8
179
Imperial Valley
USA
15/10/1979
El Centro
140
0.151
6.6
27.8
180
Imperial Valley
USA
15/10/1979
El Centro
230
0.162
6.6
27.6
181
Imperial Valley
USA
15/10/1979
El Centro
140
0.229
6.6
27.6
182
Imperial Valley
USA
15/10/1979
El Centro
230
0.275
6.6
27.3
183
Imperial Valley
USA
15/10/1979
El Centro
140
0.221
6.6
27.3
184
Imperial Valley
USA
15/10/1979
El Centro
230
0.147
6.6
36.3
185
Imperial Valley
USA
15/10/1979
El Centro
140
0.130
6.6
36.3
186
Imperial Valley
USA
15/10/1979
El Centro
360
0.137
6.6
33.9
187
Imperial Valley
USA
15/10/1979
El Centro
270
0.145
6.6
33.9
188
Imperial Valley
USA
15/10/1979
USGS 5058
S50W
0.362
6.6
27.9
189
Imperial Valley
USA
15/10/1979
USGS 5058
S40E
0.352
6.6
27.9
190
Imperial Valley
USA
15/10/1979
USGS 5115
S50W
0.365
6.6
31.5
191
Imperial Valley
USA
15/10/1979
USGS 5115
S40E
0.309
6.6
31.5
192
Imperial Valley
USA
15/10/1979
USGA 5028
S50W
0.450
6.6
27.1
193
Imperial Valley
USA
15/10/1979
USGA 5028
S40E
0.331
6.6
27.1
194
Imperial Valley
USA
15/10/1979
USGA 0942
S50W
0.429
6.6
27.1
195
Imperial Valley
USA
15/10/1979
USGA 0942
S40E
0.332
6.6
27.1
196
Imperial Valley
USA
15/10/1979
USGS 5054
S50W
0.762
6.6
6.7
197
Imperial Valley
USA
15/10/1979
USGS 5054
S40E
0.584
6.6
6.7
198
Imperial Valley
USA
15/10/1979
USGS 0958
S50W
0.438
6.6
27.2
199
Imperial Valley
USA
15/10/1979
USGS 0958
S40E
0.599
6.6
27.2
200
Imperial Valley
USA
15/10/1979
USGS 0952
S50W
0.360
6.6
27.6
201
Imperial Valley
USA
15/10/1979
USGS 0952
S40E
0.539
6.6
27.6
202
Imperial Valley
USA
15/10/1979
USGS 5165
NS
0.473
6.6
26.3
203
Imperial Valley
USA
15/10/1979
USGS 5165
EW
0.339
6.6
26.3
204
Imperial Valley
USA
15/10/1979
USGS 0955
S50W
0.349
6.6
27.2
205
Imperial Valley
USA
15/10/1979
USGS 0955
S40E
0.480
6.6
27.2
206
Livermore
USA
24/1/1980
CDMG 0364
EW
0.031
5.9
24.4
207
Livermore
USA
24/1/1980
CDMG 0364
NS
0.027
5.9
24.4
208
Mammoth Lakes
USA
27/5/1980
SER no. 3679
S34E
0.337
6.2
17.3
209
Mammoth Lakes
USA
27/5/1980
SER no. 3679
N56E
0.138
6.2
17.3
210
Mammoth Lakes
USA
27/5/1980
SER no. 1480
NS
0.389
6.2
19.8
211
Mammoth Lakes
USA
27/5/1980
SER no. 1480
EW
0.344
6.2
19.8
212
Mammoth Lakes
USA
27/5/1980
SER no. 3754
S15E
0.631
6.2
14.8
213
Mammoth Lakes
USA
27/5/1980
SER no. 3754
N75E
0.452
6.2
14.8
214
Korinthos
Greece
24/2/1981
Athens
L
0.221
6.7
69.0
215
Korinthos
Greece
24/2/1981
Korinthos
L
0.233
6.7
31.0
216
Korinthos
Greece
24/2/1981
Korinthos
T
0.299
6.7
31.0
217
Korinthos
Greece
24/2/1981
Xylokastro
L
0.289
6.7
35.0
218
Korinthos
Greece
24/2/1981
Xylokastro
T
0.114
6.7
35.0
219
Korinthos
Greece
25/2/1981
Athens
T
0.095
6.7
69.0
220
Korinthos
Greece
25/2/1981
Korinthos
L
0.117
6.7
33.0
221
Korinthos
Greece
25/2/1981
Korinthos
T
0.121
6.7
33.0
222
Lefkada
Greece
10/3/1981
Lefkada
L
0.057
5.8
53.0
223
Lefkada
Greece
10/3/1981
Lefkada
T
0.097
5.8
53.0
224
Lefkada
Greece
10/3/1981
Preveza
L
0.141
5.8
39.0
225
Lefkada
Greece
10/3/1981
Preveza
T
0.137
5.8
39.0
226
Lefkada
Greece
10/4/1981
Lefkada
L
0.023
4.7
34.0
227
Lefkada
Greece
10/4/1981
Lefkada
T
0.036
4.7
34.0
228
Westmoreland
USA
26/4/1981
WEST 2588
EW
0.354
5.6
11.2
229
Westmoreland
USA
26/4/1981
WEST 2588
NS
0.435
5.6
11.2
230
Lefkada
Greece
25/5/1981
Lefkada
L
0.057
4.7
25.0
231
Lefkada
Greece
25/5/1981
Lefkada
T
0.032
4.7
25.0
232
Lefkada
Greece
27/5/1981
Lefkada
L
0.117
5.5
3.0
233
Lefkada
Greece
27/5/1981
Lefkada
T
0.093
5.5
3.0
234
Ionian
Greece
17/1/1983
Argostoli
NS
0.162
7.2
28.7
235
Ionian
Greece
17/1/1983
Argostoli
WE
0.139
7.2
28.7
236
Heraklio
Greece
19/3/1983
Heraklio
T
0.170
5.7
38.0
237
Ionian
Greece
23/3/1983
Argostoli
NS
0.177
5.9
29.0
238
Ionian
Greece
23/3/1983
Argostoli
WE
0.234
5.9
29.0
239
Ionian
Greece
24/3/1983
Argostoli
NS
0.242
5.5
14.7
240
Ionian
Greece
24/3/1983
Argostoli
WE
0.299
5.5
14.7
241
Coalinga
USA
2/5/1983
Coalinga
360
0.595
6.5
28.1
242
Coalinga
USA
2/5/1983
Coalinga
270
0.549
6.5
28.1
243
Coalinga
USA
2/5/1983
Coalinga
360
0.264
6.5
40.6
244
Coalinga
USA
2/5/1983
Coalinga
270
0.371
6.5
40.6
245
Coalinga
USA
2/5/1983
Coalinga
360
0.189
6.5
36.4
246
Coalinga
USA
2/5/1983
Coalinga
270
0.212
6.5
36.4
247
Horasan
Turkey
30/10/1983
Horasan
NS
0.149
6.9
30.0
248
Heraklio
Greece
1/3/1984
Heraklio
L
0.106
4.7
30.0
249
Heraklio
Greece
1/3/1984
Heraklio
T
0.197
4.7
30.0
250
Morgan Hill
USA
24/4/1984
Anderson Dam
340
0.283
6.2
15.0
251
Morgan Hill
USA
24/4/1984
Anderson Dam
250
0.416
6.2
15.0
252
Morgan Hill
USA
24/4/1984
Gilroy Array
0
0.183
6.2
38.0
253
Morgan Hill
USA
24/4/1984
Gilroy Array
0
0.215
6.2
38.1
254
Morgan Hill
USA
24/4/1984
Gilroy Array
90
0.280
6.2
38.1
255
Morgan Hill
USA
24/4/1984
Gilroy Array
270
0.217
6.2
37.9
256
Morgan Hill
USA
24/4/1984
Gilroy Array
360
0.328
6.2
37.9
257
Morgan Hill
USA
24/4/1984
No. 47380
90
0.210
6.2
38.8
258
Morgan Hill
USA
24/4/1984
No. 47380
0
0.154
6.2
38.8
259
Valparaiso
Chile
3/3/1985
Llolleo
100
0.187
7.8
120.0
260
Valparaiso
Chile
3/3/1985
Llolleo
10
0.182
7.8
120.0
261
Valparaiso
Chile
3/3/1985
Llolleo
100
0.437
7.8
120.0
262
Valparaiso
Chile
3/3/1985
Llolleo
10
0.698
7.8
120.0
263
Valparaiso
Chile
3/3/1985
Vina del Mar
290
0.233
7.8
43.0
264
Valparaiso
Chile
3/3/1985
Vina del Mar
200
0.356
7.8
43.0
265
Valparaiso
Chile
3/3/1985
Valparaiso
50
0.292
7.8
36.0
266
Valparaiso
Chile
3/3/1985
Valparaiso
140
0.163
7.8
36.0
267
Valparaiso
Chile
3/3/1985
Santiago
L
0.124
7.8
130.0
268
Valparaiso
Chile
8/4/1985
Llolleo
100
0.167
7.5
120.0
269
Valparaiso
Chile
8/4/1985
Llolleo
10
0.200
7.5
120.0
270
Michoacan
Mexico
19/9/1985
SCT1
N90W
0.168
8.1
400.0
271
Michoacan
Mexico
19/9/1985
La Union
N00E
0.163
8.1
107.0
272
Michoacan
Mexico
19/9/1985
Michoacan
N90W
0.162
8.1
111.0
273
Michoacan
Mexico
19/9/1985
La Union
N90E
0.147
8.1
107.0
274
Michoacan
Mexico
19/9/1985
Caleta de Campo
N00E
0.139
8.1
21.0
275
Michoacan
Mexico
19/9/1985
Caleta de Campo
N90E
0.138
8.1
21.0
276
Michoacan
Mexico
19/9/1985
La Villita
N00E
0.121
8.1
65.0
277
Michoacan
Mexico
21/9/1985
Michoacan
S00E
0.156
7.5
91.0
278
Michoacan
Mexico
21/9/1985
Michoacan
N90W
0.140
7.5
91.0
279
Michoacan
Mexico
21/9/1985
Papanoa
S00E
0.252
7.5
130.0
280
Nahanni
Canada
9/11/1985
Slide Mountain
330
0.374
5.0
5.8
281
Nahanni
Canada
9/11/1985
Slide Mountain
240
0.451
5.0
5.8
282
Nahanni
Canada
23/12/1985
Iverson
10
1.080
6.9
8.0
283
Nahanni
Canada
23/12/1985
Iverson
280
1.319
6.9
8.0
284
Nahanni
Canada
23/12/1985
Iverson
10
0.224
6.9
8.0
285
Nahanni
Canada
23/12/1985
Slide Mountain
330
0.386
6.9
15.0
286
Nahanni
Canada
23/12/1985
Slide Mountain
240
0.534
6.9
15.0
287
Nahanni
Canada
23/12/1985
Nahanni
360
0.190
6.9
23.8
288
Nahanni
Canada
23/12/1985
Nahanni
270
0.192
6.9
23.8
289
Kalamata
Greece
13/9/1986
Kalamata
L
0.217
6.0
10.0
290
Kalamata
Greece
13/9/1986
Kalamata
T
0.291
6.0
10.0
291
Kalamata
Greece
15/9/1986
Kalamata
L
0.328
4.1
7.0
292
Kalamata
Greece
15/9/1986
Kalamata
T
0.161
4.1
7.0
293
San Salvador
San Salvador
10/10/1986
No. 90005
270
0.524
5.4
5.7
294
San Salvador
San Salvador
10/10/1986
No. 90005
180
0.391
5.4
5.7
295
San Salvador
San Salvador
10/10/1986
No. 90018
0
0.214
5.4
6.2
296
San Salvador
San Salvador
10/10/1986
No. 90018
270
0.296
5.4
6.2
297
San Salvador
San Salvador
10/10/1986
No. 90006
180
0.412
5.4
4.3
298
San Salvador
San Salvador
10/10/1986
No. 90006
90
0.681
5.4
4.3
299
San Salvador
San Salvador
10/10/1986
No. 90014
90
0.339
5.4
4.5
300
San Salvador
San Salvador
10/10/1986
No. 90014
0
0.421
5.4
4.5
301
Strazhitsa
Bulgaria
7/12/1986
V. Tarnovo
L
0.067
5.7
225.0
302
Strazhitsa
Bulgaria
7/12/1986
V. Tarnovo
T
0.074
5.7
225.0
303
Whittier Narrows
USA
1/10/1987
No. 24157
90
0.149
6.1
27.0
304
Whittier Narrows
USA
1/10/1987
No. 24157
0
0.139
6.1
27.0
305
Whittier Narrows
USA
1/10/1987
Tarzana
90
0.527
6.1
43.3
306
Whittier Narrows
USA
1/10/1987
Tarzana
360
0.397
6.1
43.3
307
Whittier Narrows
USA
1/10/1987
Hollywood
0
0.201
6.1
25.2
308
Whittier Narrows
USA
1/10/1987
Inglewood
90
0.219
6.1
25.8
309
Whittier Narrows
USA
1/10/1987
Inglewood
0
0.246
6.1
25.8
310
Whittier Narrows
USA
1/10/1987
Long Beach
90
0.233
6.1
27.4
311
Whittier Narrows
USA
1/10/1987
Long Beach
0
0.144
6.1
27.4
312
Whittier Narrows
USA
1/10/1987
Alhambra
90
0.284
6.1
6.8
313
Whittier Narrows
USA
1/10/1987
Alhambra
360
0.250
6.1
6.8
314
Whittier Narrows
USA
1/10/1987
Whittier
180
0.382
6.1
15.6
315
Whittier Narrows
USA
1/10/1987
Whittier
90
0.607
6.1
15.6
316
Whittier Narrows
USA
1/10/1987
Vernon
7
0.267
6.1
15.7
317
Whittier Narrows
USA
1/10/1987
Vernon
277
0.240
6.1
15.7
318
Whittier Narrows
USA
1/10/1987
Norwalk
360
0.201
6.1
21.9
319
Whittier Narrows
USA
1/10/1987
Norwalk
360
0.235
6.1
21.9
320
Rhodos
Greece
5/10/1987
Rhodos
L
0.056
5.6
27.0
321
Rhodos
Greece
5/10/1987
Rhodos
T
0.052
5.6
27.0
322
Superstition Hills
USA
24/11/1987
Wildlife Liq.
360
0.131
6.6
30.0
323
Superstition Hills
USA
24/11/1987
Wildlife Liq.
90
0.129
6.6
30.0
324
Lefkada
Greece
24/4/1988
Lefkada
L
0.114
5.0
33.0
325
Lefkada
Greece
24/4/1988
Lefkada
T
0.208
5.0
33.0
326
Valsamata
Greece
18/5/1988
Valsamata
L
0.165
5.8
23.0
327
Valsamata
Greece
18/5/1988
Valsamata
T
0.174
5.8
23.0
328
Valsamata
Greece
22/5/1988
Valsamata
L
0.053
5.5
21.0
329
Valsamata
Greece
22/5/1988
Valsamata
T
0.079
5.5
21.0
330
Kyllini
Greece
16/10/1988
Zakynthos
NS
0.125
6.0
36.0
331
Kyllini
Greece
16/10/1988
Zakynthos
WE
0.167
6.0
36.0
332
Vartholomio
Greece
19/10/1988
Vartholomio
L
0.047
4.4
19.0
333
Vartholomio
Greece
19/10/1988
Vartholomio
T
0.035
4.4
19.0
334
Vartholomio
Greece
31/10/1988
Vartholomio
L
0.062
4.8
17.0
335
Vartholomio
Greece
31/10/1988
Vartholomio
T
0.035
4.8
17.0
336
Nafpaktos
Greece
22/12/1988
Nafpaktos
L
0.112
5.0
10.0
337
Nafpaktos
Greece
22/12/1988
Nafpaktos
T
0.103
5.0
10.0
338
Loma Prieta
USA
17/10/1989
Anderson Dam
340
0.239
7.0
96.7
339
Loma Prieta
USA
17/10/1989
Anderson Dam
250
0.239
7.0
96.7
340
Loma Prieta
USA
17/10/1989
Hollister
255
0.281
7.0
48.1
341
Loma Prieta
USA
17/10/1989
Hollister
165
0.276
7.0
48.1
342
Loma Prieta
USA
17/10/1989
Palo Alto Hospital
302
0.341
7.0
46.3
343
Loma Prieta
USA
17/10/1989
Palo Alto Hospital
212
0.378
7.0
46.3
344
Loma Prieta
USA
17/10/1989
Hollister
180
0.217
7.0
45.0
345
Loma Prieta
USA
17/10/1989
Hollister
90
0.252
7.0
45.0
346
Loma Prieta
USA
17/10/1989
Stanford
360
0.255
7.0
51.5
347
Loma Prieta
USA
17/10/1989
Stanford
90
0.215
7.0
51.5
348
Loma Prieta
USA
17/10/1989
Calaveras
180
0.142
7.0
46.5
349
Loma Prieta
USA
17/10/1989
Calaveras
90
0.191
7.0
46.5
350
Loma Prieta
USA
17/10/1989
Apeel Array
133
0.223
7.0
64.1
351
Loma Prieta
USA
17/10/1989
Apeel Array
43
0.272
7.0
64.1
352
Loma Prieta
USA
17/10/1989
Emeryville
350
0.210
7.0
96.7
353
Loma Prieta
USA
17/10/1989
Emeryville
260
0.255
7.0
96.7
354
Loma Prieta
USA
17/10/1989
Anderson Dam
333
0.239
7.0
96.7
355
Loma Prieta
USA
17/10/1989
Anderson Dam
243
0.245
7.0
96.7
356
Loma Prieta
USA
17/10/1989
Emeryville
350
0.210
7.0
96.7
357
Loma Prieta
USA
17/10/1989
Emeryville
260
0.255
7.0
96.7
358
Loma Prieta
USA
17/10/1989
Emeryville
260
0.209
7.0
96.7
359
Loma Prieta
USA
17/10/1989
Emeryville
350
0.182
7.0
96.7
360
Loma Prieta
USA
17/10/1989
San Francisco
95
0.220
7.0
96.7
361
Aigio
Greece
17/5/1990
Aigio
T
0.203
5.0
20.0
362
Erzincan
Turkey
13/3/1992
Erzincan
NS
0.392
6.9
6.0
363
Erzincan
Turkey
13/3/1992
Erzincan
EW
0.492
6.9
6.0
364
Northridge
USA
17/1/1994
Los Angeles
166
0.257
6.4
13.7
365
Northridge
USA
17/1/1994
Los Angeles
76
0.462
6.4
13.7
366
Northridge
USA
17/1/1994
Los Angeles
348
0.133
6.4
31.0
367
Northridge
USA
17/1/1994
Los Angeles
258
0.147
6.4
31.0
368
Northridge
USA
17/1/1994
Canoga Park
90
0.334
6.4
6.0
369
Northridge
USA
17/1/1994
Los Angeles
360
0.260
6.4
42.2
370
Northridge
USA
17/1/1994
Los Angeles
270
0.154
6.4
42.2
371
Northridge
USA
17/1/1994
Malibu Canyon
360
0.173
6.4
14.9
372
Northridge
USA
17/1/1994
Malibu Canyon
270
0.161
6.4
14.9
373
Northridge
USA
17/1/1994
Littlerock
300
0.120
6.4
61.3
374
Northridge
USA
17/1/1994
Topanga
360
0.326
6.4
6.0
375
Northridge
USA
17/1/1994
Topanga
270
0.194
6.4
6.0
376
Northridge
USA
17/1/1994
Los Angeles
360
0.164
6.4
31.2
377
Northridge
USA
17/1/1994
Los Angeles
270
0.291
6.4
31.2
378
Northridge
USA
17/1/1994
Los Angeles
325
0.383
6.4
19.1
379
Northridge
USA
17/1/1994
Los Angeles
235
0.298
6.4
19.1
380
Northridge
USA
17/1/1994
Los Angeles
325
0.247
6.4
28.7
381
Northridge
USA
17/1/1994
Jensen
22
0.388
6.4
12.2
382
Northridge
USA
17/1/1994
Jensen
292
0.621
6.4
12.2
383
Northridge
USA
17/1/1994
USC no. 0003
S00E
0.444
6.4
2.2
384
Northridge
USA
17/1/1994
USC no. 0006
N90E
0.430
6.4
11.1
385
Northridge
USA
17/1/1994
USC no. 0009
S90W
0.248
6.4
12.0
386
Northridge
USA
17/1/1994
USC no. 0013
N09E
0.434
6.4
12.7
387
Northridge
USA
17/1/1994
USC no. 0014
S55E
0.434
6.4
15.5
388
Northridge
USA
17/1/1994
USC no. 0016
N90E
0.257
6.4
16.5
389
Northridge
USA
17/1/1994
USC no. 0021
N00W
0.409
6.4
26.5
390
Northridge
USA
17/1/1994
USC no. 0053
S16W
0.381
6.4
6.0
391
Northridge
USA
17/1/1994
USC no. 0054
S65W
0.324
6.4
25.2
392
Northridge
USA
17/1/1994
USC no. 0055
N90E
0.503
6.4
13.0
393
Northridge
USA
17/1/1994
USC no. 0056
N46E
0.411
6.4
21.5
394
Northridge
USA
17/1/1994
USC no. 0057
N00W
0.389
6.4
25.6
395
Northridge
USA
17/1/1994
USC no. 0091
S70E
0.420
6.4
24.9
396
Northridge
USA
17/1/1994
USC no. 0095
N90W
0.186
6.4
42.9
397
Lefkada
Greece
25/2/1994
Lefkada
L
0.171
5.8
16.0
398
Lefkada
Greece
25/2/1994
Lefkada
T
0.197
5.8
16.0
399
Kobe
Japan
17/1/1995
Kobe
EW
0.619
6.9
1.0
400
Kobe
Japan
17/1/1995
Kobe
NS
0.821
6.9
1.0Table 3
Number of accelerograms employed per country.
Country
Number of accelerograms
Bulgaria
2
Canada
9
Chile
12
Greece
54
Japan
46
Mexico
10
New Zeeland
2
Romania
4
San Salvador
6
Turkey
8
USA
247Table 4
Number of accelerograms employed per PGA range.
PGA range (g)
Number of accelerograms
0.01–0.1
23
0.1-0.2
165
0.2-0.3
103
0.3-0.4
50
0.4-0.5
32
0.5-0.6
13
0.6-0.7
7
>0.7
7Figure 1
Plot of standard seismic magnitudes versus epicentral distances.The number of the used accelerograms is essential for the accuracy of the statistical models. In addition, the utilized accelerograms must generate a whole spectrum of damage (low, medium, large, and total) for statistical completeness. However, regional (locally possible) seismic records do not exist in sufficient quantity for most of the seismically active areas. Therefore, a set of worldwide natural acceleration records has been utilized for the present study. Alternatively, artificial accelerograms compatible with the design spectrum, combined with a scaling procedure (to generate a whole spectrum of damage), could be employed. In that case, the variability of the frequency content is limited.


## 4. Damage Indices

Among the many structural response parameters, attention is focused on those that can best describe seismic damage. The attention is focused on overall structural damage indices (OSDIs) because these parameters summarily lump all existing damage in columns and beams into a single value, which can then be easily interrelated with the single-value seismic parameters. For this purpose, the modified overall damage index of Park and Ang [19, 20] is used in the present study. In this model, the global damage is obtained as a weighted average of the local damage at the ends of each element, with the dissipated energy as the weighting function. The local damage index according to Park and Ang must be calculated using the following equation:
(14)
DI
P
/
A
,
Local
=
θ
m
-
θ
r
θ
u
-
θ
r
+
β
M
y
θ
u
E
T
,
where θ
m is the maximum rotation during the load history, θ
u is the ultimate rotation capacity of the section, θ
r is the recoverable rotation at unloading, β is a constant parameter (0.1–0.15 for nominal strength deterioration [21]), M
y is the yield moment of the section, and E
T is the dissipated hysteretic energy.In this study, the numerical value of parameterβ in (14) is equal to 0.1. This value corresponds to nominal strength degradation [21]. However, the numerical value of β can vary in the range between 0 (no strength degradation) and 0.4 (severe deterioration) [21]. A low β value corresponds to well-detailed reinforced concrete or steel members [22]. In contrast, a high β value corresponds to poorly detailed reinforced concrete members or to unreinforced masonry [20].The OSDI of Park and Ang is defined by the following equation:(15)
DI
P
/
A
,
Global
=
∑
i
=
1
n
DI
P
/
A
,
Local
E
i
∑
i
=
1
n
E
i
,
where E
i is the energy dissipated at location i and n is the number of locations at which the local damage is computed.The maximum interstory drift ratio (MISDR) has been selected as a second OSDI. It is defined as the maximum interstory drift (u
max
⁡) normalized by the story height (h), as given by the relation
(16)
MISDR
=
|
u
|
max
⁡
h
100
[
%
]
.The maximum softening index of DiPasquale and Cakmak [23] has been selected as a third OSDI, which is based on the vibration parameters of the structure. It is given by the following expression:
(17)
δ
M
=
1
-
T
0
T
max
⁡
,
where δ
M is the maximum softening, T
0 is the fundamental period, and T
max
⁡ is the maximum natural period of the examined structure during the excitation. For the evaluation of T
max
⁡ the instantaneous natural period is required to be evaluated, which is accompanied by the actual time-dependent (instantaneous) tangent stiffness matrix. The latter varies with time due to nonlinear phenomena. The natural period computed for each time step of a nonlinear dynamic analysis presents high variability. The duration of these maximum values is very short, and therefore, their influence in the natural period of an equivalent linear system is not significant. Thus, a more meaningful indication of the change in the natural period can be obtained by observing a moving average of the instantaneous natural period using a sliding time window for the smoothness of the time-fundamental period curve [23, 24]. The time-fundamental period curve can be evaluated by a nonlinear dynamic procedure that calculates the fundamental period of the structure by considering the stiffness degradation in every time step. T
0 is the initial, fundamental period of the structure that corresponds to the initial stiffness of the structure.Table5 presents the range limits of the four damage classes using the different damage indices [25]. The four levels (low, medium, large, and total) correspond to undamaged or minor damage, repairable damage, irreparable damage, and partial or total collapse of the structure.Table 5
Damage classification limits.
Damage classes
Low
Medium
Large
Total
OSDIs of Park and Ang(
DI
P
/
A
,
Global
) and of DiPasquale and Cakmak (
δ
M
)
≤0.3
0.3 < OSDI ≤ 0.6
0.6 < OSDI ≤ 0.8
>0.8
MISDR (%)
≤0.5
0.5 < MISDR ≤ 1.5
1.5 < MISDR ≤ 2.5
>2.5


## 5. Application

The proposed procedures have been applied to a reinforced concrete structure, shown in Figure2, and designed in agreement with the rules of the recent Eurocodes for structural concrete and aseismic structures, EC2 and EC8 [26, 27]. The fundamental period of the frame was 1.20 s. The following loads have been considered: self-weight, seismic loads, snow, wind, and live loads. The cross-sections of the beams are treated as T-beams with a 40 cm width, 20 cm plate thickness, 50 cm total beam height, and 1.45 m effective plate width. The distances between each frame of the structure have been chosen to be 6 m. According to the EC8 Eurocode, the structure shown in Figure 2 is considered an “importance class II, ductility class M” structure with category B subsoil. The ordinate of the used elastic response spectrum of EC8 was 0.24 g.Figure 2
Examined reinforced concrete frame structure.After designing and detailing the reinforced concrete frame structure, nonlinear dynamic analyses were carried out to evaluate the structural seismic response. For this purpose, the computer program IDARC [21] has been used. The hysteretic behavior of the beams and columns has been specified at both ends of each member using a three-parameter Park model [21]. This hysteretic model incorporates stiffness degradation, strength deterioration, nonsymmetric response, slip-lock, and a trilinear monotonic envelope. The parameter values that specify the above degrading parameters have been chosen from the cyclic force-deformation characteristics from experimental results of the typical components of the studied structure [20]. Thus, nominal parameters for stiffness degradation and strength deterioration have been chosen. In contrast, no pinching has been considered. Realistic hysteretic material models result in realistic proposed statistical models. On the other hand, simplified assumptions for the hysteretic material models, like ignoring the pinching effect (default option in the utilized computer program IDARC [21]), do not affect on principle the value of the proposed statistical models that are in any case procedures for the approximate assessment of structural damage. Of the several response parameters, the focus is on the OSDIs. Consequently, the OSDIs of Park and Ang and of DiPasquale and Cakmak (maximum softening index) and the MISDR have been calculated for all of the accelerograms corresponding to the excitations given in Table 2, which were used as the seismic input for the nonlinear dynamic analyses. For the evaluation of the global damage index of DiPasquale and Cakmak according to (17), a sliding time window of 2.5 s is used to smooth the time-fundamental period curve. This window size is between 2 and 5.5 times the fundamental period T
0 of the examined structure (T
0 = 1.20 s), as suggested by Rodriguez-Gómez/Cakmak [24]. The maximum value of the smoothed time-fundamental period curve is the maximum natural period of the examined structure during each seismic excitation considered (T
max
⁡ in (17)). Finally, the hardening period T
h and yield period T
y needed for the assessment of the SIMR in (8) are evaluated by a nonlinear pushover analysis, as described in Section 2. Their numerical values are T
y
=
1.2 s and T
h
=
1.5 s.


## 6. Multilinear Regression Analysis

In general, a multilinear regression analysis [28] is used to express a dependent variable (y) by (n)-independent variables (x
i, with i
=
1
,
…
,
n) and a constant value (a) minimizing the error term (e), as shown by the relation
(18)
y
=
a
+
b
1
x
1
+
b
2
x
2
+
⋯
+
b
n
x
n
+
e
.In the present study, the dependent variable (y) represents each used OSDI (of Park and Ang, of DiPasquale and Cakmak (maximum softening index), and the MISDR), and the independent variables (x
i) are the twenty seismic parameters (n
=
20) shown in Table 1.The analyses have been carried out using the statistical software STATGRAPHICS [29]. The regression results for the OSDIs after Park and Ang, for the MISDR, and for the maximum softening index after DiPasquale and Cakmak are given in Tables 6–9 and will be presented in detail subsequently. Thus, the result for the OSDI after Park and Ang is given in Table 6. The equation of the fitted multilinear model is(19a)
DI
P
/
A
,
Global
=
-
0.032
+
0.370
PGA
+
0.084
PGV
+
0.005
PGA
PGV
+
0.007
Arias
-
0.074
RMS
a
-
0.0004
T
0.90
-
0.004
P
0.90
+
0.036
SI
H
-
0.014
SI
K
-
0.379
SI
MR
-
0.385
EPA
+
0.076
EPA
max
⁡
+
0.037
E
inp
-
0.020
CAV
-
0.202
DP
A
/
S
+
0.038
CP
-
0.110
SD
+
0.167
SV
+
0.031
SA
+
0.017
I
F
/
V
/
F
.Table 6
Multilinear regression model for the OSDI of Park and Ang.
Parameter
Estimate
Lower limit
Upper limit
Constant
−0.032
−0.063
−0.001
PGA
0.370
0.234
0.507
PGV
0.084
−0.030
0.199
PGA/PGV
0.005
−0.004
0.014
I
0
0.007
0.004
0.009
RMSa
−0.074
−0.156
0.008
T
0.90
0.0004
−0.001
0.001
P
0.90
−0.004
−0.015
0.006
SIH
0.036
−0.004
0.077
SIK
−0.014
−0.127
0.099
SIMR
−0.379
−0.543
−0.215
EPA
−0.385
−0.652
−0.118
EPAmax
0.076
0.004
0.148
E
inp
0.037
0.013
0.061
CAV
−0.020
−0.052
0.011
DP
A
/
S
−0.202
−0.437
0.032
CP
0.038
−0.022
0.098
SD
−0.110
−0.428
0.207
SV
0.167
0.074
0.260
SA
0.031
−0.092
0.154
I
F
/
V
/
F
0.017
−0.018
0.051Table 7
Multilinear regression model for the OSDI of Park and Ang without the constant term.
Parameter
Estimate
Lower Limit
Upper Limit
PGA
0.324
0.195
0.454
PGV
0.072
−0.042
0.186
PGA/PGV
−0.001
−0.008
0.005
I
0
0.007
0.005
0.009
RMSa
−0.095
−0.175
−0.016
T
0.90
−0.001
−0.002
0.000
P
0.90
−0.003
−0.014
0.007
SIH
0.028
−0.012
0.068
SIK
−0.010
−0.123
0.104
SIMR
−0.337
−0.496
−0.178
EPA
−0.370
−0.637
−0.102
EPAmax
0.076
0.003
0.148
E
inp
0.043
0.019
0.067
CAV
−0.020
−0.052
0.012
DP
A
/
S
−0.202
−0.437
0.034
CP
0.008
−0.044
0.060
SD
−0.086
−0.403
0.232
SV
0.154
0.062
0.247
SA
0.045
−0.077
0.168
I
F
/
V
/
F
0.016
−0.018
0.051Table 8
Multilinear regression model for the MISDR.
Parameter
Estimate
Lower limit
Upper limit
Constant
−0.072
−0.290
0.147
PGA
2.331
1.380
3.281
PGV
0.359
−0.438
1.156
PGA/PGV
0.046
−0.017
0.109
I
0
0.043
0.028
0.058
RMSa
−0.322
−0.893
0.249
T
0.90
0.003
−0.004
0.010
P
0.90
−0.019
−0.089
0.052
SIH
0.410
0.129
0.691
SIK
−0.520
−1.308
0.268
SIMR
−2.726
−3.864
−1.587
EPA
−0.873
−2.731
0.985
EPAmax
0.019
−0.483
0.522
E
inp
−0.048
−0.216
0.120
CAV
−0.397
−0.619
−0.175
DP
A
/
S
1.475
−0.155
3.104
CP
−0.167
−0.583
0.250
SD
−2.292
−4.501
−0.084
SV
1.802
1.155
2.449
SA
0.236
−0.619
1.090
I
F
/
V
/
F
0.024
−0.217
0.264Table 9
Multilinear regression model for the OSDI of DiPasquale and Cakmak.
Parameter
Estimate
Lower limit
Upper limit
Constant
−0.090
−0.130
−0.050
PGA
0.293
0.119
0.466
PGV
0.094
−0.052
0.239
PGA/PGV
0.017
0.005
0.028
I
0
0.001
−0.002
0.004
RMSa
−0.035
−0.140
0.069
T
0.90
0.0002
−0.001
0.001
P
0.90
−0.005
−0.018
0.008
SIH
0.019
−0.032
0.071
SIK
0.237
0.093
0.380
SIMR
−0.178
−0.386
0.030
EPA
−0.288
−0.627
0.052
EPAmax
0.027
−0.065
0.118
E
inp
0.081
0.050
0.112
CAV
−0.032
−0.072
0.009
DP
A
/
S
−0.656
−0.953
−0.358
CP
0.097
0.021
0.173
SD
0.648
0.245
1.052
SV
−0.079
−0.197
0.039
SA
−0.040
−0.196
0.116
I
F
/
V
/
F
0.011
−0.032
0.055R
2
=
0.859, which indicates that the model as fitted explains 85.9% of the variability in the Park and Ang OSDI. In addition, Table 6 presents the 95% confidence intervals for the coefficients in the multilinear model. Each variable coefficient in (19a) and in Table 6, or generally in (18), is interpreted as the change in the response based on a one-unit change in the corresponding explanatory variable keeping all other variables fixed. This interpretation is fictitious because it is not possible in a seismic accelerogram to change only one of the seismic parameters. Furthermore, in some explanatory variables (seismic parameters), a one-unit change can be easily observed (such as the SMD after Trifunac/Brady), whereas such a change cannot be easily observed in others (such as the SD). In addition, the comparison between coefficients of different explanatory variables is not possible because they are assigned quantities having different dimensions and units [30]. Finally, the constant term (“a” in (18) and “−0.032” in (19a)) has a physical meaning only in the case in which all of the explanatory variables x
i can simultaneously have zero values. Otherwise, as in the present study in which the seismic parameters with zero values are out of the observed value range, the constant term is an extrapolated value without a physical meaning. However, Table 7 presents the coefficients and their 95% confidence intervals for the multilinear regression model of the OSDI after Park and Ang, without the constant term (R
2 = 0.900218). As indicated by R
2, both models (with and without the constant term) explain a high percentage of the variance (85.9% and 90%, resp.) in the dependent variable.Similarly, Table8 presents the multilinear model for the MISDR (R
2 = 0.799279) and Table 9 presents the model for the OSDI (maximum softening index) after DiPasquale and Cakmak (R
2 = 0.763627), along with their 95% confidence intervals. The fitted multilinear models for these global structural damage indices are provided by(19b)
DI
P
/
A
,
Global
=
-
0.072
+
2.331
PGA
+
0.359
PGV
+
0.046
PGA
PGV
+
0.043
Arias
-
0.322
RMS
a
+
0.003
T
0.90
-
0.019
P
0.90
+
0.410
SI
H
-
0.520
SI
K
-
2.726
SI
MR
-
0.873
EPA
+
0.019
EPA
max
⁡
-
0.048
E
inp
-
0.397
CAV
-
1.475
DP
A
/
S
-
0.167
CP
-
2.292
SD
+
1.802
SV
+
0.236
SA
+
0.024
I
F
/
V
/
F
,
(19c)
DI
P
/
A
,
Global
=
-
0.090
+
0.293
PGA
+
0.094
PGV
+
0.017
PGA
PGV
+
0.001
Arias
-
0.035
RMS
a
+
0.0002
T
0.90
-
0.005
P
0.90
+
0.019
SI
H
+
0.237
SI
K
-
0.178
SI
MR
-
0.288
EPA
+
0.027
EPA
max
⁡
+
0.081
E
inp
-
0.032
CAV
-
0.656
DP
A
/
S
+
0.097
CP
+
0.648
SD
-
0.079
SV
-
0.040
SA
+
0.011
I
F
/
V
/
F
.The models, represented by the regression equations, are strictly valid only within the range of the data (in the present study these data are the values of the seismic parameters employed) used to develop these regression equations (training phase). Trying to predict the outcome outside the range of the data can be seriously misleading and is not advised.The initial set of explanatory variables (see (19a), (19b), and (19c)) contains a relative large number of seismic intensity parameters. Thus, in the present study twenty seismic intensity quantities have been used (Table 1) to avoid an a priori omission of important, independent (explanatory) seismic parameters. A regression model with only a few explanatory variables (e.g., two to three accidentally selected parameters) runs the risk of becoming an unreliable or invalid regression model. However, a statistically systematic reduction of the initial regression model is possible. This reduction has been realized in the present study by a stepwise backward elimination procedure combined with an elimination criterion.Thus, the multilinear regression model, which contains the twenty examined seismic parameters (Model 1), can be simplified to consider fewer parameters without losing significant model quality. This simplification is realized by successive elimination of the independent variables with aP value greater than or equal to 0.05 (elimination criterion) because that term is not statistically significant at the 95% or higher confidence level. This lack of significance is explained by the strong correlation between several seismic parameters.TheP value is the smallest level of significance that would lead to rejection of the null hypothesis H
0 with the given data. In the case of multilinear regression analysis, the null hypothesis (H
0) and alternative hypothesis (H
a
) are H
0
:
b
i
=
0 and H
a
:
b
i
≠
0, where b
i are the coefficients of the explanatory variables in (18) [30]. Thus, the importance to keep a seismic intensity parameter in the regression model is determined by the previously described explicit statistical procedure.As an example, Figure3 presents the P values for all of the terms of regression equation (19a) (Table 6) in ascending order. The SI after Kappos parameter has the maximum P value (0.809) which is greater than 0.05. Therefore, this specific parameter can be eliminated from the considered model. Subsequently, a new regression equation is evaluated without the Kappos SI parameter in its explanatory variables, and the corresponding P values are different from those of the initial equation (19a) (with all of the seismic parameters considered initially). In the new regression equation the explanatory variable (seismic parameter), with the maximum P value greater than 0.05, will be eliminated. This elimination procedure is repeatedly applied until all of the P values of the explanatory variables remain less than 0.05. Applying the described procedure to (19a), the sequence of the eliminated parameters is SI after Kappos, SA, SD, SMD, I
F
/
V
/
F, P
0.90, CP, DP
A
/
S, and PGA/PGV. The result of this procedure is the reduced regression model shown in Table 10. It is not recommended to eliminate all of the explanatory variables that have a P value greater than 0.05 in Figure 3 at once from the regression model in (19a) because the P values are different in the case of stepwise elimination of explanatory variables and in the case of their at-once elimination.Table 10
Reduced multilinear regression model for the OSDI of Park and Ang.
Parameter
Estimate
Lower limit
Upper limit
Constant
−0.015
−0.031
0.001
PGA
0.367
0.235
0.500
PGV
0.111
0.017
0.206
I
0
0.007
0.005
0.009
RMSa
−0.082
−0.149
−0.014
SIH
0.037
0.003
0.072
SIMR
−0.354
−0.496
−0.212
EPA
−0.430
−0.683
−0.176
EPAmax
0.086
0.016
0.157
E
inp
0.023
0.012
0.035
CAV
−0.028
−0.046
−0.009
SV
0.166
0.120
0.213Figure 3
P values in ascending order for all of the terms of regression equation (19a).Thus, in accordance with the proposed stepwise elimination procedure with aP value greater than 0.05 criterion, Tables 10, 11, and 12 show the reduced multilinear models (Model 2) for the OSDI of Park and Ang (R
2 = 0.856131), the MISDR (R
2 = 0.795159), and the OSDI (maximum softening index) of DiPasquale and Cakmak (R
2 = 0.757862) with their respective 95% confidence intervals.Table 11
Reduced multilinear regression model for the MISDR.
Parameter
Estimate
Lower limit
Upper limit
Constant
0.003
−0.104
0.110
PGA
2.217
1.309
3.126
I
0
0.039
0.029
0.050
SIH
0.190
0.041
0.339
SIMR
−2.542
−3.393
−1.691
CAV
−0.353
−0.475
−0.231
DP
A
/
S
0.974
0.281
1.668
SD
−2.431
−4.541
−0.320
SV
1.836
1.413
2.259Table 12
Reduced multilinear regression model for the OSDI of DiPasquale and Cakmak.
Parameter
Estimate
Lower limit
Upper limit
Constant
−0.092
−0.126
−0.059
PGA
0.289
0.131
0.448
PGA/PGV
0.015
0.005
0.024
SIK
0.191
0.073
0.310
SIMR
−0.243
−0.398
−0.087
E
inp
0.085
0.063
0.107
CAV
−0.021
−0.038
−0.005
DP
A
/
S
−0.743
−0.995
−0.491
CP
0.121
0.051
0.190
SD
0.468
0.144
0.793The independent variables in the reduced regression model are selected by the proposed statistical procedure primarily to predict the damage indicator and not to explain it physically. Therefore, it is not unexpected that different seismic parameters are used for the adequate prediction of different damage indices (Tables10–12).The multilinear regression is a robust procedure, even for the case in which two or more explanatory variables are linearly dependent or highly correlated. Perfect multicollinearity occurs when two or more explanatory variables in the multilinear regression are linearly dependent. This multicollinearity type is rare. Imperfect multicollinearity occurs when two or more regressors are highly correlated. Perfect and imperfect multicollinearity does not affect the overall predictive power or reliability of the regression model. However, multicollinearity does reduce the effectiveness of a regression analysis if its primary purpose is to determine the specific effects of the various independent variables. The goal of the present study is to predict the dependent variable (damage indices) from a set of independent variables (seismic intensity parameters); therefore, multicollinearity does not affect the quality of the proposed multilinear regression model. Nevertheless, the proposed stepwise backward elimination procedure reduces the number of independent variables and consequently provides models with less multicollinearity. An a priori omission of several independent variables includes the risk of becoming an untrustworthy model.


## 7. Testing the Multilinear Regression Model

To verify the quality of the full (Model 1) and the reduced (Model 2) regression models, in general, it is sufficient that the validation data set is a fraction of the training data set. In the present study, the validation data consists of 410 accelerograms, separated into two sets. The first one is identical to the 400 accelerograms used for the training phase of the regression model, as shown in Table2. The second set is from an entirely different set of 10 artificial accelerograms. In their original form, the latter are compatible with the design spectrum employed. Thus, set 1 is used as internal and set 2 as external validation set. This is common validation procedure in statistical analyses. Two utilized artificial accelerograms are presented in Figure 4. Because the examined frame structure is designed using the same spectrum to which the artificial accelerograms are compatible, the expected damage grade will be low. However, the proposed statistical models must also be verified for medium, large, and total damage grades. Therefore, some of the artificial accelerograms have been scaled (the ordinates of the considered seismic accelerogram have been multiplied by a factor with a value greater than one) to achieve medium, large, and total damage grades. Thus, some of these accelerograms have been scaled to produce medium, large, and total structural damages for each of the examined damage indicators. The program SIMQKE [31] has been utilized to create the aforementioned artificial accelerograms. The method used by the program for the artificial seismic accelerogram generation is the superposition of sinusoids having random phase angles and amplitudes derived from a stationary power spectral density function of the motion. The produced signals are then enveloped in a trapezoidal shape to simulate the nonstationary characteristics of the ground motion [31]. The following input data are required for the generation of the artificial accelerograms: peak ground acceleration (PGA), total duration (TD), duration of the ascending and the descending parts of the trapezoidal envelope, and design spectrum with which the artificial accelerograms must be compatible. Another option is to use natural accelerograms instead of the artificial accelerograms to verify the proposed models. Scaling an accelerogram to a greater PGA value is a simple procedure to produce accelerograms with more energy content and damage potential. The seismic intensity parameters have been evaluated also for all the utilized scaled accelerograms because they change with the scaling process.Figure 4
Two utilized artificial accelerograms.Table13 presents the prediction results of all of the examined damage indicators. In this context, correct prediction means that the evaluated damage index lies within the 95% confidence interval of the multilinear regression model, as provided in Tables 6–12. Thus, the results of Model 1 display correct predictions, from 94.75% to 97.50% and from 80% to 90% for acceleration sets 1 and 2, respectively. In addition, Figure 5 shows the regression line and the 95% confidence interval of the correct versus predicted Park and Ang OSDI values. In this case the coefficient of determination R
2 = 85.9%. This indicates that the model as fitted explains 85.9% of the variability in Park and Ang damage index. Furthermore, the percentages of correct prediction for Model 2 are from 95% to 97% and from 70% to 90% for acceleration sets 1 and 2, respectively. Model 2, with its reduced number of seismic parameters in the regression model, is not of poorer quality than Model 1, and the results of acceleration set 2 are satisfactory, particularly for the OSDI of Park and Ang and the MISDR. Finally, Table 14 shows the Pearson and the Spearman rank correlation coefficients between the correct and predicted values of all examined OSDIs based on regression analyses. The numerical results show a high correlation in all cases.Table 13
Damage prediction results for the regression models.
Regression
OSDI
Correctly predicted damage values
Set 1
Set 2
400 accelerograms
10 accelerograms
Model 1
DI
P
/
A
,
Global
386 (96.50%)
9 (90%)
MISDR
390 (97.50%)
8 (80%)
δ
M
379 (94.75%)
8 (80%)
Model 2
DI
P
/
A
,
Global
385 (96.25%)
9 (90%)
MISDR
388 (97.00%)
8 (80%)
δ
M
380 (95.00%)
7 (70%)Table 14
Correlation coefficients between correct and predicted OSDI values based on regression analyses.
Regression
OSDI
Correlation coefficients
Set 1
Set 2
400 accelerograms
10 accelerograms
Pearson correlation
Spearman rank correlation
Pearson correlation
Spearman rank correlation
Model 1
DI
P
/
A
,
Global
0.927
0.912
0.975
0.953
MISDR
0.894
0.921
0.979
0.958
δ
M
0.874
0.841
0.962
0.833
Model 2
DI
P
/
A
,
Global
0.925
0.913
0.977
0.953
MISDR
0.892
0.933
0.982
0.967
δ
M
0.871
0.846
0.982
0.827Figure 5
Regression line and the 95% confidence interval of the correct versus predicted Park and Ang OSDI values.


## 8. Discriminant Analysis

In the case for which it is sufficient to know the Postseismic damage grade as provided in Table5 (low, medium, large, and total), then a statistical discriminant analysis can be performed. The purpose of discriminant analysis is to classify objects into one or more possible groups based on a set of features that describe the objects [32]. In general, we assign an object to one of a number of predetermined groups based on observations made about the object. The discriminant analysis creates an equation which will minimize the possibility of misclassifying cases into their respective groups or categories.The principle is to find the discriminant functions for each group. The element to be classified belongs to the group with the greatest value of the discriminant functions. In the present study, four groups and twenty independent parameters are used. Thus, the discriminant functions are given by the relations(20)
F
1
=
a
0,1
+
a
1,1
X
1,1
+
a
2,1
X
2,1
+
⋯
+
a
20,1
X
20,1
,
F
2
=
a
0,2
+
a
1,2
X
1,2
+
a
2,2
X
2,2
+
⋯
+
a
20,2
X
20,2
,
F
3
=
a
0,3
+
a
1,3
X
1,3
+
a
2,3
X
2,3
+
⋯
+
a
20,3
X
20,3
,
F
4
=
a
0,4
+
a
1,4
X
1,4
+
a
2,4
X
2,4
+
⋯
+
a
20,4
X
20,4
,
where F
i are the discriminant functions, a
j
,
i are parameters to be specified, and X
j
,
i are known quantities. Here, i = 4 (four damage grade groups) and j = 20 (twenty seismic parameters). The discriminant analysis has a direct analytical solution and therefore, calculations are very fast. The produced models are concise and easily programmed. On the other hand, the discriminant analysis is sensitive to outliers.


## 9. Testing the Discriminant Analysis Model

To verify the quality of the discriminate analysis, the same two sets of accelerograms from the regression analysis have been used. Table15 presents the prediction results of all of the examined damage indicators. In this context, correct prediction means that the damage grade estimate by the discriminant analysis agrees with the corresponding evaluation of the nonlinear dynamic analysis. The results display correct predictions from 87.75% to 97.50% and from 70% to 90% for acceleration sets 1 and 2, respectively. Figure 6 shows the correct versus predicted damage grade based on the Park and Ang OSDI values. The points out of the diagonal line show the incorrect classified cases. Thus, 9 cases have been classified in damage group 2 (medium damage) instead of the correct damage group 1 (low damage) and 1 case has been classified in damage group 2 (medium damage) instead of the correct damage group 4 (total damage). The points in the diagonal line show the correct classified cases (390 cases, 97.5%). The corresponding discriminating functions have been evaluated with P values less than 0.05 and are statistically significant at the 95.0% confidence level. Thus, all points in Figure 6 are inside the 95.0% confidence interval.Table 15
Damage prediction results for the discriminant analysis.
Correctly predicted damage grade
OSDI
Set 1
Set 2
400 accelerograms
10 accelerograms
DI
P
/
A,Global
390 (97.50%)
7 (70%)
MISDR
351 (87.75%)
8 (80%)
δ
M
374 (93.50%)
9 (90%)Figure 6
Discriminant analysis results. Correct versus predicted damage grade based on the Park and Ang OSDI values.


## 10. Remarks on the Proposed Statistical Procedures

The proposed regression models (Tables6–12) and discriminant analysis model are clearly tailored to the examined reinforced concrete frame structure. For another structure, the nonlinear dynamic analyses should be repeated to determine the coefficients in the regression models (Tables 6–9) and in the discriminant analysis model. Although the proposed procedures are applied to a two-dimensional reinforced concrete frame as an example, they could be applied to any two-dimensional frame and expanded to three-dimensional frames for other building materials (e.g., steel, masonry, timber) or to other building types (e.g., bridges, shell structures). Furthermore, the set of twenty seismic intensity parameters considered here could be extended with additional quantities or partially or totally replaced by other parameters. However, the proposed procedure remains the same regardless of the structure examined and seismic intensity parameters employed. Thus, in the first step, a set of seismic intensity parameters is evaluated for a set of accelerograms. Then, nonlinear dynamic analyses furnish the damage indices of the considered structure. In the final step, the coefficients of the proposed statistical models (multivariate regression and discriminant analysis) are determined.The proposed procedures are simple and accurate tools for the estimation of the damage index of a building by a multivariate regression analysis and of its Postseismic damage grade by discriminant analysis. These multivariate statistical techniques do not replace the exact nonlinear dynamic analyses, which are necessary to determine the coefficients of the statistical models, but they do serve as a simple utility for the Postseismic damage status estimation of a specific building based on seismic intensity parameters. Evaluation of the coefficients in statistical models, in which nonlinear dynamic analyses are obligatory, must be performed by professional engineers. If the models have been defined, then they may feasibly be applied by nonspecialists. The presumption is that the seismic parameters employed are known or estimated. The nonlinear analyses and determination of the proposed statistical models are complicated and time-consuming numerical procedures that must be performed by professional engineers (e.g., this can be realized in connection with the design procedure of a considered new building and the evaluated statistical models can be announced to the building owner and the housing authority). The statistical models are determined by the evaluation of the coefficients in (18) and (20) for the regression and discriminant models, respectively. These statistical models can be easily and quickly applied once they are resolved. They could also be used by nonspecialists under the assumption that the seismic intensity parameters for an actual seismic event are provided by the responsible authorities (e.g., seismological institutes). The proposed methods convert the complicated dynamic nonlinear models to black boxes that take the ground motion information and predict the Postseismic damage levels. Thus, as mentioned in the introduction section, the proposed procedures can be applied to damage scenarios of important buildings (e.g., hospitals, schools, bridges, silos, and hangars) by public authorities. The proposed statistical models are faster and easier to use than the nonlinear dynamic analyses. Epigrammatic, nonlinear dynamic analyses can be utilized only by specialists, while on the other hand, the proposed statistical models can be utilized also by nonspecialists. Thus, nonspecialists can approximately evaluate the Postseismic damage status of buildings, ignoring the exact damage values provided by nonlinear dynamic analysis. The exact damage value needs not to be known once the statistical models are resolved. Furthermore, the proposed methods can handle probable uncertainties in the seismic ground motion and variability in structural or material properties by parametric analyses or by their combination with fuzzy logic procedures (fuzzy regression analysis). Finally, the proposed method can also be applied to a group of buildings. In that case, the damage parameters must be appropriately tailored to describe the Postseismic damage status and the proposed statistical procedures can be applied once on the group and not repeatedly on separate buildings. A proper damage status description could be a financial damage index [25].Alternative procedures for seismic damage potential classification are presented in the literature and are based on intelligent techniques. Thus, fuzzy logic, artificial neural network, neurofuzzy and adaptive neurofuzzy inference systems, and support vector machines techniques have been used for seismic damage potential classification. The efficiency of the intelligent techniques was expressed by the correct classification rates of a set of accelerograms that were not used during the training process. The intelligent techniques provided a correct classification rate between 71% and 97.5%. The results depend on the used procedure, the overall structural damage indices (MISDR orDI
P
/
A
,
Global), and the number of training samples [33]. The intelligent techniques in comparison with the proposed statistical procedures have no significant better correct classification rates and are not appropriate to be used by nonspecialists.


## 11. Conclusions

Two statistical methods have been presented for the evaluation of the Postseismic damage status of buildings based on the parameters of the seismic excitation. The first is multilinear regression analysis, and the second is discriminant analysis. Twenty seismic parameters have been extracted from the accelerograms employed. The overall damage index of Park and Ang, the MISDR, and the maximum softening of DiPasquale and Cakmak have been used to describe the damage status of the structures. The multilinear regression has been used for the explicit evaluation of the damage indices. In contrast, the discriminant analysis furnished the Postseismic damage grade of the structures. These statistical procedures, combined with seismic intensity parameters, are novel methodologies in earthquake engineering and advantageous for the estimation of the damage indices and damage grades of buildings after severe seismic excitations.The proposed statistical methods have been applied to an eight-story reinforced concrete frame structure designed in accordance with the rules of the EC2 and EC8 Eurocodes for reinforced concrete and antiseismic structures, respectively. A set of 400 natural accelerograms has been applied for the training phase of the models. In addition to the first set, a second set of synthetic accelerograms has been used to verify the statistical methods. The numerical results have shown that 94.75–97.50% and 70–90% of the first and second sets of accelerograms, respectively, were correctly classified in accordance with their damage potential for the examined structure within the confidence interval provided by the regression analysis. In addition, the regression model with a reduced number of considered seismic parameters is not of poorer quality than the model that takes all twenty seismic parameters into account. In contrast, the damage grades for 87.75–97.50% and 70–90% of the first and second sets of accelerograms, respectively, were correctly predicted by the discriminant analysis. Thus, these results led to the conclusion that the multilinear regression analysis and discriminant analysis are useful tools for the prediction of the Postseismic damage status when the seismic parameters are known. Finally, based on the regression and the discriminant analysis results of the 10 examined artificial accelerograms (set 2 in Tables13 and 14), it can be suggested to use the Park and Ang OSDI and the maximum softening index of DiPasquale and Cakmak as appropriate Postseismic damage value and damage grade estimators of a building, respectively.



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*Source: 916820-2014-02-23.xml*