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.



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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.



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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.

Documents

Robustness of Externally and Internally Post-Tensioned Bridges