AIR 000-248
AIR 000-248
May 1988
REVIEW OF THE FOURTH POWER LAW
by
D.F. KINDER
Senior Research Scientist
and
M.G. LAY
Executive Director
t
4 ..
,•.
.-
L;BRAFtY
AUSTRALIAN ROAD RESEARCH BOARD
INTERNAL REPORT
AUSTRALIAN ROAD RESEARCH BOARD
500 BURWOOD HIGHWAY
VERMONT SOUTH, VICTORIA
This document was prepared as an internal report for
circulation within the then Australian Road Research Board
and to selected stakeholders.
It may have received a level of internal review at the time of
compilation but was not intended to be an externally
published document.
For historical purposes it has now been included within the
ARRB Knowledge Base.
ARRB Group, 2015
-
AUSTRALIAN ROAD RESEARCH BOARD
REPORT SUMMARY
THE PURPOSE OF THIS REPORT
To provide a definitive current statement on the fourth power law and pavement damage caused by trucks.
THIS REPORT SHOULD INTEREST
Persons concerned with assessing the damage caused by trucks.
THE MAJOR CONCLUSIONS OF THE REPORT ARE
The 4th power law can be applied to the rutting of asphalt pavements.
An exponent of 2 is relevant to the fatigue cracking of asphalt pavements, and
FHWA, World Bank and OECD reports give exponents for other distress modes and pavement types.
AS A CONSEQUENCE OF THE WORK REPORTED, THE FOLLOWING ACTION IS RECOMMENDED
The exponent values in the conclusions be used until research provides better values.
The limitations on the use of the exponents be clearly recognised.
Further laboratory tests and associated mathematical modelling be undertaken as defined in this report.
RELATED ARRB RESEARCH
CUT OUT INFORMATION RETRIEVAL CARd -
KINDER, D.F. and LAY, M.G. (1988) : REVIEW OF THE FOURTH POWER LAW. Australian
Road Research Board. Internal Report AIR 000-248. 53 pages, including 4 figures and 4 appendices.
KEYWORDS
Pavement/deterioration/estimation/mathematical modelllony/performance/evaluation/axle Ioad/AASHTO road test/present serviceability index*Ievennessldamage/wear/four power
law*
ABSTRACT This paper provides a definitive current statement on the fourth power law and pavement
damage caused by trucks. It contains (1) a review of the major data sources of power law exponents, (2)
performance predictions for the Somersby pavement as they relate to power law modelling, and (3) a
description of some misconceptions about, and limitations on the use of, the fourth power law. The
general conclusion is that the fourth power law represents the best available single tool for estimating
pavement wear due to traffic but, it must be used with care and within its limitations. In particular it is
recommended that the fourth power law be applied to the rutting of asphalt pavements, whereas an
exponent of about 2 is relevant to fatigue cracking. FI-IWA, World Bank and OECD reports give
exponents for other distress modes and pavement types. Further advances in understanding pavement
wear will come from laboratory testing and mathematical modelling. Such advances due to work at
ARRB are imminent.
*Non IRRD Keywords
Page
EXECUTIVE SUMMARY
INTRODUCTION.
1
LITERATURE REVIEW.
3
2.1
2.2
2.3
2.4
2.5
2.6
The
The
The
The
The
The
AASHO Road test.
FHWA Report.
Nantes Test Track paper.
Paterson report.
OECD Report.
NAASRA Pavement Design Guide.
THE ALF TEST AT SOMERSBY.
9
3.1 The rutting data.
3.2 Mechanistic analyses using laboratory
data.
3.3 Mechanistic analyses using NAASRA
damage relationships.
PRACTICAL APPLICATION OF THE 4TH POWER LAW.
13
4.1 Computations of relative life and
relative damage.
4.2 Doubling the load does not necessarily
produce 16 times the damage.
4.3 The 4th power law has an upper limit.
4.4 A higher power law exponent does not
necessarily imply greater relative damage.
4.5 A higher power law exponent does not
necessarily imply a higher number of
standard axles.
4.6 The 4th power law should not be
unquestionably applied to sub- and
super-standard pavements.
4.7 The 4th power law is not unique to road
pavement damage.
4.8 The 4th power law does not apply directly
to road pavement costs.
4.9 The 4th power law does not deal directly
with dynamic loads.
4.10 The 4th power law may not be valid when
applied directly to gross vehicle mas.
SUMMARY AND DISCUSSION.
5.1
5.2
5.3
5.4
32
Recommended power law exponents
Limitations on the use of the 4th power law.
Understanding the nature of the 4th power law.
Further advances.
Page
7
FIGURES.
REFERENCES.
29
APPENDIX A. Basis of the 4th power law:
The AASHO Road Test.
31
A.l
A.2
A.3
A.4
A.5
Present serviceability concept
Performance model
Seasonal weighting factors
Design and load equations
Reinterpretation on a power law basis
APPENDIX B. Limitations on the use of the
AASHO-based 4th power law.
35
B. 1 The traffic.
B.2 Environmental factors.
B.3 Materials.
B.4 Pavement types.
B.5 Construction, maintenance and
administrative standards.
B.6 Small and Winston criticism.
B.7 Road Test results focus on PSI.
APPENDIX C. Relative damage.
38
APPENDIX D. Sensitivity of load transformations
to the exponent in the power law.
51
APPENDIX E. Pavement costs.
53
ABSTRACT
This paper provides a definitive current statement on the
fourth power law and pavement damage caused by trucks. It
contains (1) a review of the major data sources of power
law exponents, (2) performance predictions for the
Somersby pavement as they relate to power law modelling,
and (3) a description of some misconceptions about, and
limitations on the use of, the fourth power law.
The general conclusion is that the fourth power law
represents the best available single tool for estimating
pavement wear due to traffic but, it must be used with
care and within its limitations. In particular it is
recommended that the fourth power law be applied to the
rutting of asphalt pavements, whereas an exponent of
about 2 is relevant to fatigue cracking. FHWA, World Bank
and OECD reports give exponents for other distress modes
and pavement types. Further advances in understanding
pavement wear will come from laboratory testing and
mathematical modelling. Such advances due to work at ARRB
are imminent.
ACKNOWLEDGEMENT
The authors acknowledge the major contribution of their
colleague David Potter who pointed out that the
conclusions for a single load increment, rP, and a fleet
increment of rp's, would be quite different and that the
single increment would follow the fourth power law.
EXECUTIVE SUMMARY.
The general conclusion from this report is that the 4th
power law represents the best available single tool for
estimating pavement wear due to traffic. However, like
any useful tool it must be used with care and within its
limitations. Particular conclusions relating to its
specific applications follow.
1. Reconunended power law exponents.
The 4th power law can be applied to asphalt
pavements in Australia when rutting or NAASRA roughness
is used as a measure of pavement wear.
The ALF test at Somersby, NSW supports the 4th power
law when rutting is used as the damage index.
A power of about 2 is relevant to the fatigue
cracking of asphalt pavements.
The NAASRA (NAASRA, 1987) Pavement Design Guide
criterion for the fatigue of asphalt materials, and an
analysis of the ALF pavement at Somersby, give an
exponent of about 1.5 for the fatigue cracking of the
Somersby pavement. Autret et al. report a power of about
2 for an experimental asphalt test pavement at Nantes in
France.
FHWA, World Bank and OECD studies give powers
for other distress modes and pavement types, but they
should only be applied with caution, and whenever
possible a sensitivity analysis should be carried out to
determine the effect of varying the power concerned.
2. Overload effects.
The 4th power law says that N passes of a load
P will do as much damage as (P/PB ) 4 N passes of a
standard axle of load P8, and thus reduce the relative
life of the pavement (time to reach a terminal condition)
by a factor of 1/(P/P B ) 4 . Therefore, the factor (P/PB ) 4
may be used to predict the relative life of pávmehts in
a network when changing load limits, but maintaining the
same road network condition.
The power will be 2 for fatigue cracking (see 1(2)
above).
If a pavement is subjected to a single pass of
an overloaded- truck with axle loads P, this will produce
(P/P9 ) 4 times the damage caused by one pass of a truck
with axle loads equal to the standard axle load.
Therefore, when charging single overloaded trucks for
pavement damage, the relevant damage factor is (P/P9)4.
It is generally incorrect to use (P/P E,) 4 to
compute relative total damage due to an increase in the
fleet loads from P to P, when the timing of
maintenance/rehabilitation actions is not changed.
The 4th power law applies to pavement wear. It
does not apply to overload failure.
3. Practical application on the use of the 4th power law.
Pavement damage is related to truck axle loads
and not to truck gross mass, and so use of the 4th power
law will not be generally valid when it is applied
directly to gross vehicle mass.
The power law should not be unquestionably
applied to sub- and super-standard pavements.
The power law should not be used directly to
estimate the extra cost of new pavements built to carry
higher axle loads. Further, it is suggested that,
generally, there is a cost-effective attraction to
provide over- rather than under-designed pavements.
The 4th power law does not deal directly with
dynamic loads. For example, it may not be particularly
valid for very rough roads. Work at ARRB on truck
suspension and profilometry are moving us rapidly towards
solutions in this area.
4. Understanding the nature of the 4th power law.
The power law is consistent with other similar
laws applying to metal fatigue.
The number of equivalent axles (ESA) is an
important pavement design parameter. It is incorrect and
misleading to refer to them as loads (i.e. as ESAL).
A higher power law exponent does not necessarily
imply greater relative damage or a greater number of
standard axles.
5. Further advances.
Further advances in understanding pavement wear will
come from laboratory testing and mathematical modelling.
Such advances due to work at ARRB are imminent.
AIR 000-248
1
1. INTRODUCTION.
Road pavements deteriorate when used by heavy traffic.
Therefore, in order to achieve rational pavement
management, it is necessary to be able to estimate the
relative deterioration caused by trucks of different
types carrying different levels of payload. In
particular, reliable data is required for the purposes of
calculating provisions for truck cost recovery and
regulation with respect to pavement damage.
Heavy trucks benefit a community only if the advantage
they give of carrying freight more cheaply than lighter
(smaller) trucks is not outweighed by their extra costs,
particularly the cost of repairing any additional
deterioration the heavier trucks cause to the highway
network.
Pavement damage is related to the axle loadings imposed
by trucks. The terms wear, damage and deterioration are
used synonymously in this report, but the term wear is
probably preferable. When estimating pavement wear, it is
necessary to take into account the variation in magnitude
of applied axle loads, and one way of doing this is to
use a common mathematical tool known as t. he 4t.b power
law.
According to the 4th power law, N passes of a single axle
of load magnitude P, will produce the same damage as N9
passes of a standard axle with load magnitude P9. N9 is
referred to as the number of equivalent standard axles
(ESA) and is defined by the formula:
N9 = N.(P/P9 ) 4
It should be noted that as N. refers to a number of loads
rather than a load magnitude, it is incorrect and
misleading to refer to it as an equivalent standard axle
load, ESAL. Pedantically, it might be better to refer to
it as NESAL (number of equivalent standard axle loads)
rather than ESA. The exponent a is used in eqn(l) for
cases where the fourth power does not apply.
This paper contains:
A review of the research literature. In
particular it discusses the relevance and limitations of
the data base from which the 4th power law was originally
derived (the AASHO Road Test), and then looks at a number
of other major, generally complementary, data sources of
power law exponents.
A review of some recent results of a field trial
on a pavement at Somersby, NSW using the ARRB Accelerated
Loading Facility (ALF).
AIR 000-248
In particular, it discusses the results of
performance of the Somersby pavement using
the-art mechanistic computer models and
from laboratory tests on the Somersby
subgrade materials.
predicting the
both state-ofdata obtained
pavement and
It also discusses the results of predicting power law
exponents for rutting and cracking of the Somersby
pavement using the mechanistic computer models in
conjunction with the rutting criterion and fatigue damage
relationships in the new NAASRA Pavement Design guide.
This demonstrates the degree to which the NAASRA damage
relationships are consistent with the performance of the
Somersby pavement and a 4th power law transformation of
the load data.
A description of some misconceptions about, and
limitations on the use of, the 4th power law.
A summary and discussion of the conclusions,
recommendations, limitations and future advances
regarding the practical application of the 4th power law.
AIR 000-248
3
2. LITERATURE REVIEW.
It is useful to look at the origin of the 4th power law
because of the assumptions made in its derivation and the
quality and relevance of the data base from which it was
originally derived (the AASHO Road Test) and then to look
at a number of other major, generally complementary, data
sources of power law exponents. In all, there are six
major sources of information. They are:
2.1 the AASHO Road Test,
2.2 an FHWA report,
2.3 an OECD report,
2.4 a paper on the Nantes Test track,
2.5 a report by W.D.O. Paterson
2.6 the NAASRA Pavement Design Guide.
2.1 The AASHO Road Test.
Most references to the 4th power law cite the AASHO Road
Test (HRB, 1962) as the source, but in fact the AASHO
load/ damage relations derived from that Road Test are
more complex than a simple power law.
Appendix A contains a brief description of the AASHO Road
Test scheme of analysis, and shows how the 4th power law
can be extracted from the results.
The ratio N B/N in eqn(1) is sometimes called the load
equivalence factor, LEF. Thus
LEF =
(P/P B ) 4
(2)
If the AASHO Road Test performance equations are used to
compute an average LEF for both rigid and flexible
pavements, and the results are plptted on log/log axes,
the graph (Fig 1) is a straight line with a slope of
4.15. This is the exponent in the power law eqn (2) with
the 4.15 roundec off to 4.0,
Because a regression-type statisticai approach was
originally used to analyse the Road Test data, the Road
Test formulae may not be valid for use in other
situations unless the environment, traffic, materials,
pavement type and pavement construction methods are the
same as, or similar to, those in the Road Test. Clearly
the circumstances of the Road Test are not duplicated in
Australia.
Further, there has been some criticism of the particular
type of regression procedure used in the Road Test and
4
AIR 000-248
the focus on PSI as a single composite measure of damage.
These matters are discussed in some detail in Appendix B.
Studies since the Road Test indicate that the exponent in
the power law depends on pavement type and the mode of
distress.
2.2 The FHWA report.
In a cost allocation study in 1982 (FHWA, 1982) the FHWA
recognized that the Road Test results focused on
serviceability loss rather than the individual pavement
distresses contributing to that loss, and reported the
results of a research study by Brent Rauhut Engineering,
Inc. to better understand pavement deterioration under
the influence of traffic. The study is a step forward
from the Road Test approach, but it reflects the U.S
experience only.
The report gives load equivalence factors which were
determined from distress models based on both pavement
performance theory and on the performance of actual
pavements. However, there were not enough monitored
pavements with adequate traffic data to conclusively
verify the mechanistic pavement behaviour theory that was
relied upon in the development of the distress models.
Equivalence factors are given for both rigid and flexible
pavements and for different modes of distress. The
results are reported in the form:
E
=
Rm
where R is the ratio of the load of the axle of interest
to any reference axle and E is the equivalent effect of
the axle of interest with respect to the reference axle.
A summary of the results is given in Tables I and II.
TABLE I: LOAD EQUIVALENCE FACTORS FOR FLEXIBLE PAVEMENTS.
Single axles
Ta -idern axles
Serviceability
Loss
R47
(R/1.94)49
Alligator
Cracking
R' 3°
Rutting
R416
(R/1.98)483
Transverse
Cracking
R' 73
(R/1.53)' 92
AIR 000-248
5
TABLE II: LOAD EQUIVALENCE FACTORS FOR RIGID PAVEMENTS.
Single axles
Tandem axles
Serviceability
Loss
R3
(R/1.58)3-9 '
Faulting
R°67
(R/1.41)098
Pumping
R°83
(R/1.29)' 65
Loss of Skid
Resistance
R 3-74
(R/1.61)209
Joint
Deterioration
R416
(R/1.51)530
Cracking
R548
(R/1.56)681
2.3 The Nantes Test Track paper.
Autret et al. (1986) report the results of an accelerated
toot on an aophalt pavement on a circular tost track in
Nantes, France. The pavement consisted of a 50 mm
asphaltic concrete wearing course over an unstabilized
base material. The purpose of the experiment was to lead
to equivalence factors. It was found that the power for
cracking depended on the extent and severity of cracking
and it ranged from 1.3 to 2.1. This was in contrast to
the power for rutting which depended on the rut depth and
ranged from 8.2 to 9.6.
This range of values for cracking will be referred to
again in Section 3.3 which deals with the analysis of the
ALF pavement at Somersby, NSW. There it will be argued
that the Nantes test track result is consistent with the
analysis of the Somersby pavement and the Shell (1978)
fatigue law for asphalt materials,
2.4 The Paterson report.
Paterson (1985) includes a 70 page review of the relative
damaging power of different axle loads and
configurations. It argues that load equivalence factors
are distress specific, and that this is consistent with
mechanistic principles.
Three distress modes are represented by relative load
damage powers -of 0, 2 and 4 respectively:
(1) Power of 0:
Initiation and progression of ravelling, and skid
resistance (stone polishing in particular) fall into this
category. Such damage is essentially abrasive wear of the
AIR 000-248
surfacing, and being independent of the wheel loading can
be attributed uniformly across vehicle axles. The
marginal costs of damage involved are very small.
Power of 2:
The initiation and progression of load-associated
cracking, fall in this category, with the probable
exception of pavements with asphalt layers thicker than
100 mm. For pavements with surfacings thicker than 100 mm
a relative load damage power of 2.5 to 3 appears to be
appropriate. In reality there is a considerable variation
in the power value.
Power of 4:
Rut depth variation is the primary structural, loadrelated cause of roughness progression. Hence, both rut
depth and roughness may be represented by the same damage
power. The empirical evidence supports an average value
of 4. However major variations can occur with values
ranging from 0 to 6 or more.
2.5 The OECD report.
A study of pavement damage by the OECD Steering Committee
for Road Transport Research contains a detailed review of
damage relationships for flexible, rigid and semi-rigid
pavements (RTRP, 1988).
It further supports the view that equivalence
relationships are a function of pavement type and mode of
distress. In particular, it suggests powers of 11 to 33
for the fatigue of semi-rigid pavements , and powers of 5
to 13 for the fatigue of rigid pavements. The figures for
semi-rigid pavements are not based on empirical
load/damage relationships for pavements but on material
fatigue laws, and the figures for rigid pavements are
based on theoretical finite element studies. It would
therefore be necessary to know the strain/load
relationships for the pavements concerned before the
pavement's load/damage relationships could be deduced.
2.6 The NAASRA Pavement Design Guide.
The NAASRA Pavement Design Guide (NAASRA, 1987) contains
design procedures for the following pavement types:
Granular pavements with thin bituminous surfacing,
Flexible pavements containing asphalt and/or cemented
materials,
Rigid (cement concrete) pavements,
Asphalt or granular overlays on flexible pavements.
AIR 000-248
7
Lraffic is
pdVelIIeIlL Lypes (1) and (4), design
FOL
expressed in ESA and the damage exponent is 4. For
pavement type (2), three modes of distress - fatigue
cracking of asphalt, fatigue cracking of cemented
material, and permanent deformation of the subgrade - are
catered for.
Design traffic is characterised by the number of ESAs
which would cause:
the same fatigue damage in asphalt,
the same fatigue damage in cemented material,
the same permanent deformation in the subgrade.
The value of P8 is based on a single axle with dual tyres
with 80 kN load. For single axles with single tyres,
tandem axles (all dual tyres) and triple axles (all dual
tyres), the loads which produce the same damage as one
Standard Axle are 53, 135 and 181 kN respectively.
The power law exponent is 5 for fatigue damage in
asphalt, 18 for fatigue damage in cemented materials, and
7.14 for permanent deformation of the subgrade. The
exponents are derived direr.tly from t.he fatigue and
permanent deformation relationships adopted in the Guide.
These are of the form:
N
=
(3)
(k1/E)
where N is the number of repetitions of a critical strain
of magnitude E, to reach a terminal distress condition,
and k1 is a constant depending on material properties
and, in the case of asphalt, temperature and traffic
speed. E is the compressive strain at the top of the
subgrade for the rutting 'failure' criterion and the
tensile strain at the bottom of the asphalt for the
asphalt fatigue cracking criterion.
NAASRA uses the power law exponents for 'a'. This assumes
that strain is proportional to load. This is true for
linear elastic materials, provided the contact area for
the load (tyres) remains constant as the load varies.
If, for a given pavement, the material properties are
nonlinear, and/or the tyre contact area varies as the
applied load varies, it is possible to derive, using
mechanistic response models, strain/load relationships of
the form:
E
=
(4)
k2P
Substituting eqn(3) into eqn(4) gives:
N
=
(k1/(k2P''))
=
(k1/k2)(1/P)
AIR 000-248
8
Which, upon rearrangment, gives:
=
PN
(k1/k2 )
=
a constant
(5)
Therefore, by analogy with eqn(l), ab is the damage
exponent. Power law exponents as computed by this method
are given in Section 3.3 of this report.
For rigid pavements type (3), the fatigue relationship
has the form:
N
=
(k3 /(y)
where a is the magnitude of critical stress. The exponent
a varies from 15 to 24 as a varies from low to high
values. The exponent b varies from 0.8 to 0.9. The damage
exponent for load is the product a.b and hence varies
from 12 to 22.
AIR 000-248
9
3. THE ALF TEST AT SOMRSBY.
In 1984/5 the ARRB conducted a full scale field trial on
a pavement at Somersby, NSW using a the Accelerated
Loading Facility (ALF). Descriptions of ALF, the testing
program and analyses of the pavement and materials
testing results have been published in a series of ARRB
Internal Reports (Metcalf, McLean and Kadar, 1985),
(Kadar,1985a,1985b,1985c, 1986), Vuong (1986) and Kinder
(1986,1987)). The purpose of this section of the report
is to discuss:
a) the rutting data, inasmuch as it relates to the
4th power law,
predictions of the performance of the Somersby
pavement using both state-of-the-art mechanistic computer
models and data obtained from laboratory tests on the
Somersby pavement and subgrade materials, and the degree
to which the computer models are consistent with the 4th
power law, and
predictions of power law exponents for rutting
and cracking of the Somersby pavement using mechanistic
computer models in conjunction with the rutting criterion
and fatiguc damage relationships in the NAASRA Pavement
Design guide to determine the degree to which the NAASRA
damage relationships are consistent with the performance
of the Somersby pavement and a 4th power law
transformation of the load data.
3.1 The rutting data.
The composition of the Somersby pavement consists of an
asphalt surface course, a macadam basecourse, a lime
stabilized sandstone subbase and a sandstone subgrade.
The pavement was subjected to a 120 kN (dual wheel) load
over about the first metre of the test strip, and to
about 763000 applications of a 80 kN wheel load (i.e.
twice the load produced by a standard axle) over the rest
of its length. The larger force was a result of dynamic
forces being generated at the point where the wheel first
came in contact with the pavement.
Fig. 2 is a plot of the mean rut depths corresponding to
the 120 and 80 kN loads. If rutting is used as a measure
of pavement damage, one way of estimating the exponent in
a power law model is to compare, at each load level, the
number of cycles necessary to produce the same rut depth.
If this is done for the Somersby data it is found that
the 80 kN load produced a 5.4 mm rut depth after 763000
cycles, whereas the 120 kN load produced the same rut
depth after only 192000 cycles. Substituting these values
into the power law formula produces an exponent of 3.4.
AIR 000-248
10
3.2 Mechanistic analyses using laboratory data.
The mechanistic computer models used to compute the
rutting of the pavement were CIRCLY, VESYS III and a
finite element method (FEM). Unlike VESYS, CIRCLY and the
FEM do not contain rutting models but the stresses output
from CIRCLY and the FEM were used in conjunction with a
layer-by-layer rut depth model described in Kinder
(1987b) to predict rutting. CIRCLY could, and was, used
to model the wheel load as a dual and single wheel
loading whereas the other models were restricted to a
single wheel loading representation. However, it is not
the purpose of this report to describe the differences
between the models, but simply to give an indication of
how relevant the mathematical models are as far as
rutting predictions are concerned.
Fig. 3 is a plot of the measured and predicted rutting
for the 80 kN load, and Fig. 4 is a similar plot for the
120 kN load. It needs to be emphasized that the models
are based on the load/damage relations for the pavement's
component materials as measured in the laboratory: they
are not regression models. Given the uncertainties and
variabilities inherent in pavement modelling, the models
are producing realistic rutting predictions.
The outputs from the model are not in power law form, but
it is possible to estimate predicted power law exponents
by curve fitting the predicted performance. In the
present case this was done by fitting the predicted
performance data with equations of the form
D
=
cPmNm
(6)
where D is the damage caused.
Values for the exponents m and m/z are given in Table
III. c is a constant of proportionality whose value need
not be specified in the context of the present
discussion.
If N1 cycles of load P1 produce the same damage to the
pavement as N2 cycles of load P 21 then eqn(6) may be
rearranged to give:
(P2/P1)
= N1/N2
(7)
So it can be said that the pavement follows an a power
law. a values as computed for each of the model outputs
are given in Table III.
11
AIR 000-248
TABLE III: POWER LAW EXPONENTS.
CIRCLY
(Dual
wheel)
CIRCLY
(single
wheel)
VESYS III
FEM
rn
0.143
0.737
0.136
0.678
0.208
0.678
0.158
0.787
a
5.2
5.0
3.3
5.0
rn/0C
The computed values of a show that the models are
predicting powers for rutting which are consistent with
the 4th power law.
3.3 Mechanistic analyses using NAASRA damage
relationships.
Section 2.6 describes the NAASRA Pavement Design rutting
criterion and fatigue damage relationships, given there
as eqn(3). In order to convert eqn(3) into a pavement
load/damage relationship, it is also necessary to
evaluate eqn(4).
Outputs from the mechanistic analyses of the Somersby
pavement using CIRCLY, VESYS III and FEM, include strain,
load pairs (E,P) at different load levels. They thus
provide the data necessary to define the relationship
between load and strain for the pavement and distress
mode concerned.
From eqn(5) ab = a, which is the exponent in the power
law.
a values (from the NAASRA Guide) and b values (from the
mechanistic models) and the exponents a for rutting and
crackthg are given in Tables IV and V.
TABLE IV: POWER LAW EXPONENTS FOR RUTTING.
CIRCLY
(Dual
wheel)
CIRCLY
(single
wheel)
VSYS TIT
FEM
a
b
7.14
0.810
7.14
0.813
7.14
0813
7.14
0.916
a
5.8
5.8
5.8
6.5
The predicted exponents of 5.8
higher than the exponents of
computed using the data obtained
the Somersby materials. They are
value of 3.4. This suggests that
to 6.5 for rutting are
3.3 to 5.2 which were
from laboratory tests on
higher than the measured
the' exponent of 7.14 in
AIR 000-248
12
the NAASRA rutting criterion, which is based on a backcalculation using the NAASRA CBR/thickness/ESA formula
for the thickness design of unbound pavements, might be
too high, at least for the Somersby pavement.
TABLE V: POWER LAW EXPONENTS FOR CRACKING.
a
b
CIRCLY
(Dual
wheel)
CIRCLY
(single
wheel)
VESYS III
FEM
5.00
0.355
5.00
0.187
5.00
0.187
5.00
0.306
1.8
0.9
0.9
1.5
The cracking power law exponent predicted using the
CIRCLY dual wheel model gives the best result, because
the single wheel loading models do not accurately
represent the asphalt strains under the applied dual
wheel load. Cracking was not observed in the Somersby
pavement. However the predicted exponent of 1.8 is
consistent with the empirically determined exponent for
the test track at Nantes, France, referred to in Section
2.3.
The computed result is encouraging because it shows that
the mechanistic response models, in conjunction with the
NAASRA fatigue law for asphalt, predicb realistic
exponents for cracking and that an exponent of 5 in the
material damage equation is consistent with an exponent
of about 2 in the pavement load/damage relationship. Of
course it also means that if cracking is the relevant
distress mode, then an exponent of about 2 rather than 4
should be used to transform the load distribution into
ESA.
AIR 000-248
13
4. THE PRACTICAL APPLICATION OF THE 4TH POWER LAW.
Although the mathematics of the 4th power law is selfevident, it is sometimes mis-stated, misunderstood or
misinterpreted, and so an explanation of its application
is required. In particular there are ten matters to be
discussed:
Computation of relative life and relative
damage.
Doubling the axle load does not necessarily
produce 16 times the damage.
There is an upper limit on axle loads beyond
which the 4th power law does not apply.
A higher power law exponent does not necessarily
imply greater relative damage.
A higher power law exponent does not necessarily
imply a greater number of Standard Axles.
The 4th power law should not be unquestionably
applied to sub- and super-standard pavements.
The 4th power law is not unique to road
pavements.
The 4th power law does not apply to road
pavement costs.
The 4th power law does not deal directly with
dynamic loads.
The 4th power law may not be valid when applied
directly to gross vehicle mass.
4.1 Computation of relative life and relative damage.
There are at least three contexts in which the 4th power
law might be used:
When changing load limits but uaintaining the same
road network condition,
When charging single overloaded trucks for pavement
damage, and
When changing load limits but not changing the timing
of maintenance/rehabilitation actions.
(i) Relative life.
When changing load limits but maintaining the same road
network condition, the 4th power law Is usually used to
predict the ratio of ESA, and hence relative life under
two different traffic scenartos which will be referred to
AIR 000-248
14
as the base traffic and the new traffic. Here relative
life is defined as:
RL
LN.J/L.I.E = ( N9 )
( N9
is the time for the pavement to reach
where
some particular level of damage under the new(base)
traffic.
If the base traffic consists of N passes of load P, and
the new traffic consists of N passes of load r'P, eqn(1)
shows that the relative life ratio is:
RL = N/(TN)
where a is the exponent in the power law.
(ii) Relative unit-incremental damage.
When charging single overloaded trucks for pavement
damage, the relevant damage ratio is the relative unitincremental damage (RUID) ratio. RUID is defined as the
ratio of the increment in damage caused by, one pass of
load rp to the increment in damage caused by one pass of
load P. Of course, the increments in damage due to rP and
P refer to changes in damage from the same initial damage
point on the pavement damage/traffic curve.
In order to compute RUID ratios it is first necessary to
define the damage (performance/distress) model. Appendix
C contains RUID computations for four different damage
models, namely:
The Somersby pavement rutting model,
A NAASRA Road Study roughness model,
A simplified version of the NAASRA Road Study
roughness model, and
The NAASRA Pavement Design Guide cracking model.
The computations show that for the models (1), (2) and
(3):
RUID =
(9)
rM
and for model (4):
(10)
RUID =
In fact, it can be shown that, for any damage model of
the form:
D
RUID
f(N9
)
(11)
(12)
AIR 000-248
15
Therefore, for practical purposeG, the exponent in the
power law may be used to estimate the relative damage
caused by the passage of a single overloaded truck.
(iii) Relative total damage.
When changing load limits, without changing the timing of
any maintenance/rehabilitation actions, a different
damage must be considered. A relevant damage ratio is the
relative total damage (RTD) which is the ratio of the
damage caused by a number of passes of load rp to the
damage caused by the same number of passes of load P.
Here the damage due to rP and P refers to the total
damage caused by the trucks.
The computation of RTD ratios depends on the particular
models concerned and
(performance/distress)
damage
computations for the four different damage models
referred to earlier are given in Appendix C. The results
show that for models (1), (2) and (3) RTD is not even
approximated by r.
The exception is the NAASRA Pavement Design Guide
cracking model for which relative total damage and
relative unit-incremental damage both equal F. The
result for the cracking model is a direct consequence of
the definition of cracking 'damage' which is the
proportion of number of cycles to failure.
The essential point to be made is that, in general, the
relative damage caused by. a single overloaded truck
(RUID) is not the same as the relative damage due to an
increase in a distribution of loads applied to a new
pavement when the timing of maintenance/rehabilitation
actions is not changed (RTD). Whereas it is possible to
estimate relative unit-incremental damage using the
exponent a, it is generally incorrect to use the same
exponent to compute relative total damage.
4.2 Doubling the axle load does not necessarily produce
16 times the damage.
it follows from the discussion in Sectton 4.1 that, if
the applied load is double the standard axle load, then
one pass of the higher load is equivalent to (2 )4 = 16
ESA. That is, 16 passes of a standard axle will be needed
to cause the same damage to the road pavement as one pass
at the higher load level. The life of the pavement under
the doubled loads will be 1/16 of the life under the
original loads.
As discussed in Section 4.1, in most cases it is a
reasonable approximation to assume that, if a pavement
that follows the 4th power law is subjected to a single
pass of a truck with axle loads double the standard axle
load, this will produce 16 times the incremental damage
produced as a result of one pass of a truck with axle
loads equal to the standard axle load.
AIR 000-248
16
However, if relative total damage rather than relative
unit-incremental damage is being considered, it is not
always true that doubling the load produces 16 times the
damage, even if the pavement follows a 4th power law.
This can be demonstrated by way of an example.
Suppose the performance of a pavement is such that when
rutting is used as a measure of damage (D), the damage
caused by N applications of load with magnitude (P) is
given by eqn(6) with the exponent values m=0.8 and
m/a=0.2. These values are similar to the ones obtained
from an analysis of the .ALF test on the pavement at
Somersby, NSW (see Section 3.2).
Eqn(7) demonstrated that such a pavement obeys the 4th
power law. However, for the case where P = 2P9 and
N = N9, eqn(6) shows that the damage is:
D(2P9,N9) = c(2P9)m(N9 )m
= 1.74.D(P9,N9
=c2°-8(P9
)
(N9
) °8
) °2
(12)
So, N passes of an axle with twice the standard load
produces only 1.74 times more damage than N passes of a
standard axle. That is, for the example given, doubling
the load produces not 16 times the damage, but only 11%
of that amount.
Another related example is the case where P = 2P9 and,
In this case, apart from the tare load, the
N9 = 2N.
freight carried is equivalent in each situation
(2P9 x N = P9 x 2N). Under these circumstances eqn(6)
shows that the damage caused by N passes of the higher
load is only 1.52 times the damage caused by 2N passes of
a standard axle.
It follows from these examples that it is not appropriate
to estimate the increase in
to use the factor (P/P9
damage to new pavements corresponding to a general
increase in the distribution of ioa4s applied to it.
) 4
4.3 The 4th power law has an upper ltmtt.
A pavement that can withstand 106 app1ictions of a
standard axle should, according to the ',4th power law, be
just able to withstand one application of load P such
that:
(P/P9)4
i.e.
= (
10/1)
P = ( 106)0.25 P. = 31.6 P 9
It is unlikely that this would be the case as there is an
upper limit on the axle loads beyond which the strength
of the pavement is exceeded. Clearly, the 4th power law
applies to pavement wear but not pavement overload
failure. In structural engineering terms the 4th power
AIR 000-248
17
law relates to "fatigue failure" and not to a collapse
failure under a single extreme load. This link to fatigue
is discussed further in Section 4.7.
4.4 A higher power law exponent does not necessarily
imply greater relative damage.
When talking about the effect of overloading on
pavements, it is often assumed that pavements with a
higher power law exponent will suffer more relative
damage than other pavements which are subjected to the
same traffic but follow a lower power law exponent. As a
general statement it is incorrect and again this will be
demonstrated by way of example using the Somersby
pavement rutting model, but first it is necessary to
introduce some new definitions.
The exponent (a) in the power law is a property of the
pavement, and not merely a parameter which may be used to
transform load distributions into a single number (N9 );
so when talking about an increase in a, one is in fact
comparing different pavements.
Suppose there are two pavements: one obeys a power law
with exponent a1 and the other obeys a power law with
exponent a2
Let:
= the lower (higher) power law exponent,
RTD1(2) = the relative total damage of the
pavement with the a1(2) exponent,
RUID1(2) = the relative unit-incremental damage of
the pavement that obeys a power law with the a1(2)
exponent,
The damage(performance/distress) formulae are based on
eqn(6).
Two relative (as between pavements) damage ratios of
interest are RTD1/RTD2 and RUID1/RUID2 .
If each pavement is subjected to N passes of a load with
magnitude ri'9 , where F is a load multiplier and F > 1,
then:
RTD1/RTD2
=
(
rtm/rm)
(13)
That is, the damage ratio RTD1/RTD2 is independent of a,
so the pavement with the higher exponent suffers the same
relative total damage as the pavement with the lower
exponent.
AIR 000-248
18
However, if each pavement is subjected to a single pass
of load magnitude rP9 it can be shown that:
,
(14)
RUID1/RUID2
> a, the damage ratio
Because the exponent CX2
RUID1/RUID2 < 1 1 i.e. the pavement with the higher power
law exponent suffers more relative unit-incremental
damage than the pavement with the lower exponent.
Thus the statement that pavements with a higher power law
exponent will suffer more relative damage than other
pavements which are subjected to the same traffic but
follow a lower power law exponent is not correct as a
general statement, but it is approximately true if damage
is understood to mean the relative unit-incremental
damage for each pavement.
4.5 A higher power law exponent does not necessarily
imply a greater number of Standard Axles.
There are two reasons for this.
First, eqn(1) shows that if P < P, N. will, for a given
N, decrease as the exponent in the power law increases,
whereas if P > P the reverse is true.
Secondly, when considering the power law transformation
of load distributions, it can be shown that the computed
may not be monotonic in a. For example:
For the case where a pavement is subjected to nL
applications of load P, eqn(1) can be generalized to
become:
=
(15)
(P/P).n
1
Eqn (15) can be rewritten as:
(16)
N. = (P 0/P)°'.N
where N is the total number of load applications, and
1
PQ = [ E
(P)°'(n/N) ]
(1/cr)
(17)
1
By definition, N passes of P
as the set of (P r, n) pairs.
produces the same damage
Appendix D contains an application of eqns (16) and (17)
to a rural road distribution of single axles (dual tyres)
for NSW given in ERVL (1976). The computations show that
whereas P Q increases with a, the relative ESA (i.e.
Na/N) are not monotonic in a, and, for the example given,
pass through a minimum at a = 3.5. it is concluded that,
AIR 000-248
because N. is not generally monotonic in a,
appropriate measure of axle load aggressivity.
19
is not an
It is also worthwhile noting that, for the example given,
the computed N. is not particularly sensitive to a in
that a change in a from 3.5 to 8 only produces a 47%
change in the relative N E,.
4.6 The 4th power law should not be unquestionably
applied to sub- and super-standard pavements.
The 4th power law does not predict the actual life of a
pavement. It is usually used to predict the ratio of ESA,
and hence relative life (see section 4.1) under two
different traffic scenarios often referred to as the base
traffic and the new traffic.
However, the need to know the actual life of the pavement
concerned becomes acute when dealing with either
substandard or super-standard pavements. With a superstandard pavement where there is little or no damage, the
actual life of the pavement may be so great as to render
the computations of relative life meaningless.
On the other hand, with a substandard pavement the actual
life of the pavement may be so short that queotions need
to be asked as to:
Whether the pavement should have been built so
as to better withstand more frequent or heavier axle
weights?, and
Whether it is fair to penalize the trucking
industry for damage to a pavement that may have been
built to inadequate standards?, as against
(C) Is it fair to demand high public expenditures in
order to build roads designed for higher axle loads
promoted by the trucking industry?
Small and Winston (1986) argue that these questions are
best dealt with by way of an optimal investment analysis,
but this must be influenced by the current stock of
assets existing, which, for whatever reason, may not have
been built for higher axle loads.
4.7 The 4th power law is not unique to road pavement
damage.
The 4th power law of pavement damage has a foundation in
the permanent deformation properties of the materials
from which the road pavement is built. But it should not
be thought that the 4th power law is unique to road
pavement damage. Lay (1982) shows that a similar power
law applies to the fatigue of metal structures where, for
example, a power of 3 is indicated for steel fillet
welds.
AIR 000-248
20
4.8 The 4th power law does not apply directly to road
pavement costs.
A re-analysis of the material in Lay (1979) argues that
the ratio of new to old annual pavement reconstruction
costs resulting from an increase in axle loads from
magnitude P to FP, is approximated by the expression
1
1 +
r°-5
2
-
where a is the exponent in the power law. This stems from
the fact that the strains in a pavement decrease more
than linearly with an increase in pavement thickness,
whereas the material costs only increase linearly. Many
other costs do not alter at all.
For the case when a = 4 and F = 1.25, the cost ratio is
1.27. If the 4th power law was to be incorrectly applied
directly to road costs the cost ratio would be 2.44.
4.9 The 4th power law does not deal directly with dynamic
loads.
Axle loads are dynamic not static. The magnitude of the
dynamic loads depend on the condition of the pavement and
there is an interaction between dynamic loads and road
roughness. It has been suggested (RTRP, 1988) that a
"dynamic" form of the 4th power law could be written as:
N8/N =c*(P/P)4 ;
with c = 1.06 for a good (smooth) surface, 1.24 for an
average surface and 1.54 for a poor surface.
4.10 The 4th power law may not be valid when applied
directly to gross vehicle mass.
A key point to consider when dealing with pavement wear
caused by trucks is the distinction between gross vehicle
weights and axle loads. Stresses and strains induced in
pavements by moving wheels of loaded trucks are confined
to a limited area around the wheels. The stresses and
strains associated with the individual wheels ot an axle
group (tandems and triaxles) interact, the extent of the
interaction for a given axle spacing depending on the
stiffness of the pavement and the magnitude of the axle
load. The axle groups on a truck are generally separated
such that interactions do not occur between axle groups.
Pavement damage is therefore related to the axle loadings
imposed by a truck, and not to its gross vehicle mass.
Gross mass, however, affects larger span bridges.
Further, an increase in gross vehicle (combination) mass
could be accompanied by a redistribution of loads between
axle groups and/or the addition of extra axles. If, for
example, an increase in gross vehicle mass from 30t to
AIR 000-248
21
38t was met by the addition of an extra axle, the ESA
would increase by a ratio of only about (38/30) i.e 1.27,
not by the ratio (38/30) = 2.57. Hence the 4th power law
may not be valid when applied directly to gross vehicle
mass.
Load sharing between axle groups is discussed in relation
to "road efficiency" (i.e payload/Na ) in Sweatman (1988).
AIR 000-248
22
5. SUMMARY AND DISCUSSION.
The general conclusion from this report is that the 4th
power law represents the best available single tool for
estimating pavement wear due to traffic. However, like
any useful tool it must be used with care and within its
limitations. Particular conclusions relating to its
specific applications follow:
5.1 Recommended power law exponents.
(1) The 4th power law can be applied to asphalt pavements
in Australia when rutting or NAASRA roughness is used as
a measure of pavement wear.
There are difficulties in applying the results of the
AASHO Road Test in Australia because:
A regression type statistical approach was
used to analyse the results and the
circumstances of the Road Test are not
duplicated in Australia. For example, the Road
Test was conducted under freeze-thaw
conditions, unbound pavements were not studied
and the traffic mix did not include wide single
tyres.
The AASHO Road Test data focused on a
single composite measure of pavement damage
(PSI) rather than considering the individual
modes of pavement distress.
(C) The major component of PSI is roughness and
roughness in Australia is generally measured in
terms of NAASRA roughness, not slope variance
as in the Road Test model.
However, in spite of the shortcomings of the AASHO Road.
Test data, the 4th power law is still the best available
model for the wear of asphalt pavements when, as is
commonly done in Australia, pavement wear is measured in
terms of NAASRA roughness or rutting. This is because:
The AASHO Road Test data showed that the
4th power law could be applied to US asphalt
pavements when PSI was used as an index or
measure of pavement wear,
Papers by Lister (1977), and Kaesehagen et.
al. (1972) have shown that PSI, rutting and
NAASRA roughness are approximately linearly
related, and
The ALF test at Somersby, NSW supports the
4th power law when rutting is used as the
damage index.
23
AIR 000-248
(2) A power of about 2 is relevant to the fatigue
cracking of asphalt pavements.
The evidence that a power of about 2 is more appropriate
than a 4th power consists of:
the combination of the NAASRA (NAASRA,
1987) Pavement Design Guide criterion for the
fatigue of asphalt materials, and the Kinder
(1987b) analysis of the ALF pavement at
Somersby, which gives an exponent of about 1.5
for the fatigue cracking of the Somersby
pavement, and
the results of an experiment on an asphalt
test pavement at Nantes in France, as reported
by Autret, Baucheron de Boissoudy and
Gramsammer (1987), who report a power of about
2.
(3) FHWA, World Bank and OECD reports give powers for
other distress modes and pavement types.
Powers for other pavement types and distress modes, as
suggested by the US Department of Transportation, are
reproduced in Section 2.2 of this report, but they should
only be applied with caution because they relate to the
US experience and have only limited empirical support.
Whenever possible a sensitivity analysis should be
carried out to determine the effect of varying the power
concerned. This was the approach adopted in Bayley and
Kinder (1984) which was concerned with the impact on road
costs of increased transport of grain by road, and in
particular with seasonal variations of load and pavement
response. A simple example of the sensitivity of the
transformation of load distributions to different power
law exponents is given in Appendix D.
5.2 Limitations on the use of the 4th power law.
Pavement damage is related to truck axle loads and
not to truck gross mass, and so use of the 4th power law
will not be generally valid when it is applied directly
to gross vehicle mass.
The 4th power law applies to pavement wear. It does
not apply to overload failure.
The 4th power law says that N passes of a load P will
do as much damage as (P/P9 ) 4 N passes of a standard axle
of load P, and thus reduce the relative life of the
pavement (time to reach a terminal condition) by a factor
where a is
of 1/(P/P)4 . Therefore, the factor (P/P9
the exponent in the power law) may be used to predict the
relative life of pavements in a network when changing
load limits but maintaining the same road network
condition.
)
(
24
AIR 000-248
Also, for the damage models studied in this report, it is
a reasonable approximation to assume that, if a pavement
that follows the 4th power law is subjected to a single
pass of an overloaded truck with axle loads P, this will
produce (P/P)4 times the damage produced as a result of
one pass of a truck with axle loads equal to the standard
axle load. Therefore, when charging single overloaded
trucks for pavement damage, the relevant damage factor,
which may be called the relative unit-incremental damage
(RUID), is (P/P).
However, for the examples studied in this report it is
shown that, in general, the relative damage caused by a
single overloaded truck (RUID) is not the same as the
relative damage due to an increase in a distribution of
loads applied to a new pavement when the timing of
maintenance/rehabilitation actions is not changed (RTD).
Whereas it is possible to estimate relative unitincremental damage using the factor (P/P), it is
generally incorrect to use the same factor to compute
relative total damage.
The power law should not be unquestionably applied to
sub- and super-standard pavements.
The power law should not be used directly to estimate
the extra cost of new pavements built to carry higher
axle loads. Further, it is suggested that, generally,
there is a cost-effective attraction to provide overrather than under-designed pavements.
The 4th power law does not deal directly with dynamic
loads. For example, it may not be particularly valid for
very rough roads. Work at ARRB on truck suspension and
profilometry are moving us rapidly towards solutions in
this area.
5.3 Understanding the nature of the 4th power law.
The power law is consistent with other similar laws
applying to metal fatigue.
The number of equivalent axles (ESA) is an important
pavement design parameter. It is incorrect and misleading
to refer to them as loads (i.e. as ESAL).
A higher power law exponent does not necessarily
imply greater relative damage or a greater number of
standard axles.
5.4 Further advances.
Further advances in understanding pavement wear will come
from laboratory testing and mathematical modelling. Such
advances due to work at ARRB are imminent.
In particular, these advances are likely to come from
both mechanistic computer model studies and ongoing field
AIR 000-248
25
investigations such as the ARRB ALF and P357 research
programs, the ARRB laboratory based research programs
under P403 and the Long Term Pavement Performance Study
(LTPS) of SHRP (AASHTO-FHWA-TRB, 1986) program.
Mechanistic computer models will need to be used to look
at the problems of how particular variables effect
pavement performance. The models could be applied to
specific pavements and the results aggregated so as to
produce regression relationships for the network
concerned. It will be necessary to adopt this approach
because:
it is not feasible to conduct field trials for all of
the pavement types and traffic conditions likely to be
encountered in practice and
without continuing fundamental research into cause
and effect, the results of the field trials will only be
applicable to the circumstances prevailing at the
investigated road sections.
26
AIR 000-248
J/
f - cii—
R
Report written by:
Report reviewed by:
DISTRIBUTION TO INCLUDE: Directors; RTEC; NAASRA
Executive Director; NAASRA Project Groups on Australian
Legislation, Pavement Research, Heavy Vehicles and Road
Transport; Interstate Commission.
27
AIR 000-248
4
Complex AASHO formula
for load equivalence factor
for single axles
o
2
Power law fit
I
Load
Equivalence
Factor
(Ns/N)
•
4.15
Load Factor (Ns/N) = (P/18)
.2
30
20
10
Load (Kips)
Fio I - AASHO combined load equivalence factor
(LEF) for rigid and flexible pavements
18
D
80 kN load
16
120 kN load
14
12
Permanent 10
deformation
(mm) 8
Power = 3.40
6
4
2
0
200000
400000
600000
Cycles.
Fig 2 - Somersby pavement: permanent deformations.
800000
AIR 000-248
18
a
16
- - - CIRCLY: Dual wheel loading.
14
CIRCLY: Single wheel loading.
-
- - VESYS
12
- - Finite Element
Permanent 10
deformation
(mm)
Experiment (Nominal load = 80 kN)
-
-
8
6
-- ---- --a
,
4
aa
a
2
-a
a
a
0
800000
600000
400000
200000
0
Cycles.
Fig 3 - Somersby pavement: permanent deformations.
a
Experiment (Nominal load . 120kN)
- - - CIRCLY: Dual wheel loading.
- CIRCLY: Single wheel loading.
- - VESYS
- - Finite Element
20
Permanent 12
deformation
- ___
(mm)
4
U
- --
- - __n -/---
8
I
(a
t-
a
i
0
I
I
200000
I
I
400000
I
I
I
600000
Cycles.
Fig 4 - Somersby pavement: permanent deformations.
I
1
800000
AIR 000-248
29
REFERENCES.
AASHTO-FHWA-TRB (1986) Strategic Highway Research
Program. Research Plans. Final Report. Washington DC.
AUTRET, BAUCHERON DE BOISSOUDY AND GRANSANMER (1986) The
circular test track of the "Laboratoire Central Des Ponts
et Chaussees" (L.C.P.C.) Nantes - First Results. Seventh
Int. conf. on the structural design of asphalt pavements.
Deift.
BARTELSMEYER, R.R. and FINNEY, E.A. (1962) Use of AASHO
Road Test findings by the AASHO committee on highway
transport. Highway Research Board, Special Report 73.
Washington.
BAYLEY, C. and KINDER, D.F. (1984) The impact on road
costs of increased transport of grain by road. ARRB
Internal Report, AIR 1129-1A.
FHWA (1982) Final Report on the Federal Highway Cost
Allocation Study. U.S.Department of Transportation.
FRY, A., EASTON, G., KER, I., STEVENSON, J. AND WEBER, J.
(1976) NAASRA Study of the Economics of Road Vehicle
Limits. Commercial Vehicle Surveys.
HIGHWAY RESEARCH BOARD (1962) The AASHO Road Test,
Pavement research, Special Report 61E, Washington.
IRICK, P.E. and HUDSON, R.H (1964) Guidelines for
satellite studies of pavement performance. NCHRP Report
2A.
KADAR, P. (1985a) Accelerated Loading Facility (ALF) ARRB
Internal Report, AIR 415-2.
KADAR, P. (1985b) Trial program and site preparations for
the first Accelerated Loading Facility (ALF) pavement
trial at Somersby, NSW. ARRB Internal Report, AIR 415-3.
KADAR, P. (1985c) The first Accelerated Loading Facility
(ALF) pavement trial at Somersby, NSW - test results.
ARRB Internal Report 415-3.
KADAR, P. (1986) Analysis of the test results of the
Accelerated Loading Facility (ALF) trial at Somersby,
NSW. ARRB Internal Report, AIR 415-4.
KAESEHAGEN, R., WILSON, 0., SCALA, A. AND LEASK, A.
(1972) The develo ment of the NAASRA Roughometer. Proc.
6th. ARRB Conf., 6(1), 303-330.
KINDER, D.F. (1986) A studyof both the viscoelastic and
permanent deformation properties of a NSW asphalt. Proc.
13th. ARRB/REAA.A Conf. Vol.13, Part 5, ppl-ll.
30
AIR 000-248
KINDER, D.F. (1987a) A re-analysis of the creep and
dynamic data on the Somersby asphalt. ARRB Internal
Report, AIR 403-9.
KINDER, D.F. (1987b) A prediction of critical stresses,
strains, deflections and deformations of a pavement at
Somersby, NSW using CIRCLY, VESYS and a finite element
method. ARRB Internal Report AIR 403-11.
LAY, M.G. (1979) Road Deterioration and the Fourth Power
Law. ARRB Internal Report, AIR 000-146.
LAY, M.G. (1982) Source Book for the Australian Steel
Structures Code, As 1250. AISC, Sydney, 3nd. Edition.
LAY, M.G. (1986) Handbook of Road Technology (Gordon and
Breach: London and New York)
LAY, M.G. (1987) bc cit. Update of Chapter 11, Aust Rd
Res, 17(4), Dec 1987
LISTER, N.W. (1977) Heavy wheel loads and road pavements
- damage relationships. Symposium on Heavy Freight
Vehicles. Held at OECD France.
METCALF, J.B., McLEAN, J.R. and KADAR, P. (1985) The
development and implementation of the Accelerated Loading
Facility (ALF) program. ARRB Internal Report, AIR 403-1.
NAASRA (1987) Pavement Design. A Guide to the Structural
Design of Road Pavements.
NAASRA ROADS STUDY (1984) Use of NIMPAC Technical Report
T-7, April 1984.
PATERSON, W.D.O. (1985) Prediction of road deterioration
and maintenance effects: theory and quantification. Vol
III. Trans. Dept, World Bank, Washington, DC.
ROAD TRANSPORT RESEARCH PROGRAM (1988) Pavement damage
due to heavy vehicles and climate. Final Report OECD.
(1988)
SHELL (1978) Pavement Design Manual.
SMALL, K.A. and WINSTON, C.W. (1986) Efficient pricing
and investment solutions to highway infrastructure needs.
Economic Issues in US Infrastructure Investment. vol.76,
no.2, AEA Papers and Proceedings.
SWEATMAN, P.F. (1988) Heavy vehicles: The dynamics of
change. 26th ARRB Regional Symposium Bunbury, W.A.
VUONG, B.V. (1986) Mechanical response properties of road
materials obtained from the ALF pavement test section in
Somersby, NSW. ARRB Internal Report, AIR 403-8.
31
AIR 000-248
APPENDIX A.
Basis of the 4th power law: The AASHO Road Test.
Most references to the 4th power law cite the AASHO Road
Test (HRB, 1962) as the source, but in fact the AASHO
load/ damage relations are more complex than a simple
power law. This Appendix contains a brief description of
the AASHO Road Test scheme of analysis and a
demonstration 'of how a power law can be extracted from
the AASHO Road Test results.
The form of equations given apply to both rigid and
flexible pavements, but unless stated otherwise, the
parameters in the equations are for flexible pavements
only.
A.l Present serviceability concept
In the Road Test a single index, referred to as the
Present Serviceability Index (PSI), was used as a measure
of the deterioration of the experimental pavements under
traffic.
The ASHO equation for the PSI of flexible pavements is:
PSI = 5.03-1. 91*log( 1+SV)-0 . 01*(C+P)° -1. 38R2
(Al)
where SV = the mean of slope variance in the two wheel
paths,
C+P = a measure of cracking and patching in the pavement
surface (area exhibiting class 2 or class 3 cracking).
R = a measure of rutting in the wheel paths.
A.2 Performance model
The AASHO performance model represents the performance of
a pavement as its PSI history, i.e. its PSI as a function
of the applied loads.
j:3
N
PSI = CO
-
(
C0
-
( A2)
C1).[
]
where C1 :5 PSI :5 C o ;
PSI = the Present Serviceability Index;
C. = the initial PSI. For the Road Test C o was chosen as
4.5 for rigid pavements and 4.2 for flexible pavements.
C1 = the serviceability level at which a test section was
considered out of test and no longer observed. For the
Road Test C1 was chosen as 1.5.
AIR 000-248
32
and 0 are functions of design and load and they will be
discussed later.
0
N = the accumulated axle load applications at the time
when PSI is observed and may represent weighted or
unweighted applications.
A.3 Seasonal weighting factors
It was observed early in the Road Test that the rate at
which pavement damage accumulated with applications of
load was affected by seasonal changes, especially in the
case of flexible pavements. In fact 80% of the flexible
pavements failed during and immediately after the spring
thaw when the moisture conditions in the sub-grade and
sub-base made them abnormally weak. To allow for this a
weighting function based on the seasonal variation of
pavement deflection was applied to the number of load
repetitions.
The weighted applications of load were computed as:
Nw
n1 +
= q1
q2
n2
+ q3
n3
+
.........
qn t .
(
A3)
n. = the actual number of load applications during the
time period t.
q, = the value of the seasonal weighting function at time
period t,
The weighting function is defined as:
2d. - d_ 1
q=[
(A4)
d
where:
d, is the deflection as measured on a non-traffic section
of the pavement,
d_1 is the deflection in period t-1, anc
ci is the 2 y average of d.
The exponent 2 was assumed as an appropriate factor for
increasing the amplitude of q.,,, in 'periods of high
deflection relative to periods of low deflection.
A.4 Design and load equations
When N. represents weighted applications as obtained
through the use of the seasonal weighting function, the
relationship between 3, o and the design and load
variables for flexible pavements is;
r
AIR 000-248
0.081 (P + L)323
0 = 0.4 +
(SN +1)5.19 L323
105-93 (SN +1)9-36L433
KM
(P + L)479
where:
P = the nominal load axle weight in kips (18 for a
single-axle load, and 32 for a tandem axle;
L = axle code, 1 for single, 2 for tandem;
SN = structural number (a measure of pavement thickness
and strength, Lay, 1986).
A.5 Reinterpretation on a power law basis
Bartelsmeyer and Finney (1962) used the AASHO Road Test
equations to derive a series of axle load equivalence
factors from which it is possible to determine the
effects on the pavement structure of one axle load as
compared to another. It was done by rewriting eqn (A2)
as:
log N = log(ø) + (1/3)l09((C 0 -PSI)/(Co-C1 ))
(A7)
where 0 and 0 are defined by eqns (A5) and (A6)
respectively, and are functions of the axle type, axle
load and structural number.
If N18 denotes the number of standard (18 kip) load
applications to reach some terminal PSI, and N_ denotes
the number of load applications of magnitude x to reach
the same terminal PSI, then the ratio N 18/N,. defines a
load equivalence factor which when multiplied by the
number of axle loads within a given weight category,
gives the number of single standard axle load
applications (ESA) that will have an equivalent effect on
the performance of the pavement.
Bartelsmeyer and Finney solved eqns (A2), (A5) and (A6)
to produce tables of load equivalence factors (LEF) as a
function of axle load, axle type, terminal PSI and
structural number (SN). The tables show that the computed
LEF are relatively insensitive to SN, so Bartelsmeyer and.
Finney graphed LEF (averaged over a range of SNs) as a
function of axle load, terminal PSI and axle type.
Fig 1 shows the Bartelsmeyer and Finney graph of average
LEFs for both rigid and flexible pavements (and a
terminal PSI of 2.2) replotted on log/log axes. The
result is a straight line with slope 4.15, this being the
exponent in the power law equation. Trick and jiudson
34
AIR 000-248
(1964) referred to this relationship as 'the fourth
power' approximations.
35
AIR 000-248
APPENDIX B.
Limitations on the use of the AASHO-based 4th power law.
Because a regression type statistical approach was used
to analyse the Road Test data, the resulting formulae may
not be valid for use in Australia unless the environment,
traffic, materials, pavement type and pavement
construction methods are the same as, or similar to,
those in the Road Test. Clearly the circumstances of the
Road Test are not duplicated in Australia. Further, there
has been some criticism of the particular type of
regression procedure used in the Road Test and the focus
on PSI as a single composite measure of damage.
B.1 The traffic.
The AASHO Road Test traffic did not (nor could it be
expected to) cover all of the relevant traffic variables.
A full specification of traffic requires information on
axle types, axle loads and load sharing between axle
groups; tyres types, air pressures, tyre contact areas,
suspension type and the condition of the axle systems and
suspensions (old, new), etc. Axle groups may be single,
tandem, or tridem (triple) axles and tyres may be single,
dual (twin) tyres, wide base tyres, diagonal or radial
ply. There were no triaxles in the AASHO study.
As far as axle loads are concerned, they are dynamic not
static. The dynamic loading depends on the condition of
the pavement and there is an interaction between dynamic
loads and road roughness. It has been suggested (RTRP,
1988) that a "dynamic" form of the 4th power law could be
written as:
N9/N =c*(P/P9 ) 4 ;
(
Bi)
with c = 1.06 for a good (smooth) surface, 1.24 for an
average surface and 1.54 for a poor surface.
B.2 Environmental factors.
Because the AASHO Road Test was an accelerated test (it
took place over a period of about 2 years), only traffic
effects not environmental effects were studied.
Furthermore, the AASHO Road Test pavements were subjected
to frost (which thawed out in the spring) to a depth of
approx. one metre below the surface. This significantly
affected the results of the experiment because most of
the flexible pavements failed during or immediately after
the spring. These conditions generally do not exist in
Australia.
It should be noted that, although environmental and
traffic factors are being discussed separately, it is
clear that there is an interaction between traffic and
environmental variables. Temperature affects both the
AIR 000-248
36
rate of rutting and the fatigue life of asphalt
pavements, and moisture in the subgrade reduces a
pavements strength and stiffness and resistance to
traffic. The fact that cracking patterns are described as
alligator, chicken wire, fish net, block, map,
centreline, longitudinal, wheel track, pavement edge,
transverse,
shrinkage,
contraction,
meandering,
reflection, and hair-line indicates that there are
multiple causes due to a combination of both traffic and
environmental factors.
B.3 Materials.
Australian pavement materials and materials specification
standards (particularly with respect to durability) are
different from the Road Test materials. Furthermore, only
one subgrade type was employed in the Road Test and many
Australian subgrades, such as the expansive soil
subgrades in Qld, would perform differently to the Road
Test subgrade.
B.4 Pavement types.
Pavements are usually classified as flexible, semi-rigid
or rigid, and the cause of distress can be different for
each pavement type.
Flexible pavements may be either asphalt or unbound
pavements. Asphalt pavements can have a cause of distress
which is not directly related to traffic. For example
failure of asphalt can result from either thermal
cracking or low temperature cracking. Many of the
pavements in Australia are unbound pavements and there
were no unbound pavements in the Road Test.
Semi-rigid pavements have bound base layers and can have
a cause of distress different from those observed in the
Road Test. For example, in the recent ALF trial in Qid.
failure of the pavement was caused by a debonding of the
cement-treated base layer.
The classification (and types of distress) of rigid
pavements depends on whether they are constructed :
with short, long or continuous slabs
with plain, reinforced or pre-stressed concrete
with or without dowelled joints.
Not all rigid pavements in Australia are of the the same
type as those studied in the Road Test.
B.5 Construction, maintenance and administrative
standards.
The particular maintenance standards adopted with respect
to pavement sealing practice would play a part in
determining the life of a pavement and hence the
relevance of the Road Test equations. The assumption
AIR 000-248
37
inherent in the Road Test equations is that the integrity
of the road pavement will be preserved.
The degree of overloading tolerated is relevant because
an administrative discretion that allowed for some degree
of overloading would lead to more pavement failures than
predicted by the Road Test formulae.
The standard of road construction and design is reflected
in both the initial serviceability and the terminal
serviceability of a pavement. The Road Test formulae, and
in particular the 4th power law, depend on both the
initial and terminal values of PSI.
B.6 Small and Winston criticism.
Small and Winston (1986) argue that by modern statistical
standards, the statistical analysis of the Road Test data
is "totally unsatisfactory". They state that after they
reestimated the parameters in the AASHO performance
equation using a limited dependent variable model
(Tobit), their results showed that pavement lifetimes for
thick pavements, both rigid and flexible, were
substantially overestimated by the AASHO statistical
procedures.
B.7 Road Test results focus on PSI.
The Road Test analyses focused on a single composite
measure of damage, namely PSI. PSI is principally a
measure of rider comfort rather than structural
condition, but Lister (1977) has argued that PSI can be
related to structural damage by showing that PSI is
approximately linearly related to rutting, and Kaeshagen
(1972) et al. have shown that PSI is approximately
linearly related to NAASRA roughness.
Nevertheless, PSI is not a very satisfactory measure of
pavement damage because:
it is preferable to have a separate damage index for
each major cause, of distress( and the major modes of
traffic -induced distress are roughness, rutting, fatigue
cracking and loss of,skid resistance.
PSI is insensitive to cracking, and
PSI is not used in Australia. Further, the major
component of the PSI is roughness and roughness in
Australia is measured in terms of NAASRA roughness, not
slope variance as in the Road Test model.
AIR 000-248
APPENDIX C.
Relative damage.
C.l A general formula for relative incremental damage..
It is useful to look at not only the Somersby pavement
rutting model (eqn(6)), but a number of other damage
models as well. All the models to be considered can be
put in the general form:
=
D(P,N)
(Cl)
f1(N,)
where f1 is a function.
Consider a pavement which has undergone some intial
damage (D±) . This damage could have been caused by N 9
repetitions of load P 9 or N, repetitions of load FP 9, so
that:
(C2)
D = D(P9,NB) = D(rP8,N,)
Now compare the incremental damage (6D(FP 9)) caused by SN
applications of the load rP9 (eqn(C3), and the
incremental damage (8D(P 9 )) caused by SN applications of
the load P. (eqn(C4).
(C3)
D(rP9,N r ,+6N)-D(rP9,Ni-,)
5 (rP9)
=
8D(P9)
D(P91 N9+SN)-D(P91 N9
(
)
C4)
Because N repetitions of P9 causes the same damage (Dr )
it follows from eqn(l) and
as N, repetitions of rP9
eqn(Cl) that:
,
NI ,/N8
=
(
(C5)
l/)
Substitution of eqn(C5) into eqns(C3) and (C4) gives the
relative incremental damage (RID) as:
SD(rP8
)
D(rP8
, (
l/r)°N9+SN)-D(rP8
, (
l/T)N9
)
RID
SD(P9
D(P9 ,N9+SN)-D(P9 ,N9
)
)
(C6)
C.2 Definition of relative unit-incremental damage.
For the special case SN = 1, the relative incremental
damage defined by eqn(C6) will be referred as relative
unit-incremental damage (RUID). That is relative unitincremental damage is defined as:
D(FP8
, (
l/r)9N9+l)-D(rP9
, (
l/r)9N9
)
RU ID
(C7)
39
AIR 000-24 8
This damage ratio might be relevant when charging single
overloaded trucks.
It can be shown, for any damage model of the form given
by eqn(Cl) that:
RUID
The proof is as follows:
Eqns(Cl) and (1) may be combined and rearranged to give:
f1 '(D)
where f1
'
=
(P/P)°'N
is the inverse function of f1
Assuming N can be treated as a continuous variable,
differentiation of both sides of eqn(C9) gives:
d(f1 '(D))/dN
=
(d(f1 '(D))/dD)(dD/dN)
=
(P/P)
Writing d(f1 1(D))/dD
written as:
dD
-- (P,D)
dN
=
as
(ClO)
f2(D) allows eqn(Cl0) to
be
(Cli)
(P/P)°'/f2(D)
Then, using eqn(C7):
D(rP, (l/ryN 8+i)-D(rP9
, (
l/F)N)
RUID
D(P9,N9+1)-D(P8,N9
)
dD
-- (rP9,D±)
dN
(Cl2)
dD
-- (P9,D)
dN
and substitution of eqn(C1l) into egn(C12) gives:
RUID
(C13)
C.3 Definition of relative total damage.
For the special case when N = 0, the relative damage
defined by eqn(C4) will be referred to as the relative
total damage (RTD) which is defined as;
AIR 000-248
40
D(FP8, SN)-D(rP8, 0)
RTD
=
D(P9, SN)-D(P8, 0)
(C14)
The definition excludes from the total damage any initial
damage due to construction. This ratio might be relevant
when changing load limits but where the timing of
maintenance/rehabilition actions is not changed.
For the particular case when N8 = 0 and SN = 1, RUID =
RTD = RID.
In order to proceed further and compute these relative
damage ratios, it is necessary to define a damage
(performance/distress) model. In the present study the
ratios RID, RUID and RTD were computed for four different
damage models, which are:
The Somersby pavement rutting model,
A NAASRA Roads Study (1984) roughness model,
A simplified version of the NAASRA Roads Study
roughness model.
The NAASRA IGPTD cracking model.
C.4 The Somersby pavement rutting model.
In this model rut depth (RT) is used as a measure of
damage (D), and RT is a function of the applied load (P)
and the number of applications (N) of P (eqn(6)).
(C15)
RT = cPm N''where the values of the exponents are:
m
=
=
0.80
4.00
(i) Computation of relative incremental damage.
Substitution of eqn(C15) into eqn(C6) gives:
(FP8 ) m
RID
(P8)m
rm
(1+5N/N)m
_Nrn
(C16)
Eqn(C16) shows that for this particular damage model, RID
is a function of SN/N. Table C2 gives computed RID values
for a range of SN/N and r, but before discussing the
results in Table C2, it is instructive to look at the
values of RID at N = 0 and as N tends to oo,
41
AIR 000-248
When N = 0 , eqn(C8) becomes:
(C17)
rM
RID
=
N= 0
The value of RID, as N tends to co, may be computed by
differentiating both the numerator and denominator of
eqn(C16) with respect to 1/N, and letting N tend to 00
(1/N tends to 0). It follows that:
lim RID =
as N tends to
Ftm
1/N =0
=
(C18)
Eqns(C17) and (C18) show that the relative incremental
damage varies from Ftm at N = 0, to F as N tends to CO .
is a
The RID values given in Table C2 show that F
SN/N
<
10.
provided
RID
reasonable approximation of
(ii) Computation of relative-unit incremental damage.
RUID values may be computed by substituting eqn(C15) into
eqn(C7). However, the limits given in eqn(C17) and
eqn(C18) also apply to the particular case when SN =1,
hence the relative unit-incremental damage also varies
from Ftm at N = 0, to F as N tends to co• But RUID values
tend to the F limit fairly rapidly. For example, suppose
the load multiplier (F) = 2 and N takes the range of
values 1 to 106, so that SN/N varies from 1 to 10-6.
Table C2 shows that the RUID goes from a value of 5.13 to
16, but it reaches a value very close to 16 (15.90) when
SN/N = 10 3. Similar results follow for F values of 0.5,
1.25 and 3. Therefore for practical purposes it would
seem that:
RUID = F
(C19)
Therefore, for this particular damage model, the exponent
(u) in the power law may be used not only to compute
relative life, but also to estimate relative unitincremental damage.
42
AIR 000-248
(iii) Computation of relative total damage.
Substitution of eqn(C15) into eqn(C14) gives:
(6N)m
=
RTD
(8Nm
=
(C20)
Eqn(C20) shows that the relative total damage is Ftm for
all values of SN.
The essential point to be made is that, for this
particular damage model, the damage ratios RUID and RTD
are not the same; so whereas it is possible to estimate
relative unit-incremental damage using the exponent a, it
is incorrect to use the same exponent to compute relative
total damage.
C.5 A NAASRA Roads Study (1984) roughness model
In this model NAASRA roughness (R in counts/km) is used
as a measure of damage.
R = a + b (age) + c (age)2
(C21)
where the parameter values are:
a
b
c
=
=
=
52.5
1.31
0.0751
This model was converted from a roughness/time model to a
roughness/ traffic model by replacing the variable 'age'
with N9/50,000, so as to give:
R = a2 + b2 (N9) + c2 (N9
) 2
(C22)
where:
a2
b2
=
=
=
52.5
2.62F-05
3.00E-11, and
(i) Computation of relative incremental damage.
Substitution of eqn(C22) and eqn(1) into eqn(C6) gives:
(b2/c.) (FSN)+(N+F0 6N)2 _N2
RID
=
(b2/c2 ) ( SN)+(N+SN)2 -N2
=
(b2/c2)ro(6N/N)/N+(1+r0 8N/N)2 1.
(b2/c2)(5N/N)/N+(1+8N/N)2-1.
(C23)
AIR 000-248
43
Which shows that, for this particular damage model, the
relative incremental damage is a function of both SN/N
and N. Tables C3, C4, C5 and C6 give matrices of RID
values as a function of SN/N and N for F values of 0.5,
1.25, 2 and 3 respectively but, before discussing the
results in the Tables, it is instructive to first look at
the values of RID at N = 0 and as N tends to w.
When N = 0 eqn(C23) becomes:
(b2/c2)F08N+(F0SN)2
(b2/c2)SN+(SN)2
N= 0
The value of RID as N tends to ooll may be computed by
differentiating both the numerator and denominator of
eqn(C23) with respect to 1/N, and letting N tend to 00
(1/N tend to 0). It follows that:
2 (b2/c2 )r5N/N+2 ( l+rSN/N) I'
lim RID
=
as N tends to co
2(b2/c2)SN/N+2( 1+SN/N)
1/N = 0
Eqns (C24) and (C25) show that the relative incremental
damage varies from a value different from F at N = 0, to
F as N tends to w. The values of RID given in Tables C2,
C3, C4, C5 and C6 show that F is a reasonable
approximation of RID provided SN/N < 10.
(ii) Computation of relative unit-incremental damage.
RUID values may be computed by substituting eqn(C22) and
eqn(1) into eqn(C7). However the limits given in eqn(C24)
and eqn(C25) also apply when SN = 1, hence:
When N = 0, eqn(C24) becomes:
(b2/c2 ) r01r2
RUID
(b2/c2)+1
N= 0
(C26)
F (for b2/c2 = 873,000)
As N tends to w (1/N tend to 0), eqn(C25) gives:
2(b2/c2)F°'/N+2(1+F/N)F
lim RUID
as N tends to oo
2(b2/c2)/N+2(1+1/N)
1/N = 0
=
(C27)
AIR 000-248
44
Eqns (C26) and (C27) show that, for this particular
damage model, the relative incremental damage changes
from = r at N = 0, to r as N tends to oo. Thus RUID
values are sensibly constant at r for all values of N.
In particular, if the load multiplier (I') =2 and N takes
the range of values 1 to 106, so that SN/N varies from 1
to 10-6. Table CS shows that the RUID stays constant at
the value of r = 16. Similar results follow for r values
of 0.5, 1.25 and 3 (see Tables C3, C4 and C6
respectively).
Therefore, for this particular damage model, the exponent
(cr) in the power law may be used not only to compute
relative life, but also to estimate relative unitincremental damage.
(iii) relative total damage.
Substitution of eqn(C22) and eqn(1) into eqn(C14) gives:
(b2/c2 )r05N+(r0 8N)2
RTD
(b2/c2)6N+(5N)2
(C28)
When SN tends to 0, eqn(C28) becomes:
(b2/c2 ) ro.+r20.SN
lim RTD
as SN tends to 0
(b2/c2)+SN
SN = 0
(C29)
=
When SN tends to w, eqn(C28) becomes:
(b2/c2 ) r"/5N+r2°
lim RTD
as SN tends to co
=r
(b2/c2)+1.
1/SN = 0
(C30)
Eqn (C29) and eqn(C30) show that RTD varies from r to
r2 as SN goes from 0 to w. Again, the essential point to
be made is that, for this particular damage model, the
damage ratios RUID and RTD are not the same; so whereas
it is possible to estimate relative unit-incremental
damage using the exponent a, it is incorrect to use the
same exponent to compute relative total damage.
45
AIR 000-248
C.6 A simplified version of the NAASRA Roads Study
roughness model.
The form of this model is a special case of eqn(C22).
That is:
- a3 + c3 N,2, 2
where, as will be shown, a3 and c3 are constants whose
values need not be specified.
(i) Computation of relative incremental damage.
Substitution of eqn(C31) and eqn(l) into eqn(C6) gives:
((N+r0SN)2 _N2
RID
(N+SN)2-N 2
=
(1+F0SN/N)2 _1
-(1+5N/N)2-1
(C32)
Which shows that, for this particular damage model, the
relative incremental damage is a function of SN/N. Table
Ci gives computed RID values for a range of SN/N and F
but, before discussing the results in the Table, it is
instructive to first look at the values of RID at N = 0
and as N tends to w.
When N = 0eqn(C32) becomes:
(F0SN)2
RID
(SN)2
N= 0
=
(C33)
r2
The value of RID as N tends to oo, may be computed by
differentiating both the numerator and denominator of
eqn(C32) with respect to i/N, and letting N tend to CO
(i/N tend to 0). It follows that:
2 ( l-I-rSN/N) r
limRID
2(1+SN/N)
as N tends to co
i/N = 0
=
(C34)
Eqns (C33) and (C34) show that the relative incremental
damage varies from a value of r 2 at N = 0, to r as N
tends to oo. The values of RID given in Tables Cl show
that r is a reasonable approximation of RID provided
SN/N < l0.
AIR 000-248
46
Computation of relative unit-incremental damage.
RUID values may be computed by substituting eqn(C31) and
eqn(1)into eqn(C7). However, the limits given in eqn(C33)
and eqn(C34) also apply to the particular case when
SN =1, hence the relative unit-incremental damage also
at N = 0, to r= as N tends to 00. But
varies from r2
limit fairly rapidly. For
RUID values tend to the r
example, suppose the load multiplier () = 2 and N takes
the range of values 1 to 106, 50 that SN/N varies from 1
to 10-6. Table Cl shows that the RUID goes from a value
of 96 to 16, but it reaches a value very close to 16
(16.1) when SN/N = iO. Similar results follow for r
values of 0.5, 1.25 and 3. Therefore for practical
purposes it would seem that:
(C35)
RUID
Therefore, for this particular damage model, the exponent
() in the power law may be used not only to compute
relative life, but also to estimate relative unitincremental damage.
relative total damage.
Substitution of eqn(C31) and eqn(l) into eqn(C14) gives:
(rSN)2
RTD
=
(SN)2
=
r2°'
(C36)
Again, it is clear that, for this particular damage
model, the damage ratios RUID and RTD are not the same;
so whereas it is possible to estimate relative unitincremental damage using the exponent a, it is incorrect
to use the same exponent to compute relative total
damage.
C.7 The NAASRA PDG cracking model.
The NAASRA Pavement Design Guide cracking model is given
by eqn(3) with:
N
=
N,f
(C37)
where N f is the number of repetitions of the load to
produce failure.
47
AIR 000-248
Then, if damage (D)is defined as the proportion of N f
consumed, it follows from eqn(C5) that:
D
=
=
N/Nf
(k2/k1)PN
(C38)
which is a particular form of eqn(C15) with m = a = ab.
Hence, as far as relative damage is concerned:
RID=RUIDRTD
=r
(C39)
Therefore, for this particular damage model, the exponent
(a) in the power law may be used not only to compute
relative life, but also to estimate relative incremental
damage, relative unit-incremental damage and relative
total damage.
TABLE Cl: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
SIMPLIFIED NAASRA ROAD STUDY ROUGHNESS MODEL.
0.50
Load
1.25
0.0625
0.0625
0.0625
0.0625
0.0622
0.0597
0.0430
0.0137
0.0051
0.0040
0.0039
0.0039
0.0039
2.44
2.44
2.44
2.44
2.46
2.61
3.61
5.37
5.89
5.95
5.96
5.96
5.96
SN/N
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
Multiplier
2
16.0
16.0
16.0
16.1
17.2
27.4
96.0
216.0
251.3
255.5
256.0
256.0
256.0
(I')
3
81.0
81.0
81.3
84.2
113.2
389.6
2,241.0
5,481.0
6,433.9
6,548.1
6,559.7
6,560.9
6,561.0
AIR 000-248
M.
TABLE C2: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
SOMERSBY PAVEMENT RUT DEPTH FORMULA:
Load
1.25
SN/N
0.50
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
0.063
0.063
0.063
0.063
0.063
0.065
0.082
0.166
0.321
0.434
0.494
0.527
0.546
2.44
2.44
2.44
2.44
2.43
2.32
1.89
1.48
1.32
1.26
1.23
'1.22
1.21
2
Multiplier ()
3
16.00
16.00
15.99
15.90
15.12
10.94
5.13
2.86
2.22
1.99
1.88
1.82
1.79
81.00
80.97
80.74
78.53
63.25
28.85
9.51
4.58
3.33
2.88
2.67
2.56
2.50
TABLE C3: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
NAASRA ROAD STUDY ROUGHNESS MODEL (LOAD MULT. = 0.5)
SN/N
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
N
1E+03
1E+00
1E+01
1E+02
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0624
0.0618
0.0565
0.0312
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0625
0.0624
0.0618
0.0565
0.0312
0.0086
0.0625 0.0625
0.0625 0.0625
0.0625 0.0625
0.0625 0.0625
0.0625 0.0625
0.0625-0.0625
0.0625 0.0624
0.0624 0.0618
0.0618 0.0565
0.0565 0.0312
0.0312 0.0086
0.0086 0.0044
0.0044 0.0040
1E+04
1E+05
1E+06
0.0625
0.0625
0.0625
0.0625
0.0625
0.0624
0.0619
0.0566
0.0315
0.0087
0.0044
0.0040
0.0039
0.0625
0.0625
0.0625
0.0625
0.0624
0.0620
0.0575
0.0342
0.0096
0.0045
0.0040
0.0039
0.0039
0.0625
0.0625
0.0625
0.0625
0.0623
0.0605
0.0474
0.0170
0.0055
0.0041
0.0039
0.0039
0.0039
AIR 000-248
49
TABLE C4: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
NAASRA ROAD STUDY ROUGHNESS MODEL (LOAD MULT. = 1.25)
SN/N
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
1E+00
1E+01
1E+02
N
1E+03
1E+04
1E+05
1E+06
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.45
2.48
2.80
4.32
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.45
2.48
2.80
4.32
5.68
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.45
2.48
2.80
4.32
5.68
5.93
2.44
2.44
2.44
2.44
2.44
2.44
2.45
2.48
2.80
4.32
5.68
5.93
5.96
2.44
2.44
2.44
2.44
2.44
2.45
2.48
2.80
4.30
5.67
5.93
5.96
5.96
2.44
2.44
2.44
2.44
2.44
2.47
2.74
4.14
5.62
5.92
5.96
5.96
5.96
2.44
2.44
2.44
2.44
2.45
2.56
3.35
5.18
5.86
5.95
5.96
5.96
5.96
TABLE C5: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
NAASRA ROAD STUDY ROUGHNESS MODEL (LOAD MULT. = 2.00)
SN/N
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
1E+00
1E+01
1E+02
N
1E+03
1E+04
1E+05
1E+06
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.3
18.7
40.7
144.2
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.3
18.7
40.7
144.2
236.8
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.3
18.7
40.7
144.2
236.7
253.9
16.0
16.0
16.0
16.0
16.0
16.0
16.3
18.7
40.6
144.1
236.7
253.9
255.8
16.0
16.0
16.0
16.0
16.0
16.3
18.7
40.2
142.8
236.3
253.9
255.8
256.0
16.0
16.0
16.0
16.0
16.2
18.2
36.5
131.8
232.8
253.5
255.7
256.0
256.0
16.0
16.0
16.0
16.1
16.8
24.1
78.0
202.5
249.3
255.3
255.9
256.0
256.0
AIR 000-248
50
TABLE C6: RELATIVE INCREMENTAL DAMAGE COMPUTED USING THE
NAASRA ROAD STUDY ROUGHNESS MODEL (LOAD MULT. = 3.00)
N
SN/N
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06
1E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
81
81
81
81
81
81
81
81
82
88
154
748
3,543
81
81
81
81
81
81
81
82
88
154
748
3,543
6,041
81
81
81
81
81
81
82
88
154
748
3,542
6,041
6,505
81
81
81
81
81
82
88
154
746
3,539
6,040
6,505
6,555
81
81
81
81
82
88
153
734
3,506
6,030
6,504
6,555
6,560
81
81
81
82
87
141
634
3,208
5,934
6,492
6,554
6,560
6,561
81
81
81
83
103
299
1,755
5,115
6,380
6,542
6,559
6,561
6,561
51
AIR 000-248
APPENDIX D.
Sensitivity of load transformations to the exponent in
the power law.
By way of example consider the rural road distribution of
single axles (dual tyres) for NSW from in ERVL (1976).
Table Dl gives loads (interval midpoints) and
frequencies. The distribution is bimodal.
TABLE Dl: AXLE LOAD DISTRIBUTIONS
Load
(Tonnes)
Frequency
(nw/N)
1
2
3
4
5
6
7
8
9
10
11
12
0.029
0.165
0.165
0.076
0.063
0.062
0.114
0.164
0.112
0.035
0.011
0.004
Table D2 shows PEQ, and NEZ/N computed for different
values of a using eqns(6), (7) and (8).
TABLE D2: P VALUES.
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
P Q
(Tonnes)
N/N
5.84
6.14
6.41
6.64
6.84
7.02
7.18
7.32
7.44
7.56
7.67
7.77
7.86
7.95
0.600
0.562
0.541
0.532
0.531
0.537
0.549
0.566
0.587
0.614
0.646
0.684
0.728
0.780
52
AIR 000-248
The table of computed relative N I/N above, shows that,
for the example given, N E /N is not monotonic in a, nor is
it particularly sensitive to changes in a.
53
AIR 000-248
APPENDIX E.
Pavement costs
The pavement thickness, t, required is a complex function
of the applied load, P. For simple load spreading,
however,
(El)
p
and for simple bending,
(E2)
P
A check of some actual design relationships (e.g. Lay,
1979) suggests eqn(El) can be used and hence
(pavement material costs) < or =
This is conservative, given that eqn(E2) suggests a power
of 1/3 and given that many construction costs (e.g.
surface finishing) depend on pavement area rather than a
pavement thickness. Hence, the real cost increase ratio
due to rp is less than r1- 2 .
Of course, the costs must be taken over the life of the
road and increasing the thickness increases that life.
Avoiding discounting, (i.e. assuming zero interest) in
order to illustrate the effect, gives the cost increase
ratio per year as, conservatively, proportional to
nv2 x r4
=
However, increasing P to rP decreases the freight task
needed by a ratio between 1/n (no tare weight) and 1 (no
payload). Hence, the cost increase is about
n2 x1/r=n
(E4)
to an approximation reasonable enough to demonstrate that
it is well short of r4 .
A more detailed re-analysis of the material in Lay (1979)
showed that it is closer to
1
1 +
2
This suggets that, for r = 1.25, the increase was 1.27,
whereas eqn(E4) suggests 1.25. n4 would give 2.44.