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Validating the Fatigue Endurance Limit for Hot Mix Asphalt (2010)

Chapter: Chapter 4 - Beam Fatigue Test Results and Analyses

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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 27
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 28
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 29
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 41
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 42
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Suggested Citation:"Chapter 4 - Beam Fatigue Test Results and Analyses." National Academies of Sciences, Engineering, and Medicine. 2010. Validating the Fatigue Endurance Limit for Hot Mix Asphalt. Washington, DC: The National Academies Press. doi: 10.17226/14360.
×
Page 43

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20 This chapter describes the beam fatigue testing conducted to confirm the existence of the endurance limit. One of the most important aspects of this research is a practical defini- tion for the endurance limit. Asphalt mixtures simply cannot be tested for an infinite fatigue life in the laboratory. Testing at 10 Hz, approximately one million load repetitions can be applied to a beam fatigue sample in a given day. The primary goal of this testing was to confirm the existence of a fatigue endurance limit. In order to accomplish this goal, it was nec- essary to develop a method for estimating the endurance limit through accelerated testing in a reasonable period of time. A secondary goal was an estimate of the variability associated with beam fatigue testing and its potential impact on pave- ment thickness design. The first portion of Chapter 4 discusses methods for extrap- olating the fatigue failure point for strain levels that did not result in failure in less than 50 million loading cycles. The methods were applied to samples that had fatigue lives in excess of 10 million, but less than 50 million loading cycles. The second portion of the chapter presents the data collected in Phase I, as well as additional binder grades tested with the same mixture in Phase II. Evidence of the existence of a fatigue endurance limit is presented for each of these mixtures. The third portion of the chapter describes a limited round- robin conducted to assess the variability of fatigue testing and the prediction of the endurance limit. Finally, indirect tensile tests were investigated as a surrogate for beam fatigue tests. As discussed in Chapter 3, a single gradation and aggregate type was used for all of the testing. A full-factorial experiment, shown in Table 4.1, was conducted in Phase I to evaluate the existence of an endurance limit and to identify factors affect- ing the endurance limit. Two main factors were included in the experiment—binder grade and asphalt content. Binder grade was varied at two levels: PG 67-22 and PG 76-22. As noted previously, the PG 67-22 also met the requirements of a PG 64-22. Asphalt content was varied at two levels: opti- mum and optimum plus 0.7%. Optimum asphalt content was determined at Ndesign = 80 gyrations. In addition, the samples’ air void contents were reduced from 7.0 ± 0.5% to 3.3 ± 0.5% for the optimum plus samples. The voids for the optimum plus samples were reduced to simulate the expected improved densification in the field. Advantages of optimum plus or rich bottom layers are believed to include better com- pactability, greater resistance to fatigue damage, and improved moisture susceptibility. Extrapolation Methods to Predict Fatigue Life As discussed previously, it was decided prior to the start of testing that beam fatigue tests would be terminated at 50 mil- lion cycles. If a shift factor of 10 was applied to the test results from a sample tested to 50 million cycles, it would then be estimated that the pavement could withstand 500 million loading cycles at the corresponding strain level. Based on capacity analysis of a lane, this then represents a reasonable maximum number of loading cycles that might occur in a 40-year period. For the practical definition of the endurance limit, a 40-year life was considered to be indicative of a long- life pavement. It takes approximately 50 days to test a single sample to 50 million cycles. Additional analyses will be dis- cussed later to evaluate the existence of a theoretical or truly infinite life endurance limit. For samples that failed in less than 50 million cycles at 50% of initial stiffness, the number of cycles to failure was deter- mined from the data acquisition software controlling the test. However, if the test was terminated prior to reaching 50% of initial stiffness, either due to an equipment problem or to reaching 50 million cycles, an extrapolation procedure was used to estimate the number of loading cycles, Nf, correspon- ding to 50% of initial stiffness. Ideally, a method of extrap- olation would be identified that could be used to shorten the beam fatigue testing procedure used to determine the endurance limit. Then, samples could be tested to 4 million C H A P T E R 4 Beam Fatigue Test Results and Analyses

21 cycles as done in the Asphalt Institute study (38, 39) or pos- sibly 10 million cycles and the fatigue life at, or close to, the endurance limit predicted. In Phase I, testing was conducted at progressively lower strain levels until two samples at a given strain level reached 50 million cycles without reaching 50% of their initial stiff- ness (failure). Instead of testing at a lower strain level below that providing a fatigue life of at least 50 million cycles, sam- ples were tested at the strain level predicted to provide a fatigue life of 50 million cycles. The goal of this additional point was to help define the transition from “normal” strain test to tests below the apparent fatigue endurance limit or “low” strain tests. For the PG 67-22 mix at optimum asphalt content, the data from 800 through 200 ms were used to estimate the strain level that would result in a fatigue life of 50 million cycles. A linear regression was performed between the Log10 of ms and the Log10 of loading cycles to 50% initial stiffness. 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 life of 50 million cycles. This was rounded to 170 ms for testing purposes. Testing was conducted at this strain level to better define the endurance limit. where, Nf = number of cycles to 50% of initial stiffness and ε = constant strain used in beam fatigue test (ms). When testing the PG 67-22 mix at optimum asphalt content at 170 ms, the first replicate failed in 34.7 million cycles. The second replicate was at 55% of its initial stiffness at 50 million cycles. Therefore, testing was extended to see if the failure point could be determined. However, at 60 million cycles, Sample 23 still retained 53% of its initial stiffness. Therefore, testing was suspended at 60 million cycles. For the PG 67-22 mix at optimum asphalt content, samples tested at 200 and 170 ms were used to investigate extrapola- tion techniques. These strain levels and similar strain levels for the other mixes used in Phase I provided long fatigue lives (in excess of 10 million cycles) while still having a defined fail- ure point that could be used to investigate the accuracy of the extrapolation. Five techniques were investigated for extrapo- N f = × −10 120 6 5 81. . ( )ε lation: exponential model, logarithmic model, Weibull func- tion, three-stage Weibull function, and ratio of dissipated energy change (RDEC). Each of these is discussed below, and examples are provided of the predicted fatigue lives. Fatigue Life Extrapolation Using AASHTO T321 Exponential Model AASHTO T321 specifies an exponential model (Equation 2) for the calculation of cycles to 50% initial stiffness, as follows: where, S = sample stiffness (MPa), A = constant, b = constant, and n = number of load cycles. The constants are determined by regression analysis of load- ing cycles versus the natural logarithm of the flexural stiff- ness. The number of cycles to failure is determined by solving Equation 2 for 50% of initial stiffness. In this study, for sam- ples tested to less than 50 million cycles, the number of cycles reported to reach 50% of the initial stiffness are the actual number of cycles recorded by the test equipment, not the num- ber of cycles determined using Equation 2. No discussion is provided in AASHTO T321 regarding whether or not all of the data (particularly the initial data) should be used when solving for the constants in Equation 2 (52). Sample 5 of the PG 67-22 mix at optimum asphalt content, which was tested at 170 ms, was selected as an example. It was desirable to select a sample that had as long a fatigue life as pos- sible and had reached 50% of initial stiffness within 50 million cycles. It was felt that an extrapolation method that worked well at high strain levels may not prove to be as accurate at strain levels closer to the anticipated endurance limit. Fig- ure 4.1 shows fits to the loading cycle versus sample stiffness data determined using Equation 2. The coefficients for Equa- tion 2 were fit using the data up to 4 million cycles, 10 mil- lion cycles, and failure (34.7 million cycles). As can be seen from Figure 4.1, when all of the data up to failure was used to fit the model, the model provides a reasonable estimate of fatigue life. S Aebn= ( )2 Table 4.1. Experimental design. Granite NMAS (mm) Binder Content PG 67-22 PG 76-22 Optimum X X 19.0 Optimum + X X

22 Figure 4.1. Examples of fatigue life estimates using the exponential model. y = 3970 .6 e -5 E-0 8x R² = 0. 4992 y = 3927 .2 e -2 E- 08x R² = 0. 4776 y = 3890 .5 e -1 E- 08x R² = 0. 7259 0 500 1000 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 1500 2000 2500 3000 3500 4000 4500 5000 Fl ex ur al S tif fn es s, M Pa Loading Cycles 50% Initial Expon. (4.00E+06) Expon. (1.00E+07) Expon. (To Failure) PG 67-22 Sample 5 at 170 ms Figure 4.2. Logarithmic model fits for PG 67-22 at optimum, Sample 5. Data to 4 million Cycles y = -91.789Ln(x) + 4865.2 R 2 = 0.9714 Data to 10 million Cycles y = -91.483Ln(x) + 4862.4 R 2 = 0.9763 To Failure y = -106.17Ln(x) + 5005.6 R 2 = 0.9108 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Fl ex ur al S tif fn es s, M Pa Loading Cycles 50% Initial Log. (4.00E+06) Log. (1.00E+07) Log. (To Failure) 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 Fatigue Life Extrapolation Using Natural Logarithm of Loading Cycles versus Stiffness A logarithmic model (Equation 3) using the natural loga- rithm of loading cycles versus stiffness was evaluated as one alternative to the exponential model. where, S = the sample stiffness at loading cycle n, and α and β are regression constants. S n= + × ( )α β ln ( )3 When all of the fatigue data are used to fit a logarithmic 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- estimation of the fatigue life. This is illustrated in Figure 4.2 for Sample 5 of the PG 67-22 mix at optimum asphalt content. The fits to the logarithmic model using just the first 10 million cycles and using all of the data are indistinguishable on the plot. Note that in Figure 4.2 the logarithmic model provides very high R2 values, but the fitted model does not match the experimental data at a high number of cycles. However, by eliminating a portion of the early loading cycles, a good match to the data can generally be obtained,

where, α = 10 raised to the power of the intercept from regression of log (S) versus log (n), and β = the slope from regression of log (S) versus log (n). The power model has a similar shape to the logarithmic model. Shen (54) also reported the need to eliminate a number of ini- tial cycles to obtain a good fit to the slope at high numbers of cycles. Failure to eliminate some of the initial cycles results in an overestimation of the fatigue life. Additional discussion on the use of the power model and its application to the ratio of dissipated energy will be provided later in this report. Fatigue Life Extrapolation Using the Weibull Survivor Function Often, failure data can be modeled using a Weibull distri- bution. The Weibull function is commonly used in reliability engineering to estimate survival life. Tsai et al. (55) applied the Weibull survivor function to HMA beam fatigue data. The generalized equation for the Weibull function is given by Equation 5. where, R(t) = the reliability at time t where t might be time or another life parameter such as loading cycles, γ = the slope, δ = the minimum life, and θ = the characteristic life. R t t( ) = − − − ⎛⎝⎜ ⎞⎠⎟exp ( ) δ θ δ γ 5 23 Figure 4.3. Logarithmic model fits for PG 67-22 at optimum, Sample 6. y = -235.ln(x) + 7766. R² = 0.965 y = -405ln(x) + 10226 R² = 0.965 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 0 2,000,000 4,000,000 6,000,000 8,000,000 10,000,000 12,000,000 14,000,000 Fl ex ur al S tif fn es s, M Pa Loading Cycles Log. (All Data) Log. (Starting 1E+6 cycles) Logarithmic Model Starting at 1 million Cyles Logarithmic Model using All Data particularly at low strain levels. In Figure 4.3, logarithmic models were fit to the data from Sample 6 of the PG 67-22 mix at optimum asphalt content, which was tested at 200 ms. In Figure 4.3, logarithmic models are shown including all of the initial loading cycles and excluding the first million loading cycles. This provides a better fit to data, but still would tend to overestimate the fatigue life. Further, the number of early loading cycles that are not included must be determined by trial and error. Note that all of the logarithmic models shown in Figures 4.2 and 4.3 provide high R2 values, even when the fit to the data at a high number of cycles is not very good. This suggests that R2 values alone are not adequate to evaluate extrapolation models. It is believed that the poor fit at a high number of cycles results from the fact that the data are col- lected using a logarithmic progression. That is, the sampling 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 are more data points to fit in the early portion of the curve. Rowe and Bouldin (53), when examining fits from the expo- nential model, concluded that fatigue data should be taken with every 5% reduction in stiffness. Fatigue Life Extrapolation Using the Power Model The ratio of dissipated energy, developed by Shen and Carpenter (40) also requires that the number of cycles to 50% of initial stiffness be calculated in order to determine the plateau values. Shen (54) recommends a power model (Equation 4) for the extrapolation of stiffness versus loading cycles, as follows: S n= +α β ( )4

Tsai et al. (55) applied a specialized case of the Weibull func- tion where the minimum life, δ, was assumed to be 0. In this case, the characteristic life = 1/λ and the Weibull function simplifies to Equation 6. Since the beam fatigue loading cycles are applied at a constant frequency of 10 Hz, loading cycles, n, can be substituted for time, t. where, S(t) = probability of survival until time t, λ = scale parameter (intercept), and γ = shape parameter (slope). The stiffness ratio can be used to characterize fatigue dam- age. The stiffness ratio is the stiffness measured at cycle n, divided by the initial stiffness, determined at the 50th cycle. Tsai (56) states that at a given cycle n, the beam being tested has a probability of survival past cycle n equal to the SR times 100%. Thus, SR(n) can be substituted for S(t). Tsai (56) presents the derivation of Equation 7, which allows the scale and shape parameters for laboratory beam fatigue data to be determined by linear regression. where, SRn = stiffness ratio or stiffness at cycle n divided by the initial stiffness. ln ln ln ln ( )− ( )( ) = ( )+ × ( )SR nn λ γ 7 S t n( ) = − ×( )exp ( )λ γ 6 Figure 4.4 shows an example of the data from the two 100 ms samples from the PG 67-22 at optimum mixture in the form of Equation 7. Tsai et al. (55) observed that the concave down shape, exhibited by Sample 13, “implied that the fatigue damage rate is slowed down and flattens out with increased repetitions and thus causes no further damage after a certain number of repetitions.” This behavior is believed to be indicative of the endurance limit. Fatigue Life Extrapolation Using Three-Stage Weibull Function In the previous section, the Weibull survivor function was presented as a method for modeling the fatigue life of beam fatigue tests. In later sections, it will be demonstrated that the single-stage Weibull function generally provides a good esti- mate of a sample’s fatigue life and is reproducible when calcu- lated by different laboratories. There are, however, cases for which the single-stage Weibull function apparently under- predicts fatigue life. Sample 13 in Figure 4.4 is one such exam- ple (but Sample 13 is not specifically labeled on the plot). It should be noted that with the exception of two examples analyzed by Tsai, the three-stage Weibull analyses were not conducted until after the completion of all of the Phase I and II testing. To improve upon the accuracy of the single-stage Weibull function, Tsai et al. (57) developed a methodology for fitting a three-stage Weibull curve. Tsai et al. (57) theorized that a 24 Figure 4.4. Weibull survivor function for PG 67-22 at optimum 100 ms samples. y = 0.243x -5.816 R² = 0.82 y = 0.4014x -8.2141 R² = 0.72 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 8642 10 12 14 16 18 20 Ln (-L n( St iff ne ss R at io )) Ln (Cycles) Sample 4 Sample 13 Linear (Sample 4) Linear (Sample 13)

plot of loading cycles versus stiffness ratio could be divided into three stages: initial heating and temperature equilibrium, crack initiation, and crack propagation. In the case of low 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. A Weibull function is fit to each of these stages as shown in Equation 8: Coefficients α1, α2, α3, β1, β2, β3, n1, and n2 are illustrated in Figure 4.5. Using a series of mathematical manipulations (57), n1, γ1, n2, and γ2 can be calculated sequentially as follows: γ ββ1 2 1 11 10= − ⎛ ⎝⎜ ⎞ ⎠⎟ ×n ( ) n1 2 1 2 1 2 1 2 1 9= × ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢⎢ ⎤ ⎦⎥⎥ −α α β β β β β ( ) SR e n n SR e n n 1 1 1 1 2 2 1 2 = ≤ < = − ×( ) − × −( )( ) α β α γ β for 0 for for n n n SR e n n n 1 2 3 3 2 3 2 8≤ < = ≤ <− × −( )( ) ( ) α γ β n3 Tsai et al. (57) applied a genetic algorithm to resolve the six unknown parameters. A genetic algorithm requires a param- eter definition, in this case Equations 8 through 12, and a fit- ness function. Residual sum of squares between the measured and fitted ln(-ln(SR)) for each cycle is used as the fitness func- tion. A “good gene” is defined as an optimum set of param- eters to minimize the fitness function. A set of input ranges is first determined for α1, α2, α3, β1, β2, and β3. The input ranges are determined by visual inspection of the data. Simple linear regressions are performed for each stage to determine slopes and offsets. An example is shown in Figure 4.5. Tolerances are applied to the parameters determined by inspection to set ini- tial ranges for each coefficient. The input ranges and test data are entered into a FORTRAN program, N3stage.exe, devel- oped by Tsai. The program randomly generates a set of param- 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 and replaced with new genes. The cycle of producing genes, ranking genes by the residual sum of squares, mating and γ ββ β β γ2 3 2 2 3 2 11 12= − ⎛ ⎝⎜ ⎞ ⎠⎟ × + ×n ( ) n2 1 3 2 3 2 3 1 2 3 11= + × ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢⎢ ⎤ ⎦⎥⎥ − γ α α β β β β β ( ) 25 Figure 4.5. Three-stage Weibull curve definitions. y = 0.5427x -6.6613 R² = 0.547 y = 0.1769x -4.1281 R² = 0.9075 y = 0.0667x -2.6913 R² = 0.7617 -6 -5 -4 -3 -2 -1 0 0 2 4 6 8 10 12 14 16 18 20 Ln ( - Ln (S tif fn es s Ra tio )) Ln (Loading Cycles) Stage 1 Stage 2 Stage 3 Stage 1 Initial Heating Stage 2 Stage 3 LN (n1) Ln (n2) 1 Intercept = Lnα3 Intercept = Lnα2 Intercept = Lnα1 1 2 1 3 1 PG 76-22 Optimum Plus Sample 9 200 ms

26 Figure 4.6. Three-stage Weibull fit to transformed data. -6 -5 -4 -3 -2 -1 0 0 2 4 6 8 10 12 14 16 18 20 Ln (- Ln (S tif fn es s Ra tio )) Ln (Loading Cycles) PG 76-22 Optimum Plus Sample 9 200 ms Raw Data Fit Figure 4.7. Three-stage Weibull fit to stiffness data. 0 1,000 2,000 3,000 4,000 5,000 6,000 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000 St iff ne ss , M Pa Loading Cycles PG 76-22 Optimum Plus Sample 9 200 ms Raw Data 3-Stage Weibull Fit replacing genes, continues until the specified number of gen- erations is complete (58). The N3stage program typically takes 30 to 60 minutes to complete 750 generations, depending on the size of the data set. The complete calculation procedure is described in Appendix A, Proposed Standard Practice for Predicting the Endurance Limit of Hot Mix Asphalt (HMA) for Long-Life Pavement Design. The NCHRP 9-38 research team developed a Microsoft Excel spreadsheet to solve the three-stage Weibull parameters, which produces similar results to the N3stage program. Figures 4.6 and 4.7 show examples of the three-stage Weibull fit. This methodology provides a good fit to both

normal and low strain fatigue data. In some cases, only one or two stages are fit, even if three stages are initially identified. Ratio of Dissipated Energy Change (RDEC) Dissipated energy is a measure of the energy that is lost to the material or altered through mechanical work, heat gener- ation, or damage to the sample. Other researchers have used cumulative dissipated energy to define damage within a spec- imen, assuming that all of the dissipated energy is responsi- ble for the damage. The approach suggested by Ghuzlan and Carpenter (34) considers that only a portion of the dissipated energy is responsible for actual damage. Typically, three regions are observed in the RDEC analy- sis as shown in Figure 4.8. Region I represents the initial “settling” of the sample where the rate of change of dissipated energy decreases. In Region II, the rate of change of dissipated energy reaches a plateau, representing a period where the amount of damage occurring to the sample is constant. Finally, in Region III, sample instability begins as the rate of change of dissipated energy rapidly increases. A lower dissipated energy ratio (DER) plateau value implies that less damage is occurring per cycle. Therefore, a sample with a low DER plateau value would be expected to have a longer fatigue life than a sample with a high DER plateau value. Shen and Carpenter (40) refined this technique and suggested that the RDEC plateau value (PV) should be calculated at the number of cycles that produced 50% of the initial sample stiffness (Nf). A PV of 8.57E-9 was proposed by Shen and Carpenter as indica- tive of a long life pavement (40). The RDEC analysis procedure is described in Appendix C, Proposed Standard Practice for Extrapolating Long-Life Beam Fatigue Tests Using the Ratio of Dissipated Energy Change (RDEC). RDEC is the ratio of dissipated energy change between two data points divided by the number of cycles between the two data points, that is, the average ratio of dissipated energy change per loading cycle. This is written as follows: where, RDECa = the average ratio of dissipated energy change at cycle a, comparing to next cycle b; a, b = load cycle a and b, respectively (the cycle count between cycle a and b for RDEC calculation will vary depending on the data acquisition soft- ware); and DEa, DEb = the dissipated energy produced in load cycle a, and b, respectively. The dissipated energy for each loading cycle is determined by measuring the area within the stress-strain hysteresis loop for each captured load pulse. This methodology is used by the IPC Global beam fatigue device used in the study by NCAT, Asphalt Institute, and the University of Illinois. Alter- natively, the dissipated energy can be calculated according to Equation 14. where, σn = maximum tensile stress in cycle n, in kPa, εn = maximum tensile strain in cycle n, δn = 360 × f × s, f = loading frequency, Hz, and s = time lag in seconds between peak load and peak deflec- tion in seconds. Due to testing noise, as shown in Figure 4.8, the raw dissi- pated energy data are not directly usable for calculating RDEC wn n n n= × × ×π σ ε δsin ( )14 RDEC DE DE DE b a a a b a = − −( ) ( )13 27 Figure 4.8. Typical RDEC versus loading cycles plot and the indication of PV. 0 0.002 0.004 0.006 10 510 1,010 1,510 2,010 2,510 3,010 3,510 4,010 R D EC Number of Load Cycles I III PV II

where, f = the slope from the regressed dissipated energy-loading cycle curve. In the RDEC approach, PV is defined as the RDEC value at the 50% stiffness reduction failure point (Nf50). Therefore, the PV value can be obtained using Equation 16. Here, the PV value depends only on the f factor of the regressed power law DE-LC curve and the defined failure point, Nf50. For long-life tests, where Nf50 was not known, the stiffness- loading cycle curve first needed to be extrapolated to deter- mine Nf50, resulting in the calculation being based on a double extrapolation. Using this approach, Shen and Carpenter (40) demon- strated a unique PV-Nf curve for all HMA mixes at normal strain/damage level testing, regardless of the testing condi- tion, loading modes, and mixture types. The tests used for es- tablishing this relationship at all normal testing were carried to or beyond the failure point (i.e., the Nf50 values are known). Also, using the results from long-term fatigue testing, Shen and Carpenter (40) demonstrated that the unique relation- PV Nf f = − + ⎛ ⎝⎜ ⎞ ⎠⎟1 1 100 100 1650 ( ) RDEC a a f = − + ⎛⎝⎜ ⎞⎠⎟1 1 100 100 15( ) 28 Figure 4.9. DE versus LC chart for one IDOT mix with fitted curve. y = 3.4255x -0.164 R² = 0.9512 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 1000 2000 3000 4000 5000 D E, k Pa Loading Cycles Nf50 Exponential Slope k IDOT03 mix 3N704B DE vs. Loading cycles 800 microstrains Figure 4.10. Dissipated energy versus loading cycle for raw data and various power models. R^2 = 0.90 y = 0.7162x-0.1384 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0.100 Pr ed ic te d DE , k Pa Loading Cycles Predicted DE Sample 2 at 200 ms 66910 4617890 8403190 Measured Legend Number Denotes Starting Cycle and PV. A curve fitting procedure is recommended to obtain the best fit equation for the dissipated energy-loading cycle data. It is assumed that the regression equation of dissipated energy-loading cycle relationship follows a power law rela- tionship, Axf (as indicated in Figure 4.9). The key for the curve fitting process is to obtain a slope (in the power law relation plot) of the curve, f, which can best represent the original curve. In general, there are two rules for evaluating the goodness of the fitted curve: (1) a high R-square value, and (2) correct trend of the DE-LC curve. This is similar to the procedures described previously for fitting logarithmic or power models to the stiffness-loading cycle curves and is illustrated in Figure 4.10. The average RDEC for an arbitrary 100 cycles at cycle a can be simply calculated using the following equation:

ship between PV-Nf is also valid for low strain testing. In other words, the PV-Nf relationship is unique for the whole loading range including both normal and low strain/damage level. An error analysis was performed for two dissipated energy curve fitting routines using 19 Illinois DOT mixtures tested at low strain levels, and the results are presented in Figure 4.11. All the samples were tested to extended load repetitions (i.e., 5 to 30 million load repetitions). Hence, the Nf values obtained from the stiffness reduction curve are reasonable, and the comparison can be focused on the errors that could be involved due to fitting the DE-LC curve. Two PV-Nf lines are shown in Figure 4.11. One is with the PVs obtained from highest R2 fitting of the dissipated energy-loading cycle curve; the other is with the PVs from visual fitting that represents the dissipated energy-loading cycle curve’s best trend (especially the extension trend). They are closer at relatively higher PV (shorter fatigue life). With the decrease of PV (increase in fatigue life), the PV calculated from the highest R2 fitting gives a greater value compared to the values obtained from visual fitting. For low strain testing, the segment that gives higher R2 does not necessarily represent the real trend of the curve, which could induce error. Therefore, for low strain long fatigue testing, the “highest R2 fitting” rule is not best suited. For most cases, the initial segment of the DE-LC curve has to be eliminated and only the later segment that gives a good extension trend of the curve should be used, since it is more representative for the actual long-term fatigue performance. The unique PV-loading cycle curve is also illustrated in Figure 4.11. As illustrated in Appendix C, an alternative approach for extrapolating the fatigue life of a sample that does not fail is to plot the sample’s RDEC versus loading cycle on a log-log plot and determine the intersection with the unique PV-Nf line. Comparison of Fatigue Life Methods This extended discussion on fatigue life extrapolation techniques is provided as an introduction to future method- ologies for identifying the endurance limit. In this study, tests were conducted to a maximum of 50 million cycles in order to confirm the existence of the endurance limit. As noted previously, it takes approximately two months to com- plete a single test at this high number of cycles. This extended test time is not practical for routine determination of the endurance limit. One alternative to determine the strain level that corresponds to the endurance limit for a given mix- ture would be to conduct beam fatigue tests at a low strain level to a more limited number of cycles (perhaps less than 10 million or approximately 10 days) and extrapolate the data. Thus, a model would be fit to the stiffness versus load- ing cycle data and the number of cycles required to reach 50% of the initial stiffness would be extrapolated. A significant deviation from a log-log plot of strain versus cycles to failure would indicate the strain level corresponding to the endurance limit (this will be shown later in the section on Existence of 29 Figure 4.11. PV-Nf curve for IDOT03 mix at low strain with error bars indicated. Visual fitting: y = 0.3553x-1.101 R² = 0.9989 highest R^2 fitting y = 0.0708x-1.024 R² = 0.9956 1.E-39 1.E-36 1.E-33 1.E-30 1.E-27 1.E-24 1.E-21 1.E-18 1.E-15 1.E-12 1.E-09 1.E-06 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 PV a t N f5 0 Nf at 50% stiffness reduction PV-Nf from visual fitting PV-Nf from highest R^2 fitting PV-Nf unique curve Power (PV-Nf from visual fitting) Power (PV-Nf from highest R^2 fitting) Unique PV-Nf curve: PV=0.4428Nf^(-1.1102) Endurance Limit

30 Figure 4.12. Convergence of extrapolated stiffness for PG 67-22 Sample 4. 0 10 20 30 40 50 60 70 80 90 100 110 120 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000 Pr ed ic te d Pe rc en t o f A ct ua l S tif fn es s at 5 0 M ill io n Cy cl es Number of Cycles Used for Model Exponential Model Logarithmic Model Power Model Weibull Function Figure 4.13. Convergence of extrapolated stiffness for PG 67-22 Sample 13. 0 10 20 30 40 50 60 70 80 90 100 110 120 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000 Pr ed ic te d Pe rc en t o f A ct ua l S tif fn es s at 5 0 M ill io n Cy cl es Number of Cycles Used for Model Exponential Model Logarithmic Model Power Model Weibull Function the Endurance Limit). The two main requirements for this technique that need to be evaluated are (1) the appropriate form of the model and (2) the minimum number of cycles that need to be tested. The samples tested at 100 ms for the PG 67-22 at optimum asphalt content were first used to evaluate the ability of the var- ious models to predict the sample stiffness at 50 million cycles. Four models were considered: exponential (AASHTO T321), logarithmic, power, and Weibull function. All of the initial cycles were included when fitting the models. Figures 4.12 and 4.13 show the percentage of the actual measured stiff- ness at 50 million cycles for each of the models for PG 67-22 at optimum for Samples 4 and 13, respectively. The cycles shown in Figures 4.12 and 4.13 represent the total number of cycles (starting at the first cycle) used to fit the model. The stiffness at 50 million cycles extrapolated using that model

is then shown as a percentage of the measured stiffness on the y-axis. For example, if Sample 4 would have been tested to 10 million cycles and a logarithmic model fit to the data, the extrapolated stiffness at 50 million cycles would be 108.2% of the measured stiffness at 50 million cycles. Examination of Figures 4.12 and 4.13 show that the expo- nential model consistently underestimates the stiffness at 50 million cycles and is slow to converge on the measured stiffness (testing would need to be conducted to a high num- ber of cycles to even approach the measured stiffness). This would suggest that the exponential model recommended by AASHTO T321 is not a good choice for extrapolating fa- tigue data. The predicted stiffness values using the logarithmic and power models are basically the same in Figures 4.12 and 4.13. Both converge to a reasonable predicted stiffness within 10 mil- lion cycles. However, when all of the loading cycles are used, both overestimate the stiffness at 50 million cycles and, con- sequently, would overestimate the fatigue life. The single-stage Weibull function converges quickly and provides the most accurate results for Sample 4, but does a relatively poor job for Sample 13. Recall that the Weibull function for Sample 13 had the concave down shape in Figure 4.4. The accuracy of the stiffness prediction is not the only factor which will affect the accuracy of the fatigue life extrapolation. The shape of the model will also have an effect. Logarith- mic and power models can produce very flat slopes at high numbers of loading cycles that result in overestimation of the fatigue life (particularly if some of the initial cycles are not eliminated to better match the slope of stiffness versus load- ing cycles at a high number of loading cycles). Five samples tested in Phase I had fatigue lives between 20 and 50 million cycles. These samples were used to evaluate the accuracy of the extrapolation techniques. Predictions were based on models developed using the first 4 million loading cycles and the first 10 million loading cycles. A pre- vious study on the endurance limit by Peterson and Turner (38) extrapolated the fatigue life based on testing to 4 million cycles. Shen and Carpenter (40) extrapolated test results based on tests conducted to greater than 8 million cycles. Table 4.2 shows the fatigue life predictions for the five sam- ples using six different extrapolation methods: exponential model, logarithmic model, power model, single-stage Weibull, three-stage Weibull, and RDEC. The RDEC procedure consistently overestimates the sam- ples’ fatigue lives by three to seven orders of magnitude. For example, the actual fatigue life of Sample 2 of the PG 67-22 at optimum asphalt content is 2.60E+07 whereas the predicted fatigue life using the RDEC procedure for the first 10 million cycles is 3.08E+11. The power model also consistently over- estimates fatigue life by two to seven orders of magnitude. For the remaining methods, with the exception of the expo- nential model, the fatigue life of Sample 5 of the PG 67-22 at optimum asphalt content was overestimated by a larger degree than for the other samples. Sample 5 was tested at 170 ms. The logarithmic model overestimated the fatigue life of Sample 5 by five orders of magnitude, but the remaining four samples by one to two orders of magnitude. The three-stage Weibull model overestimated fatigue life of Sample 5 by three or four orders of magnitude based on the data from 4 million and 10 million loading cycles, respectively. However, for the remaining samples, the three-stage Weibull function over- estimated fatigue life by zero to two orders of magnitude. Sample 23 of the same mix was also tested at 170 ms. It was tested to 60 million cycles without reaching 50% of its ini- tial stiffness. The exponential and single-stage Weibull function pro- duced the most accurate fatigue life predictions. As shown in Figure 4.14, the exponential model consistently underesti- mates fatigue life. The fatigue life of points above the line of 31 Table 4.2. Comparison of fatigue life extrapolations. Mix Sample Cycles Used for Extrapolation (Millions) Actual Exponential Logarithmic Power Single- Stage Weibull Three- Stage Weibull RDEC 4 7.89E+06 2.54E+09 3.80E+10 3.31E+07 1.41E+08 6.78E+112 10 2.60E+07 1.87E+07 2.83E+09 3.43E+10 5.59E+07 1.89E+08 3.08E+11 4 4.94E+06 4.84E+08 4.19E+09 1.44E+07 4.43E+07 5.53E+1021 10 2.08E+07 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 PG 67-22 Optimum 5 10 3.47E+07 2.88E+07 2.27E+12 2.15E+14 7.24E+08 1.92E+10 8.02E+13 4 8.62E+06 5.57E+09 1.58E+11 3.11E+07 1.20E+09 2.26E+15PG 67-22 Optimum Plus 4 10 3.90E+07 1.84E+07 4.59E+09 2.97E+10 4.97E+07 3.39E+09 1.06E+11 4 7.69E+06 6.56E+08 7.16E+09 1.13E+07 3.28E+08 2.56E+12PG 76-22 Optimum Plus 10 10 3.96E+07 1.73E+07 7.77E+08 7.41E+09 2.01E+07 4.54E+08 1.71E+10

10 million cycles using the single-stage Weibull function, is 7.75. This indicates that the two predictions are relatively close. In summary, the Weibull functions were selected for extrap- olating fatigue tests that did not fail within 50 million cycles or when the test was interrupted prior to failure (as occurred with Sample 6 of the PG 67-22 mix at optimum asphalt content). For long-life fatigue tests, at strain levels slightly above the endurance limit, the single-stage Weibull function appears to provide the most accurate extrapolation of fatigue life. The three-stage Weibull function, however, provides the best fit to the stiffness versus loading cycle data. Fatigue extrapo- lations from both methods are shown when discussing evi- dence of the endurance limit. Based on the results from this section, an AASHTO Standard Practice for Predicting the En- durance Limit of Hot Mix Asphalt (HMA) for Long-Life Pave- ment Design was developed and is presented in Appendix A. The draft format includes extrapolation techniques using both the single- and three-stage Weibull functions. Existence of the Endurance Limit Samples tested below the fatigue endurance limit are ex- pected to have an essentially infinite fatigue life. As noted previously, testing was only conducted to 50 million cycles. Therefore, the failure point of these samples needed to be extrapolated. Two techniques were used to extrapolate the stiffness versus loading cycle data, the single- and three-stage Weibull functions. Additionally, the data were analyzed using the RDEC procedures to determine the plateau value. The fol- lowing sections present the results from the testing and dis- cuss each of the analyses. Tables 4.3 through 4.6 present the data collected in Phase I of the study. 32 Figure 4.14. Comparison of exponential and single- stage Weibull extrapolations with measured fatigue lives. 0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08 0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08M ea su re d Fa tig ue L ife , C yc le s Predicted Fatigue Life, Cycles Extrapolations from 4 Million Cycles Exponential Single-Stage Weibull Sample 5 Weibull Notes: 1Failure extrapolated. Testing suspended at 58% of initial stiffness at 57,000 due to computer problem. 2Software froze, apparently due to error writing to network drive. Sample stiffness 3,439 MPa, at 53.4% of initial stiffness. Result extrapolated using linear regression of latter cycles. 3Results extrapolated using single-stage Weibull model. 4Less than 8.57E-9 proposed by Shen and Carpenter (40) as indicative of long-life pavement. Extrapolated Cycles to 50% Initial Stiffness Beam ID Air Voids, % Initial Flexural Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull PV Cycles to 50% Initial Stiffness Average Cycles to Failure 18 6.6 5,175 800 6,000 NA NA 3.66E-5 6,000 3 6.8 4,686 800 7,130 NA NA 2.06E-5 7,130 7 7.4 4,522 800 6,000 NA NA 2.63E-5 6,000 6,377 10 6.8 5,153 400 246,220 NA NA 6.25E-7 246,220 46 7.0 5,239 400 57,000 NA NA 2.24E-7 267,8081 1 7.0 5,868 400 242,380 NA NA 3.17E-7 242,380 252,136 2 6.6 5,175 200 26,029,000 NA NA 5.33E-94 26,029,000 6 7.2 6,435 200 12,930,000 NA NA 6.19E-94 14,537,1862 21 7.4 6,240 200 20,771,580 NA NA 6.35E-94 20,771,580 20,445,922 5 6.7 4,519 170 34,724,500 NA NA 2.30E-94 34,724,500 23 6.8 5,645 170 60,000,000 1.04E+08 9.16E+07 5.37E-104 1.04E+083 69,362,250 4 6.7 6,602 100 50,000,000 5.49E+09 5.52E+09 9.25E-154 5.49E+093 13 7.4 5,059 100 50,000,000 3.00E+08 1.04E+11 6.37E-164 3.00E+083 2.90E+09 Table 4.3. Granite 19.0 mm NMAS mix with PG 67-22 at optimum asphalt. equality is underestimated and below the line of equality is overestimated. The error is greater for larger extrapolations (e.g., testing to 10 million cycles for a sample with a fatigue life of 40 million cycles). The extrapolations for the single-stage Weibull model are distributed around the line of equality. As noted previously, the fatigue life for Sample 5 is significantly overestimated. However, the single-stage Weibull function appears to give the most reasonable extrapolation of fatigue test results. When looking at the accuracy of fatigue predictions, it should be considered that strain versus fatigue life data typ- ically is looked at on a log-log plot. The log of 26 million, the measured fatigue life for Sample 2, is 7.41, while the log of 56 million, the fatigue life estimated based on the first

Notes: 1Not included in average. 2Results extrapolated using single-stage Weibull model. 3Less than 8.57E-9 proposed by Shen and Carpenter (40) as indicative of long-life pavement. 4Sample did not fail, extrapolated using single-stage Weibull model. Extrapolated Cycles to 50% Initial Stiffness Beam ID Air Voids, % Initial Flexural Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull PV Cycles to 50% Initial Stiffness Average Cycles to Failure 8 3.7 3,520 800 252,450 NA NA 2.61E-07 252,4501 5 3.0 5,451 800 32,520 NA NA 3.12E-06 32,520 14 3.3 5,764 800 63,580 NA NA 1.09E-06 63,580 48,050 1 2.8 5,532 400 2,860,000 NA NA 1.17E-08 2,860,000 4 3.0 5,532 400 9,655,000 NA NA 2.05E-093 9,655,000 6,257,500 10 3.6 4,308 300 39,624,000 NA NA 1.71E-093 39,624,000 2 3.5 5,427 300 8,811,8104 4.88E+7 5.63E+09 1.84E-133 4.88E+72 12 3.5 4,105 300 20,080,7504 1.47E+8 2.46E+10 7.33E-173 1.47E+82 11 4.0 5,162 300 50,000,000 6.75E+7 4.50E+09 1.25E-133 6.75E+72 7.57E+07 13 3.1 6,841 200 50,000,000 5.96E+9 6.79E+11 2.22E-183 5.96E+92 9 3.1 5,609 200 50,000,000 3.10E+10 1.58E+16 0.00E+003 3.10E+102 1.85E+10 Table 4.6. Granite 19.0 mm NMAS mix with PG 76-22 at optimum plus asphalt. 33 Notes: 1Results extrapolated using single-stage Weibull model. 2Less than 8.57E-9 proposed by Shen and Carpenter (40) as indicative of long-life pavement. Extrapolated Cycles to 50% Initial Stiffness Beam ID Air Voids, % Initial Flexural Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull PV Cycles to 50% Initial Stiffness Average Cycles to Failure 4 7.2 3,025 800 42,240 NA NA 4.02E-06 42,240 7 6.7 5,445 800 10,080 NA NA 1.58E-05 10,080 26,160 2 7.1 4,191 400 3,609,470 NA NA 1.66E-08 3,609,470 8 6.6 4,976 400 591,770 NA NA 2.87E-07 591,770 13 7.2 3,675 400 791,960 NA NA 2.15E-07 791,960 1,664,400 1 7.3 4637 250 14,837,450 NA NA 3.30E-08 14,837,450 11 6.8 4148 250 50,000,000 2.91E+09 1.31E+15 2.22E-182 2.91E+091 NA 5 6.7 4,460 200 50,000,000 2.75E+09 1.53E+11 0.00E+002 2.75E+091 3 7.1 4,062 200 50,000,000 2.68E+09 2.61E+21 0.00E+002 2.68E+091 2.72E+09 Table 4.4. Granite 19.0 mm NMAS mix with PG 76-22 at optimum asphalt. Notes: 1Results extrapolated using single-stage Weibull model. 2Testing conducted by Rutgers University on an IPC Global fatigue device. 3Tested on Asphalt Institute IPC Global fatigue device. 4No solution. Extrapolated Cycles to 50% Initial Stiffness Beam ID Air Voids, % Initial Flexural Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull PV Cycles to 50% Initial Stiffness Average Cycles to Failure Cox & Sons fixture in Interlaken Load Frame, except as noted 8 3.0 5,054 800 15,464 NA NA 15,464 14 3.2 5,306 800 34,500 NA NA 34,500 24,982 10 3.2 5,896 400 468,343 NA NA 468,343 15 3.3 6,698 400 338,121 NA NA 338,121 403,232 9 3.4 6,094 200 10,000,000 24,944,621 1.14E+08 24,944,6211 42 3.5 6,923 200 38,985,510 NA NA 38,985,510 13 3.8 6,219 200 50,000,000 1.23E+08 9.95E+07 1.23E+081 62,310,044 IPC Global fatigue device 6 4.7 6,862 800 5,570 NA NA 4.17E-05 5,570 3 4.1 7,472 800 5,230 NA NA 3.99E-05 5,230 5,400 7 5.1 7,675 400 131,390 NA NA 1.49E-06 131,390 4 4.9 7,653 400 57,840 NA NA 6.26E-06 57,840 94,615 2 4.7 7,512 200 3,584,740 NA NA 1.58E-07 3,584,740 3,584,740 6a 3.3 8,605 100 15,350,090 5.81E+08 NA4 NA 5.81E+081 5.81E+08 Table 4.5. Granite 19.0 mm NMAS mix with PG 67-22 at optimum plus asphalt.

PG 67-22 Mix at Optimum Asphalt Content The results for the PG 67-22 mix tested at optimum asphalt content were presented in Table 4.3. The PG 67-22 mix at opti- mum asphalt content was tested by NCAT. The extrapolations shown in Table 4.3 are based on the single- and three-stage Weibull functions, as well as the RDEC. Sample 23, tested at 170 ms, produced Nf of 1.04E+08 and 9.16E+07 using the single- and three-stage Weibull functions, respectively. Sam- ples 4 and 13, tested at 100 ms produced extrapolated Nf using the single-stage Weibull function of 5.49E+09 and 3.00E+08, respectively. Although both of these numbers rep- resent extraordinarily long fatigue lives, Table 4.3 indicates that Sample 13 would be expected to have a longer fatigue life. The single-stage Weibull function fits for Samples 4 and 13 was previously presented in Figure 4.4. As shown in Figure 4.4, the slope of the data decreases above approximately 1 million loading cycles, indicating less damage. The best fit line for the Weibull function for Sample 13 has a steeper slope, resulting in the prediction of a shorter fatigue life. Tsai et al. (57) developed a three-stage Weibull function to more accurately model the changes in slope observed in the data. As discussed previously, three regions can be observed with the Weibull function. In the third region, damage can either increase rapidly—leading to failure—or decrease as observed for Sample 13. The three-stage Weibull results for Sample 13 are shown in Figures 4.15 and 4.16. The three- stage Weibull function resulted in Nf predictions of 5.52E+09 and 1.04E+11 for Samples 4 and 13, respectively. Figure 4.17 shows a log-log plot of cycles to failure versus strain. For samples that did not fail within 50 million cycles, extrapolations are shown using single-stage Weibull, three- stage Weibull, and RDEC. The data from 800 through 170 ms were used to fit the regression line. A fatigue life of 60 million cycles was assigned to Sample 23, tested at 170 ms (the actual number of cycles tested). Ninety-five percent confidence limits are shown for the regression line and 95% prediction intervals are shown from 170 through 50 ms. Although the three-stage Weibull extrapolation of Nf for Samples 23 and 4 fall on the upper prediction limit, the fatigue life estimate at 100 ms for Sample 13 indicates a deviation from the log-log regression line, which in turn indicates the existence of an endurance limit between 100 and 170 ms. Recall that 170 ms was selected to produce a beam fatigue life of 50 million cycles, or approximately 500 million load repetitions in the field. Based on Sample 23’s deviation from the prediction limits in Figure 4.17, this strain level appears to be close to the endurance limit, but slightly high. Nf was substituted in the regression as the predictor for strain level and the regression re-run, resulting in Equation 17, as follows: Ninety-five percent prediction limit, in terms of strain, was calculated for N = 50 million cycles. The lower predic- tion limit for Equation 17 at 50 million cycles was 151 ms. ε = ×( )−10 173 54 0 170. . ( )N 34 Figure 4.15. Three-stage Weibull function for Sample 13, PG 67-22 at optimum. -12 -10 -8 -6 -4 -2 0 0 2 4 6 8 10 12 14 16 18 20 Ln (- Ln (S tif fn es s R ati o) ) Ln (Loading Cycles) Raw Data Fit

pavements. Although all three strain levels appear to provide a long fatigue life, 170 ms appears to be at, or slightly above, the endurance limit based on the other analyses. The recom- mended plateau value may not define the endurance limit, but rather a long fatigue life. Figure 4.18 shows the relationship between cycles to failure and plateau value. The relationship for low strain tests devel- oped by Shen and Carpenter (40) based on testing 602 beams 35 Figure 4.16. Three-stage Weibull fit for Sample 13, PG 67-22 at optimum. 0 1,000 2,000 3,000 4,000 5,000 6,000 0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000 60,000,000 St iff ne ss , M Pa Loading Cycles Raw Data Fit Figure 4.17. Cycles to failure versus strain for PG 67-22 at optimum. R² = 0. 99 1 10 100 1000 1 100 10000 1000000 100000000 1E+12 1E+10 1E+14 M ic ro -S tra in Cycles to Failure (50% Stiffness) Measured Single-Stage Weibull Function 3-Stage Weibull Function RDEC Confidence Limits Prediction Limits Sa mp le 13 Sa mp le 4 Sam pl e 23 By using the lower prediction limit, the strain level resulting in 50 million cycles should be below the endurance limit for the PG 67-22 mix at optimum asphalt content. The RDEC plateau values were calculated for each of the PG 67-22 at optimum samples. The results are shown in Table 4.3. The samples tested at 200, 170, and 100 ms produce plateau values lower than the critical value, 8.57E-9 recom- mended by Shen and Carpenter (40) as indicative of long-life

36 Figure 4.18. Cycles to failure versus plateau value. y = 0.2937x-1.05 R2 = 0.999 y = 0.2871x-1.0793 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 1E-09 1E-08 1E-07 1E-06 1E-05 0.0001 0.001 0.01 0.1 1 1 100 10000 1000000 10000000 0 1E+10 1E+12 1E+14 Cycles to 50% Initial Stiffness Pl at ea u Va lu e NCAT NCHRP 9-38 Shen and Carpenter Power (NCAT NCHRP 9-38) Power (Shen and Carpenter) Figure 4.19. Cycles to failure versus strain for PG 76-22 at optimum. R² = 0.92 1 10 100 1000 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 1.E+14 1.E+16 1.E+18 1.E+20 1.E+22 M ic ro -S tr ai n Cycles to Failure (50% Stiffness) Measured Three-Stage Weibull Single-Stage Weibull 95% Confidence Interval 95% Prediction Interval Sample 5 Sample 3 Sample 11 is shown for comparison. The relationships are very similar. This would support Shen and Carpenter’s proposal that there is one relationship between cycles to failure and plateau value for all mixes, regardless of the manner of testing. PG 76-22 at Optimum Asphalt Content The results for the PG 76-22 mix tested at optimum asphalt content were presented in Table 4.4. The PG 76-22 mix at optimum asphalt content was tested by NCAT. The extrap- olations shown in Table 4.4 are based on the single-stage Weibull function. Extrapolations were also conducted using the three-stage Weibull function and RDEC. Sample 2 was evaluated as a potential outlier using the repeatability data developed in Phase II. The acceptable difference between two results is estimated to be 0.69 (on a log basis), while the differ- ence between Sample 2 and Sample 13 is 0.66 on a log basis. This indicates that Sample 2 is within acceptable variation. Sample 2 increased the variability of the data, producing an R2 = 0.92 and resulting in larger prediction and confidence intervals (Figure 4.19). However, both points at 200 ms and one point at 250 ms indicate a deviation from the log-log plot

of cycles to failure versus strain, indicative of the endurance limit. Similar to the PG 67-22 results at optimum, the regres- sion was reversed to solve for the strain level that would produce 50 million and 100 million cycles. The lower 95% prediction interval indicated a strain level of 146 ms to produce 50 mil- lion cycles to failure. This strain level is slightly lower than that determined for the PG 67-22 at optimum even though the endurance limit appears to be at a higher value (between 200 and 250 ms). This is due to the increased variability in the testing. One sample at 250 ms and both samples at 200 ms pro- duced plateau values less than the critical value indicated by Shen and Carpenter to be indicative of long fatigue life. PG 67-22 at Optimum Plus Asphalt Content Testing for the PG 67-22 mix at optimum plus asphalt con- tent was initially conducted on the Asphalt Institute’s Inter- laken servo-hydraulic frame using a Cox & Sons fixture. Due to problems with the Interlaken system, low-strain beams were later tested on IPC Global beam fatigue devices operated by both the Asphalt Institute and Rutgers University. Due to concerns about possible differences caused by the various machines, it was decided to retest the cells using the Asphalt Institute’s IPC Global beam fatigue device. The initial results from the three machines and the retests using the IPC Global machine were presented in Table 4.5. The fatigue lives for the retests are significantly shorter than for the original mix. A number of factors appear to con- tribute to this difference. The initial stiffness for the original set of beams averaged 6,027 MPa; the initial stiffness of the replacement beams averaged 7,435 MPa. The beams were prepared at air voids contents outside of the tolerance for the optimum plus target (3.3 ± 0.5%) and used NCAT’s lab stock PG 67-22 binder instead of the dual graded PG 64/67-22 binder, which was used at the NCAT Test Track. The SHRP A-404 surrogate fatigue model (Equation 18) (18) was used to assess whether the differences in initial stiff- ness and voids filled with asphalt (VFA) would tend to cause the degree of observed difference in the measured fatigue lives. VFA would be lower due to the higher air voids. where, Nf = fatigue life, VFA = voids filled with asphalt, percent, ε0 = initial strain, and S″0 = initial loss stiffness, psi. Table 4.7 shows the predicted fatigue lives using Equa- tion 18 and the actual and predicted percent difference between the two sets of beams. The SHRP A-404 surrogate fatigue equation fairly accurately predicts the percentage reduction in fatigue life at 800 ms (estimated 70% versus actual 78%). It underestimated the reduction at the lower strain levels. All of the fatigue lives predicted using the SHRP A-404 surrogate model are considerably lower than the measured values. Sen- sitivity analyses indicated that most of the effect was due to the increased initial stiffness. For the PG 76-22 mixes, initial stiff- ness increased with the lower air voids determined for the optimum plus samples. Comparisons between the asphalt content and gradation of randomly selected beams indicated no significant differences. There were also no differences in the environmental chamber temperature of measured stiff- ness of a plastic test beam to support the change in initial stiff- ness. Therefore, it is felt that the difference in initial stiffness must be attributed to the different binders used to produce the beams. Figure 4.20 shows a log-log plot of cycles to failure versus strain. Sample 1 from the first set at 200 ms and Sample 6a from the second set at 100 ms show deviations indicative of the endurance limit from their respective best-fit lines. Based on the first data set, the 95% lower prediction interval for a fatigue life of 50 million cycles is 158 ms, which is essentially the same as that determined for the PG 67-22 at optimum (151 ms). PG 76-22 at Optimum Plus Asphalt Content The results for the PG 76-22 mix tested at optimum plus asphalt content were presented in Table 4.6. The PG 76-22 N VFA Sf = × ( ) ′′( )− −2 738 105 0 077 0 3 624 0 2 7. exp . . .ε 20 18( ) 37 Binder Micro- Strain Initial Stiffness, psi VFA Predicted Nf Predicted Percent Reduction Measured Percent Reduction First Set 800 751,307 76 1,659 Second Set 800 1,039,502 72 504 70 78 First Set 400 913,317 76 12,027 Second Set 400 1,111,587 72 5,180 57 77 First Set 200 929,996 76 141,164 Second Set 200 1,089,540 72 67,442 52 94 Table 4.7. Fatigue life predictions based on SHRP A-404 surrogate model.

mix at optimum asphalt content was tested by NCAT. The extrapolations shown in Table 4.6 are based on the single- stage Weibull function. Extrapolations also were conducted using the three-stage Weibull function and RDEC. Figure 4.21 shows the log-log plot of cycles to failure versus strain. Although a deviation from the regression line is first indicated at 300 ms, particularly for the three-stage Weibull and RDEC extrapola- tions, it is not clear until 200 ms. The 95% lower confidence limit for the endurance limit using the methodology described in Appendix A is 200 ms. This represents an increase as com- pared to both the PG 76-22 at optimum and PG 67-22 at opti- mum plus mixes. Summary of Phase I Observations Regarding Endurance Limit Clear indications of the endurance limit were shown for three of four mixes (not PG 67-22 at optimum plus). The strain level corresponding to the endurance limit appears to be mix dependent. Visually, the endurance limit appears to be more sensitive to binder properties than to asphalt content/air void content. An endurance limit (predicted value, not lower pre- diction interval) of approximately 170 ms was determined for the PG 67-22 mix at optimum asphalt content. The endurance limit for the PG 76-22 mixture appears to be on the order of 38 Figure 4.20. Cycles to failure versus strain for PG 67-22 at optimum plus. R² = 0.96 R² = 0.98 1 10 10 0 1000 M ic ro -S tr ai n Cycles to Failure (50% Stiffness) First Set Measured Three-Stage Weibull Single-Stage Weibull Sample 1, First Set Sample 6a, Second Set 1 100 10000 1000000 100000000 1E+10 Figure 4.21. Cycles to failure versus strain for PG 76-22 at optimum plus. R² = 0.93 1 10 100 1000 1 100 10000 1000000 100000000 1E+10 1E+12 1E+14 1E+16 M ic ro -S tra in Cycles to Failure (50% Stiffness) Measured Three-Stage Weibull Single-Stage Weibull RDEC

220 ms, and approximately 300 ms for the PG 76-22 at opti- mum plus. The plateau value criteria determined by Shen and Carpenter (40) appears to be indicative of very long fatigue life, but not necessarily the endurance limit. The Weibull function appears to be the best technique for extrapolation of low strain stiffness results. One rapid technique for determining the endurance limit may be to test three repli- cates at each of two strain levels, nominally 800 and 400 ms and then fit a log-log relationship between cycles to failure and strain. Testing additional samples at normal strain levels would potentially reduce the estimate of the standard deviation and therefore increase the confidence in the prediction. The predicted endurance limit for the PG 76-22 mix at optimum asphalt content would most likely benefit from testing addi- tional samples. The lower 95% prediction limit for a fatigue life of 50 million cycles appears to be reasonably close to the endurance limit. This technique was originally presented in the Proposed Standard Practice for Predicting the Endurance Limit of Hot Mix Asphalt (HMA) for Long-Life Pavements in Appendix A. This technique was used for the testing con- ducted in Phase II. Appendix A was later modified in an effort to obtain a more accurate estimate of the endurance limit. Phase II Testing to Investigate Additional Binder Grades Testing was conducted in Phase II to evaluate additional binder grades. The 19.0 mm NMAS granite test track mix was replicated using a true grade PG 64-22 and PG 58-28. Three beams were tested at 800 ms and three beams were tested at 400 ms for each mixture. The fatigue testing was conducted by NCAT. The endurance limit was estimated using the one- sided 95% lower prediction interval for a strain level corre- sponding to 50 million cycles according to the methodology described in Appendix A. The lower 95% prediction inter- vals were 82 and 75 ms, respectively, for the PG 58-28 and PG 64-22 mixtures. Confirmation tests for the endurance limit of the mixture using the PG 58-28 binder were carried out at 76 ms due to an error in the t-value used in determin- ing the lower prediction limit. The tests should have been carried out at 82 ms. The test data is presented in Tables 4.8 and 4.9 and shown graphically in Figures 4.22 and 4.23. For the PG 58-28 mixture, the single-stage Weibull extrap- olations for the samples tested at 76 ms indicate fatigue lives that are longer than that predicted from the log-log regression, 39 Table 4.8. Granite 19.0 mm NMAS mix with PG 58-28 at optimum asphalt. Extrapolation Cycles to 50% Stiffness Sample Air Voids, % Initial Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull Cycles to 50% Stiffness Average Cycles to Failure 4 6.8 3,216 800 12,730 NA NA 12,730 8 7.2 3,014 800 8,730 NA NA 8,730 9 7.0 2,974 800 8,830 NA NA 8,830 10,097 2 6.9 3,372 400 166,290 NA NA 166,290 6 7.4 3,424 400 148,090 NA NA 148,090 7 7.4 3,424 400 237,000 NA NA 237,000 183,793 5 7 4,217 76 12,000,000 1.01E+09 1.11E+08 1.01E+091 15 7 4,332 76 12,000,000 1.57E+09 5.30E+09 1.57E+091 16 7 4,706 76 12,000,000 5.04E+08 4.39E+09 5.04E+081 1.03E+09 Note: 1Extrapolated using single-stage Weibull model. Table 4.9. Granite 19.0 mm NMAS mix with PG 64-22 at optimum asphalt. Extrapolation Cycles to 50% Stiffness Sample Air Voids, % Initial Stiffness, MPa Micro- Strain Cycles Tested Single-Stage Weibull Three-Stage Weibull Cycles to 50% Stiffness Average Cycles to Failure 5 7.5 3,635 800 5,580 NA NA 5,580 6 7.5 3,736 800 5,060 NA NA 5,060 8 7.5 4,234 800 5,490 NA NA 5,490 5,377 2 7.3 4,666 400 98,120 NA NA 98,120 4 7.5 4,449 400 111,250 NA NA 111,250 9 7.4 4,227 400 76,760 NA NA 76,760 95,377 3 6.6 5,190 75 12,000,000 8.18E+10 1.08E+09 8.18E+101 7 6.7 4,667 75 12,000,000 3.64E+10 3.18E+08 3.64E+101 10 6.7 5,989 75 12,000,000 2.82E+08 2.82E+08 2.82E+081 3.95E+10 Note: 1Extrapolated using single-stage Weibull model.

but within the prediction limits for the extrapolation. Two of the three-stage Weibull extrapolations exceed the prediction limits, but Sample 16 indicates a shorter fatigue life. The single- and three-stage Weibull fatigue life extrapola- tions for the PG 64-22 mixture samples exceeded the fatigue lives estimated from the log-log plot of cycles to failure ver- sus strain. The extrapolated fatigue lives also exceeded the prediction limits for the log-log regression line. This is a clear indication of the endurance limit. In general, the predicted endurance limits for the PG 58-28 and PG 64-22 binders were lower than what might have been expected based on the Phase I testing. Historically, softer binders are believed to perform better in constant strain tests (see Table 2.1). 40 Figure 4.22. Cycles to failure versus strain for PG 58-28 at optimum asphalt content. R² = 0. 98 1 10 100 1000 10000 1 100 10000 1000000 100000000 1E+10 M ic ro -S tr ai n Cycles to Failure (50% Stiffness) Measured Three-Stage Weibull Single-Stage Weibull Confidence Limits Prediction Limits Figure 4.23. Cycles to failure versus strain for PG 64-22 at optimum asphalt content. R² = 0.99 1 10 100 1000 10000 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 M ic ro -S tr ai n Loading Cycles Measured Confidence Limit Series5 Three-Stage Weibull Prediction Limit Single-Stage Weibull Series7 Power (Measured)

Estimate of Precision of Beam Fatigue Tests In Phase II, a small-scale round-robin was conducted to develop precision estimates for the beam fatigue test and the extrapolation procedures. ASTM C802 recommends a mini- mum of 10 labs for a round-robin study. Due to the testing time commitments involved with beam fatigue testing, only five laboratories, three represented by the research team, and two volunteers, participated. The volunteer labs only tested one mix at three strain levels. The experimental matrix for the mini round-robin is shown in Table 4.10. Variability is expected to consist of four components: materials, sample preparation, beam testing, and analysis. To minimize materials variability, NCAT prepared and batched all of the aggregates. SEM Materials and the University of California participated on a voluntary basis. The directions provided for preparation of the test samples are presented in Appendix F. Each lab mixed and compacted their own beams for testing. The aggregate batches were ran- domized prior to shipping. In some cases, however, additional batches were required in order to obtain beams with the appro- priate air void levels. The samples were short-term oven aged for 4 h at 275°F (135°C) according to AASHTO R30 prior to compaction. The range of equipment used to compact the beams included linear kneading compactors, vibratory com- pactors, and rolling wheel compactors. Testing was conducted in accordance with the draft prac- tice for the determination of the endurance limit, which has since been modified (Appendix A). Fatigue lives were also extrapolated using the logarithmic and RDEC described in Appendix C. Each lab first tested three beams each at 800 and 400 ms. Extrapolations were conducted using each lab’s data to estimate the endurance limit. The average estimated en- durance limit for all of the participating labs was determined. Using a single strain level for the endurance limit allowed re- peatability and reproducibility calculations to be performed on the fatigue life extrapolations. Each lab then tested three beams at the average estimated endurance limit to 12 million cycles. Low strain beams were not tested for the PG 67-22 mix at optimum plus. The average estimated endurance limits were 151, 175, and 188 ms, respectively, for the PG 67-22 at optimum, PG 67-22 at optimum plus, and PG 76-22 at opti- mum asphalt content mixes. The strain levels used for the beams tested to evaluate the low strain extrapolations were 130 and 220 ms, respectively, for the PG 67-22 and PG 76-22 mixes at optimum asphalt content. The differences arose since not all labs had reported their data when the confirma- tion strain levels were selected. The round-robin data were analyzed according to ASTM E691. When analyzing round-robin data, it is desirable to have a constant standard deviation or constant coefficient of varia- tion at the various test (strain) levels. To achieve this, the pre- cision estimates are based on the log base 10 of the actual test results. This is reasonable since fatigue transfer functions uti- lized in pavement design are logarithmic. Three potential outliers were identified based on the h and k statistics utilized in ASTM E691. Only one outlier was removed from the data set, Lab 2’s results for the PG 67-22 at optimum asphalt content samples tested at 120 ms and extrapolated using the logarithmic model. The precision estimates, without the outlier, are presented in Tables 4.11 and 4.12 for the normal strain and extrapo- lated data, respectively. The complete data are presented in Appendix D. An examination of Table 4.11 suggests that the repeatability and reproducibility standard deviations for the normal strain tests are consistent for the different mixes and strain levels. Based on the data in Table 4.11, the log of two properly conducted tests by the same operator should not dif- fer by more than 0.69 with 95% confidence. Similarly, the log of two properly conducted normal strain tests by two differ- ent laboratories should not differ by more than 0.89. For the extrapolation methods shown in Table 4.12, the single-stage Weibull function produces the least variable results, followed by the logarithmic model. The RDEC pro- cedure produces the most variable results. The use of the unique plateau value versus cycles to failure line proposed in Appendix C appeared to create erroneously low fatigue lives with one of the data sets and exceptionally long lives with one of the other lab’s data. Based on the potential for overestimating fatigue life and the associated variability in calculation, the RDEC model is not recommended for extrap- olating endurance limit data. 41 Table 4.10. Mini round-robin testing matrix. Lab/Mix PG 67-22 at Optimum PG 67-22 at Optimum Plus PG 76-22 at Optimum NCAT X X X Asphalt Institute X X X University of Illinois X X X SEM Materials X University of California X

Indirect Tensile Strength as a Surrogate for Endurance Limit Determination Indirect Tensile Strength (IDT)—or, more correctly, the tensile strain at failure from the IDT test—was examined as a surrogate for beam fatigue tests to identify the fatigue endurance limit. The Asphalt-Aggregate Mixture Analysis System (AAMAS) used a mixture’s resilient modulus and ten- sile strain at failure from the IDT test to assess fatigue resist- ance (59). In the AAMAS system, testing was conducted at the following three temperatures: 5°C, 25°C, and 40°C. The mea- sured resilient modulus and tensile strain at failure were com- pared to the properties of a “standard” mix, the dense-graded mix used at the AASHO Road Test. Von Quintus (personal communication) suggested that long-life pavements be designed with tensile strains at the bottom of the asphalt layer that were < 1% of the tensile strain at failure. Maupin and Freeman (9) demonstrated satisfactory correlations between constant strain fatigue-life curves and indirect tensile test results. Samples of the PG 67-22 mixtures at optimum and opti- mum plus asphalt content and PG 76-22 mixtures at optimum and optimum plus asphalt content were compacted in the Superpave Gyratory Compactor (SGC) for IDT testing to the same air void levels used for the beam fatigue tests. The samples were tested in the IDT at 25°C. Testing was conducted according to AASHTO T322. Tensile strain was calculated as described by Kim and Wen (60). Strain was calculated at 98% of the peak stress, and the results are shown in Figure 4.24 versus the predicted and 95% lower confidence limits for the endurance limit determined from the beam fatigue tests. From Figure 4.24, it is apparent the predicted and 95% lower confident interval for the endurance limit are approximately 5 and 3%, respectively, of the indirect tensile failure strain. However, although the indirect tensile test appears to be sen- sitive to the two different binders, it does not appear to be sensitive to binder content. This procedure appears to have some potential to predict the magnitude of the endurance limit. Additional work is necessary with a broader range of materials. 42 Table 4.11. Summary of normal strain round-robin results. Code No. of Labs Log of Average of All Labs Standard Deviation between Log of Cell Averages (Sx) Repeatability Standard Deviation (Sr) Reproducibility Standard Deviation (SR) Between- Lab Standard Deviation of Log of Lab Means (SL) Within-Lab Coefficient of Variation, % Between- Lab Coefficient of Variation, % PG 67-22 Opt. at 800 ms 7 3.876 0.220 0.249 0.300 0.167 6.4 7.7 PG 67-22 Opt. at 400 ms 7 5.370 0.365 0.240 0.414 0.338 4.5 7.7 PG 76-22 Opt. at 800 ms 3 3.932 0.127 0.261 0.261 0.000 6.6 6.6 PG 76-22 Opt. at 400 ms 3 5.624 0.299 0.295 0.384 0.246 5.2 6.8 PG 67-22 Opt.+ at 800 ms 3 4.203 0.107 0.207 0.207 0.000 4.9 4.9 PG 67-22 Opt.+ at 400 ms 3 5.717 0.134 0.243 0.243 0.000 4.2 4.2 Pooled 0.234 0.248 0.318 0.164 5.4 6.8 Table 4.12. Summary of round-robin extrapolations at the estimated endurance limit. Code Number of Labs Log of Average of All Labs Standard Deviation between Log of Cell Averages (Sx) Repeatability Standard Deviation (Sr) Reproducibility Standard Deviation (SR) Between- Lab Standard Deviation of Log of Lab Means (SL) Within- Lab Coefficient of Variation, % Between- Lab Coefficient of Variation, % Logarithmic PG 67-22 Opt. at 130 ms 5 10.609 1.642 1.322 1.965 1.454 12.5 18.5 PG 76-22 Opt. at 220 ms 3 8.209 0.441 1.207 1.207 0.000 14.7 14.7 Pooled 1.192 1.279 1.681 0.909 13.3 17.1 Weibull PG 67-22 Opt. at 130 ms 5 8.794 0.915 0.586 1.032 0.850 6.7 11.7 PG 76-22 Opt. at 220 ms 3 7.563 0.309 0.719 0.719 0.000 9.5 9.5 Pooled 0.688 0.636 0.915 0.531 7.7 10.9 RDEC PG 67-22 Opt. at 130 ms 4 10.275 3.421 2.341 3.919 3.143 22.8 38.1 PG 76-22 Opt. at 220 ms 3 9.607 0.967 2.151 2.151 0.000 22.4 22.4 Pooled 2.369 2.259 3.161 1.796 22.6 31.4

43 Figure 4.24. IDT tensile strain at failure versus beam fatigue endurance limit. y = 0.047x R² = 0.650 y = 0.031x R² = 0.400 0 50 100 150 200 250 300 350 0 1000 2000 3000 4000 5000 6000 En du ra nc e Li m it, M ic ro -S tr ai n Tensile Strain at Failure, Micro-Strain Predicted Endurance Limit 95% Lower Confidence Endurance Limit Linear (Predicted Endurance Limit) Linear (95% Lower Confidence Endurance Limit)

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Validating the Fatigue Endurance Limit for Hot Mix Asphalt Get This Book
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TRB’s National Cooperative Highway Research Program (NCHRP) Report 646: Validating the Fatigue Endurance Limit for Hot Mix Asphalt explores the existence of a fatigue endurance limit for hot mix asphalt (HMA) mixtures, the effect of HMA mixture characteristics on the endurance limit, and the potential for the limit’s incorporation in structural design methods for flexible pavements.

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