Citation: Cueva, J., K. Kusunoki (2022), Estimation of the equivalent damping ratio for low energy
dissipation systems under mainshock-aftershock sequences to determine the damage level, Synopsis ofIISEE-GRIPS Master’s Thesis.

ESTIMATION OF THE EQUIVALENT DAMPINGRATIO FOR LOW ENERGY DISSIPATION SYSTEMSUNDER MAINSHOCK-AFTERSHOCK SEQUENCES
TO DETERMINE THE DAMAGE LEVELJean Cueva1
MEE21709

Supervisor: Koichi KUSUNOKI2

ABSTRACTThe assessment of Reinforced Concrete (RC) buildings, considering mainshock-aftershock (MA)sequences, has been addressed in previous studies in which the main objective is to predict the damagelevel after a maximum expected aftershock and then judge if the building can continue operating or not.
This study uses numerical simulations to determine the equivalent damping ratio for structures with low
energy dissipation and under mainshock-aftershock sequences and using several Single Degree ofFreedom (SDOF) systems with different hysteresis models.
Results indicate the variation of the damping ratio for each hysteresis model, and the higherand lower values correspond to the Degrading trilinear and Origin-oriented models, respectively. Inaddition, the parameter 𝑎2 , which is defined as the ratio of the yielding stiffness to the elastic stiffnessof the capacity curve, presents a significant influence on the equivalent damping ratio. For that reason,in the beginning, this study presented a new expression to determine the damping ratio, which considered
new reduction factor values “𝛾” and a new coefficient to quantify the additional damping ratio due tothe damage before yielding point “𝑏,” both were determined for specific values of the parameter 𝑎2 .
Due to the high values and the uncertainty of coefficient “𝑏,” the new expression was reduced
for practical purposes during the mainshock and aftershock. Additionally, the damage level (ductility)
for the maximum expected aftershock, and using the new “𝛾” values, presented a lower error percentage
than the previous studies, which considered 𝛾 equal to 0.06.
Keywords: RC buildings, damping ratio, mainshock, aftershock.
1. INTRODUCTION
Although the magnitude of an earthquake is the first indicator of the level of damage to which ourbuildings will be subjected, it is not always the main one. In many cases, the cumulative damage due tothe mainshock and subsequent aftershocks ends up being a factor that increases the risk of collapse of
the buildings and compromises their functionality. Due to those mentioned above, it is essential to know
the state of the building after each seismic event to know its residual capacity and thus decide whether
it can be occupied immediately or not. The rapid inspection of buildings becomes more critical inessential facilities, such as hospitals, where their operability must be confirmed after each specific eventdue to their service. The rapid inspection method in Japan is based on a method to evaluate the residual
seismic capacity index "R," defined as the ratio of the residual seismic capacity to the original capacity.
A new method for assessing the real-time residual seismic capacity of existing structures using
1
2Japan-Peru Center for Earthquake Engineering Research and Disaster Mitigation (CISMID), Peru.
Professor, Tokyo University.

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accelerometers is part of the recent studies based on Structural Health Monitoring (SHM). Thismethodology uses the double integral with a wavelet transform and obtains the capacity curve of thebuilding from the acceleration measurement in the installed sensors (Pan & Kusunoki, 2020). In addition,a study complementary to the previous one is the aftershock damage prediction of reinforced-concretebuildings using sensors, where the predicted maximum response during the Mainshock-aftershock
sequence is obtained from the Capacity Spectrum Method (CSM).
2. METHODOLOGY
2.1. Performance-based procedure
The Performance-based procedure or Capacity Spectrum Method (CSM) defines a demand and capacity
curve and is widely used to evaluate the performance of existing buildings as new ones (ATC-40, 1996).
The demand curve represents the earthquake ground motion, and the capacity curve is presented throughskeleton or primary curves and shows representative displacements and restoring forces undermonotonically increasing loading.
The total damping ratio (ℎ) is the sum of the equivalent and initial damping ratio ℎ0 and canbe written as Equation (1) as presented in (MLIT, 2000). In addition, the value of ℎ𝑒𝑞 is ductility
dependent and takes different values during a mainshock or aftershock. For mainshocks, the reduction
factor 𝛾 takes values of 0.25 and 0.20 (MLIT, 2000), and for aftershocks, new values of coefficient 𝛾
were proposed (Kusunoki, 2006) and are between 0.06 and 0.10.
ℎ = 𝛾 (1 −1) + ℎ0√𝜇(1)The response reduction factor 𝐹ℎ is the ratio between the maximum inelastic response and the elastic
spectral response for a specific effective period (𝑇𝑚 ), which is ductility dependent. The response
reduction factor, in terms of total damping ratio, Equation (2), is presented in Equation (3) (MLIT, 2000).
ℎ = ℎ𝑒𝑞 + ℎ01.5𝐹ℎ = 1+10∙ℎ(2)(3)2.2. Seismic evaluation for the Mainshock-Aftershock sequence
The maximum considered aftershock is the same as the mainshock for the life-safety limit state. Thiscondition is part of considering the most harmful scenario, where the energy released by the largest
aftershock is relatively close to the mainshock. Therefore, the total ground motion is composed of wave
Level-1, which corresponds to the mainshock, and wave Level-2, which corresponds to the expected
aftershock, equal to the mainshock. The procedure for the numerical simulation is divided into two parts.
First, determine the MA sequence for a ductility target during the mainshock. Second, the non-linear
time history analysis response will be determined using the MA sequence defined in the first part. Then,the equivalent damping ratio ℎ𝑒𝑞 and response reduction factor 𝐹ℎ for Mainshock (Level-1) andAftershock (Level-2) are obtained for each Single Degree of Freedom (SDOF) system and specifics
input ground motions.
3. DATA
3.1. Inelastic restoring forces models
Depending on each model, the inelastic restoring force is defined as the relation between therepresentative restoring force and displacement and following specific hysteresis rules. Four hysteresis
models were used in this investigation: Degrading trilinear model, Origin-oriented model, Takeda model,
and Takeda-slip model, and were selected to represent the systems with low energy dissipation.

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Previously, the primary curve, resistance-displacement relation, under monotonically increasing loadingis presented as a trilinear skeleton curve, as is illustrated in Figure 1.
Where:
𝛿𝑐 : Displacement at cracking resistance.
𝛿𝑐 : Displacement at cracking resistance.
𝑎1 : Ratio of the cracking resistance to theyielding resistance (𝑄𝑐 /𝑄𝑦 ).
𝑎2 : Ratio of the yielding stiffness to theelastic stiffness (𝐾𝑦 /𝐾).
Figure 1. Primary curve (Trilinear).

𝑎3 : Ratio of the post-yielding stiffness to theelastic stiffness (𝐾𝑢 /𝐾).

Source: (Diaz, 2013).
3.2. Parameters

For the numerical simulation, many parameters were defined to cover as many cases as possible, bothin the capacity and demand curve. The capacity curve was obtained from the Primary Curve, Figure 1,and for this research, some parameters are listed in Table 1, indicating the range of periods 𝑇 andductility targets 𝜇.
Table 1. List of parameters for the capacity curve.
Parameter

Values𝑇

0.2 to 1.2 @0.2s (6 combinations)𝑎1

0.2 to 0.6 @0.1 (5 combinations)𝑎2

0.25 to 0.85 @0.15 (5 combinations)𝑎30.001𝜇

1 to 5 @1 (5 combinations)

For the Demand curve, ten artificial waves (Okawa, 1998) and ten earthquake records were used as input
ground motions in the numerical simulations, and also, all used waves have 50Hz as sampling.
3.3. Numerical modelThe new proposal for calculating the equivalent damping ratio, considering new 𝛾 values, will beapplied to a numerical model, a 2D RC frame structure with six stories and two bays. The non-linear
pushover analysis will be performed using the software STERA3D (Saito, 2015). The displacement
versus shear curve for each story will be used as an output to convert the model into an SDOF system
and apply the Capacity Spectrum Method (CSM) and Non-linear Time History Analysis (NLTHA).
4. RESULTS AND DISCUSSION
4.1. Influence of parameter 𝒂𝟐 in the equivalent damping ratio
The theoretically equivalent damping ratio of each model clearly shows the influence of parameter 𝑎2 ,and it can be seen in cases where ductility is equal to one because values of ℎ𝑒𝑞 higher than zero at this3

yielding point indicates an additional damping ratio due to the damage after the cracking and before theyielding point, Figure 2; from now on, this additional damping ratio will be known as "𝑏".

New expression to curve fitting thenumerical results, ductility versus
damping ratio:ℎ = (𝜸 (1 −1) + 𝒃) + ℎ0√𝜇(4)Figure 2. Additional damping ratio "𝑏".
From those mentioned above and the theoretical values, the parameter 𝑎2 was used as a variable todisaggregate the numerical results, ductility versus damping ratio, into the range of values defined inTable 1. In each group, the data were fitted to a new expression (4), similar to the one currently used
(MLIT, 2000), but includes the parameter "𝑏, " and applied during the mainshock and aftershock.

(a) 𝑎2 = 0.25(b) 𝑎2 = 0.85

Figure 3. ℎ − 𝜇 relationship during Aftershock for 𝑎2 = 0.25 and 𝑎2 = 0.85 (Takeda slip).
Additionally, Figure 3 (a), and (b) are presented as part of the result disaggregation during the aftershock.
These graphs include a new value of 𝛾, and "𝑏" for the 50th percentile (𝛾𝑀 ), 16th percentile (𝛾𝑀−𝑆𝐷 ), andfinally, 2.5th percentile (𝛾𝑀−2𝑆𝐷 ), which corresponds to the median minus two-time standard deviation.
For the following procedure, the lower limit, which corresponds to the median minus two standard
deviations (2.5th percentile), will be used for the new expression of calculating the damping ratio. Table
2 summarizes the 𝛾, and 𝑏 values, respectively, during the aftershock and for the 2.5th percentile.
Table 2. The 𝛾 and 𝑏 values during aftershock for the 2.5th percentile.
𝑎2 = 0.25

𝑎2 = 0.40

𝑎2 = 0.55

𝑎2 = 0.70

𝑎2 = 0.85

𝛾

𝑏𝛾

𝑏𝛾

𝑏𝛾

𝑏𝛾

𝑏

Degrading trilinear

0.04

0.08

0.03

0.06

0.03

0.05

0.02

0.03

0.02

0.01

Origin oriented

0.07

0.00

0.05

0.00

0.04

0.00

0.02

0.00

0.03

0.00

Takeda

0.08

0.03

0.10

0.02

0.13

0.00

0.12

0.00

0.13

0.00

Takeda slip

0.14

0.02

0.12

0.02

0.13

0.01

0.12

0.01

0.13

0.00

Model4

4.2. Proposal of New reduction factors
Table 2 presents 𝑏 values during the aftershock; and as can be seen in some models and values ofparameters 𝑎2 , the coefficient 𝑏 takes high values. Such condition may not occur in reality, and in thatsense, it is necessary to adjust the above Expression (4), ignoring coefficient 𝑏 for practical purposes.
In summary, an already known equation will be used to determine the damping ratio but considering
new 𝛾 values for each hysteresis model as presented in Expression (5) during the aftershock and thesame during the mainshock. The 𝛾 values obtained during mainshock (𝛾𝑚𝑠 ) and aftershock (𝛾𝑎𝑠 ) wereselected for each hysteresis model to propose a linear regression and Table 3 shows the new expressions
to determine the reduction factor during the mainshock and aftershock independently.
ℎ = 𝛾𝑎𝑠 (1 −1) + ℎ0√𝜇(5)Table 3. Reduction factor during the mainshock 𝛾𝑚𝑠 and aftershock 𝛾𝑎𝑠 .
ModelMainshock

AftershockDegrading trilinear

𝛾𝑚𝑠 = −0.047 + 0.060(𝑎2 )𝛾𝑎𝑠 = −0.033 + 0.046(𝑎2 )Origin oriented

𝛾𝑚𝑠 = −0.053 + 0.075(𝑎2 )𝛾𝑎𝑠 = −0.073 + 0.082(𝑎2 )Takeda𝛾𝑚𝑠 = 0.067 + 0.133(𝑎2 )𝛾𝑎𝑠 = 0.080 + 0.068(𝑎2 )Takeda slip

𝛾𝑚𝑠 = 0.027 + 0.149(𝑎2 )𝛾𝑎𝑠 = −0.013 + 0.135(𝑎2 )4.3. Proposal of New reduction factors
The 2D RC Frame numerical model was converted into an SDOF system, and the capacity Spectrum
and Primary Curve were obtained. The SDOF model parameters are presented in Table 4, and thereduction factor during the mainshock and aftershock and for the specific 𝑎2 are shown in Table 5.
Table 4. Parameters of SDOF system.

Table 5. The reduction factor 𝛾 for theParameters
𝑇 = 0.77 𝑠
𝑄𝑐 = 261.78 𝑘𝑁
𝑄𝑦 = 858.88 𝑘𝑁
𝑎1 = 0.305, 𝑎2 = 0.276

equivalent SDOF system
Model
TakedaTakeda slip

Figure 4. Comparison of the %𝑒𝑟𝑟𝑜𝑟 “wg60”
and Takeda slip model.

Mainshock𝛾𝑚𝑠 = 0.15
𝛾𝑚𝑠 = 0.16

Aftershock
𝛾𝑎𝑠 = 0.09
𝛾𝑎𝑠 = 0.13

Figure 5. Comparison of the %𝑒𝑟𝑟𝑜𝑟 “El CentroEW” and Takeda slip model.

5Finally, the %𝑒𝑟𝑟𝑜𝑟, between the CSM and NLTHA, for the Takeda Slip model is presented for one
artificial wave “wg60” and one earthquake record “El Centro EW” in Figure 4 and Figure 5, respectively.
These graphs show the difference using 𝛾 equals to 0.06 compared with the results considering the newreduction factor 𝛾𝑎𝑠 during the aftershock.
5. CONCLUSIONSFor practical uses of the new expression to determine the damping ratio (ℎ), the coefficient 𝑏 is ignored
because it takes high values. This condition, high values of 𝑏, may not occur in reality because theaccumulative damage and additional damping ratio at the yielding point cannot be determinedtheoretically. In that sense, it is necessary to adjust the new expression, ignoring the 𝑏 coefficient due
to the uncertainty and being reduced to the expression indicated by MLIT but using the reduction factors
𝛾 determined in this study during the mainshock and aftershock.
The present study shows more accuracy in the determination of the damping ratio during the aftershockand considering the influence of the parameter 𝑎2 , which comes from the definition of a trilinear Primary
Curve and using a hysteresis model representing systems with low energy dissipation. Additionally, theprocedure is summarized as follows, after defining 𝑎2 from the Capacity Curve, the damping ratio (ℎ =𝛾𝑎𝑠 ∙ (1 − 1/√𝜇) + 0.05) is obtained using a new reduction factor during the aftershock (𝛾𝑎𝑠 ) to obtain
the response reduction factor (𝐹ℎ = 1.5/(1 + 10 ∙ ℎ)), the same expression as presented by MLIT, andfinally, determine the damage level for the maximum expected aftershock.
ACKNOWLEDGMENT
I would like to express my gratitude to my supervisor Dr. Koichi Kusunoki and advisor Dr. HaruhikoSuwada for their valuable comments and their guidance throughout the process of this investigation. Inaddition, I would like to thank Dr. Carlos Zavala and Dr. Miguel Diaz from CISMID for their support
and comments in each meeting.

REFERENCES
ATC-40. (1996). Seismic Evaluation and Retrofit of Concrete Buildings: Vol. Report No. SS96-01.
Applied Technology Council, California.
Diaz, M. (2013). Development of the Seismic Performance Estimation during an Aftershockconsidering the Equivalent Damping Ratio and the Response Reduction Ratio [Dr. Thesis].
Yokohama National University.
Diaz, M., Gibu, P., Estacio, L., & Proano, R. (2014). Implementation of Building Monitoring Networkin Peru Under SATREPS Project. Journal of Disaster Research, 9(6), 1001–1007.
https://doi.org/10.20965/JDR.2014.P1001
Kuramoto, H., Teshigawara, M., Okuzono, T., Koshika, N., Takayama, M., & Hori, T. (2000).
Predicting the earthquake response of building using equivalent single degree of freedom
system.
Kusunoki, K. (2006). Analytical Study on Estimation of Equivalent Viscous Damping Ratio foraftershock. Proceedings of the Japan Concrete Institute. In Japanese, 28(2), 1057–1062.
Ministry of Land, I. T. and T. (MLIT). (2000). Notification No. 1457-6: Technical Standard for
Structural Calculation of Response and Limit Strength of Buildings.
Pan, H., & Kusunoki, K. (2020). Aftershock damage prediction of reinforced-concrete buildings using
capacity spectrum assessments. Soil Dynamics and Earthquake Engineering, 129, 105952.
https://doi.org/10.1016/J.SOILDYN.2019.105952
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