Reliability and Redundancy Analysis
of Structural Systems
with Application to Highway Bridges
Michel Ghosn
The City College of New York / CUNY
Contributors
Prof. Joan Ramon Casas
UPC Construction Engineering
Ms. Feng Miao
Mr. Giorgio Anitori
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.
I

35W over Mississippi River (2007)
Truss bridge Collapse due to initial failure of gusset plate
I

35 Gusset Plate
I

40 Bridge in Oklahoma (2002)
Bridge collapse due to barge impact
Route 19 Overpass, Quebec (2006)
Box

Girder bridge collapse due to corrosion
Corroded Bridge Deck
Oklahoma City Bombing (1995)
Structural Redundancy
Collisions
Fatigue Fracture
Seismic Damage
Bridges survive initial damage due to
system redundancy and reserve safety
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.
Structural Performance
Deterministic Criteria
Ultimate Limit State
Functionality Limit State
Damaged Limit State
3
.
1
1
LF
LF
R
u
u
2
.
1
1
LF
LF
R
f
f
5
.
0
1
LF
LF
R
d
d
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
.
Structural Reliability
Load

Resistance
S

R
function Z
state
limit
load)
Applied
nce
Pr(Resista
)
(
collapse
of
y
Probabilit
C
P
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
2
2
S
R
Z
S
R
Z
b
b
f
P
Reliability Index,
b
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
2
S
2
R
Z
V
V
S
R
ln
Z
b
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
)
(
)
(
)
(
)
(
E
L
E
P
E
L
P
LE
C
P
C
P
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,
β
, defined as:
threshold
P
C
P
)
(
C
P
1
b
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
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
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.
Types of Failures
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
Finite Element Analysis
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)
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.
Monte Carlo Simulation (MCS
)
Probab. of failure = Number of cases in
failure domain/ total number of cases
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
f
P
)
(
b
First Order Reliability Method
Use optimization techniques to find design point = shortest
distance between Z=0 to origin of normalized space
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.
m
i
i
i
m
i
i
i
m
c
b
a
G
1
2
1
2
1
)
,...,
,
(
Response Surface Method (RSM)
Subset Simulation (SS)
If F denote the failure domain. Subset failure regions F
i
are arranged to form a decreasing sequence of failure
events:
The probability of failure P
f
can be represented as the
probability of falling in the final subset given that on the
previous step, the event belonged to subset F
m

1
:
F
F
F
F
m
...
2
1
)
(
)
(
1
1
m
m
m
f
F
P
F
F
P
P
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
.
.
m
i
i
i
m
m
m
f
F
F
P
F
P
F
P
F
F
P
P
2
1
1
1
1
Response B
Failure Probability Estimate
0
Monte Carlo Simulation
Uncertain Parameter Space
Uncertain Parameter Space
Monte Carlo Simulation
0
Failure Probability Estimate
Response B
(a) Level 0: Monte Carlo Simulation
F1
(b) Level 0: selection of first intermediate threshold level
P
0
N samples
b
1
NP
0
samples
within F
1
N(1P
0
) samples
b
i
are chosen
“adaptively”
so that the
conditional
probabilities
are
approximately
to a pre

set
value, p
0
.
(e.g. p
0
=0.1)
Illustration of
Subset
Simulation
Procedure
(c) Level 1: conditional samples generated using MH algorithm
F
1
Response B
Failure Probability Estimate
0
Monte Carlo Simulation
Uncertain Parameter Space
(d) Level 1: selection of second intermediate threshold level
P
0
b
1
Generate N(1P
0
)
more samples in F
1
to
complete N samples
NP
0
samples in F
2
b
1
P
0
Uncertain Parameter Space
Monte Carlo Simulation
0
Failure Probability Estimate
Response B
(P
0
)
2
F
2
b
2
Illustration of
Subset
Simulation
Procedure
0
1
2
3
4
5
0
1
2
3
4
5
X2
X1
LSP from Simulation
Accurate LSF
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
X2
X1
LSP from Simulation
Accurate LSF
3
2
1
0
1
2
3
3
2
1
0
1
2
3
X2
X1
LSP from Simulation
Accurate LSF
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
2
3
4
5
6
X2
X1
LSP from Simulation
Accurate LSF4
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
Input Data for Reliability
Analysis
•
Dead loads
•
Bending moment resistance:
Composite steel beams
Prestr. concrete beams
Concrete T

beams
%
25
V
D
0
.
1
D
%
10
V
D
05
.
1
D
%
8
V
D
03
.
1
D
DW
W
W
2
DC
2
C
2
C
1
DC
1
C
1
C
%
13
14
.
1
R
n
V
R
R
%
5
.
7
05
.
1
R
n
V
R
R
%
10
12
.
1
R
n
V
R
R
Live Load Simulation
Bin
I
Bin II
Repeat for N loading events
•
Maximum of N
events.
•
75

yr design life
•
5

yr rating cycle
•
ADTT = 5000
= 1000
= 100
Simulated vs. Measured
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Frequency
Normalized Moment
Single Lane
TwoLane Original
TwoLane Simulated
Single event
Two

lane
100

ft span
Cumulative Distribution
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Distribution
Moments/HL93
Oneevent
5year Extreme
10year Extreme
Maximum Load Effect
1.5
2.0
2.5
3.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Frequency
Normalized Moment
WIM Data
Ev Proj. Hist.
Max. 5

yr event
Two

lane
100

ft span
Reliability

Based Criteria for Bridges
Based on bridge member reliability
Corresponding system safety, redundancy and
robustness criteria:
5
.
3
member
b
35
.
4
ultimate
b
75
.
3
ity
functional
b
80
.
0
damaged
b
85
.
0
u
b
25
.
0
f
b
70
.
2
d
b
Deterministic Criteria
Ultimate Limit State
Functionality Limit State
Damaged Limit State
3
.
1
1
LF
LF
R
u
u
2
.
1
1
LF
LF
R
f
f
5
.
0
1
LF
LF
R
d
d
Design Criteria
Apply system factor during the design process to
reflect level of redundancy
f
s
<1.0 increases the system reliability of designs with low levels
of redundancy.
f
s
> 1.0 allows members of systems with high redundancy to have
lower capacities.
50
.
0
,
10
.
1
,
30
.
1
min
d
f
u
s
R
R
R
f
i
i
i
n
s
P
R
f
f
Example Ps/Concrete Bridge
100

ft simple span, 6 beams at 8

ft
8''
5@8ft
Truck 2
Truck 1
Example Ps/Concrete Bridge
0
10
20
30
40
50
60
70
0
500
1000
1500
2000
2500
Load (Kips)
Deflection (in)
Intact Bridge
Damaged Bridge
Example Ps/Concrete Bridge
5.75
3.69
2.85
b
ultimate
β
functionality
β
member
4.44 10

9
1.12 10

4
2.19 10

3
P
f
(ultimate
)
P
f
(functionality
)
P
f
(member
)
5.75
3.69
2.85
b
ultimate
β
functionality
β
member
4.44 10

9
1.12 10

4
2.19 10

3
P
f
(ultimate
)
P
f
(functionality
)
P
f
(member
)
∆βu = βult

βmem = 5.75

2.85 = 2.90 > 0.85
∆βf = βfunct

βmem = 3.69

2.85 = 0.84 > 0.25
Steel Truss Bridge
11'
3'
8@13'
Steel Truss Bridge
7.80
7.60
6.80
β
ultimate
β
functionality
β
member
3.1*10

15
1.45*10

14
5.13*10

12
P(ultimate
)
P(functionality
)
P(member
)
7.80
7.60
6.80
β
ultimate
β
functionality
β
member
3.1*10

15
1.45*10

14
5.13*10

12
P(ultimate
)
P(functionality
)
P(member
)
∆βu = βult

βmem = 7.80

6.80 = 1.00 > 0.85
∆βf = βfunct

βmem = 7.60

6.80 = 0.80 > 0.25
Damaged Bridge Analysis
2.42
1.90
β
damaged
7.76*10

3
0.0287
P(ultimate
)
Truss Bridge
Prestressed
Concrete Bridge
2.42
1.90
β
damaged
7.76*10

3
0.0287
P(ultimate
)
Truss Bridge
Prestressed
Concrete Bridge
∆βd = βdamaged
–
βmem = 1.90

2.85 =

0.95>

2.70 for P/C bridge
∆βd = βdamaged
–
βmem = 2.42

6.80 =

4.38<

2.70 for truss bridge
•
Truss bridge is not robust.
•
But
b
damaged is greater than 0.80 ; system safety is satisfied
•
Member reliability index of the truss is
β
member=6.8
Deterministic Analysis of
Ps/Concrete Bridge
.
.
30
.
1
44
.
1
26
.
3
68
.
4
1
k
o
LF
LF
R
u
u
.
.
10
.
1
22
.
1
26
.
3
98
.
3
1
k
o
LF
LF
R
f
f
.
.
50
.
0
56
.
0
26
.
3
82
.
1
1
k
o
LF
LF
R
d
d
Twin Steel Box Girder Bridge
Structural Analysis
0
5
10
15
20
25
0
80
160
240
320
400
Load (kips)
Deflection (in)
undamaged box
_
grillage
damaged box
_
grillage
undamaged box
_test
damaged box
_test
Reliability Analysis
Variable
Bias
COV
Distribution type
Main member Resistances
1.12
10%
Lognormal
Dead load
1.05
10%
Normal
Maximum rotation
1.0
20%
Lognormal
75

year Live load
1.89
19%
Lognormal
2

year live load
1.75
19%
Lognormal
member
b
ity
functional
b
ultimate
b
damaged
b
8.532
8.668
9.773
5.069
Redundancy Analysis
b
u
= 1.24 > 0.85 O.K.
b
f
= 0.14 < 0.25 N.G.
b
d
=

3.46 <

2.70 N.G.
LF1
LFf
LFu
LFd
Ru
Rf
Rd
Actu
al Case 12

ft
11.51
11.87
15.25
4.86
1.32
1.03
0.42
5 times bracing
11.51
12.32
15.25
4.70
1.32
1.07
0.41
No bracing
11.51
11.82
15.24
4.05
1.32
1.03
0.35
System Safety Analysis
b
ultimate
= 9.77 > 4.35 O.K.
b
functionality
= 8.67 > 3.75 O.K.
b
damaged
= 5.07 > 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
Bridge system analysis
•
Multicellular box girder deck
•
Integral design
•
4 spans (max 48 m)
Probabilistic
results
0
10000
20000
30000
40000
50000
60000
70000
80000

0.1
0.0
0.1
0.2
0.3
0.4
base shear (kN)
disp(m)

10000
0
10000
20000
30000
40000
50000
60000
70000
80000

0.1
0.0
0.1
0.2
0.3
0.4
base shear (kN)
disp(m)
85
.
0
54
.
1
u
b
25
.
0
43
.
0
f
b
70
.
2
42
.
0
d
b
Intact structure
Damaged structure
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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