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APPENDIX A
Proposed Standard Practice for
Predicting the Endurance Limit of Hot Mix
Asphalt (HMA) for Long-Life Pavement Design
AASHTO Designation: PP XX-XX
1. Scope
1.1 This practice describes methodology for predicting the endurance limit for hot mix
asphalt for long-life pavement design.
1.2 This standard may involve hazardous materials, operations, and equipment. This standard
does not purport to address all of the safety problems associated with its use. It is the
responsibility of the user of this procedure to establish appropriate safety and health
practices and to determine the applicability of regulatory limitations prior to its use.
2. Referenced Documents
2.1 AASHTO Standards
· T 321, Determining the Fatigue Life of Compacted Hot-Mix Asphalt (HMA) Sub-
jected to Repeated Flexural Bending.
· R 30, Mixture Conditioning of Hot Mix Asphalt (HMA)
2.2 Other Publications
· NCHRP 9-38, "Endurance Limit of Hot Mix Asphalt Mixtures to Prevent Fatigue
Cracking in Flexible Pavements," Draft Final Report.
3. Terminology
3.1 Endurance limit the strain level, at a given temperature, below which no bottom-up
fatigue damage occurs in the HMA.
3.2 Long-Life Pavement Design a pavement designed to last a minimum of forty years
without bottom-up fatigue failure, or need for structural strengthening.
3.3 Normal strain levels strain levels where failure (50 percent of initial stiffness) occurs
in less than 12 million cycles. For tests conducted at 20°C, strain levels of 300 micro-
strain or greater generally meet this requirement.
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3.4 Low Strain levels strain levels where failure (50 percent of initial stiffness) does not
occur by 12 million cycles. The failure point of low strain tests generally needs to be
extrapolated by one of the methods described in this document.
4. Summary of Practice
4.1 This practice describes the analysis needed to determine the endurance limit for hot-
mix asphalt concrete mixtures. It involves collecting beam fatigue test data at specified
strain rates, predicting the endurance limit based on a log-log extrapolation, and then
running tests at the predicted strain level to confirm endurance limit behavior. Since
the tests conducted at the predicted strain level should not fail, the failure point is
extrapolated from the test data by use of one of several different techniques to confirm
the endurance limit of the asphalt mixture.
5. Significance and Use
5.1 The endurance limit can be used during pavement design to determine a pavement
thickness which will prevent bottom-up fatigue cracking.
6. Apparatus
6.1 Specimen Fabrication Equipment Equipment for fabricating beam fatigue test speci-
mens as described in AASHTO T 321, Determining the Fatigue Life of Compacted Hot-
Mix Asphalt (HMA) Subjected to Repeated Flexural Bending.
6.2 Beam Fatigue Test System Equipment for testing beam fatigue samples as described in
AASHTO T 321, Determining the Fatigue Life of Compacted Hot-Mix Asphalt (HMA)
Subjected to Repeated Flexural Bending.
6.3 Analysis Software Data is collected during the test using a data acquisition system
described in section 6.2. Data analysis can be conducted using a spreadsheet program,
or a variety of statistical packages.
7. Hazards
7.1 This practice and associated standards involve handling of hot asphalt binder, aggregates
and asphalt mixtures. It also includes the use of sawing and coring machinery and servo-
hydraulic or pneumatic testing equipment. Use standard safety precautions, equipment,
and clothing when handling hot materials and operating machinery.
8. Standardization
8.1 Items associated with this practice that require calibration are included in the documents
referenced in Section 2. Refer to the pertinent section of the referenced documents for
information concerning calibration.
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9. Beam Fatigue Test Data
9.1 Test Specimen Fabrication
9.1.1 Prepare at least twelve test specimens to the target air void content in accordance
with AASHTO T 321. Samples should be short-term oven aged according to AASHTO
R 30 for four hours at 135°C prior to compaction. The target air void content should
be representative of that expected to be obtained in the field. A target air void content
of 7 percent was used for mixes produced at optimum asphalt content in the NCHRP
9-38 research. A reduced air void content would be expected for optimum plus or
so-called rich-bottom type mixes.
Note 1 A reasonable air void tolerance for test specimen fabrication is ± 0.5 %.
Note 2 The following are estimated based on a limited round robin. The coefficient
of variation of the log (base 10) of the fatigue life of a properly conducted beam fatigue
tests at normal strain levels is 5.4 and 6.8 percent, respectively, for within- and between-
lab variability. The difference between the logs of the fatigue lives (log sample 1 log
sample 2) of two properly conducted test should not exceed 0.69 for a single operator
or 0.89 between two labs.
9.2 Testing Conditions
9.2.1 Separate the test specimens into four groups with approximately equal air voids. Deter-
mine the number of cycles to failure (50 percent of initial stiffness) of each specimen in
one group using the beam fatigue test system at a strain level of 700 micro-strain. Deter-
mine the number of cycles to failure in a second group at a strain level of 500 micro-
strain. Determine the number of cycles to failure in the third group at a strain level of
300 micro-strain. The test results at the normal strain level should not be extrapolated;
all nine sample should be tested to failure.
Note 3 The original research, including round robin testing, was based on testing
three samples each at 800 and 400 micro strain. Additional samples were added to
improve the estimation of the endurance limit due to the sensitivity of pavement thick-
ness to the endurance limit.
9.3 Beam Fatigue Data Summary to Predict Endurance Limit
9.3.1 Transform the data by taking the log (base 10) of both the micro-strain levels and cycles
to failure. Research has shown that the log-log transformation of the data from tests
conducted at normal strain levels (above the endurance limit) produce a straight line.
Perform a simple linear regression on the transformed data using fatigue life as a pre-
dictor for micro-strain level. Using the regression coefficients determined from the
regression analysis, determine the micro-strain level corresponding to a fatigue life of
log 50,000,000 cycles = 7.69897. Designate this value yo. Calculate the one-sided lower
95% prediction interval according to Equation 1.
1 (x - x )
2
^o - ts 1 + + o
Lower Prediction Limit = y (1)
n Sxx
where:
t = value of t distribution for n - 2 degrees of freedom = 1.89458 for n = 9 with
= 0.05%,
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s = estimate of standard deviation from the regression analysis, also referred to in
Microsoft Excel as the standard error,
n = number of samples = 9,
n
Sxx = ( x i - x ) (Note: log of fatigue lives),
2
i =1
x0 = log 50,000,000 = 7.69897,
_
x = average of the fatigue life results determined in 9.2.1.
Note 4 A simple spreadsheet has been developed to perform the calculations described
above.
9.3.2 Conduct the beam fatigue test at the strain level corresponding to the 95% one-sided
lower prediction limit for a fatigue life of 50,000,000 cycles to 10,000,000 cycles. Research
has shown that tests conducted to a minimum of 10,000,000 cycles can extrapolate to
estimate long-life fatigue lives.
10. Data Analysis to Extrapolate Long-life
Fatigue Tests
Beam fatigue tests conducted at low strain levels are unlikely to fail in a reasonable
number of cycles. This is particularly true for tests conducted at the 95% one-sided
lower prediction limit for the endurance limit determined in section 9.3.1. In order to
confirm the existence of the endurance limit, the test data needs to be extrapolated to
predict a failure point. This section provides three procedures for extrapolating the
beam fatigue failure point for low-strain tests.
10.1 Data Extrapolation Using the Single-Stage Weibull Survivor Function
10.1.1 General Form. The general form of the Weibull Survivor function is shown as Equa-
tion 2:
t -
R ( t ) = exp - (2)
-
where:
R(t) = the reliability at time t where t might be time or another life parameter such as
loading cycles,
= the slope,
= the minimum life, and
= the characteristic life.
10.1.2 Simplified Form. A specialized case of the Weibull function assumes the mini-
mum life, , equals zero. Therefore, the hazard function would equal 1/, which
simplifies Equation 2 into Equation 3. Since the beam fatigue loading cycles are
applied at a constant frequency of 10 Hz, the loading cycles, n, can be substituted
for time, t.
S (n ) = exp ( - × n ) (3)
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where:
S(t) = probability of survival until time t,
n = number of loading cycles,
= scale parameter (intercept),
= shape parameter (slope).
10.1.3 Final Form. The stiffness ratio (SR) is used to characterize fatigue damage. The stiff-
ness ratio is the stiffness measured at cycle n, divided by the initial stiffness, determined
at the 50th cycle. Tsai reports that at a given cycle n, the beam being tested has a prob-
ability of survival past cycle n equal to the stiffness ratio times 100 percent. Thus, SR(n)
can be substituted for S(t). Equation 4 allows the scale and shape parameters for labo-
ratory beam fatigue data to be determined by linear regression.
ln ( - ln ( SR )) = ln ( ) + × ln (n ) (4)
10.1.4 Plot the left-hand side of Equation 4 versus the natural logarithm of the number of
cycles, n so a straight line regression can be determined. If the measured stiffness at a
given number of cycles is greater than the initial stiffness, e.g. SR > 1.0, ln(-ln(SR)) can-
not be computed. Eliminate these data from the regression analysis. Using the shape
(slope) and scale (intercept) parameters determined from the simple linear regression,
solve Equation 4 for n which produces an SR of 0.5. This can be readily done using a
solver function or by trial and error in a spreadsheet. This value of n is the extrapolated
fatigue life for 50 percent initial stiffness.
10.1.5 The endurance limit is confirmed by the divergence of the test data below the regres-
sion line at a high number of cycles. Figure 1 illustrates this divergence.
PG 64-22 at Optimum
0 Upper divergence, indicating endurance limit
-1
-2
Ln(-Ln(Stiffness Ratio))
-3
-4
y = 0.401x -8.214
-5 R² = 0.722
-6
-7
-8
-9
-10
0 2 4 6 8 10 12 14 16 18 20
Ln (Cycles)
Sample 13 Linear (Sample 13)
Figure 1. Weibull functions.
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10.2 Data Extrapolation Using Three-Stage Weibull Survivor Function
10.2.1 To improve upon the accuracy of the single-stage Weibull function, Tsai et al. (2)
developed a methodology for fitting a three-stage Weibull curve. Tsai et al. (2) theo-
rized that a plot of loading cycles versus stiffness ratio could be divided into three
stages: initial heating and temperature equilibrium, crack initiation, and crack propa-
gation. In the case of low strain tests (below the endurance limit), the third stage does
not appear to represent crack propagation, but rather concave down stage with a reduced
rate of damage.
In most cases, low strain fatigue tests can be most accurately extrapolated using the single-
stage Weibull function. However, in some cases, there are three distinct slopes to the
transformed data, in which case the three-stage Weibull function may be used to pro-
vide a better estimate of the fatigue life.
10.2.2 First, plot the data as described in Section 10.1.4. Visually examine the data to deter-
mine three stages, determined by groups of data exhibiting distinct slopes. Assign a data
series to each group of data, representing a single slope. Perform a linear regression on
the transformed data for each stage. The regression coefficients become seed values for
either a compiled Fortran program or Microsoft Excel spreadsheet solution. An exam-
ple is shown in Figure 2.
0
0 2 4 6 8 10 12 14 16 18 20
y = 0.0583x -2.554
R² = 0.5874
-1
y = 0.1605x -3.949
R² = 0.9193
-2
Series 1
Stage 1
Stage 2
-3 Stage 3
Linear (Stage 1)
y = 0.5158x -6.51 Linear (Stage 2)
R² = 0.6636
Linear (Stage 3)
-4
-5
-6
Figure 2. Estimation of three Weibull stages by inspection.
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10.2.3 A Weibull function is fit to each of these stages as shown in Equation 5:
SR 1 = e ( )
- 1 ×n1
for 0 n < n1 (5)
SR 2 = e ( )
- 2 ×(n- 1 )2
for n1 n < n2
SR 3 = e ( )
- 3 ×(n- 2 )3
for n2 n < n3
10.2.4 Coefficients 1, 2, 3, 1, 2, 3, n1, and n2 are illustrated in Figure 3. Using a series of
mathematical manipulations (2), n1, 1, n2, and 2 can be calculated sequentially as follows:
1
2 2 2 2 -1
n1 = × (6)
1 1
1 = 1 - 2 × n1 (7)
1
1
3 2 - 3
n2 = 1 + 3 × 3 (8)
2 2
PG 76-22 Optimum Plus Sample 9 at 200 ms
0
y = 0.0667x -2.6913
R² = 0.7617
-1
y = 0.1769x -4.1281
R² = 0.9075 3
1
Ln( -Ln(Stiffness Ratio))
-2
y = 0.5427x -6.6613 2
R² = 0.547
1
-3 Stage 1
Stage 2
1 Intercept = Ln 3
Stage 3
-4
1 Intercept = Ln 2
Intercept = Ln 1 LN (n1) Ln (n2)
-5
Stage 1 Stage 2 Stage 3
Initial Heating
-6
0 2 4 6 8 10 12 14 16 18 20
Ln (Loading Cycles)
Figure 3. Three-stage Weibull curve definitions.
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2 = 1 - 3 × n2 + 3 × 1 (9)
2 2
10.2.4 The trial values for 1, 2, 3, 1, 2, 3 and he test data are entered into a FORTRAN
program, N3stage.exe, developed by Tsai or into a spreadsheet developed as part of
NCHRP 9-38. N3stage.exe provides a robust solution, but can take up to 45 minutes
to run. The spreadsheet uses Microsoft Excel's Solver Function and takes less than a
moment, but in a few cases does not identify a solution. The N3stage.exe program can
be obtained from: BWTsai@Berkeley.edu
10.2.5 Using the results from the NStage3.exe program or NCHRP 9-38 spreadsheet, the pre-
dicted fatigue life can be calculated according to Equation 10.
3
( ln( - ln( 0.5))- ln 3
Nf = e
+ 2 (10)
11. Report
11.1 For each sample, report the following:
11.1.1 Sample air voids
11.1.2 Test Temperature
11.1.3 Initial flexural stiffness (measured at 50 cycles)
11.1.4 Normal Strain Tests
11.1.4.1 Micro-strain level
11.1.4.2 Number of cycles (measured) to 50 percent of initial stiffness
11.1.5 Low Strain Tests
11.1.5.1 Method of Extrapolation
11.1.5.2 Equation used for extrapolation and R2 value for equation
11.1.5.3 Extrapolated fatigue life Nf for 50 percent of initial stiffness
11.1.6 Endurance Limit
11.1.6.1 Log-Log plot of data with cycles to failure on x-axis and micro-strain on y-axis.
11.1.6.2 Best-fit regression line/equation of normal-strain data should be included on graph.
11.1.6.3 Predicted endurance limit from normal-strain regression with Nf = 50,000,000 cycles
11.1.6.4 Lower one-sided 95 percent confidence interval for micro-strain level corresponding
to Nf = 50,000,000 cycles, termed predicted endurance limit.
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12. Keywords
12.1 Beam fatigue, endurance limit, long-life pavement
13. References
13.1 Tsai, B-W, "High Temperature Fatigue and Fatigue Damage Process of Aggregate-
Asphalt Mixes." Report to California Department of Transportation, Pavement
Research Center, Institute of Transportation Studies, University of California at
Berkeley.
13.2 Tsai, B.-W., J. T. Harvey, and C. L. Monismith. "Using the Three-Stage Weibull Equa-
tion and Tree-Based Model to Characterize the Mix Fatigue Damage Process." In
Transportation Research Record No. 1929, Transportation Research Board, Washing-
ton, DC, 2005, Pp 227-237.
