SOME DECLENSIONS OF EISMIC ISK ANALYSIS FOR … · SOME DECLENSIONS OF SEISMIC RISK ANALYSIS FOR...

SOME DECLENSIONS OF SEISMIC RISKANALYSIS FOR PERFORMANCE BASEDANALYSIS FOR PERFORMANCE‐BASED

EARTHQUAKE ENGINEERING

Iunio IervolinoIunio Iervolino

Dipartimento di Ingegneria Strutturale, U i i à d li S di di N li F d i IIUniversità degli Studi di Napoli Federico II

Topics1. REXEL: Computer aided code‐based record selection

Topicso pu e a ded code based eco d se ec o

Disaggregation MapsConditional hazard analysis for secondary IMs

2. Early warning as Real‐Time Performance‐Based Earthquake Engineering (RTPBEE);Real‐Time Probabilistic seismic hazard analysisReal Time Probabilistic seismic hazard analysisExpected‐Loss approach to early warning

3. Near‐Source probabilistic seismic hazard analysisDirectivity evidences in near source L’Aquila earthquake ground motion

4. Seismic risk analysis of classes of buildingsElastic period of RC building classes

REXEL Conditional Haz. NSPSHARTPBEEDisag-Maps Stock Risk Assessment

REXEL: computer aided code‐basedrecord selection


An investigation about Eurocode 8 provisions for record g pselection

The set of accelerograms, regardless they are natural, artificial or simulated h ld h h f ll i i ishould match the following criteria:

- a minimum of 3 accelerograms should be used (7 to consider the mean effects on the structure);mean effects on the structure);

- in the range of periods between 0,2T1 and 2T1, the average spectrum cannot underestimate more than 10% elastic response spectrum from the codethe code.

0 ÷ 2.0 sec


At the time only three hazard levelsAt the time only three hazard levels were possible:p

Th l i th PGAThe ag values is the PGA with 475 years return period

on rock.

HazardHazard level/Zone ag

1 0.35g2 0.25g3 0.15g


What we found?What we found?

Iervolino I., Maddaloni G., Cosenza E. (2008a). Eurocode 8 compliant real record sets for seismic analysis of structures. Journal of Earthquake Engineering, 12(1):54‐90.Iervolino I., Maddaloni G., Cosenza E. (2009a). A note on selection of time‐histories for seismic analysis of bridges in Eurocode 8. Journal of Earthquake Engineering. (in press)


What we found?it is almost unfeasible for practitioners to search in large databases to find i f l d ( ll l i ) h

What we found?

suites of seven real records (eventually multi‐component) whose average matches closely the design spectral shape without a specific software tool;

the spectra based on seismic zonation may be too severe in a way that do not exist suites of unscaled records whose average has such spectral shape;

it is not easy to control the variability (very large) of individual spectra in a combination, and this, in the fortunate case combinations are found, may impair the confidence in the estimation of the seismic performance usingimpair the confidence in the estimation of the seismic performance using such set;

the requirement of selecting records consistent with the earthquake eventsthe requirement of selecting records consistent with the earthquake events dominating the hazard at the site (i.e., the design earthquakes) requires PSHA/disaggregation data and skills seldom available to practitioners;

Iervolino I., Maddaloni G., Cosenza E. (2008). Eurocode 8 compliant real record sets for seismic analysis of structures. Journal of Earthquake Engineering, 12(1):54‐90.Iervolino I., Maddaloni G., Cosenza E. (2009). A note on selection of time‐histories for seismic analysis of bridges in Eurocode 8. Journal of Earthquake Engineering. (in press)


Seismic action in the new Italian codeSeismic action in the new Italian codeThe building code links the seismic design actions on structures directly to the PSHA results. In fact, the Istituto Nazionale di Geofisica e Vulcanologia(INGV) evaluated probabilistic seismic hazard for each node of a regular grid having 5km spacing and covering the whole Italian territory with over 103g p g g ynodes.

This resulted in hazard curves in terms of PGA (also disaggregation is provided for PGA) and spectral acceleration, Sa(T), for 10 different periods from 0.1 to 2 seconds Hazard curves are lumped in 9 probabilities of exceedance in 50 yearsseconds. Hazard curves are lumped in 9 probabilities of exceedance in 50 years (from 2% to 81%). All data can be accessed at http://esse1‐gis.mi.ingv.it.

NIBC acknowledges these data defining design spectra completely sitedependent, which, although given with standard functional forms, practicallycoincide with uniform hazard spectra (UHS) on rock for the site in question.coincide with uniform hazard spectra (UHS) on rock for the site in question.


The new italian seismic code now allowsThe new italian seismic code now allowsthe use of UHS for designg


Provisions of Italian code about recordProvisions of Italian code about record selection

As for EC8, the signals can be artificial, simulated and natural accelerograms. Three is still the minimum number of records to use, and seven allows the consideration of the mean effects on the structure as design value (rather than the maxima).

For the artificial records also the main condition the set of records should satisfy is the same as per EC8.

Real records or accelerograms generated trough a physical simulation of earthquake process, may be used, provided that, again as in EC8, the samples used are adequately qualified with regard to the seismogenetic features of the source and the soil conditions appropriate to the site. Selected real records have to be scaled to approximate somehow the elastic response spectrum in a range of period of interest for the structure.


Computer aided code‐based record selectionComputer aided code based record selection

Tolerances

Definition ofthe return O tithe returnperiod ofinterest

Options

Type ofsearchsearch

Records among to search amongIervolino I., Galasso C., Cosenza E. (2009c) REXEL 2.31 (beta) e la selezione normativa dell’input sismico per l’analisi dinamica non linearedelle strutture. Proc. of XIII Convegno ANIDIS, Bologna, Italy. (in Italian)


REXEL databases


Integration of REXEL and ITACA >

AVANZAMENTI

Integration of REXEL and ITACA ‐> REXELiteREXELite


Design earthquakes mapsDesign earthquakes maps


PSHA for southern Appennines (1)PSHA for southern Appennines (1)• Official seismic zoning ZS9 (Meletti et al., 2008)g ( )

• Hystorical completeness catalogue parameters

• Attenuation from Sabetta e Pugliese (1996)Attenuation from Sabetta e Pugliese (1996)

Zona α (eventi/anno) b Mmax

925 0.17 -0.75 6.83

926 0.09 -1.38 6.14

927 0.69 -0.72 7.06

928 0.21 -0.66 5.91


PSHA for southern Appennines (2)PSHA for southern Appennines (2)

• PGA and Sa(T=1s) 10% in 50 contours.


Disaggregation for site S1Disaggregation for site S1


Disaggregation for site S2Disaggregation for site S2


Design earthquake mapsDesign earthquake mapsPGA Sa(T=1s)

I° M d II° M d I° M d II° M d( )

I° Modo II° Modo I° Modo II° Modo

Convertito V., Iervolino I., Herrero A. (2009) The importance of mapping the design earthquake: insights for southern Italy. Bulletin of the Seismological Society of America. (in press)


Modal values from disaggregationModal values from disaggregation

PGA Sa(T=1s)

Convertito V., Iervolino I., Herrero A. (2009) The importance of mapping the design earthquake: insights for southern Italy. Bulletin of the Seismological Society of America. (in press)


Conditional hazard analysis for secondaryIMs


Duration Related MeasureDuration Related Measure

Cosenza and Manfredi IM: Arias intensitydivided by PGA timesdivided by PGA timesPGV which has shown a good correlation with

li lcyclic structuralresponse.

t2( )

Eta t dt∫

0DI PGA PGV= ⋅

∫

Iervolino I., Manfredi G., Cosenza E. Ground motion duration effects on nonlinear seismic response. Earthquake Engineering and StructuralDynamics, 35:21–38, January 2006.


Attenuation for Id (1)The log of ID is a linear combination of the logs of the terms

Attenuation for Id (1)The log of ID is a linear combination of the logs of the termscomprised in its definition:

Sabetta and Pugliese (1987) attenuation for PGA, PGV and IA:


Attenuation for Id (2)Attenuation for Id (2)

a b c d σ0.668 − 0.011 1.717 − 0.039 0.197


Joint distribution for Id and PGAJoint distribution for Id and PGA

(a) (b)

ff(M, R)(M, R)ff(R)(R)ff(R)(R)


Conditional distribution of Id given PGA[ | ] [ | , * , * ]D D D DP I i P G A p g a P I i p g a M R> = >∼

Conditional distribution of Id given PGA


RTPBEE for Early Warning SystemsRTPBEE for Early Warning Systems


Basic elements of an EEWSBasic elements of an EEWS

•A seismic sensor network with real time capabilities;•A seismic sensor network with real‐time capabilities;

•An unit, local or central, to process the data of the sensor network and to eventually issue the alarm;

•A transmission infrastructure to the structure system or•A transmission infrastructure to the structure, system or community to alert;

•An automated system aimed at risk reduction for structures?


The Ignalina Nuclear Power PlantThe Ignalina Nuclear Power Plant

I th I li NPP th EEWS iIn the Ignalina NPP the EEWS is made up of a number of stations circling around the plant (Wieland et al 2000)et al. 2000).

Each station is made up of three sensors triggering at 0.025g. A 2‐

30km

out‐of‐3 logic is used to determine if a real seismic event has occurred.

Then an alarm is generated in the 30kmThen an alarm is generated in the reactor control room. Two seconds are needed to reduce the reactor capacity and prevent a meltdown p y pduring a severe accident. The false alarm issue is reduced by the

redundancy of measurement at the same geographic location.g g p


The Tohoku Shinkanzene o o u S a e

The Tohoku Shinkanzen is furnished with an EEWS which stops the trains if a

Morioka

certain acceleration treshold is exceeded at any of the accelerometers controlling aspecific segment of the track; this is to protect the trains from derailments.

The system was causing several delays due to a severe false alarm issue.

Tokio

Papadimitriou & Veneziano (1995) proposed a Tokioprobabilistic calibration of three different decisional rules strongly reducing the delay rate.


Regional Earthquake Early Warning Systems andRegional Earthquake Early Warning Systems and the ISNet Irpinia Seismic Network (Italy)

0 25 50 km0 25 50 km

Avellino

BeneventoCaserta

Napoli

SalernoPotenza

1981 – 2002 SeismicityINGV Seismic Catalogue

Earthquake M>3Earthquake M<31980 Earthquake Ms 6.9Seismogenetic faults of 1980 EarthquakeCurrent Seismic NetworkSeismic network under construction (2006)

Commonly used to give distributed estimates of the ground motion right after the

Main CityUrbanized area

Commonly used to give distributed estimates of the ground motion right after the event: SHAKEMAPS.


Site‐Specific Warning by Regional Networks:Site Specific Warning by Regional Networks: Hybrid EEWS

Seismic network

Structural/non‐structural performance/loss

iEDP (i.e. Maximum Interstory Drift Ratio)

Epicenter

Source‐to‐site distance

IM (i.e. PGA)

Ground motion atmotion at the site

Signal at the network stations

BECAUSE OF REAL‐TIME SEISMOLOGY!


RTS: Rapid estimation of event magnitudeRTS: Rapid estimation of event magnitude

MT

Seismologists (i.e. Allen & Kanamori, 2003) claim it is possible to estimate the magnitude from the predominant period (τ) of the first 4 sec of the P‐wave velocity recording


RTS: Rapid estimation of event locationRTS: Rapid estimation of event location

EpicenterEpicenter

TriggeredTriggered Stations

Other seismologists (i.e. Zollo et al., 2007) claim it is possible to locate the hypocenter with negligible uncertainty within 4 sec from the event origin time


Real‐Time Probabilistic Hazard Analysis (RTPSHA) for

PDF of distance dueOrdinary

Real‐Time Probabilistic Hazard Analysis (RTPSHA) forHybrid EEWS

PDF of distance due to rapid localization method

Ordinary Attenuation relationship

( ) [ ] ( ) ( )1 2 1 2| , ,..., | , ,...,| | , | |M R s s sf PGA f PGA m r f m f r s dr dm

ν ντ τ ττ = τ∫ ∫( ) [ ] ( ) ( )1 2 1 2| , ,..., | , ,...,M R s s s

M Rν ντ τ τ∫ ∫

PDF of magnitude Distribution of PGA at Negligible uncertainty

conditional on the measures of the seismic instruments

the site conditional on the measures of the seismic instruments

uncertainty

Iervolino I., Convertito V., Giorgio M., Manfredi G., Zollo A. (2006) Real time risk analysis for hybrid earthquake early warning systems. Journal of Earthquake Engineering, 10(6): 867–885. Convertito V., Iervolino I., Manfredi G., Zollo A. (2008) Prediction of response spectra via real‐time earthquake measurements. Soil DynEarthquake Eng, 28:492–505.


Magnitude’s distributionMagnitude s distribution

( ) ( ) ( ) ( )2 2

ilog l l2 ln τ 2

ν

τμ νμ σ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⋅ −⎜ ⎟⎜ ⎟∑

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )ilog log log

i 1

2 2ilog log log

i 12 ln τ 2

| |τ τ τ

ν

τ τ τ

μ μβ

μ νμ σβ

τ τ=

⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⋅ −⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠

∑

= =∑

∫MAX

m

Mm

e ef m f m

di 1 β=⎝ ⎠⎝ ⎠∫MIN

m

M

e e dm

Gutenberg‐RichterMeasurements gMeasurements

τ[s]

The mean of the tau network

f(M

)

The mean of the tau network measurements is all we need to estimate

h i d !the magnitude!

3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8

MMagnitudeMagnitude


M=6 R=110km Event Simulation

8 stationst = 6s 23 stationst = 9s 28 stationst = 11s 30 stationst = 12s 2 stationst = 5s


Real‐Time Probabilistic Seismic Hazard AnalysisReal Time Probabilistic Seismic Hazard Analysis(RTPSHA) ‐ Summary

NaplesEstimation of Magnitude

Estimation of Distance

Estimation of PGA at the sitePGA at the site


When to activate security measures?

1

When to activate security measures? Decisional Rules

0 8

0,9

1

[ ]Alarm if P PGA PGA P> >

0 6

0,7

0,8

]

[ ]C CAlarm if P PGA PGA P> >

0,4

0,5

0,6

P[PG

A>P

GAc]

ALARM ! Because the probability that PGA exceeds

0,2

0,3

0,4

Pcthe limit value is too high

0

0,1

,

PGAc

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7PGA [m/s2]


False and Missed Alarm Probabilities{ }{ }: True CMA no Alarm PGA PGA⎧ ∩ >⎪

⎨

False and Missed Alarm Probabilities

{ }: True CFA Alarm PGA PGA⎨

∩ ≤⎪⎩

Iervolino et al., 2006.


Event detected on 19/11/2008 – 8.17 PM


Lead time maps for the case study region can beLead‐time maps for the case‐study region can be superimposed to real‐time risk reduction actions for specific

structural systemsstructural systems.


A school classrom equipped with an EEWS terminal: howq ppto set the alarm threshold

Let’s consider a simple school class equipped with a ringer and suppose that the students are trained to shelter under the desks when the alarm is issued.

Desk

mm6 6

7 m7 m

Lighting36 m2


What causes loss?What causes loss?a) Structural collapse (DS)

b) No structural damage, but collapse of lighting (NDS)

a) No structural damage, no lighting damage (loss due to false alarm)

Desk

6 m

6 m

Lighting36 m2

7 m7 m

36 m2


R l Ti l tReal‐Time loss assessmentE t di th h d h it i ibl t d t i th t dExtending the hazard approach it is possible to determine the expectedlosses condiotioned to the measurements of the seismic network in the

case of alarming or not

[ ] ( )( ) ( ) ( ) ( )| , |L D EDP PGA

E L l f l d g f d edp f edp pga f pga dL dD dEDP dPGAτ τ τ= ∫ ∫ ∫ ∫

Expected pLoss

1. Loss probability depending on the alarmin de ision

2. Structural damage probability depending on b ildin ’s seismi

3. Seismic response probability depending

on ha ard

4. Real‐time hazard analysis

alarming decision on building’s seismic response

on hazard

Iervolino I., Giorgio M., Manfredi G. (2007a) Expected loss‐based alarm threshold set for earthquake early warning systems. Earthquake Engineering and Structural Dynamics 2007; 36:1151–1168.


Expected loss as a function of the seismic instrumentspmeasures

No alarm

oss

[€

]

Alarm

pect

ed

Lo

Exp

[s]τ̂Optimal Alarm threshold

τ

Iervolino et al., 2009.


NSPSHA: Near Source ProbabilisticSeisimic Hazard Analysis


Pulse‐like records induced by directivity

Pulse‐like records can induce comparatively large inelastic nonlinear demandh d h h f f h f h l *

Pulse like records induced by directivity

in structures having a period which is a fraction of that of the pulse*.


T t f th l th t i t dj t t t PSHA* i d iTo account for the pulse threat appropriate adjustments to PSHA* are required; in fact pulses are not always observed in near‐source conditions where directivity

effects are expected, therefore:

Att tiMAF Sa>x AttenuationMAF Sa>x(hazard)

Distribution of Magnitude and DistanceDistribution of Distribution of Magnitude and DistanceDistribution of pulse predictors

Distribution ofpulse period

*Iervolino, I, and Cornell, CA (2007). Prediction of the Occurrence of Velocity Pulses in Near‐Source Ground Motions. Unpublished manuscript.*Tothong, P., Cornell, C.A., and Baker, J.W. (2007). Explicit directivity‐pulse inclusion in probabilistic seismic hazard analysis. Earthquake Spectra, 23, 867–891.


Directivity effects’ predictors for strike‐slip (SS) and dip‐slip (DS) events*:

ParameterR [km] Closest distance of the site to fault rupture d [km] Distance to the site measured along the ruptureφ [deg] Angle between the rupture and the siteW [k ] Wid h f h f l

ParameterR [km] Closest distance of the site to fault rupture s [km] Distance to the site measured along the ruptureθ [deg] Angle between the rupture and the site[k ] h f h f l W [km] Width of the fault

Y=d/W Width ratioM MagnitudeYcos(φ) Somerville’s amplification parameter

L [km] Length of the faultX=s/L Length ratioM MagnitudeXcos(θ) Somerville’s amplification parameter

*Somerville, P.G., Smith, N.F., Graves, R.W., and Abrahamson, N.A. (1997). Modification of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seism. Res. Lett. 68 199–222.


Pulse identification is based on that of Baker 2007*.

• Baker (2007) developed a method based on Lucerne (landers, 1992)( ) pwavelets to assign a score, a real numberbetween 0 and 1, to each analyzed record andto determine the pulse period. The larger thescore determined the more likely the record is Original GMto show a pulse.

• Herein only those in the fault‐normalcomponent have been considered; in

g

particular those ground motions which have apulse score larger or equal to 0.85 have been,arbitrarily, counted as pulse‐type records.

Extracted Pulse

• Baker analyzed extensively the NGA databaseand he is the only researcher the authors areaware of who has looked systematically at allrecords in the database. Therefore, we knowhi h th l d l hi h th

Residual GMwhich are the pulses, and also which are thenon‐pulses.

* Baker, J.W. (2007). Quantitative classification of near‐fault ground motions using wavelet analysis, Bull. Seism. Soc. Am. 97 1486–1501.


Strike‐Slip Model Scores

Covariate R2 (E) R2 (MF) AIC

R [km] 0.16051 0.12467 136.3561

s [km] 0.040238 0.034712 149.958

θ [d ] 0 12381 0 12873 135 7425θ [deg] 0.12381 0.12873 135.7425

M 0.000848 0.000775 155.0894

X 0.001278 0.001522 154.9764

Xcos(θ) 0.011531 0.013597 153.1507( )

Iervolino I., Cornell C.A. (2008) Probability of occurrence of velocity pulses in near‐source ground motions. Bulletin of the Seismological Society of America, 98(5): 2262‐2277.


Non‐Strike‐Slip Model Scores

Covariate R2 (E) R2 (MF) AIC

R [km] 0.058661 0.050954 202.3675

d [km] 0.007687 0.009624 211.0061

[d ] 0 082195 0 080198 196 255φ [deg] 0.082195 0.080198 196.255

M 1.42E‐05 1.36E‐05 213.0149

Y 0.004479 0.004608 212.0546

Ycos(φ) 0.013224 0.012865 210.3287(φ)



S ik Sli M d l SStrike‐Slip Model Scores

Site

θ [deg]

Epicenter

Rupture



N S ik Sli M d l SNon‐Strike‐Slip Model Scores





Class scale seismic risk assessmentClass scale seismic risk assessment


ObjectiveObjectiveThe objective is to compute the expected fraction in a class of buildings expected toreach a conventional limit state if subjected to seismic hazard in a given time.

The classes considered are those of existing rectangular RC buildings (both pre‐code or seisimically designed) defined by a limited number of parameters as geometricalseisimically designed) defined by a limited number of parameters as geometricaldimensions, and number of storeys.

a z

L z

a z

L z

a za z

L z

ayLy

ax

ayLyayLy

axax

LxLxLxLx

Iervolino I., Manfredi G., Polese M., Verderame G.M., Fabbrocino G. (2007b). Seismic risk of R.C. building classes. Engineering Structures, 29:813–820.


FormulationThe computation of the expected number of failures within the class proceeds extendingThe computation of the expected number of failures within the class proceeds extendingthe failure probability approach for a structure‐specific problem. The final result is thefailure probability of the class which in its frequentistic interpretation provides theexpected fraction of failures.p

Failure probability Class‐Capacity Function

( ) ( ) ( )0P P Z X P C X D X⎡ ⎤ ⎡ ⎤≤ ≤( ) ( ) ( )0fP P Z X P C X D X⎡ ⎤ ⎡ ⎤= ≤ = ≤⎣ ⎦ ⎣ ⎦

Limit State Function depending on the vector of random variables varying ithi th l d fl ti l

Seismic Demand

within the class and reflecting also bulding‐to‐building variability


Class scale capacityClass‐scale capacityThe class‐scale capacity function associates a bilinear capacity curve to any vector of

t (X) d fi i t t ithi th b ildi l Th f ti

( )n

parameters (X) defining a structure within the buildings class. These functions areretreived by regression of significant cases.

( ) 01

s ss C , C ,i ii

C X a a X=

≈ + ∑C structuralC structural

( )n

C X a a X≈ + ∑

Cs collapsecapacity curve

SDOF

Cs collapsecapacity curve

SDOF( ) 01

d dd C , C ,i ii

C X a a X=

≈ + ∑kk

( ) 0

n

T , T ,i iT X a a X≈ + ∑Cd

kCd

k

( )1

, ,i=∑


Cl l it (2)Class‐scale capacity (2)A set of structures is defined to capture the variability of the structural features withinth l ( di t th di t ib ti f th i bl ) Th l d b SPOthe class (according to the distribution of the variables). These cases are analyzed by SPOand then the capacity features are obtained interpolating the results of the set ofstructures specifically analyzed.

iX

( )ji ,max XX ,μ

( )Xμ ( )Xμ

jX

( )iX j ,min,Xμ ( )iX j ,max,Xμ

Δ

iXΔ

( )ji ,min XX ,μjXΔ ( )j,


Class‐scale capacity (4){ }sycpypxyxyx f,f,OR,n,n,a,a,L,LX =

Class scale capacity (4)

Cs)X(Cs )(Cs

)X(Cd

Cd )X(T


1 75 1 75

1 00

1.25

1.50

1.75Elastic period, Tel [sec]

transverselongitudinal

1 00

1.25

1.50



T = 0.135H0.67

T = 0.076H0.93

0 25

0.50

0.75

1.00T = 0.091H0.79

T = 0.112H0.69

0 25

0.50

0.75

1.00

0.00

0.25

0 3 6 9 12 15 18 21 24 27 30

Height, H [m]

(a)

0.00

0.25

0 3 6 9 12 15 18 21 24 27 30

Height, H [m]

(b) Period‐height relationships

for the analyzed

0 75

1.25

1.50



1.25

1.50



ypopulations: (a) gravity‐loads design (b) seismic design (0.05g) (c) seismic

T = 0.098H0.75

T = 0.118H0.660.50

0.75

1.00 T = 0.107H0.70

T = 0.118H0.65

0.50

0.75

1.00 design (0.07g) (d) seismic design (0.10g).

0.00

0.25

0 3 6 9 12 15 18 21 24 27 30

Height, H [m]

(c)

0.00

0.25

0 3 6 9 12 15 18 21 24 27 30

Height, H [m]

(d)( ) ( )

Verderame G.M., Iervolino I., Manfredi G. (2009). Elastic period of sub-standard reinforced concrete moment resisting framebuildings. Bulletin of Earthquake Engineering. (submitted May 2009)


Seismic demandSeismic demand

Limit‐state function

Inelastic demand

( ) ( ) ( )d d e RZ X C S T C R ,T= −( ) ( ) ( )d d ,e R ,

ˆSpectral modification factor

Spectral elastic demand from

RR R CC C ε=PSHA

All uncertaintines in demand are explicitlty considered

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