1J. Baker

An Introduction to the Conditional Mean SpectrumConditional Mean Spectrum

Jack BakerAssistant Professor

Civil & Environmental EngineeringStanford University

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Motivation

• What does probabilistic seismic hazard analysis tell us about future ground motions at our site?

• How can we best use that information to select appropriate ground motions for structural analysis?

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Outline

• Quick overview of P b bili i i i h d– Probabilistic seismic hazard

– Uniform hazard spectrum

A l k d i i d i h hi h li d • A look at ground motions associated with high-amplitude response spectra

• Development of the Conditional (Mean) Spectrum

• Implicationsp

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Background: seismic hazard and deaggregation results

Hazard maps

fHazard deaggregationFrom USGS

Site specifichazardhazard

From USGSFrom USGS

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The Uniform Hazard Spectrum (UHS) for a given site

J. Baker 6Motivation: hazard analysis results and

the Uniform Hazard Spectrum

UHS for Riverside, California# of standard deviations above median Sa

Distribution of magnitude, distance, and ε given Sa(1s)=0.9g

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This “ε effect” is a real phenomenon

Response spectra from real ground motions having approximately magnitude = 7 and distance = 12 km

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Calculation of ε values at three periods

ln

ln ( )

ln ( ) ( , , )( ) Sa

Sa T

Sa T M R TT μεσ−

=

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ε values at 1s and 2s, from many ground motions

( ) ( )2 (2 ,1 ) 1

0.75 (1 )s s s s

sεμ ρ ε

ε

= ⋅

=

( )2 1 (2 ,1 )

0.5s s sεσ ρ= −

=

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ε values at varying periods, from many ground motions

( )

( )

2

2

0.75 (1 )

0.5s

s

sε

ε

μ ε

σ

=

=

( )

( )

0.2

0.2

0.44 (1 )

0.75s

s

sε

ε

μ ε

σ

=

=

Using these data, we can find the mean and standard deviation of ε at all periods, conditional on a target ε at T*, the period of primary interest

J. Baker 11Conditional mean values of spectral acceleration at all periods, given the target Sa(1s)

ln ( )|ln ( *) ln ln( , , ) ( , *) ( *) ( )iSa T Sa T Sa i i Sa iM R T T T T Tμ μ ρ ε σ= +

J. Baker 12Conditional distribution of spectral acceleration at all periods, given the target Sa(1s)

ln ( )|ln ( *) ln ln

2ln ( )|ln ( *) ln

( , , ) ( , *) ( *) ( )

( ) 1 ( , *)i

i

Sa T Sa T Sa i i Sa i

Sa T Sa T Sa i i

M R T T T T T

T T T

μ μ ρ ε σ

σ σ ρ

= +

= −ln ( )|ln ( ) lniSa T Sa T Sa i i

J. Baker 13We can select and scale ground motions to

match this Conditional (Mean) Spectrum

Match target mean and sigma Match target mean only (minimizing sigma)

J. Baker 14Ground motion selection at four intensities,

for the Riverside site

J. Baker 15Using a conditional mean spectrum instead of a UHS can have a large impact on fragility curves

Collapse fragility curves for a midrise building, from nonlinear dynamic analysis

1

0.8

)

Analysis using uniformhazard spectrum

0 4

0.6

Col

laps

e)

Analysis using conditional

0.2

0.4

P(C

ε Neutral Set

mean spectrum

0 0.5 1 1.5 2 2.5 30

Sa(T1 = 1.0, ζ = 0.05)

ε1.0

Set

1

Haselton C.B. and Baker J.W. (2006). "Ground motion intensity measures for collapse capacity prediction: Choice of optimal spectral period and effect of spectral shape,” 8th National Conference on Earthquake Engineering.

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Pros and cons of the conditional mean spectrum

Pros Cons

• A more “realistic” spectrum than the UHS (given Sa(T*))

• Less widely available than the UHS (so far)

• Less conservative than UHS

• Utilizes deaggregation information

• Less conservative than UHS

• Structure and site-specific: Ut es eagg egat o o at o (magnitude, distance, ε) to predictspectral shape

St uctu e a s te spec c: requires re-selection of ground motions as each case changes

• The spectrum changes in shape as you increase in amplitude, consistent with intuition

• The spectrum changes with increasing amplitude, requiring multiple ground motion setsp g

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Resource: CMS and ground motion selection algorithms

http://www.stanford.edu/~bakerjw/gm_selection.html

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Resource coming soon: CMS calculation tools from USGS

“USGS is adding output of Conditional Mean Spectrum (for seismogram selection) that is fully-consistent with selection) that is fully consistent with USGS-NSHMP hazard & deaggregation computations.”

Acknowledgement: Steve Harmsen Eric Acknowledgement: Steve Harmsen, Eric Martinez and Nicolas Luco (USGS)

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More resources

• This information in journal paper format:

Baker, J. W. (2011). "The Conditional Mean Spectrum: a tool for ground motion selection." Journal of Structural Engineering (in press).

• Beta version of user-friendly software to compute CMS and select matching ground motions:

http://peer2.berkeley.edu/peer_ground_motion_database

• Related publications and (non-user-friendly) research software: • Related publications and (non-user-friendly) research software:

http://www.stanford.edu/~bakerjw

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Conclusions

– The Conditional Mean Spectrum answers the question: “What is the expected response spectrum associated with a target Sa(T*)?” using knowledge of the magnitude, distance and ε value that caused occurrence of that S (T*)Sa(T*)

– The variability in spectral values can also be computed (“Conditional Spectrum”)Spectrum )

– For large-amplitude (ε>0) Sa levels, this spectrum has a peak at the period (T*) used for conditioning, and decays to relatively lower amplitudes at

i d h diff l f T*periods that differ greatly from T*

– This may be a useful target spectrum for ground motion selection in many applications (as the alternative Uniform Hazard Spectrum is conservative applications (as the alternative Uniform Hazard Spectrum is conservative relative to this target)

– Structural responses from ground motions matching the CMS may be f l ll h h f d h h significantly smaller than the responses from ground motions matching the

UHS and having the same Sa(T*) level