Comparison of Methods for Determining Precompression Stress Based on Computational Simulation

There are many methods for determining precompression stress (σp), whose value is affected by the slope of the soil compression curve. This study was designed to evaluate the hypothesis that for a certain compression curve all methods used to determine σp present the same value and accuracy. The aim of this study was to compare the accuracy and the relationship among seven of these methods by computational simulation of soil compression curves under nine scenarios. The following methods were used: Casagrande, Pacheco Silva, intersection of the initial void ratio with the virgin compression line (VCLzero), and the regression methods based on 2 (reg1), 3 (reg2), 4 (reg3), and 5 (reg4) points for modeling the elastic curve. Under each scenario, created by combining the swelling and the compression indices, 1,000 compression curves were computationally simulated via the Monte Carlo method. Subsequently, 95 % percentile confidence intervals were built using the 1,000 estimates of σp from each method under each scenario. Most of the differences among the methods were detected under scenarios consisting of high swelling and low compression indices. In general, Casagrande, Pacheco Silva, and reg4 were strongly correlated and presented the highest values of σp, as well as similar variability. The latter two can be considered as alternatives to the standard method of Casagrande, except for Pacheco Silva when the curve has a low compression index (≤0.2) and from medium to high swelling index (≥0.025), for which differences (p<0.05) were detected.


INTRODUCTION
Soil compaction as an effect of agricultural machinery has been one of the great challenges of modern agriculture . Vast cultivated areas have received increasingly heavy and intensive machine traffic (Mosaddeghi et al., 2007;Lima et al., 2015), especially at crop harvest. This has adverse effects on crop production and the environment (O'Sullivan et al., 1999). According to Cavalieri et al. (2008), soil compaction has been a subject of study for many years due to its implications for crop yield.
Compaction can be understood by studying soil compressibility (Dias Júnior and Pierce, 1995). Compression is characterized by a mechanical process that describes the decrease in volume when soil is exposed to a mechanical load, which is defined by a soil compression curve. Three important parameters extracted from the soil compression curve were describe by Keller et al. (2011): the swelling or recompression index (C s ), the compressibility coefficient (C c ), and the preconsolidation or precompression stress (σ p ), where C s is defined as the slope of the swelling line (SL) and C c is the slope of the Virgin Compression Line (VCL).
The C s is used as a measure of rebound and soil mechanical resilience, reflecting the first part of the curve subjected to historical stress, characterized by elastic deformation (recoverable). The second part is the VCL, for which plastic deformations are irreversible. This part can be verified by the C c value, not subject stress (Keller et al., 2011). Finally, σ p is mathematically defined as the point that divides the compression curve into the elastic and plastic parts of the soil compression curve (Casagrande, 1936).
There are many methods for determination of σ p . The most widely used was proposed by Casagrande (1936), which is based on the maximum curvature point of the soil compression curve. Nevertheless, other methods have been developed, such as the Pacheco Silva (ABNT, 1990) method, based on the intersection of the VCL and the initial void ratio. Dias Júnior and Pierce (1995) showed the procedure for determination of σ p by intersection of two linear regressions made for VCL and SL, which can have a different number of points considered for fitting VCL and SL (Cavalieri et al., 2008). Another method (VCLzero) consists of considering σ p as the value on the x-axis defined by the intersection of the VCL with a horizontal line from the initial void ratio (Arvidsson and Keller, 2004).
The studies of Arvidsson and Keller (2004), Gregory et al. (2006), Cavalieri et al. (2008), Ajayi et al. (2013), and An et al. (2015) demonstrated there are variations in the methods used for determination of the σ p and that the shape of curve is an important source of variation of indices (σ p , C s , C c ) extracted from the soil compression curve, showing that further studies are required. However, these studies have in common many soils, moisture contents, textures, and different conditions in soil physical properties, which makes it hard to define the parameters of the compression curve for a specific study and more accurate analysis. Under these conditions, simulations can help create scenarios for reproducing experimental data (Tagar et al., 2015), which formalize and analyze some error propagation methods for modeling; among them, the Monte Carlo method has general applicability and can be used in models with mathematical formulations (Ortiz et al., 2004). This procedure was used by Ortiz et al. (2004), and simulations based on other methods were used by Oliveira et al. (2013Oliveira et al. ( , 2014 for soil data in Brazil.
In this context, this study was designed to evaluate the hypothesis that for a certain compression curve all methods used to determine σ p present the same value and accuracy. Hence, the aim of this study was to compare the accuracy and relationship among seven methods used to determine σ p by simulating soil compression curves under nine scenarios.

Methods for determination and calculation of indices
Seven methods for determination of σ p were used: Casagrande, Pacheco Silva, intersection of initial void ratio with VCL (VCLzero), and linear regression methods based on 2, 3, 4, and 5 points for modeling of the elastic curve (swelling line, SL). An illustration of the regression method based on 2 points can be seen in figure 1.
The virgin compression line (VCL) was estimated through linear regression considering the last three points of the compression curve. The compressibility coefficient (C c ) was estimated as the slope of the linear regression fitted for VCL, determined as shown in equation 1, where e is the void ratio. The swelling index (C s ) was determined as the mean slope of the loading path up to 25 kPa (Equation 2), according to Keller et al. (2011). The C c and C s indices are identified on the compression curve and graphically represented in figure 1.

Scenarios of simulation
We created scenarios based on the values of the swelling (recompression index) and compression indices (Keller et al., 2011). This allowed us to reproduce different compression curves, which are associated with the results of a soil compressibility test. Simulation was based on the result of a simple uniaxial compression test with loads of 1, 12.5, 25, 50, 100, 200, 400, 800, and 1,600 kPa. In this case, the loading of 1 kPa corresponds to the initial void ratio or bulk density, which was only to calculate the swelling line associated with the initial state of the soil sample (Keller et al., 2011;An et al., 2015).  We combined three values of C c and three values of C s in order to create nine scenarios ( Figure 2). The slopes of the VCL and SL were generated on the log 10 scale of the applied loads. The original data set and the boundary of the initial void ratio and the values of C c and C s used in simulations were based on data from Ajayi et al. (2009), Keller et al. (2011), Ajayi et al. (2013), and An et al. (2015).

Simulation
Monte Carlo simulation was performed to compute mean and standard deviation. For each one of the nine compression curves described earlier, a fourth degree polynomial model was fitted, y = Xθ + ε, where y is the vector of void ratio, X is the polynomial model matrix, and ε is a random vector representing the error of the model. After that, we computed the vector of estimates (Equation 3) and its (co)variance matrix (Equation 4).
[ ] where s 2 is the estimate of residual variance.
Subsequently, we considered the vector of estimates to be normally distributed as ] in order to simulate 1,000 other vectors of estimates, say θ* . For every θ i * (i = 1, 2, ..., 1,000), a corresponding predicted vector ŷ i = Xθ i * was calculated. Finally, the pairs (ŷ i ,x) were used to determine 1,000 random estimates of precompression stress, σ * p , by each one of the seven methods.

Statistics for comparisons
We calculated the mean and coefficient of variation (%) of σ * p determined by each method. In addition, a confidence interval was built using the percentile method, i.e., taking the quantiles ) ( Furthermore, we computed Pearson's correlation matrix in order to study the relationship among the methods.

Computing
All the simulations and data analyses were made using the software R 3.1.2 (R Core Team, 2015) soilphysics package . Calculation of σ p was performed through the sigmaP() function. Simulations and percentile confidence intervals were performed using the simSigmaP() and plotCIsigmaP() functions, respectively.

RESULTS
Considering scenarios where C c = 0.2, the reg4 and Casagrande methods were similar and showed the highest values of σ p for all the C s conditions (Figure 3). The methods of Pacheco Silva, reg3, reg2, and reg1 changed more than Casagrande and reg4 with variation in C s . The Pacheco method was similar to Casagrande when considering the smallest C s , but they were statistically different (p<0.05) when C s increased. The regression method had higher values when using more points for modeling the swelling line (reg4>reg3>reg2>reg1).  For C c = 0.35, the Casagrande and reg4 methods also tended to show the highest values of σ p . The VCLzero method showed the lowest values, regardless of C s (Figure 3). The methods of Pacheco, reg3, reg2, and reg1 changed more than the Casagrande and reg4 methods with variation in C s . As C s increased, VCzero was the only method that was statistically different (p<0.05) than Casagrande.
For C c = 0.50, the methods of Casagrande, Pacheco Silva, and reg4 showed the highest σ p values. The VCLzero was statistically different (p<0.05) from Casagrande for all C s .
In general, the methods of Casagrande, Pacheco, and reg4 tended to show the largest values of σ p . Specifically for C c = 0.20, Casagrande and reg4 promoted the largest values, regardless of C s . For C c = 0.35 and 0.50, the general observation applies. The VCLzero method had the lowest values of σ p for most scenarios. The Pacheco method was closer to Casagrande as C s declined.
Under (C c = 0.20, C s ~ 0.055), we found the largest number of statistical differences (p<0.05) among the methods. In contrast, no difference (p>0.05) was found under the combination (C c = 0.35, C s ~ 0.003).
The variability shown by the methods in scenario (C c = 0.20, C s ~ 0.055) is noteworthy (Table 1). In fact, we observed ascending variability in the estimates according to the C s , whatever the value of the C c . Likewise, C c = 0.20 tended to show the highest variability in the simulated means.
In all scenarios, the standard Casagrande method was more correlated (r>0.90) with the Pacheco method and regression method using 5 (reg4) and 4 (reg3) points (Table 2). Although VCLzero always tends to show the lowest value of σ p , it had the same behavior as Pacheco and Casagrande. The reg1 and reg2 methods were related to each other in all scenarios.
A relationship between the simulated scenarios and situations of soil physical properties was observed. Ajayi et al. (2009) and Ajayi et al. (2013) obtained soil compression curves with lower values for C c for samples with low water content. When water content increased, C c also increased changing the shape of the soil compression curve.
For black and brown soils from Northeastern China under different bulk densities and water contents, An et al. (2015) obtained C c values similar to those obtained by Ajayi et al. (2009) and Ajayi et al. (2013). They found that C c was higher when water content in the soil samples increased. Variations in the shape of the curve when bulk density changed were also observed by Saffih-Hdadi et al. (2009) and An et al. (2015). They stated that C c decreases for high bulk density values. According to the results obtained by An et al. (2015), the combination of high bulk density values and low water content minimizes the value of C c .
The shape of the elastic line of a soil compression curve, represented here through C s , varies mainly with water content, as observed by O' Sullivan and Robertson (1996) and Braida et al. (2008). However, the results found by O' Sullivan and Robertson (1996) show C s values were higher in dry than in wet soils. However, Braida et al. (2008) found exactly the opposite, and attributed the results to the effects of the organic C and water content, which increasing the elastic proprieties of soil. Nonetheless, in any case, it is important to know that changing C s increases the differences among the σ p methods. This is most evident when comparing the methods under the same C c value (Figure 3).

Behavior of the methods under each scenario
The slope of the elastic curve and VCL influenced the estimate of σ p by the methods. Specifically, high values of C s (0.055) and low values of C c (0.2) increased differences among the methods (Figure 3). This can also be seen by the gradual increase in the    (Table 1). When the variation among methods is analyzed under the same C s , differences mainly occur under low C c (Table 1). Variation among methods is largely influenced by the slope of VCL, C c (Rosa et al., 2011).
Considering all scenarios, Casagrande, Pacheco Silva, reg4, and reg3 showed strong correlation (Table 2). Casagrande was compared with regression methods based on 2 (reg3) and 3 (reg4) points for modeling VCL (Arvidsson and Keller, 2004). These authors found that VCLzero was best correlated with Casagrande, but also found that the correlation between the regression method and Casagrande increased with the number of points (regression using three points>regression using two points), corroborating the results in table 2. However, Arvidsson and Keller (2004) did not test regression with four and five points (reg3 and reg4 as specified here, respectively), which would probably increase similarity with the Casagrande method, as found here. Cavalieri et al. (2008) showed medium to high correlations between regression methods and Casagrande (Cavalieri et al., 2008), at least higher than those obtained by Arvidsson and Keller (2004). The regression method using 4 and 5 points was correlated with Casagrande, as well as the Pacheco method (Table 2). Similarity between Casagrande and Pacheco in terms of σ p also was found by Rosa et al. (2011).

Applicability of the methods
The Casagrande method has been considered as standard in almost all comparison studies involving soil compressibility. However, its algorithm is relatively complex, since the point of maximum curvature of the compression curve must be determined. Regression methods are considerably simpler since they consist of intercepting two regression lines. However, evaluations of the regression methods, including comparison of their performance with the Casagrande method, can be found in the studies of Dias Júnior and Pierce (1995), Arvidsson and Keller (2004), and Cavalieri et al. (2008).

CONCLUSIONS
Most of the differences among the methods were detected under scenarios consisting of high swelling and low compression indices.
In general, Casagrande, Pacheco Silva, and reg4 were strongly correlated, showing the largest values of σ p , and similar variability. The latter two can be considered as alternatives to the standard Casagrande method, except for Pacheco Silva when the curve has a low compressibility coefficient (≤0.2) and medium to high swelling index (≥0.025), for which differences (p<0.05) were detected.