Cover Image

Not for Sale



View/Hide Left Panel
Click for next page ( 21


The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 20
20 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 40-year period. For the practical definition of the endurance the same mixture in Phase II. Evidence of the existence of a limit, a 40-year 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 full-factorial 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 experiment--binder grade and asphalt content. Binder used to estimate the number of loading cycles, Nf, correspon- grade was varied at two levels: PG 67-22 and PG 76-22. As ding to 50% of initial stiffness. Ideally, a method of extrap- noted previously, the PG 67-22 also met the requirements olation would be identified that could be used to shorten of a PG 64-22. 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

OCR for page 20
21 Table 4.1. Experimental design. NMAS (mm) Binder Content Granite PG 67-22 PG 76-22 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, three-stage 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 67-22 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 67-22 mix at optimum asphalt content Sample 5 of the PG 67-22 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 67-22 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.

OCR for page 20
22 PG 67-22 Sample 5 at 170 ms 5000 y = 3970 .6e-5E-08x y = 3927 .2e-2E-08x y = 3890 .5e-1E-08x 4500 R = 0.4992 R = 0.4776 R = 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 67-22 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 67-22 at optimum, Sample 5.

OCR for page 20
23 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 R = 0.965 R = 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 67-22 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 67-22 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.

OCR for page 20
24 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 67-22 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 Three-Stage 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. single-stage 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 single-stage 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 three-stage 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 single-stage 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 three-stage Weibull curve. Tsai et al. (57) theorized that a 0 y = 0.243x -5.816 -1 R = 0.82 -2 Ln(-Ln(Stiffness Ratio)) -3 -4 y = 0.4014x -8.2141 -5 R = 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 67-22 at optimum 100 ms samples.

OCR for page 20
25 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 76-22 Optimum Plus Sample 9 200 ms 0 y = 0.0667x -2.6913 R = 0.7617 -1 y = 0.1769x -4.1281 R = 0.9075 3 Ln( -Ln(Stiffness Ratio)) 1 -2 y = 0.5427x -6.6613 2 R = 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. Three-stage Weibull curve definitions.

OCR for page 20
26 PG 76-22 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. Three-stage Weibull fit to transformed data. replacing genes, continues until the specified number of gen- for Long-Life Pavement Design. The NCHRP 9-38 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 three-stage 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 three-stage Predicting the Endurance Limit of Hot Mix Asphalt (HMA) Weibull fit. This methodology provides a good fit to both PG 76-22 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 3-Stage Weibull Fit Figure 4.7. Three-stage Weibull fit to stiffness data.

OCR for page 20
27 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 stress-strain 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 8.57E-9 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 Long-Life 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.

OCR for page 20
28 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 energy-loading 0.6 y = 3.4255x -0.164 cycle curve. 0.4 R = 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 energy-loading 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 DE-LC curve and the defined failure point, Nf50. energy-loading cycle relationship follows a power law rela- For long-life 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 R-square value, strated a unique PV-Nf curve for all HMA mixes at normal and (2) correct trend of the DE-LC 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 stiffness-loading 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 long-term 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 = 0.7162x-0.1384 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.

OCR for page 20
29 ship between PV-Nf is also valid for low strain testing. In other The unique PV-loading cycle curve is also illustrated in words, the PV-Nf 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 log-log plot and determine the intersection with low strain levels, and the results are presented in Figure 4.11. the unique PV-Nf 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 DE-LC curve. Two PV-Nf 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 energy-loading 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 energy-loading 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 DE-LC 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 log-log plot of strain versus cycles to failure it is more representative for the actual long-term fatigue would indicate the strain level corresponding to the endurance performance. limit (this will be shown later in the section on Existence of 1.E-06 1.E-09 1.E-12 Endurance Limit 1.E-15 1.E-18 highest R^2 fitting y = 0.0708x-1.024 PV at Nf50 1.E-21 R = 0.9956 Visual fitting: 1.E-24 y = 0.3553x-1.101 1.E-27 R = 0.9989 1.E-30 Unique PV-Nf curve: PV=0.4428Nf^(-1.1102) 1.E-33 1.E-36 1.E-39 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 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) Figure 4.11. PV-Nf curve for IDOT03 mix at low strain with error bars indicated.

OCR for page 20
30 120 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 67-22 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 67-22 The samples tested at 100 ms for the PG 67-22 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 67-22 Sample 13.

OCR for page 20
31 is then shown as a percentage of the measured stiffness on loading cycles and the first 10 million loading cycles. A pre- the y-axis. 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, single-stage Weibull, stiffness (testing would need to be conducted to a high num- three-stage 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 67-22 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 67-22 at sequently, would overestimate the fatigue life. The single-stage 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 three-stage 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 three-stage 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 single-stage 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 67-22 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 67-22 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 76-22 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