revised beta criteria for the cbr airfield pavement …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
College of Engineering
REVISED BETA CRITERIA FOR THE CBR AIRFIELD PAVEMENT
DESIGN METHOD
A Thesis in
Civil Engineering
by
Eric J. Miller
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
December 2012
The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
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The thesis of Eric J. Miller shall be reviewed and approved* by the following: Prasenjit Basu Assistant Professor of Civil Engineering Peggy Johnson Department Head and Professor of Civil Engineering Mansour Solaimanian Senior Research Associate, Geotechnical and Materials Engineering Shelley M. Stoffels Associate Professor of Civil Engineering!Thesis Adviser *Signatures are to be on file in the Graduate School.
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ABSTRACT
The empirical California Bearing Ratio (CBR) design method for flexible airfield
pavements has been a mainstay for practicing engineers for decades. Its simplicity and proven
performance makes it easy to use and reliable. However, design scenarios for increasingly larger
aircraft have revealed a significant design shortfall at these extremely heavy and multi-wheeled
loading conditions. This has driven a redevelopment of the CBR method to underpin it with a
mechanistic load response of vertical stress at the subgrade surface instead of an equivalent
single wheel load at the surface of the pavement. This response is correlated to an empirical
failure model of pavement life developed from full-scale test data. This new design method has
eliminated limitations in the original procedure that resulted in overdesigned pavements for large
aircraft. However, any new method always leaves room for improvement. This thesis explored
the development of the current failure model and identified an opportunity to refine it. The
primary target of this modification was the concentration factor, an empirical modification to the
Boussinesq stress distribution theory used in calculating the load response. The test data used to
develop the current failure model was previously analyzed with a constant concentration factor
for all test points. However, a more representative relationship of the concentration factor as a
function of the subgrade CBR was available to better model soil behavior. After collecting
additional test data beyond that used to develop the current failure model, a new failure model
was developed that included the concentration factor as a function of CBR. In addition, it was
shown, through comparison of nonlinear regression statistics, that a failure model developed
using the concentration factor as a function of CBR provided failure model with a better fit to the
test data than a model developed with a constant concentration factor.
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TABLE OF CONTENTS
List of Tables ................................................................................................................................. vi!List of Figures ............................................................................................................................... vii!List of Abbreviations and Acronyms ............................................................................................. ix!1. Introduction ................................................................................................................................. 1!
1.1. Summary ...................................................................................................................... 1!1.2. Research Goals ............................................................................................................. 1!1.3. Relevance ..................................................................................................................... 2!1.4. Scope ............................................................................................................................ 4!
2. Prior Research/Literature Review ............................................................................................... 5!2.1. Breadth of Existing Research ....................................................................................... 5!2.2. Background Concepts .................................................................................................. 5!
2.2.1. Coverages ..................................................................................................... 6!2.2.2. California Bearing Ratio (CBR) ................................................................... 8!2.2.3. Equivalent Single Wheel Load ...................................................................... 9!
2.3. The Classic CBR Design Method .............................................................................. 11!2.3.1. Origins ........................................................................................................ 11!2.3.2. Definition .................................................................................................... 13!2.3.3. Shortfalls ..................................................................................................... 15!
2.4. Beta Criteria; the Revised CBR Design Method ....................................................... 16!2.4.1. Definition .................................................................................................... 16!2.4.2. Development of the Beta Criteria Model .................................................... 20!2.4.3. The n = f(CBR) Model ................................................................................ 23!
3. Research Approach and Methods ............................................................................................. 26!3.1. Data Collection .......................................................................................................... 26!3.2. Stress and Beta Calculations ...................................................................................... 27!3.3. Beta Criteria Curve Fitting ......................................................................................... 29!3.4. Model Testing and Verification ................................................................................. 31!
3.4.1. Statistical Analysis ...................................................................................... 32!3.4.2. Design Thicknesses ..................................................................................... 33!3.4.3. Test Data ..................................................................................................... 34!
3.5. Selection of Model ..................................................................................................... 34!4. Results and Analysis ................................................................................................................. 35!
4.1. Data Calculation ......................................................................................................... 35!4.2. Stress and Beta Calculations ...................................................................................... 38!
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4.3. Curve Fits ................................................................................................................... 42!4.3.1. Resulting Curves ......................................................................................... 42!4.3.2. Concentration Factor Effects ...................................................................... 47!
4.4. Model Testing and Verification ................................................................................. 50!4.4.1. Statistical Analysis ...................................................................................... 50!4.4.2. Design Thicknesses ..................................................................................... 51!4.4.3. Test Data ..................................................................................................... 60!
4.5. Model Selection ......................................................................................................... 65!5. Summary, Findings, Conclusions, and Recommendations ....................................................... 67!
5.1. Summary .................................................................................................................... 67!5.2. Findings ..................................................................................................................... 67!5.3. Conclusions ................................................................................................................ 68!5.4. Recommendations ...................................................................................................... 68!
References ..................................................................................................................................... 70!Appendix: Test Data ..................................................................................................................... 72!
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List of Tables
Table 2.1. Airfield Pavement Design Parameters versus Highway Design Parameters ................. 6!Table 2.2. CBR and Resilient Modulus Values by Soil Type ........................................................ 9!Table 3.1. Subgrade CBR Categories ........................................................................................... 31!Table 3.2. Design Thickness Comparison Aircraft Specifications ............................................... 33!Table 4.1. Collected Test Data ...................................................................................................... 36!Table 4.2. Data Sets for n = 2 and n = f(CBR), Beta versus Coverages ....................................... 39!Table 4.3. Curve Coefficients and Statistics ................................................................................. 44!Table 4.4. Subgrade Category Curve Coefficients and Statistics ................................................. 44!Table 4.5. Curve Coefficients and Statistics from Current Criteria Test Data ............................. 47!Table 4.6. NAPTF CC5 Life Prediction and Error ....................................................................... 61!
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List of Figures
Figure 2.1. Influence of Depth on ESWL ..................................................................................... 11!Figure 2.2. Alpha as a Function of Coverages and Number of Wheels ........................................ 13!Figure 2.3. F-15 CBR Flexible Pavement Design Chart .............................................................. 14!Figure 2.4. Current Flexible Pavement Design Beta Criteria ....................................................... 18!Figure 2.5. ESWL Comparison by Equivalent Response ............................................................. 21!Figure 2.6. Effects of Concentration Factor Model on Design Thickness .................................... 22!Figure 2.7. Effect of Concentration Factor on Data Points ........................................................... 23!Figure 2.8. Concentration Factor Models ..................................................................................... 25!Figure 3.1. Research Process Flow Chart ..................................................................................... 26!Figure 3.2. PEU Test Vehicle Editor Window; Example, Test Point 22 ...................................... 28!Figure 3.3. PEU Main Window; Example, Test Point 22 ............................................................. 29!Figure 3.4. FAA Mechanistic Subgrade Failure Model ................................................................ 30!Figure 3.5. DoD Mechanistic Subgrade Failure Model ................................................................ 30!Figure 4.1. Test Data Plot of Beta versus Coverages ................................................................... 40!Figure 4.2. Source Plot of Estimated Beta Values ........................................................................ 41!Figure 4.3. Beta Calculation Comparison and Reproducibility Plot ............................................ 42!Figure 4.4. Resulting n = f(CBR) Curves ..................................................................................... 43!Figure 4.5. Subgrade Category Curves ......................................................................................... 45!Figure 4.6. Combined Subgrade Category Curves ....................................................................... 45!Figure 4.7. Comparison of Curves from Current Criteria Test Data ............................................ 47!Figure 4.8. Current Form Curves .................................................................................................. 49!Figure 4.9. Mechanistic Form Curves ........................................................................................... 49!Figure 4.10. Reciprocal Logarithm Curves ................................................................................... 50!Figure 4.11. Boeing 747 Design Thickness at 100 Coverages ..................................................... 54!Figure 4.12. Boeing 747 Design Thicknesses at 10,000 Coverages ............................................. 54!Figure 4.13. Boeing 747 Design Thicknesses at 100,000 Coverages ........................................... 55!Figure 4.14. Boeing 767 Design Thicknesses at 100 Coverages .................................................. 55!Figure 4.15. Boeing 767 Design Thicknesses at 10,000 Coverages ............................................. 56!Figure 4.16. Boeing 767 Design Thicknesses at 100,000 Coverages ........................................... 56!Figure 4.17. F-15E Design Thicknesses at 100 Coverages .......................................................... 57!
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Figure 4.18. F-15E Design Thicknesses at 10,000 Coverages ..................................................... 57!Figure 4.19. F-15E Design Thicknesses at 100,000 Coverages ................................................... 58!Figure 4.20. T-38 Design Thicknesses at 100 Coverages ............................................................. 58!Figure 4.21. T-38 Design Thicknesses at 10,000 Coverages ........................................................ 59!Figure 4.22. T-38 Design Thicknesses at 100,000 Coverages ...................................................... 59!Figure 4.23. NAPTF CC5 Test Data Plot Against Model Development Points ........................... 62!Figure 4.24. Example Reliability Based Analysis of Beta Criteria .............................................. 64!Figure 4.25. Design Thickness Reliability Evaluation, Boeing 767, 10,000 Coverages .............. 64!Figure 4.26. Design Thickness Reliability Evaluation, F-15E, 10,000 Coverages ...................... 65!
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List of Abbreviations and Acronyms
ACC Asphalt Cement Concrete
ACN Aircraft Classification Number
CBR California Bear Ratio
CC5 Construction Cycle Five
CDF Cumulative Damage Function
CF Current Form
DCP Dynamic Cone Penetrometer
DoD Department of Defense
ERDC Engineer Research and Development Center
ESWL Equivalent Single Wheel Load
ICAO International Civil Aviation Organization
LED Layered Elastic Design
MF Mechanistic Form
NAPTF National Airport Pavement Test Facility
P/C Pass-to-Coverage Ratio
PCASE Pavement-Transportation Computer Assisted Structural Engineering
PCN Pavement Classification Number
PEU Pavement Engineering Utility
RL Reciprocal Logarithm
USACE United States Army Corps of Engineers
USCS Unified Soil Classification System
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1. Introduction
1.1. Summary
The California Bearing Ratio (CBR) flexible airfield pavement design procedure is an
empirical design method that has been in use since the 1940s. The empirical design input of the
subgrade CBR results in a simple design procedure requiring few resources to implement. Due
to recently identified shortcomings caused by the introduction of very large aircraft, the US
Army Corps of Engineers (USACE) Engineer Research and Development Center (ERDC)
recently redeveloped the method to account for these shortfalls, resulting in a mechanistic-
empirical failure model that retains the simplicity of the original CBR input for the design
process. The mechanistic feature of the failure model is the vertical stress at the top of the
subgrade induced by the aircraft landing gear. This stress is a function of the loading conditions
and the properties of the material through which the stress is distributed, quantified by the
concentration factor. This factor is a function of the soil type and is taken into consideration in
the current CBR design process. However, it was only partially accounted for in development of
the failure model itself. This research will further describe the development of the CBR design
process, the theory behind the redeveloped method, and then redevelop the current failure model
with full inclusion of the concentration factor.
1.2. Research Goals
The goal of this research is twofold. First, the research objective is to develop new beta
criteria for the CBR airfield pavement design procedure by including the concentration factor as
a function of subgrade CBR in the test data points used in development of the criteria curve.
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Second, the research hypothesis is that this new curve will provide a better fit to the test data, and
will subsequently allow better confidence in pavement thickness design.
1.3. Relevance
In the modern era of computational ease enabled by computers, coupled with the ever-
deepening understanding of, and capabilities to characterize, geotechnical materials, the use of
an empirical design procedure such as the CBR method may seem obsolete. However, the use of
such a method is actually considered valuable by many in the airfield pavement field, primarily
in the Department of Defense (DoD). This is due to the simplicity in application of the design
procedure as well as its ability to easily accommodate a worldwide myriad of geographical
locations, environmental conditions, and construction materials. Also, the International Civil
Aviation Organization (ICAO) has selected the CBR procedure for use in the Aircraft
Classification Numbers (ACN) and Pavement Classification Numbers (PCN) system. This
system is used to classify the relative strength of airport pavements as well as the relative
damage imparted by an aircraft in order to allow pilots and airport managers to quickly and
easily determine aircraft operability.
The US military is broadly recognized as the world’s preeminent fighting force.
Although this can be attributed to numerous factors such as shear size, technological mastery,
and superb training, one of the most significant enabling factors is air power, provided by all
service branches. The US military is alone in the ability to transmit mass amounts of military
strength, be it combat operations or disaster and humanitarian assistance, anywhere in the world,
at anytime, a concept known as global reach. This global reach is made possible through a
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network of air bases, aerial refueling, and aircraft carriers. The backbone of this network is the
presence of existing airbases throughout the United States and allied nations around the globe.
However, at times, new airfields are required where none exist, and are often needed
quickly. These locations can be austere in terms of environment and material availability, or in
terms of logistical support such as electrical supply, computer access, and material testing
capabilities. These factors lend perfectly to the use of the empirical CBR design method. From
the structural designer’s perspective, all that is needed is one piece of testing equipment, the
dynamic cone penetrometer (DCP), and a design chart for the design aircraft. All other design
aspects, primarily materials and construction methods, are considered in specifications that have
been adjusted from six decades of experience throughout the entire world. Putting the simple
structural CBR design method together with the specifications provides a method that has proven
capable of quickly, and with minimal resources, handling tropical environments, arctic
conditions, and arid deserts.
Despite this aptness to urgent and austere design scenarios, the CBR method is perfectly
suited to permanent air base design needs. Since the failure mode of pavements is rarely
catastrophic, the primary two concerns in pavement design are minimal (life-cycle) cost and
achieving the desired pavement life, functionally and structurally. As such, good design models
will neither over-design, causing excess up-front costs, nor under-design, causing early failure
and excessive maintenance, repair, or rehabilitation costs. Due to certain assumptions made in
the current empirical CBR failure model, it may be possible to produce a revised model that
more closely fits the available test data and a more reliable design thickness across the full
spectrum of the model’s potential applications.
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1.4. Scope
The scope of this research was an analytical redevelopment of the CBR design process
subgrade failure model. Due to the cost and complexity of full scale pavement testing, no actual
testing was done in this research to develop the model. Instead, historic test data was collected
to do so. The general steps required to complete this research are as follows:
1. Search for existing flexible airfield pavement test reports.
2. Evaluate and consolidate available test data for adequacy to this research.
3. Process the test data via appropriate methods to produce failure model data points.
4. Fit several general forms of potential failure models to the data points.
5. Evaluate the produced models using various model testing and verification methods.
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2. Prior Research/Literature Review
2.1. Breadth of Existing Research
Very little published research exists on this topic. Most research in regard to CBR design
methods has primarily been conducted by the Department of Defense (DoD), specifically the US
Army Corps of Engineers (USACE) Engineer and Research Development Center (ERDC). In
fact, the primary reference for beta criteria, Gonzalez et al. (2012), is a technical report produced
by the ERDC. In addition to the ERDC, the US Air Force (USAF) and the Federal Aviation
Administration (FAA) have also done research on the CBR design method. Other references
included in this thesis are primarily required for general civil and materials engineering
knowledge, rather than specific relevance to airfield CBR design methods.
2.2. Background Concepts
The field of airfield pavements follows the same engineering principles as road and
highway pavements, however the magnitudes of load and structure are vastly different from that
of the more common pavement and geotechnical application to highways. Additionally, the
unique type of traffic lends to some additional concerns that do not appear in other fields. To
compare numbers, some general magnitudes of design parameters are shown in Table 2.1. These
values are most representative of heavier load pavements with higher traffic volumes.
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Table 2.1. Airfield Pavement Design Parameters versus Highway Design Parameters
Parameter Airfields Highways
Design vehicle load 900,000 lbs 100,000 lbs
Load group 200,000 lbs per landing gear strut 12 wheels in twin delta tandem
34,000 lbs per axle group 12 wheels in dual tridem
Tire pressure 100psi - 350 psi 35psi - 150 psi
Design life 10,000 passes 100,000 passes
Wander 70 in - 140 in (Pereira 1977) 10 in (ME-PDG 2004)
Design thickness 5 ft 2 ft
In addition to these structural design parameters, airfield pavements can experience
additional distresses such as jet blast, which causes excessive raveling, or high-temperature
exhaust, which can quickly age pavements. Any distress that leads to the breakdown of
pavement materials and causes debris is of a much larger concern since the debris can cause
expensive, catastrophic damage to a high-speed aircraft and its engines. On the same note,
smoothness and grading requirements are much more stringent as significant bumps can damage
landing gear at high-speeds, and uneven grades can cause instability of the aircraft.
2.2.1. Coverages
Another significant difference of airfield pavements from highways is the
characterization of traffic. This is due to the variability in wheel group geometries of the
numerous types of aircraft and their wander. Vehicle repetitions are therefore converted to
coverages via a pass-to-coverage ratio (P/C). Coverages are essentially the measure of the
number load applications that occur at the critical transverse location of the pavement (UFC 3-
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260-03 2001). This critical location is the transverse location that experiences the most load
repetitions. The P/C is therefore the number of passes it requires to achieve one coverage for the
given aircraft. Numerically, the P/C is a function of gear geometry and wander. Testing has
determined that aircraft wander follows a normal lateral distribution, and is defined as the width
over which the centerline of aircraft traffic is distributed 75 percent of the time (Pereira 1977).
On channelized pavement features such as taxiways, aircraft wander is 70 inches, but on non-
channelized features such as runways, wander is 140 inches. Therefore, each aircraft will have a
different P/C for these two types of traffic areas. Additionally, it was determined that on flexible
pavements, a tandem wheel group will apply two stress repetitions to the structure (Pereira 1977)
as there is significant enough spacing between the tandem wheels to create two maximum stress
repetitions. As an example, a smaller, single-wheeled aircraft, such as the F-15 Eagle, has higher
P/C values of 8.0 to 9.0, where as a larger, multi-wheeled aircraft, such as the Boeing 747, have
smaller P/C values of 1.5 to 4.0.
It is evident that the coverage concept is not perfect. For example, the convention is that
a tandem gear counts as two stress repetitions. The design load considered is still that of the total
wheel group load. To be more representative, the front half of the tandem and the back half of
the tandem should be considered at a load less than that of the total wheel group. Additionally, it
does not account for the fact that the pavement is not going to be completely unloaded between
the two stress repetitions. Ideally, from a purely mechanistic perspective, the P/C would also be
a function of pavement thickness since the extent to which the two wheel loads interact is a
function of depth. Lastly, the coverage concept does not account for the overlap of loads with
depth that are offset from the lateral point of maximum coverages. These concerns are all well
founded, and are indeed carefully considered in most any modern mechanistic design approach.
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However, for the purposes of the CBR design method, it was found that the crudeness of the
concept was adequate considering the empirical nature of the method. First, an inordinate
amount of effort would be required to develop a more accurate model, which is then still being
applied to an empirical approach. Second, as the structural requirements of a pavement are
generally logarithmically related to stress repetitions, the sensitivity of final design thickness is
fairly forgiving to number of repetitions. That is to say that an order of magnitude increase in
repetitions would cause a significant change in design thickness. However, only doubling the
number of repetitions, perhaps through error in the P/C, may only result in a seven percent
increase in design thickness (Ahlvin 1991). This can also be thought of in the opposite direction,
such that a small increase in design thickness could see a doubling of the design life.
2.2.2. California Bearing Ratio (CBR)
The CBR is an empirical measure of a soil’s resistance to penetration. It is measured
with a dynamic cone penetrometer, which uses a standard weight, falling a standard height, to
penetrate a cone into the soil. Through a series of blows through the depth of the soil, a bearing
capacity profile can be developed. The quantity of inches (or millimeters) of penetration per
blow is empirically related to the soil’s CBR, expressed as a percentage. The standard of
measurement is the bearing capacity of a crushed limestone, which has a CBR of 100 percent,
though common convention drops the percent in conversation and literature. O. James Porter, an
engineer with the California Division of Highways, originally developed the measurement in the
1920s (Huang 2004). A soil’s CBR has been shown to relate to the resilient modulus (MR), in
psi, by Equation 2.1 (Huang 2004) and Equation 2.2 (PCASE 2.09.02 Significant Software
Changes Document, January 2012). The significant difference between these two models is in
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their applicability between soils above or below a CBR of 10, which generally separates fine-
grained and coarse-grained soils. Representative CBR and resilient modulus values for various
USCS soil types are provided for reference in Table 2.2.
Equation 2.1
!! ! !!!"" ! !"# when CBR < 10
Equation 2.2 !! ! !!!"# ! !"#!!!!" when CBR > 10
Table 2.2. CBR and Resilient Modulus Values by Soil Type (AC 150/5320-6E 2009)
Soil Type CBR Resilient Modulus, MR (psi)
High plastic silts and clays (CH, MH)
3 – 8 4,500 – 12,000
Low plastic silts and clays (CL, ML)
5 – 15 7,500 – 22,500
Silty and clayey sands (SM, SC)
10 – 40 15,000 – 38,000
Poorly graded sands (SP)
10 – 25 15,000 – 28,000
Well graded sands (SW)
20 – 40 24,000 – 38,000
Silty and clayey gravels (GM, GC)
20 – 80 24,000 – 60,000
Poorly graded gravels (GP)
25 – 60 28,000 – 50,000
Well graded gravels (GW)
60 – 80 50,000 – 60,000
2.2.3. Equivalent Single Wheel Load
An Equivalent Single Wheel Load (ESWL) is the load on a standard wheel that would
impart the same response on a soil at a given point of interest as the design load of a multi-wheel
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group. For flexible pavements, the point of interest is the surface of the subgrade. Either a set
contact area or a set contact pressure defines the standard wheel. The ESWL was originally
developed to handle limitations in early design methods. This is due to the fact that the earlier
mechanistic models only worked efficiently for a single load. The complexities of separate,
superimposed loads were too computationally intense to be of practical use without the
widespread availability of computers. There are two challenges in calculating ESWLs.
The first challenge is ensuring the correct depth of response is chosen since the amount to
which influence of the multi-wheel loads will overlap increases with depth. As seen in Figure
2.1, the two wheels act independently at shallow depth, but their influence overlaps at greater
depths. In the design process, this depth must be selected before knowing the actual pavement
thickness and resulting depth to subgrade.
The second challenge is in choosing the approach to model the equivalent responses of
the ESWL and the multi-wheels. Upon formal adoption by the DoD in the 1950s, responses
were modeled using Boussinesq solutions of equivalent deflections. This method was chosen
since full-scale testing at the time showed that equivalent deflection models were conservative
and predicted a larger ESWL than that measured in the tests. Conversely, equivalent stress-
based models proved to be non-conservative (Gonzalez et al. 2012).
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Figure 2.1. Influence of Depth on ESWL (Boyd and Foster 1950)
2.3. The Classic CBR Design Method
2.3.1. Origins
The CBR design method had its beginning in the California Division of Highways during
the 1920s where it was developed by O. James Porter (Huang 2004). At its simplest, it was an
empirical model that related the bearing capacity of a soil in terms of the CBR to the thickness
requirement of the pavement section. It was presented graphically, CBR versus thickness, as two
curves representing the generic categories of light traffic and medium traffic.
Upon the onset of World War II, the DoD needed a new design model for its airfields.
Pavement requirements in the past had been fairly simple as most aircraft were relatively light,
around a 12,000 lb single wheel load. However, new, heavier aircraft were being developed for
long-range transport and bombing missions, which proved too great a load for existing
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pavements. Although the pavement field wanted a mechanistic approach and knew it would
provide the most ideal design approach, it was recognized that the time and resources were not
available to develop such a method in time to support the war effort. Therefore, the CBR
method was chosen for its simplicity and proven performance (Gonzalez et al. 2012).
The original highway curves were extrapolated, and later verified through accelerated
testing, for aircraft single wheel loads up to 200,000 lbs. The curves were eventually derived
into Equation 2.3 for a single circular load.
Equation 2.3
! ! !!!! ! !"# !
!!
Where: ! = pavement thickness above subgrade ! = wheel load ! = contact area of load
As is evident by the lack of an input for traffic volume, the understanding of the fatigue
failure nature of pavements was in its infancy upon establishment of Equation 2.3. In 1949, the
DoD adopted adjustment factors to the equation to account for the amount of traffic in coverages,
establishing 5,000 coverages as the standard capacity. In 1955, ESWLs were then introduced to
accommodate many of the larger, multi-wheel aircraft put into service during and after World
War II. However, field performance indicated the ESWL was overly conservative, so an
additional adjustment factor was added to the equation to reduce pavement thickness as a
function of number of wheels in the multi-wheel group. Upon adoption of the ESWL adjustment
in the 1970s, the standard for capacity operations was redefined as 10,000 coverages due to
increased aircraft operations and greater channelization of aircraft (Gonzalez et al. 2012).
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2.3.2. Definition
The final form of the old CBR design method was established as shown in Equation 2.4.
The alpha factor includes the previously mentioned adjustments for number of coverages and
number of tires. It is modeled as a third order polynomial and can be depicted graphically as
seen in Figure 2.2 (Gonzalez et al. 2012).
Equation 2.4
! ! ! ! !"#$!!! ! !"# !
!!
Where: ! = pavement thickness above subgrade ! = alpha factor ! = contact area
Figure 2.2. Alpha as a Function of Coverages and Number of Wheels (Gonzalez et al. 2012)
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The CBR design method can be executed directly from the CBR equation, as is often
done in software design programs, or through design charts derived to the specifications and load
geometry of a given design aircraft. An example design chart for the F-15 Eagle is shown in
Figure 2.3.
Figure 2.3. F-15 CBR Flexible Pavement Design Chart (UFC 3-260-02 2001)
To use the chart, the subgrade CBR value is entered at the top, taken down to the design
load, across to the traffic, and down to the design thickness. This process would first be done for
the subgrade to determine the minimum total pavement section required above the subgrade. It
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would then be repeated for each layer of the pavement section to determine the required
thicknesses for the flexible layer, base and subbase. In practice, additional requirements are
implemented to account for drainage layer requirements and frost effects that may increase the
final design thicknesses. As these requirements are handled outside of the basic CBR design
model as a modification to the structural design thicknesses, they will not be discussed further.
Note that this structural design process only considers failure of the subgrade. Other
failure modes such as tensile strain at the bottom of the asphalt cement concrete (ACC) layer,
shearing of the ACC, or failure of the base layers are considered through two methods. First are
the material specifications. Second is the implementation of minimum ACC and base layer
thicknesses as a function of aircraft type and material type (UFC 3-260-02 2001).
2.3.3. Shortfalls
One of the largest shortfalls of the old CBR method was the use of the ESWL. As
previously mentioned, the deflection-based method of calculating ESWL was found to be overly
conservative and required the use of the alpha factor to adjust for this obvious over designing.
However, the alpha factor model was built on very limited test data and extrapolated far beyond
the limits of the available data (Gonzalez et al. 2012). This became of a great concern to the
aircraft industry as extremely large and heavy aircraft, such as the Boeing 767, came into service.
Industry had a great concern that potential over-design of pavements due to over classification of
the aircrafts’ ACNs would make the purchase of their aircraft unattractive to airlines because of
associated costs to upgrade airports beyond the true requirements for the aircraft.
Another shortfall existed in the handling of mixed traffic. As the vehicle characteristics
were handled in the CBR equation by the ESWL and contact area, only one vehicle can be
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directly used in the design model. If multiple aircraft were desired, they had to be converted into
equivalent passes of the design aircraft. This was done by summing the number of allowable
passes of the other vehicles, normalized to the number of passes of the design aircraft, on the
required pavement thickness of the design aircraft. This concept was flawed in that the mix of
aircraft will not load the pavement in the same lateral distribution as the lone design aircraft.
Also, the thickness used to calculate the equivalent passes will likely not be the final design
thickness.
2.4. Beta Criteria; the Revised CBR Design Method
2.4.1. Definition
In light of the previously mentioned shortfalls, pursuit of a revised method was sought.
The majority of this work was presented by Gonzalez et al. (2012). One of the main goals of this
revision was to introduce a more obvious mechanistic approach to the model, but still base it on
the empirical CBR input. The revised CBR design method was defined by the following three
equations (Gonzalez et al. 2012).
Equation 2.5
!"#!"!!! !!!!""#! !!!"#! ! !"#!"!!"#$!!! !!!"#$ ! !"#!"!!"#$!
Where: !"#$ = coverages
Equation 2.6 ! ! !! ! !
!"# Where: !! = vertical failure stress
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Equation 2.7
!! !! ! !
! ! ! ! !! ! !"#!!! !
Where: !! = vertical stress at depth ! and angle ! ! = concentration factor ! = point load at surface ! = depth ! = angle from z axis under load to point of interest
Equation 2.5 was defined as the beta criteria and is shown graphically in Figure 2.4. It is
an empirical fatigue failure model derived from full-scale test data. Although not obvious due to
the change in scales in Figure 2.4, the beta criteria shows the general characteristics recognized
in a fatigue failure model. The asymptotic behavior of the curve at high coverages illustrates the
theory that at a low enough stress (beta value), the structure can withstand infinite repetitions.
Equation 2.6 is the design equation used to calculate the beta value based on the design problem.
It was derived from Equation 2.7, through the design criterion that the vertical stress imparted by
the load at the top of the subgrade must be limited to the vertical failure stress. The beta value is
essentially a stress to strength ratio.
18
Figure 2.4. Current Flexible Pavement Design Beta Criteria (Gonzalez et al. 2012)
Equation 2.7 is a generalized solution for vertical stress in a half space. For practical
application, it can be numerically integrated over a (multi) wheel footprint following rules of
superposition. For simplified analysis, it can also be solved into a distributed load over a circular
area yielding Equation 2.8. This mechanistic backing resolved one of the main limitations to the
old CBR method. ESWLs and alpha factor adjustments are no longer necessary, as the
mechanistic response was implemented as the design input instead of an assumed and adjusted
input.
Equation 2.7 is a modified version of the Boussinesq solution for stresses in an elastic,
isotropic half-space. Where as the Boussinesq solution is only valid for an isotropic half space,
19
this modified version accounts for the anisotropic behavior of real soils, primarily the increase of
modulus of elasticity with depth. This was done through the introduction of the concentration
factor “n” as first proposed by Fröhlich (Jumikis 1964). This should not be confused with the
stress concentration factor “Kt” that is used in characterizing a stress increase around a geometric
discontinuity, such as a corner or crack. The concentration factor is an empirical factor and a
function of the soil properties, and, to an extent, the nature of the loading. For a concentration
factor of three, Equation 2.7 becomes the Boussinesq solution. Also, Gonzalez et al. (2012)
showed that with a concentration factor of two and for a single wheel, it could be derived into
Equation 2.9, very similar to the classic CBR design model represented by Equation 2.4.
Equation 2.8
!! ! !! ! !!!
!! !!
! !
Where: !! = vertical stress at depth ! below center of load !! = uniformly distributed circular load ! = concentration factor ! = radius of loaded area ! = depth
Equation 2.9
! ! !! ! !"# !
!!
Combining Equation 2.5, Equation 2.6, and Equation 2.7 provide the full design model,
while still maintaining simplicity to the designer. Equation 2.6 and Equation 2.7 provide a
20
mechanistic input to the empirical failure model defined by Equation 2.5. The entire relationship
can be converted into aircraft specific design charts just as with the classic CBR design method.
An additional benefit is the ability to handle multiple aircraft through the use of a cumulative
damage function (CDF). This resolved another major shortcoming of the old CBR method.
However, this process is not readily adaptable to design charts since the CDF process would
require “hunting” for the lateral position with the maximum damage. It also requires an iterative
process since each aircraft’s damage is a function of pavement thickness. However, this still
reinforces the elegance of the revised method and its suitability for the full spectrum of design.
2.4.2. Development of the Beta Criteria Model
The first area of improvement observed in the new CBR design method was within the
development of the beta criteria model. Equation 2.5 is the best-fit curve to test and field data
following the general form proposed by Gonzalez et al. (2012) of Equation 2.10 shown below.
This form was chosen because of its smooth, continuous form, and asymptotic behavior (C.
Gonzalez, personal communication, July 17, 2012).
Equation 2.10
!"#!"!!! !! ! ! ! !"#!"!!"#$!!! ! ! !"#!"!!"#$!
Where: !"#$ = coverages
In calculating data points used to fit the current criteria, the vertical stress component of
beta, Equation 2.6, was calculated using Equation 2.7, numerically integrated by software over
the wheel footprints. A concentration factor of two was used for these calculations. Gonzalez et
al. (2012) justified this with the fact that the classic CBR equation was essentially a derivation of
21
the beta method using a concentration factor of two, as described in regard to Equation 2.9.
They also showed that using n = 2, a stress-based ESWL can be calculated that was more
conservative than the Boussinesq solution with n = 3. Additionally, this n =2, stress-based
ESWL was less conservative than the deflection-based ESWL. Figure 2.5 shows an example of
this trend for the Boeing 747. According to Gonzalez et al. (2012), essentially, an n = 2 based
ESWL was a compromise between the n = 3 stress-based ESWL and the deflection-based ESWL
originally compared in development of the old CBR method, and was accordingly more
representative of the “true” ESWL.
Figure 2.5. ESWL Comparison by Equivalent Response (Gonzalez et al. 2012)
A value of n = 2 was also somewhat appropriate as it represented a soil with a CBR of 6,
a fairly commonly encountered subgrade design value for clays and silts. However, Gonzalez et
22
al. (2012) went on to observe that the concentration factor was a function of the CBR, n =
f(CBR). This was supported by Fröhlich’s original development of the concentration factor
(Jumikis 1964). He asserted that the CBR was a function of the soil type and the loaded area.
Soil type was also generally related to CBR, as presented in Table 2.2. More discussion will
follow in the next section of specific n-CBR relationships. Gonzalez et al. (2012) continued this
thought and recommend using n = f(CBR) model in design calculations of the beta value. The
impact of doing so on the design thickness is shown in Figure 2.6 for the F-15, a single wheel
aircraft. There is no difference at a CBR of 6 as this is where this n = f(CBR) model corresponds
to a concentration factor of two, however it can have an impact of three inches at other values.
Figure 2.6. Effects of Concentration Factor Model on Design Thickness (Gonzalez et al. 2012)
Despite the recognition that the n = f(CBR) relationship was pertinent to the design
method, the beta criteria itself was not calibrated to this factor. As the criteria was based on
23
relatively few data points, only 34, a shift in the data points due to a different beta value
calculated with n = f(CBR) could cause an appreciable shift in the beta criteria model. For a
basic extrapolation to the full set of test data, Gonzalez et al. (2012) compared data plots for n =
2 and n = 3 in Figure 2.7. Although there is an appreciable shift in the data, the trend remained
the same. However, an n = f(CBR) relationship would result in a shift of individual data points
in different directions dependent on the subgrade CBR and the specific n = f(CBR) model, rather
than a bulk shift of all data points in the same direction and magnitude.
Figure 2.7. Effect of Concentration Factor on Data Points (Gonzalez et al. 2012)
2.4.3. The n = f(CBR) Model
In Fröhlich’s development of the concentration factor, he summarized that the value of
the concentration factor was a function of the soil type and size of the loaded area. Sands
correspond to a value of approximately four. This value also corresponded to a solution
24
representing a medium with a modulus of elasticity that increases with depth, an identifiable
behavior of granular soils whose modulus will increase with confining pressure and/or depth
(Huang 2004). He also determined that values up to 6 were valid for small loading areas with
large contact pressures that impart a plastic deformation of the soil in the perimeter of the load
(Jumikis 1964). Since a concentration factor of three, n = 3, derives the Boussinesq solution,
which is only applicable to an isotropic body, Jumikis (1964) explained that only values of n > 3
are under consideration for practical use in foundation design. However, Ullidtz (1998)
explained that values of n > 3 correspond to a Poisson’s ratio greater than 0.5. On the contrary,
he explained that values of n < 3 correspond to a Poisson’s ratio less than 0.5. Values less than
0.5 correspond to the typical range of subgrade materials of 0.2 to 0.5.
The previous paragraph describes the limits of available literature on concentration factor
values. Considering that this discussion was rather limited and was primarily in regard to the
loads imparted to soils via foundations, a new and complete model was needed for
implementation in the revised CBR design method. Gonzalez et al. (2012) review of test data
revealed two models, both hinged around the point of a CBR of 6 corresponding to n = 2. These
models are represented by Equation 2.11, Equation 2.12, and Figure 2.8. As implemented by the
DoD, Equation 2.11 is for unsurfaced pavements and Equation 2.12 is for surfaced pavements
(C. Gonzalez, personal communication, August 3, 2012).
Equation 2.11
! ! ! ! !"!!!!!!"
Equation 2.12
! ! ! ! !"#!!!!"!#
25
Figure 2.8. Concentration Factor Models (Gonzalez et al. 2012)
Equation 2.11
Equation 2.12
26
3. Research Approach and Methods
Before discussing specific results, the following section outlines the general approach to
this research. This includes a description of the tools and methods used in gathering data,
processing the data, producing final results, and the final analysis of the results. A summary of
the process is presented below in Figure 3.1.
Figure 3.1. Research Process Flow Chart
3.1. Data Collection
Since the majority of the data used by Gonzalez et al. (2012) was from tests ranging from
the 1940s to the 1970s, with a few points in the early 20th century, additional test data was sought
to increase the number of data points available to develop new curves. The test data needed for
this research had to meet several requirements. First, it had to contain all the required testing
parameters to perform beta calculations. This included the pavement structural dimensions,
27
subgrade strength measurements in terms of CBR, weight of the test load, wheel configuration,
and tire dimensions. Second, the test pavements must have been taken to failure, and that failure
must have occurred in the subgrade. Lastly, the test traffic must have been a single gear type,
and not a mixed traffic test. This eliminates assumptions needed to characterize cumulative
subgrade damage induced by different aircraft gear geomotries at varying transverse locations.
3.2. Stress and Beta Calculations
Once data collection was completed, a repeatable method to calculate stress at the
subgrade, and subsequently beta, was required. This process produced the final data points used
to develop the new curves. The software Pavement Engineering Utility (PEU) developed by
Gonzalez et al. (2012) was used to calculate stresses from the test data. This software is a stand-
alone program for airfield and road pavement design. Its capabilities go well beyond the needs
of this research as it is capable of executing both mechanistic and empirical design and
evaluation procedures. Although not specifically intended for this research’s purpose, the
included CBR beta design method uses a numerical integration routine of Equation 2.7 to
calculate the vertical stress at the subgrade, and displays the resulting subgrade stress and the
location. It also includes a full library of in-service aircraft and the ability to input custom gear
and wheel footprints and loading conditions. A screenshot of the test vehicle input window is
shown in Figure 3.2. In this window, the wheel configuration and loading condition is entered.
In the main window, Figure 3.3, the pavement thickness and subgrade strength is entered. The
option button to calculate passes is chosen in order to evaluate the given structure, rather than to
determine a design thickness for the given load. In executing the full evaluation process, the
program determines and outputs the maximum vertical stress, as seen the bottom left corner of
28
Figure 3.3. The predicted number of passes on this structure using the current beta criteria is
also displayed.
The above process was be done both with the concentration factor equal to two, n = 2,
and the n = f(CBR) model depicted in Equation 2.12. This model is included in PEU. To
calculate stresses with n = 2, a CBR of 6 was input as the subgrade strength instead of the actual
test subgrade strength since this corresponds to n = 2 in the n = f(CBR) model. The calculated
stresses were then transferred to a spreadsheet where formulas were used to calculate the final
beta value for each test. Once a calculated beta value from a test is paired with the
corresponding number of coverages to failure, it will be referred to as a data point.
Figure 3.2. PEU Test Vehicle Editor Window; Example, Test Point 22
29
Figure 3.3. PEU Main Window; Example, Test Point 22
3.3. Beta Criteria Curve Fitting
Once each test’s data was processed into a beta versus coverages data point, a nonlinear
regression analysis was conducted to fit several general curve forms to the data points. This
process produced several potential curves in support of the research goal to redevelop the beta
criteria. These general curve forms were fit to both the n = 2 and the n = f(CBR) data set to
allow testing of the research hypothesis.
Curve fitting was done with the nonlinear regression tools in and Minitab 16. It does so
through an iterative minimization of the sum of squared residuals. It can fit standard or user-
defined models to the data sets. The first general form used was that of Equation 2.10, referred
30
to as the current form (CF). Second, review of FAA and DoD mechanistic failure models,
Figure 3.4 and Figure 3.5, respectively, revealed another general form. If stress is swapped with
strain, and material properties are considered a function of beta since beta calculation includes
the subgrade CBR, both models can be generalized to the form of Equation 3.1. Rearranging this
equation to appear in the order of the beta criteria gave Equation 3.2, referred to as the
mechanistic form (MF). Third, a simplified version of the current form was chosen, referred to
as the reciprocal logarithm (RL), Equation 3.3. In addition to fitting these general forms to the
data points, the current criteria, Equation 2.5, was reproduced in Minitab 16 in order to produce
regression statistics for analysis purposes.
Figure 3.4. FAA Mechanistic Subgrade Failure Model (AC 150/5320-6E 2009)
Figure 3.5. DoD Mechanistic Subgrade Failure Model (UFC 3-260-02 2001)
31
Equation 3.1
!"#$ ! ! ! !!
!
Equation 3.2 ! ! ! ! !
!"#$ ! !
Equation 3.3
! ! !! ! ! ! !"#!"!!"#$!
The final selected means by which to account for differences in subgrade strength in
design models was to consider entirely separate criteria for separate subgrade strength values.
This resulted in multi-curve criteria, divided into the subgrade strength categories, derived from
the ACN – PCN system, shown in Table 3.1. For simplicity of analysis, these curves were only
be fitted with n = f(CBR) and to the current form of Equation 2.10.
Table 3.1. Subgrade CBR Categories (AC 150/5335-5B 2011) Subgrade Category Strength Range High (H) CBR >13 Medium (M) 8 < CBR < 13 Low (L) 4 < CBR < 8 Ultra low (UL) CBR < 4
3.4. Model Testing and Verification
Once final curves were produced, they had to be evaluated in some manner to verify their
function, evaluate their fit to the data points, and verify that any curve is significantly different
enough from the current criteria to consider implementation.
32
Three primary techniques were used to determine the confidence of the curves in fitting
to the data and verify their function. The first method was through statistical comparison of the
curves. The second method was to compare predicted design thicknesses from each curve in a
given design scenario. The third method was to test the curves against additional test data.
3.4.1. Statistical Analysis
The regression statistics produced by Minitab 16 were used to determine the adequacy of
fit of each curve to the data points. This consisted primarily of a comparison of Standard Error
(S) for each curve. An S of zero would indicate a model with perfect prediction. This is nearly
unobtainable, and could actually indicate over-fitting. The actual magnitude of S is in units of
the dependent variable, beta in this case. As such, an upper bound indicating a poor fit is not
clearly definable. A simple way of describing of S in nonlinear regression is that it is similar to
the standard deviation in sample descriptive parameters.
Unlike linear regression, a meaningful coefficient of determination (R-squared) cannot be
calculated for nonlinear regression curves. This meant that the useful tool of rating a curve’s fit
on a scale of one (perfect prediction) to zero (no prediction) was not available for this research,
so standard error was the only statistic available. However, caution must be taken when making
comparisons. Due to the methods used in nonlinear regression, standard error values are only
meaningful if comparing curves fit to the same data. Fortunately, since the n = 2 and n = f(CBR)
data sets contain the same quantity of points at the same predictor (coverage) values, standard
errors could be compared for all curves fit to these data sets. Unfortunately, this also meant that
a direct comparison could not be made between the full test data set and a limited data set, such
as the 34 points used for the current criteria.
33
3.4.2. Design Thicknesses
Statistical analysis is a useful tool to compare the various curves directly, however it does
not reveal what differences will result in actual implementation and design execution. To
evaluate the practical application of the curves, one must compare the design pavement
thicknesses resulting from the use of each curve in the design process. The PEU software and
methods used by Gonzalez et al. (2012) was used to plot and evaluate design thickness
differences predicted by each curve for a number of representative aircraft at three different
coverage levels. The three coverage levels represented an emergency use pavement (100
coverages), a medium use pavement (10,000 coverages), and a heavy use pavement (100,000
coverages). At least one lightweight, single-wheel aircraft, one heavier single-wheel aircraft, one
medium weight, multi-wheel aircraft, and one heavy weight, multi-wheel aircraft were chosen
for the evaluation. The aircraft chosen were the T-38 Talon, F-15E Strike Eagle, Boeing 767-
200 and Boeing 747-400, respectively. Specifications for each aircraft are listed in Table 3.2.
Once design thicknesses were produced for each curve, aircraft, and coverage level, they could
be compared graphically on a plot of design thickness versus subgrade CBR to evaluate any
trends between the different curves and identify the existence of significant differences in design
thicknesses, if any existed.
Table 3.2. Design Thickness Comparison Aircraft Specifications (PEU software) Aircraft Weight (lbs) Main Gear Type T-38 Talon 12,712 Single F-15E Strike Eagle 81,000 Single Boeing 767-200 267,500 Dual Tandem Boeing 747-400 798,000 Dual Tandem
34
3.4.3. Test Data
A final method to verify the curves was to test them against an additional full-scale test
and compare each curve’s predicted pavement life (coverages) with that of the actual test. To do
so, test data points 61 and 62 were omitted from the points used in curve development so they
could be used for this analysis. These test points were from the NAPTF construction cycle 5.
3.5. Selection of Model
Once the previous testing and verification processes are completed, the results were
evaluated across all curves. Statistical data, test data verification, and design thickness
evaluation results were used in a quantitative and qualitative analysis of each curve to determine
the model that performs the best.
35
4. Results and Analysis
4.1. Data Calculation
Test data collection began with obtaining all data used by Gonzalez et al. (2012) in
development of the current beta criteria, a total of 34 data points. This included all pertinent data
available from numerous DoD tests prior to 1994, which was consolidated by Barker and
Gonzalez (1994), as well as relevant data from the National Airport Pavement Test Facility
(NAPTF). The NAPTF data included all flexible tests taken to failure in construction cycles 1
through 3 as consolidated by Hayhoe (2004).
The original data sources cited by Barker and Gonzalez (1994) were scoured for
additional, and more detailed, test information. This search led to the discovery of numerous
additional tests. However, these tests either involved mixed traffic or were not taken to failure,
so they were not adequate for this research. Additional data was sought from recent completion
of the NAPTF construction cycle 5. Construction cycle 4 was not relevant as it only tested rigid
pavements. This ultimately resulted in two additional tests. Lastly, test data was obtained from
recent ERDC full-scale testing. This testing was done specifically to validate the revised CBR
method and included 12 data points. In total, 62 data points were found. The collected test data,
limited to pertinent input values for calculations and analysis, are presented in Table 4.1. The
Appendix contains details beyond direct inputs and includes sources. Tests numbers used by
Gonzalez et al. (2012) are shown in bold. The additional test numbers 63 through 66 are
discussed in section 4.4.3.
36
Table 4.1. Collected Test Data
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37
Table 4.1 (Continued)
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IN%9:"#'9B"<&=' 47'>'48'@'48' 22./...' 016.' 44/...' -84' -..' -51..' 218' 46'IO%9:"#'9B"<&=' 47'>'48'@'48' 22./...' 016.' 44/...' -84' -..' 281..' 717' 0/...'JP%9:"#'9B"<&=' 47'>'48'@'48' 25./...' 016.' 64/...' 2-4' -..' 781..' 717' 0-/...'JG%9:"#'9B"<&=' 47'>'48'@'48' -8./...' 016.' 74/...' --4' -..' -4107' 814' 3/2..'JH%9:"#'9;#<&=' 47'>'48' --./...' 016.' 44/...' -84' -..' -51..' 712' 44'JI%9:"#'9;#<&=' 47'>'48' --./...' 016.' 44/...' -84' -..' 281..' 712' 0/-..'JJ%9:"#'9;#<&=' 47'>'48' -6./...' 016.' 64/...' 2-4' -..' 781..' 712' 08/...'JK%9:"#'9;#<&=' 77'>'43' 03./...' 016.' 74/...' --4' -..' -4107' 812' 4/3..'JL%9:"#'9;#<&=' 77'>'43' 03./...' 016.' 74/...' --4' -..' 05100' 817' 5/-..'JM%9:"#'9;#<&=' 77'>'43' -7./...' 0166' 6./...' -5.' -.8' 401..' -1.' 7.'73'!"#$%&'()&&%' *+,' 74/26.' 01-8' 74/26.' -33' 048' 061..' 071-' 37.'75'!"#$%&'()&&%' *+,' 74/26.' 01-8' 74/26.' -33' 048' -21..' 513' 4/...'4.'!"#$%&'()&&%' *+,' 74/26.' 01-8' 74/26.' -33' 048' -41..' 215' 7..'40'!"#$%&'()&&%' *+,' 74/26.' 01-8' 74/26.' -33' 048' 2-1..' 215' -/...'4-'CD;%' 7.14' 5./8-.' 01-8' 74/26.' -33' 048' 061..' 071-' -/...'42'CD;%' 7.14' 5./8-.' 01-8' 74/26.' -33' 048' -21..' 513' 06/...'47'CD;%' 7.14' 5./8-.' 01-8' 74/26.' -33' 048' -41..' 215' 0/-..'44'CD;%' 7.14' 5./8-.' 01-8' 74/26.' -33' 048' 2-1..' 215' 0/-..'46'!"#$%&'()&&%' *+,' 27/7.3' 0180' 27/7.3' 0.8' 2-0' 061..' 071-' 23.'48'!"#$%&'()&&%' *+,' 27/7.3' 0180' 27/7.3' 0.8' 2-0' -21..' 513' 0/.4.'43'!"#$%&'()&&%' *+,' 27/7.3' 0180' 27/7.3' 0.8' 2-0' -41..' 215' 0.4'45'!"#$%&'()&&%' *+,' 27/7.3' 0180' 27/7.3' 0.8' 2-0' 2-1..' 215' -/-..'LP%9B"E%&'9;#<&='
'434/...' 0164' 77/34-' 208' 07-' 0-14.' 61.' -'
38
Table 4.1 (Continued)
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4.2. Stress and Beta Calculations
Once test data was collected, the pavement structure, gear geometry, and loading were
input into PEU to calculate stress at the subgrade. This was done once using a concentration
factor of two, n = 2, and again using the concentration factor as a function of CBR model, n =
f(CBR), in the stress calculation. Beta values were then calculated from each stress using
Equation 2.6. These beta values were then matched with each test’s corresponding coverages to
failure. Final calculated stresses and beta values are shown in Table 4.2. The column “Est. Beta,
Gonzalez et al. (2012)” will be discussed shortly. The data points are also shown graphically
below in Figure 4.1. In comparing the two sets for the case of n = 2 and n = f(CBR), it can be
seen that most of the points saw a change with the different concentration factor models, except
for those points at which the CBR was 6. This is due to the fact that at a CBR of six, the n =
f(CBR) model yields n = 2.
39
Table 4.2. Data Sets for n = 2 and n = f(CBR), Beta versus Coverages
!"#$%
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0' 04.' 20176' 06178' 20176' 06178' 0214.' '' 27' 0.7' 06138' 3132' 06103' 3178' 0013.'-' 0/8..' -610.' 5100' -8154' 5186' 817.' '' 24' 0/4..' 00175' 71.0' 0.134' 2185' 315.'2' 0.' 88150' 0412.' 36135' 081.6' 021..' '' 26' 2/34.' 51.3' 0183' 3164' 018.' 810.'7' 6.' 65148' 0-107' 85173' 02138' 0.14.' '' 28' 0/4..' 06138' -157' 04158' -185' 514.'4' 26.' 6.18.' 0-12.' 63137' 02154' 0.14.' '' 23' 46' --105' 714.' -.130' 71--' 071..'6' 0/4..' 74175' 3108' 42127' 5143' 615.' '' 25' 0/...' 06176' -154' 04134' -134' 518.'8' 0/2..' -0150' 316.' -21.2' 51.7' 810.' '' 7.' 0-/...' 04144' 6100' 041.6' 4150' 514.'3' 2/86.' -4186' 0.10-' -613.' 0.14-' 315.' '' 70' 3/2..' --14-' 3137' -21-6' 5102' 810.'5' 2/86.' -4186' 3155' -81-7' 5140' 815.' '' 7-' 44' -.154' 8120' 05152' 6156' 0-1..'0.' 0' 7315.' 7014-' 741-6' 23172' 251..' '' 72' 0/-..' 07134' 0-160' 07108' 0-1.2' 518.'00' -..' -2176' 06184' --1-0' 04136' 071..' '' 77' 08/...' 02107' 5123' 0-160' 51..' 514.'0-' 0-.' -5127' -7150' -8106' -21.6' -01..' '' 74' 4/3..' --128' 03155' -21.7' 05146' 817.'02' -06' 73167' 0.150' 44188' 0-140' 51..' '' 76' 5/-..' 2213.' 8143' 27152' 8137' 310.'07' 083' 73167' -0132' 75133' --125' 031..' '' 78' 7.' 0-152' 413.' 00120' 41.3' 061..'04' -.2' 73167' -4178' 73167' -4178' -01..' '' 73' 37.' 70108' -0146' 781-3' -7186' *+,'06' 7.' 4810-' -5150' 4810-' -5150' -61..' '' 75' 4/...' -2107' 0-10-' -41--' 021-0' *+,'08' 0/4..' 4-185' 6107' 621.8' 8127' *+,' '' 4.' 7..' -.1.4' -122' 03148' -106' *+,'03' -/...' 4-185' 7187' 67135' 413-' *+,' '' 40' -/...' 0-15.' 0106' 0015-' 01.8' *+,'05' 2-3' 37132' 8160' 0.610-' 5142' *+,' '' 4-' -/...' 7-1-4' 2185' 73103' 712-' *+,'-.' -/...' 40166' 6186' 6214-' 3120' *+,' '' 42' 06/...' -7130' 21-4' -613-' 2140' *+,'-0' 8/035' 2814-' 317-' 7-137' 5160' *+,' '' 47' 0/-..' -0136' 7150' -.17.' 7143' *+,'--' 7.' 05150' 04167' 03130' 07188' 0414.' '' 44' 0/-..' 041.0' 00185' 071.-' 001.0' *+,'-2' -3.' 04164' 0-1-5' 07138' 00163' 0-1..' '' 46' 23.' 2810-' -5104' 7217-' 2710.' *+,'-7' -/...' 72136' 6135' 4018-' 310-' *+,' '' 48' 0/.4.' 051-8' 21.2' -01.5' 2120' *+,'-4' 60.' 46137' 515-' 66108' 00144' *+,' '' 43' 0.4' 06175' -133' 041--' -166' *+,'-6' -/...' 24134' 7108' 77125' 4108' *+,' '' 45' -/-..' 0.12.' 01-.' 514.' 0100' *+,'-8' 0/4..' 2-122' 6188' 281-2' 813.' *+,' '' 6.' -' 4715-' 0014.' 4715-' 0014.' -51..'-3' 5.' 2-122' -.120' 20172' 05184' *+,' '' 60' 0-/.83' 06182' 0.140' 04166' 5137' *+,'-5' 20.' 2-122' 3176' 24158' 517-' *+,' '' 6-' 5/36.' 08103' 714.' 0610-' 71--' *+,'2.' 07.' -.168' 0-155' -.10-' 0-167' *+,' '' 62' 0-/.83' 06182' 0.140' 0617-' 0.12-' *+,'20' 2/66.' 06163' 5126' 0614-' 51-8' *+,' '' 67' 5/36.' 08103' 5167' 06138' 5176' *+,'2-' 6..' -.1..' 071-3' 05104' 02168' *+,' '' 64' 0-/.83' 06182' 00154' 06182' 00154' *+,'22' 3' 2.157' -61-8' -3150' -7144' -61..' '' 66' 5/36.' 08103' 07145' 08103' 07145' *+,'
40
Figure 4.1. Test Data Plot of Beta versus Coverages
An attempt was made to validate beta value calculation procedures by reproducing the
original 34 points used by Gonzalez et al. (2012). This attempt was unsuccessful. However,
detailed input data, procedures, and assumptions used by Gonzalez et al. (2012) were not
available to determine the root cause of the differences. Beta values were pulled from Figure 4.2
for the comparison and are shown in Table 4.2. The error associated with estimating the values
from this graph was not significant enough to explain the lack of reproducibility. In fact, error of
calculated values to Gonzalez et al. (2012) values was scattered, but had an approximate mean
value of 20 percent. The only major input assumption that was not always specified in test data
was the tire shape. This parameter was approximated with an ellipse and a major/minor (b/a)
axis ratio. A sensitivity analysis of beta values to tire shape revealed an approximate error of
41
only 3 percent between the extremes of a circular tire footprint and a b/a of 1.75, a rather extreme
ellipse for a tire. In addition, the software provided by Mr. Gonzalez was checked against the
simplified circular loading case, Equation 2.8. Multiple back-calculations were done in an
attempt to identify a trend or common variable in the error, but nothing was found. Each step of
the beta value calculation was scrutinized multiple times, as well as peer-reviewed. As seen
below in Figure 4.3, the difference in beta values primarily resulted in an upward shift of the
calculated data points. An examination of the impact of this discrepancy in regard to the
resulting curves is be explored in the next section.
Figure 4.2. Source Plot of Estimated Beta Values (Gonzalez et al. 2012)
42
Figure 4.3. Beta Calculation Comparison and Reproducibility Plot
4.3. Curve Fits
4.3.1. Resulting Curves
Once the n = 2 and n = f(CBR) data sets were produced, they were fit to several models
using the nonlinear regression tool in Minitab 16. This included the current form (CF), Equation
2.10, the mechanistic form (MF), Equation 3.2, and the reciprocal logarithm (RL), Equation 3.3.
Before doing so, data points 61 and 62 were omitted so they could be used later for model testing
and verification as discussed in section 3.4.3. This means a total of 60 data points were used to
develop new beta criteria, 76 percent more than that of the current criteria.
For purposes of comparison, all curves fitted to n = f(CBR) beta values are shown in
Figure 4.4 along with the current beta criteria. As can be seen, the resulting curves show a
43
qualitative difference from the current criteria and themselves. A quantitative comparison will
be discussed later using the model coefficients and statistics in Table 4.3.
A general observation from these results show that below approximately 2,000
coverages, except for a small segment below five coverages, all of the new curves are less
conservative than the current criteria. This is because greater beta values at a given coverage
level would yield thinner cross sections in design applications. Above 2,000 coverages, the
opposite is true for all new curves except the reciprocal logarithm (RL).
Figure 4.4. Resulting n = f(CBR) Curves
44
Table 4.3. Curve Coefficients and Statistics Curve General Form Coefficients Statistics
a b c S Current, n=2 Equation 2.10 1.5441 0.2354 0.0730 4.30 Current, n=f(CBR) Equation 2.10 1.5441 0.2354 0.0730 3.80 CF, n=2 Equation 2.10 1.5824 0.0695 -0.1095 4.15 CF, n=f(CBR) Equation 2.10 1.5586 0.0778 -0.0886 3.69 MF, n=2 Equation 3.2 11.6105 5.686 N/A 4.14 MF, n=f(CBR) Equation 3.2 10.9368 6.1191 N/A 3.69 RL, n=2 Equation 3.3 0.024361 0.021491 N/A 4.20 RL, n=f(CBR) Equation 3.3 0.025996 0.020454 N/A 3.73
The results of the multi-curve, subgrade category model are shown below in Figure 4.5,
Figure 4.6, and Table 4.4. Figure 4.5 indicates that not enough data was available to develop a
meaningful model based on the original subgrade categories. This is indicated by the behavior of
the curves. In fact, the medium strength subgrade curve is inverse of the expected behavior. For
this reason, this criteria was omitted from further analysis. A second attempt was made by
merging the ultra low and low categories and the medium and high categories. This resulted in
more satisfactory curves, with the primary difference being a shift in location. However, since
there is no theoretical explanation for this occurrence, these curves will not be further analyzed.
Table 4.4. Subgrade Category Curve Coefficients and Statistics Subgrade Category General Form Coefficients Statistics
a b c S Ultra Low Equation 2.10 1.587 0.2521 0.3577 3.93 Low Equation 2.10 1.5057 0.0046 -0.1253 4.10 Medium Equation 2.10 0.9389 0.25873 0.25044 1.37 High Equation 2.10 1.3557 -0.04314 -0.16979 1.90 Ultra Low – Low Equation 2.10 1.56457 0.1425 0.00828 3.98 Medium - High Equation 2.10 1.47195 0.24219 0.07017 1.79
45
Figure 4.5. Subgrade Category Curves
Figure 4.6. Combined Subgrade Category Curves
46
Due to the inability to reproduce Gonzalez et al. (2012) beta value results, as discussed in
4.2, additional curves were fitted to the original 34 data points to enable comparison. These
results are shown in Figure 4.7 and Table 4.5. These curves were fit to three different data sets;
one using the estimates from Figure 4.2, and two more with calculated values using both n =2
and n = f(CBR). As can be seen in Figure 4.7, the major impact of the difference in calculated
beta values is a shift upward of the curve. This is seen in a direct comparison of the curve fit to
Gonzalez et al. (2012) beta values, which used n = 2, and the curve fit to the calculated values
using n = 2. It should be noted that even though the resulting curves have very similar shapes,
the individual data points did not shift consistently, so this phenomenon should only be
considered a coincidence. Another interesting observation reveals that the curve fit to the
estimated data points is actually a slightly worse fit than the current criteria. This would indicate
that Gonzalez et al. (2012) used a different regression analysis method that resulted in a better
curve. However, a detailed explanation of that method was not available.
47
Figure 4.7. Comparison of Curves from Current Criteria Test Data
Table 4.5. Curve Coefficients and Statistics from Current Criteria Test Data Curve General Form Coefficients Statistics
a b c S Current Equation 2.10 1.5441 0.2354 0.073 3.62 Fit to Gonzalez et al. (2012) values
Equation 2.10 1.5439 0.1152 -0.0627 3.66
Fit to calculated values, n=2
Equation 2.10 1.5828 0.1484 -0.0032 4.51
Fit to calculated values, n=f(CBR)
Equation 2.10 1.5565 0.1289 -0.015 4.17
4.3.2. Concentration Factor Effects
Once all curves were fitted to both the n = 2 and the n = f(CBR) data sets, the effect of
the two concentration factor models were evaluated. To simplify analysis, Figure 4.8, Figure
48
4.9, and Figure 4.10 each show only one general form of curve so that the impact of the
concentration factor on the resulting curves can be more easily seen. An observable trend exists
in all three figures. For all the models, the n = f(CBR) curves have higher beta values at higher
coverages and are therefore less conservative in the high coverage range. This would imply a
greater life, or thinner cross sections, for permanent pavements, and a resulting cost savings due
to a smaller design thickness. This is of great relevance as a primary concern in the design of
permanent pavements is cost. Conversely, all the n = f(CBR) models are more conservative at
lower coverages. This would imply a need for conservatively thicker pavements for short-term,
contingency requirements where performance is generally of a much greater concern than that of
cost. It can also be seen in Figure 4.7 and Table 4.5 that this trend even holds true for the
original 34 data points used by Gonzalez et al. (2012) in development of the current beta criteria,
though less pronounced at high coverages.
49
Figure 4.8. Current Form Curves
Figure 4.9. Mechanistic Form Curves
50
Figure 4.10. Reciprocal Logarithm Curves
4.4. Model Testing and Verification
After development of all the curves, each was evaluated in order to come to a conclusion
regarding the best curve to select as the revised beta criteria per the research objective. In
addition, the effects of the two concentration factor models could be compared to test the
research hypothesis.
4.4.1. Statistical Analysis
The first method of model verification used was the comparison of the curve’s statistics
through the standard error (S). This method evaluated which curve best describes the trend of
the data points. First, the n = f(CBR) curves were evaluated to identify the best curve for
51
selection as the redeveloped beta criteria. Second, the n = 2 curves were compared with the n =
f(CBR) curves to evaluate the effects of the two concentration factor models.
Referring back to Table 4.3, it can be seen that the CF, n = f(CBR) and MF, n = f(CBR)
curves share the best statistical fit with the smallest standard error of 3.69. Arguably, the
difference between other n = f(CBR) forms is not that great; the RL curve standard error is only
0.04 greater. It can also be seen that all of the new n = f(CBR) curves show a better fit than the
current criteria.
In comparing the effects of n = 2 versus the n = f(CBR) model, it is evident that the
concentration factor does indeed make a difference. In fact, review of Table 4.3 shows that for
any general form of curve, the one fit using the n = f(CBR) model has a lower standard error, an
average of 0.5, than the n = 2 model, and therefore fits the data better. This is likely due to the
fact that this model removes some error associate with the simplifying assumption of n = 2, and
therefore, is better aligned with real field performance. It can also be seen in Figure 4.7 and
Table 4.5 that this trend even holds true for the original 34 data points used by Gonzalez et al.
(2012) in development of the current beta criteria.
4.4.2. Design Thicknesses
The next method of curve testing and verification was to compare the design thickness
resulting from each curve in practical design applications. To evaluate this across the full
spectrum of possible design scenarios, the aircraft in Table 3.2 were selected for this evaluation.
In addition, each aircraft was evaluated against the coverage levels discussed in section 4.4.2.
52
The resulting graphs of design thickness versus CBR are shown in Figure 4.11 through
Figure 4.22. As can be inferred from Figure 4.8, Figure 4.9, and Figure 4.10, the n = 2 curves
have the same general trend in comparison to their corresponding n = f(CBR) curve, so design
thicknesses were only developed for the current form, CF, n = 2, in order to show the general
impact of concentration factor on the design thicknesses.
As expected from comparing the criteria in Figure 4.4, it can be observed that across all
aircraft and coverage levels, design thicknesses should, and indeed do, vary from the current
criteria. But what was not evident in Figure 4.4 is the magnitude of the difference. All of the
resulting design thickness figures showed that each model would have an appreciable difference
in actual designs, such that consideration of different models is significant. In the following
discussions, the only the design thicknesses at low CBR values will be compared, as this is
where the resulting design thickness is most sensitive to the different models.
For all aircraft in the 100 coverage range, the CF, n = 2 curve and the CF, n = f(CBR)
curve predicted almost the same thickness since this is where the criteria roughly match. Also in
this range, all curves resulted in a thicker design thickness than the current criteria. These results
agreed with the general observations made in section 4.3.1. However, at 10,000 coverages and
greater, this trend was reversed, except for the RL curve. The most extreme variations from the
current criteria were seen in the heavier Boeing 747 and Boeing 767. The largest difference was
found with the Boeing 767 at 100,000 coverages, Figure 4.16, where the MF curve results in
design thickness 21.7 inches greater (+50 percent) than the current criteria. Although
magnitudes are not as extreme, the lighter F-15E and T-38 saw the same trends as well. For the
F-15E at 100,000 coverages, Figure 4.20, the MF curve is 7.5 inches greater (+21 percent) than
53
the current criteria. Even at low coverage levels, significant differences occurred. For the
Boeing 747 at 100 coverages, Figure 4.11, the MF curve resulted in 3.4 inches thinner (-10
percent) than the current criteria. For the F-15E, Figure 4.17, the MF curve was 1.45 inches (6
percent) less than the current criteria.
It should be noted that for the lightest aircraft, the T-38, at high CBR values, design
thicknesses in the range of 3 inches to 5 inches were predicted. This thickness is only what was
predicted based on the stress distribution theory, and does not consider issues such as asphalt
layer failure, constructability, freeze/thaw, or drainage.
54
Figure 4.11. Boeing 747 Design Thickness at 100 Coverages
Figure 4.12. Boeing 747 Design Thicknesses at 10,000 Coverages
55
Figure 4.13. Boeing 747 Design Thicknesses at 100,000 Coverages
Figure 4.14. Boeing 767 Design Thicknesses at 100 Coverages
56
Figure 4.15. Boeing 767 Design Thicknesses at 10,000 Coverages
Figure 4.16. Boeing 767 Design Thicknesses at 100,000 Coverages
57
Figure 4.17. F-15E Design Thicknesses at 100 Coverages
Figure 4.18. F-15E Design Thicknesses at 10,000 Coverages
58
Figure 4.19. F-15E Design Thicknesses at 100,000 Coverages
Figure 4.20. T-38 Design Thicknesses at 100 Coverages
59
Figure 4.21. T-38 Design Thicknesses at 10,000 Coverages
Figure 4.22. T-38 Design Thicknesses at 100,000 Coverages
60
4.4.3. Test Data
The final method of curve testing and verification was to compare the curves against an
independent full-scale test. With an input of the independent test’s pavement structure and test
vehicle, a beta value was calculated just as was described in developing the data points. This
beta value was then input into each model to produce a predicted number of coverages to failure.
This prediction was then compared to the independent test’s actual coverages to failure in order
to evaluate the curves.
Two tests were used for the model verification. They were both from the NAPTF
construction cycle 5 (CC5). The test vehicles were 6-wheel and 10-wheel configurations with a
load of 70,000 lbs per wheel. Additional test data is available in Table 4.1 and the Appendix
under test items 61 through 66. Although there were only two tests, six data points are shown.
The as-built subgrade CBR values of the test items was 3.2, however post-test CBR
measurements were in the range of 4 to 6. Specific values and reasons for the change in CBR
were unavailable at the time the test data was provided (G. Hayhoe, personal communication,
September 19, 2012). Therefore, the two tests were treated as six different test points at three
CBR levels so the sensitivity of analysis the tests’ subgrade CBR could be evaluated as well.
The two tests were analyzed against three different CBR values due to the changes in the
subgrade over the testing period. The first CBR value used was the as-built value of 3.2, tests 61
and 62 in Table 4.1. The next CBR value of 5 represents the likely strength during testing since
it averages the post-test values. These are tests 63 and 64. Lastly, a CBR of 6 was chosen to
represent the best-case scenario of rapid strength gain before testing began and lasting through
the test. These are tests 65 and 66.
61
The resulting predicted design lives for each test are shown below in Table 4.6. Also
included is the error of the predicted life for each curve. Unless otherwise noted, all curves are
based on beta values using n = f(CBR).
Table 4.6. NAPTF CC5 Life Prediction and Error Curve Current CF, n=2 CF MF RL Test Item C
overages to Failre
Predicted C
overages
Error
Predicted C
overages
Error
Predicted C
overages
Error
Predicted C
overages
Error
Predicted C
overages
Error
6-Wheel CBR 3.2
12,078 48 -1.00 105 -0.99 103 -0.99 124 -0.99 72 -0.99
10-Wheel CBR 3.2
9,860 40 -1.00 88 -0.99 85 -0.99 104 -0.99 59 -0.99
6-Wheel CBR 5
12,078 1,594 -0.87 1,447 -0.88 1,794 -0.85 1,572 -0.87 2,925 -0.76
10-Wheel CBR 5
9,860 1,206 -0.88 1,207 -0.88 1,470 -0.85 1,334 -0.86 2,194 -0.78
6-Wheel CBR 6
12,078 10,988 -0.09 4,533 -0.62 6,379 -0.47 4,284 -0.65 20,417 0.69
10-Wheel CBR 6
9,860 7,764 -0.21 3,739 -0.62 5,143 -0.48 3,631 -0.63 14,497 0.47
This shows that at the as-built CBR, prediction was drastically incorrect for all models.
This is likely explained by the change in support conditions over the testing. Hayhoe (2011)
acknowledges this same phenomenon in his analysis with mechanistic failure models. When
considering the post-failure test data, the models showed better prediction. The best prediction
overall was actually the current criteria for the CBR 6 tests, with -9 percent and -21 percent. The
best prediction of the new models is made by the CF curve for the 6-wheel, CBR 6 at -47 percent
error, and by RL for the 10-wheel, CBR 6 at +47 percent error. At the most likely representative
CBR value, the best predictor is the RL curve at -76 percent and -78 percent. It would appear
that none of the models were very good based on this test data. However, when the test data was
plotted alongside the model development data, Figure 4.23, it can be seen that at a CBR 5 and
62
CBR 6, they plot within the scatter of the available test points. At a CBR of 5, the tests’ beta
values are approximately 2.3 away from the CF curve, well within the standard error of 3.69.
Figure 4.23. NAPTF CC5 Test Data Plot Against Model Development Points
To explore this data scatter in a logical fashion, a reliability-based approach can be
applied to the curves. This could be done by determining an appropriate magnitude by which to
shift the fitted curve downward, conservatively, to achieve a desired level of confidence that
actual required design beta values are equal to or greater than those predicted by the model. An
example of such an analysis is shown below in Figure 4.24 for the CF, n = f(CBR) curve. This
was done by treating the standard error in the same manner as the standard deviation is used in
normally distributed population analysis. This method was not completely validated through
additional research and should therefore only be considered an approximation for illustration
63
purposes only. The standard error of the curve was multiplied by the z-score for the
corresponding reliability/probability. This resulting value was then subtracted from the curve
across its entire length to produce the final reliability curve. Accordingly, the original, CF, n =
f(CBR) curve corresponds to 50 percent reliability.
The 75 percent reliability curve was used for additional design thickness comparisons for
the Boeing 767 and F-15 at 10,000 coverages, Figure 4.25 and Figure 4.26 respectively. An
additional curve is included using the mechanistic layered elastic design (LED) process included
in the USACE software Pavement – Transportation Computer Assisted Structural Engineering
(PCASE). Conservative modulus values were used for the asphalt layer, 350,000 psi, and the
base layer, 30,000. These were also the default material property values in PCASE. The
leveling out behavior of these curves is due to the minimum thickness requirements not included
in this beta criteria analysis. It can be seen that for these aircraft and this coverage level, the
conservative LED thicknesses were very similar to the developed CF, n =f(CBR) curve, further
verifying the redeveloped criteria. For the heavier aircraft, the 25 percent increase in reliability
would cause a 63 percent thickness increase for the Boeing 767 at low subgrade CBRs, but only
a 32 percent increase at high CBRs. For the F-15, the increase was about 25 percent for all
CBRs. Due to this drastic increase in design thickness, and accordingly cost, as well as the
increase over the conservatively plotted mechanistically produced thicknesses, further study and
implementation of this topic would be sensitive to the implementing agency’s risk analysis.
64
Figure 4.24. Example Reliability Based Analysis of Beta Criteria
Figure 4.25. Design Thickness Reliability Evaluation, Boeing 767, 10,000 Coverages
65
Figure 4.26. Design Thickness Reliability Evaluation, F-15E, 10,000 Coverages
4.5. Model Selection
Review of the preceding model testing and verification showed that the current form, n =
f(CBR) curve is the best selection for the redeveloped beta criteria. This is due to several
reasons. First, it was shown that n = f(CBR) based curves consistently produce a better fit to the
data than the n = 2 models. Second, a review of Table 4.3 showed that it has the best-fit statistics
of all the curves, though it did tie with the MF curve. However, at much higher coverages than
shown in Figure 4.4, the MF curve did not show a good representation of the fatigue failure trend
of infinite repetitions at low stress. It maintains a steeper slope than the CF curve and
approaches zero many orders of magnitude earlier. Third, the current form, n = f(CBR) model
showed an appreciable difference in design thicknesses relative to the current criteria along the
entire length of the curve and the full spectrum of aircraft and lifespans. This was true for the
66
lighter, single-wheel aircraft and the heavier, multi-wheel aircraft. The largest design thickness
difference of +10.1 inches occurred with the Boeing 767 at 100,000 coverages, inferring that this
model is significantly different enough to warrant considering implementation. Finally, the
current form, n = f(CBR) model was reasonably verified by the NAPTF CC5 test data,
considering the scattered nature of the model development data itself and the CC5 subgrade
behavior.
67
5. Summary, Findings, Conclusions, and Recommendations
5.1. Summary
This research had two primary goals. The first was to develop revised beta criteria by
including the concentration factor as a function of CBR model into the beta criteria development.
The second was to test the hypothesis that criteria developed from this concentration factor
model would provide a more confident model in terms of accurately representing the empirical
data used to develop it. These goals were accomplished though a process of full-scale test data
collection, processing the collected data into data points of beta versus coverages, nonlinear
regression, and finally, model testing and verification.
This process resulted in revised beta criteria by including the concentration factor as a
function of CBR in the development of the failure model. Furthermore, the results showed that
including the concentration factor as a function of CBR in the model development also provided
a model with more confidence than one built with the concentration factor equal to two.
5.2. Findings
Several observations were made through this research in addition to the actual research
objectives. First, it was found that curves produced with the n = f(CBR) model are more
conservative than n = 2 models at low coverages, approximately below 2,000 coverages. The
opposite trend is true over 2,000 coverages. Also, the inclusion of new test data into the criteria
development resulted in curves that were all less conservative than the current criteria below
approximately 2,000 coverages. Lastly, it was found that the data points developed from the
68
tests are fairly scattered, and make deterministic beta criteria verification difficult. For this
reason, reliability-based criteria may be more appropriate.
5.3. Conclusions
With the addition of new test data beyond the original data used to build the current beta
criteria, several new models were fit to the data points for several general curve forms. Through
statistical analysis, design thickness comparison, and test verification, it was shown that the
current form, n = f(CBR) model, Equation 5.1, produced the best fit to the data while still
following fatigue failure behavior. In addition, design application of the model results in
appreciably different pavement thicknesses and therefore warrants consideration for
implementation.
Equation 5.1
!"#!"!!! !!!!!"#! !!!""# ! !"#!"!!"#$!!! !!!""# ! !"#!"!!"#$!
In the case of every general curve form, it was shown that the curves fit around the n =
f(CBR) data points proved a better fit than those curves using the n = 2 data points. This trend
implies that the n = f(CBR) model, although limited, better represents real soil behavior instead
of using a simplifying assumption of n =2.
5.4. Recommendations
This research resulted in the development of two recommendations for further study.
First, would be to reevaluate the data input and procedural assumptions between this research
and that of Gonzalez et al. (2012) before any of the models are considered for implementation.
69
All data inputs and calculation outputs are consolidated in the Appendix to facilitate this analysis
and any further work on the topic.
Second, considering the scattered nature of available test data, any criteria considered for
implementation should be put through further reliability-based analysis. As illustrated in section
4.4.3, none of the potential curves came close to precisely predicting the life of a pavement.
Therefore, a reliability based approach could better account for the inherent variability in the
design method that result from material variability, simplifying assumptions, and the empirical
CBR strength. In addition to analyzing the variability between test points, it would also be
pertinent to include and analysis of the sensitivity of the model to the variability in the CBR
measurements themselves. As discussed, variability in the test points alone could result in
upwards of 60 percent increases in design thicknesses, so implementation of reliability-based
criteria would require a detailed cost and risk analysis by an implementing agency.
70
References
Advisory Circular (AC) 150/5320-6E. (2009). “Airport Pavement Design and Evaluation.” Federal Aviation Administration (FAA).
AC 150/5335-5B. (2011). “Standardized Method of Reporting Airport Pavement Strength –
PCN.” FAA. Ahlvin, R. G., Ulery, H. H., Hutchinson, R. L., and Rice, J. L. (1971). “Multiple-Wheel
Heavy Gear Load Pavement Tests,” Technical Report S-71-17, Basic Report, Volume 1. Waterways Experiment Station (WES), U.S. Army Corps of Engineers (USACE), Vicksburg, MS.
Ahlvin, Richard G. (1991). Origin of Developments For Structural Design of Pavements.
Technical Report GL-91-26. WES, USACE, Vicksburg, MS. Barker, Walter R., and Carlos R. Gonzalez. (1994). Super-Heavy Aircraft Study. Technical
Report GL-94-12. WES, USACE, Vicksburg, MS. Boyd, W. K. and Foster, C. R. (1950). “Development of CBR Flexible Pavement Design Method
for Airfields: A Symposium: Design Curves for Very Heavy Multiple Wheel Assemblies.” Transactions of the American Society of Civil Engineers, 115(1) 534-546.
Cooksey, D. L., and Ladd, D. M. (1971). “Pavement Design for Various Levels of Traffic
Volume,” Technical Report No. AFWL-TR-70-133. U. S. Air Force Weapons Laboratory, Kirtland Air Force Base, Albuquerque, NM,
Gonzalez, Carlos R., Barker, Walter R., and Bianchini, Alessandra. (2012). Reformulation of the
CBR Procedure; Report I: Basic Report. Technical Report 12-16. Engineer Research and Development Center (ERDC), USACE, Vicksburg, MS.
Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures (ME-
PDG). (2004). Part 2. Design Inputs, Chapter 4. Traffic. National Cooperative Highway Research Program (NCHRP).
Hayhoe, G. F. (2004). “Traffic Testing Results from the FAA’s National Airport Pavement
Testing Facility.” Proceedings, 2nd International Conference on Accelerated Pavement Testing, University of Minnesota, Minneapolis, MN.
Hayhoe, G. F. (2011). “Full-Scale Traffic Tests on Flexible Pavements with 6 and 10-Wheel
Gear Configurations.” Proceedings, XXIVth World Road Congress, Mexico City, Mexico.
Huang, Yang H. (2004). “Pavement Analysis and Design.” Pearson Education Inc., Upper
Saddle River, NJ.
71
Jumikis, Alfreds R. (1964). “Mechanics of Soils.” D. Van Nostrand Company, Inc., Princeton,
NJ. O. J. Porter and Company, Consulting Engineers. (1949). “Accelerated Traffic Tests at
Stockton Airfield; Stockton, California (Stockton Test No. 2),” USACE, Sacramento District, Sacramento, CA.
Pereira, Taboza A. (1977). “Procedures for Development of CBR Design Curves.” Instruction
Report S-77-1. WES, USACE, Vicksburg, MS. Ullidtz, Per. (1998). “Modeling Flexible Pavement Response and Performance.” Polyteknisk
Forlag, Lyngby, Denmark. Unified Facilities Criteria (UFC) 3-260-02. (2001). “Pavement Design for Airfields.”
Department of Defense (DoD), Washington D. C. UFC 3-260-03. (2001). “Airfield Pavement Evaluation.” DoD. Washington D. C. WES. (1947). “Flexible Pavement Behavior Studies,” Interim Report No. 2. WES, USACE,
Vicksburg, MS. WES. (1950). “Investigation of Effects of Traffic with High-Pressure Tires on Asphalt
Pavements,” Technical Memorandum No. 3-312. WES, USACE, Vicksburg, MS. WES. (1952). “Design of Flexible Airfield Pavements for Multiple-Wheel Landing Gear
Assemblies; Test Section with Lean Clay Subgrade,” Technical Memorandum No. 3-349, Report No. 1. WES, USACE, Vicksburg, MS.
WES. (1962). “A Limited Study of Effects of Mixed Traffic on Flexible Pavements,”
Technical Report No. 3-587. WES, USACE, Vicksburg, MS.
72
Appendix: Test Data
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