Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand

Demand and Capacity Factor Design: Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and AssessmentA Performance-based Analytic Approach to Design and AssessmentSharif University of Technology, 25 April 2011Sharif University of Technology, 25 April 2011

Demand and Capacity Factor Design:

A Performance-based Analytic Approach to Design and Assessment

Fatemeh Jalayer

Assistant Professor

Department of Structural Engineering

University of Naples Federico II


One of the main attributes distinguishing performance-based earthquake

engineering from traditional earthquake engineering is the definition of

quantifiable performance objectives.

Performance objectives are quantified usually based on life-cycle cost

considerations, which encompass various parameters affecting structural

performance, such as, structural, non-structural or contents damage, and

human casualties.

Probabilistic performance-based engineering can be distinguished by

defining probabilistic performance objectives.

Probabilistic Performance-Based Earthquake Engineering


Probabilistic Performance objectives

• There is uncertainty in the future ground motion that is

going to take place at the site of the engineering project.

• There is uncertainty in determining the parameters and

building the mathematical model of the real structure.


Probabilistic Performance Objective

• The performance objective can be stated in terms of the mean

annual frequency of exceeding a limit state, e.g., collapse

LS is the mean annual frequency of exceeding a limit state

P0 is the allowable frequency level

oLS P

is also known as the limit state probability or probability of failureLS


• The probabilistic performance objective can be stated in terms of the

mean annual frequency of demand exceeding capacity for structural

limit state LS

CLS is the structural capacity for limit state LS

D is the structural demand

Probabilistic Performance Objective in terms of Structural Parameters

P)( o LSLS CD


Earthquake Ground Motion the Major Source of Uncertainty

• The uncertainty in the prediction of

earthquake ground motion significantly

contributes to the uncertainty in demand

and capacity.


Alternative Probabilistic Representations of

Earthquake Ground Motion

A Direct Probabilistic Representation of the

Ground Motion

B Implicit Probabilistic Representation of

the Ground Motion


Alternative Direct Probabilistic Representations of

Ground Motion Uncertainty

A Probabilistic Representation of Ground Motion using Intensity

Measures (IM-Based, FEMA-SAC Guidelines, PEER Methodology)

B Complete Probabilistic Representation of the Ground Motion

Time History


• It is assumed that the spectral acceleration is a

sufficient intensity measure.

• A sufficient intensity measure renders the

structural response (e.g., max) independent of

ground motion parameters such as M and R.

Direct Probabilistic Representation of Ground Motion Using Intensity Measure


Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CAAttenuation law: Abrahamson and Silva, horizontal motion on soil

xIM

Direct Probabilistic Representation of Ground Motion Using Intensity Measure (IM) -- IM Hazard Curve

• A probabilistic representation of the ground motion intensity measure be stated in

terms of the mean annual frequency of exceeding a given ground motion intensity level.

This quantity is also known as the IM hazard curve.

)( xIM


Current seismic design procedures (FEMA 356, ATC-40) take into account the uncertainty in

the ground motion implicitly by defining “design earthquakes” with prescribed probabilities of

exceeding given peak ground acceleration (PGA) values in a given time period (e.g., Po=10%

probability in 50 years).

Implicit Probabilistic Representation of Ground Motion in Current Seismic Design and Assessment Procedures

Mean Annual Frequency of Exceeding PGAAlso Known as PGA Hazard Curve

PGA design

years 50in %10

)design ( PGAPGA


Choice of IM

• The spectral acceleration at the small-amplitude fundamental period

of the structure denoted by or simply, Sa is adopted as the

intensity measure (IM).

)( 1TSa

,rM)(tu

k 1m

c

)(tu

t

oscillator theof period 1 T

t coefficien damping

)(abs(u(t))T

,ξ(TSa max4

)2

2

1


Choice of Structural Response Parameter

We have chosen the maximum inter-story drift angle, , a displacement-

based structural response, as the structural response parameter.

h

h

t))(max(max

max

241241 241241 241241

157157

105105

105105

105105

106106

105105

105105

)/l(l 221

h

RM ,

maxD


Structural Limit States • The limiting states for which the assessments are done depend on

the performance objectives.

• Here, we focus on the onset of global dynamic instability in the

structure that can be considered as an indicator of imminent

collapse in the structure.

• A non-linear dynamic analysis procedure called the incremental

dynamic analysis can be used to determine the onset of global

dynamic instability.


The onset of global dynamic instability

Structural Limit State: Global Dynamic Instability

capLSC

Similar to a pushover curve that maps out the structural behavior for increasing

lateral loads, an IDA curve maps out the structural response for incrementally

increasing ground motion intensity.


max

maxmaxmax

a

a

SaSacapLS )(Sdλ)|Sp()|θP(θλ

Probabilistic Representation of Ground Motion using Intensity Measures

P)( o LSLS CDP

Probabilistic performance objective:

IM-based presentation of the probabilistic performance objective:

Seismic Hazard for IM

PDF for Structural Response given IM

CDF for Structural Capacity given Response


Seismic Hazard (Direct Probabilistic

Representation) for the Ground Motion

Intensity Measure (IM)

max

maxmaxmax

a

a



source i: San Andreas Fault

(M,R)site: Van Nuys

Faults of Los Angeles region

Ground motion and site parameters:

magnitude, distance and/or additional variablesRM ,

Seismic Hazard Model


Probabilistic Representation for IM for a given M and r

• The relation between IM and ground motion parameters, such as magnitude and

distance, can be expressed in the following generic form:

rMIMrMfIM ,|ln),(ln

The spectral acceleration for a given magnitude and distance can be described by

a log-normal distribution. The parameters of this distribution, namely, mean and

standard deviation, are predicted by the ground motion prediction relation:

)),(ln

(1],|[,|ln rMS

aa

rMfxrMxSP


N

iai

N

iaiS dMdrdrMprMxSImMxSx

a1 andr M, all

01

),,(),,|()()()(

Seismic Hazard for IM

summation over all the surrounding seismic zones

mean annual rate that an earthquake event of interest takes place at seismic zone i

all the possible earthquake event scenarios that can take place on seismic zone i and which produce spectral acceleration larger than x.

The mean annual rate of exceeding a given spectral acceleration value, also known

as spectral acceleration hazard can be calculated as follows: attenuation relation


Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CAAttenuation law: Abrahamson and Silva, horizontal motion on soil

xSa

)(xaS

Spectral Acceleration Hazard Curve


Probabilistic Representation for Structural Demand given IM

Implementing Non-Linear Dynamic Analysis Methods

max

maxmaxmax

a

a



SSaa==00..7700 gg PP00

Probabilistic Representation for Demand given Spectral Acceleration

The record-to-record variability in structural demand for a given intensity

level can be expressed by the conditional probability density function

(PDF) of for a given level.max aS

Estimating using nonlinear dynamic analyses)|( max aSp

)|( max aSp


Probabilistic Representation for Demand

)()|( )(0

maxmax aSx

a SdSyPya

The mean annual frequency of exceeding a given value of the structural

demand parameter:


Drift Hazard Curve

y

)(max

y


Probabilistic Representation for Limit State Capacity

Implementing Non-Linear Dynamic Analysis Methods

max

maxmaxmax

a

a



Incremental Dynamic Analysis (IDA)

The IDA curve provides unique information about the nature of the

structural response of an MDOF system to a ground motion record.


41.0ˆ LSC

0278.0ˆ LSC

)( maxmax y|θθP cap

Estimating using nonlinear dynamic analyses)( maxmax y|θθP cap

The record-to-record variability in structural capacity can be expressed

by the complementary cumulative distribution function (CCDF) of

capacity for a given .capmax

A Probabilistic Representation for Structural Limit State Capacity


Demand and Capacity Factored Design (DCFD)


Demand and Capacity Factor Design (DCFD)

Factored Demand (Po) Factored Capacity

The probabilistic performance objective:

After algebraic manipulations and making a set of simplifying

assumptions, an LRFD-like probabilistic design criterion for a given

allowable probability level, Po , can be derived:

P)( o LSLS CD


Main Assumptions Leading to a Closed-form Expression for (DCFD)


• The spectral acceleration hazard curve can be described by a

power-law function (a linear function in the logarithmic scale).

HSa(sa) =k0 (sa) -k


• Demand (given spectral acceleration) can be described by a lognormal

distribution with constant standard deviation and power-law median.

D=g(Sa)

This is a probabilistic model of the (conditional) distribution of demand given an intensity level.

Maximum Inter-story Drift, D

D.e-

D.e

D

D


• Median capacity is described by a lognormal distribution with

constant median and standard deviation.

41.0ˆ cao

0278.0ˆ cao


A Closed-Form Analytical Solution the Annual Frequency of Exceeding Limit State Capacity

is the spectral acceleration corresponding to median capacity.b

a aS capcap

1

22

22

|max2

2

2

1

2

1

)(capaS

cap

ab

k

b

k

aSLS eeSλ

max

maxmaxmax

a

a



Closed-Form Presentation of DCFD Format

2

|max2

0max

2

1

2

1

|cap

cap

aS

aP

b

k

b

k

See


After algebraic manipulations and making a set of simplifying assumptions,

an LRFD-like probabilistic design criterion for a given allowable probability

level, Po , can be derived:

oLS P


Where is the spectral acceleration corresponding to median

demand y.

bya a

yS

1

2|max2

2

max

2

1

)()(aS

a

b

kyaS eSyλ

A Closed-Form Analytical Solution the Annual Frequency of Exceeding Structural Demand

(Also Known as Drift Hazard)


P0

LS

F.D. F.C.

Drift hazard curve - closed form

A Graphic Presentation of DCFD format:



Displacement-based Non-Linear Beam-Column Fiber Element Model in OPENSEES

Structural Model: A Generic 8-Storey RC Frame Structure

600600 200200 600600

400400

300300

300300

300300

300300

300300

300300

300300


Approximating the Hazard Curve with a Line in the Region of Interest

years) 50in (10%

102 3oP

gSaPo 65.0

k=3


Approximating Structural Demand as a Power-Law Function of Spectral Acceleration

033.0)(|maxa

PoS S

a

gSaPo 65.0

1

b=1


0.051957.1033.0e0.033)002.0( ..

)65.0()002.0( ..

2

0|max2

0

0

max

)55.0(1

0.3

2

1

)()(

)(

2

1

|

DF

eDFa

PaS

a

sPb

Pk

S

Calculating factored demand for the tolerable probability, Po=0.002:

Factored Demand


Evaluating Structural Capacity for the Limit State of Global Dynamic Instability

50.0capaS

40.0capaS

40.0cap

03.0cap


0.02378.003.0e0.03 ..

..

2

2

)40.0(1

0.3

2

1

2

1

CF

eCFcap

capb

k

Calculating factored capacity for global dynamic instability limit

state:

Factored Capacity


Finally the “checking” moment:

?Factored Capacity Factored Demand (0.002)

0.023 0.052


In the presence of structural modeling uncertainty the statement for the performance

objective can be written as:

Where x the level of confidence in the statement of the performance objective.

LS represents the uncertainty in the limit state probability due to the presence

of structural modeling uncertainty.

DCFD Formulation Taking into Account the Structural Modeling Uncertainty

(FEMA/SAC Formulation)

oK

LS Pe LSx

x

kx


DCFD Formulation Taking into Account the Structural Modeling Uncertainty

(FEMA/SAC Formulation)• After some algebraic manipulations the DCFD format can be

presented as:

where:

..

).(.ln UT

0 xKCF

PDF

22UCUDUTβ

UT represents the uncertainty in the demand and capacity due to structural

modeling uncertainty.


Beam-column model with stiffness and strength degradation in shear and flexure

using DRAIN2D-UW by J. Pincheira et al.

241241 241241 241241

157157

105105

105105

105105

106106

105105

105105

,, RM

Structural Model: An Existing RC Frame Structure in Los Angeles Area


Approximating the Hazard Curve with a Line in the Region of Interest

k=2.7 P0=0.0084

Sa=0.70 g


0.02009.10183.0e0.0183)0084.0( ..

)()( ..

2

0|max2

0

0

0

max

)49.0(6.3

6.2

2

1

)()(

)(

2

1

|0

DF

esPDFa

PaS

a

sPb

Pk

aP

S

Estimating the factored demand for the tolerable probability,

Po=0.0084:


Factored capacity estimation for the limit state of global dynamic instability:

Getting help from the IDA's …

41.0ˆ cao

0278.0ˆ cao

38.0ˆ cao

39.0ˆ cao

0.02695.00278.0e0.0278 ..22 )41.0(

4

6.2

2

1

2

1

LSC

LS

b

k

C eCF


Finally the “checking” moment:

?Factored Capacity Factored Demand (0.0084)

0.026 0.02


If the variability in response due to structural uncertainty can be

represented by:

And the factored demand to capacity ratio is equal to:

2.00.26- 0.026

0.02ln

..

).(.ln 0

XKCF

PDF

%2022 UCUDUTβ

There is 90% confidence associated with the

statement of performance objective.

x=90%

kx=-1.31


Conclusions

• Probabilistic performance-based engineering is based on

quantifiable and probabilistic performance objectives.

• The probabilistic nature of the performance objectives is due to the

uncertainties in the prediction of the future ground motion and also

in the structural modeling.

• The uncertainty in the future ground motion input is the dominant

source of uncertainty in the performance assessments.


Conclusions (Continued)

• DCFD is an analytical format for structural performance

assessments that is based on probabilistic performance objectives.

• Non-linear dynamic analyses can be used to make structural

performance assessments in the framework of the DCFD taking into

account ground motion uncertainty.

• The uncertainty in structural model can be taken into account in the

form of a confidence factor in the statement of the probabilistic

performance objective .


This Presentation is Prepared Based on the Following References:

• Cornell C. A., Jalayer F., Hamburger R. O., and Foutch D. A. (2002), “The probabilistic basis for the 2000

SAC/FEMA steel moment frame guidelines’’, ASCE Journal of Structural Engineering, April, 2002.

• Jalayer F., Franchin P. and Pinto P.E. (2007), “A scalar decision variable for seismic reliability analysis of RC

frames”, Special issue of Earthquake Engineering and Structural Dynamics on Structural Reliability, Vol. 36 (13):

2050-2079, June 2007.

• Jalayer F., and Cornell C. A. (2009), “Alternative nonlinear demand estimation methods for probability-based seismic assessments”, Earthquake Engineering and Structural Dynamics, 38: 951-972, 2009.

• Jalayer F., and Cornell C. A. (2003), “A Technical Framework for Probability-Based Demand and Capacity Factor Design (DCFD) Seismic Formats”, PEER Report 2003/08.

• Jalayer F. (2003), “Direct Probabilistic Seismic Analysis: Implementing Non-linear Dynamic Assessments”, Ph.D. Dissertation, Department of Civil and Enviromental Engineering, Stanford University, California.

•


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Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand