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
Bridges survive initial damage due to
system redundancy and reserve safety
Collisions
Fatigue Fracture
Seismic Damage
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
Ru  LFu LF1  1.3

Functionality Limit State
R f  LFf LF1  1.2

Damaged Limit State
Rd  LFd LF1  0.5
State of the Art



New guidelines to have
redundancy in buildings.
high
levels
of
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
Probability of collapseP(C )  Pr(Resistance  Appliedload)
limit state function Z  R - S  Resistance - Load
Reliability Index, b

Reliability index, b, is defined in terms of the
Gaussian Prob. function:
Pf   b 

If R and S follow Gaussian distributions:
b
Z
Z

RS
 R2   S2
b function of means and standard deviations
Reliability Index, b
Lognormal Probability Model

If the load and resistance follow Lognormal
distributions then the reliability index is
approximately
R
ln 
S
Z

b

Z
VR2  VS2
b function of coefficients of variation:
V =stand. Dev./ mean
System Reliability

Probability of structural collapse, P(C), due to
different damage scenarios, L, caused by
multiple hazards, E:
P(C )  E L P(C LE ) P( L E ) P( E )



P(E) =probability of occurrence of hazard E
P(L|E) = probability of local failure, L, given E
P(C|LE) is probability of collapse given L due to E
Safety Criteria

The probability of bridge collapse must be
limited to an acceptable level:
P(C)  Pthreshold

Alternatively, the criteria can be set in
terms of the reliability index, β, defined as:
b  1PC 
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(L|E)



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(C|LE)

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 firstorder function.
Reliability index is the minimum distance
between the mean value to the failure function.
If limit state function is linear
(b )  Pf
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.
m
m
i 1
i 1
G (1 , 2 ,..., m )  a   bi i   ci i2

The function is obtained by perturbation of
variables near design point.
Response Surface Method (RSM)
Subset Simulation (SS)

If F denote the failure domain. Subset failure regions Fi
are arranged to form a decreasing sequence of failure
events:
F1  F2  ...  Fm  F

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:
Pf  P(Fm Fm1 )P(Fm1 )
Subset Simulation (SS)

By . recursively repeating the process, the following
equation is obtained:
Pf  PFm Fm1 PFm1   PF1  PFi Fi 1 
m
i 2

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.
Illustration of
Subset
Simulation
Procedure
Response B
Monte Carlo Simulation
N samples
0
Uncertain Parameter Space
Failure Probability Estimate
bi are chosen
“adaptively”
so that the
conditional
probabilities
are
approximately
to a pre-set
value, p0.
(e.g. p0=0.1)
(a) Level 0: Monte Carlo Simulation
Monte Carlo Simulation
NP0 samples
within F1
Response B
F1
b1
N(1-P0) samples
0
Uncertain Parameter Space
P0
Failure Probability Estimate
(b) Level 0: selection of first intermediate threshold level
Illustration of
Subset
Simulation
Procedure
Monte Carlo Simulation
Generate N(1-P0)
more samples in F1 to
complete N samples
Response B
F1
b1
P0
0
Failure Probability Estimate
Uncertain Parameter Space
(c) Level 1: conditional samples generated using M-H algorithm
Monte Carlo Simulation
NP0 samples in F2
Response B
F2
b2
b1
0
Uncertain Parameter Space
(P0)2
P0
Failure Probability Estimate
(d) Level 1: selection of second intermediate threshold level
3
5
2
4
-1
2
1
-2
-3
LSP from Simulation
Accurate LSF
3
LSP from Simulation
Accurate LSF
0
X2
X2
1
-3
-2
-1
0
1
2
0
3
0
1
2
X1
5
2.5
2.0
LSP from Simulation
Accurate LSF4
4
1.5
X2
X2
4
X1
6
5
3
3
LSP from Simulation
Accurate LSF
1.0
0.5
2
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
X1
0.0
-0.5
0.0
0.5
1.0
1.5
X1
2.0
2.5
3.0
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
DC 1  1.03 DC 1
VDC 1  8%
DC 2  1.05 DC 2
VDC 2  10%
DW  1.0 DW
VDW  25%
• Bending moment resistance:



Composite steel beams
Prestr. concrete beams
Concrete T-beams
R  1.12 Rn
VR  10%
R  1.05 Rn
VR  7.5%
R  1.14 Rn
VR  13%
Live Load Simulation
• Maximum of N
•
•
•
Bin
I
Bin II
Repeat for N loading events
events.
75-yr design life
5-yr rating cycle
ADTT = 5000
= 1000
= 100
Simulated vs. Measured
0.07
0.06
Frequency
0.05
Single Lane
Two-Lane Original
Two-Lane Simulated
0.04
0.03
0.02
0.01
Single event
Two-lane
100-ft span
0.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Normalized Moment
Cumulative Distribution
Cumulative Distribution
1.0
0.8
0.6
One-event
5-year Extreme
10-year Extreme
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Moments/HL93
2.5
3.0
Maximum Load Effect
0.12
0.10
WIM Data
Ev Proj. Hist.
Frequency
0.08
0.06
0.04
0.02
Max. 5-yr event
Two-lane
100-ft span
0.00
1.5
2.0
2.5
Normalized Moment
3.0
Reliability-Based Criteria for Bridges

Based on bridge member reliability
b member  3.5

Corresponding system safety, redundancy and
robustness criteria:
bultimate  4.35 b functionality  3.75
b u  0.85
b f  0.25
bdamaged  0.80
bd  2.70
Deterministic Criteria

Ultimate Limit State
Ru  LFu LF1  1.3

Functionality Limit State
R f  LFf LF1  1.2

Damaged Limit State
Rd  LFd LF1  0.5
Design Criteria

Apply system factor during the design process to
reflect level of redundancy
fsfRn   i Pi
i
 Ru R f Rd 

fs  min
,
,
 1.30 1.10 0.50 


fs <1.0 increases the system reliability of designs with low levels
of redundancy.
fs > 1.0 allows members of systems with high redundancy to have
lower capacities.
Example Ps/Concrete Bridge
100-ft simple span, 6 beams at 8-ft
Truck 1
Truck 2
8''
5@8ft
Example Ps/Concrete Bridge
2500
Load (Kips)
2000
1500
Intact Bridge
Damaged Bridge
1000
500
0
0
10
20
30
40
Deflection (in)
50
60
70
Example Ps/Concrete Bridge
Pf(member)
Pf(functionality)
Pf(ultimate)
2.19 10-3
1.12 10-4
4.44 10-9
βmember
βfunctionality
bultimate
2.85
3.69
5.75
∆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
3'
11'
8@13'
Steel Truss Bridge
P(member)
P(functionality)
P(ultimate)
5.13*10-12
1.45*10-14
3.1*10-15
βmember
βfunctionality
βultimate
6.80
7.60
7.80
∆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
Prestressed
Concrete Bridge
Truss Bridge
P(ultimate)
0.0287
7.76*10-3
βdamaged
1.90
2.42
∆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 bdamaged is greater than 0.80 ; system safety is satisfied
• Member reliability index of the truss is βmember=6.8
Deterministic Analysis of
Ps/Concrete Bridge
LF u 4.68
Ru 

 1.44  1.30 o.k .
LF1 3.26
LF f
3.98
Rf 

 1.22  1.10 o.k.
LF1 3.26
Rd 
LF d 1.82

 0.56  0.50 o.k .
LF1 3.26
Twin Steel Box Girder Bridge
Structural Analysis
400
Load (kips)
320
240
undamaged box_grillage
damaged box_grillage
undamaged box_test
damaged box_test
160
80
0
0
5
10
15
Deflection (in)
20
25
Reliability Analysis
Variable
Main member Resistances
Dead load
Maximum rotation
75-year Live load
2-year live load
Bias
1.12
1.05
1.0
1.89
1.75
COV
10%
10%
20%
19%
19%
Distribution type
Lognormal
Normal
Lognormal
Lognormal
Lognormal
b member
b functionality
b ultimate
bdamaged
8.532
8.668
9.773
5.069
Redundancy Analysis



bu = 1.24 > 0.85 O.K.
bf = 0.14 < 0.25 N.G.
bd = -3.46 < -2.70 N.G.
Actual Case 12-ft
5 times bracing
No bracing
LF1
11.51
11.51
11.51
LFf
11.87
12.32
11.82
LFu
15.25
15.25
15.24
LFd
4.86
4.70
4.05
Ru
1.32
1.32
1.32
Rf
1.03
1.07
1.03
Rd
0.42
0.41
0.35
System Safety Analysis




bultimate
= 9.77 > 4.35 O.K.
bfunctionality = 8.67 > 3.75 O.K.
bdamaged = 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)
80000
80000
70000
70000
60000
60000
base shear (kN)
base shear (kN)
Probabilistic results
50000
40000
30000
40000
30000
20000
20000
10000
10000
0
0
-0.1
50000
0.0
0.1
0.2
0.3
0.4
-0.1
-10000
0.0
0.1
0.2
0.3
disp(m)
disp(m)
Intact structure
Damaged structure
bu  1.54  0.85
b f  0.43  0.25
bd  0.42  2.70
0.4
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.