towards multiple model approach to bridge deterioration
TRANSCRIPT
University of Texas at El PasoDigitalCommons@UTEP
Open Access Theses & Dissertations
2019-01-01
Towards Multiple Model Approach to BridgeDeteriorationJin A. CollinsUniversity of Texas at El Paso
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TOWARDS MULTIPLE MODEL APPROACH
TO BRIDGE DETERIORATION
JIN A COLLINS
Masterâs Program in Civil Engineering
APPROVED:
Jeffrey Weidner, Ph.D., Chair
Carlos M. Ferregut, Ph.D.
Jose Espiritu Nolasco, Ph.D.
Stephen Crites, Ph.D. Dean of the Graduate School
Copyright ©
by
Jin A Collins
2019
TOWARDS MULTIPLE MODEL APPROACH
TO BRIDGE DETERIORATION
by
JIN A COLLINS, BS
THESIS
Presented to the Faculty of the Graduate School of
The University of Texas at El Paso
in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE
Department of Civil Engineering
THE UNIVERSITY OF TEXAS AT EL PASO
August 2019
iv
Table of Contents
Table of Contents ....................................................................................................................... iv
List of Tables ............................................................................................................................. vi
List of Figures ...........................................................................................................................vii
Chapter 1 : Introduction .............................................................................................................. 1
Background ........................................................................................................................ 1
Objectives .......................................................................................................................... 1
Organization of Thesis........................................................................................................ 2
Chapter 2 : Literature Review of Bridge Deterioration Modeling................................................. 3
Condition Rating ................................................................................................................ 3
Explanatory Variables ........................................................................................................ 6
Deterministic models .......................................................................................................... 7
Stochastic models ............................................................................................................... 9
Markovian process ..................................................................................................... 9
Weibull distribution ................................................................................................. 11
Artificial Intelligence models ........................................................................................... 14
Mechanistic-based models ................................................................................................ 16
Reliability-based models .................................................................................................. 17
Chapter 3 : Methodology ........................................................................................................... 20
Markov chain models ....................................................................................................... 20
Transition Probability Matrix (TPM) ................................................................................ 21
Regression Nonlinear Optimization (RNO) ...................................................................... 23
Bayesian Maximum Likelihood (BML) ............................................................................ 24
Ordered Probit Model (OPM) ........................................................................................... 25
Poisson Regression (PR)................................................................................................... 28
Negative Binomial Regression (NBR) .............................................................................. 30
Proportional Hazard Model (PHM) ................................................................................... 31
Hazard Ratio ............................................................................................................ 32
Kaplan-Meier Estimator ........................................................................................... 32
Summary .......................................................................................................................... 33
v
Chapter 4 : Case Study .............................................................................................................. 36
Data Review and Filtration ............................................................................................... 36
Results ............................................................................................................................. 39
Transition Probability Matrix (TPM) ................................................................................ 40
Deterioration Rate Curves ................................................................................................ 46
Comparison ...................................................................................................................... 48
Chi-squared goodness-of fit Test .............................................................................. 48
Modal Assurance Criterion (MAC) .......................................................................... 50
Chapter 5 : Conclusions and Future Work ................................................................................. 54
Summary .......................................................................................................................... 54
Conclusions ...................................................................................................................... 54
Future Work ..................................................................................................................... 56
References ................................................................................................................................ 57
Vita 60
vi
List of Tables
Table 2.1: Condition rating codes and descriptions for bridge......................................................5 Table 2.2: Condition status and description of bridge .................................................................5 Table 2.3: Condition state codes and description of bridge elements revised in 2011 ...................5 Table 2.4: Explanatory variables applied on bridge deterioration modeling .................................7 Table 3.1: Number of bridge decks of transition ........................................................................ 23 Table 3.2: Summary of objective functions and random variables of models. ............................ 35 Table 4.1: Mean transition probability matrices for bridge components with 2008-2010 data .... 42 Table 4.2: Transition probability matrices of bridge decks with 2008-2010 data ........................ 43 Table 4.3: Transition probability matrices of bridge superstructures with 2008-2010 data ......... 44 Table 4.4: Transition probability matrices of bridge substructures with 2008-2010 data ............ 45 Table 4.5: Results of Chi-square goodness-of-fit of bridge components ..................................... 49 Table 4.6: Results of Modal Assurance Criterion for bridge deck .............................................. 51 Table 4.7: Results of Modal Assurance Criterion for bridge superstructure ................................ 51 Table 4.8: Results of Modal Assurance Criterion for bridge substructure ................................... 52 Table 4.9: Transition probability matrices of decks using proportional hazard model ................ 52
vii
List of Figures
Figure 2.1: Weibull probability density function with different shape parameters ...................... 14 Figure 2.2: Schematic diagram of ENN process (Winn and Burgueño 2013) ............................. 15 Figure 2.3: Sketch of the BPM process (Huang 2010) ............................................................... 16 Figure 3.1: Schematic illustration of the ordered probit model (H. D. Tran 2007) .................... 28 Figure 4.1: Total number of bridges in Texas from 2000 to 2010............................................... 37 Figure 4.2: Number of bridges according to structure material in 2010 ...................................... 38 Figure 4.3: Number of bridge decks in 2004 by condition rating groups .................................... 38 Figure 4.4: Number of bridges by component rating after filtering ............................................ 39 Figure 4.5: Multiple model approach workflow chart ................................................................ 40 Figure 4.6: Deterioration rate curves for bridge deck ................................................................. 47 Figure 4.7: Deterioration rate curves for bridge superstructure .................................................. 47 Figure 4.8: Deterioration rate curves for substructure ................................................................ 48 Figure 4.9: Deterioration curves estimated using by proportional hazard model......................... 53 Figure 5.1: Actual bridge deck condition ratings in 2010 ........................................................... 55 Figure 5.2: Deterioration curves of multiple model approach and bridge deck condition ratings in 2010 .......................................................................................................................................... 56
1
Chapter 1 : Introduction
Background
The Intermodal Surface Transportation Efficiency Act of 1991 (ISTEA) marked the start
of a new era in transportation in the United States. ISTEA required transportation agencies to take
a proactive approach to planning and management of their assets. This included requirements for
managing pavement, bridges, safety, congestion, public transportation, and intermodal systems.
For horizontal transportation assets (i.e., bridges and pavement), deterioration modeling is an
essential tool (Yanev and Chen 1993). Since ISTEA, Bridge Management Systems (BMS) have
been utilized to inform decision-making regarding bridge projects such as maintenance,
rehabilitation, and replacement (MR&R) under financial limitations (Agrawal, Kawaguchi, and
Chen 2010). The goal of a BMS is to optimize the performance of bridge networks by
implementing planned MR&R events to the selected bridges in the correct manner and at the
correct time. Reliable predictions of future condition states are critical for optimizing MR&R
activities. The condition rating of bridges is the most vital variable to predict the future
performance of bridges (Jiang 2010). Deterioration models are used to estimate a future condition
ratings. There are numerous approaches to develop deterioration models such as deterministic,
stochastic, mechanistic, artificial intelligent, reliability-based approaches. Each of these
approaches is defensible but is based on different assumptions and could potentially provide
different results. A stochastic approach using Markov chains was the focus of this research.
Objectives
The objectives of this research are twofold; 1). to identify common Markovian-based
deterioration modeling approaches and apply them to a population of Texas bridges to explore the
similarities and differences between approaches 2). to generate a deterioration curve combines the
individual models, minimizing the individual basis of any single model.
2
Organization of Thesis
Chapter 2 presents a review of the literature focused on infrastructure deterioration
modeling approaches. Chapter 3 presents the individual methodologies of modeling a transition
probability matrix. The mathematical models of regression nonlinear optimization, ordered probit,
Poisson regression, negative binomial regression, Bayesian maximum likelihood, and proportional
hazard are presented. Chapter 4 presents a case study including data collection and filtering,
transition probability matrices and deterioration curves obtained from each model, comparison
with existing data, results of consistence test of each model, and a deterioration curve generated
using multiple model approach. Chapter 5 presents conclusions and recommendations for future
work.
3
Chapter 2 : Literature Review of Bridge Deterioration Modeling
The goal of a bridge management system is to support decision-making at Departments of
Transportation in order to maximize the performance of a bridge network system under financial
constraints. The condition rating of bridges is one of the most important factors considered in of
the allotment of capital bridge projects. Deterioration models can be used to estimate future
condition ratings of a bridge. If the estimated future condition rating is reliable, the selection of
bridge MR&R projects can be optimized (Agrawal, Kawaguchi, and Chen 2010; Jiang 2010).
However, it must be noted that the bridge rating system has significant constraints based on two
assumptions (Yanev and Chen 1993):
⢠The selection of structural components and corresponding weights is based on engineering
experience and reliability estimates.
⢠The lowest component condition rating is used to decide for the entire structure.
Condition Rating
Bridges are an important element in the highway transportation system. These structures
are expected to be safe. The issue of bridge safety was made prominent by the collapse of the
Silver Bridge located at the Ohio River in 1967. The Secretary of Transportation was required by
Congress to develop and implement the National Bridge Inspection Standards (NBIS) for
estimating the deficiencies of existing bridges. The NBIS requires visual inspection biennially.
The National Bridge Inventory (NBI) database contains information on every bridge in the nation.
The database is primarily populated with bridge inspection data. Bridge owners are responsible for
the inspections and for reporting the information to FHWA for inclusion in the NBI database.
(FHWA 2004).
NBI contains 116 items including the following (Ryan et al. 2012):
⢠Identification â bridges are identified by coordinates and qualitative descriptions.
4
⢠Structure material and type â bridges are classified by structural material (i.e., concrete,
steel, etc.), number of spans, and design type (i.e., truss, multi-girder, slab).
⢠Age and service â built year and functionality of a bridge.
⢠Geometric data â overall dimensions (i.e., structure length and width, skew) but not section
level geometric data
⢠Condition â inspection date and the condition ratings of bridge components.
Condition ratings are subjectively assigned by bridge inspectors using a 0 to 9 rating scale. The
inspectors, who must be trained and certified by FHWA, determine the ratings for each bridge
component (deck, superstructure, substructure) based on engineering expertise and experience
(Ryan et al. 2012). The general guideline of condition rating for bridge components are described
in the 1995 edition of the FHWA Coding Guide in Table 2.1 (Ryan et al. 2012). In 1995, FHWA
revised the standards to focus on bridge elements, a more refined discretization of bridges than
components. The Commonly Recognized (CoRe) Structural Elements manual was accepted as an
official American Association of State Highway and Transportation Officials (AASHTO) manual
in 1995.
Table 2.2 describes an initial guideline used in evaluation of element condition rating (Congress
2012). In 2011, the AASHTO Guide Manual for Bridge Element Inspection was published. It
included four standardized condition states utilizing a 1-4 rating scale. The AASHTO element-
level data can be aggregated to determine the condition ratings of bridge components in NBI
(Yanev and Chen 1993).
Table 2.3 describes the final condition ratings of bridge elements that were adopted. These
condition states provide severity and extent (i.e. total element quantity) of deterioration of bridge
elements (Congress 2012). The National Bridge Investment Analysis System (NBIAS) introduced
in 1999 models the investment needs for bridge maintenance, repair, and rehabilitation. This
system incorporates analytical approaches such as a Markovian modeling, optimization, and
simulation. Also, the NBIAS model can perform an analysis of bridge conditions using element
level data (Ryan et al. 2012). States were required to report element level data of all bridges to
5
FHWA by the Moving Ahead for Progress in the 2lst Century legislation (MAP-21) signed into
law in 2012 (Congress 2012).
Table 2.1: Condition rating codes and descriptions for bridge
Codes Descriptions N Not applicable 9 Excellent condition 8 Very good condition â no problems noted 7 Good condition â some minor problems 6 Satisfactory condition â structural elements show some minor deterioration 5 Fair condition â all primary structural elements are sound but may have minor
section loss, cracking, spalling or scour 4 Poor condition â advanced section loss, deterioration, spalling, or scour 3 Serious condition â loss of section, deterioration, spalling, or scour have
seriously affected primary structural components. Local failures are possible. Fatigue cracks in steel or shear cracks in concrete may be present.
2 Critical condition â advanced deterioration of primary structural elements. Fatigue cracks in steel or shear cracks in concrete may be present or scour may have removed substructure support. Unless closely monitored it may be necessary to close the bridge until corrective action is taken.
1 âImminentâ Failure condition â major deterioration or section loss present in critical structural components, or obvious vertical or horizontal movement affecting structure stability. Bridge is closed to traffic, but corrective action may put bridge back in light service.
0 Failed condition â out of service; beyond corrective action.
Table 2.2: Condition status and description of bridge
Status Descriptions Good Element has only minor problems. Fair Structural capacity of element is not affected by deficiencies Poor Structural capacity of element is affected or jeopardized by deficiencies.
Table 2.3: Condition state codes and description of bridge elements revised in 2011
States Descriptions 1 Good â No deterioration to minor deterioration 2 Fair â Minor to Moderate deterioration 3 Poor â Moderate to Severe deterioration 4 Severe â Beyond the limits of 3
6
Explanatory Variables
Explanatory variables are defined here as external factors that affect bridge deterioration.
Bridges are generally classified with explanatory variables prior to deterioration modeling
analysis. Morcous et al. (2003) classified concrete bridge decks in Quebec, Canada with
explanatory variables including functionality, location, average daily traffic, percentage of truck
traffic, and environments. Wellalage et al. (2015) grouped railway bridges in Australia by
explanatory variables including structure material, number of tacks, average ton passed per week,
element type, environments, and span length. Agrawal et al. (2010) classified bridge elements in
New York State, U.S.A. based on explanatory variables including design type, location, structure
material, ownership, annual average daily truck traffic, deicing salt usage, snow accumulation,
environments, and functionality. Huang (2010) identified 11 explanatory variables statistically
relevant to concrete bridge deck deterioration in Wisconsin by using the artificial neural network
approach. The explanatory variables that were identified included maintenance history, age,
previous condition, district, design load, structure length, deck dimension, average daily traffic,
skew, number of spans, and environments. Table 2.4 summarizes explanatory variables based on
application. Note that in nearly all cases, the selection of explanatory variables was based on
heuristics and engineering judgment alone. Analysis was rarely conducted to determine which
explanatory variables to consider.
7
Table 2.4: Explanatory variables applied on bridge deterioration modeling
Application Explanatory variables
Bridge elements
⢠Design type ⢠Location ⢠Structure material ⢠Ownership ⢠Annual average daily truck traffic ⢠Deicing salt usage ⢠Snow accumulation ⢠Environments ⢠Functionality
Bridge components
⢠Functionality ⢠Location ⢠Average daily traffic ⢠Percentage of truck traffic ⢠Maintenance history ⢠Age ⢠Previous condition ⢠District ⢠Design load ⢠Structure length ⢠Deck dimension ⢠Skew ⢠Number of spans ⢠Environments
Railway bridge components
⢠Structure material ⢠Number of tacks ⢠Average ton passed per week ⢠Element type ⢠Environments ⢠Span length
Deterministic approaches to deterioration modeling
Deterministic approaches are the simplest method to obtain bridge condition predictions.
In deterministic models the most possible condition rating can be estimated as a function of age
and other explanatory variables via regression. The basic assumption of deterministic approaches
is that the relationship between the future condition of bridges over time is certain. The models
8
neglect the uncertainty and randomness of deterioration processes (Ranjith et al. 2013). Since the
probabilistic nature of models is not considered, the same outcome from a deterministic model will
be obtained if the input variables are the same (Kotze, Ngo, and Seskis 2015). When an analysis
of each bridge does not influence on an bridge network analysis , a statistical method is suitable to
develop an equation in bridge conditions (Hyman and Hughes 1983). Deterioration rates can be
developed by a regression analysis of data (Agrawal, Kawaguchi, and Chen 2010). In deterministic
models the relationship between explanatory variables influencing bridge deterioration and the
condition rating are illustrated by this regression analysis. The average duration at each condition
state, the average condition rating by age, and the minimum rating of an element are the common
deterministic methods to estimate deterioration rates (Agrawal, Kawaguchi, and Chen 2010).
Yanev and Chen (1993) estimated the future condition state of bridge decks in New York City by
linear regression analysis. The rate of change for each condition rating state was obtained and
averaged. Also, the average condition rating at various ages was calculated. The non-linear
regression method was applied to formulate a third order polynomial model to describe the
relationship between the condition rating of bridge components and the bridge age in Indiana
through statistical and regression analysis by Jiang and Sinha (1989). Limitations to deterministic
models include (Agrawal, Kawaguchi, and Chen 2010):
⢠The uncertainty due to intrinsic probabilistic nature of infrastructure deterioration and
unobserved explanatory variables is not considered.
⢠The average condition of a group of bridges is calculated without consideration of the
current and historical condition of each individual bridge.
⢠Maintenance actions are not considered because it is difficult to calculate the impact of
MR&R activities.
⢠The influence of interaction between components is not considered.
⢠New deterioration rates must be calculated when new data are obtained.
9
Stochastic approaches to deterioration modeling
The condition states of bridge components/elements or the duration at each condition state
are treated as random variables in stochastic models. Probability distributions are used to develop
deterioration models (Kotze, Ngo, and Seskis 2015). Probabilistic deterioration models can be
grouped into two categories: state-based and time-based models. In stated-based models, such as
a Markov chain approach, the probability of transition from one condition state to another in a
discrete time is estimated, conditional on a set of explanatory variables. In time-based models,
such as Weibull distribution function, the probability of duration of time at a condition state is
estimated, conditional on a set of explanatory variables (Mauch and Madanat 2002).
Markovian process
A Markovian process, named after Russian mathematician Andrey Markov, is a stochastic
process that satisfies the Markov property which states that a prediction for the future of the process
only depends on the present state of a system. Markov used ârandom walkâ and âgamblerâs ruinâ
as the examples of Markovian processes in his first paper published in 1906. These examples are
the Brownian motion and Poisson processes, which are the main components of the theory of
stochastic processes. A Markov chain has a discrete state space (Leonard 2011). Markovian
deterioration models have been utilized in modeling the deterioration of infrastructure facilities
such as pavement (A. Butt et al. 1987; Carnahan et al. 1987; DeLisle, Sullo, and Grivas 2003;
Golabi, Kulkarni, and Way 1982), storm water pipe (Micevski, Kuczera, and Coombes 2002),
sewer pipe (Baik, Jeong, and Abraham 2006), bridge components (Hatami and Morcous 2015;
Jiang 2010; Jiang, Saito, and Sinha 1988), bridge elements (Ranjith et al. 2013; J. O. Sobanjo
2011), and culverts and traffic signs (Thompson et al. 2012). Markovian models are based on the
following assumptions (Ranjith et al. 2013):
⢠The deterioration process is homogenous with a constant transition probability in an
inspection period.
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⢠A condition state can transition to across multiple states. For example, the condition
rating can change to any lower condition rating in consecutive inspection period.
⢠The deterioration process is assumed a continuous process within a discrete time
interval and constant bridge population.
⢠The future condition of a bridge relies only on the present condition (Markov property)
A Markov chain is a chain of random variables ðððð , ððððâ1, ððððâ2, ⊠which are finite and which
satisfy the Markov property. The probability of transition from the current state to the next state is
ðððð{ðððð+1 = ð¥ð¥ðð+1|ðððð = ð¥ð¥ðð,⯠,ðð0 = ð¥ð¥0} = ðððð{ðððð+1 = ð¥ð¥ðð+1|ðððð = ð¥ð¥ðð} Eq. 1
The Markov chain can be described as a matrix called the transition probability matrix (TPM). The
Markov chain as applied to bridge performance prediction models is used by defining discrete
condition ratings and obtaining the probability of transition from one condition rating to another
in discrete time. This method is commonly used when prior maintenance records are not available
in BMS databases (Jiang, Saito, and Sinha 1988; Morcous 2006).
The example format of a transition probability matrix as follows:
ðð =
â£â¢â¢â¢â¡ðð9900â®0
ðð98ðð880â®0
ðð97ðð87ðð77â®0
ðð96ðð86ðð76â®0
â¯â¯â¯â¯â¯
ðð90ðð80ðð70â®ðð00âŠ
â¥â¥â¥â€ Eq. 2
The bridge condition ratings are from 9 to 0, and 9 is the maximum rating number for excellent
condition. The prediction can be estimated by:
ðð(ð¡ð¡) = ðð(0) à ðð à ððâ¯ðð = ðð(0) à ðððð Eq. 3
11
where ðð(ð¡ð¡) = the probability distribution at time, t; ðð(0) = the initial-state vector; P = the transition
probability matrix; and T = the power that is the same value with the time, t (Jiang, Saito, and
Sinha 1988). The accuracy of Markovian models is dependent on obtaining an accurate transition
probability matrix with a reliable estimation method (Wellalage, Zhang, and Dwight 2014). Jiang
et al. (1988) estimated a transition probability matrix for bridges in Indiana, USA by using
regression nonlinear optimization method with zone technique developed by Butt et al. (1987) to
improve the accuracy of prediction of future condition state of pavements. In the zone technique,
pavements are grouped into zones based on their age. The deterioration rate was assumed to be
different between zones, but it was constant in each zone. Therefore, a transition matrix was
developed for each zone. Wellalage et al. (2015) generated transition probability matrices for
railway bridges in Australia using regression nonlinear optimization (RNO), Bayesian maximum
likelihood (BML), and Metropolis-Hasting algorithm (MHA)-based Markov chain Monte Carlo
(MCMC) simulation technique methods. He concluded that transition probabilities obtained using
MHA-based MCMC and BML are similar, and they were more accurate than ones estimated using
RNO. The ordered probit model was used to estimate a transition probability matrix for storm
water pipes in the City of Greater Dandenong, Victoria, Australia by Tran (2007) and for bridges
in Indiana, USA by Madanat et al. (2002). Madanat and Ibrahim (2002) estimated a transition
probability matrix for bridges in Indiana, USA using Poisson regression and negative binomial
regression models. Cavalline et al. (2015) developed a transition probability matrix for bridges in
North Carolina, USA by the proportional hazard model combining the Kaplan-Meier estimation
and hazard ratio.
Weibull distribution approach to deterioration models
The Weibull distribution is commonly used in reliability analysis due to its flexibility. It
can take on the features of other distributions depending on the values of the shape (ðœðœ) and scale
12
(ðŒðŒ) parameters. For example, the curve which has ðœðœ = 1 and ðŒðŒ = 1 is an exponential distribution
as shown in Figure 2.1. The probability density function is
ðð(ð¥ð¥) =ðœðœðŒðŒ ï¿œ
ð¥ð¥ðŒðŒï¿œ
ðœðœâ1ððâ(ð¥ð¥ ðŒðŒâ )ðœðœ ðððððð ð¥ð¥ ⥠0. Eq. 4
Weibull survival models have been use for deterioration modeling of infrastructure facilities such
as pipe culverts, roadway lighting fixtures, pavement markings (Thompson et al. 2012), reinforced
concrete bridge decks (Mishalani and Madanat 2002), bridge elements (Agrawal, Kawaguchi, and
Chen 2010), bridge decks (J. O. Sobanjo 2011), and bridge components (J. Sobanjo, Mtenga, and
Rambo-Roddenberry 2010). Weibull models capture the effects of age and uncertainty more than
Markov chain models (Agrawal, Kawaguchi, and Chen 2010). Time series data and information
of historical MR&R activities are needed (Agrawal, Kawaguchi, and Chen 2010). The Weibull
survival function is:
ðŠðŠðð = ððð¥ð¥ððï¿œâ1.0 à (ðð ðŒðŒâ )ðœðœï¿œ Eq. 5
where ðŠðŠðð is survival probability at age ðð; ðœðœ is the shape parameter and ðŒðŒ is the scale parameter:
ðŒðŒ =ðð
(ðððð2)1/ðœðœ Eq. 6
where T is the median life expectancy from the Markov model (Thompson et al. 2012). When ðœðœ
is less than 1, the failure rate (also known as a hazard rate) is decreasing. When ðœðœ is equal to 1, the
failure rate is constant. When ðœðœ is greater than 1, the failure rate is increasing. Increasing the failure
rate indicates that a component/element has been at a condition rating for a long time, so the
component/element will transition to a lower condition rating in the next inspection period
(Agrawal, Kawaguchi, and Chen 2010). The shape parameter (ðœðœ) of 1 is equivalent to a Markov
13
deterioration model, meaning the transition probability does not change with time. A bigger shape
parameter means that the initial deterioration rate is slow, and then the deterioration rate increases
faster as the age of a facility increases. Figure 2.1 shows the effects of the shape parameter on the
resulting distribution. The average duration that a component/element stays at a condition rating,
ðžðž(ðððð) is:
ðžðž(ðððð) = ðŒðŒððð€ð€ ï¿œ1 +1ðœðœððï¿œ Eq. 7
where Î is the Gamma function defined as Î(ðð) = (ðð â 1)!; ðŒðŒðð and ðœðœðð are scale and shape
parameters at condition rating i respectively. The average durations for different condition ratings
are calculated cumulatively (Agrawal, Kawaguchi, and Chen 2010). Some or all of the components
or elements in a given condition rating can transition to the next lower condition rating within a
discrete interval. The duration or time at which p% of the components/elements will transition to
lower condition ratings, ð¡ð¡ðð is:
ð¡ð¡ðð = ðŒðŒ[âðððð(1 â ðð)]1/ðœðœ Eq. 8
where ðŒðŒ and ðœðœ are the scale and shape parameters (Agrawal, Kawaguchi, and Chen 2010).
14
Figure 2.1: Weibull probability density function with different shape parameters
Artificial Intelligence approaches to deterioration modeling
Artificial intelligence (AI) techniques have been used to model the deterioration of
infrastructure facilities such as neural network for storm water pipes (D. H. Tran, Perera, and Ng
2009), artificial neural network (ANN) for water mains, ensemble of neural network (ENN) for
bridge elements and components (Bu et al. 2014; Lee et al. 2014; Li and Burgueño 2010),
backward prediction model (BPM) for steel bridge structures and bridge elements (Lee et al. 2008;
Pandey and Barai 1995), multilayer perceptron (MLP) for abutment walls (Li and Burgueño 2010),
and combination of ANN and Case-Based Reasoning (CBR) for the Dickson Bridge (Morcous and
Lounis 2005). An ANN is a computational model derived from the study of nerve cells or neurons
in physiology. An artificial neuron contains receiving sites, receiving connections, a processing
element and transmitting connections. Neural network models are stipulated by the network
topology, node features, and training or learning processes. A multilayer perceptron has an input
x
Prob
abili
ty d
ensit
y fu
nctio
n, f(
x)
15
layer, an output layer, and a number of hidden layers (Pandey and Barai 1995). An ANN can
generate accurate response even though there is noise or uncertainty in training data and evaluate
the results of complex matters with a higher order of nonlinear behavior. An ANN learns to map
the accurate input-output by an iterative procedure. In the learning process, the weighted
connections between neurons are modified, and the error of the model reduced to produce more
accurate results. If data are trained only to focus to reduce the errors, the ANN model might not
generalize the connection between inputs and outputs. In the process, neurons receive inputs from
the previous layer, compute an output through a pre-defined function, and send the output to
neurons on the next layer. Figure 2.2 shows an ENN process diagram. An ENN consists of many
individual MLP models and predict the outcomes by combining these models (Winn and Burgueño
2013). Lee et al. (2008) used an ANN-based BPM to generate unavailable historical bridge
condition ratings for using limited bridge inspection data. Figure 2.3 illustrate the process of BPM.
The missing information such as condition ratings can be estimated by using the correlation
between explanatory variables and condition states. The correlation can be obtained from the
existing historical data by using ANN process (Huang 2010).
Figure 2.2: Schematic diagram of ENN process (Winn and Burgueño 2013)
16
Figure 2.3: Sketch of the BPM process (Huang 2010)
Mechanistic-based approaches to deterioration modeling
The methods mentioned above are based on observation of bridge condition states using
visual inspection. Mechanistic-based approaches rely on modeling the physical processes of bridge
deterioration; most commonly corrosion of rebar in reinforced concrete. A passive film (oxide
film) is formed on the surface of the steel reinforcement during concrete hydration and protects
the reinforcement from corrosion. The film can be destroyed by chloride ions that penetrate to the
reinforcement from the surface of the concrete (Roelfstra et al. 2004). Corrosion models have been
applied to predict reinforced concrete bridge deck conditions. Morcous and Lounis (2007) applied
Fickâs second law to estimate the corrosion initiation time and used the Monte Carlo simulation
technique to generate the probability density function and cumulative distribution function of the
corrosion initiation time:
ð¶ð¶(ð¥ð¥, ð¡ð¡) = ð¶ð¶ð ð ï¿œ1â ðððððð ï¿œð¥ð¥
2âð·ð·ð¡ð¡ï¿œï¿œ Eq. 9
17
where C(x,t) is the chloride concentration at depth (x) and time (t) ð¶ð¶ð ð is surface chloride
concentration, D is diffusion coefficient of chlorides, and erf is error function. Roelfstra et al.
(2004) divided the deterioration of reinforced concrete due to corrosion into two phases, the
initiation phase which chlorides penetrates through the concrete to the reinforcement from the
surface and the propagation phase which the reinforcement actively corrodes. The deterioration
curve generated from the proposed mechanistic model was compared with the curve developed
using the standard Markov chain method with the difference between two curves stemming from
explicit inclusion of the corrosion initiation time. Hu et al. (2013) and Nickless and Atadero (2017)
divided the corrosion process into three phases; corrosion initiation at the rebar surface, initiation
of cracking at the interface between concrete and rebar, and propagation of cracking to the concrete
surface. A numerical model was selected for each phase. The Monte Carlo simulation technique
was implemented to estimate cumulative deterioration of an entire deck. The damage over the deck
was mapped to condition rating scale described by the NBI (Morcous and Lounis 2007).
Reliability-based approaches to deterioration modeling
Bridge management systems are often reliability-based to obtain the spread of uncertainties
in the lifetime process. Commonly used approaches to predict bridge condition (e.g., state and
time-based approaches) do not consider the reliability of performance of elements, components,
or systems of a bridge. Bridge design, however, is based on achieving a uniform probability of
failure using the reliability index, ðœðœ. The same concept can be applied to bridge condition. A bridge
is in excellent condition where ðœðœ ⥠9.0, in very good condition where 9.0 > ðœðœ ⥠8.0, in good
condition where 8.0 > ðœðœ ⥠6.0, in fair condition where 6.0 > ðœðœ ⥠4.6, and in unacceptable
condition where ðœðœ < 4.6 (Frangopol et al. 2001). The performance function, g(t) for a given failure
mode, the probability of failure of a system with several failure modes, Psys(t), and the reliability
index, ðœðœ(ð¡ð¡) associated with the failure state of the system are
18
ðð(ð¡ð¡) = ðð(ð¡ð¡)â ðð(ð¡ð¡) Eq. 10
ððð ð ð ð ð ð (ð¡ð¡) = ðð[ðððððŠðŠ ðððð(ð¡ð¡) < 0],ðððððð ðððððð ð¡ð¡ > 0 Eq. 11
ðœðœ(ð¡ð¡) = ð·ð·â1 ï¿œ1â ððð ð ð ð ð ð (ð¡ð¡)ï¿œ Eq. 12
where ðð(ð¡ð¡) and ðð(ð¡ð¡) are the instantaneous resistance and load effect at time t, respectively, ðððð(ð¡ð¡)is
the performance function associated with the ith system failure mode, and Ί is the standard normal
cumulative distribution function (Barone and Frangopol 2014). There are three methods to
approximate the failure probability, the mean value first-order second-moment method, the first
order reliability method, and the Monte Carlo simulation method (Frangopol, Kallen, and
Noortwijk 2004). The time-dependent bi-linear and nonlinear reliability index models, ðœðœ(ð¡ð¡)
without maintenance are as follows:
ðœðœ(ð¡ð¡) = ï¿œ ðœðœ0,ðððððð 0 †ð¡ð¡ †ð¡ð¡ðŒðŒðœðœ0 â ðŒðŒ1(ð¡ð¡ â ð¡ð¡ðŒðŒ),ðððððð ð¡ð¡ > ð¡ð¡ðŒðŒ
Eq. 13
ðœðœ(ð¡ð¡) = ï¿œ ðœðœ0, ðððððð 0 †ð¡ð¡ †ð¡ð¡ðŒðŒ ,ðœðœ0 â ðŒðŒ2(ð¡ð¡ â ð¡ð¡ðŒðŒ) â ðŒðŒ3(ð¡ð¡ â ð¡ð¡ðŒðŒ)ðð,ðððððð ð¡ð¡ > ð¡ð¡ðŒðŒ ,
Eq. 14
where ðŒðŒ1,ðŒðŒ2,ðŒðŒ3 are reliability index deterioration rates, ð¡ð¡ðŒðŒ is the deterioration initiation time, and
p is a parameter related to the nonlinear effect in terms of a power law in time. An increase in p
results in an increase in the rate of reliability index deterioration. The time-dependent condition
index model, ð¶ð¶(ð¡ð¡) at time ð¡ð¡ ⥠0 is following:
ð¶ð¶(ð¡ð¡) = ï¿œ ð¶ð¶0,ðððððð 0 †ð¡ð¡ †ð¡ð¡ðŒðŒðŒðŒð¶ð¶0 â ðŒðŒðŒðŒ(ð¡ð¡ â ð¡ð¡ðŒðŒðŒðŒ),ðððððð ð¡ð¡ > ð¡ð¡ðŒðŒðŒðŒ
Eq. 15
where ð¶ð¶0 is the initial condition, ðŒðŒðŒðŒ is the condition deterioration rate, ð¡ð¡ðŒðŒðŒðŒ is the condition index
which is considered constant for a period equal to the time of damage initiation, and ð¶ð¶(ð¡ð¡) is the
19
condition at time t which is assumed to decrease with time. The reliability-based model with
maintenance is as follows:
ðœðœðð(ð¡ð¡) = ðœðœðð ,0(ð¡ð¡) + ï¿œâðœðœðð,ðð(ð¡ð¡)
ðððð
ðð=1
, Eq. 16
where ðððð is the number of maintenance actions associated with reliability index profile j, ðœðœðð,0(ð¡ð¡) is
the reliability index profile without maintenance, and âðœðœðð,ðð(ð¡ð¡) is the additional reliability index
profile associated with the ith maintenance action. The reliability index profile of the system is
obtained by combining the reliability index profiles of all individual elements and limit states
considered (Frangopol, Kallen, and Noortwijk 2004).
20
Chapter 3 : Methodology
Six different deterioration methods are presented in this chapter. The selected methods are
⢠Regression Nonlinear Optimization
⢠Bayesian Maximum Likelihood
⢠Ordered Probit
⢠Poisson Regression
⢠Negative Binomial Regression
⢠Proportional Hazard
Though different, each of these approaches has some common attributes. Each model optimizes
an objective function to obtain input parameters (Morcous, Lounis, and Mirza 2003) and then
applies a mathematical model to estimate transition probabilities. Each approach results in a unique
TPM.
Markov chain models
In the Markov chain models, finite number of states and a set of initial probabilities for all
states are needed to estimate transition probabilities. These probabilities are time-independent and
satisfy the Markov property. The Markovian property states that the transition of a condition state
to any future condition state only depends on the present condition state (Eq. 1) (I and Glagola
1994). The probability of transition from state i to state j in an interval, s is following:
ðððððð(ð ð ) = ðððð{ðð(ð¡ð¡ + ð ð ) = ðð|ðð(ð¡ð¡) = ðð} ð€ð€ððð¡ð¡â ð ð , ð¡ð¡ ⥠0 Eq. 17
The probability of being state i at time t is following:
ðððð(ð¡ð¡) = ðððð{ðð(ð¡ð¡) = ðð} Eq. 18
21
It can be a vector form representing the probability distribution of state at time t.
The expected value at time t is following:
ðžðž[ðð(ð¡ð¡)] = ï¿œðð â ðððð(ð¡ð¡)
âðð
Eq. 19
where j is the condition rating, j = 9, 8, 7, âŠ, 0 and ðððð(ð¡ð¡) is the transition probability at condition
rating, k at time t.
Transition Probability Matrix (TPM)
TPMs can be categorized into two types, one-step transition probability matrices and multi-
step transition probability matrices. In a one-step TPM, a condition state can transition only the
next lower condition state. However, in a multi-step TPM, a condition state can transition to
multiple lower states (Wellalage, Zhang, and Dwight 2014). A one-step TPM is a subset of a multi-
step TPM where all probabilities outside of the primary pair of the current state and next lower
state are equal to zero.
An example of a one-step TPM is following:
ðð
=
â£â¢â¢â¢â¢â¢â¢â¢â¢â¡ðð99000000000
1â ðð99ðð8800000000
01 â ðð88ðð770000000
00
1 â ðð77ðð66000000
000
1 â ðð66ðð5500000
0000
1 â ðð55ðð440000
00000
1â ðð44ðð33000
000000
1 â ðð33ðð2200
0000000
1â ðð22ðð110
00000000
1 â ðð111
âŠâ¥â¥â¥â¥â¥â¥â¥â¥â€
Eq. 20
An example of a multi-step TPM is following:
22
ðð =
â£â¢â¢â¢â¢â¢â¢â¢â¡ðð99000000000
ðð98ðð8800000000
ðð97ðð87ðð770000000
ðð96ðð86ðð76ðð66000000
ðð95ðð85ðð75ðð65ðð5500000
00ðð74ðð64ðð54ðð440000
000ðð63ðð53ðð43ðð33000
0000ðð52ðð42ðð32ðð2200
00000ðð41ðð31ðð21ðð110
000000ðð30ðð20ðð101
âŠâ¥â¥â¥â¥â¥â¥â¥â€
Eq. 21
It is assumed that the bridge condition rating could stay in its current state or transit to the next
lower state in one year without any repair or rehabilitation. While it is technically possible for a
condition rating to transition two states in one period, this occurs very infrequently. For example,
Table 3.1 shows the number of bridge decks in 2010 classified into condition rating groups. This
population of bridges was selected for the comparison conducted in Chapter 4 and is presented in
more depth there. The specifics of the bridge population are immaterial for the discussion of
adoption of one or multi-step TPM. The first column indicates condition rating, ðð. The first row
represents the number of bridge components in condition rating, ðððððð, the condition rating transits
from a state, ðð to a state, ðð (ðð = 9, 8, ⊠, 3). For example, in 2008, 631 bridges were in condition
rating 8. 381 of 631 bridges stayed the same condition rating, 242 bridges transited to condition
rating 7, and 8 bridges transited to condition rating 6 in the next cycle in 2010. This equates to
about 1 % of the bridges which transitioned to condition rating 6. As such a low percentage
transitioned more than one state, a one-step TPM was adopted. Also, since the lowest condition
rating in the data was 3, a truncated TPM was adopted. Based on these characteristics of the data,
the following TPM proposed by Jiang et al. (1988) was used in this research.
ðð =
â£â¢â¢â¢â¢â¢â¡ðð99000000
1 â ðð99ðð8800000
01â ðð88ðð770000
00
1 â ðð77ðð66000
000
1 â ðð66ðð5500
0000
1â ðð55ðð440
00000
1 â ðð441 âŠ
â¥â¥â¥â¥â¥â€
Eq. 22
23
Table 3.1: Number of bridge decks of transition
CR ðððð9 ðððð8 ðððð7 ðððð6 ðððð5 ðððð4 ðððð3 Sum ðð = 9 0 3 2 0 0 0 0 5 ðð = 8 0 381 242 8 0 0 0 631 ðð = 7 0 0 2,672 136 6 0 0 2,814 ðð = 6 0 0 0 413 22 0 0 435 ðð = 5 0 0 0 0 42 0 0 42 ðð = 4 0 0 0 0 0 2 0 2 ðð = 3 0 0 0 0 0 0 0 0
Regression Nonlinear Optimization (RNO)
In RNO method, a transition probability matrix is estimated by minimizing the absolute
difference between the regression curve, ðð(ð¡ð¡), that best fits the actual condition rating at age, t,
and the estimated condition rating for the corresponding age, ðžðž(ð¡ð¡,ðð) obtained by the Markovian
process. The formula of the objective function is as follows (Jiang et al. 1988):
ððððððï¿œ|ðð(ð¡ð¡)â ðžðž(ð¡ð¡,ðð)|ðð
ð¡ð¡=1
Eq. 23
ðð(ð¡ð¡) = ðŽðŽ + ðµðµð¡ð¡ + ð¶ð¶ð¡ð¡2 + ð·ð·ð¡ð¡3 Eq. 24
ðžðž(ð¡ð¡,ðð) = ðð0 Ã ððð¡ð¡ Ã ð ð Eq. 25
where:
ðð = the largest age of the bridge in a data set;
ðð(ð¡ð¡) = the average condition rating at time, t;
A, B, C, and D = coefficients;
ðžðž(ð¡ð¡,ðð) = the expected value;
ðð = transition probabilities;
ðð0 = the initial condition state, ðð0 = [1 0 0 0 0 0 0] when t = 0;
24
ððð¡ð¡ = TPM at any time, t;
ð ð = the condition rating, [9 8 7 6 5 4 3].
Bayesian Maximum Likelihood (BML)
The Bayesâ theorem states that the conditional distribution of ðð (an unknown variable set)
given by ðð (a known data set of bridge condition ratings) is the following:
ðð(ðð|ðð) â ðð(ðð)ð¿ð¿(ðð|ðð) Eq. 26
where ðð(ðð|ðð), ðð(ðð), and ð¿ð¿(ðð|ðð) are called a target distribution, prior distribution, and likelihood
distribution respectively. From Bayes-Laplace âprinciple of insufficient reason,â the prior
distribution, ðð(ðð) can be assumed to be a uniform distribution. Therefore, the target distribution
can be proportional to the likelihood distribution (Wellalage, Zhang, and Dwight 2014). From the
joint probability theory, the likelihood distribution can be expressed (H. D. Tran 2007).
ð¿ð¿(ðð|ðð) = ᅵᅵ(ð¶ð¶ððð¡ð¡)ððððð¡ð¡
9
ðð=1
ðð
ð¡ð¡=1
Eq. 27
To simply computations, the distribution function can be transformed into a logarithm likelihood
function and applied to bridges as follows:
ðððððð[ð¿ð¿(ðð|ðð)] = ᅵᅵððððð¡ð¡ðððððð(ð¶ð¶ððð¡ð¡)9
ðð=1
ðð
ð¡ð¡=1
Eq. 28
where T is the largest age in the data set, ððððð¡ð¡ is the number of bridge components in condition state
i at year t, and ð¶ð¶ððð¡ð¡ is the probability of condition state ðð at year ð¡ð¡. Since only transition probabilities,
25
ð¶ð¶ððð¡ð¡ are random variables, a vector of transition probabilities, ð¶ð¶ððð¡ð¡ can be obtained by maximizing
the log likelihood function, ðððððð[ð¿ð¿(ðð|ðð)].
ð¶ð¶ððð¡ð¡ = [ð¶ð¶9ð¡ð¡ ð¶ð¶8ð¡ð¡ ð¶ð¶7ð¡ð¡ ð¶ð¶6ð¡ð¡ ð¶ð¶5ð¡ð¡ ð¶ð¶4ð¡ð¡] Eq. 29
where 9, 8, âŠ, and 4 are condition rating.
Ordered Probit Model (OPM)
The ordered probit model is most commonly used to model unobservable features of a
population in social sciences. Applied to infrastructure, facility deterioration can be considered
unobservable, while condition rating is observed. However, the model can still be used because of
the assumption that condition rating is a constant unobservable variable. An incremental
deterioration model can be generated from the observed condition rating as an âindicatorâ of the
unobserved deterioration. The variable, ðððð is the number of transitions of condition state of bridge
n between two successive inspection periods. It is assumed that the deterioration is constant within
the same condition rating group, and each group has different deterioration mechanism. Therefore,
a different deterioration model is needed for each condition rating group. Madanat et al. (2002)
categorized the deterioration process of bridge decks into two steps. The transitions from condition
rating 9 to 7 were determined by the change of the electrical potential intensity and the chloride
content amount. The transition from condition rating 6 to 5 is determined by the amount of spall
of concrete. Based on the deterioration process of reinforced concrete defined by Hu et al. (2013)
and Nickless and Atadero (2017), the first stage (condition rating 9 to 7) is related to the phase
from corrosion initiation to cracking initiation. The second stage (condition rating 6 to 5) is related
to the phase of cracking propagation to the surface. The unobserved deterioration of a bridge, n in
given condition state, i, ðððððð can be expressed as a function of explanatory variable, ðððð.
ðððððð(ðððððð) = ðœðœððâ²ðððð + ðððððð Eq. 30
26
where ðœðœððâ² is a set of parameters in condition state i, ðððð is a set of explanatory variables for bridge
n; ðððððð is an error term (Madanat, Mishalani, and Ibrahim 2002). Figure 3.1 schematically illustrates
a ordered probit model (H. D. Tran 2007). There are two y-axes showing pipe deterioration in
different variables. One is threshold, ðð and the other is state and ranges from 1 to 3. The graph has
one curve line which is actual deterioration curve of a pipe and a straight line, ðððð which consists
of two parts, deterministic (ðœðœððâ²ðððð) and random (ðððððð) parts. ðð1 and ðð2 are thresholds and 1, 2, and 3
indicate condition states (1 is corresponding to condition rating 9 in NBI). Since ðððððð is
unobservable, the relationship can be expressed by using ðððððð and between two thresholds, ðŸðŸðððð and
ðŸðŸðð(ðð+1) as follows (Madanat, Mishalani, and Ibrahim 2002):
ðððððð = ðð ðððð ðŸðŸðððð †ðððððð < ðŸðŸðð(ðð+1); Eq. 31
ðððððð = ðð ðððð ðððððð ðŸðŸðððð â ðœðœððâ²ðððð †ðððððð < ðððððð ðŸðŸðð(ðð+1) â ðœðœððâ²ðððð; Eq. 32
for j = 0, âŠ, i. Based on the assumption that ðððððð is a normal cumulative distribution, ð¹ð¹(ðððððð), the
transition probability from condition state i to condition state i-j for a bridge in an inspection period
can be written as follows:
ðð(ðððððð = ðð) = ð¹ð¹ï¿œð¿ð¿ðð(ðð+1) â ðœðœððððððï¿œ â ð¹ð¹(ð¿ð¿ðððð â ðœðœðððððð) Eq. 33
where ð¿ð¿ðððð = ðððððððŸðŸðððð . The parameter ðœðœðð and thresholds ðŸðŸðð1 ,ðŸðŸðð2 , ⊠, ðŸðŸðððð can be obtained by optimizing
the logarithm objective function as follows:
ð¿ð¿ððâ = ᅵᅵðð(ðððððð = ðð)ððððððððâ1
ðð=0
ðððð
ðð=1
Eq. 34
27
ðððððð(ð¿ð¿ððâ) = ᅵᅵðððððððððððð[ðð(ðððððð = ðð)]ððâ1
ðð=0
ðððð
ðð=1
Eq. 35
where:
ð¿ð¿ððâ is the likelihood function of the ordered probit model for condition state ðð
ðððð is total number of bridges in condition state ðð in the data set
ðððððð is equal to 1 if ðððððð = ðð, and 0 otherwise.
After the parameter and thresholds are obtained, the transition probabilities, ï¿œÌï¿œð(ðð|ðððð , ðð) for each
bridge can be computed as follows:
ï¿œÌï¿œð(ðð = 0|ðððð , ðð) = ð¹ð¹ï¿œð¿ð¿ðð1 â ï¿œÌï¿œðœððâ²ððððï¿œ
ï¿œÌï¿œð(ðð = 1|ðððð , ðð) = ð¹ð¹ï¿œð¿ð¿ðð2 â ï¿œÌï¿œðœððâ²ððððï¿œ â ð¹ð¹ï¿œð¿ð¿ðð1 â ï¿œÌï¿œðœððâ²ððððï¿œ
ï¿œÌï¿œð(ðð = 2|ðððð , ðð) = ð¹ð¹ï¿œð¿ð¿ðð3 â ï¿œÌï¿œðœððâ²ððððï¿œ â ð¹ð¹ï¿œð¿ð¿ðð2 â ï¿œÌï¿œðœððâ²ððððï¿œ
â®
ï¿œÌï¿œð(ðð = ðð|ðððð , ðð) = 1 â ð¹ð¹ï¿œð¿ð¿ðððð â ï¿œÌï¿œðœððâ²ððððï¿œ Eq. 36
for j = 0, 1, 2, âŠ, i. Then the probabilities of each bridge are grouped to estimate the mean value
of transition probabilities, ï¿œÌï¿œððð(ððâðð)ðð as follows:
ï¿œÌï¿œððð(ððâðð)ðð =
1ðððð
ï¿œ ï¿œÌï¿œð(ðð|ðððð , ðð)
ðððð
ðð=1
; ðð = 0, ⊠, ðð;ðð = 1, ⊠,ðºðº Eq. 37
where ðððð is the total number of bridges in group ðð and G is the total number of groups (Madanat,
Mishalani, and Ibrahim 2002).
28
Figure 3.1: Schematic illustration of the ordered probit model (H. D. Tran 2007)
Poisson Regression (PR)
The Poisson regression model describes events that occur randomly and independently
over time. Incremental bridge deterioration can be modeled as a function of the number of
transitions in an inspection period by using the Poisson Regression method. From the model,
transition probabilities of a data set of bridge components/elements can be obtained. Different
deterioration models are needed for each condition state because mechanistic deterioration
procedures are different. The model estimates the deterioration of a bridge in an inspection period
using the change in the condition states between two successive inspections. In a continuous-time
Markov process, a deterioration rate is a negative exponential distribution, so the Poisson
distribution is applicable for the deterioration model. The Poisson mass function is as follows
(Madanat, Mishalani, and Ibrahim 2002):
ðð(ðððððð = ðð) =ðððððððððððððð
ðð
ðð! , ðð = 0,1,2, ⊠, ðð; ðð = 1,2, ⊠, ðð â 1 Eq. 38
29
where ðððððð is a deterioration rate in condition state ðð; ðð is the number of transitions in condition state
in an inspection period; ðð is the highest condition state. The deterioration rate, ðððððð as a function of
explanatory variables is as follows.
ðððððð = ðð(ðœðœðððð) Eq. 39
where ðœðœ is a set of parameters and ðððð is a set of explanatory variables for a bridge, ðð. By optimizing
the objective function (log of the likelihood distribution), the ðœðœðð can be obtained. The likelihood
distribution and log of the likelihood are as follows.
ð¿ð¿ððâ = ï¿œððâðððððððððððð
ðððððð
ðððððð!
ðððð
ðð=1
Eq. 40
ðððððð(ð¿ð¿ððâ) = ï¿œâðððððð + ðððððð ðððððð(ðððððð)ðððð
ðð=1
Eq. 41
Substitute Eq. 39, then
ðððððð(ð¿ð¿ððâ) = ï¿œðððððð(ðœðœðððð) â ðð(ðœðœðððð)
ðððð
ðð=1
Eq. 42
ðððððð is the number of transitions of condition states in an inspection period. It is assumed the
maximum value of ðððððð is equal to ðð. The transition probabilities for each bridge, ðð is as follows
(Madanat, Mishalani, and Ibrahim 2002):
ðð(ðððððð = ðð|ðððð , ðð) =ððâðððððððððððð
ðð
ðð! , ðð = 0,1,2, ⊠, ðð Eq. 43
30
For network-level, bridges in a data set are grouped into condition states. The average transition
probability for each group is computed as follows:
ðððð(ððâðð)ðð =
1ðððð
ï¿œðð(ðððððð = ðð|ðððð , ðð)
ðððð
ðð=1
, ðð = 0,1, ⊠, ðð;ðð = 1, ⊠,ðºðº Eq. 44
where ðððð is the total number of bridges in group ðð and G is the total number of facility groups
(Madanat, Mishalani, and Ibrahim 2002).
Negative Binomial Regression (NBR)
In Negative Binomial Regression models, a disturbance term in the parameters of the
Poisson distribution is introduced leading to an âoverdispersionâ situation where the variance of
data is larger than the mean (Madanat and Ibrahim 2002). ððððððâ is a random variable and a function
of the selected explanatory variables:
ððððððâ = ðð(ðœðœðððð+ðððð) Eq. 45
where ðððð is a random error term. A negative binomial probability distribution is as follows:
ðð(ðððððð = ðð) =ð€ð€ ï¿œ 1
ðŒðŒðð+ ððï¿œ
ð€ð€ ï¿œ 1ðŒðŒððï¿œ ðð!
ᅵ1
1 + ðŒðŒ1ððððððâï¿œ1 ðŒðŒððâ
ï¿œ1â1
1 + ðŒðŒððððððððâï¿œðð
Eq. 46
where Î( ) is gamma function, ðŒðŒðð is rate of âoverdispersion,â and ðððððð is the number of transitions
of condition states in an inspection period. The likelihood distribution function of the negative
binomial for condition state is as follows:
31
ð¿ð¿ððâ = ï¿œð€ð€ï¿œ 1
ðŒðŒðð+ ððððððï¿œ
ð€ð€ ï¿œ 1ðŒðŒððï¿œðððððð!
ðððð
ðð=1
ᅵ1
1 + ðŒðŒððððððððâï¿œ1 ðŒðŒððâ
ï¿œ1 â1
1 + ðŒðŒððððððððâï¿œðððððð
Eq. 47
By optimizing the log of the objective function, ððððððâ and ðŒðŒðð can be estimated.
ðððððð(ð¿ð¿ððððâ ) = ï¿œððððððï¿œð€ð€ ï¿œ1ðŒðŒðð
+ ððððððᅵᅵâ ððððððï¿œð€ð€ ï¿œ1ðŒðŒððᅵᅵâ ðððððð(ðððððð!)
ðððð
ðð=1
+1ðŒðŒðððððððð ï¿œ
11 + ðŒðŒððððððððâ
ï¿œ + ðððððððððððð ï¿œ1â1
1 + ðŒðŒððððððððâï¿œ
Eq. 48
Transition probabilities of condition states for each bridge can be estimated by applying the
obtained parameter to the negative binomial probability distribution function, and a transition
probability matrix can be obtained by the same process in the Poisson regression method (Madanat
and Ibrahim 2002).
Proportional Hazard Model (PHM)
In the proportional hazard model, the effects of explanatory variables can be explicitly
expressed as hazard ratio in a transition probability matrix. This is the only approach that explicitly
addresses the effects of the explanatory variables in the development of the TPM. The example
format of a transition probability matrix is following (Cavalline et al. 2015).
ðð =
â£â¢â¢â¢â¢â¢â¢â¡ðð99
ð»ð»ð»ð»9
000000
1 â ðð99ð»ð»ð»ð»9
ðð88ð»ð»ð»ð»8
00000
01 â ðð88
ð»ð»ð»ð»8
ðð77ð»ð»ð»ð»7
0000
00
1â ðð77ð»ð»ð»ð»7
ðð66ð»ð»ð»ð»6
000
000
1 â ðð66ð»ð»ð»ð»6
ðð55ð»ð»ð»ð»5
00
0000
1 â ðð55ð»ð»ð»ð»5
ðð44ð»ð»ð»ð»4
0
00000
1 â ðð44ð»ð»ð»ð»4
1 âŠâ¥â¥â¥â¥â¥â¥â€
Eq. 49
where ðð99, ðð88, ⊠,ðð44 are transition probabilities and ð»ð»ð ð 9,ð»ð»ð ð 8, ⊠,ð»ð»ð ð 4 are hazard ratios.
32
In the model, the transition probability matrix consists of two parts:
(1) Transition probabilities that are obtained by the simplified Kaplan-Meier method
(2) Hazard ratios associated with the selected explanatory variables
Transition probabilities and hazard ratios are estimated for each condition state. If a hazard ratio
is less than 1, the deterioration process is slower. If a hazard ratio is greater than 1, the deterioration
process accelerates.
Hazard Ratio
Hazard rate is the instantaneous rate of change from a condition state to another condition
state. It can be expressed as a function of explanatory variables as follows:
â(ð¡ð¡, ð§ð§) = â0(ð¡ð¡)ððð§ð§ðœðœï¿œï¿œâ = â0(ð¡ð¡)ðð(ð§ð§1ðœðœ1+ð§ð§2ðœðœ2+â¯+ð§ð§ðððœðœðð) Eq. 50
â(ð¡ð¡, ð§ð§) = â0(ð¡ð¡)ððð§ð§1ðœðœ Eq. 51
where ðœðœ is the regression coefficient associated with the hazard rate z and â0(ð¡ð¡) is the baseline
hazard function. Hazard ratio is the ratio of the hazard rates, the relative risk of failure ï¿œâ(ð¡ð¡, 1)ï¿œ
to the value of baseline hazard function ï¿œâ(ð¡ð¡, 0)ï¿œ.
ð»ð»ð ð = â(ð¡ð¡, 1)â(ð¡ð¡, 0) = ðððœðœ(1â0) = ðððœðœ Eq. 52
Kaplan-Meier Estimator
Kaplan and Meier (KM) method is used to estimate non-parametric cumulative transition
probability corresponding to transition times and events, ðððð(ð¡ð¡ð¥ð¥). This is assumed as a one-step
transition as follows (Archilla, DeStefano, and Grivas 2002):
33
ðððð(ð¡ð¡ð¥ð¥) = 1â ð ð ï¿œ(ð¡ð¡ð¥ð¥) Eq. 53
ð ð ï¿œ(ð¡ð¡ð¥ð¥) = [(ððð¥ð¥ â 1)/ððð¥ð¥] à ð ð ð¥ð¥â1 Eq. 54
where ð ð ï¿œ(ð¡ð¡ð¥ð¥) is the reliability at ð¡ð¡ð¥ð¥ equal to ðððððððððð , ððð¥ð¥ is order of times observed in a dataset. ðððððððððð
is the time associated with one-step transition of bridge ðð and component ðð to condition rating ðð.
ðððððððððð =ᅵᅵð¹ð¹ð¶ð¶ð ð ðð â ð¿ð¿ð¶ð¶ð ð ðð"ï¿œ + (ð¿ð¿ð¶ð¶ð ð ððâ² â ð¹ð¹ð¶ð¶ð ð ððâ²)ï¿œðððð
2 Eq. 55
where ð¿ð¿ð¶ð¶ð ð ðð" is last date a component was observed in a prior condition rating ðð". ð¹ð¹ð¶ð¶ð ð ðð/ð¿ð¿ð¶ð¶ð ð ððâ² is
first/last date a component was observed in initial condition rating ððâ² (Archilla, DeStefano, and
Grivas 2002).
Summary
From the literature review, six models related to the Markovian process were collected and
studied. These included regression nonlinear optimization (RNO), Bayesian maximum likelihood
(BML), ordered probit model (OPM), Poisson regression (PR), negative binomial regression
(NBR), and proportional hazard model (PHM). Some observations about the individual methods
are included herein:
⢠In the RNO method the third order polynomial was used to describe the deterioration rate of a
bridge. However, the polynomial does not always fit well to the data. The ð ð 2 value for the
polynomial was less than 0.1 for all data sets considered in Chapter 4.
⢠In OPM, PR, and NBR methods, the explanatory variable of age is explicitly applied to
estimate the TPM for each condition rating (CR) group. Some CR groups show a wide range
of age. For example, in the group of CR 8 or 7 the age ranges from about 10 to 60 years.
34
⢠The assumption that deterioration rates are constant in each group is not realistic. For example,
the condition rating can be transition to multiple lower condition ratings according to the data
sets.
Table 3.2 summarizes the objective functions and parameters for each model with some additional
comments.
35
Table 3.2: Summary of objective functions and random variables of models
Models Objective Functions Parameters Comments
Regression Nonlinear
Optimization (RNO)
ððððððï¿œ|ðð(ð¡ð¡) â ðžðž(ð¡ð¡,ðð)|ðð
ð¡ð¡=1
ðð Explanatory variables are not explicitly expressed.
Bayesian Maximum Likelihood
(BML)
ðððððð[ð¿ð¿(ðð|ðð)] = ᅵᅵððððð¡ð¡ðððððð(ð¶ð¶ððð¡ð¡)9
ðð=1
ðð
ð¡ð¡=1
ð¶ð¶ððð¡ð¡ Explanatory variables are not explicitly expressed.
Ordered Probit Model
(OPM)
ðð(ðððððð = ðð) = ð¹ð¹ï¿œð¿ð¿ðð(ðð+1) â ðœðœððððððï¿œ â ð¹ð¹(ð¿ð¿ððððâ ðœðœðððððð)
log(ð¿ð¿ððâ) = ᅵᅵðððððððððððð[ðð(ðððððð = ðð)]ððâ1
ðð=0
ðððð
ðð=1
ð¿ð¿ðððð, ðœðœðð
Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.
Poisson Regression
(PR) log(ð¿ð¿ððâ) = ï¿œðððððð(ðœðœðððð)â ðð(ðœðœðððð)
ðððð
ðð=1
ðœðœ
Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.
Negative Binomial
Regression (NBR)
log(ð¿ð¿ððððâ ) = ï¿œððððððᅵΠᅵ1ðŒðŒðð
+ ððððððᅵᅵðððð
ðð=1
â ððððððᅵΠᅵ1ðŒðŒððᅵᅵ
â ðððððð(ðððððð!)
+1ðŒðŒðððððððð ï¿œ
11 + ðŒðŒððððððððâ
ᅵ
+ ðððððððððððð ï¿œ1
â1
1 + ðŒðŒððððððððâï¿œ
ðŒðŒðð ,ððððððâ
Explanatory variables are explicitly expressed. A transition probability of each bridge is estimated.
Proportional Hazard Model
(PHM) Baseline probability function Hazard
ratio
Explanatory variables are explicitly expressed in hazard ratio. Baseline probability function can be any continuous probability function.
36
Chapter 4 : Case Study
Data Review and Filtration
Data describing Texas bridges spanning 2000 to 2010 year were collected from the
National Bridge Inventory (NBI) database and reviewed. Figure 4.1 shows the total number of
bridges in Texas. The number of bridges increased from 2000 to 2008 and it decreased in 2010.
Though these fluctuations are substantial, they are likely do issues with data management at
TxDOT. Most of the bridges that âdisappearâ between 2008 and 2010 are likely duplicates in the
earlier dataset. The bridges in Texas were grouped according to structure material and design type
in this research.
⢠Structure material â concrete (non-prestressed)
⢠Design type â stringer/multi beam or girder; simple span
Figure 4.2 shows the number of bridges categorized by structure material in 2010. The pairs or
years that define the three data sets were randomly selected, and the environment was assumed to
be similar throughout the state. This assumption falls into the general category of âexplanatory
variableâ and is not the focus of this study. The data were paired as two consecutive inspection
periods for the following reasons:
⢠To check whether a bridge was subject to the MR&R activities. If there is an increase of
condition rating, the component may have been repaired. All models applied to estimate a
TPM are based on âdo nothingâ condition, meaning there can be no MR&R activities on
the bridges.
⢠Some models required bridges to be divided by condition rating groups. For example, in
OPM, PR, and NBR the data in past year such as 2000, 2004, or 2008 was grouped
according to condition ratings and transition probabilities within the groups in present
inspection year such as 2002, 2006, or 2010 were calculated.
Set A consisted of 2000 and 2002 data, set B included 2004 and 2006 data, and set C contained
2008 and 2010. The 2002, 2006, and 2010 data were considered as the present and the 2000, 2004,
and 2008 data were considered as the past in this research. Figure 4.3 shows the number of bridge
37
decks in 2004 grouped by condition ratings. The total number of sample bridge decks in 2004 were
3,870. The condition rating 8 group has about 1,000 decks and the condition rating group 7 has
near 2,500 decks. The selected three data sets were utilized for the consistence test. The
reconstructed bridges (defined by a designation as reconstructed in NBI) were filtered if their
condition ratings were not 9. These bridges were considered to be subject to the MR&R activities,
not considered as new bridges. The bridges more than 60 years were eliminated because the
probability of major repair, retrofit or rehabilitation is much greater. After bridge level filtering,
the data were split by components and filtered again. The bridge components were eliminated if
they showed any increases in condition rating within two consecutive inspection periods. Figure
4.4 shows the total number of bridges before filtering and total number of bridge components after
filtering. The bars hatched with dot indicate decks, the bars filled indicate superstructures, and the
bars hatched with angled straight line indicate substructures. Overall, less than 10 % of the total
number of bridges in the state of Texas were utilized in this research. This is the direct effect of a
explanatory variable selection (i.e., simple span, reinforced concrete multi-girder bridges).
Figure 4.1: Total number of bridges in Texas from 2000 to 2010
0
10,000
20,000
30,000
40,000
50,000
60,000
70,000
2000 2002 2004 2006 2008 2010
Num
ber o
f brid
ges
Year
38
Figure 4.2: Number of bridges according to structure material in 2010
Figure 4.3: Number of bridge decks in 2004 by condition rating groups
0
5,000
10,000
15,000
20,000
25,000
30,000N
umbe
r of b
ridge
s
Structure material
0
500
1,000
1,500
2,000
2,500
3,000
9 8 7 6 5 4 3
Num
ber o
f brid
ges
Condition rating
39
Figure 4.4: Number of bridges by component rating after filtering
Results
A TPM was calculated for every method described in Chapter 3 using a combination of
Microsoft Excel 2018 and Matlab 2017b. Figure 4.5 describes the proposed framework for a
multiple model approach. This research focuses on one branch of analysis - Markovian models
within the umbrella of stochastic approaches. The other arms, and the combination of these
approaches with the Markovian models is considered future work. Texas bridges collected from
the NBI database were classified into groups and filtered. Parameters were obtained by regression
analysis and optimization of objective functions. These parameters were applied to estimate
transition probabilities as described in Chapter 3. Each deterioration rate curve was obtained using
the transition probabilities by the Markovian process. The mean transition probability matrix was
computed by averaging the transition probabilities estimated from each model. The deterioration
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
53,520 53,904 54,898 57,278 58,709 51,454
2000 2002 2004 2006 2008 2010
Num
ber o
f brid
ges
Total number of bridgesYear
deck superstructure substructure
40
rate curve was obtained using the mean transition probability matrix by the Markovian process.
Then the upper and lower boundaries were estimated by combining the 95% confidence intervals
of each individual model.
Figure 4.5: Multiple model approach workflow chart
Transition Probability Matrices (TPM)
Table 4.1 shows the combined transition probability matrices for the three bridge
components estimated by averaging each transition probability matrix developed using the six
identified methods. The probabilities in each component are similar values. For example, the
probabilities of a bridge deck with a CR of 9 staying at the same condition rating (P99) are 0.22,
0.24, and 0.26 for bridge deck, superstructure, and substructure respectively. These TPMs
41
represent a basic approach to multiple model deterioration models wherein the TPM from various
methods, a common point in all analyses, are combined prior to developing a deterioration model.
Table 4.2, Table 4.3, and Table 4.4 show transition probability matrices for bridge components
estimated by each model with 2008-2010 data. These values are used to build the TPMs in Table
4.1. In regression nonlinear optimization (RNO), Bayesian maximum likelihood (BML), and
proportional hazard model (PHM), most condition states have appreciable probabilities to either
transition or to stay at the current condition. However, in ordered probit model (OPM), Poisson
regression (PR), and negative binomial regression (NBR), many of the probabilities are 1, except
the probabilities staying at a condition rating 9 or 8. This is true across components.
These findings indicate that three of the approaches to deterioration modeling selected
(OPM, PR, NBR), would not predict any changes from condition states 7 or 6 ever for a bridge
that falls into the family of bridges as defined by the explanatory variables identified. This runs
counter to our understanding of deterioration but appears to be consistent with the data.
42
Table 4.1: Mean transition probability matrices for bridge components with 2008-2010 data
Bridge Component Transition Probability Matrix
Deck ðð =
â£â¢â¢â¢â¢â¢â¡0.22
000000
0.780.82
00000
00.180.99
0000
00
0.010.99
000
000
0.010.91
00
0000
0.090.79
0
00000
0.241 âŠâ¥â¥â¥â¥â¥â€
Superstructure ðð =
â£â¢â¢â¢â¢â¢â¡0.24
000000
0.760.69
00000
00.310.99
0000
00
0.010.98
000
000
0.020.82
00
0000
0.180.95
0
00000
0.051 âŠâ¥â¥â¥â¥â¥â€
Substructure ðð =
â£â¢â¢â¢â¢â¢â¡0.26
000000
0.740.87
00000
00.130.98
0000
00
0.020.99
000
000
0.010.82
00
0000
0.180.97
0
00000
0.031 âŠâ¥â¥â¥â¥â¥â€
43
Table 4.2: Transition probability matrices of bridge decks with 2008-2010 data
Model Transition Probability Matrix
Regression Nonlinear
Optimization ðð =
â£â¢â¢â¢â¢â¢â¡0.84
000000
0.160.78
00000
00.220.99
0000
00
0.011000
0000
0.4900
0000
0.510.56
0
00000
0.441 âŠâ¥â¥â¥â¥â¥â€
Bayesian Maximum Likelihood
ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.93
00000
00.070.99
0000
00
0.010.99
000
000
0.01100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Ordered Probit Model ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.71
00000
00.29
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Poisson Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.76
00000
00.24
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Negative Binomial
Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.76
00000
00.24
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Proportional Hazard Model ðð =
â£â¢â¢â¢â¢â¢â¡0.5000000
0.50.97
00000
00.030.97
0000
00
0.030.93
000
000
0.07100
0000000
0000011âŠâ¥â¥â¥â¥â¥â€
44
Table 4.3: Transition probability matrices of bridge superstructures with 2008-2010 data
Models Transition Probability Matrix
Regression Nonlinear
Optimization ðð =
â£â¢â¢â¢â¢â¢â¡0.82
000000
0.180.75
00000
00.250.99
0000
00
0.010.99
000
000
0.010.95
00
0000
0.050.90
00000
0.11 âŠâ¥â¥â¥â¥â¥â€
Bayesian Maximum Likelihood
ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.800000
00.2
0.990000
00
0.010.99
000
000
0.01100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Ordered Probit Model ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.53
00000
00.47
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Poisson Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.56
00000
00.44
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Negative Binomial
Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.56
00000
00.44
10000
0001000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Proportional Hazard Model ðð =
â£â¢â¢â¢â¢â¢â¡0.63
000000
0.37091
00000
00.090.96
0000
00
0.040.9000
000
0.1000
00001
0.80
00000
0.21 âŠâ¥â¥â¥â¥â¥â€
45
Table 4.4: Transition probability matrices of bridge substructures with 2008-2010 data
Models Transition Probability Matrix
Regression Nonlinear
Optimization ðð =
â£â¢â¢â¢â¢â¢â¡0.83
000000
0.170.77
00000
00.230.99
0000
00
0.010.97
000
000
0.030.94
00
0000
0.060.88
0
00000
0.121 âŠâ¥â¥â¥â¥â¥â€
Bayesian Maximum Likelihood
ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.900000
00.1
0.980000
00
0.020.99
000
000
0.010.99
00
0000
0.010.99
0
00000
0.011 âŠâ¥â¥â¥â¥â¥â€
Ordered Probit Model ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.85
00000
00.050.99
0000
00
0.011000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Poisson Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.84
00000
00.160.99
0000
00
0.011000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Negative Binomial
Regression ðð =
â£â¢â¢â¢â¢â¢â¡0000000
10.87
00000
00.130.99
0000
00
0.011000
0000100
0000010
0000001âŠâ¥â¥â¥â¥â¥â€
Proportional Hazard Model ðð =
â£â¢â¢â¢â¢â¢â¡0.71
000000
0.290.96
00000
00.040.97
0000
00
0.030.96
000
000
0.04000
00001
0.950
000000.1âŠâ¥â¥â¥â¥â¥â€
46
Deterioration Rate Curves
Deterioration rate curves were plotted for each component in Figure 4.6, Figure 4.7 and
Figure 4.8. The graphs show nine individual curves. Grey lines depict deterioration curves of each
of the six models and black solid line is the deterioration curve estimated with the mean transition
probability matrix. The black dash line is the upper boundary (UB) and black dash-dot line is the
lower boundary (LB). To generate a deterioration rate curve applying a multiple model approach,
the individual models were combined. Using Eq. 25 with the mean transition probability matrix
(Table 4.1), a deterioration rate curve was generated.
The upper and lower boundary envelopes were estimated by:
1. checking the goodness-of-fit for each curve
2. selecting the third order polynomial for all curves based on the value of ð ð 2, closer to 1
3. applying the polynomial to each curve and obtaining the lower and upper boundary with
95% confidential interval
4. comparing the boundary values of each curve and
5. determining the lowest and highest values at any time.
If the value showed lower than zero, the lower limit value was zero. If the value increased, the
upper limit value was the value before increasing. Therefore, the estimated future condition rating
is within the boundaries in 95% certainty. For example, the CR of a bridge deck at 50 years old
can be 4, 5, 6, 7 or 8.
The bridge components deteriorate fast up to about 20 years and then the deterioration slows down
after 20 years shown in most of the models, except in PHM. In the lower boundary curve, the
condition rating reaches to zero at 58 years and 54 years in deck and superstructure, respectively.
47
Figure 4.6: Deterioration rate curves for bridge deck
Figure 4.7: Deterioration rate curves for bridge superstructure
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60 70
Cond
ition
ratin
g
Age
PR PHMOPM BMLRNO NBRMean LB
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60 70
Cond
ition
ratin
g
Age
PR PHMOPM BMLRNO NBRMean LB
48
Figure 4.8: Deterioration rate curves for substructure
Comparison
The estimated probability distribution obtained from each individual model was compared
with the actual distribution by using the chi-squared goodness-of-fit test to find which model best
estimated the probability distribution of the actual value. The consistency between TPMs obtained
using three different data sets in each model was measured using the modal assurance criterion test
as well as the deterioration curves obtained from the TPMs.
Chi-squared goodness-of fit Test
This test measures the closeness between the predicted probability distribution and actual
probability distribution. The formulation of chi-square goodness-of-fit is
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60 70
Cond
ition
ratin
g
Age
PR PHMOPM BMLRNO NBRMean LB
49
ðð2 = ï¿œ(ð ð ðð â ðžðžðð)2
ðžðžðð
ðð
ðð=1
Eq. 56
where ðð is number of observations; ð ð ðð is actual condition state of the ith observation; and ðžðžðð is
expected value of the ith observation and computed by using Eq. 25. The smaller chi squared value
means the estimated value is closer to actual value. For a simple computation, set the initial
condition probability distribution, ðð0 (Eq. 57) using 2008 data. The probability distribution of
time, t equal to 2 was estimated (Eq. 58). The results are present in Table 4.5. The most values
are less than 0.02, except the value of RNO substructure, which is 0.10. Overall, the models predict
future condition state distributions of bridge components well.
ðð0 = [0.001 0.161 0.715 0.112 0.011 0.001 0] Eq. 57
The expected value of condition state distribution, ðð(2) at time, t=2 is computed as follows:
ðð(2) = ðð0 Ã ðð2 Eq. 58
Table 4.5: Results of Chi-square goodness-of-fit of bridge components
Model Deck Superstructure Substructure
RNO 0.02 0.01 0.10
BML 0.01 0.01 0.02
OPM 0.02 0.01 0.01
PR 0.01 0.01 0.01
NBR 0.02 0.01 0.01
Mean 0.02 0.02 0.01
50
In this test, the proportional hazard model was not included because a probability distribution of
bridge components cannot be estimated with this model. However, the future condition rating of
each bridge component can be predicted by using this model. An example is following.
If a bridge deck is 3 years old and we want to know what condition rating of the deck after 5 years.
ðžðž = ðð8 Ã ð ð Eq. 59
where E is the estimated condition rating; P is the transition probability matrix by using PHM; and
R is the condition rating vector, [9 8 7 6 5 4 3].
Modal Assurance Criterion (MAC)
MAC is used as a statistical indicator to compare modes quantitatively in modal analysis.
The MAC value represents the consistency of modes of two structures. This concept was applied
to compute the consistency of TPMs and deterioration curves between the three data sets used in
this research. The value of the MAC is between 0 and 1. A value larger than 0.9 represents that the
modes are consistent and a value closer to 0 represents that the modes are less consistent (Pastor
et al. 2012). The calculation of the MAC is as follows:
ðððŽðŽð¶ð¶(ðŽðŽ,ðð) =ï¿œâ {ðððŽðŽ}ðð{ðððð}ðððð
ðð=1 ï¿œ2
ï¿œâ {ðððŽðŽ}ðð2ðððð=1 ᅵᅵâ {ðððð}ðð2ðð
ðð=1 ï¿œ
Eq. 60
where {ðððŽðŽ} and {ðððð} are the two sets of vectors. The results are shown in Table 4.6 and Table 4.7.
The first observation of note is that all of the MAC values for deterioration curves are 1. Second,
the values in RNO, BML, OPM, PR, and NBR are 0.9 or greater. This means estimated transition
probabilities using these models with different data sets are similar. Some MAC values in PHM
are lower than 0.9. In the deck, the values of A&B and A&C are 0.73 and 0.67 respectively. In
other words, TPMs estimated with different data set using PHM are not consistent. In Table 4.9,
51
the transition probability matrices of bridge decks using PHM with set A, set B, and set C are
shown. However, the deterioration rate curves using these TPMs are consistent. Therefore, a
deterioration rate does not change by different data sets.
Table 4.6: Results of Modal Assurance Criterion for bridge deck
Models TPM Deterioration curve
A&B A&C B&C A&B A&C B&C RNO 0.97 0.96 1.00 1.00 1.00 1.00 BML 1.00 0.98 0.98 1.00 1.00 1.00 OPM 0.97 0.97 0.91 1.00 1.00 1.00 PR 0.99 0.97 0.94 1.00 1.00 1.00
NBR 0.99 0.97 0.92 1.00 1.00 1.00 PHM 0.73 0.67 0.95 1.00 1.00 1.00 Mean 0.99 0.97 0.98 1.00 1.00 1.00
Table 4.7: Results of Modal Assurance Criterion for bridge superstructure
Models TPM Deterioration curve
A&B A&C B&C A&B A&C B&C
RNO 0.99 1.00 0.98 1.00 1.00 1.00
BML 1.00 0.98 0.98 1.00 1.00 1.00
OPM 0.95 0.98 0.89 1.00 1.00 1.00
PR 0.97 0.97 0.92 1.00 1.00 1.00
NBR 0.96 0.97 0.89 1.00 1.00 1.00
PHM 0.97 0.88 0.91 1.00 0.98 0.98
Mean 0.99 0.99 0.98 1.00 1.00 1.00
52
Table 4.8: Results of Modal Assurance Criterion for bridge substructure
Table 4.9: Transition probability matrices of decks using proportional hazard model
Set A ðð =
â£â¢â¢â¢â¢â¡0.77
000000
0.230.98
00000
00.020.96
0000
00
0.040.81
000
000
0.190.98
00
0000
0.020.98
0
00000
0.021 âŠâ¥â¥â¥â¥â€
Set B ðð =
â£â¢â¢â¢â¢â¡0.24
000000
0.760.98
00000
00.020.97
0000
00
0.030.75
000
000
0.25000
00001
0.330
00000
0.671 âŠâ¥â¥â¥â¥â€
Set C ðð =
â£â¢â¢â¢â¢â¡0.50
000000
0.500.97
00000
00.030.97
0000
00
0.030000
000
0.93000
000
0.07100
0000011âŠâ¥â¥â¥â¥â€
Models TPM Deterioration curve
A&B A&C B&C A&B A&C B&C
RNO 1.00 1.00 1.00 1.00 1.00 1.00
BML 1.00 0.99 0.98 1.00 1.00 1.00
OPM 0.94 0.99 0.92 1.00 1.00 1.00
PR 0.98 0.99 0.95 1.00 1.00 1.00
NBR 0.98 0.99 0.95 1.00 1.00 1.00
PHM 0.97 0.84 0.83 1.00 1.00 1.00
Mean 0.99 0.99 0.98 1.00 1.00 1.00
53
Figure 4.9: Deterioration curves estimated using by proportional hazard model
0123456789
10
0 10 20 30 40 50 60 70
Cond
ition
Rat
ing
Age
Bridge Deck
set Aset Bset C
54
Chapter 5 : Conclusions and Future Work
Summary
This research presented an effort to produce a multiple model approach to deterioration
modeling in an effort to combat the inherent biases of selecting a single model for a task. Transition
probability matrices and deterioration curves obtained from six different models, including
regression nonlinear optimization, Bayesian maximum likelihood, ordered probit model, Poisson
regression, negative binomial regression, and proportional hazard model, were combined. For data
analysis and computation of parameters for each model, Microsoft Excel 2018 and Matlab 2017b
were utilized. The lower and upper boundaries estimated with 95% confidence for each
deterioration model by regression analysis were combined to generate bounds. By including all
models, and establishing reasonable bounds of confidence, the decision-maker has more
confidence in their deterioration modeling results. Thus, the multiple model approach is more
robust and flexible than a single approach to deterioration modeling. Finally, chi-square goodness-
of-fit and modal assurance criterion tests were presented to measure the closeness and consistence
of models. Conclusions and future work recommendations are presented herein.
Conclusions
Models are a useful tool to understand future behavior of systems. In many fields, like weather
forecasting, multiple models are used to develop predictions. In civil engineering, typically a single
model is used, and deterioration modeling is no exception. Any single deterioration modeling
approach has idiosyncrasies and differences that can cause uncertainty and false confidence in the
results in certain circumstances. The proposed multiple model approach provides an average future
condition state of bridge components at each age. However, all bridges at the same age in actual
data are not the same condition rating. Figure 5.1 shows the actual data of bridge decks in 2010.
The plot presents three sets of data. The black dots (Mean) are the mean values of condition ratings
at any age. The dark grey dots (UB) are the upper limit values at any age and the light grey dots
55
(LB) are the lower limit values at any age. From single model approach, for example, the condition
rating of bridge deck at 50 years might be 7. However, the condition rating of some decks shows
8 or 4. Figure 5.2 presents actual bridge deck data in 2010 and a deterioration curve (black solid
line) with upper (black dash line) and lower (black dash-dot line) boundaries obtained by using
multiple model approach. This approach provides an average condition rating and other possible
condition ratings of decks at each age. For example, the average condition rating of decks at 50
years is about 6.5 and the range of possible condition ratings is from about 8 to 4. What is evident
from this analysis is that the selected data set, inclusive of reinforced concrete multi-girder bridges
in Texas, does not provide enough variation to highlight the differences in any individual model.
As such, it is difficult to demonstrate the potential benefits of the multiple model approach. For
this population of bridges, there is little difference between any single model nor between any of
the single models and the multiple model approach. This is partially an effect of the data itself,
which is subjective and is also inexorably tied to financial, logistical, and political forces that use
condition data to make administrative decisions. If the data does not demonstrate enough
variability, then model choice is irrelevant.
Figure 5.1: Actual bridge deck condition ratings in 2010
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70
Cond
ition
Rat
ing
Age
MeanLBUB
56
Figure 5.2: Deterioration curves of multiple model approach and bridge deck condition ratings in 2010
Future Work
Some future works were identified during this research to improve reliability of predicting
future condition states of bridges. The recommendations for future work are followings:
⢠Identification of a different research dataset which includes more inherent variability in
deterioration.
⢠Establish the importance and influence of explanatory variables in development of
deterioration models.
⢠Integration of more models such as mechanistic or artificial intelligence approaches into the
multiple model approach.
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60 70
Cond
ition
Rat
ing
Age
MeanLBUBestimated Meanestimated LB
57
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Vita
Jin A Collins earned a Bachelor of Science in Civil Engineering from California State
University Long Beach in 2007. She worked for the County of Los Angeles before attending the
University of Texas El Paso for a Bachelor of Science in Physics which she obtained in 2017.
She then began pursuing her Master of Science in Civil Engineering at The University of Texas
El Paso, and plans to continue working the field of infrastructure deterioration analysis with a
focus on bridges while she pursues her Ph.D.