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Reliability and Redundancy Analysis of Structural Systems with Application to Highway Bridges

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Title: Reliability and Redundancy Analysis of Structural Systems with Application to Highway Bridges


1
Reliability and Redundancy Analysis of Structural
Systemswith Application to Highway Bridges
  • Michel Ghosn
  • The City College of New York / CUNY

2
Contributors
  • Prof. Joan Ramon Casas
  • UPC Construction Engineering
  • Ms. Feng Miao
  • Mr. Giorgio Anitori

3
Introduction
  • Structural systems are designed on a member by
    member basis.
  • Little consideration is provided to the effects
    of a local failure on system safety.
  • Local failures may be due to overloading or loss
    of member capacity from fatigue fracture,
    deterioration, or accidents such as an impact or
    a blast.
  • Local failure of one element may result in the
    failure of another creating a chain reaction that
    progresses throughout the system leading to a
    catastrophic progressive collapse.

4
I-35W over Mississippi River (2007)
Truss bridge Collapse due to initial failure of
gusset plate
5
I-35 Gusset Plate
6
I-40 Bridge in Oklahoma (2002)
Bridge collapse due to barge impact
7
Route 19 Overpass, Quebec (2006)
Box-Girder bridge collapse due to corrosion
8
Corroded Bridge Deck
9
Oklahoma City Bombing (1995)
10
Structural Redundancy
Bridges survive initial damage due to system
redundancy and reserve safety
Collisions
Fatigue Fracture
Seismic Damage
11
Definitions
  • Redundancy is the ability of a system to continue
    to carry loads after the overloading of members.
  • Robustness is the ability of a structural system
    to survive the loss of a member and continue to
    carry some load.
  • Progressive Collapse is the spread of an initial
    local failure from element to element resulting,
    eventually, in the collapse of an entire
    structure or a disproportionately large part of
    it.

12
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13
Structural Performance
14
Deterministic Criteria
  • Ultimate Limit State
  • Functionality Limit State
  • Damaged Limit State

15
State of the Art
  • New guidelines to have high levels of redundancy
    in buildings.
  • Criteria are based on deterministic analyses.
  • Uncertainties in estimating member strengths and
    system capacity as well as applied load intensity
    and distribution justify the use of probabilistic
    methods.

16
Structural Reliability
17
Reliability Index, b
  • Reliability index, b, is defined in terms of the
    Gaussian Prob. function
  • If R and S follow Gaussian distributions
  • b function of means and standard deviations

18
Reliability Index, b
19
Lognormal Probability Model
  • If the load and resistance follow Lognormal
    distributions then the reliability index is
    approximately
  • b function of coefficients of variation
  • V stand. Dev./ mean

20
System Reliability
  • Probability of structural collapse, P(C), due to
    different damage scenarios, L, caused by multiple
    hazards, E
  • P(E) probability of occurrence of hazard E
  • P(LE) probability of local failure, L, given E
  • P(CLE) is probability of collapse given L due to
    E

21
Safety Criteria
  • The probability of bridge collapse must be
    limited to an acceptable level
  • Alternatively, the criteria can be set in terms
    of the reliability index, ß, defined as

22
Option 1 to Reduce Risk
  • Reduce exposure to hazards lower P(E)
  • Protect columns from collisions through barriers
  • Set columns at large distances from roadway to
    avoid crashes
  • Increase bridge height to avoid collisions with
    deck
  • Build away from earthquake faults
  • Use steel connection details that are not prone
    to fatigue and fracture failures
  • Increase security surveillance to avoid
    intentional sabotage

23
Option 2 to Reduce Risk
  • Reduce member failure given a hazard P(LE)
  • Increase reliability of connection details by
    using different connection types, advanced
    materials, or improved welding, splicing and
    anchoring techniques
  • Strengthen columns that may be subject to
    collisions or sabotage using steel jacketing or
    FRP wrapping
  • Increase capacity of columns and critical members
    to improve their ability to resist unusual loads

24
Option 3 to Reduce Risk
  • Avoid collapse if one member fails P(CLE)
  • Use structural configurations that have high
    levels of redundancy.
  • Appropriately spaced large number of columns
  • Trusses that are not statically determinate
  • Ensure that all the members contributing to a
    mode of failure are conservatively designed
  • to pick up the load shed by member that fails in
    brittle mode
  • to pick up additional load applied if member that
    initiates sequence fails in a ductile mode.

25
Types of Failures
26
Issues with Reliability Analysis
  • Realistic structural models involve
  • Large numbers of random variables
  • Multiple failure modes
  • Low probability of failure for members, 10-4
  • Probability of failure for systems, 10-6
  • Computational effort

27
Finite Element Analysis
28
Reliability Analysis Methods
  • Monte Carlo Simulation (MCS)
  • First Order Reliability Method (FORM)
  • Response Surface Method (RSM)
  • Latin Hypercube Simulation (LHS)
  • Genetic Search Algorithms (GA)
  • Subset Simulation (SS)

29
Monte Carlo Simulation (MCS)
  • Random sampling to artificially simulate a large
    number of experiments and observe the results.
  • Can solve problems with complex failure regions.
  • Needs large numbers of simulations for accurate
    results.

30
Monte Carlo Simulation (MCS)
  • Probab. of failure Number of cases in failure
    domain/ total number of cases

31
First Order Reliability Method
  • First Order Reliability Method (FORM)
    approximates limit-state function with a
    first-order function.
  • Reliability index is the minimum distance between
    the mean value to the failure function.
  • If limit state function is linear

32
First Order Reliability Method
Use optimization techniques to find design point
shortest distance between Z0 to origin of
normalized space
33
Response Surface Method (RSM)
  • RSM approximates the unknown explicit limit state
    function by a polynomial function.
  • A second order polynomial is most often used for
    the response surface.
  • The function is obtained by perturbation of
    variables near design point.

34
Response Surface Method (RSM)
35
Subset Simulation (SS)
  • If F denote the failure domain. Subset failure
    regions Fi are arranged to form a decreasing
    sequence of failure events
  • The probability of failure Pf can be represented
    as the probability of falling in the final subset
    given that on the previous step, the event
    belonged to subset Fm-1

36
Subset Simulation (SS)
  • By recursively repeating the process, the
    following equation is obtained
  • During the simulation, conditional samples are
    generated from specially designed Markov Chains
    so that they gradually populate each intermediate
    failure region until they cover the whole failure
    domain.

.

37
Illustration of Subset Simulation Procedure
bi are chosen adaptively so that the
conditional probabilities are approximately to a
pre-set value, p0. (e.g. p00.1)
38
Illustration of Subset Simulation Procedure
39
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40
Development of Reliability Criteria
  • Analyze a large number of representative bridge
    configurations.
  • Find the reliability indexes for those that have
    shown good system performance.
  • Use these reliability index values as criteria
    for future designs
  • Find the corresponding deterministic criteria

41
Input Data for Reliability Analysis
  • Dead loads
  • Bending moment resistance
  • Composite steel beams
  • Prestr. concrete beams
  • Concrete T-beams

42
Live Load Simulation
  • Maximum of N events.
  • 75-yr design life
  • 5-yr rating cycle
  • ADTT 5000
  • 1000
  • 100

Bin I
Bin II
Repeat for N loading events
43
Simulated vs. Measured
Single event Two-lane 100-ft span
44
Cumulative Distribution
45
Maximum Load Effect
Max. 5-yr event Two-lane 100-ft span
46
Reliability-Based Criteria for Bridges
  • Based on bridge member reliability
  • Corresponding system safety, redundancy and
    robustness criteria

47
Deterministic Criteria
  • Ultimate Limit State
  • Functionality Limit State
  • Damaged Limit State

48
Design Criteria
  • Apply system factor during the design process to
    reflect level of redundancy
  • fs lt1.0 increases the system reliability of
    designs with low levels of redundancy.
  • fs gt 1.0 allows members of systems with high
    redundancy to have lower capacities.

49
Example Ps/Concrete Bridge
  • 100-ft simple span, 6 beams at 8-ft

50
Example Ps/Concrete Bridge
51
Example Ps/Concrete Bridge
?ßu ßult - ßmem 5.75-2.85 2.90 gt
0.85 ?ßf ßfunct - ßmem 3.69-2.85 0.84 gt
0.25
52
Steel Truss Bridge
53
Steel Truss Bridge
?ßu ßult-ßmem 7.80-6.80 1.00 gt
0.85 ?ßf ßfunct-ßmem 7.60-6.80 0.80 gt 0.25
54
Damaged Bridge Analysis
?ßd ßdamaged ßmem 1.90-2.85 -0.95gt-2.70
for P/C bridge ?ßd ßdamaged ßmem 2.42-6.80
-4.38lt-2.70 for truss bridge
  • Truss bridge is not robust.
  • But bdamaged is greater than 0.80 system
    safety is satisfied
  • Member reliability index of the truss is
    ßmember6.8

55
Deterministic Analysis of Ps/Concrete Bridge
56
Twin Steel Box Girder Bridge
57
Structural Analysis
58
Reliability Analysis
59
Redundancy Analysis
  • Dbu 1.24 gt 0.85 O.K.
  • Dbf 0.14 lt 0.25 N.G.
  • Dbd -3.46 lt -2.70 N.G.

60
System Safety Analysis
  • bultimate 9.77 gt 4.35 O.K.
  • bfunctionality 8.67 gt 3.75 O.K.
  • bdamaged 5.07 gt 0.80 O.K.
  • Although the system is not sufficiently
    redundant, the bridge members are so overdesigned
    by about a factor of 3 that all system safety
    criteria are satisfied

61
Bridge system analysis
  • Multicellular box girder deck
  • Integral design
  • 4 spans (max 48 m)

62
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63
Probabilistic results
Intact structure
Damaged structure
64
Conclusions
  • A method is presented to consider system
    redundancy and robustness during the structural
    design and safety evaluation of bridges.
  • The method is based on structural reliability
    principles and accounts for the uncertainties in
    evaluating system strength and applied loads.
  • The goal is to ensure that structural systems
    meets minimum levels of system safety in order to
    sustain partial failures or structural damage.
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