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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
This report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWARD02095 
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Performancerelated specifications require mathematical models to link construction quality to expected life and, ultimately, to value expressed in the form of payment schedules. Although ongoing research efforts continue to advance the state of the art, the type of data needed to develop accurate and precise models may not become available for years. In the interim, present engineering and mathematical knowledge can be used to create rational and practical models that can perform effectively until better models are available. Examples are presented to illustrate how both analytical data and survey data can be used to develop realistic performance models useful for the development of payment schedules for QA specifications. The issue of the proper method to combine the effects of multiple deficiencies is also addressed.
The RQL is essentially defined as a severely deficient level of quality at which the agency reserves the option to require removal and replacement of the construction item. In the HMAC pavement specification for one agency, the RQL for both air voids and thickness had been defined in terms of percent defective as PD > 75. In other words, if either air voids or thickness exhibited this level of quality, the lot could be declared rejectable. However, this leads to the inconsistency shown in table 44.
Case  Quality Level  Rejectable?  

Air Voids 
Thickness 

1 
PD = 0 (Excellent) 
PD = 75 (RQL) 
Yes 
2 
PD = 75 (RQL) 
PD = 0 (Excellent) 
Yes 
3 
PD = 74 (Almost RQL) 
PD = 74 (Almost RQL) 
No 
Clearly, case 3 is far worse than the other two but, under the existing system, it would not trigger the RQL provision whereas the first two cases would. Defining the RQL in a more appropriate way rectified this inconsistency. Intuitively, if both air voids and thickness reach some intermediate value less than PD = 75, say PD_{VOIDS} = PD_{THICK} = 50, for example, then that might logically be just as detrimental as PD_{VOIDS} = 0, PD_{THICK} = 75 or PD_{VOIDS} = 75, PD_{THICK} = 0. To illustrate how a more suitable RQL provision can be developed, suppose the agency has determined that the conditions listed in table 45 are all likely to severely shorten the life of the pavement and, therefore, are appropriate RQL points.
Quality Level 
Rejectable? 


Air Voids 
Thickness 

PD = 75 
PD = 10 
Yes 
PD = 10 
PD = 75 
Yes 
PD = 50 
PD = 50 
Yes 
By plotting these three points on a graph, as illustrated by model #1 in figure 61, it can be seen that the model must be in the form of a curve that is concavedownward.
Since the purpose of this model is to account for the combined effect of air voids and thickness, a simple way to accomplish this is to include the crossproduct term in the RQL provision given by equation 86.
C_{1}(PD_{VOIDS}) + C_{2}(PD_{THICK}) + C_{3}(PD_{VOIDS} x PD_{THICK}) > 100  (86) 
where: PD_{VOIDS} = air voids percent defective.
PD_{THICK}= thickness percent defective.
C_{i} terms = coefficients to be determined.
The threshold value of 100 on the righthand side of equation 86 that triggers the rejection provision is chosen arbitrarily and could be any convenient value. To determine the three coefficients C_{1}, C_{2}, and C_{3}, the three predetermined points in table 45 are substituted into equation 86 to obtain equations 87 through 89, thus providing three equations with three unknowns.
75 C_{1} + 10 C_{2 }+ 750 C_{3 } = 100  (87) 
10 C_{1} + 75 C_{2} + 750 C_{3 } = 100  (88) 
50 C_{1} + 50 C_{2} + 2500 C_{3 } = 100  (89) 
Solving these simultaneous equations, and substituting the values of the coefficients back into equation 86, produces equation 90, which is plotted as model #1 in figure 61.
1.273 PD_{VOIDS} + 1.273 PD_{THICK}  0.0109 (PD_{VOIDS} x PD_{THICK})>100  (90) 
To demonstrate that the model can be made to bend the other way, if desired, and that greater weight can be put on one property, air voids for example, table 46 presents a slightly different set of assumptions that might have been used.
Quality Level 
Rejectable? 


Air Voids 
Thickness 

PD = 75 
PD = 10 
Yes 
PD = 10 
PD = 90 
Yes 
PD = 40 
PD = 40 
Yes 
Solving the simultaneous equations generated by this data set produces equation 91, which has been plotted as model #2 in figure 61. By defining the rejectable level for thickness at a lower level of quality than the rejectable level for air voids, the coefficient of the thickness term in equation 91 has been reduced, thus giving air voids greater weight in this example.
1.076 PD_{VOIDS} + 0.847 PD_{THICK} + 0.0144 (PD_{VOIDS} x PD_{THICK})>100  (91) 
Note that the coefficient of the cross product term in equation 90 is negative, producing a model that is concavedownward, while the positive cross product coefficient in equation 91 produces a concaveupward model. If there were no cross product term, i.e., if coefficient C_{3} in equation 86 were zero, the model would plot as a straight line. Because an equation of this form can produce any of these three shapes, it can be very effective as a performance model when two quality characteristics are involved. The specific application will dictate which shape is appropriate.
The method that is discussed in this and following sections is applied to the example of using air voids and thickness as acceptance measures for HMAC pavements. However, the concept that is presented is appropriate for both HMAC and PCC and for other acceptance measures, provided a method exists for estimating the pavement lives for various levels of the quality measure.
A highway agency can use whatever model or other method with which it is comfortable to arrive at the estimated lives for the asconstructed pavements. The methods used herein are only examples of possible approaches that can be used. If a performance model has been developed by the agency, then it may be possible to use this model to directly arrive at the expected pavement life for any combination of values for the variables included in the model.
Returning to the example using air voids and thickness, to derive a mathematical
performance model it will be necessary to have reasonably accurate values of
expected life for the four conditions indicated in table 47. The values in this
table were obtained by an agency using a simplified computer model that it has
developed. The first value is obtained by assuming that the expected life of
the pavement will equal the design life of 20 years if both air voids and thickness
are at the AQL of 10 PD. Next, using the results obtained with the agency's
computer model, expected lives of approximately 10 years each are obtained for
the cases in which either air voids or thickness is at the indicated poor level
of quality while the other measure is at the AQL. Finally, a method must be
found to estimate the joint effect of poor quality in both air voids and thickness
to be able to complete the table. In the absence of actual data obtained under
controlled field conditions, the agency decided that a survey of experienced
pavement engineers would be necessary to estimate this missing piece of information.
Table 47. Preliminary Performance Matrix of Expected Life Values for HMAC Pavement Under NJDOT Conditions
Air Voids Quality

Thickness Quality 


PD = 10 
PD = 90 

PD = 10 
20 yrs 
10 yrs 
PD = 75 
10 yrs 
? 
Figure 62 shows a completed survey questionnaire of the type that was sent by the agency to the chief engineer (or equivalent position) of all State transportation departments. A brief cover letter described the purpose of the survey and requested that it be forwarded to those individuals having extensive experience in the performance of HMAC pavements. Respondents were asked to estimate the expected life for seven different combinations of pavement quality under the assumption that acceptable quality in all three measures would result in the pavement providing the design life of 20 years.
Responses were received from 35 States, of which 4 indicated that this information was not available. Of the remaining responses, another five were excluded because some of the estimates of expected life were inconsistent with the assumption that, in a rational model, a large decrease in quality of any one parameter with the other parameters held constant would result in a corresponding decrease in expected life (provided it was not already zero). This left a total of 26 responses for the analysis, the averages of which appear on the survey questionnaire in figure
62.
The two matrices on the survey questionnaire provide two opportunities to examine how the effects of deficiencies in air voids and thickness should be combined based on the responses. Three different approaches were examined  additive, average, and product models  and the results are presented in table 48. It can be seen for case 1 in figure 62 that the effect of a change from good to poor in airvoids quality, with the other quality levels held constant, can be expressed as a decrease of 20  11.6 = 8.4 years, or as a ratio of expected life to design life of 11.6 / 20 = 0.58. Similarly, the effect due to thickness alone in case 1 is 5.0 years or a ratio of 0.75. Similar results are obtained for case 2.
If the effects were truly additive, the predicted life resulting from poor quality in both air voids and thickness for case 1 would be 20  8.4  5.0 = 6.6 years, as indicated in the sixth column in table 48. If the effects were averaged, the predicted life would be 20  (8.4 + 5.0) / 2 = 13.3 years, which appears in the seventh column of table 48. By the product method, the predicted life would be the product of the individual ratios and the design life, or 0.58 x 0.75 x 20 = 8.7 years, which appears in the eighth column of table 48. For case 2, the average response for good quality of 16.1 is used in place of the design life.

Figure 62. Average Results Obtained with Survey Questionnaire for HMA Pavement Performance 
Case 
Effect on Expected Life Due to Change from Good to Poor Quality ^{1} 
Combined Predicted Life, yrs (Three Combining Methods) 
Expected Life Based on Survey, yrs.  

Air Voids 
Thickness  
Years 
Ratio 
Years 
Ratio 
Add 
Average 
Product  
1 
8.4 
0.580 
5.0 
0.750 
6.6 
13.3 
8.7 
8.7 
2 
6.8 
0.578 
4.2 
0.739 
5.1 
14.5 
6.9 
6.8 
^{1 }Computed from survey results in figure 62
To judge which method is most appropriate, the last column of this table lists the average values estimated by the respondents of the survey. By comparing the values for predicted life with those in the last column, it is seen that the product method produces an almost exact agreement with the survey values, indicating that this method provides a good approximation of the manner in which experienced engineers believe the effects of multiple deficiencies manifest themselves. The average model greatly underestimates the expected loss of service life in this example, while the additive model, although overestimating the expected loss of service life, produces estimates that are reasonably close to the survey results.
These results suggest how a reasonable estimate for the missing value in the performance matrix in table 47 can be obtained. The values in table 47 indicate that poor quality in either air voids or thickness results in a ratio of expected life to design life of 10 / 20 = 0.50. Therefore, by the product method, the expected life when both air voids and thickness are at the indicated poor values is 0.50 x 0.50 x 20 = 5 years, completing the performance matrix as shown in table 49. Based on these four values for expected life, it is possible to develop a realistic performance model, and also to determine the appropriate equation form for the RQL provision discussed earlier.
It is once again stressed that the above approach is simply one example of how these estimated pavement lives could be obtained. Any method with which the agency is comfortable can be used to develop the values for estimated pavement resulting from various levels of the quality measure. For example, if a performance model is available, and the highway agency has confidence in the predictive capability of the model, then it could be used to develop the estimated pavement expected lives. As noted, such performance models are under development but may be a number of years away from widespread use. These models tend to be quite complicated, and will not likely use typical quality measures such as PWL or PD as input variables.
Air Voids Quality 
Thickness Quality 


PD = 10 
PD = 90 

PD = 10 
20 yrs 
10 yrs 
PD = 75 
10 yrs 
5 yrs 
Equation 92, patterned after the general form for the RQL provision in equation 86, is a practical model for a performance equation based on two quality characteristics. The expected pavement life in years is designated by EXPLIF, and all other terms are as previously defined.
EXPLIF = C_{0} + C_{1} (PD_{VOIDS}) + C_{2} (PD_{THICK}) + C_{3} (PD_{VOIDS} x PD_{THICK})  (92) 
The values for expected life in table 49 are used to develop four simultaneous equations that can be solved to provide the four equation coefficients. These are then substituted back into equation 92 to produce the performance model given by equation 93.
EXPLIF = 22.9  0.163 PD_{VOIDS}  0.135 PD_{THICK} + 0.000961 (PD_{VOIDS} x PD_{THICK})  (93) 
It can be seen by inspection that this equation predicts that excellent quality (PD_{VOIDS} = PD_{THICK} = 0) will extend pavement life beyond its design life of 20 years to almost 23 years. It can also be readily calculated that the worst possible quality level in terms of percent defective (PD_{VOIDS} = PD_{THICK} = 100) produces an expected life of 2.7 years. Both of these are reasonable results based on the information that is available. As a further check of the derivation of equation 93, the four combinations of quality levels listed in table 49 can be entered into this equation to demonstrate that it returns the table 49 values for expected life.
For a clearer picture of the operation of this performance model, figure 63 has been plotted. Here it can be seen that, based on the assumptions listed in table 49, the appropriate shape for the family of curves is concavedownward. This indicates that, of the two possible models for an RQL provision shown in figure 61, the concavedownward model should be selected. If, for example, the agency decided that an expected life of 10 years, or less, was sufficiently detrimental that it warranted outright rejection, then the value of EXPLIF = 10 would be substituted into equation 93, and the results scaled accordingly, to produce the RQL provision given by equation 94. (Equation 94 is very similar to the RQL provision given by equation 90 and plotted as model #1 in figure 61. It is represented by the 10yearlifeline in figure 63.)
1.264 PD_{VOIDS} + 1.047 PD_{THICK}  0.00745 (PD_{VOIDS} x PD_{THICK})>100  (94) 
A method was presented illustrating how analytical and/or survey data can be used to develop a mathematical model to predict pavement performance as a function of acceptance test results. The method involved developing a simple matrix of expected life values that are used to construct a set of simultaneous equations that are solved to derive a simplified practical performance model.
This appendix also contains a summary of a nationwide survey conducted to determine the appropriate way to estimate the combined effect of multiple deficiencies. For this particular combination of quality characteristics (air voids and thickness), the analysis suggests that the combined effect is close to the sum of the individual effects, and appears to be best represented by the product of the ratios of the individual effects. A third method, based on the average of the individual effects, substantially underestimated the expected loss of service life.