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CHAPTER 4
Beam Fatigue Test Results and Analyses
This chapter describes the beam fatigue testing conducted was determined at Ndesign = 80 gyrations. In addition, the
to confirm the existence of the endurance limit. One of the samples' air void contents were reduced from 7.0 ± 0.5% to
most important aspects of this research is a practical defini 3.3 ± 0.5% for the optimum plus samples. The voids for the
tion for the endurance limit. Asphalt mixtures simply cannot optimum plus samples were reduced to simulate the expected
be tested for an infinite fatigue life in the laboratory. Testing improved densification in the field. Advantages of optimum
at 10 Hz, approximately one million load repetitions can be plus or rich bottom layers are believed to include better com
applied to a beam fatigue sample in a given day. The primary pactability, greater resistance to fatigue damage, and improved
goal of this testing was to confirm the existence of a fatigue moisture susceptibility.
endurance limit. In order to accomplish this goal, it was nec
essary to develop a method for estimating the endurance limit
Extrapolation Methods
through accelerated testing in a reasonable period of time. A
to Predict Fatigue Life
secondary goal was an estimate of the variability associated
with beam fatigue testing and its potential impact on pave As discussed previously, it was decided prior to the start of
ment thickness design. testing that beam fatigue tests would be terminated at 50 mil
The first portion of Chapter 4 discusses methods for extrap lion cycles. If a shift factor of 10 was applied to the test results
olating the fatigue failure point for strain levels that did not from a sample tested to 50 million cycles, it would then be
result in failure in less than 50 million loading cycles. The estimated that the pavement could withstand 500 million
methods were applied to samples that had fatigue lives in loading cycles at the corresponding strain level. Based on
excess of 10 million, but less than 50 million loading cycles. capacity analysis of a lane, this then represents a reasonable
The second portion of the chapter presents the data collected maximum number of loading cycles that might occur in a
in Phase I, as well as additional binder grades tested with 40year period. For the practical definition of the endurance
the same mixture in Phase II. Evidence of the existence of a limit, a 40year life was considered to be indicative of a long
fatigue endurance limit is presented for each of these mixtures. life pavement. It takes approximately 50 days to test a single
The third portion of the chapter describes a limited round sample to 50 million cycles. Additional analyses will be dis
robin conducted to assess the variability of fatigue testing and cussed later to evaluate the existence of a theoretical or truly
the prediction of the endurance limit. Finally, indirect tensile infinite life endurance limit.
tests were investigated as a surrogate for beam fatigue tests. For samples that failed in less than 50 million cycles at 50%
As discussed in Chapter 3, a single gradation and aggregate of initial stiffness, the number of cycles to failure was deter
type was used for all of the testing. A fullfactorial experiment, mined from the data acquisition software controlling the test.
shown in Table 4.1, was conducted in Phase I to evaluate the However, if the test was terminated prior to reaching 50%
existence of an endurance limit and to identify factors affect of initial stiffness, either due to an equipment problem or
ing the endurance limit. Two main factors were included in to reaching 50 million cycles, an extrapolation procedure was
the experimentbinder grade and asphalt content. Binder used to estimate the number of loading cycles, Nf, correspon
grade was varied at two levels: PG 6722 and PG 7622. As ding to 50% of initial stiffness. Ideally, a method of extrap
noted previously, the PG 6722 also met the requirements olation would be identified that could be used to shorten
of a PG 6422. Asphalt content was varied at two levels: opti the beam fatigue testing procedure used to determine the
mum and optimum plus 0.7%. Optimum asphalt content endurance limit. Then, samples could be tested to 4 million
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Table 4.1. Experimental design.
NMAS (mm) Binder Content Granite
PG 6722 PG 7622
19.0 Optimum X X
Optimum + X X
cycles as done in the Asphalt Institute study (38, 39) or pos lation: exponential model, logarithmic model, Weibull func
sibly 10 million cycles and the fatigue life at, or close to, the tion, threestage Weibull function, and ratio of dissipated
endurance limit predicted. energy change (RDEC). Each of these is discussed below, and
In Phase I, testing was conducted at progressively lower examples are provided of the predicted fatigue lives.
strain levels until two samples at a given strain level reached
50 million cycles without reaching 50% of their initial stiff
Fatigue Life Extrapolation Using
ness (failure). Instead of testing at a lower strain level below AASHTO T321 Exponential Model
that providing a fatigue life of at least 50 million cycles, sam
ples were tested at the strain level predicted to provide a fatigue AASHTO T321 specifies an exponential model (Equation 2)
life of 50 million cycles. The goal of this additional point was for the calculation of cycles to 50% initial stiffness, as follows:
to help define the transition from "normal" strain test to
tests below the apparent fatigue endurance limit or "low" S = Ae bn (2)
strain tests.
For the PG 6722 mix at optimum asphalt content, the where,
data from 800 through 200 ms were used to estimate the S = sample stiffness (MPa),
strain level that would result in a fatigue life of 50 million A = constant,
cycles. A linear regression was performed between the Log10 b = constant, and
of ms and the Log10 of loading cycles to 50% initial stiffness. n = number of load cycles.
The R2 = 99.6 for Equation 1. Using Equation 1, it was deter
mined that a strain level of 166 ms should produce a fatigue The constants are determined by regression analysis of load
life of 50 million cycles. This was rounded to 170 ms for ing cycles versus the natural logarithm of the flexural stiff
testing purposes. Testing was conducted at this strain level ness. The number of cycles to failure is determined by solving
to better define the endurance limit. Equation 2 for 50% of initial stiffness. In this study, for sam
ples tested to less than 50 million cycles, the number of cycles
N f = 1020.6 × 5.81 (1) reported to reach 50% of the initial stiffness are the actual
number of cycles recorded by the test equipment, not the num
where, ber of cycles determined using Equation 2. No discussion is
Nf = number of cycles to 50% of initial stiffness and provided in AASHTO T321 regarding whether or not all of
= constant strain used in beam fatigue test (ms). the data (particularly the initial data) should be used when
solving for the constants in Equation 2 (52).
When testing the PG 6722 mix at optimum asphalt content Sample 5 of the PG 6722 mix at optimum asphalt content,
at 170 ms, the first replicate failed in 34.7 million cycles. The which was tested at 170 ms, was selected as an example. It was
second replicate was at 55% of its initial stiffness at 50 million desirable to select a sample that had as long a fatigue life as pos
cycles. Therefore, testing was extended to see if the failure point sible and had reached 50% of initial stiffness within 50 million
could be determined. However, at 60 million cycles, Sample 23 cycles. It was felt that an extrapolation method that worked
still retained 53% of its initial stiffness. Therefore, testing was well at high strain levels may not prove to be as accurate at
suspended at 60 million cycles. strain levels closer to the anticipated endurance limit. Fig
For the PG 6722 mix at optimum asphalt content, samples ure 4.1 shows fits to the loading cycle versus sample stiffness
tested at 200 and 170 ms were used to investigate extrapola data determined using Equation 2. The coefficients for Equa
tion techniques. These strain levels and similar strain levels tion 2 were fit using the data up to 4 million cycles, 10 mil
for the other mixes used in Phase I provided long fatigue lives lion cycles, and failure (34.7 million cycles). As can be seen
(in excess of 10 million cycles) while still having a defined fail from Figure 4.1, when all of the data up to failure was used
ure point that could be used to investigate the accuracy of the to fit the model, the model provides a reasonable estimate
extrapolation. Five techniques were investigated for extrapo of fatigue life.
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PG 6722 Sample 5 at 170 ms
5000
y = 3970 .6e5E08x y = 3927 .2e2E08x y = 3890 .5e1E08x
4500 R² = 0.4992 R² = 0.4776 R² = 0.7259
4000
Flexural Stiffness, MPa
3500
3000
2500
2000
1500
1000
500
0
0
10
20
30
40
50
,0
,0
,0
,0
,0
00
00
00
00
00
,0
,0
,
,0
,
00
00
00
00
00
0
0
Loading Cycles
50% Initial Expon. (4.00E+06) Expon. (1.00E+07) Expon. (To Failure)
Figure 4.1. Examples of fatigue life estimates using the exponential model.
Fatigue Life Extrapolation Using Natural When all of the fatigue data are used to fit a logarithmic
Logarithm of Loading Cycles versus Stiffness model, the slope of the fitted line at higher numbers of loading
cycles may be flatter than the actual data. This leads to an over
A logarithmic model (Equation 3) using the natural loga estimation of the fatigue life. This is illustrated in Figure 4.2 for
rithm of loading cycles versus stiffness was evaluated as one Sample 5 of the PG 6722 mix at optimum asphalt content.
alternative to the exponential model. The fits to the logarithmic model using just the first 10 million
cycles and using all of the data are indistinguishable on the
S = + × ln (n ) (3) plot. Note that in Figure 4.2 the logarithmic model provides
very high R2 values, but the fitted model does not match the
where, experimental data at a high number of cycles.
S = the sample stiffness at loading cycle n, and However, by eliminating a portion of the early loading
and are regression constants. cycles, a good match to the data can generally be obtained,
5000
Data to 4 million Cycles Data to 10 million Cycles
4500 y = 91.789Ln(x) + 4865.2 y = 91.483Ln(x) + 4862.4
R2 = 0.9714 R2 = 0.9763
4000
Flexural Stiffness, MPa
3500
3000
2500
2000
1500 To Failure
1000 y = 106.17Ln(x) + 5005.6
R2 = 0.9108
500
0
0
10
20
30
40
50
,0
,
,0
,0
,0
00
00
00
00
00
0,
,0
,0
,0
,0
0
0
00
0
00
00
0
0
Loading Cycles
50% Initial Log. (4.00E+06) Log. (1.00E+07) Log. (To Failure)
Figure 4.2. Logarithmic model fits for PG 6722 at optimum, Sample 5.
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8,000
Logarithmic Model using All Data Logarithmic Model Starting at 1 million Cyles
7,000 y = 235.ln(x) + 7766. y = 405ln(x) + 10226
R² = 0.965 R² = 0.965
6,000
Flexural Stiffness, MPa
5,000
4,000
3,000
2,000
1,000
0
0 2,000,000 4,000,000 6,000,000 8,000,000 10,000,000 12,000,000 14,000,000
Loading Cycles
Log. (All Data) Log. (Starting 1E+6 cycles)
Figure 4.3. Logarithmic model fits for PG 6722 at optimum, Sample 6.
particularly at low strain levels. In Figure 4.3, logarithmic where,
models were fit to the data from Sample 6 of the PG 6722 mix = 10 raised to the power of the intercept from regression
at optimum asphalt content, which was tested at 200 ms. In of log (S) versus log (n), and
Figure 4.3, logarithmic models are shown including all of the = the slope from regression of log (S) versus log (n).
initial loading cycles and excluding the first million loading
cycles. This provides a better fit to data, but still would tend The power model has a similar shape to the logarithmic model.
to overestimate the fatigue life. Further, the number of early Shen (54) also reported the need to eliminate a number of ini
loading cycles that are not included must be determined by tial cycles to obtain a good fit to the slope at high numbers of
trial and error. Note that all of the logarithmic models shown cycles. Failure to eliminate some of the initial cycles results
in Figures 4.2 and 4.3 provide high R2 values, even when the in an overestimation of the fatigue life. Additional discussion
fit to the data at a high number of cycles is not very good. This on the use of the power model and its application to the ratio
suggests that R2 values alone are not adequate to evaluate of dissipated energy will be provided later in this report.
extrapolation models. It is believed that the poor fit at a high
number of cycles results from the fact that the data are col Fatigue Life Extrapolation Using
lected using a logarithmic progression. That is, the sampling the Weibull Survivor Function
rate is high, every 10 cycles, when the test is initiated but may
be every million cycles at a high number of cycles. Thus, there Often, failure data can be modeled using a Weibull distri
are more data points to fit in the early portion of the curve. bution. The Weibull function is commonly used in reliability
Rowe and Bouldin (53), when examining fits from the expo engineering to estimate survival life. Tsai et al. (55) applied
nential model, concluded that fatigue data should be taken the Weibull survivor function to HMA beam fatigue data.
with every 5% reduction in stiffness. The generalized equation for the Weibull function is given by
Equation 5.
Fatigue Life Extrapolation Using
t 
R ( t ) = exp

the Power Model (5)
The ratio of dissipated energy, developed by Shen and
Carpenter (40) also requires that the number of cycles to 50% where,
of initial stiffness be calculated in order to determine the plateau R(t) = the reliability at time t where t might be time or
values. Shen (54) recommends a power model (Equation 4) for another life parameter such as loading cycles,
the extrapolation of stiffness versus loading cycles, as follows: = the slope,
= the minimum life, and
S = + n (4) = the characteristic life.
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Tsai et al. (55) applied a specialized case of the Weibull func Figure 4.4 shows an example of the data from the two
tion where the minimum life, , was assumed to be 0. In this 100 ms samples from the PG 6722 at optimum mixture in
case, the characteristic life = 1/ and the Weibull function the form of Equation 7. Tsai et al. (55) observed that the
simplifies to Equation 6. Since the beam fatigue loading cycles concave down shape, exhibited by Sample 13, "implied that
are applied at a constant frequency of 10 Hz, loading cycles, n, the fatigue damage rate is slowed down and flattens out with
can be substituted for time, t. increased repetitions and thus causes no further damage after
a certain number of repetitions." This behavior is believed
S ( t ) = exp (  × n ) (6) to be indicative of the endurance limit.
where,
S(t) = probability of survival until time t, Fatigue Life Extrapolation Using
= scale parameter (intercept), and ThreeStage Weibull Function
= shape parameter (slope).
In the previous section, the Weibull survivor function was
The stiffness ratio can be used to characterize fatigue dam presented as a method for modeling the fatigue life of beam
age. The stiffness ratio is the stiffness measured at cycle n, fatigue tests. In later sections, it will be demonstrated that the
divided by the initial stiffness, determined at the 50th cycle. singlestage Weibull function generally provides a good esti
Tsai (56) states that at a given cycle n, the beam being tested mate of a sample's fatigue life and is reproducible when calcu
has a probability of survival past cycle n equal to the SR lated by different laboratories. There are, however, cases for
times 100%. Thus, SR(n) can be substituted for S(t). Tsai (56) which the singlestage Weibull function apparently under
presents the derivation of Equation 7, which allows the scale predicts fatigue life. Sample 13 in Figure 4.4 is one such exam
and shape parameters for laboratory beam fatigue data to be ple (but Sample 13 is not specifically labeled on the plot). It
determined by linear regression. should be noted that with the exception of two examples
analyzed by Tsai, the threestage Weibull analyses were not
ln (  ln ( SRn )) = ln ( ) + × ln (n ) (7) conducted until after the completion of all of the Phase I
and II testing.
where, To improve upon the accuracy of the singlestage Weibull
SRn = stiffness ratio or stiffness at cycle n divided by the function, Tsai et al. (57) developed a methodology for fitting
initial stiffness. a threestage Weibull curve. Tsai et al. (57) theorized that a
0
y = 0.243x 5.816
1
R² = 0.82
2
Ln(Ln(Stiffness Ratio))
3
4
y = 0.4014x 8.2141
5
R² = 0.72
6
7
8
9
10
0 2 4 6 8 10 12 14 16 18 20
Ln (Cycles)
Sample 4 Sample 13 Linear (Sample 4) Linear (Sample 13)
Figure 4.4. Weibull survivor function for PG 6722 at optimum 100 ms samples.
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plot of loading cycles versus stiffness ratio could be divided 1
into three stages: initial heating and temperature equilibrium, 3 3 3 2  3
n2 = 1 + × (11)
crack initiation, and crack propagation. In the case of low 2 2
strain tests (below the endurance limit), the third stage does
not appear to represent crack propagation, but rather con
cave down stage with a reduced rate of damage. 2 = 1  3 × n2 + 3 × 1 (12)
2 2
A Weibull function is fit to each of these stages as shown in
Equation 8:
Tsai et al. (57) applied a genetic algorithm to resolve the six
unknown parameters. A genetic algorithm requires a param
SR1 = e ( )
 1 ×n1
for 0 n < n1 eter definition, in this case Equations 8 through 12, and a fit
SR2 = e ( )
 2 ×(n 1 )2 ness function. Residual sum of squares between the measured
for n1 n < n2 (8) and fitted ln(ln(SR)) for each cycle is used as the fitness func
SR3 = e ( )
 3 ×(n 2 )3 tion. A "good gene" is defined as an optimum set of param
for n2 n < n3
eters to minimize the fitness function. A set of input ranges is
first determined for 1, 2, 3, 1, 2, and 3. The input ranges
Coefficients 1, 2, 3, 1, 2, 3, n1, and n2 are illustrated in are determined by visual inspection of the data. Simple linear
Figure 4.5. regressions are performed for each stage to determine slopes
Using a series of mathematical manipulations (57), n1, 1, and offsets. An example is shown in Figure 4.5. Tolerances are
n2, and 2 can be calculated sequentially as follows: applied to the parameters determined by inspection to set ini
tial ranges for each coefficient. The input ranges and test data
1 are entered into a FORTRAN program, N3stage.exe, devel
2 2 2 2 1 oped by Tsai. The program randomly generates a set of param
n1 = × (9)
1 1
eters or "genes" within the input ranges. The fitness parameter
is calculated for each set of genes and the sets of genes are
ranked. Good genes are mated and bad genes are discarded
1 = 1  2 × n1 (10) and replaced with new genes. The cycle of producing genes,
1 ranking genes by the residual sum of squares, mating and
PG 7622 Optimum Plus Sample 9 200 ms
0
y = 0.0667x 2.6913
R² = 0.7617
1
y = 0.1769x 4.1281
R² = 0.9075 3
Ln( Ln(Stiffness Ratio))
1
2
y = 0.5427x 6.6613 2
R² = 0.547 1
3 Stage 1
Stage 2
1
Intercept = Ln3
Stage 3
4
1 Intercept = Ln2
Intercept = Ln1 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 4.5. Threestage Weibull curve definitions.
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PG 7622 Optimum Plus Sample 9 200 ms
0
1
Ln (Ln(Stiffness Ratio))
2
3
Raw Data
Fit
4
5
6
0 2 4 6 8 10 12 14 16 18 20
Ln (Loading Cycles)
Figure 4.6. Threestage Weibull fit to transformed data.
replacing genes, continues until the specified number of gen for LongLife Pavement Design. The NCHRP 938 research
erations is complete (58). The N3stage program typically takes team developed a Microsoft Excel spreadsheet to solve the
30 to 60 minutes to complete 750 generations, depending on threestage Weibull parameters, which produces similar results
the size of the data set. The complete calculation procedure to the N3stage program.
is described in Appendix A, Proposed Standard Practice for Figures 4.6 and 4.7 show examples of the threestage
Predicting the Endurance Limit of Hot Mix Asphalt (HMA) Weibull fit. This methodology provides a good fit to both
PG 7622 Optimum Plus Sample 9 200 ms
6,000
5,000
4,000
Stiffness, MPa
3,000
2,000
1,000
0
0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000
Loading Cycles
Raw Data 3Stage Weibull Fit
Figure 4.7. Threestage Weibull fit to stiffness data.
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normal and low strain fatigue data. In some cases, only one two data points, that is, the average ratio of dissipated energy
or two stages are fit, even if three stages are initially identified. change per loading cycle. This is written as follows:
DEa  DEb
Ratio of Dissipated Energy Change (RDEC) RDECa = (13)
DEa (b  a )
Dissipated energy is a measure of the energy that is lost to
the material or altered through mechanical work, heat gener where,
ation, or damage to the sample. Other researchers have used RDECa = the average ratio of dissipated energy change at
cumulative dissipated energy to define damage within a spec cycle a, comparing to next cycle b;
imen, assuming that all of the dissipated energy is responsi a, b = load cycle a and b, respectively (the cycle count
ble for the damage. The approach suggested by Ghuzlan and between cycle a and b for RDEC calculation will
Carpenter (34) considers that only a portion of the dissipated vary depending on the data acquisition soft
energy is responsible for actual damage. ware); and
DEa, DEb = the dissipated energy produced in load cycle a,
Typically, three regions are observed in the RDEC analy
and b, respectively.
sis as shown in Figure 4.8. Region I represents the initial
"settling" of the sample where the rate of change of dissipated
The dissipated energy for each loading cycle is determined
energy decreases. In Region II, the rate of change of dissipated
by measuring the area within the stressstrain hysteresis loop
energy reaches a plateau, representing a period where the
for each captured load pulse. This methodology is used by
amount of damage occurring to the sample is constant. Finally,
the IPC Global beam fatigue device used in the study by
in Region III, sample instability begins as the rate of change
NCAT, Asphalt Institute, and the University of Illinois. Alter
of dissipated energy rapidly increases. A lower dissipated
natively, the dissipated energy can be calculated according
energy ratio (DER) plateau value implies that less damage
to Equation 14.
is occurring per cycle. Therefore, a sample with a low DER
plateau value would be expected to have a longer fatigue life wn = × n × n × sin n (14)
than a sample with a high DER plateau value. Shen and
Carpenter (40) refined this technique and suggested that the where,
RDEC plateau value (PV) should be calculated at the number n = maximum tensile stress in cycle n, in kPa,
of cycles that produced 50% of the initial sample stiffness (Nf). n = maximum tensile strain in cycle n,
A PV of 8.57E9 was proposed by Shen and Carpenter as indica n = 360 × f × s,
tive of a long life pavement (40). f = loading frequency, Hz, and
The RDEC analysis procedure is described in Appendix C, s = time lag in seconds between peak load and peak deflec
Proposed Standard Practice for Extrapolating LongLife Beam tion in seconds.
Fatigue Tests Using the Ratio of Dissipated Energy Change
(RDEC). RDEC is the ratio of dissipated energy change between Due to testing noise, as shown in Figure 4.8, the raw dissi
two data points divided by the number of cycles between the pated energy data are not directly usable for calculating RDEC
0.006
0.004 III
RDEC
I
II
0.002
PV
0
10 510 1,010 1,510 2,010 2,510 3,010 3,510 4,010
Number of Load Cycles
Figure 4.8. Typical RDEC versus loading cycles plot and the indication of PV.
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IDOT03 mix 3N704B f
100
DE vs. Loading cycles 800 microstrains 1 1+
1.4 a
RDECa = (15)
1.2 100
1.0
where,
DE, kPa
0.8
Exponential Slope k f = the slope from the regressed dissipated energyloading
0.6
y = 3.4255x 0.164 cycle curve.
0.4 R² = 0.9512
0.2 In the RDEC approach, PV is defined as the RDEC value at
Nf50
0.0 the 50% stiffness reduction failure point (Nf50). Therefore,
0 1000 2000 3000 4000 5000
the PV value can be obtained using Equation 16.
Loading Cycles
Figure 4.9. DE versus LC chart for one IDOT mix 100
f
1 1+
Nf50
with fitted curve.
PV = (16)
100
and PV. A curve fitting procedure is recommended to obtain
the best fit equation for the dissipated energyloading cycle Here, the PV value depends only on the f factor of the regressed
data. It is assumed that the regression equation of dissipated power law DELC curve and the defined failure point, Nf50.
energyloading cycle relationship follows a power law rela For longlife tests, where Nf50 was not known, the stiffness
tionship, Axf (as indicated in Figure 4.9). The key for the loading cycle curve first needed to be extrapolated to deter
curve fitting process is to obtain a slope (in the power law mine Nf50, resulting in the calculation being based on a double
relation plot) of the curve, f, which can best represent the extrapolation.
original curve. In general, there are two rules for evaluating Using this approach, Shen and Carpenter (40) demon
the goodness of the fitted curve: (1) a high Rsquare value, strated a unique PVNf curve for all HMA mixes at normal
and (2) correct trend of the DELC curve. This is similar to strain/damage level testing, regardless of the testing condi
the procedures described previously for fitting logarithmic tion, loading modes, and mixture types. The tests used for es
or power models to the stiffnessloading cycle curves and is tablishing this relationship at all normal testing were carried
illustrated in Figure 4.10. to or beyond the failure point (i.e., the Nf50 values are known).
The average RDEC for an arbitrary 100 cycles at cycle a can Also, using the results from longterm fatigue testing, Shen
be simply calculated using the following equation: and Carpenter (40) demonstrated that the unique relation
Predicted DE Sample 2 at 200 ms
0.100 Legend Number Denotes
0.095 Starting Cycle
0.090 R^2 = 0.90
y = 0.7162x0.1384
Predicted DE, kPa
0.085 66910
0.080
4617890
0.075
8403190
0.070
0.065 Measured
0.060
0.055
0.050
Loading Cycles
Figure 4.10. Dissipated energy versus loading cycle for raw data and
various power models.
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ship between PVNf is also valid for low strain testing. In other The unique PVloading cycle curve is also illustrated in
words, the PVNf relationship is unique for the whole loading Figure 4.11. As illustrated in Appendix C, an alternative
range including both normal and low strain/damage level. approach for extrapolating the fatigue life of a sample that
An error analysis was performed for two dissipated energy does not fail is to plot the sample's RDEC versus loading
curve fitting routines using 19 Illinois DOT mixtures tested at cycle on a loglog plot and determine the intersection with
low strain levels, and the results are presented in Figure 4.11. the unique PVNf line.
All the samples were tested to extended load repetitions
(i.e., 5 to 30 million load repetitions). Hence, the Nf values
Comparison of Fatigue Life Methods
obtained from the stiffness reduction curve are reasonable,
and the comparison can be focused on the errors that could This extended discussion on fatigue life extrapolation
be involved due to fitting the DELC curve. Two PVNf lines techniques is provided as an introduction to future method
are shown in Figure 4.11. One is with the PVs obtained from ologies for identifying the endurance limit. In this study,
highest R2 fitting of the dissipated energyloading cycle curve; tests were conducted to a maximum of 50 million cycles in
the other is with the PVs from visual fitting that represents the order to confirm the existence of the endurance limit. As
dissipated energyloading cycle curve's best trend (especially noted previously, it takes approximately two months to com
the extension trend). They are closer at relatively higher PV plete a single test at this high number of cycles. This extended
(shorter fatigue life). With the decrease of PV (increase in test time is not practical for routine determination of the
fatigue life), the PV calculated from the highest R2 fitting endurance limit. One alternative to determine the strain
gives a greater value compared to the values obtained from level that corresponds to the endurance limit for a given mix
visual fitting. For low strain testing, the segment that gives ture would be to conduct beam fatigue tests at a low strain
higher R2 does not necessarily represent the real trend of the level to a more limited number of cycles (perhaps less than
curve, which could induce error. Therefore, for low strain 10 million or approximately 10 days) and extrapolate the
long fatigue testing, the "highest R2 fitting" rule is not best data. Thus, a model would be fit to the stiffness versus load
suited. For most cases, the initial segment of the DELC curve ing cycle data and the number of cycles required to reach 50%
has to be eliminated and only the later segment that gives of the initial stiffness would be extrapolated. A significant
a good extension trend of the curve should be used, since deviation from a loglog plot of strain versus cycles to failure
it is more representative for the actual longterm fatigue would indicate the strain level corresponding to the endurance
performance. limit (this will be shown later in the section on Existence of
1.E06
1.E09
1.E12
Endurance Limit
1.E15
1.E18 highest R^2 fitting
y = 0.0708x1.024
PV at Nf50
1.E21 R² = 0.9956
Visual fitting:
1.E24
y = 0.3553x1.101
1.E27 R² = 0.9989
1.E30 Unique PVNf curve:
PV=0.4428Nf^(1.1102)
1.E33
1.E36
1.E39
1.E+05 1.E+08 1.E+11 1.E+14 1.E+17 1.E+20 1.E+23 1.E+26 1.E+29 1.E+32 1.E+35
Nf at 50% stiffness reduction
PVNf from visual fitting PVNf from highest R^2 fitting
PVNf unique curve Power (PVNf from visual fitting)
Power (PVNf from highest R^2 fitting)
Figure 4.11. PVNf curve for IDOT03 mix at low strain with error bars indicated.
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120
Predicted Percent of Actual Stiffness at 50 Million Cycles
110
100
90
80
70
60
50
40
30
20
10
0
0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000
Number of Cycles Used for Model
Exponential Model Logarithmic Model Power Model Weibull Function
Figure 4.12. Convergence of extrapolated stiffness for PG 6722 Sample 4.
the Endurance Limit). The two main requirements for this logarithmic, power, and Weibull function. All of the initial
technique that need to be evaluated are (1) the appropriate cycles were included when fitting the models. Figures 4.12
form of the model and (2) the minimum number of cycles and 4.13 show the percentage of the actual measured stiff
that need to be tested. ness at 50 million cycles for each of the models for PG 6722
The samples tested at 100 ms for the PG 6722 at optimum at optimum for Samples 4 and 13, respectively. The cycles
asphalt content were first used to evaluate the ability of the var shown in Figures 4.12 and 4.13 represent the total number
ious models to predict the sample stiffness at 50 million cycles. of cycles (starting at the first cycle) used to fit the model. The
Four models were considered: exponential (AASHTO T321), stiffness at 50 million cycles extrapolated using that model
120
Predicted Percent of Actual Stiffness at 50 Million Cycles
110
100
90
80
70
60
50
40
30
20
10
0
0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000
Number of Cycles Used for Model
Exponential Model Logarithmic Model Power Model Weibull Function
Figure 4.13. Convergence of extrapolated stiffness for PG 6722 Sample 13.
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is then shown as a percentage of the measured stiffness on loading cycles and the first 10 million loading cycles. A pre
the yaxis. For example, if Sample 4 would have been tested vious study on the endurance limit by Peterson and Turner
to 10 million cycles and a logarithmic model fit to the data, the (38) extrapolated the fatigue life based on testing to 4 million
extrapolated stiffness at 50 million cycles would be 108.2% of cycles. Shen and Carpenter (40) extrapolated test results
the measured stiffness at 50 million cycles. based on tests conducted to greater than 8 million cycles.
Examination of Figures 4.12 and 4.13 show that the expo Table 4.2 shows the fatigue life predictions for the five sam
nential model consistently underestimates the stiffness at ples using six different extrapolation methods: exponential
50 million cycles and is slow to converge on the measured model, logarithmic model, power model, singlestage Weibull,
stiffness (testing would need to be conducted to a high num threestage Weibull, and RDEC.
ber of cycles to even approach the measured stiffness). This The RDEC procedure consistently overestimates the sam
would suggest that the exponential model recommended by ples' fatigue lives by three to seven orders of magnitude. For
AASHTO T321 is not a good choice for extrapolating fa example, the actual fatigue life of Sample 2 of the PG 6722 at
tigue data. optimum asphalt content is 2.60E+07 whereas the predicted
The predicted stiffness values using the logarithmic and fatigue life using the RDEC procedure for the first 10 million
power models are basically the same in Figures 4.12 and 4.13. cycles is 3.08E+11. The power model also consistently over
Both converge to a reasonable predicted stiffness within 10 mil estimates fatigue life by two to seven orders of magnitude.
lion cycles. However, when all of the loading cycles are used, For the remaining methods, with the exception of the expo
both overestimate the stiffness at 50 million cycles and, con nential model, the fatigue life of Sample 5 of the PG 6722 at
sequently, would overestimate the fatigue life. The singlestage optimum asphalt content was overestimated by a larger degree
Weibull function converges quickly and provides the most than for the other samples. Sample 5 was tested at 170 ms. The
accurate results for Sample 4, but does a relatively poor job logarithmic model overestimated the fatigue life of Sample 5
for Sample 13. Recall that the Weibull function for Sample 13 by five orders of magnitude, but the remaining four samples
had the concave down shape in Figure 4.4. by one to two orders of magnitude. The threestage Weibull
The accuracy of the stiffness prediction is not the only factor model overestimated fatigue life of Sample 5 by three or
which will affect the accuracy of the fatigue life extrapolation. four orders of magnitude based on the data from 4 million
The shape of the model will also have an effect. Logarith and 10 million loading cycles, respectively. However, for the
mic and power models can produce very flat slopes at high remaining samples, the threestage Weibull function over
numbers of loading cycles that result in overestimation of estimated fatigue life by zero to two orders of magnitude.
the fatigue life (particularly if some of the initial cycles are not Sample 23 of the same mix was also tested at 170 ms. It was
eliminated to better match the slope of stiffness versus load tested to 60 million cycles without reaching 50% of its ini
ing cycles at a high number of loading cycles). tial stiffness.
Five samples tested in Phase I had fatigue lives between The exponential and singlestage Weibull function pro
20 and 50 million cycles. These samples were used to evaluate duced the most accurate fatigue life predictions. As shown in
the accuracy of the extrapolation techniques. Predictions Figure 4.14, the exponential model consistently underesti
were based on models developed using the first 4 million mates fatigue life. The fatigue life of points above the line of
Table 4.2. Comparison of fatigue life extrapolations.
Mix Sample Cycles Used Actual Exponential Logarithmic Power Single Three RDEC
for Stage Stage
Extrapolation Weibull Weibull
(Millions)
4 7.89E+06 2.54E+09 3.80E+10 3.31E+07 1.41E+08 6.78E+11
2 2.60E+07
10 1.87E+07 2.83E+09 3.43E+10 5.59E+07 1.89E+08 3.08E+11
PG 6722 4 4.94E+06 4.84E+08 4.19E+09 1.44E+07 4.43E+07 5.53E+10
21 2.08E+07
Optimum 10 9.88E+06 3.17E+08 1.78E+09 1.91E+07 5.04E+07 1.92E+09
4 1.27E+07 2.13E+12 2.15E+14 3.87E+08 7.23E+11 1.66E+14
5 3.47E+07
10 2.88E+07 2.27E+12 2.15E+14 7.24E+08 1.92E+10 8.02E+13
PG 6722 4 8.62E+06 5.57E+09 1.58E+11 3.11E+07 1.20E+09 2.26E+15
Optimum 4 3.90E+07
Plus 10 1.84E+07 4.59E+09 2.97E+10 4.97E+07 3.39E+09 1.06E+11
PG 7622 4 7.69E+06 6.56E+08 7.16E+09 1.13E+07 3.28E+08 2.56E+12
Optimum 10 3.96E+07
Plus 10 1.73E+07 7.77E+08 7.41E+09 2.01E+07 4.54E+08 1.71E+10