Calibration to LTPP pavements

The new transfer function for BSMs was calibrated to describe the deformation trends of in field pavement structures. Fourteen long-term pavement performance study road sections were identified and analysed. These analyses aimed to find input values for the transfer function (retained cohesion, density, stress ratio and permanent strain) and relate them to the number of standard axles the pavement had been subjected to. By comparing the development of permanent strain observed in the field with that of the new transfer function, the transfer function could be calibrated to accurately predict the number of load repetitions required to reach a certain level of permanent deformation.

Three of the four input parameters were determined for each of the LTPP pavements and are shown in Table 4.10. These three parameters were kept constant for each pavement throughout its predicted life. Although this is not the case in field, too little information was available to accurately predict the reduction or increase of these factors over time.

The final parameter used in the transfer function is permanent strain (PS). Permanent strain was set as the controlling factor for this function. PS in this function is a measure of the remaining deformation that the layer can undergo before it is classified as failed. During the calibration of the transfer function, each BSM layer was limited to a PS of 10% of the BSM layer thickness. Therefore, at the start of a pavement’s life it would have accumulated zero deformation and would have 10% deformation remaining before failure. At the end of the BSM’s life, it would have accumulated 10% PS and would have 0% remaining. The use of remaining PS as well as relating it to accumulated PS is discussed in Section 3.7.1and Section3.7.2.

4.5.3.1 Initial constant analysis

The remaining life of each pavement structure was calculated with the new transfer function for each of the pavement structures at over range of PS levels. The PS values from 10% remaining to

0% remaining to failure. This provided an initial set of results for the transfer function which could be compared to the actual deformations (extrapolated with the Huurman model) at a number of points in the life of the pavement. The values obtained from the transfer function were compared to the actual values at specific points for each pavement for the initial values of the constants:

log N = A + B(DSR)3+ C(Pmod.RetC) + D(P S) (4.1)

With: A = 1 B = -60 C = 0.001 D = 0.1

These initial values showed that the transfer function predicted the overall trend of plastic strain accumulation in BSMs. However, the function did not predict the life of the material accurately. Figure 4.31shows the comparison between the NActual and NT F, where NT F was calculated using

the original estimates for the four constants in the function. The regression line through the data points is described by the function y = 2.949x + 2.2732 with an R2

value of 0.9165. The high coefficient of determination indicates that the form of the transfer function predicts the actual data very well. However, the slope of the regression line indicates that in all cases the transfer function overestimates the life of the pavement as can be seen by the deviation from the line of equality. This represents a transfer function with a very low level of reliability.

In Figure 4.31 a red circle indicates non representative values. These values are obtained when PS remaining is set to zero in the transfer function. Due to the logarithmic form of the transfer function and the other non zero variables, it still returns a positive N for zero PS remaining. From a practical point of view, this situation determines the number of load repetitions to failure after the layer has failed. Therefore, it does not represent a practical situation for design.

These values have an effect on the calibration of the transfer function as they raise the interception point of the trendline and in turn reducing the slope. These values do not represent a relevant situation in design, therefore, they were excluded from the calibration process with the further calibration as explained in the next section.

Figure 4.31: NT F vs NActual using initial constant assumptions for LTPP pavements

4.5.3.2 Calibrated constants analysis

The green line in Figure4.31indicates a 1:1 relationship. The aim of the calibration is to change the function’s constant to reach a state where NT F and NActualhave a 1:1 relationship. This relationship

would indicate that the transfer function describes the known data and does not overestimate or underestimate it.

Excel solver was used to find values for the constants A, B, C and D in order to obtain this 1:1 relationship. The linear estimate tool was used to determine the slope, R2

and the residual sum of squares of NT F vs NActual. Solver’s objective was set to find the minimum value for the residual

sum of squares by changing the variables A, B, C and D. These four constants were constrained to be able to use the evolutionary solving method. The slope of the line was controlled by setting a constraint on the slope between 0.975 and 0.125.

Figure 4.32 shows NT F compared to NActual calibrated to fit the 1:1 relationship indicated by the

green line. The trendline plotted through the data points is described by y = 1.0006x + 1.0572 (the dotted line) with a correlation coefficient of R2

= 0.927. The values obtained for the constants used in the transfer function using Excel solver were:

A = 1.204378 B = -57.286 C = 0.0009159 D = 0.086753

Figure 4.32: NT F vs NActual calibrated to 1:1 relationship for LTPP pavements

The slope of the trendline indicates that the values obtained from the transfer function predict the actual data accurately. It was significantly reduced from the value obtained using the initial constants for the function. By adjusting the transfer function to describe the observed data with a one to one relationship, the function predicts the life of the pavement as accurately as possible. This is not applicable for design purposes as pavement design reliability levels are typically around 90%. This initial calibration only aims to describe the observed data and not produce a function to be used for designs. The function was further calibrated to meet different levels of reliability for the purpose of design, the results of which are given in Section4.6.

The correlation coefficient obtained for the relationship in Figure 4.32 of R2

= 0.927 shows a very strong relationship between the actual PS development and the new transfer function. The correlation coefficient obtained after calibration is not significantly higher than the value obtained using the initial constants. The correlation coefficient was therefore used only as an indication of the precision of the transfer function, and not its accuracy or reliability.

The intercept point shown in Figure 4.32 was also significantly reduced from the value obtained using the initial constants. The value reduced from 2.273 to 1.057, which is a significant improvement in the function’s ability to describe the actual data. This reduction was in part caused by the removal of the values obtained from the function for a PS remaining value of 0%.

By inspecting the development of permanent strain using the Huurman model and the new transfer function for individual pavements, the intercept point can be explained. Figure 4.33 shows the

development of PS over number of load repetitions obtained for actual data and extrapolated using the Huurman model, compared to the new transfer function. The values obtained during the early life of the pavement (PS induced between 0% and 2%) show a significant difference between the actual data and the transfer function. This is due to the nature of the function, as it does not start at zero load repetitions for zero PS remaining. This explains why the intercept of the relationship in Figure 4.32 is not at zero but rather at 1.057.

Figure 4.33: NT F vs NActual for LTPP pavements (1)

Permanent strain development obtained from actual data and the Huurman model was compared to that of the transfer function for each of the LTTP pavement structures. Figure 4.33shows this relationship for two of the pavements, while Figures J.1toJ.5in AppendixJ show this for the rest

of the LTPP pavements.

These figures, illustrating the relationship between the transfer function results and the observed data, highlight the difference in short-term results. However, the slope of transfer function’s predicted life is very similar to the slope obtained from the actual data. This indicates that the transfer function predicts the long term rate of permanent strain accumulation accurately. The design function for BSM aims to provide a means of predicting the long-term in-field performance of BSMs based on material and structural properties. The function describes the long-term behaviour of BSMs and was therefore calibrated to different levels of reliability to produce a design function that accurately predicts the permanent strain accumulation rate of a BSM for specific design reliability criteria.

4.5.3.3 DSR power investigation

The effect of the power influencing the deviator stress ratio in the transfer function (Equation 4.1) was investigated. The transfer function was calibrated using the same method used previously in this section with the power influencing the DSR set to 2 and 4 and compared to the optimal results for the standard function. This was done to check if the initial assumptions to raise the significance of DSR on the function’s results were correct. Figure 4.32 shows the relationship between the transfer function’s prediction and the observed data for the power 3. Figure K.1 and FigureK.2

in Appendix K show the same comparison, with the power influencing DSR changed to 2 and 4 respectively.

Table4.11shows a summary of the linear regression information obtained during this investigation. The table shows the slope, interception point and correlation coefficient for each of the 3 powers considered. The slopes of the three sets showed a satisfactory degree of precision, with the slope for the power 4 being slightly too high. The interception point for the power 4 was very high and it showed a very poor coefficient of correlation of 0.355. By increasing this power to 4, the term reduces significantly as it is effectively timed by a fraction (usually smaller than 0.4). This causes the function to overestimate the life of the materials and reduce the effect of DSR.

By subjecting the DSR to the power 2, the function generates a large value for this term. This causes the function’s intercept point to drop from 1 to -0.59. The reduction in slope indicates that the function overestimates the effect of DSR and produces a conservative function. The correlation coefficient is also significantly reduced by applying the power 2. Therefore, DSR was subjected to the power 3 for the calibration and the final transfer function for this study.

Table 4.11: DSR power investigation results

4.5.3.4 Calibrated transfer function

The transfer function was calibrated to describe the plastic strain development observed for the LTPP pavements. Values were obtained for the four constants which control the effect the variables have on the transfer function’s results. The transfer function, calibrated to predict the observed data without a reliability adjustment, is given in Equation 4.2.

log N = 1.204378 − 57.286(DSR)3

+ 0.0009159(Pmod.RetC) + 0.086753(P S) (4.2)

The significance of the different variables in the transfer function was investigated. The results of this investigation are given in Section 5.3.