A Three-Point Hyperbolic Combination Model for the Settlement Prediction of Subgrade Filled with Construction and Demolition Waste.

Using construction and demolition waste (CDW) as road subgrade filling materials is an excellent way to solve the disparity between increased demand and road construction aggregate shortages. However, a key quality control problem is predicting the subgrade settlement, primarily because the CDW subgrade settlement prediction methods are not yet mature. To go some way in overcoming this problem, in this paper we developed a three-point hyperbolic combination model to predict CDW subgrade settlement, in which three appropriate points for the measured settlement curve were selected in the prediction samples to improve the hyperbolic model. Then, common prediction models—namely, the hyperbolic model, the three-point model, and the Hushino model—were compared with the proposed combination model to assess its viability. Finally, the three-point hyperbolic combination prediction accuracy was analyzed for different start points t0 and time intervals Δt. The analyses found that the proposed model was in good agreement with the measured data, had a high correlation coefficient, and had only small errors. However, the time interval Δt needed to be greater than 80 days and the start point t0 needed to be selected at the beginning of the stable post-filling period, that is, t0 = 90–100 days. The application parameters were also determined to provide a reference for the large-scale application and settlement predictions of CDW subgrade.


Introduction
As a result of continued industrial and urban growth, there has been a commensurate increase in construction and demolition waste (CDW), which now accounts for 30-40% of city waste in China and more than 40% of all municipal waste in Europe [1][2][3]. Therefore, interest in recycling and reusing CDW has increased, and there have been significant achievements in CDW subgrade application. For example, it was found that CDW could be used as the raw material for the production of recycled aggregate concrete (RAC) for road embankments, subgrades, and foundations [4][5][6][7][8]. Li et al. [9] found that after a relatively simple treatment, CDW had high strength and was suitable for urban road embankments; it had an unconfined compressive strength under optimal moisture content from 0.85 to 0.62 MPa, with a mean value of 0.74 MPa, an average CBR value of at least 34.7%, an average embankment deflection of 0.66 mm, and a resilient modulus of 162.7 MPa-all of which met the relevant specifications.
The Xi'an-Xianyang North Ring Expressway is a key connection line for the "2367" expressway network in Shaanxi Province. To protect the environment and to reduce the construction waste in Xi'an and Xianyang, about 4 million tonnes of recycled CDW, or around 28% of the whole road subgrade Therefore, to meet the CDW subgrade settlement prediction requirements for the Xi'an-Xianyang North Ring Expressway, monitoring settlement data from the AK0+980 and AK1+130 CDW subgrade sections were used as the input data for the prediction model. Using the CDW subgrade combination forecasting settlement summing law, a three-point hyperbolic combination model was developed, in which the hyperbolic model was respectively optimized with the three-point model point selection. Four CDW subgrade settlement prediction methods were then compared: The proposed three-point hyperbolic combination model, the three-point model, the hyperbolic model, and the Hushino model; additionally, the practicability, limitations, and applicability of the proposed model were analyzed. Finally, to select the point characteristic parameters for the practical application of the three-point hyperbolic combination model, the start point t 0 and the time span ∆t were studied to provide a theoretical basis for practical engineering embankment settlement and deformation predictions.

Three-Point Hyperbolic Combination Model for CDW Subgrade Settlement Prediction
Pan et al. [47] examined the theoretical bases, applicable conditions, and advantages and disadvantages of the three-point, hyperbolic, Poisson curve, and Asaoka models for foundation settlement predictions. It was found that as the three-point model accorded with Terzaghi's one-dimensional consolidation theory, the physical quantity meant that the consolidation parameters could be easily obtained and parameter β had a clear physical meaning [48]. Therefore, the three-point model was found to have simple calculation advantages and a good adaptability to different curve shapes. While the hyperbolic model's use of the graphic method to solve the parameters was found to be suitable for large deformation consolidation and settlement analysis, it was difficult to determine the consolidation parameters for the foundation. Zhao et al. [49] demonstrated that the actual settlement observation data-time curve for CDW subgrade conformed to the hyperbolic curve, with long-term observations from 1 to 5 years finding that the prediction settlement errors were only 0-2% in the later period.
Therefore, based on the foundation characteristics of CDW subgrade, for the first time, this paper combined the characteristics of the three-point method and the hyperbolic model to propose a three-point hyperbolic combination model for CDW subgrade settlement predictions, for which relevant equations were designed and the prediction effects analyzed using case studies.

Three-Point Model
The three-point model is also known as the logarithmic curve model of consolidation degree, the basic equation [28,29] for which is: where t is the consolidation time, S t is the settlement value at t, S d is the instantaneous settlement value, and S ∞ is the final settlement value. Here, α takes an approximate value for the one-dimensional consolidation theory, that is, α = 8/π 2 , and β is the coefficient obtained from the regression of the measured data. Three points (S 0 ,t 0 ), (S 1 ,t 1 ), (S 2 ,t 2 ) are chosen from the measured settlement-time curve (obtained from the measured data), with ∆t = t 2 − t 1 = t 1 − t 0 . The values for the three points are then brought into Equation (1), and the simultaneous equations in Equations (2) and (3) are obtained:

Hyperbolic Model
The hyperbolic model assumes that the foundation settlement development law after the full load accords with the hyperbolic function, the equation for which is shown in Equation (4) [26]: where t 0 is the start time and S 0 is the settlement value at t 0 , with the start point being set after the full load. Therefore, S t is the settlement value at time t, and α' and β' are the parameters that need to be determined. Equation (4) is thus rewritten as Equation (5): from which it can be seen that α' and β' are the intercept and the slope. The diagram for Equation (5) was determined using the linear fitting function in MATLAB R2016a software. By substituting the values for α' and β' into Equation (4), S t and S ∞ can be calculated.

Three-Point Hyperbolic Combination Model
Using the three-point method, (t 0 , S 0 ), (t 1 , S 1 ), (t 2 , S 2 ) were taken from the measured data and ∆t = t 2 − t 1 = t 1 − t 0 . Therefore, with t 0 as the start point, the selected points t 1 and t 2 can be calculated using Equation (5), from which Equations (6) and (7) are obtained: Equations (8) and (9) are then obtained by solving the simultaneous Equations (6) and (7): Substituting the calculated values for α' and β' into Equation (4), the prediction settlement value at t is obtained, as shown in Equation (10):

Actual Measured Settlement Data
The CDW subgrade settlement and deformation characteristics for the Xi'an-Xianyang North Ring Expressway were studied and predicted. The subgrade CDW, which came from a building waste recycling material factory in Xianyang, was fully utilized as embankment filler with a fill height of 5-6 m for the Xi'an-Xianyang North Ring Expressway. The main CDW aggregate composition was 63% concrete, 35% brick, 1% ceramic tile, and 1% other materials. The CDW aggregate particle size was 0-30 mm and the ratio of fine (passing through a 4.75 mm sieve) to coarse (retained by the 4.75 mm sieve) material was approximately 0.6 before the compaction. The subgrade filling comprised a 4:6 ratio of CDW and soil mixture that had a maximum dry density of 1.94 kN/m 3 and an average moisture content of 12.4%, with the unconfined compression strength (UCS) and the California bearing ratio (CBR) under the average moisture content being 1.87 Mpa and 63.7%.
All of the CDW material properties fully met the subgrade filler requirements of the subgrade construction technical specifications. After tamping, the ultimate bearing capacity of the CDW subgrade foundations was 315.9 kPa under a consolidation degree condition of 55% and a consolidation pressure of 150 kPa; therefore, the subgrade foundation was rigid. In this study, two continuous sections (AK0+980 and AK1+130) in the experimental section were selected to measure the completed subgrade settlement, which had a subgrade fill height of 552.4 cm and a construction period from 21 March to 20 June 2016. The cumulative settlement for the actual settlement observations was measured at 23.9 mm and 23.8 mm after 360 days, the specific observations for which are shown in Table 1.

Settlement Characteristics Analysis
From Table 1, the settlement data trend diagrams for AK0+980 and AK1+130 were drawn ( Figure 1). The settlement curves increased rapidly in the early stages, slowed after a certain time, and finally tended toward the limit value; that is, the settlement went through four main process changes: occurrence, development, stability, and limit. Therefore, the soft soil embankment "S" growth model was found to also adequately describe the CDW subgrade settlement.
From the settlement observation data in Table 1, the CDW subgrade settlement characteristics were identified as having a small settlement deformation order of magnitude and a large relative fluctuation. For example, while the total settlement value for AK0+980 was only 23.9 mm, the maximum settlement value variations in the two adjacent observations were 5 mm; for example, the settlement value was 11.3 mm on the 70th day and 16.3 mm on the 80th day-a difference of 5 mm or 30.67% of the 16.3 mm cumulative observation. Therefore, it was concluded that the CDW subgrade settlement prediction algorithm should accurately predict the settlement trends by avoiding the influences caused by the fluctuations.
Similarly, the t0, t1, and t2 for AK1+130 were set at 90, 170, and 250 days, respectively, with S0, S1, and S2 being 17.8, 21.8, and 22.7 mm. Using Equations (8) and (9), it was found that α' = 5.83675 and β' = 0.159599. Then, substituting α' and β' into Equation (10), the three-point hyperbolic model for AK1+130 was: After calculating t = 360 days, the predicted settlement for was 23.7033 mm, which, compared to the observation data (23.8 mm), indicated a prediction error of 0.41%. Figure 2 shows the comparisons between the settlement prediction fitting curve and the actual observation settlement curve, from which it can be seen that the prediction curve was consistent with the measured data curve, and the three-point hyperbolic combination model had high fitting accuracy.

Evaluation Indexes
References [50,51] evaluated the applicability of several conventional settlement prediction models in terms of their correlation coefficients (R), sum of square prediction errors (ISSE), and relative deviations (δ), with the R being calculated using a correlation function, the ISSE being calculated as shown in Equation (13), and δ being calculated as shown in Equation (14).
Similarly, the t 0 , t 1 , and t 2 for AK1+130 were set at 90, 170, and 250 days, respectively, with S 0 , S 1 , and S 2 being 17.8, 21.8, and 22.7 mm. Using Equations (8) and (9), it was found that α' = 5.83675 and β' = 0.159599. Then, substituting α' and β' into Equation (10), the three-point hyperbolic model for AK1+130 was: After calculating t = 360 days, the predicted settlement for S 360d was 23.7033 mm, which, compared to the observation data (23.8 mm), indicated a prediction error of 0.41%. Figure 2 shows the comparisons between the settlement prediction fitting curve and the actual observation settlement curve, from which it can be seen that the prediction curve was consistent with the measured data curve, and the three-point hyperbolic combination model had high fitting accuracy.

Evaluation Indexes
References [50,51] evaluated the applicability of several conventional settlement prediction models in terms of their correlation coefficients (R), sum of square prediction errors (I SSE ), and relative deviations (δ), with the R being calculated using a correlation function, the I SSE being calculated as shown in Equation (13), and δ being calculated as shown in Equation (14).

Comparison with Other Conventional Models
To verify the validity and feasibility of the proposed combination model, the actual observation data in Table 1 were used as the prediction samples. As the Hushino model [52] has been widely applied to fit foundation settlement-time curves, the Hushino model, the three-point model, and the hyperbolic model were chosen for comparison with the developed three-point hyperbolic combination model.
The Hushino model is based on Terzaghi's consolidation principle, which states that the consolidation degree (U) is the consolidation degree achieved after the external load is applied to the foundation. Therefore, as the consolidation degree is used to determine the relationships between the settlement and time, it is an important parameter for settlement prediction. The consolidation degree at t time (U t ) is calculated using Equation (15) [52]: where S t is the settlement value at time t for the foundation soil, and S ∞ is the final foundation settlement value. The Terzaghi consolidation principle states that when U is less than 60%, it is proportional to the square root of time. By studying the measured settlement values, it was concluded that the total settlement, including the shear deformation settlement, was proportional to the square root of time [53]. The Hushino formula is shown in Equation (16) [51]: where S 0 is the instantaneous settlement value, S t is the settlement value at time t, and A and K are the undetermined coefficients. Figure 3 compares the fitting data for the four settlement prediction models and the measured data, and Table 2 compares the settlement prediction evaluation indexes for the four models.
From Figure 3 and Table 2, the following could be concluded: (1) The I SSE values for the Hushino model and the three-point model were larger than for the hyperbolic model and the three-point hyperbolic combination model, and the final settlement values also deviated significantly from the actual measured settlement curve. Therefore, as the Hushino model and the three-point model were found to have low CDW subgrade settlement predictive accuracy, they were deemed unsuitable, which was consistent with the results in Pan [51].
(2) The I SSE for the hyperbolic model was small and the R was high, which indicated that the hyperbolic model was also suitable for CDW subgrade settlement predictions.
(3) The predicted settlement value for the three-point hyperbolic combination model was the closest to the actual measured value, the I SSE was the smallest, and the R was the largest. Therefore, the three-point hyperbolic combination model was shown to be the best model for CDW subgrade settlement predictions as it had the highest accuracy. Furthermore, because the three-point hyperbolic combination model only used three of the measured values, data fluctuation was avoided, thereby making this model more feasible for CDW subgrade settlement predictions.
(4) As the three-point hyperbolic combination model is based on the selected three points, the calculations are simpler. However, as the CDW subgrade settlement observations had relatively large fluctuations, the selection of these three points is very important to ensure the best parameters and the most precise predictions. Therefore, to obtain optimum CDW subgrade prediction values and to avoid the errors caused by poor point selection, in the following, the three-point hyperbolic combination model prediction accuracy was examined using different time start points t 0 and time intervals ∆t.  From Figure 3 and Table 2, the following could be concluded: (1) The ISSE values for the Hushino model and the three-point model were larger than for the hyperbolic model and the three-point hyperbolic combination model, and the final settlement values also deviated significantly from the actual measured settlement curve. Therefore, as the Hushino model and the three-point model were found to have low CDW subgrade settlement predictive accuracy, they were deemed unsuitable, which was consistent with the results in Pan [51].
(2) The ISSE for the hyperbolic model was small and the R was high, which indicated that the hyperbolic model was also suitable for CDW subgrade settlement predictions.
(3) The predicted settlement value for the three-point hyperbolic combination model was the closest to the actual measured value, the ISSE was the smallest, and the R was the largest. Therefore, the three-point hyperbolic combination model was shown to be the best model for CDW subgrade settlement predictions as it had the highest accuracy. Furthermore, because the three-point hyperbolic combination model only used three of the measured values, data fluctuation was avoided, thereby making this model more feasible for CDW subgrade settlement predictions.
(4) As the three-point hyperbolic combination model is based on the selected three points, the calculations are simpler. However, as the CDW subgrade settlement observations had relatively large fluctuations, the selection of these three points is very important to ensure the best parameters and the most precise predictions. Therefore, to obtain optimum CDW subgrade prediction values and to avoid the errors caused by poor point selection, in the following, the three-point hyperbolic combination model prediction accuracy was examined using different time start points t0 and time

Influence of Time Intervals (∆t)
The predicted three-point hyperbolic combination model results when the t 0 was the same (t 0 = 90 days, which was the subgrade filling completion date of June 21, 2016) and ∆t was set at 60, 70, 80, and 90 days are shown in Table 3, with the comparison between the prediction fitting curves and the measured data for the two sections (AK0+980 and AK1+130) shown in Figure 4.
From Table 3 and Figure 4, the following observations could be made: (1) When the start point was t 0 = 90 days, the smaller the ∆t, the larger the I SSE , the higher the δ 360d , and the lower the R. In contrast, the greater the ∆t, the closer the three-point hyperbolic model fitting curve was to the actual measured curve, the higher the R, the smaller the I SSE , and the lower the δ 360d .
(2) For AK0+980, when t 0 was set at 90 days and ∆t was set at 60, 70, 80, and 90 days, the correlation coefficients were 0.97183, 0.98195, 0.99197, and 0.99205, respectively, and the respective I SSE values were 10.90904, 5.95313, 1.4755, and 0.85920. Therefore, to improve the prediction accuracy, ∆t should be no less than 80 days (3) Similarly, when t 0 was set at 90 days and ∆t was set at 60, 70, 80, and 90 days, the correlation coefficients of AK1+130 were 0.97995, 0.98336, 0.98713, and 0.98793, respectively, and the respective I SSE values were 13.06036, 8.14553, 3.32137, and 2.817681. Therefore, the best ∆t should be no less than 80 days, which is consistent with the parameter for AK0+980.

Influence of Time Intervals (∆ )
The predicted three-point hyperbolic combination model results when the t0 was the same (t0 = 90 days, which was the subgrade filling completion date of June 21, 2016) and ∆t was set at 60, 70, 80, and 90 days are shown in Table 3, with the comparison between the prediction fitting curves and the measured data for the two sections (AK0+980 and AK1+130) shown in Figure 4.  From Table 3 and Figure 4, the following observations could be made: (1) When the start point was t0 = 90 days, the smaller the ∆t, the larger the , the higher the δ360d, and the lower the R. In contrast, the greater the ∆t, the closer the three-point hyperbolic model fitting curve was to the actual measured curve, the higher the R, the smaller the , and the lower the δ360d.
(2) For AK0+980, when t0 was set at 90 days and ∆t was set at 60, 70, 80, and 90 days, the correlation coefficients were 0.97183, 0.98195, 0.99197, and 0.99205, respectively, and the respective ISSE values were 10.90904, 5.95313, 1.4755, and 0.85920. Therefore, to improve the prediction accuracy, ∆t should be no less than 80 days (3) Similarly, when t0 was set at 90 days and ∆t was set at 60, 70, 80, and 90 days, the correlation coefficients of AK1+130 were 0.97995, 0.98336, 0.98713, and 0.98793, respectively, and the respective ISSE values were 13.06036, 8.14553, 3.32137, and 2.817681. Therefore, the best ∆t should be no less than 80 days, which is consistent with the parameter for AK0+980.  Table 4 shows the three-point hyperbolic combination model fitting calculations for when the time interval ∆t was the same (set to ∆t = 100 days) and t 0 was set at 90 days (completion date), 100, 110, and 120 days. The comparisons between the settlement prediction curves for the two sections (AK0+980 and AK1+130) and the measured data curves are shown in Figure 5.

Influence of Different Start Points (t 0 )
From Table 4 and Figure 5, the following could therefore be concluded: (1) At ∆t = 100 days, the larger the t 0 , the larger the δ 360d and I SSE , and the lower the R. On the contrary, the smaller the t 0 , the closer the three-point hyperbolic model fitting curve was to the measured curve, the higher the R, and the smaller the δ 360d and I SSE .
(2) For AK0+980, when ∆t = 100 days and t 0 was set at 90, 100, 110, and 120 days, the correlation coefficients were 0.99049, 0.98889, 0.98614, and 0.98217, respectively, and the respective I SSE values were 1.18017, 1.29816, 1.31485, and 1.67211. Therefore, it is recommended that the start point t 0 be at the beginning of the stable period after the subgrade has finished.
(3) Similarly, when ∆t = 100 days and t 0 was set at 90, 100, 110, and 120 days, the correlation coefficients of AK1+130 were 0.98825, 0.98688, 0.98383, and 0.98361, respectively, and the respective I SSE values were 1.57992, 3.21601, 6.69725, and 7.201311. Therefore, the best t 0 should be at the beginning of the stable period after the subgrade has finished, which is consistent with the parameter for AK0+980.  Table 4 shows the three-point hyperbolic combination model fitting calculations for when the time interval ∆ was the same (set to ∆ = 100 days) and t0 was set at 90 days (completion date), 100, 110, and 120 days. The comparisons between the settlement prediction curves for the two sections (AK0+980 and AK1+130) and the measured data curves are shown in Figure 5.  From Table 4 and Figure 5, the following could therefore be concluded: (1) At ∆ = 100 days, the larger the t0, the larger the δ360d and , and the lower the R. On the contrary, the smaller the t0, the closer the three-point hyperbolic model fitting curve was to the measured curve, the higher the R, and the smaller the δ360d and .
(2) For AK0+980, when ∆ = 100 days and t0 was set at 90, 100, 110, and 120 days, the correlation coefficients were 0.99049, 0.98889, 0.98614, and 0.98217, respectively, and the respective ISSE values were 1.18017, 1.29816, 1.31485, and 1.67211. Therefore, it is recommended that the start point t0 be at the beginning of the stable period after the subgrade has finished.
(3) Similarly, when ∆ = 100 days and t0 was set at 90, 100, 110, and 120 days, the correlation coefficients of AK1+130 were 0.98825, 0.98688, 0.98383, and 0.98361, respectively, and the respective ISSE values were 1.57992, 3.21601, 6.69725, and 7.201311. Therefore, the best t0 should be at the beginning of the stable period after the subgrade has finished, which is consistent with the parameter for AK0+980.

Conclusion
(1) The CDW subgrade consolidation settlement was found to be in accordance with the soft soil embankment "S" growth model, except that it had a smaller settlement deformation order of magnitude and a larger relative fluctuation.

Conclusion
(1) The CDW subgrade consolidation settlement was found to be in accordance with the soft soil embankment "S" growth model, except that it had a smaller settlement deformation order of magnitude and a larger relative fluctuation.
(2) A three-point hyperbolic combination model was proposed to predict CDW subgrade settlement. Compared with other traditional models-namely, the three-point model, the hyperbolic model, and the Hushino model-the three-point hyperbolic combination model had the highest R at 0.99197, the smallest I SSE at 1.477547 mm 2 , and the closest predicted settlement value S 360d , at 23.61826 mm, to the actual measured value (23.90 mm). As the comparison curves showed that the prediction curve for the three-point hyperbolic combination model was in good agreement with the measured data, it was concluded that the proposed combination model had the best adaptability for CDW subgrade settlement prediction.
(3) To improve the prediction accuracy of the proposed combination model, the effects of different time intervals ∆t and start points t 0 on the prediction accuracy were analyzed, and it was found that ∆t should be set at no less than 80 days and t 0 needs to be selected at the beginning of the stable period after the subgrade fill completion (that is, t 0 = 90 or 100 days).
(4) The three-point hyperbolic combination model provides a theoretical basis for CDW subgrade settlement and deformation predictions, and especially for collapsible loess foundations, as it is able to provide a scientific basis for the application of CDW subgrade filling materials. The application of the combination model, therefore, can promote large-scale CDW applications and can support sustainable development aims.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.