robustness of externally and internally post-tensioned bridges
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Robustness of
Externally and Internally
Post-Tensioned Bridges
Anna ORZE
Marzena RODZE
KBI sem. III
2012/2013
Damages and Catastrophes of Structures
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CONTENT OF THE PRESENTATION
Introduction to the topic
Event tree formulation
Risk calculation and the index of robustness
Application of the framework
Modeling of the system exposures Modeling of the system vulnerability
Modeling of the system robustness
Results
Conclusions
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INTRODUCTION
Structural robustness is defined by the structures ability to withstand any
unforeseen loading as well as initiating damage scenarios without a
disproportionate response.
More specifically, a robust structure has the ability to redistribute load in
the event that a loadbearing member suffers a loss of strength or stiffness,
and characteristically exhibits ductile rather than brittle global failure
modes.
Eurocode 1 describes robustness as:
the ability of a structure to withstand events like fire, explosions, impact
or the consequences of human error without being damaged to an extent
disproportionate to the original cause.
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INTRODUCTION
A New risk framework was developed which differentiates between damagestates and failure states in the systems.
The consequences model differentiates between:
Direct consequences
Indirect consequences (follow-up consequences)
Eurocode 2specifies the requirement for structural robustness as theconsequences of structural failure should not be disproportional to the effectcausing the failure.
Assesing the robustness of structures considers accidental exposures, suchas impact, explosion or fire.
Some of the exposures can be accounted in the design procedures directlybut for others it is not practical.
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EVENT TREE FORMULATION According to the generic risk assessment a system can be represented as a
spatial and temporal representation of all constituents required to describe
the interrelations between all relevant exposures and their consequences.
Damaged
Undamaged
Direct
consequences
Direct
consequence
s
Failure
Without system
failure
Direct and
indirect
consequence
s
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RISK CALCULATION
AND THE INDEX OF ROBUSTNESS
The total risk is defined as the expected value of the total consequences in a
given time period.
Direct risk formula:
Indirect risk formula:
Total risk is a sum of the direct and indirect risks.
Index of robustness formula:
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APPLICATION OF THE FRAMEWORK
Introduced framework is applied to a general type of post-tensioned box
girder bridge typical for roadways in german.
Bridge consistns of six spans 40-50m lengths
16 tendons in top flange, 8 in the bottom flange
Concrete class: C30/35 (according to EC 2)
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APPLICATION OF THE FRAMEWORK
The considered load cases in the design are:
Dead Loads
Traffic Loads
Temperature loads
Loads due to post-tensioning
The bridge is designed according to Eurocodes for the ULS and SLS, as well
as for decompression aiming at un-cracked concreteduring bridge lifetime
(100 years).
The Eurocode requires minimum structural reinforcement to prevent a brittle
failure and to ensure ductility.
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MODELING OF THE SYSTEM EXPOSURE
All loads are considered uncertain and modeled probabilistically: Dead loads follow the Gaussian distribution with a coefficient of variation of 0.1.
Live loads are modeled probabilistically.
Creep and shrinkage are modeled according to nonlinear effects.
Stress losses due to relaxation follow the Gaussian distribution with a coefficient of
0.3.
CHLORIDE INDUCED CORROSION:
The process of the chloride through the concrete
cover to the reinforcement is modeled accordingto Ficks law.
Monte-Carlo simulation is applied to determine
the probability of propagation at t0.
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MODELING OF THE SYSTEM EXPOSURE
STRESS CORROSION CRACKING:
Stress corrosion cracking under high post-
tensioning stresses is significantly important.
The highest tensile stresses occur in the top
flange of the box girder, so only those
16 tendons are affected to stress corrosion
cracking in the considered lifetime of the
structure.
A wire is considered that has failed when half of its diameter is corroded.
A strand is considered failed if four of its wires have failed.
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Approximations that reduces the amout of simulations necessary
to estimate the robustness:
For the probabilistic modeling of the structural system each material resistance has to
be represented with its density function.
Presented corrosion exposures are assumed to act simultaneously and independently
from each other.
Probability for a corrosive environment is dependent on the location. Effects due to creep and shrinkage are considered and exposures due to loads are
applied.
For the corrosion exposure random values are selected to calculate its probability.
Effects due to creep and shrinkage and exposure due to loads are considered
simultaneously. Damage states are integrated by using calculated loss of post-tensioning wires and the
corroded reinforcement area.
Probability of failure includes the probability of each damage state.
2.2 MODELINGOFTHESYSTEMVULNERABILITY
Probability for the damage state
Probability of failure of structural system
Direct consequences
Indirect consequences
All random variables related to the applied material resistances
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The expected values of the different kinds of consequences.
o The expected value of the total direct consequences for planning, traffic
organization, safety measures and repair maintenance costs sum up to1,203,800 EUR.
2.2 MODELINGOFTHESYSTEMVULNERABILITY
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For the risk based analysis the specific probabilities of failure depending on the
considered damage states at any given time are calculated.
For this analysis limit state function is formulated for the cross section at a location
just above the secon support.
Failure state is reached if the internal moment due to exposures exceeds the internal
moment of the resistance.
Model uncertainties are approximated by a lognormal distibuted random variable
with a mean value ofl and CoV of 0.2.The indirect consequences are defined as all consequences beyond the direct
consequences which are associated with the failure state.
Taking into account all failure modes where evacuation action could be
performed, the expected number of fatalities given failure in this example is
estimated to 2.62
2.3 MODELING OF THE SYSTEM ROBUSTNESS
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A consistent basis to include aspects of life safety into the decision making is
provided by the Life Quality Index.
Using LQI for Germany the value of a life is estimated to be 3.38 million
Idea ofLQI is to model the preferences of a society as an indicator comprised
by a relationship between GDP per capita, the expected life at birth and the
propotion of life spend for earning and living.
2.3 MODELING OF THE SYSTEM ROBUSTNESS
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Summary of the consideted indirect consequences.
Another type of the indirect cosequences is user costs for the highway user.
Additionaly the costs for the reconstruction have to be taken into account:- deconstruction,
- planning and design of a new bridge,
- traffic organisation,
- safety precautions,
- new construction for an assumed construction time of a one year.
2.3 MODELING OF THE SYSTEM ROBUSTNESS
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2.4 RESULTS
Result of the Monte Carlo Simulation for the annual probability of failure
Due to creep and
shrinkage.
Due to chloride
corrosion.
Insignificant
increment.
At the end of design lifetime 102 of 196
wires of the internal post-tensioning
tendons in the top flange are ruptured and
approx. 23% of the reinforcement is
corroded.
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2.4 RESULTS
Based on calculations of the probability of failure under consideration of
the different damage states and the direct and indirect consequences,
index of robustnesscan be calculated.
Significant influence
of post tensioning
wire losses.
Coefficient of reliability indicates that the damages
states and system effects contribute to total risk.
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CONCLUSIONS
The damage of the internal tendons have a large influence on the failure
probability and on the robustness of the system.
By changing post tensioning system the corrosion effects can be minimized
and robustness improved.
The robustness decreases rapidly if the conditional probability of failureincreases in the system.
To reduce this measure could be aimed to reduce the indirect consequences.
The index of robustness is more a characteristic of the system than of the
structure.
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REFERENCES
ROBUSTNESSOFEXTERNALLYANDINTERNALLYPOST-
TENSIONEDBRIDGES, BERNARDVONRADOWITZ, MATTHIAS
SCHUBERTANDMICHAELHAVBROFABER. BETON- UND
STAHLBETONBAUROBUSTNESSANDSAFETYOFCONCRETESTRUCTURES, 2008.
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