10CUEE CONFERENCE PROCEEDINGS
10th International Conference on Urban Earthquake Engineering
March 1-2, 2013, Tokyo Institute of Technology, Tokyo, JapanRESIDUAL SEISMIC PERFORMANCE OF SCALED THREE-STORY STEEL FRAMES TESTEDON A SHAKING TABLE USING THE SUBSTITUTE DAMPING MODEL

Miguel A. Diaz1), Koichi Kusunoki2), and Akira Tasai3)
1) PhD. Candidate, Department of Architecture and Building Science, Yokohama National University, Japan
2) Associate Professor, Department of Architecture and Building Science, Yokohama National University, Japan
3) Professor, Department of Architecture and Building Science, Yokohama National University, Japanmiguel.ad17@gmail.com, kusunoki@ynu.ac.jp, tasai@ynu.ac.jp

Abstract: This paper compares the experimental and estimated seismic performance using the results of scaled
three-story steel frames tested on a shaking table, excited with artificial and earthquake waveforms, and formulations
based on the capacity spectrum method and the substitute damping model, such as the equivalent damping ratio and the
response reduction ratio. For each specimen, the elastic and inelastic behavior is reproduced; in the last, the specimen was
excited by the mainshock, and then the maximum expected aftershock in order to analyze the residual seismic
performance.

1. INTRODUCTION
The collapse of structures, in many cases, is not causedby the mainshock, but it may be caused by the subsequent
aftershocks due to the seismic capacity degradation of thebuilding during the mainshock. Entry into a damagedbuilding at earliest is often indispensable for different
emergency reasons. Thus the estimation of the seismic
performance due to the maximum expected aftershock isvery much important in order to determine if the building
may or may not survive.
This paper presents an instrument to conduct thisestimation; it is based on the capacity spectrum method andthe substitute damping model; by means of formulations,such as the equivalent damping ratio and response reduction
ratio given for the life-safety limit state. This limit isestablished for earthquakes motions whose return period isapproximately 500 years, as prescribed the Japanese
Building Standard Law Enforcement Order.
The equivalent damping and response reduction ratios
are prescribed by Notification No. 1457-6 (2000) of
Ministry of Land, Infrastructure, Transport and Tourism ofJapan (MLIT). They are used to estimate the seismic
performance due to the mainshock.
On the other hand, the equation of the equivalent
damping ratio for the seismic performance estimation due toaftershock proposed in previous work by Kusunoki et al.
(2006) and Diaz et al. (2012) is used in this paper.
Additionally, a new equation of the response reduction ratio
proposed by Diaz et al. (2012) which holds good formainshocks and aftershocks is also used.

Thus the seismic performance due to mainshocks andaftershocks from the shaking table tests are compared withthe estimated seismic performance using the formulations ofequivalent damping and the response reduction ratios. Ten
specimens were tested under one artificial waveform andtwo earthquake waveforms; properties such as period,
equivalent damping and response reduction ratios due tomainshocks and aftershocks are analyzed.

2. RESIDUAL SEISMIC PERFORMANCE
2.1 Aftershock assumption
The aftershock is defined as the subsequent shakes after
a significant earthquake with a magnitude less than the given
earthquake (mainshock).
Generally, a magnitude of the largest aftershock issmaller by 1 than that of the mainshock. The largest
aftershock, in many cases, occurs within 3 days after theoccurrence of the mainshock in the case of inland
earthquake. As for those occurred in sea area, the largest
aftershock generally occur within about 10 days (Japan
Meteorology Agency).
The energy released by earthquake is proportional to its
magnitude. Thus, a rough approximation may suppose thatthe energy released by the maximum expected aftershock isrelatively close to that by the mainshock.
Therefore, if a given earthquake and its subsequent
aftershocks are considered as one-long duration earthquake
from the beginning of the mainshock toward the end of the
maximum expected aftershock; the motion of the maximum
expected aftershock can be supposed same as that of the- 1203 -mainshock, then neglect the inner shakes, because they do
not produce larger responses than the maximum expected
aftershock.
Using this assumption, the total motion to conduct theresidual seismic performance analysis is given by theearthquake inputted twice, mainshock and aftershock, and a
gap between both. The duration of this gap is set in themeasure that the system vibration converges to zero, asshown in Figure 1.

Eq. (2) is also prescribed by Notification No. 1457-6(2000), based on the substitute structural method (Shibata
and Sozen, 1976). It estimates the equivalent damping ratioof an equivalent SDOF system as the weighted average
respect to the strain energy with a viscous damping ratio of
0.05 for the first-mode at the damaged-initiation limit state,since at this stage the building behavior remains elastic.
𝑚 𝑊𝑖 is the strain energy dissipated in member 𝑖.
ℎ𝑒𝑞 =∑ 𝑚ℎ𝑒𝑞−𝑖 ∙𝑚𝑊𝑖
∑ 𝑚 𝑊𝑖1.00+ 0.05

(2)

0.75
0.50
0.25
0.00

2.3 Response reduction ratio
The response reduction ratio reduces the elastic spectral
response to the inelastic response. Notification No. 1457-6(2000) also prescribes the response reduction ratio as Eq. (3),which guarantees the life-safety limit state.

-0.25
-0.50
-0.75
-1.00

Mainshock

Inner shakes

Maximum expected
Aftershock

1.00
0.75
0.50
0.25
0.00
-0.25

𝐹ℎ =

-0.50
-0.75
-1.00

1.5
1+10∙ℎ𝑒𝑞(3)

Gap until system reaches rest

Figure 1 Aftershock assumption
2.2 Equivalent damping ratio
The equivalent damping ratio for yielding structures
can be determined by the hysteresis damping in terms of theelastic strain energy of a structure by means of the geometricstiffness method (Jennings, 1968).
Notification No. 1457-6 (2000) prescribes theequivalent damping ratio 𝑚ℎ𝑒𝑞−𝑖 of a structural member 𝑖in Eq. (1) for the life-safety limit state.
𝑚ℎ𝑒𝑞−𝑖 = 𝛾 ∙ (1 − 1⁄√𝜇 )(1)

The coefficient 𝛾 in Eq. (1) is assumed as 0.25 in caseof material which constitutes the member, and the joint
connected to the adjacent member are rigid; and as 0.20 incases of members or braced members where the bucklingstrength is degraded by the compressive forces when seismic
forces acts (member exhibits a slip-type characteristic), as
prescribed Notification No. 1457-6 (2000).
The coefficient 𝛾 in Eq. (1) may hold good for theestimation of the equivalent damping ratio due tomainshocks; otherwise the energy dissipation (or equivalent
damping) due to the aftershock is less than or equal to that
due to its corresponding mainshock. Therefore, thiscoefficient can be conveniently reduced in order to obtainlarger responses, such as responses during an aftershock,since the response reduction ratio in Eq. (3) is inversely
proportional to the equivalent damping ratio in Eq. (1). Thus,
the coefficient 𝛾 is reduced to 0.12 and 0.08 by anappropriate curve fitting based on a series of nonlinear
simulations obtained for systems under aftershocks (Diaz etal., 2012).

Additionally, previous work by Diaz et al. (2012)
proposes a new equation of the response reduction ratio,
given by Eq. (4). It is developed solving the equation of
motion under the stationary vibration, and then adapting it to
the non-stationary vibration, such as earthquake motions, bymeans of curve fitting to analytical response of a series ofnonlinear simulations.
𝐹ℎ ∗ = √1.1+𝛼∙ℎ𝑒𝑞 2

1+(40+𝛼)∙ℎ𝑒𝑞 2(4)

The Japanese Building Standard Law Enforcement
Order requires that spectral acceleration of a structure at alimit state should be higher than the corresponding
acceleration of the reduced demand spectrum using theequivalent damping ratio at the same limit state.
2.4 Seismic performance evaluation
During a damaging earthquake, some buildings may
survive, but the subsequent aftershocks may or may notcause the building collapse; that’s why, it is desirable torecognize the building state (seismic performance) after amainshock, and to estimate the seismic performance due tothe maximum expected aftershock in order to anticipate if
the building may or may not survive, and so safeguard life.
The seismic performance of a building due to a given
earthquake motion is examined by comparing the capacity
curve and demand spectrum in terms of 𝑆𝑎 − 𝑆𝑑 relations,as shown in Figure 2. The intersection between the capacity
curve and the demand spectrum for an appropriate
equivalent damping ratio which represents the inelastic
response under the given earthquake, is called theperformance point (ATC-40, 1996).

- 1204 -Representative restoring force (Sa)

Demand spectrum(A)

Yield point
Performance point due tomainshockPerformance point due toaftershock
Performance point at
ultimate state
Curve-1 (for mainshock)
Curve-2 (for aftershock)(B) (C)Capacity curve(D)

Representative displacement (Sd)Figure 2 Scheme of the seismic performance evaluation
The concept of the seismic performance evaluation
method is represented by the scheme shown in Figure 2. The
performance point due to the mainshock is represented by
the point (B) on the capacity curve, in this figure; thedemand spectrum is reduced by increasing the viscousdamping ratio until it intersects the point (B). Thus the
reduced demand spectrum is represented as curve-1 inFigure 2, and the converged viscous damping ratio is definedas the equivalent damping ratio for mainshock.
The maximum response during the maximum expected
aftershock, represented by the performance point (C) on thecapacity curve in Figure 2, is larger than or equal to thatduring the mainshock, represented by the performance point(B) in the same figure. Thus the equivalent damping ratio foraftershock is less than or equal to that for mainshock, as
observed in Figure 2, because the energy dissipation (orequivalent damping) during the maximum expected
aftershock is less than or nearly equal to energy dissipation
during its corresponding mainshock. Then the demand
spectrum is again reduced until it intersects the performancepoint (C) in Figure 2; the reduced demand spectrum isrepresented as curve-2 and the converged viscous damping
ratio is defined as the equivalent damping ratio foraftershock.

3.1 Tests specimens
The specimen consists in a scaled three-story andone-bay steel plane frames; their members are connected bybolts to a rigid joint, as shown in Figure 3.
The bay width is 1000 mm, the first-story height varies
from 805 mm to 1005 mm, and the second and third stories
heights are 700 mm. The beams are rectangular bars of
100×6 mm widened at the middle to support theaccelerometer on each level. The first-story is constituted byrectangular bars as columns without braces; the second andthird stories are braced frames with rectangular bars of
100×6 mm as columns, and circular bars M10 as braces (apair in each front).
The design of the specimen supposed that the columnsof the first-level would be only affected during the tests, due
to the soft-story behavior of this structure. The sections ofthese columns were reduced in 50% at the bottom in order toprovide to the specimen a plastic hinge to control the failure
mechanism.
Thus, the properties of the specimen depend on thedimensions of first-story columns (thickness and height); thesecond, third braced frames are preserved for all tests.
However, the second and third braced frames are also
observed by the measurement system.
Accelerometer

Rigid
joint

Displacementtransducer

BracesFix supports

Device for

To be replacedrockingby stiff springs
base

U

Shaking table
E

3. SHAKING TABLE TESTS

Figure 3 Configuration of specimens
The experimental tests are based on the assumption
described in Section 2.1. The testing program consists infour series of excitation. The first excitation was producedby the white-noise in order to evaluate the dynamic
properties. The second excitation was the test using the input
motion with small acceleration amplitude to induce anelastic response. The third excitation was the test using large
acceleration amplitude, so that the specimen was within the
inelastic range (mainshock). And, the fourth excitation was
the test using the previous acceleration amplitude (or slightlyless) to produce the maximum expected aftershock.

The specimens Frame1a, Frame2a,b and S-F01,02,03
are fixed support; while specimens S-S01,02,03,04 are also
fixed support, but the base (ground) includes a small rocking
effect due to the adding of stiff springs (Kspring=500 N/mm)
between supports and base, as shown Figure 3. The initial
compressive deformations of springs were 1 mm and 7 mm
for S-S01,02 and S-S03,04, respectively. Members of these
specimens consisted of SS400 (326 MPa). Table 1 presentscharacteristics of these specimens.

- 1205 -Table 1 Characteristics of specimens
Thickness
Height Total Natural Yield.
1st F
Specimen
1st F weight period Disp.
column(mm) (kN)
(s)
(mm)
(mm)
Frame1 a
4.5805
1.450 1.03
66
Frame2 a, b
4.51005 1.464 1.66
101
S-F 01, 02, 03 6.01000 1.485 0.93
71
S-S 01, 02
6.01000 1.485 0.99
73
S-S 03, 04
6.01000 1.485 0.9873a) Velocity response spectra

The measurement system was given by accelerometersat the base and beams of each level, and displacementtransducers connected at each eastern rigid joint, as shown in
Figure 3.
3.2 Input waveforms
The input motions to conduct this study were one
artificial wave: the WG60, and two earthquake records: the
KOBE-NS (Kobe, 1995) and the MYG013 (Tohoku, 2011).
The input motions were scaled in order to induce different
performance levels on the specimen within elastic andinelastic ranges, both mainshocks and aftershocks. Figure 4
shows the WG60, the KOBE-NS and the MYG013
waveforms.

b) Displacement and acceleration response spectra
Figure 5 Response spectra
3.3 Equivalent SDOF
In order to conduct the residual seismic performance
analysis, it is necessary to transform the capacity curve of
the specimen in terms of 𝑆𝑎 − 𝑆𝑑 relations.
The probable value of the maximum response is usually
given by the square root of the sum of square (SRSS) of the
maximum modal response components. Then the maximum
displacement at 𝑖-th story and the base shear force can be
expressed approximately by Eq. (5) and Eq. (6) (Shibata,2010), respectively; where 𝑠𝑆𝑑 , 𝑠𝑆𝑎 is the spectral
displacement and spectral acceleration for the 𝑠-th mode.
The base shear force of 𝑁-DOF is given by Eq. (7).

|δi |max ≈ √∑N
S=1| sβ ∙ sui ∙ sSd |2NQ ≈ √∑N
S=1*∑i=1 mi ∙ sβ ∙ sui ∙ sSa +Figure 4 Input waveforms

Figure 5 shows their respective a) normalized velocity
response spectra respect to the maximum spectral velocity,and b) normalized response spectra respect to peak groundacceleration (PGA) and maximum spectral displacement.
The maximum spectral response with viscous damping ratioof 0.05 was obtained at periods of 1.27 seconds, 0.87
seconds and 0.67 seconds for the WG60, the KOBE-NS andthe MYG013 waveforms, respectively (Figure 5a).

Q = ∑Ni=1 mi ∙ (ẍ i + ẍ g )

(5)

(6)
(7)

Particularly, the specimens can be assumed as 3-DOF
systems. Their configuration allows that second and third
participation factor can be neglected, and the first-mode
distribution can be assumed as the unit { 1𝑢 } ≈ *1+. Then
Eq. (5) is rewritten as Eq. (8) to calculate the spectral
acceleration, and Eq. (7) into Eq. (6) is rewritten as Eq. (9) to
calculate the spectral displacement.

- 1206 -1Sd ≈

1Sa =|δi |max

(8)

su i

∑3
i=1 mi ∙(ẍ i +ẍ g )(9)∑3
i=1 mi

3.4 Tests results
The representative response or spectral response iscalculated by Eq. (8) and Eq. (9), using experimental datasuch as the displacement from transducers and the absolute
acceleration from accelerometers on each level. The
responses during mainshocks and aftershocks, in terms of
𝑆𝑎 − 𝑆𝑑 relations, with different maximum acceleration
amplitudes of waveforms are plotted in Figure 6, Figure 7
and Figure 8, where 𝑔 is the acceleration of gravity.
Figure 6 shows the response during the WG60
waveform for four specimens, namely: Frame1a, S-F01,
S-S01 and S-S04 with maximum acceleration amplitude of
waveform of 0.36g, both mainshock and aftershock.

a) Frame2a (0.17g)b) Frame2b (0.33g→0.25g)c) S-S02 (0.38g)d) S-S03 (0.38g)e) S-F02 (0.38g)a) Frame1a (0.36g)

Figure 7 Response during the KOBE-NS waveformb) S-F01 (0.36g)

Figure 8 shows the response during the MYG013
waveform for the specimen S-F03 with maximum
acceleration amplitude of waveform of 0.39g, bothmainshock and aftershock.

c) S-S01 (0.36g)d) S-S04 (0.36g)

Figure 6 Response during the WG60 waveform
Figure 7 shows the response during the KOBE-NS
waveform for five specimens, namely: Frame2a withmaximum acceleration amplitude of waveform of 0.17g(mainshock and aftershock), Frame2b with 0.33g formainshock and 0.25g for aftershock; and S-F02, S-S02 andS-S03 with 0.38g, both mainshock and aftershock.
In Figure 7b, a negative slope arose in the
representative acceleration; it is because the specimen
Frame2b is slender and long-period frame (see Table 1), andsuffered the P-Δ effect (its capacity is reduced in front of thegravity effect). The maximum response within the inelastic
range was larger than the elastic spectral response with
damping ratio of 0.05.

a)S-F03 (0.39g)

Figure 8 Response during the MYG013 waveformIn the previous experimental study (Diaz, Kusunoki andTasai, 2012), the equivalent damping ratio was estimatedusing the maximum response amplitude, negative or positive,
to estimate the equivalent damping ratio; by increasing thedamping ratio until the resulting demand spectrum isintersected with this maximum response amplitude, as
described Section 2.4 in Figure 2.
Those results showed that some responses such as thosefrom specimens S-F02, S-S02 and S-S03 were unsafely- 1207 -estimated using the formulations of the equivalent damping
and response reduction ratios presented in this paper. It isbecause the fundamental period of these specimens are close
to the predominant period of the corresponding input
waveform (KOBE-NS). Also, the specimen Frame2b could
not be used because it suffered the P-Δ effect and its
maximum response amplitude is larger than the elastic
spectral response with damping ratio of 0.05.
On the other hand, the equation of equivalent damping
ratio given in Eq. (1) is developed by the geometric stiffness
method which supposes that the positive and negative
amplitudes of the hysteresis loop are equal. Although, asshown in Figure 6a, 6c, 7b, 7c, 7d and 7e, the hysteresis
loops are shifted in many cases toward negative or positive
direction, and amplitudes (negative and positive) turn out tobe quite different; this shifting is intensified during
aftershocks, where a residual deformation due to themainshock may occur.
Thus the energy dissipation, or equivalent damping, isrepresented much closer to the true value when both
amplitudes, negative and positive, are used to estimate anequivalent period (peak-to-peak), and then defining theequivalent amplitude.

An important characteristic is that the envelope of thespectral response during the mainshock maintains the same
shape during the aftershock, but moved toward one direction
or another, and the residual deformation becomes the new
origin. It means the capacity curve preserves the same shapeuntil the structure sustains major damage (large deformation).
This phenomenon can be observed in Figure 9, the envelopeof the response during mainshock looks like moved toward
positive direction during the aftershock, and the new origin
is relocated at 3.2 cm.
Therefore, the equivalent damping and response
reduction ratios are calculated after defining the equivalent
amplitude as the intersection between the line parallel to theresulting line of joining the peak amplitudes (peak-to-peak)which crosses the origin and the negative or positive branchof the envelope of the spectral response (Figure 9a). In caseof the response during the aftershock (Figure 9b), theparallel line crosses the new origin (residual deformation).
After defining the equivalent amplitude, the equivalent
damping ratio is calculated by reducing the demand curve
toward the equivalent amplitude, as describe in Section 2.4.
Then the response reduction ratio is calculated as the ratio of
the equivalent amplitude to the elastic response. The elastic
response corresponds to the intersection between the parallelwith the demand spectrum with damping ratio of 0.05, asshown in Figure 9a. In case of aftershocks, the demand
spectrum is moved toward the residual deformation asshown in Figure 9b.

4. ANALYSIS OF RESULTS
Figure 10 shows the relation between the ductility
factors due to mainshocks and aftershocks, using theabsolute maximum displacement (maximum amplitude),and the negative and positive maximum displacement(equivalent amplitude).

a) Response during mainshock

Figure 10 Relation between ductility factorsb) Response during aftershock
Figure 9 Estimation of parameters using the equivalent
amplitude

Table 2, Table 3, Table 4 and Table 5 show the ductility
factor (μ), the equivalent damping ratio (ℎ𝑒𝑞 ) and the
response reduction ratio (𝐹ℎ ) obtained from the experimental
tests, and the estimated response reduction ratio. In order to- 1208 -estimate the response reduction ratio, the ductility factor isevaluated into Eq. (2) to estimate the equivalent damping
ratio and then it is evaluated into Eq. (3) and Eq. (4), 𝐹ℎand 𝐹ℎ ∗ , respectively.
Table 2 and Table 3 show results due to mainshocks andaftershocks, respectively, using the maximum absolute
displacement. As shown in these tables, the equivalent
damping and response reduction ratios are not calculated forthe specimen Frame-2b, because the inelastic response ismuch larger than the spectral response with damping ratio of
0.05, both mainshock and aftershock.
Table 2 Results due to mainshock using the maximum
absolute amplitude
Experimental
Specimen waveform
Frame-1a
Frame-2a
Frame-2b
S-F01
S-F02
S-F03
S-S01
S-S02
S-S03
S-S04

WG60
KOBE-NS
KOBE-NS
WG60
KOBE-NS
MYG013
WG60
KOBE-NS
KOBE-NS
WG60

Estimated

𝜇

ℎ𝑒𝑞

𝐹ℎ

𝐹ℎ

𝐹ℎ ∗

1.73
0.82
1.66
1.72
2.18
1.37
1.98
1.90
1.99
1.76

16.57% 0.540
5.19% 0.968
11.24% 0.651
4.42% 1.028
7.57% 0.770
10.62% 0.672
6.48% 0.907
5.57% 0.959
10.91% 0.665

0.714
1.081
0.915
0.879
0.935
0.895
0.890
0.882
0.897

0.874
1.018
0.974
0.960
0.981
0.966
0.964
0.961
0.967

Table 3 Results due to aftershock using the maximum
absolute amplitude
Experimental
Specimen waveform
Frame-1a

WG60

Frame-2a KOBE-NS
Frame-2b KOBE-NS
S-F01

WG60

S-F02

KOBE-NS

S-F03

MYG013

S-S01

WG60

S-S02

KOBE-NS

S-S03

KOBE-NS

S-S04

WG60

Estimated

𝜇

ℎ𝑒𝑞

𝐹ℎ

𝐹ℎ

𝐹ℎ ∗

1.80
0.83
2.23
1.76
2.59
1.39
1.87
2.23
2.39
1.83

16.41% 0.542
4.99% 0.976
10.83% 0.671
5.19% 1.002
7.37% 0.778
10.04% 0.699
5.95% 0.981
5.60% 0.987
10.07% 0.691

0.831
1.056
0.905
0.858
0.929
0.901
0.872
0.855
0.884

0.939
1.013
0.970
0.951
0.978
0.968
0.957
0.950
0.962

Table 4 Results due to mainshock using the equivalent
amplitude
Experimental
Specimen waveform
Frame-1a
Frame-2a
Frame-2b
S-F01
S-F02
S-F03
S-S01
S-S02
S-S03
S-S04

WG60
KOBE-NS
KOBE-NS
WG60
KOBE-NS
MYG013
WG60
KOBE-NS
KOBE-NS
WG60

Estimated

𝜇

ℎ𝑒𝑞

𝐹ℎ

𝐹ℎ

𝐹ℎ ∗

1.63
0.56
1.44
1.66
2.01
1.36
1.79
1.89
1.95
1.72

17.95% 0.525
16.09% 0.639
7.83% 0.885
11.82% 0.642
7.70% 0.892
7.83% 0.757
10.99% 0.613
6.49% 0.912
6.19% 0.946
11.29% 0.648

0.735
1.305
0.984
0.927
0.892
0.939
0.909
0.894
0.887
0.901

0.887
1.044
0.996
0.978
0.965
0.982
0.971
0.966
0.963
0.968

Table 5 Results due to aftershock using the equivalent
amplitude
Experimental
Specimen waveform
Frame-1a
Frame-2a
Frame-2b
S-F01
S-F02
S-F03
S-S01
S-S02
S-S03
S-S04

WG60
KOBE-NS
KOBE-NS
WG60
KOBE-NS
MYG013
WG60
KOBE-NS
KOBE-NS
WG60

Estimated

𝜇

ℎ𝑒𝑞

𝐹ℎ

𝐹ℎ

𝐹ℎ ∗

1.61
0.75
1.13
1.71
2.03
1.39
1.80
1.98
1.98
1.77

16.50%
6.93%
16.14%
11.24%
8.38%
7.61%
10.80%
5.98%
6.70%
10.84%

0.498
0.886
0.698
0.663
0.870
0.772
0.644
0.955
0.927
0.669

0.855
1.084
0.985
0.924
0.890
0.935
0.907
0.888
0.882
0.896

0.950
1.019
0.996
0.977
0.964
0.980
0.971
0.963
0.961
0.966

Figure 11 shows the relationship between the ductility
factor and the equivalent damping ratio from theexperimental test using the maximum absolute amplitude
and the equivalent amplitude due to a) mainshocks and b)
aftershocks.

Table 4 and Table 5 show results due to mainshocks andaftershocks, respectively, using the equivalent amplitudedefined by the positive and negative peak amplitudes(peak-to-peak). Contrasting with the maximum absolute
amplitude, the equivalent amplitude allows calculating theequivalent damping and response reduction ratios for the
specimen Frame-2b, which suffered P-Δ effect.

a) mainshocksb) aftershocks

Figure 11 Estimation of equivalent damping ratio
Results in Figure 11 are compared with the estimatedequivalent damping ratio which is defined by the curve
heq-0.25 and heq-0.12, for mainshock and aftershock,- 1209 -respectively. It corresponds to coefficients γ 0.25 and 0.12
in Eq. (1), because the hysteretic behavior is quiet close to
the perfect elasto-plastic bilinear model.
The equivalent damping ratio becomes larger using theequivalent amplitude than using the maximum absolute
amplitude, it is observed in Figure 11. It means that theequation of the equivalent damping ratio can be estimatedmore safely these values using the equivalent amplitude.
In order to compare the experimental and estimatedresponse, the response reduction ratio is calculated from theshaking table tests and formulations. The response is safely
estimated when the actual response reduction ratio(experimental) is less than or equal to the estimated response
reduction ratio using the formulations presented in this paper,
both mainshock and aftershock, whichever is applicable.
𝐹ℎ 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ≤ 𝐹ℎ 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑(10)

The comparison between the estimated response
reduction ratio and the experimental response reduction ratio
due to mainshocks and aftershocks is presented in Figure 12
and Figure 13, using Eq. (3) and Eq. (4), respectively. Thesefigures show this comparison using a) the maximum
absolute amplitude and b) the equivalent amplitude. Here,the response safely estimated are represented by points
above the line (y=x).

a)maximum amplitudeb) equivalent amplitude

Figure 12 Response reduction ratio by 𝐹ℎ , Eq. (3)a)

maximum amplitudeb) equivalent amplitude

Figure 13 Response reduction ratio by 𝐹ℎ ∗ , Eq. (4)5. CONCLUSIONS
From the analysis conducted for the experimental andestimated results, the following conclusion can be drawn:o Ten scale three-story steel frames were tested on ashaking table, seismic performance is determined fromthe experimental results and compared with theestimated values.
o The definition of the equivalent amplitude allows
calculating responses where the maximum absolute
amplitude exceeds the elastic spectral response, evenfor the specimen which suffered the P-Δ effect.
o The equivalent damping ratio can be more safely
estimated when the equivalent amplitude is defined
instead of the maximum absolute amplitude.
o The formulations estimated safely the equivalent
damping ratio and then the response reduction ratio due
to mainshocks and aftershocks using the peak
amplitudes to define the equivalent amplitude. The
response reduction ratio is more safely estimated using
Eq. (4) than Eq. (3).
However, this technique would become safer for theseismic performance estimation if an equation to estimate
the shifting displacement is incorporated.

Acknowledgements:
Authors acknowledge Miho Yamashita, Yuichi Hatanaka, Daiki
Hinata, and Yuki Hattori for their great contributions during the tests.
The installation and maintenance of the measurement system has
been supported by Mr. Masayuki Araki, Mr. Takamori Ito and otherstaffs of aLab Co. Also, the authors would like to express their
gratitude to the Japan Science and Technology Agency (JST)through the Science and Technology Research Partnership for
Sustainable Development (SATREPS) project “Enhancement ofearthquake and tsunami disaster mitigation technology in Peru” forthe fellowship during this research.
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