Performance-Based Seismic Assessment for Loss Estimation of Isolated Bridges

1 P. E. SEBASTIANI - Ph.D. Student - [email protected]

Dept. of Structural and Geotechnical Engineering - Sapienza University of Rome, Italy

Ph.D. Student Paolo Emidio Sebastiani

Advisors Prof. Franco Bontempi Dr. Francesco Petrini

a.a. 2014/2015 – Seminario intermedio XXVIII Ciclo



1.1 – TOPICS, KEYWORDS AND TOOLS

Topics

Seismic vulnerability assessment (design, retrofitting)

Strategic structures: bridges, (demand, performance, capacity, loss)

Seismic retrofitting (aging, life-cycle cost)

Modern technologies (bearings, isolation devices)

Tools

Full probabilistic approach (uncertainties, flexibility)

Finite element modelling (nonlinear analysis, no time consuming)



1.2 – STATE OF ART AND MOTIVATIONS

References on lifecycle costs (LCC) and aging

“The time-dependency of risk (seismic) in a lifecycle context is a quite

new area to be explored. In seismic analysis, aging consideration has

started to be included in seismic performance prediction models”

(Decò and Frangopol 2013, Ghosh and Padgett 2010)

Decò A. and Frangopol D.M. (2013). Life-Cycle Risk Assessment of Spatially Distributed Aging Bridges under Seismic and Traffic Hazards. Earthquake Spectra: February 2013, Vol. 29, No. 1, pp. 127-153.

Ghosh J. and Padgett J.E. (2010). Aging considerations in the development of time-dependent seismic fragility curves, Journal of Structural Engineering 136, 1497–1511



1.3 – FULL PROBABILISTIC APPROACHES IN THE PERFORMANCE-BASED EARTHQUAKE ENGINEERING (PBEE) FRAMEWORK

Franchin P. (2009) Research Within The Framework Of Performance-based Earthquake Engineering, Earthquake Engineering by the Beach Workshop, July 2-4, 2009, Capri, Italy

Cornell C.A. and Krawinkler H. (2000). Progress and Challenges in Seismic Performance Assessment. PEER Center News Spring 2000, 3(2).

Unconditional probabilistic methods

FORM, SORM

Simulation methods (Monte Carlo, Subset Simulation)

Conditional probability methods (IM-based)

SAC/FEMA

PEER method (Cornell, 2000)

random vibration problem

classical structural reliability methods

closed-form

more flexible

decomposition in conditional probabilities

not closed-form



Structural Engineers Assn. of California (SEAOC), (1995) Vision 2000 Committee. April 3, 1995. Performance Based Seismic Engineering of Buildings. J. Soulages, ed. 2 vols. [Sacramento, Calif.]

Pinto P.E., Bazzurro P., Elnashai A., Franchin P., Gencturk B., Gunay S., Haukaas T., Mosalam K. & Vamvatsikos, D. (2012). Probabilistic Performance-Based Seismic Design. fib Bulletin 68

1.4 – STATE OF PRACTICE AND MOTIVATIONS

Italian and european codes

DM 14-01-08, Eurocodes

Other codes

SEAOC Vision 2000 (1995), FEMA273 (1997)

ATC-40 (1989)

References on PBEE for the state of practice

“The (conditional probability approaches) have a distinct practice-oriented

character, they are currently employed as a standard tool in the

research community and are expected to gain ever increasing

acceptance in professional practice” (Pinto et al., 2012)

Semi-probabilistic approach

Safety coefficient – limit states

Quantifiable confidence

Many performance levels



Cornell C.A. and Krawinkler H. (2000) Progress and Challenges in Seismic Performance Assessment. PEER Center News Spring 2000, 3(2).

1.5 – PEER FORMULATION

Random variables

Decision Variable DV (repair cost, down time)

Damage Measure DM (cracking)

Engineering Demand Parameter EDP (drift)

Intensity Measure IM (Peak ground acceleration)

Probabilistic models

G(DV|DM) loss or performance model

G(DM|EDP) capacity model

G(EDP|IM) demand model

l(x) mean annual

frequency of x



*

STATE O

F A

RT

1.6 – PEER FRAMEWORK

Krawinkler H. and Miranda E. (2004) Chapter 9: Performance-based earthquake engineering. In: Bertero V.V., Bozorgnia Y.(eds) Earthquake engineering: from engineering seismology to performance-based engineering. CRC Press, Boca Raton



2.1 – THE CASE STUDY “MALA RIJEKA VIADUCT”



Bridge data

The bridge was built in 1973 as the highest railway bridge in the World

It has a continuous five-span steel frame carried by six piers of which the

middle ones have heights ranging from 50 to 137.5 m

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

Andrews M. (2008) Analysis of the Mala Rijeka viaduct. Proceedings of Bridge Engineering 2nd Conference 2008, 16 April 2008, University of Bath, Bath, UK



2.2 – OTHER SIMILAR CASES

VIADOTTO “RAGO” - A3 SA-RC – MORANO CALABRO (CS) 1969 VIADOTTO “VACALE” - GIOIA TAURO (RC) 2011

VIADOTTO “CATTINARA” CATTINARA (TS) 2005 AUTOSTRADA SALERNO-REGGIO, POLLA (SA) 2006



VIADOTTO “MUCCIA” ASSE VIARIO MARCHE-UMBRIA (MC) VIADOTTO “FORNELLO” S.G.C. ORTE-RAVENNA E45 2003

VIADOTTO IALLÀ AUTOSTRADA MONTE BIANCO-AOSTA 1992 VIADOTTO FRAGNETO - S.S. N.95 "DI BRIENZA"(PZ). 1990



The seismic hazard can be quantified in terms of an intensity measure (IM) which should define the seismic input to the structure.

What is the best IM in case of isolated system?

Does one have hazard data for that IM?

*

3.1 – HAZARD ANALYSIS



IM selection

Type of variable (scalar, vector)

Nature of variable (structure dependent)

Linear equivalent model to approximate the nonlinear behavior

of the structure

Type of isolation

Elastomeric bearings (ERB), Friction pendulum system (FPS)

PGA

Sa(T1)*

* Issue on the evaluation of T1 in case of complex structure with nonlinear devices



Probabilistic Seismic Hazard Analysis (PSHA)

H(a) is the annual probability of exceeding a seismic hazard intensity

measure “a” in a given seismic hazard environment

Field E.H., Jordan T.H. and Cornell C.A. (2003) “OpenSHA: A Developing Community-Modeling Environment for Seismic Hazard Analysis”. Seismological Research Letters, 74, no. 4, p. 406-419



Ground Motion selection

Nature of the signal (Simulated, recorded, spectrum-compatible

Elaboration of the signal (bin groups, scaled or unscaled)

Baker, J.W., Lin, T., Shahi, K.S. and Jayaram, N. (2011). New ground motion selection procedures and selected motions for the PEER Transportation Research Program. PEER Report 2011/03, Pacific Earthquake Engineering Research Center, Berkeley, California, USA. 106 pp.

First set 40 recorded GMs, Magnitude = 6 Source-to-site distance = 25 km Range of Sa is between 0 to 0.6 g

Second set 40 recorded GMs, Magnitude =7 Source-to-site distance = 10 km Range of Sa is up to 1.5g



*

3.2 – STRUCTURAL ANALYSIS

Inputs

Signals

Hazard curve



λE0 Ets

ft

(U, fpcU)

(c0, fpc) E0=2fpc/c0

EpEtsfy

16.5 m

16.5 m

cross section of the pier

materials

Computational F.E. model

Material and geometric nonlinearities

Specific elements for device modelling

Element with fiber section

Deck mass (120 m for the 3th pier) : 870 kNs2/m

Pier mass (distributed along the pier) : 7166 kNs2/m



INITIAL STIFFNESS k1 STRENGTH fy POST-YIELDING

STIFFNESS k2

FPS k1=75 k2=160000 kN/m fy=mW= 256.1 kN k2=W/R=2134.5 kN/m

ERB k1=10 k2=50200 kN/m fy= k1dy =301.2 kN k2=5020 kN/m

Computational F.E. model

Zhang J. and Huo Y. (2009) Evaluating effectiveness and optimum design of isolation devices for highway bridges using the fragility function method. Engineering Structures, 31, 1648-1660



EDP selection

Type of EDP

Local

Intermediate

Global

Nonlinear Time-Hystory Analysis

Probabilistic Seismic Demand Model (PSDM)

Stress and strain of concrete and steel

Column moment and curvature (mc)

Pier’s top displacement (dc)

Mosalam K.M. (2012) Probabilistic Performance-based Earthquake Engineering, University of Minho, Guimarães, Portugal, October 3-4, 2012



NO

ISO

LATIO

N

FPS



Probabilistic Seismic Demand Model (PSDM) in all cases



*

3.3 – DAMAGE ANALYSIS

Krawinkler H. and Miranda E. (2004) Chapter 9: Performance-based earthquake engineering. In: Bertero V.V., Bozorgnia Y.(eds) Earthquake engineering: from engineering seismology to performance-based engineering. CRC Press, Boca Raton



23

SEC 107

SEC 105

SEC 103

SEC 101

Capacity curves (pushover analysis)

Concrete cracking achievement

st = 5.2 N/mm2 is the ultimate tensile strength



24

SEC 107

SEC 105

SEC 103

SEC 101

Capacity curves (pushover analysis)

Steel yielding achievement

ss = 440 N/mm2 is the steel yield strength



Choi E., DesRoches R., Nielson B. (2004) “Seismic fragility of typical bridges in moderate seismic zones”. EngStruct 2004;26:187

Limit states definition

Three damage states DS namely slight, moderate and complete damage are adopted in this study and their concerning limit values are shown above

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 rebars

A comparison between the values adopted by Choi et al. (2004) and the ductility factors defined in the EC8 for piers, provides the limit values referred to the collapse



Fragility curves



Annual probability of exceeding each damage state

The seismic fragility can be convolved with the seismic hazard in

order to assess the annual probability of exceeding each damage

state:



PTf 𝑖 = 1− 1− PA𝑖 T

T-year probability of exceeding each damage state

T-year probability of exceeding a damage state

The probability of at least one event that exceeds design limits

during the expected life T (i.e. T=75 years) of the structure is the

complement of the probability that no events occur which exceed

design limits

Padgett J.E., Dennemann K. and Ghosh J. (2010) Risk-based seismic life-cycle cost–benefit (LCC-B) analysis for bridge retrofit assessment. Structural Safety 2010; 32(3):165–173.

Benefits of isolation devices in terms of probability of damage



T-YEARS PROBABILITY OF DAMAGE

SLIGHT DAMAGE MODERATE DAMAGE

NO ISOLATION 23% 1.3%

ERB 7% 0.3%

FPS 3% 0.04%

Sebastiani P.E., Padgett J.E., Petrini F., Bontempi F. (2014) Effectiveness Evaluation of Seismic Protection Devices for Bridges in the PBEE Framework. Proceedings of ASCE-ICVRAM-ISUMA 2014 - second International Conference on Vulnerability and Risk Analysis and Management (ICVRAM) Liverpool, 13th-17th July 2014



* http://peer.berkeley.edu/publications/annual_report/old_ar/year6/yr6_projects/ta1/1222002.html

*

3.4 – LOSS ANALYSIS

Inputs

T-year probability of exceeding a damage state



Definition of the nominal cost of restoration

Type of repair strategy, nominal cost Ci of restoration

Evaluation of life-cycle costs due to seismic damage

The expected value of the life-cycle costs due to seismic damage in present day dollars can be expressed as follows:

Where j is the damage state, T is the remaining service life of the bridge, Cj is the cost associated with damage state j, and PTfj is the T-year probability of exceeding damage state j

Slight damage Moderate damage Complete damage

Repair cost estimate ($) 2.00E05 5.00E05 2.00E06

Wen Y.K. and Kang Y.J. (2001) Minimum building life-cycle cost design criteria. I: methodology. J Struct Eng 2001;127(3):330–7.



Time-dependent fragility curves

Time-dependent mean annual rate of failure

The mean annual rate of failure, li,m(t), due to occurrence of a

particular damage state i, can be approximated by the annual probability of damage due to damage state i as

Ghosh J. and Padgett J.E. (2010) Aging considerations in the development of time-dependent seismic fragility curves. Journal of Structural Engineering 2010

Melchers R.E. (1999) Structural Reliability Analysis and Prediction (2nd edn). Wiley: New York

3.5 – AGING IN THE FRAGILITY STEP



Non-homogeneous Poisson process

In probability theory, a counting process is called a non-

homogeneous Poisson process with rate l(t) if the following relation

holds for

The time between events in a non-homogeneous Poisson process with

a time dependent rate can be modeled by an exponential distribution

with the cumulative density function (CDF) and the probablity density

function (PDF) following the equations

CDF

PDF



Seismic losses corresponding to a damage state i

The present value of total seismic losses corresponding to a damage

state i along the service life of the bridge is given by (Beck et al. 2002)

Where d is the discount ratio to convert future costs into present

values and T is the service life of the bridge.

The present value of total seismic losses corresponding to a damage

state i along the service life of the bridge is given by

Beck J.L., Porter K.A., Shaikhutdinov R.V., Au S.K., Mizukoshi K., Miyamura M. et al. (2002) Impact of seismic risk on lifetime property values. Monograph, Technical Report: CaltechEERL:2002.EERL-2002-04, California Institute of Technology, 2002.

3.6 – AGING IN THE LOSS STEP



Expected cost value

Moreover the expected value can be evaluated as

Where Ci,m is the nominal cost associated with damage state ith to

restore the bridge and P[Ci,m(t)] is the probability of incurring the

cost Ci,m

The probability can be approximated by the summation of its PDF

values calculated from t=0 to t=T in the discrete space as follows

Ghosh, J. and Padgett, J.E. (2011) Probabilistic seismic loss assessment of aging bridges using a component-level cost estimation approach. Earthquake Engng Struct. Dyn.



Expected cost and variance

Assuming a damage state i, a nominal cost Ci,m=2.0E06 $, a discount

factor 0.03, T=75 years

35%



Flowchart of the loss estimation

Rate of failure li,m(t) due to

occurrence of a damage state i

Probability of at least one event during {0,t}

Restoration cost Ci,m for the damage state i

Probability of incurring a hypothetical cost Ci,m in {0,t}

Expected seismic loss corresponding to a damage state i during {0,t}

Time-dependent cost C(t) formulation (discount)

Total expected cost across all damage states during {0,t}



4 – CONCLUSIONS

Topic

PBEE for loss estimation of isolated bridges with aging effects

Contributions

Application to a real case study, implementing the whole PEER

procedure in Matlab environment

Working on a recent formulation to evaluate expected cost and

variance in case of aging effects, with a contribution in the

discount factor implementation



Works for the next year

Complete results with the full model of the bridge (already done)

Effectiveness evaluation of seismic protection devices in terms of LCC

Application to a second type of more common bridges





PBEE (PEER METHOD)

BRIDGES

ISOLATION

AGING

LCC

PBEE = PERFORMANCE-BASED EARTHQUAKE ENGINEERING LCC = LIFE-CYCLE COST ANALYSIS ISOLATION = SEISMIC ISOLATION SYSTEMS AGING = EFFECTS OF AGING ON THE STRUCTURE

1.7 – TARGET AND CONTRIBUTION



Mosalam K.M. (2012) Probabilistic Performance-based Earthquake Engineering, University of Minho, Guimarães, Portugal, October 3-4, 2012











Ghosh J. and Padgett J.E. (2010) Aging considerations in the development of time-dependent seismic fragility curves. Journal of Structural Engineering 2010