CHAPTER 1INTRODUCTION:

1.1. General:

Prediction of multi-layered pavement performance under the combined action

of highway traffic and environmental conditions provides valuable information to the

highway agencies. This information is very useful for proper planning of maintenance

and rehabilitation activities, budget estimation and allocation of resources [Prozzi &

Madanat, 2004]. For Pavement Management System (PMS), which is regarded as an

integral input part of infrastructure management, and defined by Guyilliamaumot et al.

(2003) as “ the process through which agencies collect and analyze data about

infrastructure systems and make decisions on the maintenance, repair and reconstruction

of facilities over a planning horizon”, pavement failure is a highly variable event which

not only depends upon material properties, environmental and subgrade conditions and

traffic loading, but also on the specific definition of failure adopted by the agencies.

Again, performance models dealing with pavements (which are supposed to last for the

entire service life) have to be developed as risk based models to deal with the random

process of failure and the uncertainty [Sanchez et al. 2005]. With to regard PMS,

pavement performances have to be accessed separately for both the following

requirements:

1. Functional performance: a subjective measure of the quality of the riding

conditions of the road from the users’ point of view.

2. Structural performance: a more objective measure which takes into account

the appearance of various forms of distress such as cracking, rutting, raveling,

faulting etc.

Correlating these two performance criteria is one of the most coveted

breakthroughs of performance prediction process and its subsequent action plan. This not

only would give PMS a cost optimum solution but also increase the sustainability of the

pavement.

Now, regarding the process of design of pavement, the Empirical design

approach and the Mechanistic-Empirical design approach are the two basic approaches.

Empirical design approach is based solely on AASHO road test results conducted way

back in 1950’s .These test results were used to establish a correlation between input and

the outcome of the process - e.g., pavement design and performance. But these

relationships do not have firm scientific basis (e.g. - structural analysis of pavement

layers) and their validity is vary much limited. On the other hand, the M-E design

approach is based on estimation of critical stress-strain, obtained from structural analysis

of pavement and linking this mechanistic response to an observed distress by empirical

relationship

Fund allocations in PMS largely depend upon improved prediction of field

performance of design solution on a seasonal basis. For this, different distress criteria

have to be estimated considering their uncertainty. Moreover, design procedure should

provide a consistent pavement performance level considering inherent variability

associated with design input parameters. [Kim, 2006]

1.2. Scope and Objective of the Work:

The Mechanistic-Empirical (ME) approach of pavement design uses the

theory of mechanics to analyze the structural behavior of pavements under external

effects like traffic load, temperature effects etc. and correlate this mechanistic response to

the pavement distress observed in the field by empirical relationship. Hence, by use of

rational method this approach has largely remove the drawbacks of the age old Empirical

approach and at the same time it is capable of giving design solutions for different kinds

of field conditions in terms of new materials, construction procedures, changed traffic

characteristics viz. wheel load and axle type, new climatic conditions etc. [Hong, 2004].

But, from the literature review (presented in Chapter 5), it is apparent that still the ME

design approach has failed to give consistent field performance due to the inherent

variability in the design process.

From earlier woks it has been observed that past studies it has been found that

field behavior of pavements have been given priority in design analysis. Research efforts

have given to formulate them in accordance with the design input parameters in one way

or other. In this regard, use of the parameter PSI (Present Serviceability Index) was a

2

more subjective application. PSI estimates the combined effects of several failure criteria

(fatigue, rutting, roughness etc.).But, in the new ME method these distresses are

considered individually so that the design would stand on each criteria. Among all these,

traffic load induced fatigue distress is the predominating cause of pavement failure and

hence it has been considered as a major design criterion in many flexible pavement

design guidelines [Zhang et al., 2003]. Now, fatigue distress is more rationally described

as fatigue cracking which is evaluated in terms of percentage (%C) of cracked area with

respect to total lane area or wheel path area. The estimation of percentage fatigue

cracking (%C) of pavement on a time frame considering its design life is a much needed

provision the pavement management system as well as pavement design guideline.

However, in Indian context, not much work on prediction of pavement

performance by use of percentage fatigue cracking criteria has been reported. Majhi

(2003) and Ghosh (2005) have drawn up computational schemes to find the fatigue

reliability of a design section using the Mean value First Order Second Moment Method

of probabilistic estimation. However, in their work also the fatigue cracking has not been

addressed. Keeping the above in view, in the present thesis, an attempt has been made to

formulate a computational scheme to predict the fatigue performance of pavements

during their service life. Further, to investigate the relative degree of influence of various

uncertain design parameters on estimation of fatigue a sensitivity analysis has been done

as a part of this study.

1.3. Organizing the Thesis:

The present thesis is divided into 7 chapters. Chapter 1, Introduction,

discusses the basic framework of the study including the scope and motivation of the

work. Chapter 2 describes an overview of the pavement design methods with basic

principals of the study. Chapter 3 deals with the uncertainties associated with pavement

engineering along with different methods of reliability analysis. Chapter 4 depicts the

concept of fatigue distress in pavement. Chapter 5 presents a review on literature on

reliability analysis of pavements. Chapter 6 deals with probabilistic formulation of

fatigue damage. Finally, Chapter 7 presents a summary on the study with conclusions

and scope of further work.

3

CHAPTER 2. METHODS OF PAVEMENT DESIGN:

AN OVERVIEW

2.1. GENERAL:

The development of flexible pavement design process started way back in

1920s. At that time, design was consisted of finding the thickness of layered materials

that would provide sufficient strength and protection to a soft, weak subgrade. Pavements

were designed against subgrade shear failure. Depending upon the experience of success

and failures of previous projects several design methods were developed based upon the

subgrade shear strength criteria. [Carvalho, 2006]

The CBR method, developed during 1928-29 was the first of those empirical

methods which developed after the soil classifications were first published. It involves

determination of CBR value of subgrade for the most critical moisture condition and

subsequently finding the thickness of different pavement layers from a design chart.

Based upon that design chart, in 1940 the U. S. Corps of Engineers adopted the CBR

method of design for airfield pavement which is still being used for runways in airport.

During the same time California (Hveem) Method was developed which considered the

traffic load parameters along with strength of subgrade material and other construction

materials [Chakraborty & Das, 2003]. In 1945 the HRB modified the soil classifications

to categorize the same in 7 separate groups with indexes to differentiate soil within each

group. The classifications were applied to estimate the subbase quality and total

pavement thickness.

Several methods, based on subgrade shear failure criteria were developed

after the CBR method. Barber (1946) used Terzaghi’s bearing capacity formula to

compute pavement thickness, while McLeod in 1953, applied logarithmic spirals to

determine bearing capacity of pavement. This approach had been used in South African

pavement design method. But with the increase in traffic volume and vehicular speed,

shear failure no longer be the sole criteria of pavement design.

4

In 1947, Kanas State Highway Commission did structural analysis of soil by

theory of elasticity and applied Boussineq’s equation to limit the vertical deflection of

subgrade upto 2.54mm. Later in 1953, the U.S. Navy applied Burimisters’s two-layer

elastic theory and limited the surface deflection to 6.35 mm. More recently, resilient

modulus has been used to establish relationships between the strength and deflection

limits for determining thickness of new pavement structures and overlays (Preussler and

Pinto, 1984). [Carvalho, 2006]

Since, design criteria have changed. As important as providing subgrade

support, it became equally important to evaluate pavement performance through riding

quality and other surface distresses that increase the rate of deterioration of pavement

structure. Performance became the focus point of pavement design. Methods, based on

serviceability were developed using test track experiments. After 1950, test tracks started

to be used for gathering more data related to performance of pavement. Regression

models were developed linking the performance data to design inputs. The AAHTO

empirical design method (AASHTO, 1993), based on the AASHO Road Tests (1960), is

still the predominating design method of pavement. The AASHTO design equation co-

relates the pavement performance in terms of serviceability to repetition of traffic load &

pavement structural capacity. But, like any other empirical methods AASHTO design

method is valid only for selected material and climate condition in which they were

developed.

Meanwhile, new materials started to be used in pavement structure but with

their own failure mechanism (e.g. fatigue cracking and rutting in case of asphalt concrete)

due to traffic loading and environmental effects. Kerkhoven & Dorman (1953) first

suggested the use of vertical compressive strain on top of subgrade as a failure criterion

and Sall & Pell (1960) published the use of horizontal tensile strain at the bottom of

asphalt layer to minimize fatigue cracking. The AI method (1982, 1981) and Shell

method (1977, 1982) incorporated the strain-based criteria in linear-elastic theory and

predicted the no. of traffic loads to failure in combination with empirical models. Later

various state organizations in USA viz. WSDOT, NCDOT, MNDOT developed their own

M-E procedures. In 1990, the NCHRP 1-26 project report (Calibrated Mechanistic

Structural Analysis Procedures for Pavements) provided the basic framework of all these

5

efforts by incorporating environmental variables (e.g. asphalt concrete temperature to

determine stiffness) and cumulative damage model using Miner’s Law with fatigue

cracking criterion.

Vary recently published NCHRP 1-37A (2004) project report incorporates the

traffic load estimation in terms of load-spectrum of different vehicular class, a step

forward from ESAL concept and gives distinct distress model for traffic load and

environmental effect. It also gives the provision of pavement performance evaluation on

seasonal basis.

However, the success of M-E design practice, suggested by various agencies

depends upon the following factors [Kim, 2006]:

The accuracy of the pavement structural model to obtain primary responses of the

pavement.

The accurate characterization of the material properties in the different pavement

layers.

The provision for better characterization of climate and aging effects on materials.

The accuracy of load-spectrum data for a site specific condition.

Better definition of the role of construction by identifying the parameters which

are most influential over pavement performance.

The accurate characterization of the uncertainties in preparing design inputs.

The adoption of realistic approach for performance prediction of pavement.

The appropriate selection of reliability model to treat uncertainties of the design

inputs.

However in Indian context, IRC brought out its first guideline in 1971 (IRC:

37-1971) for structural design of bituminous pavements, which was subsequently revised

in 1984. These guidelines were based on empirical relationships, mentioned earlier. Later

on with the advancement of computational facility, available field performance data ,

analytical design approach has enabled to introduce M-E design concept (shown in Fig.

2.1) in recently published guideline IRC: 37- 2001. It provides a design thickness chart

from field performance and incorporating fatigue and rutting model from mechanistic

analysis.

6

Figure 2.1: Flow chart for Mechanistic Empirical Flexible Pavement Design (Carvalho

& Schwartz, 2005)

2.2. BASIC DESIGN PRICIPAL OF IRC: 37-2001:

In this approach, the pavement is idealized as a layered elastic structure

consisting of three to four layers made up of bituminous surfacing, base, sub-base and the

subgrade. Each layer is characterized by its elastic modulus (E), Poisson’s ratio () and

thickness. Figure.2.1 shows a layered flexible pavement structure subjected to a set of

standard dual wheel load system on top of the surface layer. The horizontal tensile strain

(t) at the bottom of the bituminous layer and the vertical compressive strain (z) on the

subgrade are identified as the critical parameters for fatigue and rutting failures,

respectively. The concept of fatigue failure has been discussed in chapter 4. The

mechanistic pavement design consists of the selection of a thickness combination of

asphalt concrete (AC) surfacing and granular base so that t and z are limited to the

predetermined values depending upon the design life of the pavement.

Using Burmister’s (1945) basic approach, Verstraeten (1967) presented

explicit equations in integral forms for evaluation of stress, strain, and displacement for a

7

layered elastic pavement subjected to a uniformly distributed vertical pressure on a

circular area on the surface of a pavement.

Some standard computer programs such as CHEVRON, BISAR, ELSYM,

EVERSTRESS, FPAVE, etc. had been developed by different organizations and

institutions for the computation of stress and strain at a given point in a multilayer

pavement structure.

8

Bituminous Surfacing

Granular base

Or

Subbase

t

z

Subgrade

Tyres

Figure 2.2: Schematic Diagram of a Layered Pavement Structure

Figure 2.3: Schematic Diagram Showing Development of Pavement Design Process

Over The Time

Boussinq’s Eqn

Terzaghi’s Eqn.

Mechanistic Analysis

Accelerated Test Tracks

ResilientModulus

Theory of Elasticity

Failure Criteria Analysis InputsDesign Approaches

9

Subgrade Shear Failure

Bearing capacity of subgrade

Vertical Deflection of subgrade

SubgradeStrength & Deflection

Riding Quality of pavement surface

Serviceability of Pavement

Cracking Rutting RavelingFriction loss

CBR Method

Empirical Model

ME Design Approach

CHAPTER 3 CONCEPT OF RELIABILITY:

3.1. GENERAL

Reliability emerges into the frame of technical analysis due to the inherent

uncertainty in engineering solutions associated with it. Uncertainty results undesirable

performance, unsafe operation, low standard of durability of the solution system. Now all

these uncertainties approach the system in different manners.

Generally uncertainties encountered in the engineering problem may be of

three types:

(i) Natural variability associated with the inherent randomness of natural

process, manifesting as temporal variability, spatial variability or as variability over both

time & space.

(ii) Knowledge uncertainty is attributed to lack of quality data, limited

information about the event or process , or lack of understanding of physical laws which

limit our ability to model the real world i.e., statistical uncertainty. Knowledge

uncertainty is just a more common description of epistemic uncertainty.

In addition to these uncertainties, two practical types of uncertainty also

enter risk and reliability analysis. These are Operational uncertainties which include:

(A) The unpredictability of (a) loads on a structure during its life, (b) in-

place material strengths and (c) human errors.

(B) Structural idealizations in forming the mathematical model of the

structure to predict its response or behavior and

(C) The limitations in numerical methods; and

Decision uncertainties, which describe our inability to know social

acceptability, the length of a planning horizon, desirable temporal consumption –

investment or the social aversion to risk.

The concerning uncertainties which have to deal in engineering analysis are

given in Figure.3.1.

10

Figure 3.1: Schematic Diagram Showing Various Types of Uncertainties

Now the reliability against all these uncertainties can be defined as the

probability of an item performing its intended function over a given period of time under

the operating conditions encountered. It is important to note that the above definition

stresses four significant elements, viz. (i) probability, (ii) intended function, (iii) time and

(iv) operating conditions. Because of the uncertainties, the reliability is a probability

which is the most important element in the definition. All uncertainties, weather they are

associated with inherent variability or with the prediction errors, may be accessed in

statistical terms, and the evaluation of their significance on engineering design

accomplished using concepts and methods that are embodied in the theory of probability.

For engineering purpose the dual concept of probability –as-frequency in a

long or infinite no. of trials and probability–as-belief objective or subjective are

complements to each other. Thus, probability is the over-arching framework for

grappling with the dual nature of uncertainty: probability-as-frequency is used to grapple

with natural variations in the world, while probability-as- belief is used to grapple with

limited knowledge

11

Knowledge Uncertainty Decision ModelUncertainty

Risk Analysis

Natural Variability

Temporal

Spatial

Model

Parameters

Time

Objectives

Values

3.2. SOURCES OF UNCERTAINTIES ASSOCIATED WITH

PAVEMENT DESIGN PROCESS:

Regarding the flexible pavement, the design factors have always some sort of

uncertainty either due to the dispersion of their values or errors associated with

estimation of these factors. An example of stochasticity is the lateral wander of traffic.

Since wheel paths of different vehicles are not identical, lateral distribution of wheel path

should be considered in formulating the design traffic [Sun Le]. Uncertainties associated

with key pavement input factors and models which affect pavement performance can be

grouped into the following four categories:

1. Spatial variability that includes a real difference in the basic properties of

materials from one point to another in what are assumed to be homogeneous

layers and a fluctuation in the material and cross-sectional properties due to

construction quality;

2. Variability due to the imprecision in quantifying the parameters affecting

pavement performance, i.e., random measurement error in determining the

strength of subgrade soil, and estimation of traffic volume in terms of average

daily traffic;

3. Model bias due to the assumption and idealization of a complex pavement

analysis model with a simple mathematical expression;

4. Statistical error due to the lack of fit of the regression equations.

Now the first sources of uncertainties can be combined into uncertainties of

design parameters, which represent the variability from site to site and inconsistent

estimation of the parameters, and the third and fourth sources of uncertainty into

systematic errors, which consistently deviate from predicted actual pavement

performance. The uncertainties of design parameters cause the variation within the

probability distribution of the performance function, whereas systematic errors cause the

variation in possible location of probability distribution of the performance function.

Therefore, design parameters describe the scatter of the pavement properties and the

12

variation of traffic estimation and systematic errors associated with the uncertainty in the

location of the trend of predicted pavement performances.

In early research the uncertainties were curbed down by using sufficient

safety factors and arbitrary taking decision based on experience. However the use of

safety factors, with little consideration given to the uncertainty of design factors has

resulted in few failures (Hudson 1975). In order to access the effects of uncertainty

comprehensively, probabilistic concepts need to be applied in an explicit, non arbitrary

way.

3.3. THEORY OF PROBABILITY

In engineering practice many random phenomena of interest are associated

with numerical outcomes; however in all phenomenons the outcomes may not be

numerical. Events of this type may also be identified numerically by artificially assigning

numerical values to each of possible alternative events. In any case, an out-come or event

may be identified through the value(s) of a function; such a function is a random variable,

which is usually denoted with a capital letter. The value (or range of values) of a random

variable then represents a distinct event; In short, a random variable is a device (cooked

up when necessary) to identify events in numerical terms.

3.3.1. Probability Distribution of a Random Variable

The rule for describing the probability measures associated with all the values

of a random variable is a “Probability Distribution” or “Probability Law”. If is a

random variable, its probability distribution can always be described by its Cumulative

Distribution Function (CDF), which is

for all x (3.1)

Here is a discrete or a continuous random variable. If is continuous,

probabilities are associated with intervals on the real line (since events are defined as

intervals on the real line); consequently, at a specific value of , such as , only the

density of probability is defined. Thus, for a continuous random variable, the probability

13

law may also be described in terms of a Probability Density Function(PDF),so that, if

is the PDF of ,the probability of in the interval [a, b] is

(3.2)

It is to be noted that itself does not give the probability. It is only a

measure of the density of probability at the point. Probabilities are given by integrals

only.

It follows, then, that the corresponding distribution function is

(3.3)

According, if has a first derivative, then,

is not a probability; is the CDF of

However,

(3.4)

is the probability that values of will be in the interval .

3.4. PROBABILITY DISTRIBUTIONS:

Probability distributions are used to describe the nature of uncertainty of a

random variable. These are derived on certain physical assumption and are the result of

an underlying physical process. However there is no. of discrete and continuous

probability function, some of them are mentioned in following section.

14

3.4.1. Normal (Gaussian) Distribution

Figure 3.2. Normal Density Function

(3.5)

= (3.6)

3.4.2. The Standard Normal Distribution

Figure.3.3. Standard Normal Distribution

For the special case of , and

-∞ 0 a μ b x

N (μ ,σ)

fX(x)

Area= P (a<X ≤b)

-3 -2 -1 0 sp 1 2 3 s(-3σ) (-2σ) (-σ) (σ) (2σ) (3σ)

N(0,1)

fS(s)

Probability=p

15

(3.7)

Shaded area,

Special Notation:

is the cumulative probability of a standard normal variate

is tabulated in normal distribution chart standard text books.

3.4.3 Lognormal Distribution

Figure 3.4. Lognormal Distribution

(3.8)

Where, and are, respectively, the mean and standard

derivation of , and are the parameters of the distribution.

Note: As , i.e., decreases

Lognormal →Normal

It can be shown that

ξ=0.1

ξ=0.3

ξ=0.5

0 1 2

xf X

4

2

0

16

(3.9)

& (3.10)

Now,

= (3.11)

Let,

Then , ,and

= (3.12)

Thus, probabilities associated with a log-normal variate can also be

determined using the table of standard normal probabilities.

3.5. METHODS OF RELIABILITY ANALYSIS:

3.5.1. Mean-value First Order Second Moment (MFOSM) Method

In this method, the random variables are characterized by their first and

second moments. In evaluating the first and second moments of the failure function (i.e.

say, the mean and variance of M which is a nonlinear function of the basic variables), the

first order approximation is used. That is why these methods are called first-order second-

moment methods (Ranganathan 1990). In the case of nonlinear failure functions,

linearization is performed using Taylor’s series expansion in the reliability analysis.

Consider the fundamental case with only two basic variables R and S:

pF = P (R S)

M = g(R, S) = R - S (3.13)

The failure surface equation is

17

R – S = 0 (3.14)

Cornell (1969) first defined the reliability index as

(3.15)

Where, M and M are the mean value and standard deviation of M. That is,

is the reciprocal of the coefficient of variation in M. The concept of is illustrated in

Figure 2.3 which shows the PDF of M for the fundamental case-two variable problem.

The safety is defined by the condition M 0 and therefore, failure by M 0. The

reliability index may be thought of as the distance from the origin (M = 0) to the mean M

measured in standard deviation units. As such, is a measure of the probability that M

will be less than zero. If

M = M 0 (3.16)

Then the reliability in terms of safety index is at least .

18 0 μM R - S 0 μM ln(r/s)

pF pF

fM(m)fM(m)

M=(R-S)M = ln (R/S)

M>0 M<0M<0 M>0

βμM

Figure 3.5. Concept of reliability index (a) M = R - S; (b) M = ln (R/S)

When both R and S are normal and independent,

(3.17)

(3.18)

So,

(3.19)

When both R and S are lognormal and independent, the alternative

formulation for failure is taken as

(3.20)

(3.21)

The failure surface equation is

(3.22)

Using the small variance approximations,

(3.23)

19

and (3.24)

So, (3.25)

It is to be mentioned here that the reliability index () defined by Equation

(3.19) is not invariant with regard to the choice of failure function. If the failure function

is linear in nature, the point of shortest distance (from the origin) to the failure surface

will be the mean point. However, for nonlinear failure function, it is not the same. In such

a situation mean point can be an approximation of the shortest distance. Thus the

computed value of with nonlinear failure function is approximate. Level of accuracy

depends on nonlinearity of the failure curve. Thus, for highly nonlinear failure function,

Hasofer and Lind (1974) method should be used for computing which is basically

solving an optimization problem for finding the position of the point of shortest distance.

3.5.2. Second Order Second Moment method (SOSM)

This technique uses the terms in Taylor series up to the second order. The

computational difficulty is more, and the improvement in accuracy is not always worth

the extra computational effort. SOSM methods have not found wide use in engineering

application.

3.5.3. Advanced Second Moment (ASM) Method (Hasofer and Lind, 1974)

Hasofer and Lind (1974) proposed an improvement on the FOSM method

based on a geometric interpretation of the reliability index as a measure of the distance

dimensionless space between peak of the multivariate distribution of the uncertain

parameters and a function defining the failure condition. In this context and for the

purpose of a generalized formulation, we define a performance function, or state

function,

g(X) = g( X1, X2, X3,……., Xn ) (3.26)

20

where, X = ( X1, X2, X3,……., Xn ) is a vector of basic state (or design)

variables of the system, and the function g(X) determines the performance or state of the

system.

The following steps summarize a simple numerical algorithm (Ang and Tang,

1984) (Rackwitz, 1976) he above mentioned method.

(1) Assume initial values of xi* ; i = 1, 2, ……, n and obtain

xi* = (xi* - xi) / xi (3.27)

(2) Evaluate (g / Xi) and i* at xi*

(3) Form xi* = xi - i* xi (3.28)

(4) Substitute above xi* in g(x1*, x2*, xn* ) = 0 and solve for .

(5) Using the obtained in Step 4, reevaluate xi* = -i (3.29)

(6) Repeat Steps 2 through 5 until convergence is obtained.

3.5.4. Point Estimate Methods

The variance of a function- or any of its moments-is essentially the result of

integration. Rosenbleuth (1975, 1981) proposed that an accurate approximation is

obtained by evaluating the function M at a set of discrete points and using those values to

compute the desired moments. In practice, for uncorrelated variables, the points are

usually taken at plus or minus one standard deviation from the mean of each of the

variables. Other schemes can be used, especially when the variables are correlated or

skewed. The method is in a form of Gaussian quadrate (Christian and Baecher, 1999).

3.5.5. The Monte Carlo Simulation Method

The Monte Carlo method is a simple simulation technique. One of the usual

objectives in using the Monte Carlo technique is to estimate certain parameters and

probability distributions of random variables whose values depend on the interactions

21

with random variables whose probability distributions are specified. Provided high-speed

digital computing facilities are available, a simple Monte Carlo technique can often be

useful in obtaining the distribution FR (r). Let R be a function of n independent random

variables Yi.

(3.30)

The technique consists of three steps:

1) Generating a set of values yik for the material properties and geometric parameters Yi

in accordance with the empirically determined or assumed density functions fYi. The

suffix i is used to denote the i th variable and suffix k is used to represent the kth set

of values (y1k, y2k,…, yik,…, ynk ) of the corresponding variables

.

2) Calculating the value rk corresponding to the set of values yik obtained in step 1, by

means of the appropriate response equation for resistance of the section. That is

(3.31)

3) Repeating steps 1 and 2 to obtain a large sample of the values of R and therefore,

estimating FR (r).

This method can also be used to obtain distributions for M and Z where,

(3.32)

(3.33)

Here, R is the resistance and S is the action. It is then only necessary to

obtain additional sample values for S in accordance with the density function fS and

to combine the equation for resistance with Equation 3.32 or Equation 3.33 to provide

the direct means of calculating the means of M or Z.

CHAPTER 4. FATIGUE CRACKING: 4.1. GENERAL:

22

Fatigue is a mode of failure under a repeated or varying load, never reaching

a high enough level to cause failure in a single application. The fatigue process embraces

two basic domains of cyclic stressing or straining, differing distinctly in character.

Low-cycle fatigue - where significant plastic straining occurs. Low-cycle

fatigue involves large cycles with significant amounts of plastic deformation and

relatively short life. The analytical procedure used to address strain-controlled fatigue is

commonly referred to as the Strain-Life, Crack-Initiation, or Critical Location approach.

High-cycle fatigue- where stresses and strains are largely confined to the

elastic region. High-cycle fatigue is associated with low loads and long life. The Stress-

Life (S-N) or Total Life method is widely used for high-cycle fatigue applications. While

low-cycle fatigue is typically associated with fatigue life between 10 to 100,000 cycles,

high-cycle fatigue is associated with fatigue life greater than 100,000 cycles.

Figure 4.1. Different Types of Cyclic Stress Responsible for Fatigue Cracking.

Regarding pavement this fatigue strain has been defined by IRC as:

‘Horizontal tensile strain at the bottom of the bituminous layer. Large tensile strains

cause fracture of the bituminous layer during the design life’. [IRC: 37-2001, 3.2.2(ii)] –

This fracture of the bituminous layer is termed as ‘Fatigue Cracking’.

23

4.2. FATIGUE MECHANISM:

Fatigue cracking is a mechanism of failure results from cyclic stresses. The

name “fatigue” is based on the concept that a material becomes “tired” and fails at a

stress level below the nominal strength of the material. It involves initiation and growth

of a crack under applied stress which may be well within the static capacity of the

material. Discontinuities such as changes in section or material flaws are favored sites for

fatigue initiation. During subsequent propagation, the crack grows in microscopic amount

with each load cycle. The crack so-formed often remains tightly closed, and thus difficult

to find by visual inspection during the majority of the life. If cracking remains

undiscovered, there is a risk that it may spread across a significant portion of the load-

bearing cross section, leading to final separation by fracture of the remaining ligament.

Hence fatigue occurs in three stages – crack initiation; slow, stable crack growth; and

lastly rapid fracture- causing fatigue failure.

The development of fatigue cracking regarding pavement can be expressed as

three stage process - a) initiation of hairline cracking at the bottom of bituminous layer,

(b) widening of the crack and formation of crack network; and (c) formation of visible

cracks. But under certain conditions the cracks may originate at other locations like from

the top of the bituminous layer or within the layer. For a specific experiment, it has been

shown that the number of load repetitions required to reach stage (b) is about 4 times

larger than that required to reach stage (a), whereas the number of load repetitions

necessary to reach stage (c) is more than 20 times larger than that required to reach stage

(a).

Fatigue distress is also analyzed by a mechanistic parameter called damage

index. But for field estimation of fatigue distress, percentage cracking is a more effective

parameter. Percentage cracking is defined as the ratio of cracked area to the total lane

area.

4.3. TYPES OF FATIGUE CRACKING:

24

More detailed definition of the different fatigue cracking is given below.

4.3.1. Bottom-up Fatigue Cracking or Alligator Cracking:

This type of fatigue cracking first shows up as short longitudinal cracks in the

wheel path that quickly spread and interconnected to form a chicken wire/alligator

cracking pattern- these cracks initiate at the bottom of the bituminous layer and propagate

to the surface under repeated load applications. These cracks are the result of repeated

bending of the pavement layer under traffic load and are measured by the ratio of the

cracked area to the total lane area. Following are the some of the reasons of alligator

cracking:

Relatively thin bituminous layer for the magnitude and repetition of the traffic

load.

Higher wheel loads and higher tyre pressure.

Soft spots or areas in unbound aggregate base materials or in the subgrade soil.

Weak aggregate base/subbase layers caused by inadequate compaction or increase

in moisture content or presence of extremely high ground water table (GWT).

Figure 4.2. Schematic Diagram of Bottom-Up Fatigue Cracking.

25

Figure 4.2.1. Low Cracking Zone Figure .4.2.2. Moderate Cracking Zone

Figure 4.2.3. High Cracking Zone

4.3.2. Surface Down Fatigue cracking or Longitudinal Cracking:

In few cases load initiated cracks do occur on top and propagate downward.

This type of cracking is measured by the length of crack per km of road stretch. One of

the suggested mechanisms of surface-down fatigue cracking is:

High stiffness near the top surface due to severe aging of the bituminous layer and

the high tire contact pressure near the edge of tyre cause crack initiation and crack

propagation. This occurs due to the shearing of the surface mixture.

Figure 4.3 Schematic Diagram of Longitudinal Cracking

26

Figure 4.3.1. Longitudinal Cracking at Figure 4.3.2. Longitudinal Cracking at

Wheel Path Non-Wheel Path

4.3.3. Thermal Fatigue Cracking:

Cracking in flexible pavements due to cold temperature or temperature

cycling is commonly refereed to as thermal cracks. Thermal cracks typically appear as

transverse cracks on the pavement, surface roughly perpendicular to the pavement

centerline. These cracks can be caused by the shrinkage of the asphalt surface due to low

temperatures, hardening of the asphalt or daily temperature cycles.

Thermal fatigue cracking has two-different patterns viz. transverse cracking

and block cracking. Transverse cracks usually occur first and are followed by the

occurrence of block cracking as the asphalt ages and becomes more brittle with time.

Transverse cracking is usually predicted by design models whereas block cracking is

handled by material and construction variables.

Figure 4.4.1. Low Thermal Cracking Figure 4.4.2. Transverse Cracking

27

Figure 4.4.3. Block Cracking

4.4. CONCEPT OF FATIGUE LIFE

Generally 50% reduction of the initial stiffness of the asphalt beam under

strain controlled test is considered as ‘fatigue failure’ and the no. of load repetition

required for that is commonly termed as fatigue life. Here strain controlled test has been

adopted due to the consideration that crack initiation in asphalt mixes is mainly

dependent on the magnitude of the applied strain for a controlled strain testing. But as

per IRC: guidelines pavement having 20% of ‘fatigue cracking’ are considered to be

failed. The number of traffic load repetition required to reach this predefined magnitude

of fatigue cracking for a pavement is termed as the fatigue life (Nf) of the pavement. It is

actually the no. of load repetition required to initiate the micro cracks in the surface layer.

The mathematical model for finding Nf is given below.

(4.1)

ε = tensile strain

I = initial mix stiffness.

k1, k 2, k 3 = experimentally determined coefficients.

Crack-propagation is controlled by two different modes: Mode-I: Opening

of initiated crack, Mode-II: Shearing of the crack tip. Due to Mode-I the crack only open

up to a certain length (about half to two-thirds of the AC layer) and resist crack

28

propagation when crack approaches the top of the layer. Whereas Mode-II contribute

crack propagation through the entire thickness of the AC layer. Therefore a combination

of these and of a mixed mode would be more appropriate to describe the crack

propagation. Fatigue damage growth under the tire wall due to the shear stress is also an

important factor.

The different influencing factors of fatigue life (Nf) are as follows:

Thickness of bituminous layer, base layer.

Elastic modulus of subgrade, base layer, bituminous surface.

Average daily traffic, traffic growth rate.

Vehicle damage factor, lane distribution factor, tire contact pressure.

29

CHAPTER 5. LITERATURE REVIEW

5.1. GENERALl

Substantial research work has been done to analyze the variability and

uncertainty associated with pavement design process to estimate their effects on the

design process. Lemer and Moavenzadeh (1971), and Darter and Hudson (1973) were

among the first to introduce the reliability concept to pavement design and management.

[022_Sun] Later on various researchers has use the probabilistic approach to strengthen

the design process in different aspect. Here few of these research works have discussed

briefly under following sub-headings:

5.2. APPLICATION OF RELIABILITY ANALYSIS TO

ESTIMATE THE VARIATION OF DESIGN PARAMETERS:

Lemer and Moavenzadeh (1971) developed one of the first models dealing

with reliability of pavements. Lemer attempted to apply the Monte Carlo Simulation

method to a complex pavement design process for FHWA, USA. Computation time

required for the simulation proved to be excessive and impractical for practical

application. On the other hand Darter and Hudson (1973) consider two major factors viz.

Traffic effects and Environmental effects for the cause of ‘loss of serviceability’ or

‘failure of pavement’. Darter defined reliability parameter R, mathematically as,

R = P [Nt > NT ]

where, Nt =No. of 18- kip ESAL withstand by the section before serviceability reaches

limiting value. NT = No. of 18 kip. ESAL loading applied to the pavement during its

service life. VESYS model developed by Kennis (1977) uses the same concept to

estimate reliability in terms of serviceability index as follows:

R = P [pf > pt ]

Where, pf = present serviceability index at time t ; pt = terminal serviceability index.

30

George et al (1988) computed the reliability of continuously reinforced

concrete pavement calculating pf at a specified time and comparing that pt. RAPP-I –

the developed computation model estimates pavement reliability as well as expected life.

Here pf and pt are assumed to be normally distributed with known mean and standard

deviation and reliability, R can be estimated using the following equation:

Where, - standard normal cumulative probability density function ; µpf , µ pt –

mean values of pf , pt ; σpf , σpt - standard deviation of pf , pt , z0 – standard normal variate.

Killingsworth and Zollinger(1995) carried out sensitivity analysis of input

parameters for pavement design and reliability- which indicates that with low traffic and

a weak subgrade, the flexible pavement design is moderately sensitive to changes in

subgrade modulus, allowable traffic, and surface modulus; however, it is much less

sensitive to changes in surface thickness. But, Portland cement concrete pavements are

not sensitive to subgrade modulus and allowable traffic, but are sensitive to the input

surface thickness, and less sensitive to PCC surface modulus. At high traffic and a

moderately strong subgrade, the flexible pavement shows the opposite trend. Whereas,

the analysis of the PCC design for higher traffic indicates that all design parameters are

somewhat sensitive to variations in their design values.

Kenis and Wang (1997) used reliability concept to examine the effect of the

variation of selected variables on pavement performance based on information obtained

from the accelerated loading test. The most important aspect of their research was to

distinguish between the developments of pavement distress resulting from initial

variations in material properties/layer thickness and from variations in the dynamic wheel

forces imposed on the pavement due to tire-suspension dynamics. The analysis shows

that initial structural profile has little influence on final profile of pavement regardless the

type of suspension. The reliability analysis revealed that: a) Reducing pavement

structural variability will increase the reliability of pavements serviceable life; and

b) That for a given pavement, decreasing dynamic wheel force will increase pavement

reliability if all other variables are kept unchanged; hence to increase pavement

31

reliability, a thicker or stiffer pavement is required for higher dynamic wheel forces.

Jiang et al. (2003) have done the analysis of variability of in-situ pavement layer

thickness. Two types of variability: spatial variability within the section and the extent of

the deviations between the as-designed thickness and as-constructed thickness are

considered. Regarding the spatial variation in layer thicknesses following observation

were found:

Thickness variations within a layer indicate a normal distribution based on

combined test for skew ness and kurtosis for the majority of pavement layer (86%

of 1034 layers studied) in the LTPP program.

Actual mean thicknesses are within 0.5 in. (12.7 mm) for 74% of the layer and

within 1 in. (25.4) for 92% of the layers.

For the same layer and material type, the mean constructed layer thickness tends

to be above the designed value for the thinner layers and below the design value

for the thicker layers.

The mean constructed layer thickness for PCC layers and LC based layers are

generally above the designed value.

This result help to estimate the in situ variability of pavement thickness

resulting from construction and the extent of mean thickness deviation from the design

values – both being vary accountable for reliability based Mechanistic Empirical design

of pavement.

Timm et al. (2005) has proposed mixed distribution for traffic load spectra

consist of log-normal and normal distribution for a site specific condition based on WIM

data. The proposed model works satisfactorily in characterizing both single and tandem

axle load distribution, as evident by the high R2 values.

5.3. APPLICATION OF PROBABILISTIC APPROACH TO

CBR EQUATION OF PAVEMENT DESIGN:

Potter (1987) developed a probabilistic approach, providing more reliable

designs at potentially lower costs, from the current design procedure if the reliability of

the CBR curve is known. His study was undertaken to establish the reliability of the

32

current CBR-based flexible pavement design model using existing data from accelerated

traffic tests. He defined reliability of the CBR equation as the probability that the actual

test section thickness (t) is less than the design thickness (tCBR). That is,

Reliability = Probability

Then, assuming a normal distribution for the ratio of the thickness, the reliability of the

design model was found to be about 50 percent, excluding the difference between the

performance of the accelerated traffic test section, long term performance of actual in-

service pavements and effects of conservative estimates of design parameters.

Bourdeau (1990) accessed the reliability of the pavement section by

considering the Shook and Finn design equation as a function of two random variables –

the expected no. of traffic load and the California Bearing Ratio (CBR) of the sub grade

soil. The formulation is a second order, second – moment development of this equation.

A sensitivity analysis indicates that the CBR variability has large effect on the pavement

reliability. The uncertainty of the expected traffic loads has little influence on the

reliability for a large no. of axle load. Secondly an analytical model has developed for the

co-efficient of equivalence of the (unbound) granular materials of the base and sub base

courses, using the theory of stochastic stress propagation in particular media. These

co-efficient reflect the ability of the granular course to spread the applied load in a

diffusion process. They are expressed as functions of angel of internal friction (ǿ) of the

material and a modified formulation is derived for the equivalent thickness of the

pavement and can be integrated in reliability based model.

In order to get better statistical analysis Divinsky et al. (1996) has simplified

the conventional CBR equation (both for CBR/ pe < 0.22 and CBR/ pe > 0.22) based

upon the analysis for three different categories of wheel load conFigureuration- Single

assembly, Twin tandem assembly and 12-wheel assembly. It has been found the

weighting factor for A is vary much less (about 0.01 compared with 0.34 for CBR),

therefore dropped from the generalized CBR equation of following form.

t g = α m. ( ) 0.594 (5.19)

33

tg - design thickness ,αm - Load conFigureuration factor ,L –assembly load , pe – tire

pressure. A- Area of contact area between tire and pavement.

It has been found from the analysis that the difference between field thickness

and empirical thickness obey the pattern of normal distribution curve. Later Divinsky et

al. (1998) has done the supplementary analysis of previously developed generalized CBR

equation in order to estimate the thickness design of flexible pavements at a given

probability level. They compared the generalized CBR equation with the conventional

airport design method (FAA-1995) which includes an additional analysis of the

generalized CBR equation in order to verify the equation parameters, to determine the

confidence intervals of the model transformation, and to approximate the constructed

confidence limits for the original form of the generalized CBR equation. They have

estimated the values of a and b for different level of design reliability for the following

equation:

tg = a [gm .L. (log C)/ CBR]b (5.2)

Reliability Level Parameter a Parameter b

0.99 4.73 0.599

0.95 4.31 0.597

0.90 4.11 0.596

[C- no. of coverage of the design life, gm – function of no. of wheel]

5.4. RELIABILITY CONCEPT IN AASHTO DESIGN

EQUATION

To access the variation of pavement thickness as a function of different

design factors viz. traffic and soil support value (SSV) Basma et al. (1989) applied a

linear first order approximation on AASHTO design equation. It had been found that

deterministic traffic prediction for various high-volume freeways in US shows poor co-

relation with actual field traffic. It is one of the reasons behind the failure of pavement,

designed by conventional AASHTO methods within the 8-12 years instead of 20 years of

34

design life. Where as the soil support value also varies spatially and temporally. A

nomographic solution in-corporation with least-cost concept has been prepared for giving

design thickness for a given reliability and vice-versa to curb-down the ‘over design’

aspect in case of high reliability.

Based upon the stipulation made by Darter et al(1995)., the pavement design

equation of AASHTO (1993) was formulated as:

(5.3)

where, W18 = predicted number of 18 kip equivalent single axle load repetition; ZR =

standard normal deviate corresponding to reliability level R; S0= combined standard error

due to traffic prediction and performance prediction; PSI= difference between the initial

design serviceability index, p0 and the design terminal serviceability index, pt; MR =

resilient modulus (psi); SN = structural number required for the total pavement thickness.

This formulation has made it possible to design pavements with a given reliability R, and

nomographs for design can be seen in the AASHTO Guide (1993). But one drawback of

this model is that the design equation considers the total variance (attributable to all

design parameters) in the form of a correction factor, S0. To what extent a single feature

variation affects the reliability/performance of pavement cannot, therefore, be

investigated using the AASHTO model.

Using the following equation from above AASHTO design model of flexible

pavement Noureldin et al. (1996) experiments were done on actual pavement section to

find the safety factors in reliability-based design in Saudi Arabia.

SF=10-Zr.So (5.4)

Zr = standard normal deviate for any selected reliability level.

So = combined standard deviation of both pavement performance and traffic predictions.

Variation of different pavement performance factors viz. layer modulus of

different pavement layer , traffic parameters , climate factors , design failure criteria were

35

found out based on experiments incorporating with AASHTO guidelines. But it has been

found that the values of SN and S0 exhibited larger variability in case of thinner

pavement than thicker pavement , whereas AASHTO guides recommends a specific

range of values SN , S0 and S w regardless of the total pavement thickness. Moreover the

safety factor found out has a larger value than the recommended by the AASHTO guide.

For design of roads having low volume traffic in Kerala Joseph et al. (2004)

has prepared a design chart for a local soil having CBR 7 .The reliability aspect of the

design has been estimated based upon AASHTO design equation. They found a

correlation between the total pavement thickness and the cumulative equivalent standard

axles for a particular reliability level. For lesser traffic intensities the pavement thickness

doubles as the reliability level is increased from 50 to 95 %. For higher traffic intensities

the pavement thickness increases 1.5 times as reliability level is increased from 50 to

95%.

Hong & Wang (2004) has developed a probabilistic performance prediction

model for flexible pavement based on nonhomogenous continuous Markov chain process.

The model has been applied in conjunction with the pavement deterioration model

suggested by OPAC and AASHTO. For both the case the proposed approach has been

able to mimic the degradation process with respect to the corresponding guidelines.

Prozzi et al. (2005) has done the reliability analysis on current AASHTO

EMPERICAL DESIGN METHODS and forthcoming MECHANISTIC DESIGN

METHODS in order to find the design reliability and also to reduce the uncertainty in

pavement performance prediction estimation. They pointed out that the two parameter

that influence the pavement performance significantly are the surface asphalt thickness

and the model error. This supports the idea that the most simulation approaches that do

not account for model error are ignoring an important component of the overall

performance variability. Their analysis, based upon a non-linear model shows the actual

reliability of a pavement is far better than implied design reliability of the AASHTO

method. This discrepancy is attributed to the more accurate performance behavior of the

model in conjunction with a more accurate traffic characterization by use of axle load

spectra instead of ESAL data.

36

In order to get better reliability based design solution Zhang and

Damnjanovic (2006) has applied Method of Moments using following limit state

function considering fatigue failure criteria:

G (X, t) = Strength – stress (t)

Using AASHTO design equation mentioned earlier (Eq.5.3) Second Moment

(2M), Third Moment (3M) and Fourth Moment (4M) of reliability index-ß with

corresponding failure probability has been found out. Assuming the random variables as

independent to each other with normally distributed values the analysis shows that

Second Moment (2M) of ß predicts lowest failure probabilities whereas Third Moment

(3M) of ß predicts highest failure probabilities within 5 years of construction of

pavement. A comparative study between the Method of Moments and Monte Carlo

Simulation indicates that the Fourth Moment (4M) method yields the accurate prediction

of failure probability; in general the quality of estimation improves as the order of

moments increase. The sensitivity analysis shows that SN (Structural Number) variation

has a significant on reliability; hence it is suggested to implement stricter quality control

than to design a pavement with a higher level of capacity.

5.5. INCORPORATION OF RELIABILITY ANALYSIS IN ME

DESIGN APPROACH

Timm et al. (2000) incorporated reliability analysis into the Mechanistic-

Empirical (M-E) design procedure for Minnesota Department of Transportation, USA.

They used the definition of reliability proposed by Kulkarni (1994) which is given by

(5.5)

where, N is the number of allowable traffic loads, and n is the actual number of applied

traffic loads. Monte Carlo simulation was chosen for reliability analysis and was

incorporated into a computer pavement design tool, ROADENT. Sensitivity analysis

conducted by using the data collected from the Minnesota Road Research Project and the

literature showed that the traffic weight variability exerts the largest influence on

predicted performance variability. It also established a minimum number (5000) of

37

Monte Carlo cycles for design and characterized the predicted pavement performance

distribution by an extreme value Type I function. Finally, a comparative analyses studied

between ROADENT, the 1993 AASHTO pavement design guide, and the existing

Minnesota design methods showed the ROADENT produced comparable designs for

rutting performance but somewhat conservative for fatigue cracking.

Kim and Buch (2003) has given the priority to the selection of an

appropriate reliability assessment technique and careful characterization of design input

variability for probabilistic estimation of pavement performance and determination of

the reliability-based safety factor of the pavement design procedure. In addition, a

reliability analysis model for pavement design using Load and Resistance Factor Design

(LRFD) format were introduced.They defined pavement design reliability in terms of rut

depth as

(5.6)

Where, SMrut = safety margin between maximum allowable and predicted rut-depth;

RDmax = maximum allowable rut depth in the design period, and RDpredict = predicted rut

depth in the design. To quantify the systematic errors of the design procedure, a

professional factor concept, defined as a representative ratio has been introduced. The

professional factor, P, reflects uncertainties of the assumptions and simplifications used

in design models. These uncertainties could be the result of using approximations for

theoretically exact formulas. When this suggested reliability model be applied to design

the pavement with rutting failure criterion, the limit-state function of the model

incorporating the professional factor can be expressed as follows:

(5.7) Where,

RDmeasure = Measured Rut-Depth, RDpredict = Predicted Rut Depth by the Transfer Function.

Carvalho & Schwartz (2006) has done comparative study between the

applicability of 1993 AASHTO Guide guideline of pavement design and NCHRP 1-37A

mechanistic-empirical approach. The analysis results suggest that relative to the NCHRP

1-37A predicted performance 1993 AASHTO Guide overestimates the performance (i.e.,

38

underestimates distress) for pavements in warm condition as well as for pavements

designed for high traffic volume. It has been also that rutting and fatigue cracking

performances predicted by NCHRP 1-37A are relatively insensitive to the reliability

level.

5.6. PROBABILISTIC ESTIMATION OF PAVEMENT

PERFORMANCE MODEL:

For the prediction of fatigue performance Long Fenella et al. (1996) has

conducted HVS tests (fatigue beam test) on two different test specimen of asphalt

concrete pavement – one with conventional AC pavement structure and the other one

with drained asphalt concrete structure. Four different performance model viz. SHRP

Laboratory Testing method, SHRP Surrogate Model, AI model and Shell Model were

evaluated. In all cases the SHRP Surrogate Model gives the longest fatigue life prediction

which is 5-25 times the AI model whereas Shell model predict the shortest fatigue life.

Un-drained pavement appears to be more susceptible to changes in the modulus of all

pavement layers than the drain pavement. A relative damage factor n = 2.5 is evaluated

for different values of load.

During the same time Ayres et al. (1998) has done several analysis to

develop distress model like fatigue cracking , permanent deformation and low-

temperature cracking based upon rational mechanistic approach and includes the

fundamental concept of probability that is inherent to the variables affecting the

performance such as material properties, test procedure and construction techniques.

Here the major input random variables are taken regarding environmental conditions,

traffic characteristics and material properties and pavement geometry.

Prozzi & Madanat (2004) has developed a pavement performance model in

terms of roughness quality of the pavement, using ordinary least square method (OLS)

and random effects (RE) approach. They use AASHO Road Test (1962) data along with

field data collected from in-service pavement in Minnesota (USA). The standard error of

the OLS regression is quite less than the original linear model of serviceability developed

39

by AASHO. Secondly, a joint estimation model combining in-service data and test data

along with the variability of the parameters is considered to modify the roughness model.

According to the estimated model, the rate at which roughness increases is dependent on

the gradient of frost penetration along with cumulative traffic and asphalt thickness of

pavement. With the increase of cumulative traffic the rate of roughness decreases.

Ghosh et al. (2005) developed a computer program FPAVE-DET for

building up a flexible pavement design chart based on Mechanistic-Empirical approach

as per IRC: 37 -2001 for a given set of basic input parameter like CBR value , traffic load

(in MSA) and average annual temperature (AAPT) . By use of this chart design thickness

of granular sub base and bituminous surface can be selected depending upon various

safety levels against fatigue and rutting. Alternatively a given flexible pavement section

can be checked for its safety status with respect to the allowable fatigue and rutting

strains. In a particular case, a pavement section which is over safe from rutting & fatigue

by deterministic analysis is found to have probability of failure of 15.46 % and 16.59%

with respect to fatigue and rutting analysis.

5.7. PROBABILISTIC ESTIMATION OF FATIGUE

DISTRESS

Fatigue distress is usually controlled by the maximum tensile stress at the

bottom of the bituminous layer. a number of predictive model has been developed to

characterize the traffic load induced fatigue cracking. In general predictive models relate

the no. of load repetitions to a certain response of pavement structure. Other approaches

for predicting fatigue cracking involve establishing an empirical regression equation for

cracking directly. The fatigue cracking prediction model suggested by Jackson et al.

(1996) for South Dakota Department of Transportation is in the form of following

equation,

Fatigue cracking Index = 100 – 0.11726.AGE2.2 (5.8)

The Fatigue Cracking Index ranges from 0 to 100, depending on the current

age of pavement and is determined by expert opinion and regression analysis.

40

Aliand and Tayabji (1998) studied no. of regression model based on field

data to predict fatigue cracking. They correlated damage ratio with percentage cracking

using growth curves. It is found that percentage cracking increases slightly , usually far

less than 20%, before the damage index reaches 1, then goes up very quickly when

damage index approaches 10 or more, tends to be stationary at a level of 78%. An

obstacle imposed by this method is the requirement for field observation of percentage

cracking on seasonal basis. In most cases, field data are not readily available if one wants

to predict fatigue cracks rather than to evaluate them.

Abu-Lebedh et al. (2003) has developed performance models based on

autoregression for various types of Freeway and Non-freeway pavements in Michigan

(USA). Here, surface distresses are evaluated by Distress Index (DI) parameter on 0-50

scale at a certain age of pavement. The model, shown below includes pavement age as

one of the input parameters.

Predicted DI (Present) = K x DI (2 year before) + M x Age (2 year before) + C

where, values of K, M, and C for several types of pavement are given below,

K - 1.02 to 1.49, M- 0.05 to 0.25, C- 0.63 to 3.8

This model is capable of find the pavement age when DI value reaches 50

and at that age of service life rehabilitation is recommended.

41

42

43

CHAPTER 6.

FORMULATION OF FATIGUE DAMAGE

6.1. GENERAL

The most primitive form of damage concept in pavement design was used by

AASHO design model. The model estimates pavement deterioration based on the

definition of a dimensionless parameter g referred to as damage. The damage parameter

was defined as the loss in the value of serviceability index at any given time t.

(6.1)

where, gt = dimensionless damage parameter, pt = serviceability at time t,

p0 = initial serviceability at time t, pf = terminal serviceability.

But with the changes of design approaches the concept of damage has been

modified. For ME design approach separate damage criteria are evaluated based upon

different types of pavement distresses. For the present study, different mathematical

models associated with fatigue damage are analyzed.

6.2. ANALYSIS OF FATIGUE MODEL:

Generally, cracking in flexible pavements can be classified into three

categories (1) traffic loading cracking; (2) low temperature cracking; and (3) thermal

fatigue cracking. Many flexible pavement design methods consider traffic load induced

fatigue cracking as a major design criterion. Traffic load induced cracks are generally

surface-down cracking. A number of predictive models of fatigue cracking have been

developed over the past three decades to characterize the traffic load induced fatigue

cracking. These models relate the fatigue life (in terms of the number of load repetitions)

to the tensile strain at the bottom of the bituminous layer. A universal form of the ‘fatigue

law’ used to predict fatigue life of flexible pavements (Finn 1973; Finn et al. 1973, 1977)

is given below

N = k1 ε t-k2 E1

-k3 (6.2)

44

Where, ε = maximum tensile strain at the bottom of the bituminous layer.

E= elastic modulus of the bituminous layer

ki = parameters of fatigue law.

N = total no. of load repetitions to failure.

The coefficients k2 and k3 are calibrated through beam-type fatigue testing. A

number of significant differences exist between laboratory fatigue testing and field

observations. Due to these differences the laboratory fatigue life of a bituminous material

is usually lower than that observed in the field. Laboratory fatigue life therefore must be

adjusted by a shift factor to obtain the field fatigue life such that appropriate design

criteria can be developed. This process may be considered as the modification of the

laboratory result by the field- calibrated fatigue and is reflected by the coefficient k1 of

the above mentioned equation.

Various major institutes have done research work on the ‘fatigue law’ to

accommodate it with respect to their local conditions and hence provided different values

of k1, k2 and k3 as shown in Table 6.1, as obtained from Sun et al. (2003).

Table 6.1: Values of k i for Different Fatigue Model

Models k1 k2 k3

AI Model 0.0796 3.291 0.854

Shell Model 0.0685 5.671 2.363

UC- Barkley Model 0.636 3.291 0..854

Illinois Model 5x10-6 3 0

Minnesota Model 2.83 x 10-6 3.21 0

Indian Model

(IRC 37: 2001)

2.21x10-4 3.89 0.854

6.3. ESTIMATION OF FATIGUE DAMAGE:

For the present study, fatigue damage due to traffic load has been estimated

by two different mathematical approaches viz.

(i) Computation of fatigue damage in terms of ‘Damage Ratio (D f)’ and

(ii) Computation of fatigue damage by ‘Percentage Cracking (%C)’.

45

For the first one Eq.5.2 has been used considering the mean and variations of

required design input parameters, whereas, for the last one probabilistic estimation of

critical damage ratio has been done using FOSM method. A computational program has

been developed with FORTRAN code for both the cases.

6.3.1. Concept of Fatigue ‘Damage Ratio (Df )’: Now the fatigue transfer function indicate that the allowable number of

applications of any axle load/combination is a function of the strain (horizontal tensile

strain at the bottom of the AC layer) caused by the load application. Hence pavements are

designed to allow a specific no. of load repetitions depending upon its layer

characteristics before fatigue cracking failure occurs. But in the field, it has been found

that actual no. of repetitions which causes crack initiation may be less or more than the

design repetitions. With the increase of actual no. of repetitions, hairline cracks will

propagate to form widespread network of visible cracks. Now, to relate the percentage

fatigue cracking with load repetitions, concept of damage ratio (Df) had been introduced

according to Miner’s Law of cumulative damage. The general form of the Miner’s law

is

(6.3)

Where, Df = damage ratio,

Xi = actual no. of load repetition during period i,

Ni = allowable no. of load repetition during period i.

From the Eq. (5.2) it is evident that with the increase of i, the value of

damage ratio will be accumulated and hence will increase for i from 1 to n. Therefore, to

make simple the application of Miner’s Law in fatigue damage analysis of pavement it is

better to break up the entire design life of pavement into the number of analysis periods.

This division of design life can be done on the basis of different level of tensile strain or

on the basis of a few analysis periods of equal duration. But if the design life is broken up

on the basis of strain level it would not be convenient to estimate the fatigue damage to

use it in pavement management system as per desired way. On the other hand some

periodical estimation of fatigue damage would be handful for the pavement management.

46

For the present study, 15 years of design life has been divided into 5 analysis periods of 3

years equal duration, hence the value of i ranges from 1 to 5.

Depending upon the value of initial tensile strain under the bituminous

surface at the starting of each analysis period viz. i=1, 2,…, 5, there will be separate

values of fatigue life for each analysis period. Now, from the stiffness vs. load repetition

curve, the stiffness of bituminous binder will reduce with the increase in load repetition

following a certain parametric equation (which is discussed in Annexure: 1) - hence as

the stiffness of bituminous surface will decrease with subsequent analysis period- i, the

value of initial tensile value will increase accordingly which in turn will decrease the

value of fatigue life subsequently. On the other hand design traffic will increase as per

traffic growth rate for subsequent analysis period. Hence it is clear that the values of Di

(i.e., Xi /Ni ) will increase for successive analysis periods and accordingly the amount of

fatigue damage as well as percentage fatigue cracking (%C) will increase for successive

analysis periods. Until and unless the value of accumulated damage ratio reach the

critical value (i.e., 1), the pavement section will be on the safe side considering fatigue

damage.

6.3.2. Computation of Damage Ratio (D f ):

From the Eq. 5.2. it is evident that estimation of fatigue life as well as traffic

load are vary much necessary for computation of D f . Now for the probabilistic

estimation of D f all the design inputs of fatigue life and traffic load have been estimated

separately with proper mathematical equations. These have been discussed in following

sections.

6.3.2.1. Estimation of Fatigue Life

Significant experimental evidences, presented in the literature indicate that

the distribution of fatigue lives at a particular stress level is lognormal. Hence the Eq.6.1.

has been converted to logarithmic scale to give the following expression

(6.4)

Hence the mean and variance of N would be given by

47

E Ni= exp [µN + σN2/ 2] (6.5)

VAR Ni = exp [2(µN + σN2)] - exp [2µN + σN

2] (6.6)

Here, (6.7)

σN = total variance of the fatigue model. The expression of σN has been given

by equation 5.10.

Now it is evident that Ni is a function of εt, E1, k1, k 2 and k 3 .The fatigue

tensile strain (εt) at the bottom of bituminous surface is depend upon the elastic modulus,

Poisson’s ratios , thickness of different layers , the wheel spacing and tire contact

pressure . Thus,

(6.8)

Where,

E1 = Elastic modulus of bituminous surface layer

E2 = Elastic modulus of granular base layer

E3 = Elastic modulus of subgrade

1 = Poisson’s ratio of bituminous surface layer

2 = Poisson’s ratio of granular base layer

3 = Poisson’s ratio of subgrade

h1 = Thickness of bituminous surface layer

h2 = Thickness of granular base layer

ws = Wheel spacing

tp = Tyre contact pressure of vehicle

t = Maximum tensile strain at the bottom of bituminous layer

It follows, therefore

(6.9)

Here, in the fatigue performance analysis, all the variables except k3 have

been treated as random variables. The fatigue equation (eq.6.1) is calibrated from field

performance study and hence, it has some inherent variability. For that reason the

regression coefficients k1 and k2 have been included in the set of random variables. Only

these two variables are chosen because they govern the intercept and slope of the fatigue

48

equation. It is accepted that the variation of other coefficient k3 will not significantly

influence the result. Hence the vector of the random variables is therefore given by

N (6.10)

All of the random variables are assumed to be normally distributed and

mutually independent i.e. uncorrelated.

Now,

+ (6.11)

The derivatives of the terms used in the Equation 6.11 are given in the ANNEXURE-1 (as deduced by Maji, 2004).

Where,

(6.12)

Where,

= Mean value of the parameter in subscript

and

+

…..+ (6.13)

49

where, = Standard deviation of the parameter in subscript

The derivatives of the parameters used in the Equation 5.12 are obtained

through numerical analysis using the centered finite divided difference formulation

(Chopra and Canale, 1998).

6.3.2.2. Estimation of Traffic Load:

Traffic load is assumed to be normally distributed with the mean and variance

estimated directly from the field observation. Hence,

E [X] = µX (6.14)

VAR X = σx2

(6.15)

where,

(6.16)

Equation 6.16 is the Taylor Series expansion up to second order terms, of the

equation given by IRC: 37-2001(3.3.6.1.) for computation of design traffic which is

mentioned below.

X=365 x A x D x F [(1+r)n -1] / r

A = Present traffic in terms of number of commercial vehicles per day

D = Lane distribution factor.

F = Vehicle damage factor.

r = Traffic growth factor.

Here, in traffic prediction analysis all the variables except average daily

traffic-A which is taken as deterministic, are taken as random variables. So the mean and

standard deviation of cumulative traffic loading can be computed as

(6.17)

50

(6.18)

6.3.2.3. Estimation of Damage Ratio:

For analytical simplicity damage is considered as the ratio of total traffic

loading over allowable fatigue load repetition for each individual analysis period (i.e.

i=1,2,.., 5). Hence by Taylor’s expansion mean and variance of damage can be obtained

by means of Cornell’s first-order, second- moment method. Under the assumption of

independence of X and N, we have

(6.19)

(6.20)

Here, EN, VarN, µX, and σX are given by eq. (6.14), (6.15), (6.17) and (6.18)

respectively.

6.3.3. Computation of Percentage Cracking (%C):

From the experimental investigation by many researchers shows that the

critical value of Df at failure is not always close to 1 .00 but have a wide distribution.

For the present study, it is assumed to be normally distributed. Now with the value of. Df

getting pass the critical value, the probability of crack initiation and propagation will

increase.. Hence it can be concluded that the probability of a pavement surface getting

cracked under traffic load is depend upon the probability of damage ratio Df reaches or

exceeds the value 1. So percentage of fatigue cracking can be expressed by

%C=100.Prob (6.21)

Here %C represents percentage cracking.

51

6.4.SENSITIVITY ANALYSIS:

It is often necessary to identify the ‘dominant parameters’ as would have

relatively strong influence on failure, not only for the sake of computation and lack of

data, but also for the fact that once such parameters are identified for several situations,

efforts can then be concentrated on making a more reliable estimates of such parameters

in similar field situations leading to a more authentic reliability calculation. The usual

technique of identification of the ‘dominant parameters’ referred to above be a thorough

parametric study wherein each parameter is varied and the resulting change in the values

of the probability of failure noted. However, such a procedure often consumes

inordinately large computation as to render the procedure unattractive, if not altogether

impractical. In recent times, especially in structural reliability analysis, sensitivity of a

random variable is expressed in terms of its ‘Importance Factor’ defined as follows

(Adhikary and Langley 2002):

(6.22)

where,

IFi is the importance factor for the ith random parameter.

6.5. INPUT PARAMETERS:

6.5.1.Mean Values of The Random Variables

The mean values of the normally distributed design parameters considered in

this reliability analysis are presented in Table 5.2. For the mean values, IRC: 37-2001

guidelines have been followed as far as possible.

6.5.1.1.Average Annual Pavement Temperature (AAPT)

52

Considering Indian conditions, for this study, the Average Annual Pavement

Temperature (AAPT) has been taken as 35C.

6.5.1.2. Grade of Bitumen

The grade of bitumen is taken as 60/70 as par the Table “Criteria for the

selection of grade of bitumen for bituminous courses” of Annexure-6, IRC: 37-2001, for

any climate, for heavy roads, expressways, urban roads traffic and for DBM, SDBC and

BC bituminous course.

6.5.1.3. Elastic Modulus of Bituminous Surface Layers:

The Elastic Modulus value of Bituminous Material (E 1) is taken as 1695 MPa

from the Table “Elastic Modulus (MPa) values of bituminous materials” of Annexure-1,

IRC: 37-2001, for BC and DBM 60/70 bitumen,

6.5.1.4. Modulus of Elasticity of Subgrade:

As per IRC: 37-2001, Annexure-1, Modulus of Elasticity of Subgrade (E3),

E3 (Mpa) = 10 CBR for CBR 5 and

= 17.6 (CBR) 0.64 for CBR>5 (6.23)

6.5.1.5. Modulus of Elasticity of Granular Sub-base and Base layer:

As per IRC: 37-2001, Annexure-1, Modulus of Elasticity of Granular Sub-base and Base,

E2 (Mpa) = E3*0.2*(h2)0.45 (6.24)

where, E2 = Composite elastic modulus of granular Sub-base and Base (Mpa)

E3 = Elastic Modulus of Subgrade (Mpa)

h2= Thickness of granular layers (mm)

6.5.1.6. Poisson’s ratio of different layers of Pavement:

As per IRC: 37-2001, Annexure-1, the Poisson’s ratio of Bituminous layer

(1) may be taken as 0.50 for pavement temperature of 35C and 40C. Poisson’s ratio for

53

both the Granular Sub-base and Base layer (2) as well as Subgrade layer (3) are taken

as 0.4.

6.5.1.7. Tyre Contact Pressure:

As reported by Chakroborty and Das (2003), a survey on tyre pressures of

commercial vehicles in India indicated that the pressure ranges from 0.77 to 0.84 Mpa.

Accordingly, in the present study a mean value of 0.8 MPa has been adopted.

6.5.1.8. Wheel Spacing:

For Indian condition, a wheel spacing of a dual wheel system has been

adopted as 310 mm (Chakroborty and Das,2003).

6.5.1.9. Regression Coefficients of Fatigue Equations:

The mean values for the regression coefficients of fatigue equations have

been adopted from Annexure-1 of IRC: 37-2001.

6.5.1.10. Vehicle Damage Factor:

As per IRC: 37-2001, Section 3.3.4, Table 1, the indicative VDF values have

been adopted for the present study.

6.5.1.11. Lane Distribution Factor:

As per IRC: 37-2001, Section 3.3.5, for two-lane single carriageway roads,

the design should be based on 75 percent of the total number of commercial vehicles in

both directions. So, a Lane Distribution Factor (LDF) of 0.75 has been adopted for the

present study.

6.5.1.12. Traffic Growth Rate:

Traffic growth rate has been taken as 7.5% as per IRC: 37-2001, Section

3.3.6.

54

Table 6.2: Mean Values of the Design Parameters Considered In the Study

Parameters Mean values Parameters Mean values

VDF 4.5 10.5

LDF 0.75 2 0.4

k1f 2.2110-43 0.4

k2f 3.89 tp 0.8 Mpa

k3f 0.854 ws 310 mm

E1 1695 Mpa r 7.5%

6.5.2. Coefficient of Variation (COV) of the Random Variables

The values of coefficient of variation (COV) are based on published literature

wherever available. The range of values of coefficient of variation (COV) as obtained

from various literature and the values adopted in the present study are listed in the

following table (Table 5.3). However, for the regression coefficients in the fatigue

equation namely k1f, k2f and and also LDF, VDF, 1, 2, 3, ws, r, no such coeffiecient of

variation (COV) could be obtained from the literature survey. Hence these coeffiecients

of variation (COV) values have been suitably assumed and also have been presented in

Table 6.3. ( These values were earlier used by Maji (2003) andGhosh (2005))

55

Table 6.3: COV Values of the Design Parameters considered in the Study

ParametersType of distribution

Range of COV (%)

COV values adopted(%)

References

LDF Normal 10 Assumed

VDF Normal 10 Assumed

E1

Log-normal 10 to 40

10

Timm et al.(1998)

Normal 10 to 20 Noureldin et al.(1994)

Log-normal 5-70 Bush, D( 2004)

Log-normal 10-40 Timm et al.(1999)

E3

Normal 10 to 30

20

Noureldin et al.(1994)

Log-normal 5 to 60 Bush, D( 2004)

Log-normal 5 to 60 Timm et al.(1991)

10 Assumed

10 Assumed

3 10 Assumed

ws 10 Assumed

tp 15 10 Timm et al.

r 10 Assumed

h1

Normal 3-1210

Noureldin et al.(1994)Timm et al.(1998)Timm et al.(1999)

Normal 3-25 Bush D.(2004)Normal 10 Darter et al.(1973)

h2

Normal 10-1510

Timm et al.(1999)Normal 5-35 Bush D.(2004)

6.6. RESULTS AND DISSCUSSION:

Various design pavement sections based on M-E approach given by IRC: 37-

2001 has been analyzed by using the developed computer program. Designed pavement

sections for three different design traffic viz. 50 MSA, 100 MSA & 150 MSA have been

taken for analysis. For each level of design traffic there are nine alternative design

solutions recommended under PLATE-1 CATALOGUE and PLATE-2 CATALOGUE in

IRC: 37-2001. For the present study, design solutions for subgrade strength CBR 5 to

CBR 10 against mentioned level of design traffic given in PLATE-2 CATALOGUE have

56

been taken for analysis. For the sake of ready reference, the IRC: 37-2001 design sections

for the above mentioned cases are reproduced in Tables 6.4(A), 6.4(A) and 6.4(A)

respectively. Results of analysis for the above mentioned design sections, as obtained

using the developed computer program, are presented in Tables 6.4(B), 6.4(B) and 6.4(B)

respectively. From these Tables the following observations can be made:

Damages (Di) found at the end of the last two time domain i.e. 9- 12 year and 12-

15 year, are quite high than that of the first two time domain. Quantitatively it is 3

to 5 times more than that occurring during 1st time domain which is 0-3 years. It

indicates that by the time pavement’s service life passed 2/3 rd of its design life it

reaches a very much vulnerable condition to damage severely under fatigue for

the rest of its design life, if any kind of rehabilitation work is not done to it.

For most of the IRC design sections considered in the study it has been found

that the accumulated damage ratio (Df) crosses the critical value i.e. 1 during 4 th

time domain which means theoretically, between 9-12 years of its service life the

pavement would reach the limit state of failure.

Further, during the rest of its service life the value of Di will continue to

increase till the end of design life. For most of the design sections the final value

of Df lies between 1.6 to 1.9. It indicates that due to various uncertainties in the

design process the pavement can suffer the effect of load repetition which is 1.6

to 1.9 times of actual traffic load repetition. However, this conclusion is valid

only under the given set of values of input parameters, used in the present

analysis.

Looking at the cracking criteria, a few design sections (6 no.) have shown limited

damage with cracking well below 20% at the end of design life. However 2 out

of 18 selected design sections prematurely crossed the permissible limit of 20%

fatigue cracking (as given by IRC: 37-2001) during the 4th time domain i.e.

during 9-12 years of service life. Along with them other sections will exhibit

moderate to higher percentage of cracking (between 20%-54%).

For most of the design sections (15 out of 18) more than 60% of the total damage

in terms of cracking occurs at the last time domain i.e. during 12-15 years of

service life. It implies progressive failure occurs with age of the pavement.

57

Table 6.4(A)Various Layer Thicknesses for Design Traffic 50 MSA from IRC:37-2001

             

CBR Bituminous Surfacing BaseSub-Base h1(mm) h2(mm)

(%) BC(mm) DBM(mm) (mm) (mm) (BC+DBM) (Base+Sub-Base)

5 40 140 250 300 180 5506 40 125 250 260 165 5107 40 130 250 230 170 4808 40 120 250 200 160 4509 40 115 250 200 155 450

10 40 110 250 200 150 450

Table 6.4(B)Result of Analysis for Various Design Sections of 50 MSA Traffic Shown in Table 6.4(A)

CBR h1(mm) h2(mm) iNi(year

) E1(MPa) e x 10-4 N f Nd D i %C i D f %C      1 0-3 1695 2.3255 52.6 7.05 0.104  0 0.1 0.00      2 3-6 980 3.1687 25.2 8.34 0.256 0.02 0.36 0.02

5 180 550 3 6-9 930 3.258 23.7 9.87 0.323 0.59 0.68 0.61      4 9-12 898 3.317 22.7 11.7 0.397 3.44 1.08 1.02      5 12-15 874 3.365 22 13.8 0.485 10.2 1.57 11.23      1 0-3 1695 2.527 38.1 7.05 0.145  0 0.15 0.00      2 3-6 980 3.411 18.9 8.34 0.344 1 0.49 1.00

6 165 510 3 6-9 930 3.04 17.8 9.87 0.431 5.37 0.92 6.37      4 9-12 898 3.566 17.1 11.7 0.53 14 1.45 20.37      5 12-15 874 3.615 16.6 13.8 0.645 25.5 2.1 45.83      1 0-3 1695 2.373 48.6 7.05 0.113  0 0.11 0.00      2 3-6 980 3.19 24.6 8.34 0.263 0.04 0.38 0.04

7 170 480 3 6-9 930 3.275 23.2 9.87 0.33 0.71 0.71 0.75      4 9-12 898 3.332 22.3 11.7 0.405 3.92 1.11 4.67      5 12-15 874 3.377 21.7 13.8 0.492 10.8 1.6 15.42      1 0-3 1695 2.49 39.8 7.05 0.139  0 0.14 0.00      2 3-6 980 3.333 20.7 8.34 0.314 0.4 0.45 0.40

8 160 450 3 6-9 930 3.42 19.6 9.87 0.392 2.94 0.85 3.34      4 9-12 898 3.48 18.9 11.7 0.479 9.34 1.32 12.68      5 12-15 874 3.52 18.4 13.8 0.583 19.5 1.91 32.17      1 0-3 1695 2.503 39.5 7.05 0.14  0 0.14 0.00      2 3-6 980 3.32 21.1 8.34 0.308 0.31 0.45 0.31      3 6-9 930 3.4 20 9.87 0.383 2.56 0.83 2.87

9 155 450 4 9-12 898 3.46 19.3 11.7 0.468 8.53 1.3 11.40      5 12-15 874 3.5 18.8 13.8 0.565 17.9 1.86 29.28      1 0-3 1695 2.52 38.4 7.05 0.144  0 0.14 0.00      2 3-6 980 3.322 21 8.34 0.306 0.36 0.45 0.36

10 150 450 3 6-9 930 3.4 19.9 9.87 0.384 2.62 0.83 2.98      4 9-12 898 3.46 19.3 11.7 0.468 8.53 1.3 11.51      5 12-15 874 3.501 18.8 13.8 0.566 17.9 1.87 29.39

Table 6.5(A)

58

Various Layer Thicknesses for Design Traffic 100 MSA from IRC:37-2001             

CBR Bituminous Surfacing BaseSub-Base h1(mm) h2(mm)

(%) BC(mm) DBM(mm) (mm) (mm) (BC+DBM)(Base+Sub-

Base)

5 50 150 250 300 200 5506 50 140 250 260 190 5107 50 145 250 230 195 4808 50 140 250 200 190 4509 50 135 250 200 185 450

10 50 130 250 200 180 450

Table 6.5(B)Result of Analysis for Various Design Sections of 50MSA Traffic Shown in Table 6.5(A)

CBR h1(mm) h2(mm) iNi(year

) E1(MPa) e x 10-4 N f Nd D i %C i D f %C      1 0-3 1695 2.0306 89.10 14.13 0.122 0 0.12 0      2 3-6 968.38 2.7968 41.37 16.72 0.31 0.44 0.43 0.44

5 200 550 3 6-9 920.52 2.8733 38.9 19.78 0.389 3.29 0.82 3.73      4 9-12 890.15 2.9247 37.36 23.4 0.479 10.1 1.3 13.86      5 12-15 866.85 2.9658 36.2 27.69 0.586 20.3 1.89 34.19      1 0-3 1695 2.1204 75.31 14.13 0.145 0 0.15 0      2 3-6 968.38 2.8982 36.02 16.72 0.357 1.62 0.5 1.62

6 190 510 3 6-9 920.52 2.9754 33.96 19.78 0.447 7.22 0.95 8.84      4 9-12 890.15 3.027 32.68 23.4 0.55 16.7 1.5 25.52      5 12-15 866.85 3.068 31.72 27.69 0.67 28.1 2.17 53.61      1 0-3 1695 1.999 94.73 14.13 0.115 0 0.12 0      2 3-6 968.38 2.719 46.71 16.72 0.277 0.11 0.39 0.11

7 195 480 3 6-9 920.52 2.7902 43.61 19.78 0.347 1.37 0.74 1.48      4 9-12 890.15 2.8379 42.01 23.4 0.425 5.75 1.16 7.23      5 12-15 866.85 2.8757 40.81 27.69 0.518 13.9 1.68 21.13      1 0-3 1695 2.0256 89.97 14.13 0.122 0 0.12 0      2 3-6 968.38 2.7394 44.86 16.72 0.286 0.16 0.41 0.16

8 190 450 3 6-9 920.52 2.8094 42.46 19.78 0.357 1.71 0.76 1.87      4 9-12 890.15 2.8561 40.98 23.4 0.438 6.55 1.2 8.42      5 12-15 866.85 2.8933 39.86 27.69 0.532 15.2 1.73 23.57      1 0-3 1695 2.0256 89.96 14.13 0.121 0 0.12 0      2 3-6 968.38 2.7209 46.05 16.72 0.278 0.11 0.4 0.11

9 185 450 3 6-9 920.52 2.7886 43.7 19.78 0.346 1.36 0.75 1.47      4 9-12 890.15 2.8338 42.24 23.4 0.423 5.48 1.17 6.95      5 12-15 866.85 2.8697 41.15 27.69 0.514 13.6 1.68 20.52      1 0-3 1695 2.0357 88.24 14.13 0.124 0 0.12 0      2 3-6 968.38 2.7185 46.21 16.72 0.277 0.11 0.4 0.11

10 180 450 3 6-9 920.52 2.7846 43.95 19.78 0.344 1.32 0.75 1.43      4 9-12 890.15 2.8287 42.55 23.4 0.42 5.37 1.16 6.8      5 12-15 866.85 2.8637 41.48 27.69 0.509 13.1 1.67 19.94

Table 6.6(A)

59

Various Layer Thicknesses for Design Traffic 150 MSA from IRC:37-2001             

CBR Bituminous Surfacing BaseSub-Base h1(mm) h2(mm)

(%) BC(mm) DBM(mm) (mm) (mm) (BC+DBM)(Base+Sub-Base)

5 50 170 250 300 220 5506 50 160 250 260 210 5107 50 165 250 230 215 4808 50 160 250 200 210 4509 50 155 250 200 205 450

10 50 150 250 200 200 450

Table 6.6(B)Result of Analysis for Various Design Sections of 150MSA Traffic Shown in Table 6.6(A)

CBR h1(mm) h2(mm) iNi(year

) E1(MPa) e x 10-4 N f Nd D i %C i D f %C      1 0-3 1695 1.7889 145.9 21.54 0.11 0 0.111 0      2 3-6 964.51 2.478 66.48 25.03 0.29 0.2 0.398 0.25 220 550 3 6-9 917.96 2.5449 62.52 29.62 0.36 2.02 0.758 2.22      4 9-12 888.43 2.5897 60.75 35.04 0.45 7.35 1.203 9.57      5 12-15 865.78 2.6253 58.23 41.47 0.54 16.4 1.743 25.92      1 0-3 1695 1.8618 124.9 21.54 0.13 0 0.131 0      2 3-6 964.51 2.5594 58.63 25.03 0.33 0.8 0.457 0.86 210 510 3 6-9 917.96 2.6269 55.26 29.62 0.41 4.65 0.866 5.45      4 9-12 888.43 2.6719 53.2 35.04 0.5 12.5 1.368 17.96      5 12-15 865.78 2.7077 51.64 41.47 0.21 15.4 1.575 33.35      1 0-3 1695 1.7599 155.4 21.54 0.1 0 0.104 0      2 3-6 964.51 2.4075 74.38 25.03 0.26 0.04 0.36 0.0387 215 480 3 6-9 917.96 2.4695 70.29 29.62 0.32 0.71 0.681 0.752      4 9-12 888.43 2.511 67.73 35.04 0.39 3.75 1.074 4.502      5 12-15 865.78 2.5439 65.82 41.47 0.48 10.6 1.552 15.06      1 0-3 1695 1.7806 148.5 21.54 0.1 0 0.098 0      2 3-6 964.51 2.422 72.66 25.03 0.26 0.06 0.361 0.0568 210 450 3 6-9 917.96 2.4831 68.79 29.62 0.33 0.89 0.688 0.946      4 9-12 888.43 2.5238 66.41 35.04 0.4 4.27 1.089 5.216      5 12-15 865.78 2.5562 64.61 41.47 0.49 11.3 1.577 16.53      1 0-3 1695 1.778 149.4 21.54 0.11 0 0.109 0      2 3-6 964.51 2.4022 75.02 25.03 0.25 0.03 0.362 0.0349 205 450 3 6-9 917.96 2.4611 71.22 29.62 0.32 0.64 0.678 0.674      4 9-12 888.43 2.5005 68.85 35.04 0.39 3.36 1.064 4.036      5 12-15 865.78 2.5317 67.07 41.47 0.47 9.68 1.533 13.72      1 0-3 1695 1.784 147.4 21.54 0.11 0 0.11 0      2 3-6 964.51 2.3964 75.73 25.03 0.25 0.03 0.362 0.029

10 200 450 3 6-9 917.96 2.4538 72.04 29.62 0.31 0.57 0.673 0.599      4 9-12 888.43 2.4921 69.75 35.04 0.38 3.14 1.054 3.743      5 12-15 865.78 2.5224 68.04 41.47 0.46 9.18 1.515 12.92

Notations of the symbols used in the Table 6.4, 6.5, and 6.6

60

h1- Thickness of bituminous surface layer

h2- Thickness of combined base and subbase layer

i - Number of analysis periods

Ni— service life corresponding to each analysis period

E1- elastic modulus of bituminous surface layer at the start of each analysis period

εi- tensile strain at the bottom of bituminous surface layer

Nf- Fatigue life of the pavement section for the corresponding εi

Nd- Design traffic for the period i

Di- damage occurs during the period i

Df Value of accumulated damage ratio after period i

%Ci- Cracking occurred during the period i

%C Percentage cracking after i

For each level of design traffic it has been found out that design solution given

for subgrade strength CBR 6 exhibit most severe fatigue damage in terms of

accumulated damage ratio as well as cracking. It indicates a lack of uniformity in

design thicknesses provided against various subgrade strength for specific level

of design traffic. For 50 MSA and 100 MSA of design traffic the design sections

for subgrade CBR of 6 would cross the 20% cracking limit at the end of 12 th and

11th year respectively; finally it would exhibit 45.83% and 53.61% cracking at the

end of design life respectively. However, for 150 MSA of design traffic the

corresponding design section show somehow less damage with 33.35% cracking

at the end of design life.

For higher level of design traffic viz. 150 MSA, design sections for subgrade

strength CBR 7, 8, 9, and 10 have shown very safe performances.

For a typical case of 100 MSA design traffic two performance curves -

(i) Percentage fatigue cracking (%C) vs. Service life of pavement and

(ii) Accumulated damage ratio (DJ) vs. Service life of pavement

are shown in Fig.6.1 (A) and Fig.6.1 (B). From Fig.6.1 (A) design pavement

sections having subgrade CBR 7, 8, 9 and CBR 10 have shown the desired

performances. From Fig.6.1 (B) it is seen that trends of accumulation of fatigue

61

damage ratio are similar for different subgrade strength except for CBR 6, which

has a very unusual trend of sharp increase.

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16

Service Life (in year)

CBR 5

CBR 6

CBR 7

CBR8

CBR9

CBR10

Figure 6.1 (A): Performance Curve -I (for %C vs. Service life) for

Design traffic of 100 MSA

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

Service Life(in year)

Dam

age

Rat

io (

Df) CBR 5

CBR 6

CBR 7

CBR 8

CBR 9

CBR 10

Figure 6.1 (B): Performance Curve -II (for Df vs. Service life) for100 MSA Traffic

62

The shown performance curves can be utilized in PMS in terms of pavement

rehabilitation work. For the design sections for which the percentage cracking goes

beyond the permissible limit (i.e. 20%) during a certain period of their service life, the

critical period can be found out from the performance curves and accordingly the

maintenance program can be planned in advance. Now this maintenance work can be

done by applying better quality of materials with same design specification or by

modifying the geometrical design specification (i.e. increasing the thickness of layers).

But whatever alternative is selected for execution it should be cost effective as well as

structurally safe with optimum performance. For example, for the design section having

subgrade strength CBR of 6 for level of traffic 100 MSA two maintenance alternatives

can be applied as follow:

(i) Increasing the thickness of bituminous surface layer by 20 mm at the

end of 9 year of service life.

(ii) Increasing the thickness of bituminous surface layer by 10 mm at the

end of 9 year and 12 year of service life successively.

These have been obtained by trial and error as shown in table 6.7.

Now Fig.6.2 shows the original fatigue performance curve (solid line) along with

modified fatigue performance curve due to application of two maintenance alternatives,

mentioned above. Blue line for 1st alternative and the Orange line for other alternative.

Here 1st alternative shows satisfactory performance whereas the 2nd alternative shows

quite safe performance. But for economic point of view 1st alternative is preferable.

Figure 6.2.: Performance Curve- I for 100 MSA Traffic with Rehabilitation Plans

63

64

Table 6.8 presents the results of sensitivity analysis for fatigue damage in terms of

the importance factors for all the random variables. The sensitivity of a parameter

is expressed by its importance factor. The parameters have been ranked in

accordance with the magnitude of its importance factor. For the sake of

convenience, the design parameters have been grouped as traffic, Material and

Regression parameters. Relative rank of a parameter indicates its position in its

group whereas, its absolute rank indicates its position in the entire vector.

Table 6.8: Results of Sensitivity Analysis for Fatigue Damage

Parameter Importance Factor Relative Rank

Absolute Rank

Traffic ParameterLane Distribution Factor 2.879x10-3 2 8Vehicle Damage Factor 2.879x10-3 3 9Traffic growth rate 9.91x10-4 5 12Tyre contact pressure 3.778x10-3 1 7Wheel Spacing 2.325x10-3 4 10Material ParametersElastic modulus of surface layer 1.097x10-1 4 4Elastic modulus of base layer 1.6324x10-2 6 6Elastic modulus of subgrade 8.09x10-5 15 10Poisson’s Ratio of surface layer 3.17x10-4 8 13Poisson’s Ratio of base layer 2.4504x10-1 3 3Poisson’s Ratio of subgrade 2.7209x10-2 5 5Thickness of surface layer 8.162x10-3 1 1Thickness of base layer 5.081x10-3 2 2Regression coefficientsK1 2.87x10-4 9 14K2 1.519x10-3 7 11

The amount of fatigue cracking for the two design sections which have 49.37%

and 36.77% fatigue reliability (Ghosh, 2005) are 13% and 3.46 % respectively at

the end of service life.

65

66

CHAPTER 7SUMMARY & CONCLUSION:

7.1. CONCLUSIONS:

On the basis of the studies carried out in this thesis, the following concluding

remarks can be made.

The developed computer program ‘FATEVA’ is capable of estimating damage

ratio (Df) and percentage cracking (%C) considering fatigue distress of the

pavement. Damage ratio (Df) is a parameter which is more useful for analysis

point of view, whereas, percentage cracking (%C) is more applicable for visible

detection of fatigue failure and field estimation of the same. Besides these the

developed program can also estimate the changes of elastic modulus of

bituminous surface and fatigue strain at the bottom of same surface due to

repeated application of traffic load.

FATEVA can be suitably utilized in design process. In the ME design approach

pavements are designed following iterative procedure. A trial section is taken

based upon experience and analyzed mechanistically for given the traffic load

and environmental conditions. If the distresses (fatigue, rutting etc.) are found to

be within the permissible limits then trial section is treated as design section.

Otherwise the section will be modified iteratively till the distresses come within

the permissible limit. Here, FATEVA can be used to check the fatigue

performance of a trial section in terms of both ‘Df’ and ‘%C’ criteria as per local

requirements.

Besides its usefulness in the design process, FATEVA can be utilized in

Pavement Management System (PMS) to prepare different rehabilitation plans as

per desired fatigue performance of the design section. For in service pavements

with large extent of fatigue damage rehabilitation plans can be drawn up by

either of the following two ways:

1. Rehabilitation by increasing the thickness bituminous layer.

2. Rehabilitation by use of better quality of materials.

67

In such situations, FATEVA can be utilized to determine iteratively the

necessary increment in thickness of the same material or to select an improved

material with high stiffness for the same thickness.

For the failed pavement sections, maximum fatigue damage occurs during the

last 1/3rd of the service life. During this period the rate of cracking increased

sharply. To be more specific, in a service design life of 15 years severe damage

occurs during 12-15 of service life. (Refer Section 6.6)

With the increase of service life the magnitudes of fatigue strain increases,

however the rate of increase of fatigue strain decrease with the pavement age.

(Refer Section 6.6)

From the result of sensitivity analysis it has been found that, thickness of surface

layer , granular base layer and stiffness of bituminous concrete are the three

major design parameters which influence the fatigue damage of the pavement in

a large extent.

7.2. SCOPE OF FURTHER WORK:

The present study can be extended along the following lines:

Probabilistic values of very limited no. of design inputs are available in literature.

Hence estimation of these parameters based upon local conditions instead of

assumptions would make the computation more reliable.

Use of traffic load spectrum instead of ESAL data can be utilized to rationalize

the wheel load estimation.

Inclusion of temperature model to estimate the stiffness of bituminous layer

would make the study capable of evaluating the temperature effects on pavement

performance.

For more accurate probability analysis more rigorous method of reliability

analysis such as ASM method should be employed instead of MFOSM method,

which is applied for the present study with following assumptions:

68

1. The performance function, g(X) is non-linear, significant errors may be

introduced by neglecting higher order terms in the Taylor’s series

expansion.

2. This method fails to be invariant to different mechanically equivalent

formulation of the same problem

It is also desirable to verify the results by direct Monte-Carlo simulation.

For present study, all the variables have been taken as normally distributed and

un-correlated to each other. Other types of distributions viz. beta distribution,

Poisson’s distribution should be applied for the variables along with their

correlation.

Non destructive tests can be done on pavement sections to measure the changes of

materialistic properties like stiffness, plastic strain etc. of bituminous concrete due

to repletion of traffic load.

69

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75

ANNEXURE 1

Generally, Flexure Fatigue Beam Test is done on bituminous concrete sample

in the laboratory to find the fatigue life of the same. Ohio Research Institute of

Transportation and Environment (ORITE) (Report No.FHWA/OH-2001/14; Title

Pavement Performance Testing; December, 2001) had done similar tests on number of

bituminous concrete sample and produced a typical graph of ‘Flexural stiffness versus

Logarithm of Load repetition’ for the samples, as shown below in Fig. A-1.1

Figure A.1.1: Typical graph of stiffness vs. log (no. of load cycle)

(Taken from Report No.FHWA/OH-2001/14; Title Pavement Performance Testing;

December, 2001)

From the Figure A-1.1 it is evident that flexural stiffness of bituminous

concrete varies linearly with the logarithmic value of load repetition. Based on this, linear

equations having variables stiffness (E1) and log (no. of load cycle) (log N) have been

developed to get ideal relation between flexural stiffness of bituminous concrete and

traffic load repetition as follows.

E1= M x log N + C (A.1.1)

76

From the Eq. A.1.1 , using known values of E1 (in GPa) and N, values of M

and C have been calculated for different design traffic (N) are presented in Table A.1.1.

Table A.1.1: Values of M and C for Different Design Traffic

Design Traffic (MSA) M C

50 -0.064 1.9983

100 -0.613 1.9776

150 -0.596 1.9698

77

ANNEXURE 2

The derivatives of the terms used in the Equation 6.11 (Section 6.2.1) are obtained as

follows:

78