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62 All design methods that accept the endurance limit design Survival curves are typically based on age but can also be premise assume that the endurance limit is independent of based on traffic loadings or the probability of exceeding a spe- the mixture and temperature. That endurance limit is the cific level of distress. The age or condition at failure must be tensile strain below which no fracture damage occurs. If an based on a clearly defined condition. Mathematical models endurance limit value was an input into the MEPDG or are best fitted to the points in the survival curves to predict the used within other M-E based design methods, the question probability of survival or failure as a function of age, thick- becomes, what value should be used as the endurance limit? ness, cumulative traffic, or some other pavement feature. The The purpose of this section is to use the LTPP database to try general form of these models for use in life cycle cost analysis and answer three questions related to the endurance limit is as follows (67, 68): design premise, as follows: a Probability of Failure = b( Age -c ) +d (25) 1. Do field observations of alligator cracking support the exis- 1+ e tence of an endurance limit as an HMA mixture property? 2. If the field observations support the endurance limit the- a Probability of Failure = b( ESAL -c ) +d (26) ory or hypothesis, what is the tensile strain below which no 1+ e more alligator cracking has been exhibited? 3. Is the endurance limit independent of mixture type and where, dynamic modulus? Failure = Existing pavement is overlaid or reconstructed, or a specified level of distress has been exceeded; Age = Number of years since construction (new pave- Defining the Endurance Limit-- ment or overlay); A Survivability Analysis ESAL = Cumulative equivalent single 18-kip axle loads since construction (new pavement or overlay), A survivability analysis was used to try to answer the above millions; and questions using the LTPP database. The survivability analysis a, b, c, d = Regression coefficients determined from the completed within this project is an expansion of work com- analysis. pleted using the LTPP database in the mid-1990s. This section of the report describes the use of survival curves in determin- The probability of survival is 1 minus the probability of fail- ing the thickness or level of tensile strain at which only lim- ure. Optimization is typically used to determine the regression ited cracking has occurred over long periods of time. coefficients that best fit the survival points. A survival analysis also can be completed using a specific level of distress and Development and Application pavement response value. In other words, the survival curves of Survival Curves can be used to define the probability that a specific area of alligator cracking will be less than some specified amount for Survival or probability of failure analyses have been used for different HMA thicknesses or tensile strains at the bottom of decades in actuarial sciences. They have also been used in the the HMA layer. pavement industry to determine the expected service life of It is important to note that survival curves for pavements pavement structures for use in life cycle cost analysis, and to are necessarily based on previously built designs, materials, compare the mean and standard deviation of the expected construction, and maintenance. The data used to develop the service life for different design features and site factors in eval- survival rates or probability of failure represent typical con- uating the adequacy of the design procedure (65, 66). Survival struction, materials, mixture designs, and thicknesses that curves are uniquely useful because every point on the curve have been built by agencies within the past time period rep- represents the probability that a given pavement section will resented by the data. These can be defined as "benchmark" be rehabilitated or exceed a specific level of distress. survival curves. Survival analysis is a statistical method for determining the The reliability of a pavement depends on the length of time distribution of lives or "Life Expectancy," as well as the occur- it has been in service and design features and site factors that rence of a specific distress for a subset of pavements. Since not are not properly accounted for in a thickness design proce- all of the pavements included in the analysis have reached the dure. Thus, the distribution of the time to failure of a pave- end of their service life or a specific level of distress, mean ment type or thickness level is of fundamental importance in values can not be used. The age or amount of alligator crack- reliability studies. A method used to characterize this distribu- ing and probability of occurrence are computed considering tion is the failure rate, or rate of occurrence, for a specific level all sections in the subset using statistical techniques. of distress. The failure rate can be defined as follows.
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63 If f(t) is the probability density of the time to failure of a 0.45 given pavement type and thickness, that is, the probability 0.4 Premature Failure Part that the pavement will fail between times t and t + t is given 0.35 Failure Rate by f(t)* t, then the probability that the pavement will fail on 0.3 0.25 the interval from 0 to t is given by 0.2 t 0.15 F ( t ) = f ( t ) dx Wear Out (27) 0.1 Failure Part 0 Chance Failure Part 0.05 0 The reliability function, expressing the probability that it 0 10 20 30 survives to time t, is given by Age of Pavement, years Figure 6.5. Typical failure rate relationship for R (t ) = 1 - F (t ) (28) pavement structures. Thus, the probability that the pavement will fail in the inter- val from t to t + t is F(t + t) - F(t), and the conditional first part, and when failure is a result of multi-distresses probability of failure in this interval, given that the pavement as related to a combination of parameters over time (for survived to time t, is expressed by example, exponential growth increases in traffic, past the design period from which thickness was determined). F ( t + t ) - F ( t ) (29) R (t ) The failure rate can be determined by organizing the per- formance data in terms of the distribution of pavement age Dividing by t, one can obtain the average rate of failure in exceeding a critical level (failure) versus the distribution of the interval from t to t + t, given that the pavement survived age for those pavements exhibiting a value lower than the to time t by critical value. Figure 6.6 shows a typical probability of failure relationship from actual data included in the LTPP database F ( t + t ) - F ( t ) 1 for roughness measured on flexible pavements in the general R (t ) (30) pavement studies (GPS-1 and GPS-2) and special pavement t studies (SPS-1) experiments. GPS-1 sections consist of HMA For small t, one can get the failure rate, which is on granular base. GPS-2 sections consist of HMA on bitumi- nous, hydraulic cement, lime, fly ash, or other pozzolan bound f (t ) f (t ) or stabilized base. SPS-1 sections are part of a strategic study Z (t ) = = (31) of structural factors for flexible pavements. R (t ) 1 - F (t ) Given the above definition of each part of the probabil- ity of failure relationship with time, the failure rate can be The failure rate is expressed in terms of the distribution of defined as failure times. A typical failure rate curve is composed of three parts or can be grouped into three areas, as shown in Figure 6.5 - t Z (t )dx and defined as follows: f ( t ) = Z ( t ) e 0 (32) 1. The first part is characterized by a decreasing failure rate with time and is representative of the time period during Assuming that the failure rate is constant within the second which early failure or premature failures occur. This area or part, and replacing Z(t) with , the distribution of failure times time typically represents pavements that were inadequately is an exponential distribution as shown below. designed or built, using inferior materials. 2. The second part is characterized by a constant failure rate. f ( t ) = [ e - t ] (33) A constant failure rate represents the time period when chance failures occur, or the failure occurs at random with Many survival curves or, conversely, the probability of fail- pavement age. In some survival methods, this area is referred ure, are based on the above relationships and assumptions. to as the useful life of a pavement. Unfortunately, the failure rate within the second part is not 3. The third part is characterized by an increasing failure rate usually constant, and the failure rates for the first and third with time. This area or time represents the reverse of the parts are not inversely proportional to one another. For these
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64 120 100 Probability of Exceeding Roughness Level, % Low Roughness, >1.5 m/km 80 Moderate Roughness, >2.0 m/km 60 40 Excessive Roughness, >2.5 m/km 20 0 0 10 20 30 40 Age, years Figure 6.6. Illustration of results from a survivability analysis or probability of exceeding a critical roughness magnitude. cases, which are typical for pavements, the failure rate can be result in rehabilitation of the roadway (69). The test sections estimated by the following relationship: used in the survival analysis were from the GPS-1 and GPS-2 experiments. Figure 6.7 shows the distribution of pavement Z ( t ) = ( t ) -1 (34) age for the GPS-1 and GPS-2 test sections (LTPP database version 13.1/NT3.1 released in January 2002), and Figure 6.8 Thus, shows the number of test sections with different areas of alli- gator (fatigue) cracking. As shown in Figure 6.8, many of the f ( t ) = ( t ) -1 - t e (35) LTPP test sections have no alligator cracking. Figures 6.9 and 6.10 show the survival curves from the This density function is termed the Weibull distribution, LTPP data for different levels of alligator cracking that would and is typically used in failure analyses. cause some type of rehabilitation activities. As shown, the average life (50% probability) to crack initiation and a low cracking amount (less than 10% of wheel-path area) is 19 LTPP Database to Establish the Initial and 23 years, respectively. Survival Curve A similar survivability analysis was completed by Von A survivability analysis was completed by Von Quintus Quintus in 1995 for a subset of the test sections included in et al. for the Asphalt Pavement Alliance to determine the the GPS-1 and GPS-2 experiments. The test sections were expected age for an amount of fatigue cracking that would randomly selected from the LTPP program for the thicker 50 40 30 No. of Test Sections 20 10 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 = 0 9 27 38 17 29 35 46 39 23 25 27 16 16 17 7 1 Age, years Figure 6.7. Frequency histogram of pavement age at the time of the distress survey for the LTPP GPS test sections included in the Asphalt Pavement Alliance study (69).
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65 180 160 140 120 No. of Test 100 Sections 80 60 40 20 0 50- 100- 150- 200- 300- 400- 0 0-10 13-50 100 150 200 300 400 500 = 177 65 45 27 16 15 14 9 1 Area of Fatigue Cracking, sq. m. Figure 6.8. Histogram of the number of test sections with different levels of alligator cracking (69). 120 Probability of Cracking Level, % 100 Crack Initiations 80 Low Cracking, >10% 60 Moderate Cracking, >20% 40 Excessive Cracking, >50% 20 0 0 10 20 30 40 Age, years Figure 6.9. Graphical illustration of the probability of failure or exceeding a specified area of alligator cracking (69). 60 Area of Fatigue Cracking, % 50 50% Probability of 40 Occurrence 25% Probability of 30 Occurrence 20 75% Probability of Occurrence 10 0 0 10 20 30 40 Age, years Figure 6.10. Probability of occurrence for alligator cracking (69).