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2/94

A Thesis

entitled

Prediction of Remaining Service Life of Pavements

by

Chaitanya Kumar Balla

Submitted to the Graduate Faculty as partial fulfillment of the

requirements for the Master of Science Degree in Engineering

Dr. Eddie Yein Juin Chou, Committee Chair

Dr. Azadeh Parvin, Committee Member

Dr. George J. Murnen, Committee Member

Dr. Patricia Komuniecki, Dean

College of Graduate Studies

The University of Toledo

August 2010


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Copyright 2010

This document is copyrighted material. Under copyright law, no parts of this document

may be reproduced without the expressed permission of the author.


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iii

An Abstract of

Prediction of Remaining Service Life of Pavements

by

Chaitanya Kumar Balla

Submitted as partial fulfillment of the requirements for the

Master of Science in Engineering

The University of Toledo

August 2010

Pavement management is a process that helps to maintain a pavement network in a safe

and serviceable condition in a cost effective manner. A key component of an effective

pavement management system is its ability to predict the remaining service life of

pavements. Remaining service life of pavements can be predicted using the present

pavement condition and the latest rehabilitation action performed on that particular

pavement. Survival curves are often developed to obtain remaining service life of a

pavement family. The objectives of this study are to determine the average service life of

pavements and to predict their remaining service life. Remaining Service Life is defined

as the projected number of years until rehabilitation is required.


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The pavement condition data in the form of Pavement Condition Rating (PCR) were used

to develop Kaplan-Meier survival curves for different PCR thresholds. PCR 60 was

considered as the terminal condition and the average service life of pavement network

was calculated as the area under PCR 60 survival curve. Derived performance curves for

all the survival probabilities were developed between pavement age and PCR using the

Weibull approximation of the Kaplan-Meier survival curves. Derived performance

curves were employed to determine the remaining service life of individual pavements

based on current age and PCR. PCR curves were also developed for individual PCR

thresholds between RSL and pavement age by using the Weibull approximation of the

Kaplan-Meier survival curves to better understand the relationship between RSL, PCR

and pavement age. Average service life of the pavement network and remaining service

life of individual pavements obtained from this study can be used to assist in pavement

rehabilitation decision making and budget allocation.


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This thesis is dedicated to my mother, Tejovathi Perisetti, to my father, Late

Venkateswarlu Balla, and to my sister, Dr. Purnima Sobha Balla


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ACKNOWLEDGEMENTS

I would like to take this opportunity to express my deepest sense of gratitude to my

advisor, Dr. Eddie Yein Juin Chou for his invaluable guidance, motivation, constant

encouragement, tolerance, and financial support without which this dissertation could not

have this shape. I would also like to thank Dr. Azadeh Parvin, and Dr. George J.

Murnen for agreeing to be my committee members for their inputs, support, and

guidance. I would also like to acknowledge the City of Toledo for funding this study and

for providing rehabilitation data.

I would like to thank my colleagues and friends Debargha Datta, Dr. Haricharan

Pulugurta, and Praneeth Nimmatoori for their generous suggestions, inputs, and

encouragement. I would also like to thank all my friends for their support, special thanks

to Abdul, Amanesh, Anil, Ashok, Bivash, Ishan, Jatin, Jun, Madhura, Parth, Prabhu,

Shravan, Sri Hari, Shuo, Thihal, and Varun.

Finally, I would like to thank my mother and my sister; for their unceasing support,

morale, love, and encouragement they provided me in course of my thesis. They are

always been my moral support in every sphere of my life. I could not have made it this

far without them.


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TABLE OF CONTENTS

Abstract............................................................................................................................. iii

ACKNOWLEDGEMENTS ............................................................................................ vi

TABLE OF CONTENTS ............................................................................................... vii

LIST OF TABLES.............................................................................................................x

LIST OF FIGURES......................................................................................................... xi

1. INTRODUCTION..........................................................................................................1

1.1 Introduction..............................................................................................................1

1.2 Statement of Problem:.............................................................................................5

1.3 Objectives of the study: ...........................................................................................7

2. LITERATURE REVIEW .............................................................................................8

2.1 Pavement Management System..............................................................................8

2.2 Prediction Levels in Pavement Management: .......................................................9

2.3 Pavement Condition ..............................................................................................10

2.3.1 Factors That Could Affect Pavement Condition...............................................12

2.3.2 Treatment Type.................................................................................................12

2.3.3 Materials ...........................................................................................................12

2.3.4 Traffic Loading .................................................................................................13

2.3.5 Pavement Thickness..........................................................................................13

2.3.6 Climate..............................................................................................................14


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2.3.7 Condition Prior to Treatment ............................................................................14

2.4 Prediction Methods................................................................................................14

2.5 Remaining Service Life (RSL):.............................................................................15

3. DATA AND METHODOLOGY ................................................................................21

3.1 Introduction............................................................................................................21

3.3 Calculation of Pavement Condition Rating (PCR) .............................................25

3.4 Methodology ...........................................................................................................29

3.4.1. Survival Curve .................................................................................................29

3.4.2 Kaplan-Meier method .......................................................................................29

3.4.2.1 Example.................................................................................................................................30

3.4.3 Extrapolation of incomplete survival curve using Weibull distribution function34

3.4.3.1 Example.................................................................................................................................36

3.4.4 Derived Performance Curve .............................................................................39

3.4.5 Remaining Service Life ....................................................................................40

3.4.5.1 Example.................................................................................................................................40

4. RESULTS AND DISCUSSIONS................................................................................42

4.1 Introduction............................................................................................................42

4.2 Survival Curve .......................................................................................................43

4.3 Calculation of Survival Probability......................................................................44

4.3.1 Kaplan Meier method ....................................................................................45

4.3.2 Weibull approximation of Kaplan-Meier method.............................................49

4.4 Remaining Service Life .........................................................................................55

4.4.1 Median Remaining Service Life .......................................................................59

4.4.2 Remaining Service Life by PCR and Age ........................................................61


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4.5 PCR Curves ............................................................................................................62

4.6 Results .....................................................................................................................65

4.7 Conclusions.............................................................................................................67

5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ............................70

5.1 Summary.................................................................................................................70

5.2 Conclusions.............................................................................................................71

5.3 Recommendation ...................................................................................................72

5.4 Future Recommendations .....................................................................................73

APPENDIX A...................................................................................................................74

APPENDIX B ...................................................................................................................75

References :.......................................................................................................................76


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LIST OF TABLES

Table 3.1 Number of PCR collected Pavement Miles at different ages ......................24

Table 3.2 Pavement Lane Miles reached PCR 60......................................................31

Table 3.3 Calculation of Pt and S(t) for PCR 60...........................................................33

Table 3.4 Kaplan - Meier Survival Curve and Weibull Curve Data for PCR 60 ......38

Table 4.1 Pavement Lane Miles reached to each PCR Threshold at different ages..44

Table 4.2 Pavement Lane Miles that were not reached to each PCR Threshold at

any respective ages...........................................................................................................46

Table 4.3 Kaplan Meier Survival Curve Data, Calculation of tp and S (t) .............47

Table 4.4 Kaplan Meier Survival Curve Data for PCR 60 .......................................49

Table 4.5 Linear Regression Solution in Microsoft Excel............................................50

Table 4.6 Calculated S (t) values by using Weibull distribution .................................52

Table 4.7 Remaining Service Life...................................................................................57

Table 4.8 Pavement Age at different PCR values ......................................................58


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LIST OF FIGURES

Figure 2.1 Calculating RSL for an Individual Condition Index..................................20

Figure 3.1 Toledo City and Major Streets.....................................................................22

Figure 3.2 Pavement miles with rehabilitation data.....................................................23

Figure 3.2 PCR collected Frequency Plot of Pavement Age Vs Number of

Pavements/miles ...............................................................................................................25

Figure 3.3 Pavement Condition Rating (PCR) Scale.................................................28

Figure 3.4 Kaplan Meier, Survival Probability Vs Pavement Age for PCR 60 ......34

Figure 3.5 Weibull Survival Curve, Pavement Age Vs Survival Probability.............39

Figure 3.6 Example Figure to find Remaining Service Life at 6 years. ......................41

Figure 4.1 Kaplan Meier Survival Curve ...................................................................48

Figure 4.2 Weibull approximation of Kaplan Meier Survival Curve......................53

Figure 4.3 Average service Life of a pavement lane mile to reach PCR 60................54

Figure 4.4 Probable Life Curve ......................................................................................56

Figure 4.5 Derived Performance Curve for different percentile of pavement sections59

Figure 4.6 Calculating Median Remaining Service Life ...........................................60

Figure 4.7 Calculating Remaining Service Life by PCR and Age............................62

Figure 4.8 RSL variation for PCR 65..........................................................................63

Figure 4.9 Pavement Condition Rating Curves .........................................................64


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Figure 4.10 2009 PCR and RSL Miles ........................................................................65

Figure 4.11 Visual representation of 2009 PCR data for Toledo City .....................66

Figure 4.12 Visual representation of 2009 RSL data for Toledo City......................67


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CHAPTER 1

INTRODUCTION

This chapter introduces the need for the prediction of remaining service life for

pavements and states the objectives of the study.

1.1 Introduction

Transportation contributes to the economic, industrial, social and cultural development of

any country. It plays a vital role for the economic development of any region or nation,

since development of transportation facilities raises living standards, and improves the

aggregate community values. The major goal of any transportation system is the safe,

rapid, and convenient movement of people and goods from one place to another in order

to enhance economic activity and development (Gedafa D. B. 2008). In the United States,

transportation over the course of its historical development has been fundamentally

influenced and shaped by legislation (Gedafa D. B. 2008). Whereas technical advances

have made it possible to transport people and goods in a more efficient manner, major

improvements in the transportation industry have been shaped by the larger institutional

systematic frame work that determines present and future needs and seeks to give them

cost effective yet far-reaching solutions (Gedafa D. B. 2008). Because, human beings are

surrounded by three basic mediums i.e. land, water and air; the modes of transportation


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are also connected with these three mediums for the movement. Among the major modes

of transportation, transportation by road is the only mode, which provides maximum

service in terms of accessibility and mobility.

Roads are the dominant means of transportation in many countries today (Mitchell and

Maree 1994). As roads play an essential role in achievement of governments overall

social, economic, security, and developmental goals, much capital has been expended in

developing extensive road networks worldwide. The United States road network of

major highways includes almost four million miles of pavement (FHWA 1993). This

pavement network forms a significant portion of the national transportation infrastructure

and represents a cumulative investment of hundreds of billions of dollars over several

decades (Gedafa D. B. 2008). To preserve the investment spent on this huge network of

pavement, extensive maintenance and repair activities are necessary, with the intention of

using funds optimally. With a large network of highways in place, a highway engineers

concern is shifted from construction to maintenance (T.S. Vepa et al1996). It has been

said that one dollar invested in preventive maintenance at the appropriate time in the life

of a pavement can save $3 to $4 in future rehabilitation costs (Geoffroy 1996). For

facilitating the management of the existing network, pavement management systems

(PMSs) have evolved over the last three decades. With increasingly limited national

funds for transportation infrastructure preservation and renewal, there has been a growing

need for strategic management of the national pavement network to preserve this large

capital investment.


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Initially this strategic need led to the concept of increasing pavement life with the help of

Maintenance and Rehabilitation (M&R) activities. M & R activities are the activities that

are primarily concentrated on sealing surface cracks and potholes on pavement surfaces

so that they will be less likely to propagate further to endanger the stability of the

structure of the pavement (Joseph E. Ponniah et al 1996). The future performance of a

highway depends upon the suitability of the applied treatment, timing of treatment, and

quality of maintenance treatment it receives. Effective sealing of cracks and joints is

necessary to reduce the amount of water entering the pavement structure and causing

accelerated damage (Joseph E. Ponniah et al1996).

To address the problem of managing M&R activities, the Intermodal Surface

Transportation Efficiency Act (ISTEA) was passed in 1991. ISTEAs mandates include

the development and implementation of various infrastructure and monitoring systems;

pavement, bridge, highway safety, traffic-congestion, public transportation facilities and

equipment, and intermodal facilities and management systems. The goal is to optimize

available funds in preserving the national transportation infrastructure. Consequently, in

order to qualify for federal funds, states and their local jurisdictions were to implement

working infrastructure management systems, consisting of all seven mandated categories

(Amekudzi and Attoh-Okine 1996).

Proper management of the system requires the collection, analysis, and interpretation of

factual data relating to construction and maintenance activities. Prediction of the future


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condition of each pavement as well as that of the entire network is an essential element of

a management system (T.S. Vepa et al1996).

A Pavement management system requires prediction of pavement life. Pavement life can

be defined by two terms, service life and remaining service life. Service life can be

defined as a measure in years from construction to first rehabilitation or from the last

completed work to the next. Rehabilitation work may be defined as the reconstruction or

resurfacing of present pavement. Service life of a pavement is the time elapsed between

two successive constructions performed on a particular pavement. Both service life and

remaining service life of pavement can serve as tools for PMS. The purpose of remaining

service life of a pavement is to help pavement management system assessing pavements

current and projected condition, determine budget needs to maintain the average

condition of pavement above an accepted level, prioritize projects, and optimize spending

of maintenance funds. The evaluation of remaining service life is necessary to make

optimal use of the structural capacity of the in-service pavement. It simply represents the

useful life left in the pavement until a failure condition is reached. Knowledge of

remaining service life facilitates decision making in regard to strategies for

reconstruction-rehabilitation of roads, thereby leading to the efficient use of existing

resources. Prediction of remaining service life is important because prediction of future

pavement condition is one of the most important functions of a PMS, i.e. when the

pavement will reach its terminal condition which requires rehabilitation.


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The measurement and prediction of pavement condition is a critical element of any

pavement management system (PMS). Pavement condition rating (PCR), a composite

statistic derived from functional and structural conditions, is used as one measure of

serviceability (George et al1989).

Pavement serviceability or ride quality indices have also been widely applied to monitor

pavement performance and deterioration for pavement rehabilitation, design, and other

purposes. It is known that the ride quality or serviceability index of roads can be

explained mainly by the vertical jerk experienced by raters sitting in a moving vehicle

(Chiu Liu et al1998).

1.2 Statement of Problem:

Remaining service life (RSL) is the number of years that a pavement will be functionally

and structurally in an acceptable condition with only routine maintenance. This

combines severity and extent of different distresses and rates of deterioration. RSL also

requires development of a performance model and establishment of a threshold value for

each distress measurement. Based on the threshold value, current distress level, and

deterioration model, time for each distress to reach the threshold value can be computed

(Baladi 1991). Calculating remaining life has been a complex task, to say the least.

Existing methods rely on various concepts from purely empirical to truly mechanistic.

The lack of adequate performance prediction models has been the major impediment in

predicting remaining life (T.S. Vepa et al 1996). Calculating RSL has been a complex

task due to lack of adequate performance prediction models required for determining


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timing of the rehabilitation project. The failure of a pavement can be categorized as

structural or functional failure. In the functional failure-based approach, the remaining

life is computed on the basis of the performance of the pavement (for example,

serviceability or ride ability). A structural failure-based approach requires the structural

stability of pavement, such as pavement deflection data and visual condition data (PCR).

In the functional failure-based approach for estimating remaining life, the decrease in the

performance index with age or traffic is charted in conjunction with a functional failure

criterion. Alternatively, the structural failure-based approach makes use of fatigue

principles, which requires the effective thickness or modulus derived from in situ

measurements (T. S. Vepa et al1996).

The structural failure method and performance of the pavement (functional failure

method) requires historical data, which is not always available. Most pavements have the

current pavement condition data. Statistical models are based on data collected from test

roads located at diverse geographical locations. The LTTP test project is a rather extreme

example, with pavement sections monitored throughout the entire United States.

However, due to the enormous cost to construct and monitor pavements, the number of

LTPP sites in Ohio is rather limited. Therefore, a more practical and sensible approach is

to be developed in predicting remaining service life of pavement in a region. Since, it is

difficult to maintain the historical pavement performance data for each pavement section,

it is required to establish a performance model for individual pavement sections. Thus,

this study was initiated to assess the feasibility in predicting remaining service life by


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using survivor curves of groups of pavements based on its age and current condition

rating.

1.3 Objectives of the study:

The main objectives of this study are:

1. To develop a remaining service life (RSL) model using the survivor curve method.

2. Analyze the average service life of pavements from RSL obtained from the

survivor curve.

3.

To develop PCR curves to establish a relationship between pavement age,

condition rating, and remaining service life.


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CHAPTER 2

LITERATURE REVIEW

This chapter describes the past research work that was performed to predict the remaining

service life of pavements and the possible method that can be used for the current dataset.

2.1 Pavement Management System

The pavement management system (PMS) was first conceived in the late 1960s to 1970s

as a result of pioneering work by Hudson et al (1968)and Finn et al(1977)in the United

States, and by Haas (1977) in Canada. AASHTO (1990) defines PMS as follows: A

PMS is a set of tools or methods that assist decision makers in finding optimum strategies

for providing, evaluating, and maintaining pavements in a serviceable condition over a

period of time. The products and information that can be obtained and used from a PMS

include planning, design, construction, maintenance, budgeting, scheduling, performance

evaluation, and research (Hugo et al 1989; AASHTO 1990). The goal of a PMS is to

yield the best possible value for available funds in providing and operating smooth, safe,

and economical pavements (Lee and Hudson 1985). The functions of a PMS is to

improve the efficiency of decision making, to expand the scope and provide feedback on

the consequences of the decisions, to facilitate coordination of activities within the


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agency, and to ensure consistency of decisions made at different levels within the same

organization (Haas et al 1994). A PMS provides a systematic, consistent method for

selecting maintenance and rehabilitation (M&R) needs and determining priorities and the

optimal time of repair by predicting future pavement conditions (Shahin 2005).

2.2 Prediction Levels in Pavement Management:

To determine the direction and specificity of project development and planning, decisions

can be carried out at two management levels depending on the choice of the decision

maker. Those two management levels are network level and project level (Panigrahi

2004).

Network-level management focuses on determination and allocation of funds to maintain

pavement above a specified minimum operational standard. So, at the network level,

prediction model uses include condition forecasting, budget planning, inspection

scheduling, and work planning. One of the most important network uses of prediction

models is to conduct what if analyses, to study the effects of various budget levels on

future pavement condition (Shahin 1994).

Project-level management decides which specific road to repair, and the timing and

method of repair. So, prediction models at the project level are used to select specific

rehabilitation alternatives to meet expected traffic and climatic conditions (Shahin 1994).

Detailed consideration is also given to alternative conditions, M&R assignments, and unit

costs for a particular section of project within the overall program. This level of


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management involves assessing causes of pavement deterioration, determining potential

solutions, assessing effectiveness of alternative repair techniques, and selecting solution

and design parameters. The purpose of project level management is to provide the most

cost-effective feasible original design, maintenance, and rehabilitation or reconstruction

strategy possible for a selected section of pavement for the available funds (AASHTO

2001).

Different management levels will need different condition prediction models. Since the

main purpose of network-level management is to maintain the overall road network

above a specified minimum operational standard with limited budget, it does not focus on

how a specific road deteriorates. Therefore, a survival time analysis based on historical

condition data is often employed to predict the remaining service life of pavement. The

need to reasonably allocate funds requires the factors that affect pavement deterioration

be considered. Such consideration can be accomplished by introducing these factors as

parameters in prediction models.

For any pavement management system, prediction of pavement condition is the first and

foremost thing to be determined. Pavement condition can be defined by various indices.

2.3 Pavement Condition

Pavement condition is a generic phrase to describe the ability of a pavement to sustain a

certain level of serviceability under given traffic loadings. It is usually represented by

various types of condition indices such as Present Serviceability Index (PSI), Present


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Serviceability Rating (PSR), Mean Panel Rating (MPR), Pavement Condition Index

(PCI), Pavement Condition Rating (PCR), Ride Number (RN), Profile Index (PI), and

International Roughness Index (IRI). These indices can be classified into two categories:

roughness-based and distress-based.

Roughness is defined as the variation in surface elevation that induces vibrations in

traversing vehicles in ASTM E867. It has been recognized as an important measure of

road performance since the 1940s and can be measured using either direct or indirect

methods (Huang 1993). Several commonly used roughness measures are IRI, RN, and PI.

Distress-based condition ratings, for example, PCI and PCR, evaluate the comprehensive

condition of a road by categorizing a pavements surface distresses by type, frequency,

and extent. Each distress is manually inspected for representative pavement sections. A

score is assigned to each distress found according to its frequency and severity.

Distresses are weighted according to their importance to the pavement. A PCI or PCR

pavement condition is obtained by subtracting the sum of all distresses from 100 (Shahin

1994). Thus, both PCI and PCR are numerical ratings of the pavement condition that

range from 0 to 100, with 0 being the worst possible condition and 100 being the best

possible condition. The Ohio Department of Transportation has been employing PCR as

the condition index for its highway systems since 1985 (Morse and Miller 2004).


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2.3.1 Factors That Could Affect Pavement Condition

There are a number of variables that affect the deterioration of pavements. These major

factors affecting pavement performance are considered in pavement design procedures.

For accurate prediction, the same factors should be considered in condition prediction

models (Lytton 1987). These factors include treatment type, materials, traffic loading,

pavement structure, climates and pavement condition prior to the treatment. In practice,

the choice of factors is also limited by data availability. Prediction models can only be

developed based on available data. Those factors are discussed briefly in the following

sections.

2.3.2 Treatment Type

Various treatments can be performed on a pavement. Major treatments usually are new

construction and minor treatments are overlays on existing pavement. There are three

types of new constructions: rigid, flexible, and composite. Rigid pavement uses concrete

as the main pavement material. Flexible pavement uses asphalt concrete as the main

surface material. Composite pavement typically consists of asphalt overlays on an

existing rigid pavement. Each treatment type is considered to have its own deterioration

behavior and forms a unique pavement family.

2.3.3 Materials

Pavement performance is affected by material characteristics. Pavements with the same

design structure in different geographical areas may have different performances due to

the following reasons: (1) different specifications may be applied during design; (2) the


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aggregate type and its mechanical property may vary regionally; (3) the subgrade

modulus may vary from section to section. In addition, the construction quality may also

affect the strength and durability of a structure. These factors should be considered not

only in the pavement design but also in condition prediction. However, for this study,

very limited information about materials is available. Thus analysis of their effects on

pavement performance cannot be performed in this study.

2.3.4 Traffic Loading

Three types of traffic data; Average Daily Traffic (ADT), Average Daily Truck Traffic

(ADTT), and Equivalent Single Axle Load (ESAL) are usually collected by

transportation agencies. ESAL value converts all traffic into an equivalent damage done

by the passing of a single 18,000-pound axle. Since higher traffic loadings will cause

more damage to the pavement, design procedures generally account for increased traffic

loading with increased pavement thickness. The effect of traffic loading on pavement

performance should be considered whenever appropriate.

2.3.5 Pavement Thickness

Pavement thickness is a major factor that could affect pavement performance. The

thickness of a treatment is usually determined by specific design procedure and should be

considered in pavement condition prediction models.


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2.3.6 Climate

Temperature, snowfall, and precipitation affect pavement performance as pavement

materials may deteriorate faster in more severe climatic conditions. Climate effects

should be included in pavement condition prediction models.

2.3.7 Condition Prior to Treatment

Pavement condition prior to rehabilitation may affect pavement performance. It may be

hypothesized that pavements in better condition prior to overlay may perform better than

those pavements with worse prior condition. Pavement condition rating (PCR), which

accounts for various distresses, represents the overall pavement condition. STRD, which

stands for structural deduct, is an indicator of the overall remaining structural capacity of

a pavement. These two quantities may be considered during condition prediction

modeling.

2.4 Prediction Methods

Methods for predicting pavement conditions can be generally classified into three

categories according to the format of the mathematical representation: deterministic,

probabilistic and other methods such as neural network method.

Deterministic regression is perhaps the most popular prediction model in pavement

condition prediction studies. It is usually expressed as a regression equation with the

dependent variable being the condition index and independent variables being the age of

the pavement, pavement type, and other influential factors. Several regression equations


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might be developed - one for each family of pavements. A family is a group of

pavements that have similar characteristics and thus are expected to deteriorate similarly.

According to the need, family determination can be subjective or based on potential

explanatory variables such as pavement type, repair alternative, and traffic loading

(Shahin 1994). Linear and non-linear regression analysis is often used in developing

deterministic prediction models (Lytton 1987). Power curve (Chan et al 1997) and

sigmoidal curve (Sadek et al 1996) are the most popular non-linear regression formats in

predicting pavement conditions. B-spline approximation was also employed to seek

potential improvements for condition prediction (Shahin et al 1987). However, most

non-linear models used in pavement condition prediction can be converted into linear

models by variable transformation (Laird and Ware 2004). Prior knowledge of the

factors that affect performance is essential in developing reasonable empirical models.

Unlike deterministic models, probabilistic models predict the pavement condition with

certain probability. The result from such a model is usually a probability distribution but

not a fixed number. A probabilistic model can easily take the previous condition into

account for the current condition prediction (Lytton 1987). Thus, it has some advantages

over a deterministic model especially for overlays on an old pavement.

2.5 Remaining Service Life (RSL):

The remaining service life (RSL) is the anticipated number of years that a pavement is in

acceptable condition to accumulate enough functional or structural distress under normal

conditions, given that no further maintenance is performed or distress points equal to an


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as defined threshold value (Baladi 1991). RSL is calculated from the condition of the

asset during that year and the projected number of years until rehabilitation is required.

Once RSL is estimated for each pavement section in the network, the sections are

grouped into different categories (Dicdican et al 2004). It combines the severity and

extent of different distresses and the rate of deterioration. It requires development of a

performance model and establishment of a threshold value for each distress type. Based

on these threshold values, the current distress level and deterioration model for each

particular distress, and time for each distress to reach the threshold value, can be

computed. The shortest of these time periods is the RSL of the pavement section (Baladi

1991). The definition of the threshold values depends on the criteria used to control long-

term network conditions (Kuo et al 1992). Existing methods rely on various concepts

from purely empirical to truly mechanistic. Lack of adequate performance prediction

models has been the major impediment in predicting remaining life (Vepa et al1996).

Remaining service life (RSL) can be estimated in many different ways. Some researchers

tried to estimate pavement remaining service life from fatigue test (Witczak and Bell

1978, Carson and Rose 1980, Huang 1993, McNerney et al 1997). Other researchers

correlated the number of punchout failures per mile to the remaining ESALs,

equivalently the remaining life, for continuously reinforced concrete (Dossey et al1996).

Artificial Neural Network was also applied in estimating the RSL by researchers in Texas

(Ferregut et al 1999, Abdallah et al 2001). In 1986, AASHTO proposed an important

method to estimate the RSL for overlay design. In this method, the RSL of the existing

pavement is estimated using the Non-destructive Test (NDT). The current pavement


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layer elastic modulus is back-calculated from the deflection data. Then, the existing

pavement condition is related to its initial structural capacity by a condition factor, Cx.

The RSL of overlaid pavements, which is expressed as a function of the value of C x, was

calculated based on the projected future traffic applications and the ultimate number of

repetitions to failure time (Zhou et al1989). The advantage of this mechanistic method is

that historical traffic data are not required. A main drawback of this method is that it

requires the back calculation of the subgrade moduli, which is highly variable; therefore,

a very large number of deflection data would be required.

Yet another commonly used method to estimate the RSL of a pavement is to use the

performance regression model. By predefining the terminal condition, for example, PCR

of 70, it is possible to back calculate the age to reach that condition. Then, the RSL is

determined by subtracting the current age from the back calculated age.

The most popular method to estimate RSL is the survival time analysis, which is

considered a probabilistic model. This method was employed to obtain the RSL for

pavements in the United States as early as in 1940s (Winfrey and Farrell 1941).

Survival curves were developed for pavements built in each calendar year from 1903 to

1937 in 46 states using the life table method. According to Winfrey and Farrell (1941),

the distribution of survival times is divided into a certain number of equal intervals, e.g. 1

year or half a year. For each interval, the mileage of pavement sections still in service at

the beginning of the respective interval, the mileage of pavement sections that were out

of service at the end of the respective interval, and the mileage of pavement sections that


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were lost (for example, a road was completely out of service) during the respective

interval are counted. Survival probability of each interval is calculated by dividing the

remaining mileage by the total mileage entered for the respective interval. Survival curve

is formed by drawing the probability versus the time interval in a chronological order.

The remaining service life can be estimated by extrapolating the survival curve to zero

percent survival. The life table method has been extensively used in the analysis of

pavement RSL (Gronberg et al1956, Winfrey and Howell 1967).

The Kaplan-Meier method, which is also called the product-limit method (Kaplan et al

1958) is another procedure often used to generate survival curves. In the Kaplan-Meier

method, the probability of survival to time t is expressed as the product of the survival of

each year till time t. The Kaplan-Meier method and the life table method are identical if

the intervals of the life table contain at most one observation.

Survival curves method, which assumes an underlying failure distribution of the data, is

an alternative to analyze RSL (Prozzi and Madanat 2000). Because survival function is

now expressed explicitly in terms of a certain parametric distribution function, it is

possible to estimate the coefficients of those parameters, or in other words, the effects of

influential factors. However, the need to assume the underlying distribution introduces

another problem, that is, the shape of the data may not be described by a known

distribution. This is the major limitation of this method.


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RSL is used for future planning and budgeting purposes. This is not only useful for

timing a major rehabilitation but also assists managers in forecasting long-term needs of

the network. The evaluation of RSL is necessary to make optimal use of the structural

capacity of in-service pavements. Knowledge of RSL facilitates decision making in

regard to strategies for reconstruction-rehabilitation of roads, thereby leading to efficient

use of existing resources (Vepa et al1996). Accurate RSL models improve the process

of allocating funds and resources for maintenance and rehabilitation of asphalt pavements

(Romanoschi and Metcalf 2000).

To calculate RSL for a pavement section, the agency needs its current condition, a

definition of serviceable condition, and a mechanism to predict deterioration of the

pavement condition. Figure 2.1 shows the information required to calculate RSL.


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Figure 2.1 Calculating RSL for an Individual Condition Index

Threshold

Value

Condition Index

Present Condition

Serviceable

Condition

Performance Curve

Time (Years)

Remaining Service Life


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CHAPTER 3

DATA AND METHODOLOGY

This chapter describes the data used and the methodology adapted to predict the

remaining service life of pavements.

3.1 Introduction

Monte Carlo (or stochastic) modeling techniques have long been used for exploring the

impact of uncertainty. In its purest sense, Monte Carlo simulation employs a

mathematical model that interjects randomness between limits to determine a

probabilistic or likely outcome. Typically, this result is in the form of a probability

distribution, the shape of which lends insight into what is likely to occur if the modeled

course of action is pursued.

In this chapter data used in this study is described and the methodology used to predict

the remaining service life is also discussed.


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3.2 Data

The City of Toledo is divided into six districts. Its road system has a total of 1121 miles

and is divided into Major Streets and Residential Streets according to their importance,

location and traffic carried. There are a total of 356 miles of major streets and 765 miles

of residential streets. Major streets of Toledo are further divided into state routes and

county routes.

Figure 3.1 Toledo City and Major Streets

The data used to demonstrate the methodology are detailed pavement condition and

project history of Major streets of Toledo. Detailed Pavement condition is obtained in


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the form of Pavement Condition Rating (PCR) from the Ohio Department of

Transportation (ODOT) for both state routes and county routes. ODOT has maintained

state routes Pavement Condition Rating data and project history data since 1985 and

county routes data for every alternate year since 2003 i.e. for years 2003, 2005, 2007 and

2009. For county routes, the City of Toledo maintains the project history data. Figure

3.1 shows the City of Toledo and its Major streets.

In order to know the PCR data variation with age the pavement age must be known. Out

of 356 miles of major streets pavement rehabilitation data is available for 199 miles.

Figure 3.2 shows the number of unique miles with the rehabilitation data in major streets

category of City of Toledo.

157 Miles,

45%199 Miles,

55%

Rehabilitated

Others

Figure 3.2 Pavement miles with rehabilitation data


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In order to know the PCR data variation with age we need to know the pavement age.

This is possible by taking the construction year for a particular pavement as zero. Table

3.1 shows the tabulated number of pavement miles with PCR data at each age.

Table 3.1 Number of PCR collected Pavement Miles at different ages

Age Miles

0 79

1 74.92 69.7

3 69.2

4 70.9

5 52.2

6 58.8

7 43.6

8 65.5

9 60.6

10 66.611 55.5

12 51.3

13 44.7

14 34

15 32.1

16 26.1

17 21

18 13.6

19 6.2

20 5.1

21 2.4

22 1.3

23 1.2

24 0.6

25 0.6


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Figure 3.2 shows the frequency plot of number of pavement miles with PCR data

according to their age.

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Pavement Age

NumberofPavem

entMiles

Figure 3.2 PCR collected Frequency Plot of Pavement Age Vs Number of

Pavements/miles

3.3 Calculation of Pavement Condition Rating (PCR)

Pavement Condition Rating (PCR) is a distress based rating which evaluates the

comprehensive condition of a road by categorizing 14 different distresses: raveling,

bleeding, patching, surface disintegration / debonding, rutting, map cracking, base failure,

settlements, transverse cracks, wheel track cracking, longitudinal cracking, edge cracking,

pressure damage / upheaval, and crack sealing deficiency.


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The rating method is based upon the visual inspection of pavement distress. The rating

method provides a procedure for uniformly identifying and describing, in terms of

severity and extent, pavement distress. The mathematical expression for pavement

condition rating (PCR) provides an index reflecting the composite effects of varying

distress types, severity, and extent upon the overall condition of the pavement.

The model for computing PCR is based upon the summation of deducts points for each

type of observable distress. Deduct values are a function of distress type, severity, and

extent. Deduction for each distress type is calculated by multiplying distress weight

times the weights for severity and extent of distress. Distress weight is the maximum

number of deductible points for each distress type. The mathematical expression for PCR

is as follows.

PCR = 100 - =

n

I

iDeduct1

Eq (3.1)

Where, n = number of observable distresses, and

Deduct = (Weight for distress) (Weight for severity) (Weight for Extent)

The Appendix A & Appendix B describe various distresses for local flexible pavement

adopted by ODOT for establishing their severity and extent. Three levels of severity

(Low, Medium and High) and three levels of extent (Occasional, Frequent, and Extensive)

are defined.


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To illustrate the method for calculating PCR, consider the distress raveling in a

hypothetical local asphalt pavement. If the severity of this distress in the pavement is

Medium and extent is Frequent, then, the deduct points for Raveling in the

pavement would be equal to [(10) (0.6) (0.8)] or 4.8. If an extensive amount of medium

severity Surface Disintegration is also observed the deduct points for this distress

would be equal to [(5) (0.6) (1)] or 3.0. According to equation 3.1, PCR for the pavement

based upon these two distresses would be equal to [100 (4.8+3.0)] or 92.2.

To know the pavement behavior with age, the PCR values must be plotted according to

age of the pavement. To get the age of pavement the construction year of the pavement

must be known. After subtracting the latest constructed year from the present year the

age of the pavement can be obtained. In the current study, construction year of the

pavements in City of Toledo was obtained from Ohio Department of Transportation

(ODOT) and City of Toledo. Table 3.1 shows the number of pavement sections

reconstructed in each year.

Figure 3.3 illustrates the PCR scale adopted in this study and the descriptive condition of

a pavement associated with the various ranges of the PCR values. The scale has a range

from 0 to 100; a PCR of 100 represents a perfect pavement with no observable distress

and a PCR of 0 represents a pavement with all distress present at their High levels of

severity and Extensive levels of extent.


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Figure 3.3 Pavement Condition Rating (PCR) Scale


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3.4 Methodology

This section describes the method used to predict the remaining service life (RSL) for the

pavement data described in section 3.2.

3.4.1. Survival Curve

Changes in the pavement age can be described by using survival curves. Survival curves

give the percentage of pavement sections that last a certain number of years before a

terminal event. Survival curves can be constructed for an individual pavement or a

pavement population. An individual survival curve plots the probability that an

individual pavement section will remain in service as a function of age. A population

survival curve plots the fraction of pavement population that remains in service as a

function of age. When a pavement section fails, then it will be out of the system. The

height of the curve begins at one and declines as age and time increase. The slope of the

curve depends on the rate of pavement failures, with steeper decreases in periods of

higher rate of pavement failures.

3.4.2 Kaplan-Meier method

In this study, the Kaplan-Meier method is used to calculate the survival probability using

available historical PCR data. According to Kaplan-Meier, the survival probability at a

given year is a multiplication of each of the preceding years conditional probability of

failure. The conditional probability of failure in a particular year is calculated by

dividing the number of failures occurring in that year by the number of pavements at risk


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of failure at the beginning of that year. The conditional probability tp of surviving

th

t year after having survived 1t years is calculated as:

yearofbeginingfailureofriskatpavementsofNumber

year1andyearintervalinfailedpavementsofNumber1

th

thth

tt

ttp

+= , Eq (3.2)

The probability of survival to time t, )(tS is calculated as:

tppptS = ...)( 21 . Eq (3.3)

The graph of )(tS versus the tgives the Kaplan-Meier survival curves. In the current

study, survival analysis is used to determine: (1) average age of pavement network before

its terminal condition; and (2) the remaining service life of pavements.

3.4.2.1 Example

Let the PCR of 60 be the criterion for failure of a pavement. That means when a

pavement reaches PCR of 60, that pavement has failed and will be out of the system.

Table 3.2 shows the number of miles of pavements in City of Toledo that reached PCR

60 at each age.

Total number of lane miles of pavements in the data set is 199.02. If in year 6, the

number of lane miles of pavements falling to a PCR 60 is 1.40 and the cumulative

number of lane miles of pavements that reached PCR 60 at this age is 2.65 miles (0.44

miles + 0.81 miles + 1.40 miles). That means, total number of lane miles of pavements

that have not reached PCR 60 at this age is 199.02 2.65 = 197.77.

Using equation 3.2, the conditional probabilitytp can be calculated as described below.


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year6ofbeginingfailureofriskatpavementsofNumber

year7andyear6intervalinfailedpavementsofNumber16 th

thth

p =

Table 3.2 Pavement Lane Miles reached PCR 60

Age

Pavement

Lane Miles

reached PCR

60

0 0.00

1 0.00

2 0.00

3 0.00

4 0.44

5 0.81

6 1.40

7 1.41

8 2.33

9 6.05

10 11.34

11 12.2512 17.05

13 17.34

14 15.86

15 14.96

16 16.23

17 17.97

18 9.61

19 3.03

20 3.9721 2.04

22 0.32

23 0.81

24 0.58

25 0.58

26 0.00

27 0.00


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77.197

65.216 =p = 0.99

Similarly,

37.19606.417 =p =0.99

Probability of survival to time t for year 6 can be calculated by using equation 3.3

== 99.00.10.10.10.10.10.1)6(S 0.99

By using the above illustration, S (t) values for all the pavement ages are obtained and are

given in Table 3.3.


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Table 3.3 Calculation of Pt and S(t) for PCR 60

Age

Pavement

Lane Miles

reached PCR

60

Pavement

Lane Miles

at risk of

failure

Pt S(t)

0 0.00 199.02 1.00 1.00

1 0.00 199.02 1.00 1.002 0.00 199.02 1.00 1.00

3 0.00 199.02 1.00 1.00

4 0.44 199.02 1.00 1.00

5 0.81 198.58 1.00 0.99

6 1.40 197.77 0.99 0.99

7 1.41 196.37 0.99 0.98

8 2.33 194.96 0.99 0.97

9 6.05 192.63 0.97 0.94

10 11.34 186.58 0.94 0.88

11 12.25 175.24 0.93 0.82

12 17.05 162.99 0.90 0.73

13 17.34 145.94 0.88 0.65

14 15.86 128.60 0.88 0.57

15 14.96 112.74 0.87 0.49

16 16.23 97.78 0.83 0.41

17 17.97 81.55 0.78 0.32

18 9.61 63.58 0.85 0.27

19 3.03 53.97 0.94 0.26

20 3.97 50.94 0.92 0.2421 2.04 46.97 0.96 0.23

22 0.32 44.93 0.99 0.22

23 0.81 44.61 0.98 0.22

24 0.58 43.80 0.99 0.22

25 0.58 43.22 0.99 0.21

26 0.00 42.64 1.00 0.21

27 0.00 42.64 1.00 0.21


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Figure 3.4 shows the plot of Kaplan-Meier curve for the data given in Table 3.3.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Pavement Age

SurvivalProbability,

S(t)

Figure 3.4 Kaplan Meier, Survival Probability Vs Pavement Age for PCR 60

3.4.3 Extrapolation of incomplete survival curve using Weibull distribution function

A Kaplan-Meier survival curve cannot be completed for the incomplete pavement

condition data. To get unbiased estimates from a stub survival curve, the tail of the

survival curve should be extrapolated to reach zero survival probability (Reilly 1998). A


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Kaplan-Meier survival curve can be extrapolated to zero survival probability by using the

Weibull survival function.

The Weibull survival function that is given by:

B

A

Age

eS(t)

= . Eq (3.4)

Where )(tS is the survival probability at an age,Age in years. A is the scale parameter

that determines the spread of the Weibull curve; and B is the shape parameter, which

determines the shape of the Weibull curve.

The parameters A and B are estimated by reducing the residual sum of squares. Then the

Weibull survival function becomes

B

A

Age

elnS(t)ln

= Eq (3.5)

B

A

AgetS

=)(ln Eq (3.6)

B

A

Age

tS

=

)(

1ln Eq (3.7)

B

A

Age

tS

=

ln

)(

1lnln Eq (3.8)

( ) ( ) ( )[ ]AAgeB

tSlnln

1lnln =

Eq (3.9)


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( ) ( )ABAgeBtS

lnln)(

1lnln =

Eq (3.10)

It is in linear form and is similar to,

y = mX + C Eq (3.11)

Where,

( )AgeX ln=

m = B

( ) ( ) ( ) mC

eAm

CAAmABC

=

=== lnlnln Eq (3.12)

( )

yy eey etSetS

etStS

y ===

= )(

)(

1

)(

1ln

1lnln Eq (3.13)

By using the linear regression option in Microsoft Excel, the solutions for B and C are

found and A obtained from the relation in equation 3.12.

3.4.3.1 Example

In Table 3.3, the calculated conditional probability of survival to time t, is shown by

using the Kaplan-Meier method described in section 3.4.2. This example shows how to

extrapolate the incomplete survival curve using Weibull distribution.

By using the X and y values from the equation (3.11) and equation (3.13) respectively,

the linear equation is solved by using the Regression option in Microsoft Excel as

illustrated below.

Tools > Data Analysis. Then regression option is selected in the pop up window.


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Regression analysis gives the result of the right hand side of the linear equation. This is

the y column in Table 3.4. Table 3.4 shows the calculated Weibull curve data for the

corresponding Kaplan-Meier survival curve data for PCR 60.

Weibull probability of survival to time, S(t) is found by using equation (3.13). Figure 3.5

shows the generated Weibull curve for the data and calculation shown in this example.

From Figure 3.5 it is observed that Weibull distribution function closely approximates the

Kaplan-Meier Survival Curve.


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Table 3.4 Kaplan - Meier Survival Curve and Weibull Curve Data for PCR 60

Kaplan - Meier Survival Curve Data Weibull Curve Data

AgePavement

Lane Miles

reached PCR

60

Pavement

Lane Miles

at risk of

failure

Pt S(t) ln(Age) y S(t)

0 0.00 199.02 1 1 1.00

1 0.00 199.02 1 1 0.00 11.94 1.00

2 0.00 199.02 1 1 0.69 -9.00 1.00

3 0.00 199.02 1 1 1.10 -7.28 1.00

4 0.44 199.02 1.00 1.00 1.39 -6.06 1.00

5 0.81 198.58 1.00 0.99 1.61 -5.12 0.99

6 1.40 197.77 0.99 0.99 1.79 -4.35 0.99

7 1.41 196.37 0.99 0.98 1.95 -3.69 0.98

8 2.33 194.96 0.99 0.97 2.08 -3.13 0.96

9 6.05 192.63 0.97 0.94 2.20 -2.63 0.93

10 11.34 186.58 0.94 0.88 2.30 -2.18 0.89

11 12.25 175.24 0.93 0.82 2.40 -1.78 0.84

12 17.05 162.99 0.90 0.73 2.48 -1.41 0.78

13 17.34 145.94 0.88 0.65 2.56 -1.07 0.7114 15.86 128.60 0.88 0.57 2.64 -0.76 0.63

15 14.96 112.74 0.87 0.49 2.71 -0.46 0.53

16 16.23 97.78 0.83 0.41 2.77 -0.19 0.44

17 17.97 81.55 0.78 0.32 2.83 0.07 0.34

18 9.61 63.58 0.85 0.27 2.89 0.31 0.26

19 3.03 53.97 0.94 0.26 2.94 0.54 0.18

20 3.97 50.94 0.92 0.24 3.00 0.76 0.12

21 2.04 46.97 0.96 0.23 3.04 0.96 0.07

22 0.32 44.93 0.99 0.22 3.09 1.16 0.04

23 0.81 44.61 0.98 0.22 3.14 1.35 0.02

24 0.58 43.80 0.99 0.22 3.18 1.53 0.01

25 0.58 43.22 0.99 0.21 3.22 1.70 0.00

26 0.00 42.64 1 0.21 3.26 1.87 0.00

27 0.00 42.64 1 0.21 3.30 2.03 0.00


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0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Pavement Age

S

urvivalProbability,

S(t)

Kaplan-Meier Survival CurveWeibull Curve

Figure 3.5 Weibull Survival Curve, Pavement Age Vs Survival Probability.

3.4.4 Derived Performance Curve

Derived performance curves were drawn between pavement age and PCR for different

survival probabilities for different PCR values of 95, 90, 85, 80, 75, 70, 65, and 60 by

using the Weibull approximation of Kaplan-Meier survival curves which was described

in sections 3.4.2 and 3.4.3.


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3.4.5 Remaining Service Life

Remaining service life (RSL) is another important parameter for network level pavement

management. Generally, a RSL distribution of a road network is constructed to analyze

the impact of M&R actions on the future condition, to optimize and prioritize the M&R

actions, to determine life-cycle cost, and to obtain the feedback on current M&R

strategies. A uniform RSL distribution is an indication of an ideal M&R policy. RSL is

defined as the amount of time in years from a specified time (usually the latest survey

year) to the year when the pavement reaches a threshold or requires the next treatment.

Since the pavements will be at different ages at the latest condition survey year, the RSL

changes at each age. According to Winfrey (1967) and Reilly (1998), the RSL is

determined as the ratio of area under the complete survival curve to the right of an age to

survival probability at that age. For example, the remaining service life of a pavement

that is currently x years old can be calculated as:

yearsat xprobablitySurvival

yearsxofrightthetocurvesurvivalunderAreaRSL =x . Eq (3.14)

3.4.5.1 Example

By using the Weibull survival curve data from Example 3.3.3.1 and using the Remaining

Service Life equation as given in equation 3.14 the remaining service life curve is

established.

By using the equation 3.14, Remaining Service Life for a pavement network to reach

PCR 60 at the age of 6 years can be established as explained below.


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years6atprobablitySurvival

3.6)Figureinarea(shaded

years6ofrightthetocurvesurvivalunderArea

RSL6 =

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Pavement Age

SurvivalProbabi

lity,

S(t)

Figure 3.6 Example Figure to find Remaining Service Life at 6 years.

Area under Weibull Survival Curve to the right of 6 years is 10.22 units and 0.99 is the

corresponding survival probability. From equation (3.14) remaining service life for

pavement network to reach PCR 60 at the age of 6 years is0.99

10.22i.e. 10.35 years.


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CHAPTER 4

RESULTS AND DISCUSSIONS

In this chapter, the remaining service lives (RSL) of the City of Toledo major streets were

determined by the derived performance curves as described in chapter 3.

4.1 Introduction

Remaining service life is an important parameter in making decisions regarding

pavement rehabilitation. As discussed in chapter 3, the Kaplan-Meir method with the

Weibull approximation can be used to determine the pavements remaining service life

by using survival curves. In this chapter, by using the data described in chapter 3,

remaining service life is estimated for an entire pavement network by taking PCR 60 as

the terminal condition and individual pavement RSL is estimated by using derived

performance curves. PCR curves were also derived from Weibull approximation of

Kaplan-Meier survival curves for individual PCR values between pavement age and RSL

to better understand the relationship between PCR, pavement age, and RSL.


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4.2 Survival Curve

In the current study, survival curves were developed for different PCR thresholds based

on the Kaplan-Meier method using available historical pavement rehabilitation data, and

PCR data described in chapter 3. Survival curves give the percentage of pavement lane

miles that last a certain number of years before a terminal event.

Using the method explained in section 3.4 for each individual PCR threshold, pavement

lane miles were separated based on their age and the details are given in Table 4.1 and the

frequency plot between pavement age and number of miles is given in Figure 4.1.


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Table 4.1 Pavement Lane Miles reached to each PCR Threshold at different ages

Pavement Lane Miles ReachedAge PCR

95

PCR

90

PCR

85

PCR

80

PCR

75

PCR

70

PCR

65

PCR

60

0 2.60 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1 17.82 1.40 0.54 0.00 0.00 0.00 0.00 0.00

2 22.29 5.88 3.01 0.00 0.00 0.00 0.00 0.00

3 52.03 20.86 13.29 5.04 1.71 0.00 0.00 0.00

4 55.03 38.18 19.76 10.95 4.56 1.85 1.06 0.44

5 50.17 40.14 27.50 8.53 4.91 2.58 0.81 0.81

6 54.41 47.74 37.38 19.85 9.13 3.94 3.67 1.40

7 44.20 39.80 36.34 24.99 14.82 7.87 2.46 1.418 65.33 59.93 54.10 47.74 35.83 23.10 11.86 2.33

9 60.58 59.45 57.50 51.00 37.13 28.52 14.20 6.05

10 65.89 65.07 61.16 54.67 47.64 34.33 17.27 11.34

11 55.22 55.22 54.59 52.49 45.99 36.68 24.16 12.25

12 51.97 51.97 51.76 49.86 46.15 38.18 28.62 17.05

13 44.28 44.28 44.07 43.20 41.67 37.96 26.50 17.34

14 33.54 33.54 33.32 33.07 31.68 30.17 26.47 15.86

15 31.96 31.96 31.96 31.78 29.57 27.77 23.53 14.96

1626.10 26.10 26.10 26.10 25.05 24.58 23.28 16.23

17 21.00 21.00 21.00 21.00 20.13 20.13 20.07 17.97

18 13.56 13.56 13.56 13.56 12.69 12.69 12.62 9.61

19 6.24 6.24 6.24 6.24 6.24 6.24 6.17 3.03

20 5.13 5.13 5.13 5.13 5.13 5.13 5.13 3.97

21 2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.04

22 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0.32

23 1.17 1.17 1.17 1.17 1.17 1.17 1.17 0.81

24 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58

25 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58

4.3 Calculation of Survival Probability

This section describes the calculation of survival probability for different PCR thresholds

by using the method described in chapter 3.


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4.3.1 Kaplan Meier method

In this study, the Kaplan-Meier method is used to calculate survival probability. In order

to calculate the survival probability according to Kaplan-Meier method, by using the data

shown in Table 4.1, conditional probability needs to be calculated. Conditional

probability at any age is the percentage of pavement lane miles that has not reached a

corresponding PCR threshold in a particular year. Before calculating conditional

probability it is necessary to calculate the number of lane miles of pavement length that

has survived by not reaching a particular PCR threshold. Table 4.2 shows the number of

pavement miles that were not reached to each PCR threshold at any respective age.

The Probability of survival to time S(t) is the product of present year conditional

probability and the previous year probability of survival to time.


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Table 4.2 Pavement Lane Miles that were not reached to each PCR Threshold at

any respective ages

Pavement Lane Miles Not ReachedAge PCR

95

PCR

90

PCR

85

PCR

80

PCR

75

PCR

70

PCR

65

PCR

60

0 199.02 199.02 199.02 199.02 199.02 199.02 199.02 199.02

1 196.42 199.02 199.02 199.02 199.02 199.02 199.02 199.02

2 178.60 197.62 198.48 199.02 199.02 199.02 199.02 199.02

3 156.31 191.74 195.47 199.02 199.02 199.02 199.02 199.02

4 104.28 170.88 182.18 193.98 197.31 199.02 199.02 199.02

5 49.25 132.70 162.42 183.03 192.75 197.17 197.96 198.586 0.00 92.56 134.92 174.50 187.84 194.59 197.15 197.77

7 0.00 44.82 97.54 154.65 178.71 190.65 193.48 196.37

8 0.00 5.02 61.20 129.66 163.89 182.78 191.02 194.96

9 0.00 0.00 7.10 81.92 128.06 159.68 179.16 192.63

10 0.00 0.00 0.00 30.92 90.93 131.16 164.96 186.58

11 0.00 0.00 0.00 0.00 43.29 96.83 147.69 175.24

12 0.00 0.00 0.00 0.00 0.00 60.15 123.53 162.99

13 0.00 0.00 0.00 0.00 0.00 21.97 94.91 145.94

14 0.00 0.00 0.00 0.00 0.00 0.00 68.41 128.60

15 0.00 0.00 0.00 0.00 0.00 0.00 41.94 112.74

16 0.00 0.00 0.00 0.00 0.00 0.00 18.41 97.78

17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 81.55

18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 63.58

19 0.00 0.00 0.00 0.00 0.00 0.00 0.00 53.97

20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50.94

21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 46.97

22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 44.93

23 0.00 0.00 0.00 0.00 0.00 0.00 0.00 44.61

24 0.00 0.00 0.00 0.00 0.00 0.00 0.00 43.8025 0.00 0.00 0.00 0.00 0.00 0.00 0.00 43.22

The conditional probability tp and probability of survival to time S (t), calculated by

using equations 3.3 and 3.4 and are given in Table 4.3.


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Table 4.3 Kaplan Meier Survival Curve Data, Calculation of tp and S (t)

PCR 95 PCR 90 PCR 85 PCR 80 PCR 75 PCR 70 PCR 65 PCR 60Age

Pt S(t) Pt S(t) Pt S(t) Pt S(t) Pt S(t) Pt S(t) Pt S(t) Pt S(t)

0 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1 0.91 0.90 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2 0.88 0.79 0.97 0.96 0.98 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

3 0.67 0.52 0.89 0.86 0.93 0.92 0.97 0.97 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00

4 0.47 0.25 0.78 0.67 0.89 0.82 0.94 0.92 0.98 0.97 0.99 0.99 0.99 0.99 1.00 1.00

5 0.00 0.00 0.70 0.47 0.83 0.68 0.95 0.88 0.97 0.94 0.99 0.98 1.00 0.99 1.00 0.99

6 0.00 0.00 0.48 0.23 0.72 0.49 0.89 0.78 0.95 0.90 0.98 0.96 0.98 0.97 0.99 0.99

7 0.00 0.00 0.11 0.03 0.63 0.31 0.84 0.65 0.92 0.82 0.96 0.92 0.99 0.96 0.99 0.98

8 0.00 0.00 0.00 0.00 0.12 0.04 0.63 0.41 0.78 0.64 0.87 0.80 0.94 0.90 0.99 0.979 0.00 0.00 0.00 0.00 0.00 0.00 0.38 0.16 0.71 0.46 0.82 0.66 0.92 0.83 0.97 0.94

10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.48 0.22 0.74 0.49 0.90 0.74 0.94 0.88

11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.62 0.30 0.84 0.62 0.93 0.82

12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.37 0.11 0.77 0.48 0.90 0.73

13 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.72 0.34 0.88 0.65

14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.61 0.21 0.88 0.57

15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.44 0.09 0.87 0.49

16 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.83 0.41

17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.78 0.32

18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.85 0.27

19 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.94 0.26

20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.92 0.24

21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.96 0.23

22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.99 0.22

23 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.98 0.22

24 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.99 0.22

25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.99 0.21

The graph of )(tS versus the tgives the Kaplan-Meier survival curves. Figure 4.1 shows

the Kaplan Meier survival curves generated for each PCR threshold by using PCR data

and is obtained by plotting the values given in Table 4.3.

One can observe that survival curve for PCR 60 is incomplete because of the incomplete

data. According to Reilly (1998), average service life is the service life of a group of


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pavements which is calculated as the area under the complete survival curve. In this case,

since the survival curve for PCR 60 is incomplete, the area under this survival curve is

infinity, which means the average service life of a particular pavement to reach PCR 60 is

infinity.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 5 10 15 20 25 30

Pavement Age

SurvivalProbability

60

65

70

75

80

85

90

95

Figure 4.1 Kaplan Meier Survival Curve

According to Reilly (1998), to get unbiased estimates from a Kaplan - Meier survival

curve, the tail of the survival curve should be extrapolated to reach zero survival

probability. The Kaplan-Meier survival curve can be fitted with a curve to extrapolate it

to zero survival probability. In the current study, the Weibull distribution function is

used to complete the curve.


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4.3.2 Weibull approximation of Kaplan-Meier method

Table 4.4 shows the Kaplan Meier survival curve data for pavement lane miles reaching

PCR 60.

Table 4.4 Kaplan Meier Survival Curve Data for PCR 60

Age

Pavement

Lane Miles

reached PCR

60

Pavement

Lane Miles

at risk of

failure

Pt S(t) ln(Age) ln(ln(1/S(t)))

0 0.00 199.02 1.00 1.00

1 0.00 199.02 1.00 1.00 0.002 0.00 199.02 1.00 1.00 0.69

3 0.00 199.02 1.00 1.00 1.10

4 0.44 199.02 1.00 1.00 1.39 -6.11

5 0.81 198.58 1.00 0.99 1.61 -5.07

6 1.40 197.77 0.99 0.99 1.79 -4.31

7 1.41 196.37 0.99 0.98 1.95 -3.88

8 2.33 194.96 0.99 0.97 2.08 -3.42

9 6.05 192.63 0.97 0.94 2.20 -2.74

10 11.34 186.58 0.94 0.88 2.30 -2.06

11 12.25 175.24 0.93 0.82 2.40 -1.61

12 17.05 162.99 0.90 0.73 2.48 -1.17

13 17.34 145.94 0.88 0.65 2.56 -0.83

14 15.86 128.60 0.88 0.57 2.64 -0.57

15 14.96 112.74 0.87 0.49 2.71 -0.34

16 16.23 97.78 0.83 0.41 2.77 -0.11

17 17.97 81.55 0.78 0.32 2.83 0.13

18 9.61 63.58 0.85 0.27 2.89 0.27

19 3.03 53.97 0.94 0.26 2.94 0.31

20 3.97 50.94 0.92 0.24 3.00 0.3721 2.04 46.97 0.96 0.23 3.04 0.40

22 0.32 44.93 0.99 0.22 3.09 0.40

23 0.81 44.61 0.98 0.22 3.14 0.41

24 0.58 43.80 0.99 0.22 3.18 0.42

25 0.58 43.22 0.99 0.21 3.22 0.43

26 0.00 42.64 1.00 0.21 3.26 0.43

27 0.00 42.64 1.00 0.21 3.30 0.43


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Table 4.3 also shows the values of ln (Age) and ln (ln (1/S(t))), which is useful in

calculating the scale parameters A and B in the Weibull survival function given in

equation 3.9.

Table 4.5 shows the linear regression solution for equation (3.11) and equation (3.13) by

using the method described in section 3.3.3.

Table 4.5 Linear Regression Solution in Microsoft Excel

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.996

R Square 0.99

Adjusted RSquare 0.99

StandardError

0.19Observations 17

ANOVA

df SS MS FSignificance F

Regression 1 67.95 67.95 1803.2 4.80E-17

Residual 15 0.57 0.04

Total 16 68.51

Coefficients

Standard

Error t Stat P-value

Lower

95%

Upper

95%

Lower

95.0%

Upper

95.0%

Intercept -11.94 0.24 -49.2 5.3E-18 -12.5 -11.42 -12.46 -11.42

X Variable 1 4.24 0.10 42.5 4.8E-17 4.03 4.45 4.03 4.45

By using the intercept and X variable from Table 4.4 and applying them in equations 3.11,

3.12 and 3.13, the scale parameters A, B and modified S (t) are obtained. Table 4.6

shows the calculated S (t) by using the parameters stated in Table 4.5.


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The parameters A and B for equation 3.4 are estimated by using the regression function

in Microsoft Excel. The Weibull survival function is:

4.24

16.73Age

eS(t)

= . Eq (4.1)


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Table 4.6 Calculated S (t) values by using Weibull distribution

Kaplan - Meier Survival Curve Data Weibull Curve Data

AgePavement

Lane Miles

reached PCR

60

Pavement

Lane Miles

at risk of

failure

Pt S(t) ln(Age) y S(t)

0 0.00 199.02 1 1 1.00

1 0.00 199.02 1 1 0.00 -11.9 1.00

2 0.00 199.02 1 1 0.69 -9.00 1.00

3 0.00 199.02 1 1 1.10 -7.28 1.004 0.44 199.02 1.00 1.00 1.39 -6.06 1.00

5 0.81 198.58 1.00 0.99 1.61 -5.12 0.99

6 1.40 197.77 0.99 0.99 1.79 -4.35 0.99

7 1.41 196.37 0.99 0.98 1.95 -3.69 0.98

8 2.33 194.96 0.99 0.97 2.08 -3.13 0.96

9 6.05 192.63 0.97 0.94 2.20 -2.63 0.93

10 11.34 186.58 0.94 0.88 2.30 -2.18 0.89

11 12.25 175.24 0.93 0.82 2.40 -1.78 0.84

12 17.05 162.99 0.90 0.73 2.48 -1.41 0.78

13 17.34 145.94 0.88 0.65 2.56 -1.07 0.71

14 15.86 128.60 0.88 0.57 2.64 -0.76 0.63

15 14.96 112.74 0.87 0.49 2.71 -0.46 0.53

16 16.23 97.78 0.83 0.41 2.77 -0.19 0.44

17 17.97 81.55 0.78 0.32 2.83 0.07 0.34

18 9.61 63.58 0.85 0.27 2.89 0.31 0.26

19 3.03 53.97 0.94 0.26 2.94 0.54 0.18

20 3.97 50.94 0.92 0.24 3.00 0.76 0.12

21 2.04 46.97 0.96 0.23 3.04 0.96 0.07

22 0.32 44.93 0.99 0.22 3.09 1.16 0.0423 0.81 44.61 0.98 0.22 3.14 1.35 0.02

24 0.58 43.80 0.99 0.22 3.18 1.53 0.01

25 0.58 43.22 0.99 0.21 3.22 1.70 0.00

26 0.00 42.64 1 0.21 3.26 1.87 0.00

27 0.00 42.64 1 0.21 3.30 2.03 0.00


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Figure 4.2 shows the aforementioned Weibull approximation to the Kaplan Meier

survival curves for different PCR thresholds such as 95, 90, 85, 80, 75, 70, 65, and 60. It

can be seen that the Weibull fit shows more reasonable estimates to the survival

probabilities as it closely follows the Kaplan Meier survival curve.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Pavement Age, Years

SurvivalProbability

95

90

85

80

7570

65

60

95-St

90-St

85-St

80-St

75-St

70-St

65-St

60-St

Figure 4.2 Weibull approximation of Kaplan Meier Survival Curve

The R-square ( 2R ) is a statistical term expressing how good the regression equation is at

predicting the dependent variable. If2

R is 1.0 then given the value of one term, you can

perfectly predict the value of another term. If 2R is 0.0, then knowing one term doesn't

not help you know the other term at all.


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2R is most often used in linea

prediction of remaining service life of pavements.pdf

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