Proceedings of the 8 th U.S. National Conference on Earthquake Engineering April 18-22, 2006, San Francisc o, California, US A AN APPLICATION OF PEER PERFORMANCE-BASED EARTHQUAKE ENGINEERING METHODOLOGY T.Y. Yang 1 , J. Moehle 2 , B. Stojadinovic 3 and A. Der Kiureghian 3 ABSTRACT Earthquake engineering has evolved from a set of prescriptive provisions, indirectly aimed at providing life safety, to performance-based approaches with direct consideration of a range of performance objectives. Performance-based approaches have several advantages, including more comprehensive consideration of the various performance metrics that might be of interest to stakeholders, more direct methods for computing performance, and involvement of stakeholders in deciding acceptability. Whereas engineers are familiar with performance measures such as drift, acceleration, strain, and perhaps damage state, many decision-makers prefer performance metrics that relate more directly to business decisions, such as downtime or repair costs. An engineering challenge has been to consistently consider seismic hazard, structural response, and resulting damage and consequences, so that a fully probabilistic statement of expected performance can be made. A rigorous yet practical approach to performance-based earthquake engineering has been pursued and demonstrated through an example building. The approach considers the seismic hazard, structural response, resulting damage, and repair costs associated with restoring the building to its original condition, using a fully consistent, probabilistic analysis of the associated parts of the problem. The approach could be generalized to consider other performance measures such as casualties and down time, though these have not been pursued at this time. The procedure is organized to be consistent with conventional building design, construction, and analysis practices so that it can be readily incorporated as a design approach. Sample results demonstrate the expected repair costs and their distribution among various building components, illustrating how the results could be used to guide decisions about investment or about structural design. Introduction Traditional structural engineering design focuses on adherence to a set of technologies and prescribed means as an indirect way of achieving acceptable building performance. The means usually are prescribed by the governing building code, using engineering criteria that relate to quantities such as strain, strength, and lateral drift. Future performance is largely an undefined byproduct of the design. Performance-based earthquake engineering (PBEE) focuses 1 Graduate Research Assistant, Pacific Earthquake Engineering Research Center, Univ. of California, Berkeley. 2 Professor and Director, Pacific Earthquake Engineering Research Center, University of Cal ifornia, Berkeley. 3 Professor, Pacific Earthquake Engineering Research Center, University of Cali fornia, Berkeley. Paper No. 1448
U.S. National Conference on Earthquake Engineering
April 18-22, 2006, San Francisco, California, USA
AN APPLICATION OF PEER PERFORMANCE-BASED EARTHQUAKE
ENGINEERING METHODOLOGY
T.Y. Yang1, J. Moehle2, B. Stojadinovic3 and A. Der Kiureghian3
ABSTRACT
Earthquake engineering has evolved from a set of prescriptive provisions,indirectly aimed at providing life safety, to performance-based approaches withdirect consideration of a range of performance objectives. Performance-basedapproaches have several advantages, including more comprehensive considerationof the various performance metrics that might be of interest to stakeholders, moredirect methods for computing performance, and involvement of stakeholders indeciding acceptability. Whereas engineers are familiar with performancemeasures such as drift, acceleration, strain, and perhaps damage state, many
decision-makers prefer performance metrics that relate more directly to businessdecisions, such as downtime or repair costs. An engineering challenge has been toconsistently consider seismic hazard, structural response, and resulting damageand consequences, so that a fully probabilistic statement of expected performancecan be made.
A rigorous yet practical approach to performance-based earthquake engineeringhas been pursued and demonstrated through an example building. The approachconsiders the seismic hazard, structural response, resulting damage, and repaircosts associated with restoring the building to its original condition, using a fullyconsistent, probabilistic analysis of the associated parts of the problem. Theapproach could be generalized to consider other performance measures such as
casualties and down time, though these have not been pursued at this time. The procedure is organized to be consistent with conventional building design,construction, and analysis practices so that it can be readily incorporated as adesign approach. Sample results demonstrate the expected repair costs and theirdistribution among various building components, illustrating how the resultscould be used to guide decisions about investment or about structural design.
Introduction
Traditional structural engineering design focuses on adherence to a set of technologies
and prescribed means as an indirect way of achieving acceptable building performance. Themeans usually are prescribed by the governing building code, using engineering criteria thatrelate to quantities such as strain, strength, and lateral drift. Future performance is largely anundefined byproduct of the design. Performance-based earthquake engineering (PBEE) focuses
1 Graduate Research Assistant, Pacific Earthquake Engineering Research Center, Univ. of California, Berkeley.2 Professor and Director, Pacific Earthquake Engineering Research Center, University of California, Berkeley.3 Professor, Pacific Earthquake Engineering Research Center, University of California, Berkeley.
frames, semi-rigid connections are modeled by reducing the beam stiffness to half the beamflexural stiffness and limiting the strength to one-fifth of the beam moment strength. Gravityload is uniformly distributed throughout the building. The P- ∆ effect is accounted for in thenonlinear dynamic analysis.
Major structural and nonstructural components of the building are identified and
separated into different performance groups. Each performance group consists of one or more building components whose performance is similarly affected by a particular engineeringdemand parameter. For example, one performance group might consist of all similarnonstructural components whose performance is sensitive to inter-story drift between the secondand third floor. Table 1 shows the performance groups used in this study.
The structural components are assigned to performance groups whose performance isassociated with inter-story drift ratio in the story where the components are located. Thenonstructural components and contents of the building are subdivided into displacement-sensitive and acceleration-sensitive groups. The displacement-sensitive groups use inter-storydrift to define the performance of the group, while the acceleration-sensitive groups use absoluteacceleration at the different floor and roof levels to define the performance.
Table 1. Summary of the performance groups (PG) identified in the prototype building.
where ∆ui = inter-story drift at the ith
story and ai = absolute acceleration at the ith
floor.
Ground Shaking Hazard
Ground motions that represent the hazard at the University of California, Berkeleycampus are selected from the U.C. Berkeley Seismic Guidelines (UCB 2003). Tables 2 andTable 3 summarized the list of the ground motions used for the dynamic analysis. For thisexample, ground motions are scaled to match the target spectrum at the first-mode period of the
PG # PG Name Location EDP Components
1 SH12 between levels 1 and 2 ∆u1
2 SH23 between levels 2 and 3 ∆u2
3 SH3R between levels 3 and R ∆u3
Structural: lateral load resistingsystem
4 EXTD12 between levels 1 and 2 ∆u1
5 EXTD23 between levels 2 and 3 ∆u2
6 EXTD3R between levels 3 and R ∆u3
Exterior enclosure: panels,glass, etc.
7 INTD12 between levels 1 and 2 ∆u1 8 INTD23 between levels 2 and 3 ∆u2
structure. Alternative approaches to scaling ground motions can be used. Fig. 2 shows the scaledresponse spectra and the scaling factors for the ground motions used in this analysis.
Response Quantification
With the selected ground motions, a series of nonlinear dynamic analyses are used todetermine engineering demand parameters (EDPs) that will be used with fragility relations todefine performance of the different performance groups in Table 1. From the results of thenonlinear dynamic analysis, peak structural responses are identified and summarized into EDPmatrices, one for each hazard level. Because each row of the EDP matrix is calculated bydynamic analysis of the building for a single ground motion, the EDPs in each row arecorrelated. The EDP matrix can be extended by considering any number of EDPs and anynumber of ground motions. Table 4 shows the peak building response for the prototype buildingfor the three hazard levels considered.
Table 2. Ground motions representing the 50% in 50 years hazard level.
Earthquake Mw Station Distance Site RecordCoyote Lake, Dam Abutment 4.0 C CL_clydCoyote Lake,
1979/6/85.7
Gilroy # 6 1.2 C CL_gil6
Temblor 4.4 C PF_temb
Cholome Array # 5 3.7 D PF_cs05
Parkfield,
1996/6/276.0
Cholome Array # 8 8.0 D PF_cs08
Livermore,
1980/1/275.5 Morgan Territory Park 8.1 C LV_mgnp
Coyote Lake, Dam Abutment 0.1 C MH_clyd
Anderson Dam, Downstream 4.5 C MH_andd
Morgan Hill,
1984/4/24 6.2 Hall Valley 2.5 C MH_hall
Table 3. Ground motions representing 10% in 50 years and 5% in 50 years hazard levels.
Earthquake Mw Station Distance Site Record
Los Gatos Present Center 3.5 C LP_lgpc
Saratoga Aloha Ave 8.3 C LP_srtg
Corralitos 3.4 C LP_cor
Gavilan College 9.5 C LP_gav
Gilroy Historic Building C LP_gilb
Loma Prieta,1989/10/17
7.0
Lexington Dam Abutment 6.3 C LP_lex1Kobe, Japan1995/1/17
4Response Spectrum 50% in 50 y ears (SMRFX T1 = 1.139 sec)
Period [Sec]
S a
[ g ]
Target (maxAg,SF)
CLclyd (0.75g,2.68)
CLgil6 (0.27g,0.60)
PFtemb (0.53g,1.43)
PFcs05 (0.52g,1.57)
PFcs08 (0.55g,2.25)
LVmgnp (0.88g,2.96)
MHclyd (0.44g,0.50)
MHandd (0.88g,2.01)
MHhall (0.22g,0.73)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5Response Spectrum 10% in 50 y ears (SMRFX T1 = 1.139 sec)
Period [Sec]
S a
[ g ]
Target (maxAg,SF)
LPlgpc (0.51g,0.79)
LPsrtg (0.46g,1.28)
LPcor (0.81g,1.67)
LPgav (1.11g,3.79)
LPgilb (0.66g,2.35)
LPlex1 (0.21g,0.47)
KBkobj (0.42g,0.49)
TOhino (0.59g,0.56)
EZerzi (0.59g,1.23)
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6Response Spectrum 5% in 50 years (SMRFX T1 = 1.139 sec)
Period [Sec]
S a
[ g ]
Target (maxAg,SF)
LPlgpc (0.66g,1.02)
LPsrtg (0.60g,1.65)
LPcor (1.04g,2.16)
LPgav (1.44g,4.88)
LPgilb (0.85g,3.03)
LPlex1 (0.28g,0.61)
KBkobj (0.55g,0.64)
TOhino (0.77g,0.72)
EZerzi (0.76g,1.59)
Figure 2. Scaled response spectra for the three different hazard levels.
Computing additional EDP realizations using nonlinear dynamic analysis is hampered byin the paucity of recorded strong ground motions. Therefore, instead of running additionalnonlinear dynamic analyses, a joint lognormal distribution is fitted to the EDP matrix. Additionalcorrelated EDP vectors are generated using the correlation matrix and artificially generatedstandard normal random variables (u). To do this, the EDP matrix (as shown in Table 4), X, isfirst assumed to have joint lognormal distribution. The EDP matrix is then transformed to anormal distribution, Y, by taking the natural log of the EDP matrix. The mean vector, standard
deviations and the correlation coefficient matrix are then sampled from Y. With the generatedstandard normal random variables, u, additional correlated EDP vectors, y, are generated usingEq. 5. Finally, the generated EDP vector is transformed to the lognormal distribution, x, bytaking the exponential of y. Eq. 1, 2, 3 and 4 shows the formulas used for the statistical sample.Fig. 3 shows the process of obtaining additional correlated EDP vectors.
Standard Deviation Matrix: ˆ ( ( )) D diag std Y = (2)
Correlation Coefficient Matrix: ˆ ( ) R corrcoef Y = (3)
Cholesky Factorization of correlation coefficient matrix:
( )
ˆ( )T
L chol R= (4)
ln(EDP) vector: ˆ ˆ* * y D L u M = + (5)
X
x y
ln
exp
u
R D M ˆ,ˆ,ˆY X
x y
lnln
expexp
u
R D M ˆ,ˆ,ˆY
X = EDP matrix for a given IM level.
Y = ln(X).
u = generated standard normal random variables.
y = simulated ln(EDP data).
x = exp(y) = simulated EDP data.
Figure 3. Correlated EDP generator.
Damage Assessment
Different damage states are defined for each performance group in Table 1. The damagestates are defined in relation to the repair actions. For each damage state, a damage model(fragility relation) defines the probability of damage being equal to or greater than the thresholddamage given an EDP. Fig. 4 shows an example of the fragility curves defined for the first three performance groups. Depending on the EDP values, the probabilities of the performance group being in each damage state can be computed. A uniformly distributed random number generatoris used to select the damage state for the performance group, given the EDP. Once the damagestate is identified, the repair quantities for each of the performance groups are located from alookup table, as shown in Fig. 4.
Figure 4. Performance group fragility curves and the associated repair quantities.
The process is repeated for each performance group to identify the total repair quantitiesfor each item in the building. In the methodology adopted here, the simulation procedure
described is used to generate a single realization of total repair quantities for the scenarioearthquake. The process is repeated a sufficiently large number of times to obtain arepresentative sample of the total repair quantities. For this simplified example, the performancegroups are assumed to be statistically independent.
Loss Calculation
Once the total repair quantities are identified, the total repair cost for the building iscomputed by multiplying the total repair quantity by the unit price obtained from a look-up table.The price uncertainty is represented by using a random number generator, based on the tabulated“beta” factors for the cost functions, to adjust base unit costs up or down before multiplying bythe total quantities associated with each repair measure. Fig. 5 shows the tri-linear cost functionthat represented the relation between unit cost and quantity. This is the repair cost for onerealization of EDPs. The process is repeated for all the calculated total repair quantities to obtaina distribution of costs given the hazard level represented by the IM.
Unit Cost, $
Quantity
C i
Qi
Uncertainty
Figure 5. Cost function model for repair measures for the example building.
Fig. 6 shows the fitted lognormal distribution of the building repair cost for four differentIM levels. (Repair cost for 36-yr return period was obtained by assuming linear response of thestructure and scaling the EDP vectors according to the ratio of the hazard curves at the first mode period, for the 36-yr and 72-yr return periods.) Curves such as these can be used to quantify theannual frequency of the total repair cost exceeding a given threshold as follows: the complementof each CDF (cumulative distribution function) curve presented in Fig. 6 is multiplied by theslope of the hazard curve at the corresponding IM level; the resulting curves are integrated acrossIM levels. Fig. 7 shows the annual rate of exceeding various total repair cost thresholds for allthe IM levels. Furthermore, the mean cumulative annual total repair cost can be obtained byintegrating the loss curve shown in Fig. 7 along the range of repair cost thresholds (Der
Kiureghian 2005). For this simplified example, the expected mean cumulative annual total repaircost is approximately $US 31,300.
Loss Curve (Mean Cumulative Annual Total Repair Cost = $ 3.13e+004)
In addition, with the procedure presented above, the total repair cost can be deaggregatedto identify which performance groups contribute most to the total repair cost. Such information
can be useful in making design decisions. Fig. 8 shows an example of the deaggregation of thetotal repair cost for the 975 return period hazard level. It shows performance groups 1 to 6contribute most to the total repair cost. As these performance groups are displacement-sensitive,this suggests a more economical design (considering downstream consequences) might beachieved by a design that reduces lateral drifts. Analyses for a stiffer structure likely would showreduced costs for displacement-sensitive performance groups but increased costs foracceleration-sensitive performance groups. Therefore, tradeoffs between stiff and flexible building systems can be considered.
12
34
56 7 8
910
1112
1314
1516
0
5
10x 105
0
0.5
1
PG(s)
Distribution of repair cost for each performance group(s)
Total repair cost ($C)
p r o b a b i l i t y
Figure 8. Cost function model for repair measures for example building.
Figure 7. Loss curve.Figure 6. CDF for P(TC<=$C|IM).
Other information, such as the expected repair cost for a M7 earthquake scenario and probability that repair cost exceeds a given threshold for a 975-yr return period hazard level, can be obtained using the procedure presented above.
Conclusion
Performance-based earthquake engineering can be extended to express performanceusing metrics that are of direct interest to engineers and to non-engineer stakeholders. Amethodology is illustrated through a simplified example, in which a prototype building near theUniversity of California, Berkeley campus is designed and analyzed. Suites of ground motionsthat represent the hazard are used in a series of nonlinear dynamic analyses to determine peak building responses. Major structural and nonstructural components are identified and separatedinto different performance groups. Depending on the building response, different damage statesand the corresponding repair costs for all performance groups are identified. Simple statisticalsimulation procedures are used to efficiently generate large numbers of cost realizations, makingit possible to describe probabilities of repair costs exceeding threshold values. The proceduresdescribed are rigorous and can be extend to other structural types and performance measures.
Acknowledgment
R. Hamburger and A. Dutta (SGH-San Francisco) developed the building design. C.Comartin, A. Whittaker, B. Bachman and G. Hecksher (ATC-58 project team, funded by FEMA) provided fragility relations, materials quantities, and unit costs. Seismic hazard data weredeveloped by URS for the University of California, Berkeley; Capital Projects Office is thankedfor allowing their use. This work was supported in part by the Earthquake Engineering ResearchCenters Program of the National Science Foundation under award number EEC-9701568through the Pacific Earthquake Engineering Research Center (PEER). Any opinions, findings,and conclusion or recommendations expressed in this material are those of the authors and do not
necessarily reflect those of the National Science Foundation.
References
Der Kiureghian, A. 2005. Non-ergodicity and PEER’s framework formula. Earthq. Engrg. Struct. Dyn.
International Code Council. 2003. International Building Code, International Code Council, Fall Church,VA, USA .
UCB. 2003. U.C. Berkeley seismic guideline, University of California at Berkeley , Berkeley, California,USA
