Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment

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Demand and Capacity Factor DesignA
Performance-based Analytic Approach to Design and
Assessment

• Fatemeh Jalayer
• Assistant Professor
• Department of Structural Engineering
• University of Naples Federico II

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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.
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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.

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Probabilistic Performance Objective

• The performance objective can be stated in terms
  of the mean annual frequency of exceeding a limit
  state, e.g., collapse
• lLS is the mean annual frequency of exceeding a
  limit state
• P0 is the allowable frequency level

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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
• CLS is the structural capacity for limit state
  LS
• D is the structural demand

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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.

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• Alternative Probabilistic Representations of
  Earthquake Ground Motion
• Direct Probabilistic Representation of the
  Ground Motion
• Implicit Probabilistic Representation of the
  Ground Motion

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• Alternative Direct Probabilistic Representations
  of Ground Motion Uncertainty
• Probabilistic Representation of Ground Motion
  using Intensity Measures (IM-Based, FEMA-SAC
  Guidelines, PEER Methodology)
• Complete Probabilistic Representation of the
  Ground Motion Time History

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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.

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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.

Spectral acceleration hazard curve for
T0.85sec - Van Nuys, CA Attenuation law
Abrahamson and Silva, horizontal motion on soil
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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., Po10
probability in 50 years).
PGA design
Mean Annual Frequency of Exceeding PGAAlso Known
as PGA Hazard Curve
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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).

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Choice of Structural Response Parameter
We have chosen the maximum inter-story drift
angle, , a displacement-based structural
response, as the structural response parameter.

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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.

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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.
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Probabilistic Representation of Ground Motion
using Intensity Measures
Probabilistic performance objective
IM-based presentation of the probabilistic
performance objective
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Seismic Hazard (Direct Probabilistic
Representation) for the Ground Motion Intensity
Measure (IM)
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Seismic Hazard Model
Ground motion and site parameters magnitude,
distance and/or additional variables
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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

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
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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
attenuation relation
summation over all the surrounding seismic zones
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
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Spectral Acceleration Hazard Curve
Spectral acceleration hazard curve for
T0.85sec - Van Nuys, CA Attenuation law
Abrahamson and Silva, horizontal motion on soil
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Probabilistic Representation for Structural
Demand given IM
Implementing Non-Linear Dynamic Analysis Methods
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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.
Estimating using nonlinear
dynamic analyses
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Probabilistic Representation for Demand
The mean annual frequency of exceeding a given
value of the structural demand parameter
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Drift Hazard Curve
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Probabilistic Representation for Limit State
Capacity
Implementing Non-Linear Dynamic Analysis Methods
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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.
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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 for a given .
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Demand and Capacity Factored Design (DCFD)
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Demand and Capacity Factor Design (DCFD)
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

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Main Assumptions Leading to a Closed-form
Expression for (DCFD)
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• The spectral acceleration hazard curve can be
  described by a power-law function (a linear
  function in the logarithmic scale).

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• Demand (given spectral acceleration) can be
  described by a lognormal distribution with
  constant standard deviation and power-law median.

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• Median capacity is described by a lognormal
  distribution with constant median and standard
  deviation.

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A Closed-Form Analytical Solution the Annual
Frequency of Exceeding Limit State Capacity
is the spectral acceleration corresponding to
median capacity.
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Closed-Form Presentation of DCFD Format
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
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A Closed-Form Analytical Solution the Annual
Frequency of Exceeding Structural Demand (Also
Known as Drift Hazard)
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A Graphic Presentation of DCFD format
Drift hazard curve - closed form
P0
lLS
F.C.
F.D.
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Structural Model A Generic 8-Storey RC Frame
Structure
Displacement-based Non-Linear Beam-Column Fiber
Element Model in OPENSEES
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Approximating the Hazard Curve with a Line in the
Region of Interest
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Approximating Structural Demand as a Power-Law
Function of Spectral Acceleration
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Factored Demand
Calculating factored demand for the tolerable
probability, Po0.002
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Evaluating Structural Capacity for the Limit
State of Global Dynamic Instability
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Factored Capacity
Calculating factored capacity for global dynamic
instability limit state
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Finally the checking moment
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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 Where x the
level of confidence in the statement of the
performance objective. blLS represents the
uncertainty in the limit state probability due to
the presence of structural modeling uncertainty.
kx
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DCFD Formulation Taking into Account the
Structural Modeling Uncertainty(FEMA/SAC
Formulation)

• After some algebraic manipulations the DCFD
  format can be presented as
• where

bUT represents the uncertainty in the demand and
capacity due to structural modeling uncertainty.
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Structural Model An Existing RC Frame Structure
in Los Angeles Area
Beam-column model with stiffness and strength
degradation in shear and flexure using DRAIN2D-UW
by J. Pincheira et al.
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Approximating the Hazard Curve with a Line in the
Region of Interest
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Estimating the factored demand for the tolerable
probability, Po0.0084
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Factored capacity estimation for the limit state
of global dynamic instability Getting help from
the IDA's
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Finally the checking moment
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If the variability in response due to structural
uncertainty can be represented by And the
factored demand to capacity ratio is equal to
x90
There is 90 confidence associated with the
statement of performance objective.
kx-1.31
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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.

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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 .

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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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