Title: Reliability and Redundancy Analysis of Structural Systems with Application to Highway Bridges
1Reliability and Redundancy Analysis of Structural
Systemswith Application to Highway Bridges
- Michel Ghosn
- The City College of New York / CUNY
2Contributors
- Prof. Joan Ramon Casas
- UPC Construction Engineering
- Ms. Feng Miao
- Mr. Giorgio Anitori
3Introduction
- 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.
4I-35W over Mississippi River (2007)
Truss bridge Collapse due to initial failure of
gusset plate
5I-35 Gusset Plate
6I-40 Bridge in Oklahoma (2002)
Bridge collapse due to barge impact
7Route 19 Overpass, Quebec (2006)
Box-Girder bridge collapse due to corrosion
8Corroded Bridge Deck
9Oklahoma City Bombing (1995)
10Structural Redundancy
Bridges survive initial damage due to system
redundancy and reserve safety
Collisions
Fatigue Fracture
Seismic Damage
11Definitions
- 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.
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13Structural Performance
14Deterministic Criteria
- Ultimate Limit State
- Functionality Limit State
- Damaged Limit State
15State 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.
16Structural Reliability
17Reliability 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
18Reliability Index, b
19Lognormal 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
20System 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
21Safety 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
22Option 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
23Option 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
24Option 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.
25Types of Failures
26Issues 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
27Finite Element Analysis
28Reliability 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)
29Monte 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.
30Monte Carlo Simulation (MCS)
- Probab. of failure Number of cases in failure
domain/ total number of cases
31First 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
32First 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.
34Response Surface Method (RSM)
35Subset 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
36Subset 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.
.
37Illustration 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)
38Illustration of Subset Simulation Procedure
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40Development 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
41Input Data for Reliability Analysis
- Dead loads
- Bending moment resistance
- Composite steel beams
- Prestr. concrete beams
- Concrete T-beams
42Live 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
43Simulated vs. Measured
Single event Two-lane 100-ft span
44Cumulative Distribution
45Maximum Load Effect
Max. 5-yr event Two-lane 100-ft span
46Reliability-Based Criteria for Bridges
- Based on bridge member reliability
- Corresponding system safety, redundancy and
robustness criteria
47Deterministic Criteria
- Ultimate Limit State
- Functionality Limit State
- Damaged Limit State
48Design 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.
49Example Ps/Concrete Bridge
- 100-ft simple span, 6 beams at 8-ft
50Example Ps/Concrete Bridge
51Example 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
52Steel Truss Bridge
53Steel 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
54Damaged 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
55Deterministic Analysis of Ps/Concrete Bridge
56Twin Steel Box Girder Bridge
57Structural Analysis
58Reliability Analysis
59Redundancy 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.
60System 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
61Bridge system analysis
- Multicellular box girder deck
- Integral design
- 4 spans (max 48 m)
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63Probabilistic results
Intact structure
Damaged structure
64Conclusions
- 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.