Bridge Deck Condition Rating Forecast—SurvivalBased Models
Raka Goyal, Ph.D., P.E.
Introduction
The survivalbased models on Federal Highway Administration (FHWA) InfoBridge^{TM} (FHWA, 2020) are probabilistic bridge deterioration models based on a methodology that combines survival analysis and Markov chain theory. The modeling framework was first developed and implemented on bridge deck, superstructure, and substructure components of the North Carolina State bridge inventory (Goyal, 2015; Cavalline et al., 2015). In the current research, the model’s implementation has been substantially expanded and improved for the more complex application to the entire nationwide bridge inventory.
Survival analysis is a statistical approach that analyzes the time until death or failure—in the case of bridge components, which is the time spent in a condition rating until it deteriorates to a lower condition rating. Survival analysis allows determination of the survival probability of the component in that condition rating at any point of time. The main advantage of survival analysis is that it can account for incompletely recorded durations commonly found in durationbased data, such as conditionrating observations truncated at the beginning and end of the recording period. Instead of discarding these observations, a mechanism called “censoring” is used to include these observations in the analysis, which permits a more realistic estimate of duration compared to other statistical approaches.
In this study, the Cox proportional hazards model (PHM) (Cox, 1972) has been used for the analysis of conditionrating durations. Cox PHM is a semiparametric approach that does not make any assumptions about the shape of the distribution and can be used to analyze unimodal hazard functions associated with some infrastructure components. For the survivalbased models developed in this study, the transition probabilities of the Markov chain are calculated from the PHM survival functions at each condition rating, carrying the advantage of survival analysis probabilistically over the entire lifecycle. Additionally, multivariable effects at each condition rating are quantified in terms of PHM hazard ratios and used to modify the Markov chain transition probabilities. In this way, the effects of time dependence and exogenous factors on deterioration, as analyzed through survival analysis, are incorporated in the Markov chain to develop a probabilistic lifecycle deterioration model.
ProportionalHazards Bridge Deterioration Model
The survival function (S(t)) associated with a bridgecomponent condition rating is the cumulative survival rate of bridge components in the condition rating. The instantaneous risk of transitioning to a lower rating at time (t), conditional to survival until that time, is the hazard rate or the hazard function. In the Cox PHM, the hazard rate (h(t,z)) is defined as the product of a timedependent nonparametricbaseline hazard function (h_{0}(t)) and a timeindependent exponential function, representing the effects of design variables (z) through regression coefficients (β):
(1)
In this function, the timeindependent exponential function represents the effects of covariates, or explanatory factors, on the hazard rate. The baseline hazard rate is associated with the baseline variables, which are assigned a value of zero.
Proportionalhazards lifecycle deterioration modeling
The proportionalhazards bridge deterioration model (Goyal, 2015) can be represented in terms of structurespecific transition probability matrices (P_{i}) associated with each year (i) of the prediction period,
(2)
The matrix elements P_{ij} represent the probability of the bridge component transitioning from condition rating i to condition rating j, assuming there is no improvement in condition rating and that the component deteriorates by no more than a single rating in one year. In each row, P_{kk, }on the diagonal is the baseline staythesame transition probability for condition rating k. If S_{k} (t,z) is the baseline survival function of a bridge deck associated with condition rating k for a bridge described by the vector of covariates z, the baseline transition probability (Goyal, 2015; Goyal et al., 2020) of staying at the same condition rating over the next annual reporting cycle at any time t is given by
(3)
Since the sum of probabilities in each row should be equal to one, the baseline transition probability of deteriorating to a lower rating is 1 ‒ P_{kk}. The baseline transition probabilities are uniquely modified for each bridge or category of bridges using structurespecific hazard ratios HR_{k}, calculated by multiplying the PHM hazard ratios for the factors associated with a bridge that are identified as significantly influencing deterioration at condition rating k. A stationary implementation of the proportionalhazards deterioration model is used in this study, in which stationary or constant transition probabilities (P_{kk}) are obtained by averaging the yearly transition probabilities across the duration of the PHMbaseline survival functions for each condition rating k. The resulting stationarytransition probability matrix (P) is used in a homogeneous Markov chain to predict the future conditionstate vector Z_{n} of a bridge component after n years if its present conditionstate vector Z_{0} is known using
(4)
The conditionstate vectors comprise of the probabilities of the bridge component being at all possible condition ratings at any given time. The future state vector Z_{n} can be multiplied by the column vector (R) of condition ratings to produce the expected condition rating, E, of the bridge component after n years.
(5)
This document describes briefly the implementation of the proportionalhazards deterioration model to develop deck deterioration models for the nationwide bridge inventory with more details to follow in a journal publication currently under preparation. Detailed background and theoretical development of the proportionalhazards deterioration model can be found elsewhere (Goyal, 2015; Goyal et al., 2017; Goyal et al., 2020).
Data Structuring
The nationwide survivalbased models are based on the FHWA National Bridge Inventory (NBI) data spanning the years from 1983 to 2018 for highway bridges nationwide (FHWA, 2019). The NBI files store historical inspection records of all bridges in the United States with a span greater than 20 ft. with more than 100 data items per bridge per year of recorded service (FHWA, 1995). The first step in organizing the data for modeling was to identify, query, and extract the NBI fields relevant to deterioration modeling from the yearly inspection records of individual bridges, and assemble a continuous record of bridge conditionrelated data from 1983 to 2018.
Bridges with concrete decks constitute an overwhelming percentage (>80 percent) of the NBI and are a focus area of the LongTerm Bridge Performance Program. For survivalbased deterioration modeling, bridges were classified based on deck material, mainstructure material, and design type. Further, since durationbased analysis is best served by long duration records, and in the interest of optimizing the use of computational resources, researchers decided that only bridges with continuous records of 25 years or more be used for model development. The bridge categories for survivalbased models with the number of bridges in the historical database selected for model development are provided in Table 1.
Table 1. Bridge categories for survivalbased deterioration models. © Raka Goyal (2019).
Deck Type (NBI Item 107) 
Main Structure Material (NBI Item 43A) 
Main Structure Design (NBI Item 43B) 
No. Bridges Included in Model Development (1983‒2018)* 
Concrete (CastinPlace and Precast Panels) 
Steel and Steel Continuous 
Stringer/Multibeam or Girder 
33409 
Prestressed Concrete and Prestressed Concrete Continuous 
Stringer/Multibeam or Girder 
23296 

Box Beam or Girders – Multiple and Single or Spread 
15731 

Concrete and Concrete Continuous 
Slab 
24884 

Stringer/Multibeam or Girder, Tee Beam, and Channel Beam 
24951 
* Rebuilt and reconstructed bridges separated based on NBI items “Year Built” and “Year Reconstructed.”
Design variables for proportionalhazards survival analysis
Data was further preprocessed to extract all observations of the response variable, which is the observed continuous duration at each NBI condition rating analyzed. For each observed duration, corresponding censoring information was compiled in a separate vector of the same size as the response variable, but stored as a binary variable of either 0 or 1 depending on whether the observations were classified as completely observed or censored. In this study, all continuous observations that were truncated at the beginning year (1983) or end year (2018) of the NBI database were classified as right censored, presuming that the actual duration of the condition rating was longer than observed due to the limited time span of the datarecording period. Further, all observations where an increase in condition rating was observed, rather than deterioration, were considered as complete, or uncensored, under the assumption that observed improvements in condition rating reflected maintenance action performed because of an unrecorded transition to the lower rating.
For proportionalhazards analysis, the descriptive information on each structure, such as its functional classification, traffic characteristics, design parameters, and other details contained within the NBI historical records that could potentially influence the deterioration rates of specific bridge components were considered as potential explanatory factors. Some factors were derived from two or more NBI fields, for example, “age” refers to the age of the deck at the beginning of the observed conditionrating duration and is calculated based on the corresponding data year and the year when the bridge was last built or reconstructed. Additionally, average daily traffic per lane (ADTL) and average daily truck traffic per lane (ADTTL) were calculated by dividing the ADT and ADTT, respectively, by the number of throughtraffic lanes on a bridge also recorded in the NBI (ADTT mentioned here refers to the number of trucks, which was first calculated from the percentage ADTT recorded in the NBI). ADTL and ADTTL are new variables that were introduced in the nationwide study to account for the width of the roadway in studying the impact of traffic on deck deterioration.
Each explanatory factor is organized into categories designated by one or more design variables to which bridges are classified based on either binary or referencecell coding. The variables of ADTL, ADTTL, age, and maximum span length, which are continuously recorded, are divided into categories of approximately equal frequency of occurrence for each component based on weighted averages computed across the available bridge records. The categorical ranges for these variables differ for disparate bridge categories (Table 1) depending on the different statistical distributions of variables associated with the various bridge categories. The design variables included in the development of survivalbased models for concrete decks nationwide are listed in Table 2. Category ranges developed for steel stringer bridges are provided for illustration. The subsequent steps involving multivariable proportionalhazards regression were performed individually on each of the distinct conditionrating specific sets of dependent and independent PHMregression inputs, extracted in this way, for each category of bridges.
Table 2. Design variables included in proportionalhazards survival analysis of concrete decks nationwide.
© Raka Goyal (2019).
Factor 
Baseline Category* 
Design Variable* 
Deck Type 
CastinPlace 
Precast Panels 
Span Type 
Simple 
Continuous 
Functional Class 
NonInterstate 
Interstate 
Average Daily Traffic /Lane (ADTL) 
ADTL (≤ 112) 
ADTL2 (112‒978) 
ADTL3 (978‒3941) 

ADTL4 (>3941) 

Age (years) 
Age (≤ 19) 
Age2 (19‒30) 
Age3 (30‒43) 

Age4 (> 43) 

Skew 
No Skew 
Skew 
Reconstruction 
Original/Rebuilt 
Reconstructed 
Average Daily Truck Traffic /Lane (ADTTL) 
ADTTL (≤ 3) 
ADTTL2 (3‒60) 
ADTTL3 (60‒329) 

ADTTL4 (> 329) 

Wearing Surface 
No Wearing Surface 
Monolithic Concrete 
Integral Concrete 

Latex Concrete 

Low Slump Concrete 

Epoxy Overlay 

Bituminous 

Timber 

Gravel 

Other 

Deck Membrane 
No Membrane 
Deck Membrane 
Deck Protection 
No Protection 
Deck Protection 
Maximum Span (m) 
Max Span (≤13) 
MaxSpan2 (13–20) 
MaxSpan3 (20–28) 

MaxSpan4 (>28) 

Number of Spans 
Single Span 
Multiple Spans 
*Category values in parentheses are for concretedeck steel stringer bridges.
ProportionalHazards Deterioration Model Development
The variables that were found to be statistically significant at each condition rating using PHM regression were further processed through a best subsetselection algorithm to optimize the size of the model without compromising its reliability. The survival functions developed using the best subset model incorporate the effect of the most significant explanatory variables on the deterioration rate over individual condition ratings. The conditionratingdependent best subsets and associated hazard ratios obtained for the significant explanatory factors identified in the proportionalhazards deck model for steel stringer bridges nationwide are summarized in Table 3.
Table 3. Best subset factors and hazard ratios in proportionalhazards deck deterioration model for concrete deck steel stringer bridges nationwide. © Raka Goyal (2019).
Best Subset Factor 
Hazard Ratios at Condition Rating 

9 
8 
7 
6 
5 
4 
3 

Precast Panels 
1 
1 
1 
0.7953 
1 
1 
1 
Interstate 
1 
1.1639 
1 
1.0827 
1.1456 
1.1310 
1 
ADTL2 
1.2683 
1.4195 
1.2799 
1.0798 
1.2953 
1.2559 
1 
ADTL3 
1.6802 
1.6660 
1.3127 
1 
1.4448 
1.3810 
1 
ADTL4 
1.9113 
1.7603 
1.2776 
0.9150 
1.3848 
1.3806 
1 
Age2 
1.4447 
1.7797 
1.2833 
1 
1 
1 
1 
Age3 
1.3013 
2.2195 
1.5088 
0.9236 
0.9486 
0.9254 
1.3984 
Age4 
1 
2.0239 
1.4753 
0.8674 
0.7488 
0.6912 
1 
Skew 
1.0922 
1.0789 
1 
1 
1 
1 
1 
Reconstructed 
1.4090 
1.2995 
1 
0.9464 
0.9349 
0.8630 
1 
ADTTL2 
1 
0.9491 
0.9125 
1.1597 
1 
1 
1 
ADTTL3 
1 
1 
1 
1.2684 
1 
1 
1.1221 
ADTTL4 
1 
1 
1 
1.3005 
1.0237 
1 
1 
Monolithic Concrete 
0.8621 
0.7182 
1.1010 
1 
1 
1 
1 
Integral Concrete 
1 
1 
1.6265 
1.4731 
1.2577 
1 
1 
Latex Concrete 
1 
1 
1.6754 
1.3233 
1.3418 
1.2933 
1 
Low Slump Concrete 
1 
1 
1.4070 
1.1531 
1.1586 
1.2530 
1 
Epoxy Overlay 
1 
1 
1.5182 
1.3659 
1 
1 
1 
Bituminous 
1 
0.8046 
1 
1 
1 
1 
1 
Gravel 
1 
0.6353 
1.1543 
1 
1 
1 
1 
Other 
1 
1 
1.6209 
1.4052 
1.5825 
1 
1 
Deck Membrane 
1.6699 
1.2428 
1.1864 
1 
1 
1 
1 
Deck Protection 
1.1245 
1.1221 
1 
1 
1 
1 
1 
MaxSpan2 
1 
1.1192 
1 
1.1806 
1 
1 
1 
MaxSpan3 
1 
1.2391 
1.1036 
1.2122 
1 
1 
1 
MaxSpan4 
1 
1.2250 
1.0871 
1.2558 
1 
1 
1 
Number of Spans 
1 
1.0520 
0.9401 
0.9302 
1 
1.1776 
1.1859 
An HR value of 1 signifies a lack of influence on the deterioration rate, and indicates that the factor was not included in the best subset for that rating. For example, in Table 3, deck type is included only in the best subset associated with condition rating 6. Absence of span type in Table 3 indicates that it was not included in the best subsets associated with any of the deck condition ratings throughout the lifecycle of concretedeck steel stringer bridges. An HR value less than 1 indicates that the factor is associated with a reduced rate of deterioration, and a value greater than 1 indicates that the factor is associated with an increased rate of deterioration. As seen in the table, revealing the varying effects of the same factors at different condition ratings across the bridgedeck lifecycle is a unique aspect of the proportional hazards deterioration model.
The final step in the proportionalhazards deteriorationmodel development is the calculation of baselinetransition probabilities at each condition rating, and assembly of the Markovchain transitionprobability matrix, as shown in equation 2, for future condition forecasting. Sufficient historical conditionrating data was available in the nationwide NBI database to develop survivalfunctionbased transition probabilities for all the condition ratings from 9 to 3. The staythesame transition probability for condition rating 2 was prescribed as 0.85 to prevent the deterioration models from converging abruptly to condition rating 2. Condition rating 1 is the lowest or absorbing state with a staythesame transition probability of 1. Figure 1 shows deck deterioration models illustrating the combined effects of the variables, deck type, age, ADTL, and maximum span length on deckdeterioration rates of concretedeck steel stringer bridges, illustrating the depth of information revealed by the proportionalhazards deterioration models based only on the variables available within the NBI database.
© Raka Goyal (2019). 
Figure 1. Proportionalhazards deck deterioration models for concretedeck steel stringer bridges showing the effect of deck type, age, ADTL, and maximum span over the bridge lifecycle 
Proportionalhazards deteriorationmodel plots for individual bridges in each bridge category (Table 1) are available on InfoBridge (FHWA, 2020). In addition to the meanpredicted condition rating, prediction curves associated with cumulative probabilities of 75 percent (lower bound), 50 percent (median), and 25 percent (upper bound) are also displayed as shown in Figure 2.
Source: FHWA. 
Figure 2. Survivalbased deckcondition forecasting curves on LTBP InfoBridge (FHWA, 2020). 
References
Cavalline, T. L., Whelan, M. J., Tempest, B. Q., Goyal, R., and Ramsey, J. D. (2015). Determination of Bridge Deterioration Models and Bridge User Costs for the NCDOT Bridge Management System. Final Report No. FHWA/NC/201407, North Carolina Department of Transportation, Raleigh, NC.
Cox, D. R. (1972). Regression Models and LifeTables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), pp. 187–220.
FHWA. (1995). Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s Bridges. Report FHWAPD96001, Office of Engineering, Bridge Division, FHWA, Washington, DC.
FHWA. (2020). LTBP Infobridge Data, InfoBridge, obtained from: https://infobridge.fhwa.dot.gov/data, last accessed January 2020.
FHWA. (2019). Historical bridge inspection records with span greater than 20 ft., 100 data items/bridge/year, National Bridge Inventory, 1983 to 2018 datasets, obtained from: https://www.fhwa.dot.gov/bridge/nbi.cfm, last accessed March 18, 2020.
Goyal, R. (2015). Development of SurvivalBased Framework for Bridge Deterioration Modeling with LargeScale Application to the North Carolina Bridge Management System. PhD. Dissertation, University of North Carolina at Charlotte. Charlotte, North Carolina.
Goyal, R., Whelan, M., and Cavalline, T. L. (2017). Characterizing the Effect of External Factors on Deterioration Rates of Bridge Components Using Multivariate Proportional Hazards Regression. Structure and Infrastructure Engineering, 13(7): 894–905.
Goyal, R., Whelan, M., and Cavalline, T. L. (2020). Multivariable Proportional HazardsBased Probabilistic Model for Bridge Deterioration Forecasting. Journal of Infrastructure Systems, 26(2): 04020007, DOI: 10.1061/(ASCE)IS.1943555X.0000534.