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1
Effect of Column to Beam Plastic Modulus Index on
Seismic Damage Mechanism of Steel Moment Frames
Arman Kamalzadeha, Reza Karami Mohammadib*
a PhD Candidate, Civil Engineering Department, University of Auckland, Auckland, New Zealand.
bAssociate Professor, Civil Engineering Department, K.N. Toosi University of Technology, Tehran,
Iran.
*Corresponding Author: rkarami@kntu.ac.ir
Abstract
In order to secure adequate ductile behavior and high seismic damage capacity of steel mo-
ment frames during an earthquake, AISC 341 represents a strength criterion. According to
this criterion, the moment capacity of columns must be more than the moment capacity of
beams in each joint (Strong Column/ Weak Beam concept). In this study, nonlinear time his-
tory analyses have been carried out on several steel moment frames in which, frame design is
based on plastic modulus ratio of column to beam section. Modified Park and Ang damage
index has been utilized as seismic damage indicator. It is concluded for satisfying SC/WB
concept, plastic modulus ratio approximately should be in the range of 1.1 to 1.3 for different
frame designs and configurations. This design could consider more uniform distribution of
seismic energy in structure and could be simpler for assigned as a rule of thumb for prelimi-
nary design of beams and columns in steel moment frames.
Keywords: structural strength; plastic section modulus; damage mechanism; strong column/
weak beam; ductile structural performance
1. Introduction
First generation of building design standards only represented design criteria for structural el-
ements under gravitational loads. Afterwards, cases with serious seismic damages in buildings and
immense casualties such as Italy 1908 Messina-Reggio, Japan 1923 Kanto and USA 1925 Santa
Barbara earthquakes [1], led to formation of UBC code [2].
The design philosophy on the basis of element strength had been dominantly used until 1994
Northridge and 1995 Kobe ground motions. Despite the fact that, the existed buildings guaranteed
the life safety of residents, yet the financial losses were heavy. These incidents shed light on a new
philosophy called “Performance Based Seismic Design”. The first codes and standards were found-
ed and broadly utilized on this new philosophy are SEAOC Vision 2000 [3], ATC 40 [4] and FE-
MA 273 [5].
In recent steel moment frame codes, mostly structure designing is element strength based at
first step. Then, global and local deformations will be checked to meet the desired performance. In
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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order to design a desired ductile frame, seismic damage must be concentrated in beams rather than
columns. Therefore, AISC 341 [6] presents the following strength criterion:
(1)
In which, and are the sum of moment capacity of columns and beams in a joint,
respectively. According to AISC 341 when Eq. 1 is satisfied in a joint, the earthquake damage
should be accumulated mostly in beams and the ductile behavior of frame will be secured.
In this research, a simple index is introduced on the basis of plastic modulus of structural sec-
tions to satisfy strong column/weak beam (SC/WB) concept. In order to assess the concurrence of
SC/WB more accurate than what is done before, a more convenient measurement of seismic dam-
age rather than structural deformation is required taking into account seismic energy dissipation of
structures. In recent decades, researchers presented several damage indices to represent seismic
damage of structures. These damage indices can be local or global. Some developed for local dam-
age Park and Ang [7], Bozorgnia and Bertero [8] and Kamaris et al. [9] and some for global indices
are Park et al. [10], Bracci et al [11] and Bojorquez et al. [12].
In the present study, modified Park and Ang damage index [13] is utilized which takes dissi-
pated seismic energy of the members into account. The effect of plastic modulus of column to beam
sections index (Section 2.2) on the state and distribution of seismic damage and the behavior of
frames is investigated. As a result, a numerical range for plastic modulus index is recommended for
meeting SC/WB along with AISC 341 criterion Eq. 1. At the end, contribution and the sequence of
possible plastic hinges in structural seismic damage of a nine-storey frame is illustrated to test the
efficiency of using plastic ratio index to satisfy SC/WB for investigated steel moment frames.
2. Method of the study
2.1 Modeling
This study investigates 2D, one-bay frames with one, two, three, five and nine floors. The bay
length and the floor height of each frame are 6 and 3.5 meters, respectively. Loading spans of the
frames are assumed to be 6 meters long from both sides perpendicular to the plane. Dead and live
loads are considered to be 400 and 150 kg/m2, respectively. Also, the participation factor of mass
during an earthquake is considered the sum of dead load and 25% of live load masses [6].
The beam and column sections are considered to be Stahlbou profiles [14] IPE and HEB, re-
spectively. In this research the discrete classification of Stahlbou sections are set to be continuous
regarding to plastic section modulus (Z) so that for instance a section can be chosen between HEB
270 and HEB 300. Therefore, plastic modulus (Z) is used as an indicator to identify section proper-
ties. The moment frame design is based on plastic modulus index (ZRatio), being discussed through
Section 2.2. At the first step, one, two and three-storey frames are designed for gravitational loads
while five and nine-storey ones are designed based on simple equivalent static analysis of IBC [15].
The results of this initial designs are summarized in Table 1.
Beam sections and as a result their plastic modulus (Zb) remained unchanged during analyses while,
column sections altered with application of plastic modulus ratio of column to beam section
( ). In addition, for five and nine storeys frames, the columns sections were not
constant in the height of frame (note “Column HEB” at Table 1). In Table 1 the beam sections are
presented while column sections are defined in terms of plastic modulus (Zc). The OpenSees [16],
an open source program for analyzing structures, has been utilized to model and analyze these
frames.
The behavior of steel is modelled by bilinear hysteretic material (uniaxialMaterial Hysteretic)
with the following properties.
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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Table 1. Initial Design of Steel Moment Frames
Table 2. Steel Properties
Ey (Pa)
Fy (Pa)
Rayleigh Damping
2.48e8
3.7e8
0.001143
5%
2.2 Frame Design
As it is mentioned in Section 2.1, the beams sections remained unaltered and designs changed
by using Eq. 2.
(2)
In which ZRatio index is the plastic modulus ratio of column to beam sections, Zc and Zb are
plastic modulus of column and beam section, respectively. While Zb is constant (since beam sec-
tions remained unchanged), by altering ZRatio, only Zc varies so and column section was changed.
The fundamental periods of frames for different ZRatio values of each frame are presented in Table 3.
Table 3. Fundamental Periods of Frames
Fundamental Period (s)
ZRatio
1 Storey
2 Storeys
3 Storeys
5 Storeys
9 Storeys
0.50
-
0.499
0.719
1.055
1.552
0.75
0.220
0.415
0.608
0.909
1.311
0.90
0.200
0.385
0.569
0.859
1.227
1.00
0.190
0.370
0.549
0.831
1.184
1.10
0.181
0.357
0.532
0.808
1.149
1.20
0.174
0.346
0.518
0.788
1.119
1.30
0.167
0.335
0.504
0.772
1.094
1.50
0.154
-
-
-
-
1.60
-
0.308
0.470
0.732
1.033
1.75
0.142
-
-
-
-
2.00
0.132
0.285
0.438
0.694
0.979
As can be seen from Table 2, there were nine designs for each type of frames, ranging from
more ductile frame design to stiffer frame design.
2.3 Earthquake Records
In this study, nonlinear time history analysis has been utilized using 12 far-field earthquake
acceleration records. These records are chosen from PEER [17] and ITACA [18] databases. At-
tempts have been carried out so that, the chosen records be destitute of any strong pulse. For this
purpose, with the aim of SeismoSignal [19], a signal processing program, the velocity record, power
spectra and fast Fourier spectra were obtained and the records that had no strong pulses (this type of
9
0.322Zc
IPE 450
8
0.322Zc
IPE 450
7
0.322Zc
IPE 450
6
0.559Zc
IPE 450
5
0.559Zc
IPE 550
0.6845Zc
IPE 450
4
0.559Zc
IPE 550
0.6845Zc
IPE 450
3
Zc
IPE 550
Zc
IPE 450
Zc
IPE 450
2
Zc
IPE 550
Zc
IPE 500
Zc
IPE 450
Zc
IPE 450
1
Zc
IPE 550
Zc
IPE 500
Zc
IPE 450
Zc
IPE 450
Zc
IPE 450
Storey No.
Column
HEB
Beam
Column
HEB
Beam
Column
HEB
Beam
Column
HEB
Beam
Column
HEB
Beam
Element
9
5
3
2
1
Storey
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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pulse represents near-field earthquakes) were chosen. Also, the classification of Mohraz [20] for
far-field region was considered to filter ground motions (further than 20 km from quake source). In
addition, for much more resemblance of the records, the classification of region soil was set to be C
( ) according to FEMA 450 [21]. The earthquake magnitude (Mw) and the peak
ground acceleration (PGA) of all the chosen records are more than 5.6 and 0.125g, respectively.
More characteristics of these records are shown in Table 4.
Table 4. Characteristics of Acceleration Records
Earthquake/Station
Component
Magnitude (Richter)
PGA (g)
ElCentro
180
6.95
0.313
Friuli 1st Shock
NSC
6.40
0.346
Friuli 3rd Shock
NSC
5.90
0.264
Friuli 4th Shock
NSC
5.90
0.349
Kern County
21
7.36
0.156
Irpinia Bagnoli Irpinio
NSC
6.90
0.129
Irpinia Brienza
NSC
6.90
0.218
Irpinia Sturno
NSC
6.90
0.225
Northridge
180
6.69
0.245
San Fernando
205
6.61
0.323
Sierra Madre
1155
5.61
0.302
Tabas
LN
7.35
0.836
For analysis in this research, the PGA values of records are scaled to cover the range of possi-
ble earthquakes exerting serious damages to studied frames. These ranges of scaled PGA values are
summarized in Table 4.
Table 5. Scaled PGAs for Analysis of Used Frames
Storey
Scaled PGAs (g)
1
0.5
0.75
1.00
1.25
1.50
2.00
2.50
3.00
4.00
2,3 5, 9
0.20
0.40
0.60
0.80
1.00
1.25
1.50
2.00
2.50
2.4 Appropriate Damage Index
In the present study, modified Park and Ang damage index [13] is utilized to represent the
seismic damage of frames, which is presented in Eq. 3.
(3)
in which, is maximum curvature of the element obtained from nonlinear time history
analysis, is ultimate curvature of the element acquired from pushover analysis and is yield
curvature of the element obtained from static relations or pushover analysis. is strength reduction
coefficient (energy participation factor), which is assumed to be 0.025 for steel moment frames ac-
cording to the Cosenza and Manfredi [22] and Castiglinai and Pucinotti [23] studies. is dissipat-
ed hysteretic energy of a member in a loading cycle.
The major reasons of choosing of modified Park and Ang were:
It is so common in research studies
Contribution of dissipated hysteretic energy in seismic damage
Although, this index is established for RC frames, it is appropriately capable for using in
steel frames [23, 24].
This index is calibrated for more than 250 empirical samples [7].
This index is more consistent with real behavior of member during an excitation than other
well-known indices, according to research of Kunnath and Jenne [25].
Accordingly, the index value less than 0.1 represents minor damage and linear behavior of el-
ement and the index value more than 1.0 shows collapse of the structure.
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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3. Results and Discussion
In this section the maximum damage of possible hinges is characterized by this symbol:
ELNs, Where E is representing type of the member (B/C for beams or columns), L is representing
the location of plastic hinge in column (T/B for top or bottom end) and Ns is representing the storey
that plastic hinge is located. ELNs shows the damage value of a plastic hinge (in a specific storey)
which is more damaged. For example CB2 shows the maximum damage value of plastic hinges
located in the column bottom of second floor, CT5 represents the maximum damage value of plastic
hinge in column top of fifth floor and B4 manifests the maximum damage index of two plastic
hinges in the beam of fourth floor.
3.1 Single Storey Frame
The average of damage index for all 12 records scaled to PGA=1.5g is illustrated in Fig. 1.a.
In this figure DIMPA is the value of modified Park and Ang damage index.
a
b
Figure 1. One storey frame a) average damage index when PGA=1.5g, b) max DI for scaled PGAs
According to Fig. 1.a, at lower values of ZRatio (weak column/ strong beam) the seismic dam-
age of the single storey frame had its highest values concentrated in columns. Hereafter, with the
increase of ZRatio value, the beam initiated to participate in enduring seismic damage. From
ZRatio=1.0 and ZRatio=1.3 beam damage was more than of column top and column bottom, respec-
tively. Also, with ZRatio almost more than 1.2 there is no damage in column top. Furthermore, at first
the damage value of beam soared with increase of ZRatio then, remained constant approximately.
After ZRatio=1.3 the beam was more damaged. From this point on, the ideal behavior of
SC/WB concept is achieved and the beam had the major role in seismic damage mechanism of the
frame. It should be born in mind the increase of ZRatio can lead to larger sections and growth of
costs. Therefore, a balanced ZRatio must be chosen for satisfying the SC/WB concept and being eco-
nomical. The red dashed line in Fig. 1.a is the 1.5g line in Fig. 1.b. This line indicates before
ZRatio=1.3 damage is controlled by column and after this point by beam.
Fig. 1.b shows the damage mechanism of the single floor frame for different values of PGA
(the maximum damage value for each ZRatio of the frame). Choosing ZRatio more than the ones for
specified spots on the curves, could result in concentrating seismic damage in the beam. As it can
be seen from Fig. 1.b, the increase in ZRatio led to decrease in maximum damage value of the frame.
3.2 Two-Storey Frame
Similar to 1 storey frame by increasing ZRatio the damage values in columns and the damage in
beams decreased and increased, respectively. According to Fig. 2.a, damage of column top of first
and second floor became inconsiderable at the ratios of 0.9 and 1.1, respectively. Considering low
damage of B2, choosing ZRatio more than 0.7 can lead to SC/WB concept and economical design.
By taking a ZRatio more than corresponding ones of specified points on curves in Fig. 2.b, seismic
damage concentrated more in beams. These points are representative of ZRatio values that damage
mechanism was shifted to first floor beam. The second floor beam damage is neglected mostly be-
cause seismic damage was eligible.
Damage Concentrated
in Column
Damage Concentrated
in Beam
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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a
b
Figure 2. Two-storey frame a) average damage index when PGA=0.8g, b) DI for scaled PGAs
3.3 Three-Storey Frame
The specified point in Fig. 3 shows the lowest ZRatio from which, the damage value of one of
the beams became more than all columns damage values. As it can be seen, for majority values of
PGA, after the ZRatio corresponding to intersection point, the maximum damage value of the frame
did not decrease or increase tangibly. One can see that, the intersection point is a local minimum in
the range of ZRatio values around it and by increasing PGA value the corresponding ZRatio value of
intersection point is increased.
Figure 3. Maximum DI of 3-storey frame for scaled PGAs
The points in Fig. 3 only show the lowest ZRatio from which one of the beam (for this case
CB1) play the major role in damage mechanism. Hence, the ZRatio values from which damage of
each beam exceeded all the columns are summarized in Table 7. By increasing PGA values, the
ZRatio values of B1, B2 and B3 with CB1 became closer. Also, after ZRatio values corresponding to
B3 and CB1 intersection, the design led frame to ideal SC/WB concept. However, the economical
point of view must be considered in choosing appropriate ZRatio since the damage value of B3 is
rather low.
Table 6. The ZRatio Values in Which, Beam Damage Value is Equal to Column for 3 Storeys Frame
PGA (g)
ZRatio
PGA (g)
ZRatio
PGA (g)
ZRatio
B1
B2
B3
B1
B2
B3
B1
B2
B3
0.2
0.65
0.9
-
0.8
0.685
0.862
1.712
1.5
0.734
0.978
1.523
0.4
0.675
0.885
1.75
1.0
0.7
0.87
1.595
2.0
0.755
1.085
1.415
0.6
0.683
0.855
1.61
1.25
0.717
0.912
1.575
2.5
0.765
1.145
1.395
3.4 Five-Storey Frame
The ZRatio values corresponding to intersections between beams and CB1 are listed in Table 8.
Table 7. The ZRatio Values in Which, Beam Damage Value is Equal to Column for 5 Storeys Frame
PGA (g)
ZRatio
PGA (g)
ZRatio
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
0.2
0.737
0.737
0.727
1.711
-
1.25
0.674
0.724
0.692
0.804
1.581
0.4
0.641
0.683
0.613
0.836
-
1.5
0.662
0.727
0.705
0.851
1.581
0.6
0.662
0.687
0.627
0.752
-
2.0
0.678
0.730
0.722
0.969
1.476
0.8
0.677
0.708
0.651
0.726
1.804
2.5
0.692
0.743
0.734
1.079
1.463
1.0
0.667
0.713
0.662
0.745
1.553
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
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It can be noted from Table 8, before PGA=1.0g the damage of B1 was dominant and after
that, B3 was dominant. After ZRatio corresponding to damage exceedance of B5 and CB1 (the most
damage concentrated column) the frame design was ideal for satisfying SC/WB concept. However,
because of lighter loads on fifth floor, low damage value of B5 and economical viewpoint, it is bet-
ter to choose ZRatio values close to B4 damage exceedance point. The lowest ZRatio values in which,
first to fourth floor beams played the major role in frame damage mechanism, would be the most
appropriate ones.
3.5 Nine-Storey Frame
The average of maximum damage index values for twelve records of Table 4 scaled to
PGA=2.5g is illustrated in Fig. 4.a. In order to make Fig. 4.a clearer, the plastic hinges that had
damage values less than 0.1 (in elastic range) are eliminated. Also, damage values are omitted at
ZRatio=0.5 for PGA values 2.0g and 2.5g since, the frame collapsed with very high damage values
for five out of twelve acceleration records. As can be seen in Fig. 4.a, the concentration of seismic
damage was in the 7th floor columns (CB7 and CT7) rather than, bottom of 1st floor column (CB1)
due to participation of higher modes. From ZRatio values 1.05 to 1.25, most of the beams damage
values (1st to 8th floor) was more than CT7 and CB7. Also, from ZRatio=2.1 the damage value of B9
was more than the damage value of other columns. It is important to note that the damage of CB1
was more than columns of 7th floor from ZRatio=1.5 meaning mode participation is changing with
ZRatio.
a
b
Figure 4. Nine-storey frame a) average damage index when PGA=0.8g, b) max DI for scaled PGAs
The averages of maximum damage index values for all 12 records scaled to each PGA values
are shown in Fig. 4.b. As can be noted in Fig. 4.b, the minimum damage index of the frame, oc-
curred at the ZRatio corresponding to first beam damage exceedance point. However, as the ZRatio
increased after the particular points, the frame behavior was closer to SC/WB concept.
Table 8. The ZRatio Values in Which, Beam Damage Value Equal to Column for 9 Storeys Frame
PGA (g)
ZRatio
PGA (g)
ZRatio
B1
B2
B3
B4
B5
B6
B7
B8
B9
B1
B2
B3
B4
B5
B6
B7
B8
B9
0.2
-
-
-
-
-
0.93
-
-
-
1.25
1.14
1.08
1.14
1.27
1.26
1.43
1.07
1.12
2.06
0.4
1.14
0.99
1.09
1.18
1.15
0.95
1.06
1.26
-
1.5
1.15
1.10
1.15
1.28
1.27
1.06
1.09
1.15
2.03
0.6
1.19
1.15
1.18
1.28
1.27
0.94
1.02
1.11
-
2.0
1.11
1.08
1.11
1.27
1.27
1.06
1.11
1.21
2.06
0.8
1.27
1.11
1.17
1.31
1.28
0.93
1.05
1.09
2.03
2.5
1.12
1.07
1.12
1.27
1.26
1.06
1.13
1.26
2.10
1.0
1.17
1.08
1.11
1.28
1.26
1.01
1.06
1.08
2.09
The ZRatio value from which, each beam damage exceeded columns (in this case columns of
7th floor) is listed in Table 9. According to Table 9, B2 and B6 are plastic hinges damaged more
than plastic hinges in columns at lower values of ZRatio. It can be noted that, in the range of ZRatio
from 0.93 to 1.26, the damage was concentrated in most of the beams (1st floor to 8th floor) rather
than columns. Due to low damage value of B9, the light load on the 9th floor and economical point
of view it is better to choose the ZRatio close to 1.3.
The damage mechanism process and the contribution of possible plastic hinges in seismic
damage of the nine-storey frame are illustrated in Fig. 5. The PGA values of earthquake accelera-
tion records exerted to this frame were scaled to 1.5g and this process is shown at ZRatio values 0.74,
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
8
0.89, 1.11 and 1.33. The size of the possible plastic hinges shows how damaged were plastic hinges,
qualitatively. Also, the magnitude consequence of damaged plastic hinges and damage values are
manifested right beside the hinges. The hinges that had damage value less than 0.1 are eliminated in
Fig. 5. Columns had the major role in damage in Fig. 5.a. By increasing ZRatio values to 0.89 the
beams started to participate in damage mechanism of the frame and at ZRatio=1.11 the beams of 6th
and 7th floor played the major role although, the columns had still part in damage mechanism. Final-
ly, at ZRatio=1.33 beams of 1st to 8th floor, the behavior of the frame was almost satisfied the SC/WB
concept although.
(a) ZRatio=0.74
(b) ZRatio=0.89
(c) ZRatio=1.11
(d) ZRatio=1.33
Figure 5. Damage mechanism of 9-storey frames subjected to PGA=1.5g
4. Concluding Remarks
The results obtained from carrying out 972 nonlinear time history analyses on different kinds
of one-bay frames in Table 1, led this study in conclusions to design ductile steel moment frames:
According to maximum damage index of frame for all scaled PGA values, the minimum
damage value occurred at the intersection point of beam and column damage for 9 and 5 sto-
reys frames. However, by increasing ZRatio there was no significant increase in modified
Park and Ang damage index. Also, the minimum damage occurred at the highest ZRatio value
for 2 and 3 storeys frames but, in majority of PGA values the difference between damage
values corresponding to intersection point and the highest ZRatio value is negligible. There-
fore, maximum damage index could not be enough for anticipation of the frame behavior
under seismic excitation.
The ZRatio index whereabouts that led the design of frame to meet Strong Column/Weak
Beam behavior considering economical point of view, for 1, 2, 3, 5 and 9 storeys frame was
1.3, 1.3, 1.1, 1.0 and 1.3 respectively.
Overall, it can be concluded that the most appropriate range of ZRatio index values to satisfy
SC/WB concept and considering economical viewpoint for studied frame is between 1.1 and
1.3 and it is recommended to choose 1.3 to have more conservative margin. This ZRatio index
value range can be used along with AISC 341 criterion ( ). As it can be seen in the
case of 9 storeys, this range could meet the SC/WB concept despite of higher vibration
modes effect.
In comparison with AISC 341 criterion, application of ZRatio index as a preliminary design
factor, in addition to elements deformation, seismic energy dissipation of possible plastic
hinges are taken into account and this ratio can be more applicable for engineers because of
7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
ogy, Iran, 28-29 Nov. 2017
9
its simplicity. The seismic damage distribution of plastic hinges illustrates that the imple-
mentation of an appropriate ZRatio index has resulted in SC/WB structural behavior.
5. References
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book for state earthquake and mitigation managers” FEMA, Volume 313 (1998).
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neers Association of
California, Sacramento, CA, 1995.
4. ATC Committee, “Seismic evaluation and retrofit of concrete buildings”, Applied Technology
Council, report ATC-40, Redwood City, 1996.
5. NEHRP Committee, “Guidelines and Commentary for the Seismic Rehabilitation of Buildings:
FEMA 273–274”, Federal Emergency Management Agency (FEMA), Washington DC, 1997.
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Motion”, 96th Annual Meeting of Seismological Society of America, San Francisco, CA (2001).
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exhibiting strength and stiffness degradation under seismic motion”, Engineering Structures
Jornal, Volume 46, pp. 727-736 (2013).
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7th International Conference on Acoustics & Vibration (ISAV2017), Sharif University of Technol-
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