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.
FHWAARE
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.
FHWAARE
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.
FHWAARE
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.
FHWAARE
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.
FHWAARE
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
FHWAARE
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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