MURANA, ABDULFATAI ADINOYI

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1


RELIABILITY
-
BASED PAVEMENT PERFORMANCE MODEL




BY




MURANA, ABDULFATAI ADINOYI

M.Sc./ENG/06840/2006
-
07




AN M.Sc SEMINAR PRESENTATION SUBMITTED TO THE DEPARTMENT OF CIVIL
ENGINEERING, FACULTY OF ENGINEERING, AHMADU BELLO UNIVERSITY, ZARIA,
NIGERIA



SUPERVISORS: ENGR. (DR.) A.T.OLOWOSULU

DR. I. ABUBAKAR



SEPTEMBER, 2010

2


ABSTRACT

In Mechanistic
-
empirical (M
-
E) pavement design, the Monte Carlo method has proven to
be an effective

means of determining reliability.
T
he
Pavement Performance models propose
d by

Craus et al., Finn et al., FHWA
-
ARE and
that for
rutting were tested.

Craus and Finn

Pavement
Performance

model
s result

in
best fit for the
damage

reliability

relationship

in terms of reliability
values as a result of increased axle load application
.
Craus and Finn

Pavement Performance

model
s are not that sensitive to increase in axle load application

compared to

FHWA
-
ARE

model
for all levels of
number of Monte Carlo simulation studied
.
Craus and Finn

Pavement
Performance

models show non
-
conservative
estimation of the fatigue life in pavement

and the

modulus term for the asphalt layer
included in their equation is

to capture the relationship between
stiffness and fatigue cracking
.
The

Finn

Pavement Performance
model shows the most promise in
terms of d
eveloping a quick predictive model for pavement reliability relationship between
stiffness and cracking.

FHWA
-
ARE
Pavement Performance

variability is most affected by the
inputs closer to the pavement surface i.e. asphalt modulus.

Axle weight has an overwh
elming
effect on the output variability in terms of fatigue and rutting therefore deserve careful
characterization.
T
he results from the parametric study demonstrated
that the
minimum number of
Monte Carlo simulation cycles that should be used for most pra
ctical design scenarios to provide
enough sufficient repeatability for damage reliability relationship is 2, 000 cycles.


INTRODUCTION

1.1

Preamble

The empirical
-
mechanistic
(
E
-
M
)
based method of pavement design

is based on the mechanics of
materials, which
relates input

such as a wheel loads to output such as pavement response. The

response is then used to predict pavement distress
(
including

cracking
)
and performance based on
laboratory experiments and

field testing

(
Huang 1993
)
. In the E
-
M based

methods of

pavement
design, a number of failure criteria related

to specific distress and pavement performance must be
established

based on theory and field observations

(
Aliand

and Tayabji 1998
)
.

As one of the
important factors of E
-
M based pavement design,

fatigue

cracking has been reported as the most
prevalent

form of structure distress of flexible
(
i.e., asphalt concrete
)
Pavements

(
Finn 1973
)
.
Among the many factors

causing flexible pavement cracking, traffic loading, subgrade

characteristics, and the environme
ntal factors are the primary elements.

Many flexible pavement
3


design methods consider traffic

load induced fatigue cracking as a major design criterion
(
Huang

1993
)
.

In view of the fact that predicted pavement distress varies a

great deal at the end of the

designed servicing period, it is more

reasonable to introduce probabilistic approaches to pavement
design

and management, since there is significant variability in predicting

traffic loading,
environmental conditions, and construction

quality

(Lu sun et a
l., 2003)
.

Lemer and Moavenzadeh
(
1971
)
, and

Darter and Hudson
(
1973
)
were among the first to
introduce the

reliability concept to pavement design and management. Reliability

concepts were
also incorporated in the Texas flexible pavement

design systems
(
Irick et al. 1987; Uzan et al.
1990
)
and in

the AASHTO Design Guide
(
AASHTO 1993
)

(
Lu sun et al., 2003)
.

Flexible pavement fatigue cracking is usually controlled by the

maximum tensile stress at
the bottom of the asphalt layer. A number

of predictive mode
ls of fatigue cracking have been
developed

over the past three decades to characterize traffic load induced

fatigue cracking. To
predict fatigue cracking of flexible pavement, damage

needs to be cumulated according to certain
rules. The most popular

rule o
f these rules is the Miner’s law. Cumulated damage is

interpreted as
degree of fatigue deterioration of flexible pavement

due to traffic loading
(Lu sun et al., 2003)
.

It is well known that rutting increases at an increasing rate during the initial years
of
operation, and then stabilizes with time (Monismith and Tayebali 1988; Roberts et al. 1996).
Research has shown that the pavement response under traffic loading

consists of both recovery
and irrecovery components (Perl et al. 1983; Sousa and Weissman 19
94; Uzan 1996). The
recovery part is elastic in nature, while the permanent part is related to plasticity. In general,
elastic strain remains constant, while plastic strain decreases with increasing load application
numbers (Jian
-
Shiuh et al., 2004).

The r
ole of reliability in pavement design is to quantify
the probability that a pavement
structure will perform, as intended, for the duration of its design life.

Many of the parameters
associated with pavement design and construct
ion exhibit natural variabili
ty
(Timm
et al, 1999).

D
e
t
e
r
mi
n
i
ng

r
e
li
ab
ili
t
y

r
equ
i
r
es qu
a
n
t
i
f
y
i
ng

t
he

v
a
r
i
ab
ili
t
y

of

t
h
e

i
n
put

v
a
l
ues
(s
u
ch

a
s

l
a
y
er
t
h
i
c
k
ne
s
s

and

m
o
du
l
u
s
)

and

t
h
en

u
s
i
ng

t
ho
s
e

v
a
l
ues

t
o

e
s
t
im
a
t
e

t
he

v
a
r
i
a
b
ili
t
y

of
t
he

ou
t
p
u
t

(
exp
e
c
t
ed

pa
v
e
m
e
n
t
li
f
e
)

(Bruce, 2001).

Development of analytical (mechanistic) methods for
resilient response model of pavements may be traced back to Burmister
,

(1943), who presented
a method to determine stresses, strains and displacements in a two layer elastic system based on
axi
-
symmetric
analysis of verti
cal load using stress function
(Huang, 1993).

4


1.
2

Statement of Problem

A mechanistic
-
empirical (M
-
E) pavement analysis and design has been proposed for use
in Nigeria (Olowosulu, 2005). A brief summary of the process is described below:

Pa
rt 1
consists of the development of input values, which include traffic, climate and material. Part 2
of the design process is structural/performance analysis. It also shows the step
-
by
-
step
procedure of M
-
E, starting with an assumed initial layer thicknes
s through selection of the
optimum layer thickness. The analysis is an iterative trial
-
and
-
error solution. Initially with
assumed layer thickness, the critical stresses and strains are computed using the ELYSM 5
computer program. These are then compared wi
th relevant failure criteria. When any criterion is
exceeded, the thicknesses are adjusted. This procedure is repeated until all failure criteria are
satisfied.

There is statistical variation in the input parameters. Consequently, there is variability in
t
he calculated stresses and strains that lead to variations in the number of allowable loads. There
is also variability in the number of expected loads during the design period. Finally, there is
variability in regard to the transfer functi
ons that predict
pavement life.

The
component of
concern was the pavement performance model

that predicts pavement life in flexible pavement
thickness design
.

There are several pavement performance models proposed by researchers in
which there is need to propose that for N
igeria.

1.
3

Aim and Objectives

1.
3
.1

Aim
:

To
propose

the

transfer functions
that predicts

pavement life for Nigerian Empirical
Mechanistic Pavement Analysis and Design System.

1.
3
.2

Objectives:



To assess the different fatigue transfer functions and
proposes the one that best fit
NEMPAD
S
.



To generate variability of design parameters using
Monte Carlo method with the aid of
MATLAB.



Assess the effects of
axle loads application on pavement

analysis and design reliability.



To establish the minimum number
of Monte Carlo simulation required to provides
enough sufficient repeat
ability for damage reliability relationship.

5


1.
4

Location of Study

The
study area
belongs to

region one (
i.e. Kano

and Kaduna

states
)

of Claros et al
., 1986
designation
. The Master Test Section
(MTS)
number for this location was 1
-
1. It has a

route
number A236
. The distance in kilometer from the first node is 35 km.
the study route is from
node
number 241 and node name Zaria (A126) to node number 394 and

name
A236/state
road
.

This zone has mean annual rainfall of less than 1, 000 mm with a rainy season of two to three
months. Also, wMAAT

for this region were 24 for Kaduna and 22 for Jos.



LITERATURE REVIEW

2.
1

Transfer

F
unctions

The empirical component of M
-
E design is
pavement life equation, known as a transfer
function. Transfer function use pavement responses calculated by the mechanistic model and
predict the life of pavement in terms of fatigue cracking or rutting.
It acts

as a chain between the
pavement reactions and appeared damages in the pavements (Ameri and Khavandi, 2009).

It
relates

the

pavement

responses determined

from

mechanistic

models

to

pavement

performance

as
measure
d

b
y

th
e

typ
e

an
d

severit
y

o
f

distres
s

(rutt
ing
,

cracking,
roughness,

and

so

forth)

(
Thompson

and Nauman,

1993).


It is generally recognized that the allowable number of traffic

load repetitions is closely
related to tensile strain at the bottom of

the asphalt layer. A universal form of the fatigue
law used
to

predict fatigue
-
cracking life of flexible pavements
(
Finn 1973;

Finn et al. 1973, 1977
)

is














(2.
1
)

where
ϵ =
maximum tensile strain at bottom of asphalt layer;

E

=
resilient modulus
(
i.e.,
stiffness
)

of the asphalt layer;
k
i

(
i

=
1,2,3) are parameters of fatigue law; and
N

=
total number of

load repetitions to failure.

2.
2

Reliability Analysis

On
e of the methods cited by Harr,
(1987) is the exact methods which includes numerical
integration and Monte C
arlo simulation. Briefly, the Monte Carlo method involves artificially
reproducing each input distribution, entering the values into the function, and obtaining the
output distribution. The primary advantage of an exact method is that the complete probabil
ity
distribution of the dependent random variable is determined (Harr,

1987).

6


2.
2.1

Techniques for incorporating the variation of the input parameters

There are two generally accepted techniques for accommodating the variation of the input
parameters in t
he design model. These are the Monte
-
Carlo (
Jooste
,

1999
) and Rosenblueth
(
Eckmann
, 1997
) techniques.

The Monte
-
Carlo simulation technique randomly generates huge numbers of input data
sets from the known distributions of the input parameters while
adhering to the distribution
characteristics of the individual input parameters. These input data sets serve as input to the
structural analysis model and by running the structural analysis model successively using the
different input data sets, a distribu
tion of the resilient pavement response parameters is generated.
The distribution of the pavement response parameter in turn serves as the input to the pavement
performance model

(
Jooste
,

1999
).

Monte Carlo simulation is a type of simulation that relies on

repeated random sampling
and statistical analysis to compute the results. Monte Carlo simulation can be considered as a
methodical way of doing so
-
called
what
-
if

analysis
. In Monte Carlo simulation, we identify a
statistical distribution which we can use
as the source for each of the input parameters. Then, we
draw random samples from each distribution, which then represent the values of the input
variables. For each set of input parameters, we get a set of output parameters. The value of each
output param
eter is one particular outcome scenario in the simulation run. We collect such output
values from a number of simulation runs. Finally, we perform statistical analysis on the values of
the output parameters, to make decis
ions about the course of action.
We

can use the sampling
statistics of the output parameters to characterize the output variation (
Samik, 2008)
.

The Rosenblueth technique is actually a point estimate approximation technique whereby
the continuous distribution of a particular input parameter

is approximated by a discrete
distribution of two adequately chosen values of that input parameter. The criteria for selecting the
two discrete values are that the first three statistical moments of the continuous and discre
te
distributions must be equal
(
Eckmann
, 1997
)
.

2.
2
.
2

Methods for incorporating variation and reliability in pavement design

Figure
2.1

below
shows a simplified diagrammatic representation of a mechanistic
-
empirical design procedure. The two highlighted blocks of the diagram represents
the components
of the process where measured data are input. Every single input parameter that is measured
empirically and entered into the system has a certain variation associated with it because of the
7












natural variability of the parameter and error in t
he measurement technique which is hopefully
small. Assuming that the variation in the model is a true reflection of the actual variation of the
physical system, there are therefore two entry points for introducing variability in the design
process namely i
n the input data which characterize the system and in the performance models
which model the distress or deterioration of the system in response to loading.


1
.


System

geometry

input

2
.


Load

characterization

3
.


Material

input

parameters: Resilient

properties
Strengt
h

properties



Structura
l

analysi
s

model:
Pavemen
t

response

F

an
d

,




Pavement

performance

model:
Transfer

function




Pavemen
t

bearin
g

capacit y
estimate




N
o


Adequate

?



Yes


Final

pavement

design


Figur
e

2.1
:

Schemati
c

diagra
m

o
f

a

mechanistic
-
empirica
l

desig
n

procedure


METHODOLOGY

3.1

Analysis of Pavement Structure

In
mechanistic design method, the pavement is idealized as a layered elastic structure
consisting of various sub
-
layers of
asphalt concrete

surfacing, granular base, sub
-
base, and the
sub
-
grade. Materials are assumed to be homogeneous and isotropic. The layers are horizontally
infinite with each layer characterized by its resilient modulus
M
R

and Poisson’s ratio
μ
. Horizontal
tensile strain
ε
t

at the bottom of the
asphalt concrete

layer and vertical compressive strain
ε
z

on the
subgrade are identified as the causative factors for fatigue and rutting failures, respectively (
Shell
8


1978;
Thickness
1981). Based on the algorithm for the analysis of a layered elastic system
for
Nigeria
(
Claros et al., 1986
), a computer program
NEMPADS

was developed by
a researcher

(
Olowosulu, 2005
) for computation of stresses in a pavement structure
using the concept o
f
standard computer programs ELSYM
5

developed by other organization
.

NEMPADS

uses a modified version the Monte Carlo method described by Timm et al.
(
1999
). A flow cha
rt representation of the NEMPADS

procedure is shown in Figure
3
. T
he steps
in the NEMPADS

Monte Carlo simulation are as follows:

1.

Randomly select input values from their respective probability distributions.

2.

Calculate the damage using Miner’s Hypothesis.

3.

Perform enough cycles to generate a repeatable output distribution.

4.

Determine the number of

cycles that resulted in
Damage
< 1.

5.

Calculate reliability according to Equation
3.1
:














































3.
2

Characterization

of Pavement Material

3.2.1

Pavement layer thickness

Pavement layer thicknesses

were taken as normally distributed

with coefficient
of
variation of 5%, 8% and 15%
for asphalt concrete, granular base and granular subbase
respectively. The respective pavemen
t layer thicknesses for asphalt concrete, granular base,
granular subbase and subgrade were 2.5, 5.5, 2.8, and 300 inches respectively.

3.2.2

Layer modulus

The resilient modulus (M
R
) is a measure of the elastic property of a soil recognizing
certain non
-
li
near characteristics. Resilient modulus can be used in a mechanistic analysis using
multi
-
layer elastic systems for prediction of cracking, rutting, etc (Claros et al., 1986).
The
resilient modulus of all the materials (asphalt concrete, granular base, gra
nular subbase, and
subgrade) w
ere

taken to be log
-
normally distributed

(Timm et al., 1999)
.

3.2.
2.1

Subgrade materials

Soils usually display stress dependent resilient behavior characterized by a resilient
modulus M
R
.
S
oils were collected from several locations on various National Highways, and the
9


resilient modulus test was conducted in a triaxial device equipped for repetitive loading in
which the confining pressure and deviator load are varied. The load and deformatio
n were
recorded during the recommended 200 repetitions at each load
setting and confining pressure.

The subgrade modulus obtained after adjusting the modulus for representative stresses
by
the
Nigerian overlay design methodology research

for the route sele
cted for study was
26,
000 PSI. Based upon literature review

(Timm et al., 1999)
, the modulus of the unbound
materials

was taken to be

log
-
normal distribution
and

a practical C
oefficient of
V
ariation

of
40% was
adopted

for the unbound materials of the sele
cted route.

3.2.
2
.2

Granular base and subbase

The resilient modulus of granular materials is stress dependent, and it varies both in
radial and vertical directions because of different levels of confinement and traffic loading
(
Animesh and Pandey,
1999
)
.
The promising factors which represent the in
-
situ

conditions are
water content, density, load duration, and stress state

(Claros et al., 1986)
.


T
he Nigerian overlay design methodology estimate
d

the elastic modulus of bases and
subbases
to be
90, 000 PSI a
nd 45, 000 PSI

respectively (Claros et al., 1986)
. Based on the
conclusion of Timm et al. (1999) and other researchers, a log
-
normal distribution was also used
for the granular base and subbase materials. A practical
modulus
C
oefficient of
V
ariatio
n

of
30% was adopted
for granular base and subbase materials
respectively for the route selected

Timm et al. (1999)
.

3.2.
2.3

Asphalt concrete

For asphalt concrete samples the dynamic indirect tensile test (ASTM D
-
4123) was used
to estimate modulus. The most
significant factors that represent the in
-
situ conditions are
temperature, load duration, and stress state.
The estimate
of
the elastic modulus of
asphalt
concrete

from the resilient modulus test using the standard test AASHTO T
-
274

for the study
area was
taken to be 900,000 PSI (Claros et al., 1986).
Based upon the several research carried
out, the practical modulus C
oefficient of
V
ariation

of 20%
was adopted
for asphalt concrete of
the selected route.

3.2.
3


Poisson’s ratio

Poisson’s ratio is the ratio of

transverse strain to axial strain when a material is axially

loaded. Yoder and Witczak (
1975
) cite that, for most pavement material, the influence of many
factors on Poisson’s ratio is generally small. According to Pavement Evaluation Unit of
10


Nigerian Fed
eral Ministry of Works and Housing, asphalt concrete is highly dependent upon
temperature, where Poisson’s ratio varies between 0.15 at cooler temperatures (less than 30
0

F)
to 0.45 at warmer temperatures (120
0

F plus), with a typical value of 0.35, cement

stabilized
bases tends to increase Poisson’s ratio value towards 0.30 from sound (crack free) to value of
0.15 as a result of degree of cracking in stabilized layer, with a typical value of 0.20, granular
base/subbase uses lower Poisson’s ratio value of 0
.30 for crushed material and high Poisson’s
ratio value of 0.40 for unprocessed rounded gravels/sands, with a typical value of 0.35, and
subgrades Poisson’s ratio value depends on the type of subgrade soil i.e. Poisson’s ratio value of
0.30 is use for cohe
nsionless soils while P
oisson’s ratio value of 0.50 is
for very plastic clays
(cohesive soils), with a typical value of 0.40.

3.2.
4


Traffic input

Currently, there is a choice between using Equivalent Single Axle Loads (ESALs) or load
spectra. An ESAL is
defined as an 80 kN (18 kip) dual tire axle load. The load spectrum consists
of a combination of single and dual tires in single, tandem, or tridem axle configurations

(
Bruce
,

2001).
The t
raffic
was

modelled

as an equivalent number of standard s
ingle axle
loads of 80 kN
with a contact stress of
55
0 kN/m
2

and a loaded radius of 1
52

mm.

3.
3

Layered
-
Elastic Analysis Output

The LEA model calculates normal stresses, strains, and deflections a
s well as shear
stresses at any
point in the pavement structure. In
NEMPAD
, critical strains are used to determine
damage and reliability. The critical strains are the tensile strain at the bottom of the asphalt layer
and the compressive strain at the top of the subgrade

(
Olowosulu, 2005
)
.

3.
4

Monte Carlo Simulation and Re
liability Formulation

When a distribution
is characterized by a well
-
known function (e.g., normal or
lognormal), it is possible to work directly with equations to artificially
generate

the distribution
Timm et al. (1999). Box and Muller (
1958
) have shown t
hat if U1
1

and U
1
2 are two
independent standard uniform variates, then

S11 = sqrt (
-
2

×
log

(U11))

×
sin

(2

×

pi

×

U12)


Equation 3.
2

S12 = sqrt (
-
2

×
log

(U11))

×
cos

(2

×

pi

×

U12)


Equation 3.
3

are a pair of statis
ti
cally indepen
den
t standard normal
variates. Therefore, a pair of random
numbers from a normal distribution (N(μ,σ)) may be obtained by:

H1

=

[M1

+

(D1*S11)





Equation 3.
4

11


H2

=

[M2

+

(D2*S12)





Equation 3.
5

Equations

3.
2

to

3.
5

can then be repeated for layers 3 and 4.
For lognormally
distr
ibuted
modulus values,
e
quation
s

3.
2

and 3.
5

can again be used to generate
S
11 and
S
12. For a
lognormal variable
E
and transformed variable
Y
= ln(
E
),
e
quation
s

3.
6

and 3.
7

can be used to
calculate the standard deviation and mean of the transformed variable, respectively.

D1

=

sqrt

(log

(CV^2+1))




Equation 3.
6

M1

=

log

(M)

-

((D1^2)/2)




Equation 3.
7

Finally, two E values (log
-
normally distributed) are calculated by:

E
1
=

exp

(M1

+

(D1*S11))




Equation 3.
8

E2 = exp

(M1

+

(D1*S1
2
))




Equation 3.
9

The various values obtained from

the above concepts were incorporated into the existing
computer program,
NEMPAD
S
. The program enables the designer

to
generate the horizontal
tensile strain at the bottom of the existing asphalt concrete layer and vertical compressive strain at
the top of the subgrade so as to set levels of input variability and evaluate their effects on the
design reliability using Microsoft Excel
.

3.5

Pavement

Performance Model

3.5
.1

Fatigue

Claros et al. (
1986
) used a version developed by Craus et al., (1984) that was calibrated
using data from the AASHO Road test for thin pavements and a failure criterion of thirty percent
class II cracking. This model includ
es a modulus term for the asphalt layer in order to capture the
relationship between stiffness and fatigue cracking.

The Nigerian version is shown in Equation
3.1
0
.










(





)




(



)

































































(



)

where

N
f
= Number of allowable 8200 kg ESAL applications,

ε
t
= Horizontal tensile strain at the bottom of the asphalt layer, and


E = dynamic modulus of the asphalt concrete in PSI

A
fatigue model developed by Finn et al. (
1977
) and similar the one developed by Craus et
al., (1984) was also evaluated

which was the version used by Olowosulu, (2005)
. This model
12


(
e
quation
3.1
1
) also includes a modulus term for the asphalt layer in order t
o capture the
relationship between stiffness and fatigue cracking.










(





)




(



)

































































(



)

where

N
f
= Number of allowable 8200 kg ESAL applications,

ε
t
= Horizontal tensile
strain at the bottom of the asphalt layer, and


E = dynamic modulus of the asphalt concrete in PSI

The FHWA
-
ARE equation which was also developed using the AASHO road test data
(FHWA report, 1975) was also evaluated. It was observed that this equation is c
onsidered a
conservative estimation of the fatigue life in pavement. The equation has the following
expression:











(



)

































































(



)

3.5
.2

Rutting

Equation
3.1
3

shows the rutting transfer function used in Nigerian Overlay Pavement
Design. This equation was also calibrated for NEMP
AD

as described by Claros et al (
1986
).











(



)

































































(



)

whe
re

N
r

= Number of allowable 8, 200 kg ESAL application.

ε
v

= Vertical compressive strain at the top of the subgrade

3.5
.3

Miner’s hypothesis

In the simplest case (single load configuration and no seasonal variations in material
properties), the damage over the life of the pavement can be characterized by
e
quation
3.1
4
:


































































where

Damage
= an index indicating the expected level of damage after
n
load applications
(
Damage
≥ 1 indicates pavement failure)

n
= applied number of loads

N
= number of loads required to cause failure (based on empirical transfer functions)
.

13



Table
3
.1: Reliability values for fatigue models with 1, 000 cycles

ESALs

CRAUS ET AL.

FINN ET AL.

FHWA
-
ARE

337750

99.6

99.7

85.4

506625

99.5

99.6

74.7

675500

98.9091

99.2

67.8636

844375

97.84

99

60.2

1013250

95.15

98.04

53.4

1182125

92.35

96.52

48.5

1351000

89

93.8

43.7727

1519875

85.4

91.4

40.7

1688750

81.2

88.95

36.9

1857625

77.8

85.9

34.7

2026500

73.7

82.9

32


Table
3
.2: Reliability values for fatigue models with 1, 500 cycles

ESALs

CRAUS ET AL.

FINN ET AL.

FHWA
-
ARE

337750

99.6667

99.7333

85.7

506625

99.5333

99.6667

75.3

675500

98.92

99.2667

67.9

844375

97.9

99

60.8636

1013250

95.2

98.1

54.6364

1182125

92.4

96.6

49.6818

1351000

89.2

93.9

43.96

1519875

85.7

91.45

40.7727

1688750

81.75

89.2

37.0909

1857625

78.0909

86.25

34.7273

2026500

74.4667

83.25

32.1818


Table
3
.3: Reliability values for fatigue models with 2, 000 cycles

ESALs

CRAUS ET AL.

FINN ET AL.

FHWA
-
ARE

337750

99.75

99.8

85.8667

506625

99.6

99.7

75.3333

675500

98.95

99.4

68.05

844375

97.9545

99.05

61.12

1013250

95.28

98.1364

54.9

1182125

92.6

96.6818

49.85

1351000

89.3333

94.08

44

1519875

85.8667

91.7333

41.08

1688750

81.8667

89.2667

37.4

1857625

78.1

86.4667

35

2026500

74.55

83.3333

32.56


14


Table
3
.4: Reliability values for fatigue model
s

with 2
,
200
cycles

ESALs

CRAUS ET AL.

FINN ET AL.

FHWA
-
ARE

337750

99.76

99.8

85.9545

506625

99.6364

99.72

75.3636

675500

99.0667

99.4

68.24

844375

98.3

99.0667

61.35

1013250

95.3636

98.3

54.92

1182125

92.6667

96.8

49.88

1351000

89.3636

94.1333

44.05

1519875

85.9545

91.76

41.2

1688750

81.9545

89.3182

37.55

1857625

78.16

86.5455

35.15

2026500

74.5909

83.4091

32.7


Table
3
.5: Reliability values for fatigue model
s

with 2
,
500 cycles

ESALs

CRAUS ET AL.

FINN ET AL.

FHWA
-
ARE

337750

99.7727

99.8182

86.08

506625

99.64

99.7273

75.4

675500

99.1

99.4091

68.3333

844375

97.84

99

61.12

1013250

95.4667

98.3333

55

1182125

92.6818

97.0667

50.2

1351000

89.44

94.1364

44.5333

1519875

86.08

91.8182

41.6667

1688750

82.04

89.4

37.6

1857625

78.2

86.64

35.7333

2026500

74.64

83.48

33.0667


Table
3
.6: Reliability values for
rutting

model with different cycles.

ESALs

2, 500 Cycles

2, 200 Cycles

2, 000 Cycles

1, 500 Cycles

1, 000 Cycles

337750

99.7

99.5333

99.5

99.48

99.4545

506625

98.9

98.8

98.8

98.75

98.9

675500

98.4

98.3333

98.25

98.16

98.1364

844375

97.36

97.4667

97.45

97.3636

97.36

1013250

97

96.8

96.8

96.7273

96.68

1182125

96.1

95.9333

95.9

95.7727

95.68

1351000

95.3

95.2

95.0667

94.9545

94.88

1519875

94.9

94.75

94.6

94.4545

94.4

1688750

94.5

94.05

93.9333

93.7727

93.68

1857625

93.8

93.4

93.2

93.1364

93.12

2026500

93

92.65

92.4091

92.4

92.32

15


Table
3
.7:
Reliability values for Craus et al. model with different cycles

ESALs/CYCLES

1000

1500

2000

2200

2500

337750

99.6

99.6667

99.75

99.76

99.7727

506625

99.5

99.5333

99.6

99.6364

99.64

675500

98.9091

98.92

98.95

99.0667

99.1

844375

97.84

97.9

97.9545

98.3

98.3333

1013250

95.15

95.2

95.28

95.3636

95.4667

1182125

92.35

92.4

92.6

92.6667

92.6818

1351000

89

89.2

89.3333

89.3636

89.44

1519875

85.4

85.7

85.8667

85.9545

86.08

1688750

81.2

81.75

81.8667

81.9545

82.04

1857625

77.8

78.0909

78.1

78.16

78.2

2026500

73.7

74.4667

74.55

74.5909

74.64


Table
3
.8:
Reliability values for Finn et al. model with different cycles

ESALs/CYCLES

1000

1500

2000

2200

2500

337750

99.7

99.7333

99.8

99.8

99.8182

506625

99.6

99.6667

99.7

99.72

99.7273

675500

99.2

99.2667

99.4

99.4

99.4091

844375

99

99

99.05

99.0667

99.1

1013250

98.04

98.1

98.1364

98.3

98.3333

1182125

96.52

96.6

96.6818

96.8

97.0667

1351000

93.8

93.9

94.08

94.1333

94.1364

1519875

91.4

91.45

91.7333

91.76

91.8182

1688750

88.95

89.2

89.2667

89.3182

89.4

1857625

85.9

86.25

86.4667

86.5455

86.64

2026500

82.9

83.25

83.3333

83.4091

83.48


Table
3
.9:
Reliability values for FHWA
-
ARE et
al. model with different cycles

ESALs/CYCLES

1000

1500

2000

2200

2500

337750

85.4

85.7

85.8667

85.9545

86.08

506625

74.7

75.3

75.3333

75.3636

75.4

675500

67.8636

67.9

68.05

68.24

68.3333

844375

60.2

60.8636

61.12

61.35

61.5333

1013250

53.4

54.6364

54.9

54.92

55

1182125

48.5

49.6818

49.85

49.88

50.2

1351000

43.7727

43.96

44

44.05

44.5333

1519875

40.7

40.7727

41.08

41.2

41.6667

1688750

36.9

37.0909

37.4

37.55

37.6

1857625

34.7

34.7273

35

35.15

35.7333

2026500

32

32.1818

32.56

32.7

33.0667

16


Table
3
.10:
Reliability values for rutting model with different cycles

ESALs

1000

1500

2000

2200

2500

337750

99.4545

99.48

99.5

99.5333

99.7

506625

98.7273

98.75

98.8

98.8

98.9

675500

98.1364

98.16

98.25

98.3333

98.4

844375

97.36

97.3636

97.45

97.4667

97.5

1013250

96.68

96.7273

96.8

96.8

97

1182125

95.68

95.7727

95.9

95.9333

96.1

1351000

94.88

94.4545

94.6

94.75

94.9

1519875

94.4

94.4545

94.6

94.75

94.9

1688750

93.68

93.7727

93.9333

94.05

94.5

1857625

93.12

93.1364

93.2

93.4

93.8

2026500

92.32

92.4

92.4091

92.65

93




ANALYSIS AND DISCUSSION OF RESULT

The number of Monte Carlo simulations was set at 1000, 1500, 2000, 2200 and 2500
cycles respectively in each of the phase of study.

Phase one allow the designer to know which of the fatigue model
that best fit and the
effect of axle load on pavement structure. The reliability values obtained on this phase of study
has been presented in Tables 3.1
-

3.5 for the fatigue models at different cycles and axle load
application. Table 3.6 presented the rel
iability obtained at different number of Monte Carlo
simulation for the rutting model with different axle load application.

The second
phase
allows the designer to ascertain the required number of Monte Carlo
simulation that could yield approximate better result. The reliability values obtained in each of the
study for this phase have been presented in Tables 3.7
-

3.10.

The

p
e
r
f
o
r
m
a
n
c
e

of

the

s
e
l
ec
t
e
d
p
a
v
e
m
e
nt st
r
u
c
t
u
r
e

wa
s

e
v
a
lu
a
t
e
d usi
n
g

M
onte

C
ar
lo simul
a
tions. T
a
ble
s

3.1
-

3.5

d
e
pi
c
ts the

numb
e
r

of

simul
a
t
ions
carr
i
e
d out

a
nd the

c
o
rre
spondi
n
g
re
li
a
biliti
e
s of

the

p
a
v
e
m
e
nts
f
or

the

v
a
r
ious
ca
s
e
s

c
onsid
e
re
d.
Figures 4.1
-

4.5 illustrate the
effects of each input parameter’s
variability on output variability in terms of each of the fatigue model studied at different cycles
ranging from 1, 000 to 2, 500 cycles.

17





0
20
40
60
80
100
120
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.1: Graph of ESALs Vs Reliability values for fatigue with

1, 000 cycles

Craus et al.
Finn et al.
FHWA-ARE
0
20
40
60
80
100
120
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.2: Graph of ESALs Vs Reliability values for fatigue with

1, 500 cycles

Craus et al.
Finn et al.
FHWA-ARE

18




0
20
40
60
80
100
120
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.3: Graph of ESALs Vs Reliability values for fatigue with


2, 000 cycles

Craus et al.
Finn et al.
FHWA-ARE
0
20
40
60
80
100
120
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.4: Graph of ESALs Vs Reliability values for fatigue with


2, 200 cycles

Craus et al.
Finn et al.
FHWA-ARE

19



4.1

FHWA
-
ARE

fatigue

model

Studying f
igure
s

4.1
-

4.5
, it is clear that the
FHWA
-
ARE

fatigue model resulted in t
he
best fit for the
damage

reliability

relationship

in terms of sensitivity to increased axle load
application
.
FHWA
-
ARE

fa
tigue is highly sensitive to increase in axle load application

than the
ot
her two models for all levels of
number of Monte Carlo simulation studied
.
This affirms the

conservative estimation of the fatigue life in pavement

as

was observed
b
y Claros et al.

(
1986).

4.2

Craus and
Finn fatigue

model

From

f
igure
s

4.1
to

4.5
, it
can be

concluded

that
both Craus and Finn

fatigue model
s
result

in
best fit for the
damage

reliability

relationship

in terms of reliability values as a result of
increased axle load application
.
Craus and Finn

fatigue model
s are not that sensitive to increase
in

axle load application

(Table
s

3
.1
to

3
.5
)
compared to

FHWA
-
ARE

model for all levels of
number of Monte Carlo simulation studied
. This indicates that
both

equation
s

are

good predictor
for
NEMPAD
S

fatigue

when considering high level of reliability
.

Both mod
els
show

non
-
conservative estimation of the fatigue life in pavement

and
include a modulus term for the
asphalt layer in order to capture the relationship between
stiffness and fatigue cracking

(
Claros et
al.
,

1986).

0
20
40
60
80
100
120
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.5: Graph of ESALs Vs Reliability values for fatigue with


2, 500 cycles

Craus et al.
Finn et al.
FHWA-ARE

20



92
93
94
95
96
97
98
99
100
101
0
337750
675500
1013250
1351000
1688750
2026500
Reliability %

Equivalent Single Axle Load (ESALs)

figure 4.6 : Graph of ESALs Vs Reliability value for rutting with

2, 500 cycles

rutting
73.6
73.8
74
74.2
74.4
74.6
74.8
0
500
1000
1500
2000
2500
3000
Reliability %

Number of cycles

Figure 4.7 : Reliability value Vs No. of Cycles for 2026500 ESALs using
Craus et al. Model

Craus et al.

21





82.8
82.9
83
83.1
83.2
83.3
83.4
83.5
83.6
0
500
1000
1500
2000
2500
3000
Reliability %

Number of cycles

Figure 4.8 : Reliability Vs No. Of Cycles for 2026500 ESALs using Finn et
al. Model

Finn et al.
31.8
32
32.2
32.4
32.6
32.8
33
33.2
0
500
1000
1500
2000
2500
3000
Reliability %

Number of cycles

Figure 4.9 : Reliability Vs No. Of Cycles for 2026500 ESALs using
FHWA
-
ARE Model

FHWA-ARE

22


Figures 4.7,

4.8
,

4.
9

and 4.10 illustrates the effect of number of Monte Carlo simulation
cycles required on different axle load application for Craus fatigue, Finn fatigue, FHWA
-
ARE
fa
tigue and NEMPAD rutting models
respectively.

Studying
these
figures
,
the designer noticed a
n increased in reliability values as the
number of Monte Carlo cycles increases from 1, 000 to 2, 000 and recorded

no much
difference in reliability values
for

2, 200 and 2, 500 Monte Carlo
simulation cycles.
So, it can
be observed that Monte Carlo simulat
ion cycles less than 2, 000 cycles would not provides
enough sufficient repeatability for damage
reliability relationship (Bruce
, 2001) and

Monte
Carlo simulation cycles
of 2, 000 up to 5, 000 (Timm et al., 1999) as the case may be could

enough to provide
sufficient repeatability for damage reliability relationship.


5.1

Conclusions



Axle
weight has

an overwhelming effect on the output variability in terms of fatigue and
rutting therefore deserve careful characterization.




The
minimum
number of
Monte

Carlo

simulation cycles that should be used
for most
practical design scenarios

to provide enough sufficient repeatability for damage reliability
relationship is 2, 000 cycles.


92.2
92.3
92.4
92.5
92.6
92.7
92.8
92.9
93
0
500
1000
1500
2000
2500
3000
Reliability %

Number of cycles

Figure 4.10: Reliability Vs No. of Cycles for 2026500 ESALs using
Rutting Model

Rutting

23



FHWA
-
ARE fatigue variability is most affected by the inputs closer to the pavement
surface i.e. asphalt modulus.




Monte Carlo simulation is an effective means of incorporating reliability analysis into the
M
-
E design process for flexible pavements.




Reliability may be defined as the probability that the allowable number of loads exceeds
the expected actual number of loads (R = P

[N>n]) which is in consistent with other
definitions of reliability.




The
NEMPAD
S

(Finn)

fatigue model shows the most promise in terms of developing a
quick predictive model

for pavement reliability

relationship
between stiffness an
d
cracking is to be considered.

5.2

Recommendation


Many more simulations are required to encompass the full range of pavements types and
CV values for

both thickness and modulus.

A neural network may prove useful in developing a
comprehensive predictive model for pavement

reliability.

A quick method of calculating
reliability will greatly speed up the process of evaluating preliminary

pavement designs.


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(
1998
):
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c

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lireza
, (2009):
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93:

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