Ph.D. Thesis project of Paolo E. Sebastiani PBEE - Mala Rijeka Viaduct

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P. E. Sebastiani – Ph.D. Student
Sapienza University of Rome - a.a. 2013/2014
paolo.sebastiani@uniroma1.it
francesco.petrini@uniroma1.it
franco.bontempi@uniroma1.it www.francobontempi.org
1. PERFORMANCE-BASED EARTHQUAKE ENGINEERING
CONDITIONAL PROBABILISTIC APPROACH
(IM-BASED METHODS)
APPROACHES
UNCONDITIONAL PROBABILISTIC
APPROACH
SEMI-PROBABILISTICAPPROACH (DM2008)
PHILOSOPHY
PBEE
DETERMINISTICAPPROACH
EVALUATION, DESIGN AND
CONSTRUCTION OF STRUCTURESTO
MEET SEISMIC PERFORMANCE
OBJECTIVES
1.1 – PBEE Outline

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1.2 – Definition
Performance-based earthquake engineering (PBEE) consists of the evaluation,
design and construction of structures to meet seismic performance objectives
(expressed in terms of repair costs, downtime, and casualties) that are specified
by stakeholders (owners, society, etc.)
It is based on the premise that performance can be predicted and
evaluated with quantifiable confidence to make, together with the client,
intelligent and informed trade-offs based on life-cycle considerations rather than
construction costs alone.
Krawinkler, H. and Miranda, E. (2004). “Performance-Based Earthquake Engineering”. Chapter 9 of
Earthquake Engineering: From engineering seismology to performance-based engineering, edited by Y.
Bozorgnia, andV. Bertero, CRC press.
1.3 – Assumptions

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1.4 – Development of PBEE
Most of the concepts that are implemented in the context of PBEE are not new.
In various forms they have been explored, tried and partially
implemented in past design/evaluation guidelines and standards of
various countries and industries
In the United States various efforts were initiated during the early 1990s which
faced up to the many challenges of performance-based seismic design. The most
widely known ones are Vision 2000 (SEAOC, 1995), FEMA 273 and FEMA 274
(1996) and ATC-40 (1996)
1.5 – PBEE concepts in the Codes

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1.6 –Vision 2000 report (SEAOC 1995)
One of the many strong points of theVision 2000 document is that it proposes
a comprehensive design/assessment/build process that incorporates important
aspects of:
• Selection of a suitable site
• Selection of suitable structural materials and systems
• Configuration and continuity of load path
• Quality of detailing
• Strength and stiffness
• Consideration of nonstructural and content systems
• Quality and consistency of design
• Quality of design review
• Quality of construction
• Quality of inspection

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Performance objectives for buildings, recommended in SEAOC (1995).

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CONDITIONAL
PROBABILISTIC
APPROACHES
PEER
METHOD
It is not in closed form but it allows
more flexibility and generality in
the evaluation of the desired so-
called “decision variable”, not
necessarily coinciding with Pf.
SAC/FEMA
It has the advantage of providing a
closed-form expression for the
failure probability (Pf), that can also
be put in a partial factor format.
1.7 – Conditional Probabilistic Approaches
In the middle of the 90’s very promising results started to materialize (i.e.
Bazzurro and Cornell 1994, Cornell 1996). The problem was posed in terms of
a direct (probabilistic) comparison between demand and capacity, with the
demand being the maximum of the dynamic response of the system to a seismic
action characterized in terms of a chosen return period
Fib (2012) Probabilistic performance-based seismic design. Bulletin n°68, Fédération internationale du
béton, Lausanne, Switzerland

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2. PEER METHOD: A PROBABILISTIC
DESIGN/ASSESSMENT APPROACH TO PBEE
2.1 – Definition
The Pacific Earthquake Engineering Research (PEER) Center has focused for
several years on the development of procedures, knowledge and tools for
a comprehensive seismic performance assessment of buildings and
bridges
In the approach, decision variables are identified whose quantification, together
with an assessment of important uncertainties, will make it feasible to
characterize and manage economic and societal risks associated with
direct losses, downtime and collapse and life safety.

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2.2 – Components of PEER method: PerformanceTargets
It is assumed that a performance target can be expressed in terms of a
quantifiable entity and, for instance, its annual probability of
exceedance. For instance, l$(y), the mean annual frequency (MAF) of the loss
exceeding y dollars, could be the basis for a performance target

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2.3 – Components of PEER method: DecisionVariables
The quantifiable entities, on which performance assessment is based, are
referred to as decision variables (DVs). Examples of DVs of primary interest are
the existence of collapse, the number of casualties, dollar losses and the length
of downtime

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To compute DVs and their uncertainties, other variables have to be evaluated to
define:
1. The seismic hazard
2. The demands imposed on the structural systems by the hazard
3. The state of damage

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2.4 – Components of PEER method: Seismic Hazard
The seismic hazard is quantified in terms of a vector of intensity measures (IMs),
which should define the seismic input to the structure. This vector could have a
single component, such as spectral acceleration at the first mode period of the
structure, Sa(T1), or could have several. If a single component is used, such as
Sa(T1), the hazard is usually defined in terms of a hazard curve. The outcome of
hazard analysis, which forms the input to demand evaluation, is usually
expressed in terms of an MAF of IMs, i.e., l(IM),

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2.5 – Components of PEER method: Engineering Demands Parameters
Given the ground motion hazard, a vector of engineering demand parameters
(EDPs) needs to be evaluated, which defines the response of the structure in
terms of parameters that can be related to DVs. Interstory drift is an example
of a relevant EDP for buildings

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2.6 – Components of PEER method: Damage Measures DM
In most cases an intermittent variable, called a damage measure (DM), has to be
inserted between the EDP and the DV simply to facilitate the computation of
DVs from EDPs. A DM describes the damage and consequences of damage to a
structure or to a component of the structural, nonstructural or content system,
and the term G (DM|EDP) can be viewed as a fragility function for a specific
damage (failure) state (probability of being in or exceeding a specific damage
state, given a value of EDP).

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The assessment problem has been “de-constructed” into the four basic
elements of
1. hazard analysis
2. demand prediction
3. modeling of damage states
4. failure or loss estimation
by introducing the three intermediate variables, IM, EDP and DM
To close the loop…
1. EDPs have to be related to IMs (Probabilistic Seismic Demand Analysis
PSDA)
2. DMs have to be related to EDPs (Damage Analysis)

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Relationships between EDPs and IMs (also called Probabilistic Seismic Demand
Models PSDM) can be obtained through simulations, which should incorporate
the complete structural, geotechnical and SFSI (soil–foundation–structure
interaction) systems
2.7 – Probabilistic Seismic Demand Analysis in the PEER method
So the first step is the evaluation of a PSDM
The outcome of this process, which may be referred to PSDA, can be
expressed as G(EDP|IM) or more specifically as G[EDP ≥ y | IM = x], which is the
probability that the EDP exceeds a specified value y, given (i.e., conditional) that
the IM is equal to a particular value x

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Synthetic or recorded
accelerograms
SEISMIC LOADS
STRUCTURE
Geometry
Meterials
Method of design
Ductility
Isolation system
INPUTS
Type
Range of values
Scalar or vector
One or more
EDP
IM
SETTINGS
Type
Range of values
Scalar or vector
One or more
COMMON APPROACHES
BIN
APPROACH
CLOUD
METHOD
STRIPE
METHOD
IDA SIMPLIFIED
METHOD
IMPROVED
CLOUD
METHOD
MCS - LHS
EMPIRICAL
METHOD
PSDM
2.8 – PSDM in the PSDA
Two values for each level of
IM:
- mIM , Median of EDP
- z, Standard deviation of
EDP

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A regression analysis can be used to obtain the mean (mIM) and the standard
deviation (z) by assuming the logarithmic correlation between median EDP and
an appropriately selected IM:
where the parameters "a" and "b" are regression coefficients obtained for
example by the nonlinear time history analyses
ln⁡(𝐸𝐷𝑃) = ln 𝑎 + 𝑏 𝑙𝑛(𝐼𝑀)

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Assuming a log-normal distribution of EDP at a given IM, the probability that the
EDP exceeds a specified value y, given (i.e., conditional) the IM, can be written
as:
where F is the standard normal distribution function
𝐺 𝐸𝐷𝑃 ≥ 𝑦|𝐼𝑀 = 1 − Φ
ln 𝑦 − ln⁡(𝑎 𝐼𝑀 𝑏
)
𝜁
The remaining variability in ln(EDP) at a given IM is assumed to have a constant
variance for all IM range, and the standard deviation can be estimated:
𝜁 =
ln 𝐸𝐷𝑃𝑖 − (ln 𝑎 + 𝑏 ln 𝐼𝑀𝑖) 2𝑛
𝑖=1
𝑛 − 2

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2.9 – Damage analysis and fragility functions in the PSDA
The specified value y of EDP, defined previously, can be related to a Damage
Measure DM. Consequently it is possible to define some Limit States LS
according to different damaging.
STRUCTURAL CAPACITY
(DAMAGEANALYSIS)

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Nielson, B. G. (2005). “Analytical fragility curves for highway bridges in moderate seismic zones.”
PhDThesis, Georgia Institute ofTechnology,Atlanta, Georgia.

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Therefore G (DM|EDP) can be viewed as a fragility function for three
different damage states

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2.10 – Loss analysis
The seismic fragility can be convolved with the seismic hazard in order to assess
the annual probability PAi of exceeding the ith damage state:
where H(a) is the hazard curve that quantifies the annual probability of
exceeding a specific level of IM at a site.
Additionally, under the assumption of time-invariant structural resistance, it is
possible to evaluate the T-year probability PTfi of exceeding the damage state ith,
estimated as:
𝑃𝐴𝑖 = 𝑃 𝐷𝐼 ≥ 𝐿𝑆|𝐼𝑀
𝑑𝐻(𝑎)
𝑑𝑎
𝑑𝑎
PTf 𝑖 = 1 − 1 − PA𝑖
T

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For each damage state it is possible to define a nominal cost of restoration
which depends on the repair strategy. So the probability of exceeding a damage
state can be related to the probability of exceeding a “cost” to close the loop
of PEER method
Fib (2012) Probabilistic performance-based seismic design. Bulletin n°68, Fédération internationale du
béton, Lausanne, Switzerland

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At the same time it is possible to calculate the expected value of the life-cycle
costs due to seismic damage in present day dollars can be expressed as follow:
where i is the damage state, T=50 years is the remaining service life of the
bridge, Ci is the cost associated with damage state i. An inflation adjusted
discount ratio, a=0.03, is used for converting future costs into present values
Wen,Y. K., and Kang,Y. J. (2001a). “Minimum Building Life-Cycle Cost Design Criteria. I:
Methodology.” Journal of Structural Engineering, 127(3), 330–337.
E LCC =
1
αT
1 − e−αT
−C𝑖
3
𝑖=1
ln 1 − PTf𝑖 − ln⁡1 − PTf𝑖+1

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3. APPLICATIONTO THE “MALA RIJEKA”VIADUCT
3.1 –The case study

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3.2 –The case study: geometry
The case study bridge "Mala Rijeka" is one of the most important bridges on
the Belgrade - Bar International Line. The bridge was built in 1973 as the highest
railway bridge in the World (Worlds Record Lists) and it is a continuous five-
span steel frame carried by six piers of which the middle ones have heights
ranging from 50 to 137.5 m measured from the foundation interface. The main
steel truss bridge structure consists in a continuous girder with a total length
L=498.80 m. Static truss height is 12.50 m, and the main beams are not parallel,
but are radially spread, in order to adjust to the route line

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3.3 – Uncertainties
AVAILABLE DATA
•Geometry
•Site Characteristics*
•Damage inspection made in
the 2007
NOT AVAILABLE DATA
•Pier’s section
•Materials
•Devices
Z. Radosavljevid and O, Markovic. (1976) Some Foundation Stability Problems of the Railway Bridge over
the Mala Rijeka. Rock Mechanics 9, 55--64

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3.4 – Modeling of the highest pier
The response of the pier III in figure 1 is evaluated via non-linear dynamic
analyses run in OpenSees 2.2.2 (McKenna, 1997).The column is modelled with a
nonlinear element with fiber-section distributed plasticity

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3.5 – Material assumptions
Deck (120 m for the 3th pier) : 870 kNs2/m
Pier (distributed along the pier) : 7166 kNs2/m
3.6 – Mass assumptions

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3.7 – Section assumptions
•Tube 6.5 16.5 m x 6.516.5 m
(variable)
•0.5 m of thickness
•Rebar F26 / 0.2 m
SECTION A

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3.8 – Pushover analysis to define limit states
First Cracking
st = 5.2 N/mm2 is the concrete tensile strength SEC 107
SEC 105
SEC 103
SEC 101P. E. Sebastiani – Ph.D. Student
franco.bontempi@uniroma1.it www.francobontempi.org 32

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SEC 107
SEC 105
SEC 103
SEC 101
Yielding
ss = 440 N/mm2 is the steel yield strength

Three damage states DS namely slight, moderate and complete damage are
adopted in this study and their concerning limit values are shown in tab. 6.
Through the pushover analysis presented previously, the slight damage has been
associated to the achievement of maximum tensile strength of concrete, while
the moderate one to the yielding of the steel rebar.
A comparison between the values adopted by Choi et al. (2004) and the
ductility factor defined in the EC8 for piers, has allowed us to define also limit
values referred to the collapse.

3.11 – PSDM results with different type of retrofit
NOT ISOLATED ISOLATED (FRICTION-PEDULUM SYSTEM) ISOLATED (ELASTOMERIC BEARINGS)
Using the curvature ductility at the pier base mc and displacement ductility dc as
EDPs, for different type of IMs and type of retrofit, the PDSM results are the
following:
Sebastiani P.E., Padgett J.E., Petrini F., Bontempi F. (2014). Effectiveness Evaluation of Seismic
Protection Devices for Bridges in the PBEE Framework. In proceeding of: ASCE-ICVRAM-ISUMA
2014 - second International Conference on Vulnerability and Risk Analysis and Management
(ICVRAM) Liverpool, 13th-17th July 2014

3.12 – PSDM results with different type of retrofit

3.12 – Fragility results

3.12 – Fragility results
In terms of damage probability, choosing the example of slight damage and
referring to the curvature ductility as EDP, the probability of damage during a
period of 50 years is: 23% for the structure without isolation, 7% for the
structure equipped with ERB, and 3% for the structure equipped with FPS
isolation.

3.12 – Example of expected cost calculation

4. FUTURE WORK
1) Full 3D model of the Mala Rijeka Bridge
2) Loss estimation taking into account aging effects
3) Effectiveness Evaluation of different Seismic Protection Devices
4) Application to two representative categories of Highway bridges:
- Most common bridge type, i.e. short span, simply supported deck
- Less common bridge type, i.e. high piers, long span, continuous deck
DEMAND CAPACITY FRAGILITY
COMPONENT
SYSTEM
NETWORK
LOSS
?

KEYWORDS
PBEE
PRACTICE-ORIENTED
TARGET
OPENSEES
SEISMIC
ADJUSTMENTAGING
LIMIT STATES
ISOLATION
(MFPS, ERB)
LCC
4. FUTURE WORK

Ph.D. Thesis project of Paolo E. Sebastiani PBEE - Mala Rijeka Viaduct

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Ph.D. Thesis project of Paolo E. Sebastiani PBEE - Mala Rijeka Viaduct