Weibull-Based Bridge
Deterioration Models for Iowa
Bridges
Dimitrios Bilionis
Basak Aldemir Bektas
outline
 Introduction
 Data
 Methodology
 Refinement
 Results
 Example
 Implementation
introduction
 Purpose
 Predict future condition
introduction
Deterioration models
Deterministic models
Stochastic models
state-based
e.g. Markov chains
time-based
e.g. Weibull
methodology
 Survival analysis (failure time analysis)
 Occurrence and timing of events
 Hazard base models investigate the conditional probability that duration of time
ends at a specific time t:
𝐹 𝑡 = 𝑃(𝑇 < 𝑡)
Here F(t) is the c.d.f. of T
 The conditional probability that an event will occur between time t and t+dt, is given
by the hazard function:
𝑓(𝑡)
ℎ 𝑡 =
1 − 𝐹(𝑡)
In other words, the hazard function gives the rate at which a duration terminates at
time t
methodology
• the probability that a duration is greater than or equal to a specific
time t is given by the survivor function:
𝑆 𝑡 = 𝑃(𝑇 ≥ 𝑡) = 1 − 𝐹 𝑡
• Weibull
Survival function: 𝑆 𝑤 = exp − 𝑒𝑥𝑝
Probability density function: 𝑓 𝑤 =
𝑤−𝜇
𝜎
1
𝑤−𝜇
exp
𝜎
𝜎
exp −𝑒𝑥𝑝
𝑤−𝜇
𝜎
where w=log(t), 𝜇 = 𝑥 ′ 𝛽 , µ is the location parameter and σ is the scale
parameter
methodology
 Censoring
 T=a, uncensored
 T<b, right censored
 c<T<d, interval censored
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
8
7
7
7
7
7
7
7
7
7
6
6
data
• NBI ratings
• Deck
• Superstructure
• Substructure
• 1983-2011 data
• Process:
•
•
•
•
Eliminate increases
Gaps
Explanatory variables
Time-in-state
Code
Description
N
NOT APPLICABLE
9
EXCELLENT CONDITION
8
VERY GOOD CONDITION ( No problems noted)
7
GOOD CONDITION (Some minor problems)
6
SATISFACTORY CONDITION (Minor deterioration in structural
elements)
5
FAIR CONDITION (Sound structural elements with minor section loss)
4
POOR CONDITION (Advanced section loss)
3
SERIOUS CONDITION (Affected structural elements from section loss)
2
CRITICAL CONDITION (Advanced deterioration of structural elements)
1
0
“IMMINENT” FAILURE CONDITION (Obvious movement affecting
structural stability)
FAILED CONDITION (Out of service)
results
Deck
Rating
9
8
7
6
5
4
# observations
Uncensored
Right
Censored
751*
280*
145*
166*
63*
21*
138*
314
425
239
202
145
Variables
Median TIS
Uncensored Right
sample
censored
sample
ADT
4
3
AGE, TR_ADT 8
6
AGE
17
5
AGE
10
4
AGE
5
5.5
4
6
Model
4.6
7.3
13
9.1
5.6
4.8
results
Substructure
Rating
9
8
7
6
5
4
# observations
Uncensored
Right
Censored
Variables
882*
181*
83*
172*
67*
16
AGE, ADT
AGE
149
603
399
265
238
102
Median TIS
Uncensored
sample
5
8
16
8
4
2.5
Right
censored
sample
9
6
10
8
5
4
Model
6.2
7.6
13.7
7.6
5.2
2.8
results
Superstructure
Rating # observations
Uncensored Right
Censored
9
836*
236
8
133*
608
7
6
5
89
124
33
259
199
157
4
10
50
Variables
AGE, ADT
AGE, TR_ADT,
CNRCSL
DLSLCNR
DMONCNR,
SSMG
Median TIS
Uncensored
sample
5
9
Right censored
sample
7
5
Model
15
8
4
6
6
4
12.7
8.2
4.2
3.5
4.5
3.8
5.7
9
example
Deck NBI CR=8
AGE
TR_ADT
2
2
2
20
20
20
2
2
2
20
20
20
Prob Survival Median Time
636
0.5
11.08
100
0.5
12.15
9000
0.5
2.62
636
0.5
0.98
100
0.5
1.08
9000
0.5
0.23
636
0.5
11.08
100
0.5
12.15
9000
0.5
2.62
636
0.5
0.98
100
0.5
1.08
9000
0.5
0.23
example
refinement
Implementation
Yearly time-in-state update
Emphasis on models for ratings 4-7
Network level prioritization based on median time-instate estimates
Thank you!
Questions?