1. Preliminary Investigations on Post-earthquake Assessment of Damaged RC Structures Based on Residual Drift
Jianze Wang
Supervisor: Assoc. Prof. Kaoshan Dai
State Key Laboratory of Disaster Reduction in Civil Engineering
May 2015
The 5th Tongji-UBC Symposium on Earthquake Engineering

2. Outline Background and Motivations
Seismic assessment methods based on residual drifts
Application to E-Defense shaking table model
Discussion and Conclusion

3. 1.Background and motivations
Performance-based Assessment
Performance Indicators
Element deformation
Damage Indices
(e.g. Park & Ang)
…………..
Roof drift
Interstory drift
(GB. FEMA-356, ATC-58,
Eurocode-8….)

4. 1.Background and motivations
Roof drift
Interstory drift
(GB. FEMA-356, ATC-58, Eurocode-8….)
Maximum drift took place during earthquake
Residual drift
Unknown
after main-shock
Measurable
Maximum displacement
Residual displacement

5. 2.Seismic assessment methods based on residual drifts
a). Empirical Relations between Maximum and Residual drifts
b). Probabilistic Estimation
………….

6. 2.Seismic assessment methods based on residual drifts
a). Empirical Relations
C 𝑟 =[ 1 𝜃 𝑇 𝜃 2 ]𝛽
𝛽= 𝜃 3 [1−exp(− 𝜃 4 𝑅− 1) 𝜃 5 ]
(Garcia, 2006)
𝑢 𝑚𝑎𝑥 =( 𝑎 1 𝑇+ 𝑎 2 𝑢 𝑟𝑒𝑠 + 𝑎 3 𝑢 𝑟𝑒𝑠 2 + 𝑎 4 𝑇 𝑢 𝑟𝑒𝑠 )×(1+ 𝑎 5 𝐻+ 𝑎 6 𝐻 2 )
(Hatzigeorgiou et.al, 2011)
𝑑 𝑅 =( 𝑎 g )( exp 10 1−𝑟 𝑑 𝑚 𝑑 𝑦 𝑑 𝑚 +(𝑟−1) 𝑑 𝑚 𝑑 𝑦 𝑑 𝑦 −1)
(Takeda)
𝑑 𝑅 =( 𝑎 g )( exp −𝑟 𝑑 𝑚 𝑑 𝑦 𝑑 𝑚 +(𝑟−1) 𝑑 𝑚 𝑑 𝑦 𝑑 𝑦 −1) (Kinematic)
 (Zhang et.al,2013)
d 𝑅 = d 𝑇𝑃 −0.069 𝑎 g 𝑎 g × 10 2 𝑟+3.58
(Gong et.al,2011)
………….

7. 2.Seismic assessment methods based on residual drifts
b). Probabilistic estimation (Yazgan and Dazio, 2012)
Step 1:
Modeling of the structure
Step 2:
Estimation the prior probabilistic distribution of the maximum drift ratio
Step 3:
Updating the maximum drift ratio distribution based on visible damage
Step 4:
Updating the maximum drift ratio distribution based on known residual drift

8. 3.Application to E-Defense shaking table model
A full-scale four-story RC structure model
(Design and instrumentation of the 2010 E-Defense Four-Story Reinforced Concrete and Post-Tensioned Concrete Buildings, Peers, 2011)
Longitudinal Direction (X) : Moment frame system
Transverse Direction (Y): Frame-Shear wall system
Story Height: 3m;
Overall Height: 12m;
All data were download from

9. 3.Application to E-Defense shaking table model
A full-scale four-story RC structure model
Ground motions: JMA-Kobe motions (1995) , scaled by 25%, 50%, 100%
Table. Roof displacements after each scenario
Excitation Input
X direction
Displacements (mm)
Drifts
Maximum
Residual
KOBE-25% 22
0.2
0.184%
0.001%
KOBE-50%
141
2.6
1.181%
0.022%
KOBE-100%
272
9.6
2.269%
0.080%

10. 2.Seismic assessment methods based on residual drifts
Method: a) Empirical Relations
C 𝑟 =[ 1 𝜃 𝑇 𝜃 2 ]𝛽
𝛽= 𝜃 3 [1−exp(− 𝜃 4 𝑅− 1) 𝜃 5 ]
Eq.1
(Garcia, 2006)
Eq.2
𝑢 𝑚𝑎𝑥 =( 𝑎 1 𝑇+ 𝑎 2 𝑢 𝑟𝑒𝑠 + 𝑎 3 𝑢 𝑟𝑒𝑠 2 + 𝑎 4 𝑇 𝑢 𝑟𝑒𝑠 )×(1+ 𝑎 5 𝐻+ 𝑎 6 𝐻 2 )
(Hatzigeorgiou et.al, 2011)
Eq.3
𝑑 𝑅 =( 𝑎 g )( exp 10 1−𝑟 𝑑 𝑚 𝑑 𝑦 𝑑 𝑚 +(𝑟−1) 𝑑 𝑚 𝑑 𝑦 𝑑 𝑦 −1)
(Takeda)
𝑑 𝑅 =( 𝑎 g )( exp −𝑟 𝑑 𝑚 𝑑 𝑦 𝑑 𝑚 +(𝑟−1) 𝑑 𝑚 𝑑 𝑦 𝑑 𝑦 −1) (Kinematic)
 (Zhang et.al,2013)
Eq.4
Eq.5
d 𝑅 = d 𝑇𝑃 −0.069 𝑎 g 𝑎 g × 10 2 𝑟+3.58
(Gong et.al,2011)
………….

11. Calculation result (mm)
2.Seismic assessment methods based on residual drifts
Method: a) Empirical Equtions
In JMA-Kobe-100% test,
the maximum roof displacement(drift) is 272mm(2.26%) in X direction
With the equations, the results are:
Calculation result (mm)
Difference
Eq.1
54.2
80%
Eq.2
16.1
94%
Eq.3
38.5
86%
Eq.4
70.5
74%
Eq.5
140
49%

12. 3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 1:Modeling of the structure
Comparisons between tests and simulations
Perform-3D Nonlinear simulation:
Beam: plastic hinges at member ends
Column: fiber sections
Actual material properties obtained from specimen in tests
Cyclic degradation and strength loss were considered
Kobe-25%-X
Kobe-50%-X
Kobe-100%-X

13. 3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 2: Estimation the prior probabilistic distribution of the maximum drift ratio
Assumption:
Certainties: Structure properties
Uncertainties: Ground motions
Prior probabilistic distribution:
Pr ( M 𝑖 )≈ 𝑛=1 𝑁 Pr M 𝑖 𝑆 𝑛 Pr ( S 𝑛 )
S 𝑛 −𝑛𝑡ℎ 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑖𝑚𝑒 ℎ𝑖𝑠𝑡𝑜𝑟𝑦 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
One structure model
A set of 50 ground motion records
(PEER-NGA database,
Earthquake Name
Year
Station Name Mw
Rjb (km)
Rrup (km)
Vs30 (m/sec)
Kobe Japan
1995
KJMA
6.9
0.94
0.96
312

14. 3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 2: Estimation the prior probabilistic distribution of the maximum drift ratio
Uncertainties extent:
Case 1:
A reliable record is available. (JMA-Kobe)
Sa(T1,ζ=0.05) of JMA-Kobe
Sa(T1,ζ=0.05) of 50 records
Case 2:
MCE Response spectrum is available (USGS)
Sa(T1,ζ=0.05) of spectrum
Sa(T1,ζ=0.05) of 50 records
Case 3:
Just fundamental properties of the seismic event are known.(Mw, Rjb, Site….)
GMPM model (Attenuation relationship)
(Campbell and Bozorgnia, 2007)
Median:0.82g
σln(Sa)=0.58

15. 3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 3: Updating the maximum drift ratio distribution based on visible damage
Damage description: (After JMA-Kobe-100%)
2.5mm shear crack width in interior beam-column joints
1.1mm shear crack width in exterior beam-column joints
250mm height of cover concrete spalled in first story
(Nagae et.al, 2012)

16. Structural Performance Levels
3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 3: Updating the maximum drift ratio distribution based on visible damage
Performance levels taken from FEMA-356
Element
Type
Structural Performance Levels
Collapse Prevention
Life Safety
Immediate Occupancy
Primary
Extensice cracking and hinge formation in ductile elements. Limited cracking and/or splice failure in some nonductile columns. Severe damage in short columns
Extensive damage to beams. Spalling of cover and shear cracking (<0.32mm) for ductile columns. Minor spalling in nonductile columns. Joint cracks <0.32mm wide.
Minor hairline cracking. Limited yielding possible at a few locations. No crushing (strains below 0.003).
Secondary
Extensive spallings in columns and beams. Severe joint damage. Some reinforcing buckled.
Extensive cracking and hinge formation in ducttile elements. Limited cracking and/or splice failure in some nonductile columns. Severe damage in short columns.
Minor spalling in a few places in ductile columns and beams. Flexural cracking in beams and columns. Shear cracking in joints<0.16mm.
Drift
4% transient
2% transient
1% transient
Assume uniform distribution : 𝐼= 𝐼 1 ∩ 𝐼 2 = 2%≤𝑀𝐴<4%
Updated maximum drift ratio distribution: Pr 𝑀 𝑖 𝐼 1 ∩ 𝐼 2 = Pr 𝐼 2 𝑀 𝑖 ∩ 𝐼 1 Pr⁡ 𝑀 𝑖 𝐼 1 𝑗 Pr 𝐼 2 𝑀 𝑗 ∩ 𝐼 1 Pr⁡ 𝑀 𝑖 𝐼 1

17. Joint probability of max and residual drift given on visible damage:
3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Step 4: Updating the maximum drift ratio distribution based on known residual drift
Pr M 𝑖 ∩ R 𝑗 𝐼 = Pr 𝐼 𝑀 𝑖 ∩ R 𝑗 Pr (M 𝑖 ∩ R 𝑗 ) 𝑖 𝑗 Pr 𝐼 𝑀 𝑖 ∩ R 𝑗 Pr (M 𝑖 ∩ R 𝑗 )
Joint probability of max and residual drift given on visible damage:
Case 1:
Case 3:
Case 2:

18. 3.Application to E-Defense shaking table model
Method: b) Probabilistic Estimation
Prior distribution
Updated distribution based on visible damage
Updated distribution based on residual drift
Roof Drift After JMA-Kobe-100% X direction
Test: %
Estimation (Maximum Probability):
Case 1: in the range 2.25%-2.5%
Case 2: in the range 2%-2.25%
Case 3: in the range 2.75%-3%

19. Post-earthquake assessment
3.Application to E-Defense shaking table model
Post-earthquake assessment
(Residual seismic capacity Sa,cap)
Residual roof drift
Maximum roof drift
With the help of SPO2IDA spreadsheet tool, IDA curves could be derived.
(Vamvatsikos amd Cornell, 2002.)
IDA curves
Pushover for intact and damaged structures

20. (considered residual drift)
3.Application to E-Defense shaking table model
Seismic capacity Sa,cap(T1) (Median,50th)
Structure
Sa,cap(T1)(g)
INTACT
2.93
DAMAGED
2.04
(considered residual drift)
2.86
Aleatory uncertainty (βR): 0.45
Epistemic uncertainty (βU): 0.40
16th
Median(50th)
84th
Fragility Curves：

21. 4.Discussion and Conclusion
Empirical relations between maximum and residual drift derived from simulations are unapplicable to result of the E-Defense shaking table test. The dispersion of residual drift should be considered.
Probabilistic estimation could predict the maximum roof drift more accurately if residual roof drift and visible damage are available.
In order to get better assessment results, the uncertainties of structure and ground shaking should be evaluated and quantified reasonably.
The maximum drift distribution of structural damage used in step 3 of probabilistic estimation should be developed to a more accurate model.

22. Thanks