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
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Performance-Based Earthquake
Engineering
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Performance Objective
• The performance objective can be stated in terms of the mean
annual frequency of exceeding a limit state, e.g., collapse
l LS  Po
lLS is the mean annual frequency of exceeding a limit state
P0 is the allowable frequency level
l LS is also known as the limit state probability or probability of failure
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Performance Objective in terms of
Structural Parameters
•
The probabilistic performance objective can be stated in terms of the
mean annual frequency of demand exceeding capacity for structural
limit state LS
l LS  l ( D  C LS )  Po
CLS
is the structural capacity for limit state LS
D
is the structural demand
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Alternative Probabilistic Representations of
Earthquake Ground Motion
A
Direct Probabilistic Representation of the
Ground Motion
B
Implicit Probabilistic Representation of the
Ground Motion
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Direct Probabilistic Representation of
Ground Motion Using Intensity Measure
• It is assumed that the spectral acceleration is a
sufficient intensity measure.
• A sufficient intensity measure renders the
structural response (e.g., qmax) independent of
ground motion parameters such as M and R.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
l ( IM  x )
IM  x
Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA
Attenuation law: Abrahamson and Silva, horizontal motion on soil
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Implicit Probabilistic Representation of Ground Motion in
Current Seismic Design and Assessment Procedures
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).
l ( PGA  PGA design ) 
10 % in 50 years
PGA design
Mean Annual Frequency of Exceeding PGA
Also Known as PGA Hazard Curve
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Choice of IM
• The spectral acceleration at the small-amplitude fundamental period
of the structure denoted by S a (T1 ) or simply, Sa is adopted as the
intensity measure (IM).
u (t )
M ,r
c
k
T1  period
m 1
of the oscillator
  damping
u (t )
t
S a (T 1 ,ξ ) 
4
T
coefficien t
2
2
max (abs(u(t)) )
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Choice of Structural Response Parameter
D  q max
We have chosen the maximum inter-story drift angle, q max , a displacementbased structural response, as the structural response parameter.
105
  q h
105
105
105
h
105
M ,R
106
157
241
241
q
241
(l 1  l 2 )/ 2
q max  max( q ( t ))
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
h
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Structural Limit State: Global Dynamic Instability
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.
T h e o n set o f g lo b a l d y n a m ic in sta b ility
C LS  q cap
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Representation of Ground Motion using
Intensity Measures
Probabilistic performance objective:
l LS  P ( D  C LS )  Po
IM-based presentation of the probabilistic performance objective:
PDF for Structural Response given IM
λ LS 
  P( θ max
 q cap |θ max )  p( q max |S a )  dλ S (S a )
a
q max S a
CDF for Structural Capacity given Response
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Seismic Hazard for IM
Seismic Hazard (Direct Probabilistic
Representation) for the Ground Motion
Intensity Measure (IM)
λ LS 
  P( θ max
 q cap |θ max )  p( q max |S a )  d λ S (S a )
q max S a
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
a
Seismic Hazard Model
Ground motion and site parameters:
M ,R
magnitude, distance and/or additional variables
source i: San Andreas Fault
site: Van Nuys
(M,R)
Faults of Los Angeles region
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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:
ln IM  f ( M , r )     ln IM | M , r
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:
P[ S a  x | M , r ]  1   (
ln x  f ( M , r )
 ln S a | M , r
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
)
Seismic Hazard for IM
The mean annual rate of exceeding a given spectral acceleration value, also known
as spectral acceleration hazard can be calculated as follows:
N
lSa ( x) 
 li ( S a
i 1


 x )   li ( M  m 0 ) 

i 1
 all
N
summation over all the
surrounding seismic
zones
 I ( S a
M , r and 
attenuation relation


 x | M , r ,  ) p ( M , r ,  ) dMdrd  


all the possible earthquake event
scenarios that can take place on seismic
zone i and which produce spectral
acceleration larger than x.
mean annual rate that an
earthquake event of interest
takes place at seismic zone i
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Spectral Acceleration Hazard Curve
lSa ( x)
Sa  x
Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA
Attenuation law: Abrahamson and Silva, horizontal motion on soil
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Representation for Structural
Demand given IM
Implementing Non-Linear Dynamic Analysis Methods
λ LS 
  P( θ max
 q cap |θ max )  p( q max |S a )  dλ S (S a )
q max S a
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
a
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 q max for a given S a
level.
p (q max | S a )
P0
S a = 0 .7 0 g
Estimating p (q max | S a ) using nonlinear dynamic analyses
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Representation for Demand
The mean annual frequency of exceeding a given value of the structural
demand parameter:

lq max ( y ) 

P (q max  y | S a )  d l S ( S a )
a
x0
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Drift Hazard Curve
l q max ( y )
y
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Probabilistic Representation for Limit State
Capacity
Implementing Non-Linear Dynamic Analysis Methods
λ LS 
  P( θ max
 q cap |θ max )  p( q max |S a )  dλ S (S a )
q max S a
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
A Probabilistic Representation for Structural Limit
State Capacity
The record-to-record variability in structural capacity can be expressed
by the complementary cumulative distribution function (CCDF) of
capacity q cap for a given q
.
max
P ( θ max  q cap |θ max  y )
ˆ C
LS
 0 .0 2 7 8
ˆC L S  0 . 41
Estimating
P ( θ max  q cap |θ max  y ) using nonlinear dynamic analyses
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Demand and Capacity Factored Design (DCFD)
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Demand and Capacity Factor Design (DCFD)
The probabilistic performance objective:
l LS  l ( D  C LS )  Po
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:
Factored Demand (Po)

Factored Capacity
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Main Assumptions Leading to a Closed-form
Expression for (DCFD)
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
• The spectral acceleration hazard curve can be described by a
power-law function (a linear function in the logarithmic scale).
H S a (s a ) = k 0 (s a )
-k
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
• Demand (given spectral acceleration) can be described by a
lognormal distribution with constant standard deviation and powerlaw median.
 D .e
-
D

 D = g (S a )
 D .e
T h is is a p ro b ab ilistic m o d el o f th e
(co n d itio n al) d istrib u tio n of d em an d
g iv en an in ten sity lev el.
M a xim u m In te r-s to ry D rift, D
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011

D

• Median capacity is described by a lognormal distribution with
constant median and standard deviation.
ˆq ca o  0 . 0 2 7 8
ˆ q ca o  0 . 4 1
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
A Closed-Form Analytical Solution the Annual
Frequency of Exceeding Limit State Capacity
  P( θ max
λ LS 
 q cap |θ max )  p( q max |S a )  d λ S (S a )
a
q max S a
2
q
λ LS  l S ( S a
a
 q cap
Sa
  q cap
 
 a

cap
1 k
2

q
max
2
)e2 b
2
|S a
1 k
2

q
cap
2
e2 b
1
b
 is


the spectral acceleration corresponding to median capacity.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Closed-Form Presentation of DCFD Format
l LS  Po
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:

q max
P
| 0S
1 k 2
   q max
e2 b
|S a
a
Factored Demand (Po)
 q

cap
e

1k
2b
2
cap
q
Factored Capacity
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
A Closed-Form Analytical Solution the Annual
Frequency of Exceeding Structural Demand
(Also Known as Drift Hazard)
2
1
Where S ay
demand y.
 y b
 
a
λq
max
( y)  l S
a
y
(S a
)e
1 k
2

 q max
2 b2
|S a
is the spectral acceleration corresponding to median
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
A Graphic Presentation of DCFD format:
Drift hazard curve - closed form
P0
lLS
F.C.
F.D.
Factored Demand (Po)

Factored Capacity
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Structural Model: A Generic 8-Storey RC Frame Structure
300
300
300
300
300
300
300
400
600
200
600
Displacement-based Non-Linear Beam-Column Fiber Element Model in OPENSEES
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Approximating the Hazard Curve with a Line in the
Region of Interest
Po  2  10
(10%
in
3
k=3
50 years)
Po
S a  0 . 65 g
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Approximating Structural Demand as a Power-Law
Function of Spectral Acceleration
Po
S a  0 . 65 g
b=1
1
 q max
|S a
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
(
Po
S a )  0 . 033
Factored Demand
Calculating factored demand for the tolerable probability, Po=0.002:
F . D . ( 0 . 002 )   q max | S a ( 0 . 65 )  e
F . D . ( 0 . 002 )  0.033  e
1 k ( P0 )


2 b ( P0 )
1 3 .0
2

 ( 0 . 55 )
2 1
2
q max | S a
(
P0
s
a
)
 0 . 033  1 . 57  0.0519
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Evaluating Structural Capacity for the Limit
State of Global Dynamic Instability
S
S
a cap
a cap
 0 . 40
 0 . 50
 q cap  0 . 03
 q cap  0 . 40
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Factored Capacity
Calculating factored capacity for global dynamic instability limit
state:
F .C .   q
cap
1 k 2
    q cap
e 2 b
F .C .  0.03 
1 3 .0
2
 
( 0 . 40 )
e 2 1
 0 . 03  0 . 78  0.023
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Finally the “checking” moment:
?
Factored Capacity

Factored Demand (0.002)
0.023

0.052
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
DCFD Formulation Taking into Account the
Structural Modeling Uncertainty
(FEMA/SAC Formulation)
In the presence of structural modeling uncertainty the statement for the performance
objective can be written as:
l LS  e
K xl
LS
 Po
Where x the level of confidence in the statement of the performance objective.
lLS represents the uncertainty in the limit state probability due to the presence
of structural modeling uncertainty.
x
kx
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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
F . D .( P0 )
F .C .
β UT 
 UD
  K x  UT
2
  UC
2
UT represents the uncertainty in the demand and capacity due to structural
modeling uncertainty.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Structural Model: An Existing RC Frame Structure in Los Angeles Area
105
105
105
105
105
M , R,q
106
157
241
241
241
Beam-column model with stiffness and strength degradation in shear and flexure
using DRAIN2D-UW by J. Pincheira et al.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Approximating the Hazard Curve with a Line in the
Region of Interest
P 0 = 0 .0 0 8 4
k = 2.7
S a = 0 .7 0 g
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Estimating the factored demand for the tolerable probability,
Po=0.0084:
F . D . ( P0 )   q max | S a ( s a )  e
P0
1 k ( P0 )


2 b ( P0 )
F . D . ( 0 . 0084 )  0.0183  e
2
q max | S a
1 2 .6
2

 ( 0 . 49 )
2 3 .6
(
P0
s
a
)
 0 . 0183  1 . 09  0.020
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Factored capacity estimation for the limit state of global dynamic instability:
Getting help from the IDA's …
ˆ q ca o  0 . 38
ˆ q
cao
 0 . 39
ˆq cao  0 . 0278
ˆ q ca o  0 . 41
F .C .   C LS  e
1 k
2
   C
LS
2 b
 0.0278  e

1 2 .6
2 4
( 0 . 41 )
2
 0 . 0278  0 . 95  0.026
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
Finally the “checking” moment:
?
Factored Capacity

Factored Demand (0.0084)
0.026

0.02
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
If the variability in response due to structural uncertainty can be
represented by:
βUT 
2
 UD   UC
2
 20 %
And the factored demand to capacity ratio is equal to:
ln
F . D .( P0 )
F .C .
 ln
0.02
0.026
 - 0.26   0 . 2 K X
There is 90% confidence associated with the
x=90%
statement of performance objective.
kx=-1.31
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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 .
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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.
•
Demand and Capacity Factor Design:
A Performance-based Analytic Approach to Design and Assessment
Sharif University of Technology, 25 April 2011
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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