Transcript Weibull-Based Bridge Deterioration Models for Iowa Bridges
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 Probability density function: 𝑓 𝑤 = 𝑤−𝜇 Survival function: 𝑆 𝑤 = exp − 𝑒𝑥𝑝 𝜎 1 𝜎 exp where w=log(t), 𝜇 = 𝑥 ′ 𝛽 parameter 𝑤−𝜇 𝜎 exp −𝑒𝑥𝑝 𝑤−𝜇 𝜎 , µ is the location parameter and σ is the scale
methodology
Censoring T=a, uncensored T
data
• NBI ratings • Deck • Superstructure • Substructure • 1983-2011 data • Process: • Eliminate increases • Gaps • Explanatory variables • Time-in-state 6 5 4 3 2
Code
N 9 8 7 1 0
Description
NOT APPLICABLE EXCELLENT CONDITION VERY GOOD CONDITION ( No problems noted) GOOD CONDITION (Some minor problems) SATISFACTORY CONDITION (Minor deterioration in structural elements) FAIR CONDITION (Sound structural elements with minor section loss) POOR CONDITION (Advanced section loss) SERIOUS CONDITION (Affected structural elements from section loss) CRITICAL CONDITION (Advanced deterioration of structural elements) “IMMINENT” FAILURE CONDITION (Obvious movement affecting structural stability) FAILED CONDITION (Out of service)
9 8 7 6 5 4
results Deck
Rating # observations
Uncensored Right Censored 751* 280* 145* 166* 63* 21* 138* 314 425 239 202 145
Variables
ADT 4 AGE, TR_ADT 8 AGE 17 AGE AGE 10 5 4
Median TIS
Uncensored sample Right censored sample 3 6 5 4 5.5
6 Model 4.6
7.3
13 9.1
5.6
4.8
results
9 8 7 6 5 4
Substructure
Rating # observations
Uncensored Right Censored 882* 181* 83* 172* 67* 16 149 603 399 265 238 102
Variables
AGE, ADT AGE 5 8 16 8 4 2.5
Median TIS
Uncensored sample 8 5 4 Right censored sample 9 6 10 Model 6.2
7.6
13.7
7.6
5.2
2.8
results
7 6 5 4
Superstructure
Rating # observations
Uncensored Right
9 8
836* 133* Censored 236 608 89 124 33 10 259 199 157 50
Variables
AGE, ADT AGE, TR_ADT, CNRCSL
Median TIS
Uncensored sample 5 9 DLSLCNR DMONCNR, SSMG 15 8 4 3.5
6 6 4 Right censored sample 7 5 4.5
Model 5.7
9 12.7
8.2
4.2
3.8
example
Deck NBI CR=8 AGE
2 2 20 20 20 20 20 20 2 2 2 2
TR_ADT
636
Prob Survival Median Time
0.5
11.08
100 9000 0.5
0.5
12.15
2.62
636 100 9000 636 0.5
0.5
0.5
0.5
0.98
1.08
0.23
11.08
100 9000 636 100 9000 0.5
0.5
0.5
0.5
0.5
12.15
2.62
0.98
1.08
0.23
example
refinement
Implementation
Yearly time-in-state update Emphasis on models for ratings 4-7 Network level prioritization based on median time-in state estimates