# Seminar - UPC

Urban and Civil

Nov 25, 2013 (4 years and 7 months ago)

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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.

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
members.

Robustness

is the ability of a structural system
to survive the loss of a member and continue to

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

intensity

and

distribution

justify

the

use

of

probabilistic

methods
.

Structural Reliability

-

Resistance

S

-

R

function Z

state
limit

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

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(L|E) = probability of local failure, L, given E

P(C|LE) 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(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

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

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(1-P
0
) samples
b
i

are chosen

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 M-H 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(1-P
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

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

Bin
I

Bin II

Maximum of N
events.

75
-
yr design life

5
-
yr rating cycle

= 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

Two-Lane Original

Two-Lane 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

One-event

5-year Extreme

10-year Extreme

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
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
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

1.05

10%

Normal

Maximum rotation

1.0

20%

Lognormal

75
-

1.89

19%

Lognormal

2
-

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.