Conditional survival analysis for concrete bridge decks
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Abstract
Bridge decks are a significant factor in the deterioration of bridges, and substantially affect long-term bridge maintenance decisions. In this study, conditional survival (reliability) analysis techniques are applied to bridge decks to evaluate the age at the end of service life using the National Bridge Inventory records. As bridge decks age, the probability of survival and the expected service life would change. The additional knowledge gained from the fact that a bridge deck has already survived a specific number of years alters (increases) the original probability of survival at subsequent years based on the conditional probability theory. The conditional expected service life of a bridge deck can be estimated using the original and conditional survival functions. The effects of average daily traffic and deck surface area are considered in the survival calculations. Using Wisconsin data, relationships are provided to calculate the probability of survival of bridge decks as well as expected service life at various ages. The concept of survival dividend is presented and the age when rapid deterioration begins is defined.
Keywords
Survival analysis Conditional survival Bridge decks Hypertabastic distribution Expected life Reliability Service life Remaining service life1 Introduction
As the average age of bridges in the U.S. continues to increase (ASCE 2018), there is a growing need to develop field performance-based probabilistic tools that could help guide bridge maintenance decisions. Large-scale data obtained from biannual bridge inspections can be particularly informative and beneficial by providing the necessary input for the development of observation-based probabilistic models for bridge deterioration with time.
Bridge decks are a significant factor influencing the deterioration of bridges, and substantially affect long-term bridge maintenance decisions (Tabatabai et al. 2015, 2016 and Brown et al. 2014). Strength-based reliability of bridge decks has been studied for many years providing important measures of structural safety. However, the end of service life is typically a result of serviceability issues and not due to local or global structural failure. Survival analysis is a set of observation-based and data-driven reliability techniques that has been commonly used in biomedical and cancer research (Tabatabai et al. 2007; Bursac et al. 2008). More recently, survival analysis techniques have been employed in reliability and remaining service life assessments of bridges (Tabatabai et al. 2011, 2015, 2016, 2018).
Parametric survival analysis of bridge decks based on the National Bridge Inventory (NBI) records for the State of Wisconsin was introduced by Tabatabai et al. (2011). Using the 2005 NBI data, the authors investigated the best fit survival model among Weibull, log-logistic, lognormal, and hypertabastic survival functions. Based on the Akaike Information Criteria (AIC), the authors concluded that the hypertabastic accelerated failure time model would best fit the data. Later, the same researchers extended the survival analysis of bridge decks to six states in northern United States (Tabatabai et al. 2016) and subsequently to all fifty states and Puerto-Rico (Tabatabai et al. 2015). In these studies, the covariates considered (obtained from NBI records) were the age of bridge, deck surface area (DA), average daily traffic (ADT), and type of superstructure (steel or concrete). It is important to note that, as actual field observation data, the NBI information analyzed necessarily includes the combined effects of all factors influencing the outcome (service life of decks) in these field bridges. The covariates selected can be used to explicitly isolate and quantify (probabilistically) the influence of those selected covariates on the outcome. The effects of all other influencing factors are still embedded in the results, even though they are not explicitly quantified through inclusion as covariates.
Nabizadeh (2015) and Nabizadeh et al. (2018) conducted parametric survival analyses of bridge superstructures in the state of Wisconsin. In these studies, the covariates used were the age of bridge, maximum span length (MSL), ADT, and the type of superstructure (steel or concrete). Other studies on survival analysis of bridges include Beng and Matsumoto (2012), Yang et al. (2013), and Mauch and Madanat (2001).
Based on the results of a study by Tabatabai et al. (2011) and using Eq. 4, the expected life of Wisconsin bridge decks is approximately 45 years, when ADT and deck areas are held at mean values for the state. It is important to note, however, that the expected life would change at different ages (i.e., different from EL_{0}). Furthermore, estimates of life expectancy at a given age would vary depending on whether that age has been successfully achieved (i.e., without failure) or when the estimate for life expectancy at that age is made at t = 0.
In the medical field, prognosis of a disease is generally predicted based on the information at the time of diagnosis. However, it has been shown that disease prognosis improves with every additional year that the patient survives (Baade et al. 2011; Zabor et al. 2013). Therefore, life expectancy based on the information at the time of diagnosis (t = 0) would need to be updated as time progresses. For example, if a patient were given 5 years to live at the time of diagnosis of a disease, the life expectancy at 2, 3, or 4 years past diagnosis would not remain the same. This statement stems from the conditional probability theory and would remain true even if conditions (including treatments) remained unchanged during the elapsed time. Similarly, a bridge deck that may have an initial life expectancy of 45 years would have changing expected life with age. There would be an increasing expected life trend as survival is confirmed at various ages. If a bridge deck has already reached 30 years of life without “failure” (defined as end of service life), then the expected survival age would be higher than the originally estimated 45 years. In this paper, this additional expected life is termed “survival dividend”. The fact that a bridge deck has already survived t_{s} years adds an important piece of information that should be used to update the initial life expectancy based on the conditional probability theory. The objective of this paper is to introduce the concept of conditional survival for the assessment of service life in bridge decks, and to demonstrate its application to assessment of bridge deck service life in Wisconsin based on the 2016 NBI data.
2 Bridge deck data
The 2016 National Bridge Inventory (NBI) data were used to assess conditional survival of bridge decks in Wisconsin. First, using a procedure described in Tabatabai et al. (2011), the overall survival functions were determined. An NBI bridge deck rating of 5 (on a numerical scale of 0–9) was selected as the end of service life. The justification for this choice is provided by Nabizadeh et al. (2018) and is primarily based on the fact that a deck rating of 4 would automatically designate a bridge as “structurally deficient” (or “poor” in the new designation by the Federal Highway Administration) with important policy implications. Therefore, bridge owners take steps (rehabilitation or replacement) before reaching that rating level to avoid a “poor” designation.
The independent covariates used in the analyses were the bridge age, deck area, and average daily traffic (ADT). The average daily truck traffic (ADTT) was found to be correlated with ADT and, therefore, ADT alone was used as a covariate. Parameters such as deck rating (NBI Item 58), bridge age (NBI Items 90 and 27), deck area (NBI Items 49 and 51), and ADT (NBI Item 29) were used as covariates for parametric survival analysis. The complete data extraction and data analysis procedures are presented by Tabatabai et al. (2011).
3 Conditional survival
The conditional survival, which is symbolically represented by CS (t, t_{s}), gives the probability of surviving t years (or t′ additional years), given that the bridge deck has already survived t_{s} years where t′ = t − t_{s}. As stated earlier, the additional “knowledge” gained because of the continuing survival alters the original survival function for the future, and thus the expected life would vary with survived age. The importance of using conditional survival to arrive at a meaningful measure of prognosis or expected remaining life has not been broadly understood (Zabor et al. 2013).
As bridges age, benchmark (initial) estimates of bridge survival (made at the time of start of service life, t = 0) can provide inaccurate prediction of remaining service life. Over time, the additional information on bridge survival would alter the survival estimates relative to the initial survival estimates. A relevant question that may arise is: “If a bridge deck has already survived \(t_{1}\) years, what is the probability that it would survive another \(t_{2}\) years?”
Conditional survival analyses can directly address this question and provide estimates for the probability of survival (of bridge decks) that have already survived to a certain age. This would require knowledge of the basic reliability curves such as that shown in Fig. 1. Probabilistic estimates of conditional survival can provide important additional information that could be used by bridge maintenance engineers to support decision-making and budget allocations for the management of bridge networks. This would be an important tool to assess bridge condition with respect to service life in a probabilistic manner.
Conditional survival has seen increasing interest and has been extensively studied in medical research over the last 20 years (Merrill et al. 1999; Kato et al. 2001; Wang et al. 2007; Fuller et al. 2007; Chang et al. 2009; Janssen-Heijnen et al. 2010; Xing et al. 2010; Merrill and Hunter 2010; Zamboni et al. 2010; Parsons et al. 2011; Baade et al. 2011; Yu et al. 2012; Harshman et al. 2012; Zabor et al. 2013; Hieke et al. 2015). These studies address changes in the probability of survival or expected life for various diseases and conditions.
While some studies have investigated the survival of bridges around the world, the authors of this paper have not identified any work that addresses conditional survival of bridges with respect to the end of service life, which is typically reached due to serviceability conditions. There are works that address changes in the reliability index over time (Zhu et al. 2017; Sun and Hong 2001; Akgul and Frangopol 2003; Kong and Frangopol 2003; Barone and Frangopol 2014). These studies, however, are not typically related to field observation-based end of service life resulting from serviceability conditions.
The above equations indicate that, given the fact that survival has been achieved up to time \(t_{\text{s}}\), the conditional probability of survival would be equal to 1 (100%) at or before time \(t_{\text{s}}\). The originally estimated survival probabilities are then adjusted using Eq. (5). The change in the probability of survival at times greater than \(t_{\text{s}}\) (as reflected in Eq. 5) also changes the expected life beyond time \(t_{\text{s}}\).
The probability of failure defined here is based on conditional survival analysis. Probability of failure indicates reaching the end of service life at a specific age of bridge (1 minus probability of survival at a specific age). CS is then calculated relative to varying survival times.
The hypertabastic accelerated failure time model, first introduced by Tabatabai et al. (2007), was used for the analyses. This model was determined to be the best fit model for bridge deck and superstructure data when compared with Weibull, log-logistic and lognormal models (Tabatabai et al. 2011; Nabizadeh et al. 2018). The hypertabastic distribution has been used in several studies including biomedical and engineering survival analyses (Tabatabai et al. 2007; Tran 2014; Nikulin and Wu 2016; Tahir et al. 2017).
In Eq. (8), parameter \(t_{\text{g}}\) is defined as a mathematical function of t (age of bridge in years), DA (bridge deck surface area in m^{2}), and ADT. Parameters α, β, c, and d are all determined for bridges in Wisconsin, using the procedures proposed by Tabatabai et al. (2011). Functions sech and coth are hyperbolic secant and hyperbolic cotangent, respectively.
Several different survival analyses were performed in this study using different subsets of the extracted NBI data. The full analysis (D_{0}) involved all bridge data that were extracted from the NBI records. In addition, several subsets of the full data were chosen for analysis. These data subsets included all bridge records that had survival times greater than specified ages from 20 to 50 years at 5-year increments (t_{s} = 20, 25, 30, 35, 40, 45, and 50 years). Each dataset analyzed is designated “D_{ts}” with “t_{s}” indicating the length of time (years) that bridge deck has already survived. Thus, the full data set (D_{0}) as well all other subsets (D_{20,}D_{25}, D_{30}, D_{35}, D_{40}, D_{45}, and D_{50}) were analyzed separately to determine the survival functions associated with them. This was done to assess the accuracy of the proposed CS models with available data as the survival age increases. The conditional survival function for bridge decks can be calculated by solving Eq. (5) using survival function of Eq. (6).
4 Analysis of NBI data
Statistical information on the entire Wisconsin NBI dataset used in the analyses
Bridges with deck rating 5 | |||
---|---|---|---|
ADT | Area (m^{2}) | Age (years) | |
Mean | 3843 | 458 | 53 |
Median | 809 | 223 | 50 |
Mode | 47 | 82 | 46 |
Standard deviation | 7314 | 807 | 20 |
Count | 1065 | 1065 | 1065 |
Median and mean ages of Wisconsin bridge decks at the end of service life
Dataset | Survived age (t_{s})(years) | Median age of decks (years) | Mean age of decks (years) | St. dev. of age (years) | Number of bridges |
---|---|---|---|---|---|
D_{0} | 0 | 50.0 | 53.2 | 20.2 | 1065 |
D_{20} | 20 | 50.0 | 54.2 | 19.5 | 1040 |
D_{25} | 25 | 51.0 | 55.2 | 18.9 | 1008 |
D_{30} | 30 | 52.0 | 56.7 | 18.2 | 954 |
D_{35} | 35 | 53.0 | 58.5 | 17.5 | 889 |
D_{40} | 40 | 55.0 | 61.1 | 16.8 | 793 |
D_{45} | 45 | 57.0 | 63.6 | 16.3 | 702 |
D_{50} | 50 | 64.0 | 68.3 | 15.4 | 549 |
Parameter and standard error estimates of the hypertabastic AFT model
Parameter | Estimate | Standard error | Wald | P value |
---|---|---|---|---|
α | 1.29E−03 | 2.11E−04 | 37.42 | 9.51E−10 |
β | 1.90E+00 | 4.48E−02 | 1796.7 | 1.34E−392 |
c | 5.70E−05 | 1.00E−05 | 32.08 | 1.48E−08 |
d | 6.93E−06 | 1.43E−06 | 23.39 | 1.32E−06 |
5 Results and discussion
Estimated baseline survival curves (static prediction) and conditional survival curves (dynamic prediction) were determined for t_{s} values ranging from 20 to 50 years at 5-year increments as shown in Fig. 6. Again, it should be emphasized that the values reported are for the values of covariates assumed above. Other covariate values may be used as applicable.
Expected service life (EL_{0}) for various ADT and DA
Expected service life (years) | |||||
---|---|---|---|---|---|
Deck area (m^{2}) | ADT (vehicles) | ||||
500 | 1000 | 5000 | 10,000 | 20,000 | |
200 | 54.8 | 54.6 | 53.2 | 51.4 | 48.0 |
500 | 53.9 | 53.7 | 52.3 | 50.5 | 47.2 |
1000 | 52.4 | 52.3 | 50.9 | 49.2 | 45.9 |
1500 | 51.0 | 50.8 | 49.5 | 47.8 | 44.6 |
Rapid deterioration age (t_{rd}) for various ADT and DA
t_{rd} (years) | |||||
---|---|---|---|---|---|
Deck area (m^{2}) | ADT (vehicles) | ||||
500 | 1000 | 5000 | 10,000 | 20,000 | |
200 | 35.2 | 35.1 | 34.5 | 33.8 | 31.8 |
500 | 34.8 | 34.8 | 34.2 | 33.3 | 31.2 |
1000 | 33.6 | 33.5 | 32.7 | 31.7 | 29.0 |
1500 | 34.2 | 34.2 | 33.5 | 32.5 | 30.2 |
From the data presented, it is evident that both ADT and DA influence the unconditional and conditional expected life as well as the probability of survival at various ages. An evaluation of the results shows that changes in ADT have larger influence on the expected life than the DA. Changing ADT from 500 to 20,000 would reduce the expected life EL_{0} by roughly 6–7 years. On the other hand, changing the DA from 200 to 1500 m^{2} reduces EL_{0} by less than 2 years.
A detailed example illustrating the calculation processes for conditional survival and expected service life estimates is provided in Appendix.
6 Summary and conclusions
Probabilistic assessment of service life of bridge decks is an important consideration in support of effective long-term maintenance of bridges. In this study, conditional survival analysis techniques are applied to bridge decks to evaluate the end of service life using the NBI bridge records. A recorded NBI deck condition rating of five was considered the end of service life of bridge decks. The survival analysis procedures developed by Tabatabai et al. (2011) were extended to include conditional survival considerations. The covariates considered were the bridge age, ADT, and DA.
As bridge decks age without failure (reaching the end of service life), the survival probabilities change. The additional knowledge gained from the fact that a bridge deck has survived several years alters (increases) the original (bridge construction) probability of survival at subsequent years based on the conditional probability theory. The fact that survival has been achieved at any given time t_{s} changes the probability of survival from S(t_{s}) to 1.0. The probability of survival at subsequent times would also change by dividing the original survival function by S(t_{s}). A conditional survival function can thus be calculated for different t and t_{s} values.
The expected service life of a bridge deck, without or with achieved survival years, can be estimated using the original and conditional survival functions, respectively. The area under the original survival function is equal to the expected life (EL_{0}) based on the state of knowledge at t = 0. Relationships are provided to calculate the expected service life at various ages of the bridge deck. These estimates can be unconditional, based on knowledge at the time of construction of bridge deck, or conditional when t_{s} years of service have been achieved without reaching the end of service life.
The average daily traffic and deck surface area affect the survival probabilities and the expected life. As an example, changing ADT from 500 to 2000 would reduce the expected life (EL0) by 6–7 years based on Wisconsin data (when DA = 500 m^{2}). A parameter representing the age when the unconditional expected life begins to suffer rapid reduction (t_{rd}) has been defined and calculated for various ADT and DA values. This parameter can be another tool in planning of maintenance, repair, and replacement operations for bridge decks. In addition, an example is provided to illustrate calculations of overall survival, conditional survival, initial expected life, and conditional expected life.
It should be noted that although the procedures described in this paper apply to bridge decks in all geographic locations, the parameters (α, β, c, and d) that are used in the calculations are determined based on the 2016 Wisconsin NBI data. For all other states, these parameters have been determined by Tabatabai et al. (2015) and can be used in a similar fashion.
The information provided in this paper can be combined with cost information to determine the optimum maintenance interventions during the service life of bridge decks. Changes in the probability of survival during the life of a bridge deck should be considered to be a dynamic and changing value that must be assessed based on the conditional probability theory.
Notes
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