Soil Science Society of America Journal 65:678-684 (2001)
© 2001 Soil Science Society of America
DIVISION S-1 - SOIL PHYSICS
An Improved Empirical Equation for Uniaxial Soil Compression for a Wide Range of Applied Stresses
D.D. Fritton
Dep. of Agronomy, Pennsylvania State Univ., University Park, PA 16802
Corresponding author (ddf{at}psu.edu)
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ABSTRACT
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The response of soil to compaction forces is nonlinear and not completely described by existing statistical equations. The objective of this study was to find a better empirical equation for uniaxial soil compression. Disturbed and undisturbed samples from three to five horizons of four soils, and from soil mixed with four different amounts of sand, were subjected to applied stresses ranging from 0 to 2971 kPa at one to four initial water contents. Data from individual samples representing the three resulting curve shapes were used to evaluate existing and new empirical equations. A new equation was found that fit all three curve shapes better than any of the existing equations. The new equation fit data points of representative data sets with an average difference of 0.002 to 0.009 Mg m-3, compared with an average difference for two existing equations of 0.011 to 0.033 and 0.014 to 0.060 Mg m-3. The new equation was then fit to all 120 sets of experimental data, using nonlinear regression procedures. Regression relationships were established between three parameters that have traditionally been used to characterize soil compression (preconsolidation stress, compression index, and elastic rebound/recompression parameter) and the parameters of the new equation.
Abbreviations: Ci, compression index • K, elastic rebound/recompression parameter • pC, preconsolidation stress
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INTRODUCTION
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AT TIMES, SOIL COMPACTION significantly reduces crop yield. There is no routine procedure, though, to predict this effect. The response of soil to an applied stress is an important aspect of this problem. Koolen and Kuipers (1983) conclude that the uniaxial soil-compression test is a sufficient representation of soil compaction for agricultural activities. This work started with the intent of evaluating the relationship between the compression index, a parameter derived from uniaxial soil-compression data, and the clay content reported by Gupta and Larson (1982). The extraction of a compression index from the data, however, required arbitrary decisions due to deviations from linearity between bulk density and applied stress at both the low- and high-stress ends of each data set. A better description of uniaxial soil-compression data is needed. It is the purpose of this article to present a new empirical equation for uniaxial soil compression that is capable of fitting soil bulk-density data for the entire range of applied stresses for both disturbed and undisturbed soil at any fixed initial water content. Three material properties used by Kirby (1994), who demonstrated the adequacy of uniaxial soil-compression tests for measuring soil material properties using a critical-state model, can be calculated from the coefficients of the new empirical equation described in this article.
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Prior Representations
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The relationship between bulk density or another measure of soil compaction and the applied stress has been represented by many equations. Koolen and Kuipers (1983) and Gupta and Allmaras (1987) discussed a number of these equations including the logarithmic equation used by Gupta and Larson (1982). The logarithmic equation in not able to fit data at applied stresses less than the preconsolidation stress—the point where the stress exceeds any previously experienced by the soil. Bailey et al. (1986) introduced an equation,
 | (1) |
where 
is soil bulk density (Mg m-3) at an applied stress of 
(kPa), o is the initial soil bulk density (Mg m-3), and a, b, and c are empirical parameters that fit compression data at stresses, including zero stress, below the preconsolidation stress. McNabb and Boersma (1993) extended the Bailey et al. (1986) equation to represent multiple soil samples that varied in initial bulk density. McNabb and Boersma (1996) further extended this approach to represent multiple soil samples that varied in initial water content as well as initial bulk density.
Assouline et al. (1997) point out that the logarithmic equation and the equations based on the Bailey et al. (1986) equation predict an ever-increasing bulk density as the applied stress increases. This is contrary to the observation that bulk density reaches an upper limit as the applied stress increases. Assouline et al. (1997) introduced an equation,
 | (2) |
where max is an empirical parameter (Mg m-3) and other symbols retain their previous definitions, to overcome this deficiency and demonstrated that it fit their data as well as the Bailey et al. (1986) equation.
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Objective and Proposed Equation
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Curve shapes generated from uniaxial soil-compression data range from a nearly straight line to a partially s-shaped curve (see Fig. 1
for examples) on a plot of bulk density vs. the logarithm of applied stress plus one. The objective of this study was to find a single equation with enough flexibility to fit the full range of uniaxial soil-compression curve shapes. The equation chosen is analogous to the equation extensively used to fit water-retention data (van Genuchten and Nielsen, 1985). With soil bulk density replacing volumetric water content, and applied stress plus one replacing soil water pressure head, the equation becomes
 | (3) |
where m is the maximum soil bulk density, the particle density (Mg m-3), 
is an empirical parameter (kPa-1), n and m are unitless empirical parameters, and other symbols retain their previous definitions.
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Fig. 1. Bulk density of three soils plotted as a function of the applied stress plus one. Points represent experimental data. The smooth lines are best-fit nonlinear regression curves based on the three-parameter version of Eq. [3]
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MATERIALS AND METHODS
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Soil Properties
Four deep, well-drained soils were sampled for this study. The soils were Rayne silt loam (fine-loamy, mixed, mesic Typic Hapludult) formed in gray shale residuum, Bucks silt loam (fine-loamy, mixed, mesic Typic Hapludult) formed in red shale residuum, Glenelg silt loam (fine-loamy, mixed, semiactive, mesic Typic Hapludult) formed in mica schist residuum, and Hagerstown silt loam (fine, mixed, semiactive, mesic Typic Hapludalf) formed in limestone residuum. For one set of measurements, a commercial white quartz sand was mixed with the sieved (<2 mm) Hagerstown (B horizon, 0.85 to 1.00 m) soil in dry mass (sand/soil) ratios of 1:4, 2:3, 3:2, and 4:1 to extend the range of particle-size distributions studied.
Particle-size distribution was determined according to Kilmer and Alexander (1949). Organic C was determined consistent with Young and Lindbeck (1964). The particle-size distribution and organic-C content of mixed samples were calculated from the properties of the two components, assuming the white quartz sand had no organic C.
Soil Sampling and Preparation
At each site, bulk samples were collected from each of three to five horizons and stored moist in airtight containers until needed. In addition, approximately six 63.5-mm-diam. undisturbed-soil cores were taken to a depth of 1 m with a hydraulically driven soil sampler at each site. The soil cores were placed moist in plastic or aluminum tubes and sealed until needed.
When needed, bulk soil was air-dried and peds were crushed to pass through a 2-mm sieve. Air-dried rock fragments were weighed and used to calculate rock-fragment content as a fraction of the whole sample. The sieved soil was poured into two 63.5-mm-diam. rings, which were 
5 mm taller than the sample height (25.4 mm) needed for compression, and leveled gently. The soil samples were placed on a wet ceramic water extraction plate and satiated from the bottom by ponding water on the plate surface. After wetting for 1 to 2 d, the samples were placed in a pressure apparatus and equilibrated for 1 to 2 d at pressures of 10 to 500 kPa to obtain one to four different water contents. One of the two samples was oven-dried at 105°C to estimate the initial water content and the other was used for compression determinations.
Undisturbed-soil core samples (21 of the 120 samples) were prepared for compression determinations in a similar manner. A location on the 1-m soil core was selected visually that matched the horizon depths at which the bulk samples were collected, contained no obvious distortions from sampling, and had no displaced rock fragments. A 63.5-mm-diam. ring was slipped onto the core and then the soil was trimmed with a wire saw until the ends were flush with the ring. These cores were then placed on the ceramic plate, satiated, and equilibrated at selected pressures intended to give two different water contents. As with the disturbed samples, a second core was prepared in the same manner and oven-dried to estimate the initial water content. Water content was determined on the sample after compression, but could not be used in most cases to characterize the initial water content since water was squeezed from most samples during the compression process.
Once either the disturbed or undisturbed soil sample was equilibrated, the sample was weighed and transferred to the compression cylinder (63.5-mm diam. and 25.4-mm length). The samples were slid undisturbed from the sample ring into the compression cylinder. Excess soil was then trimmed until the sample was level with the top of the compression cylinder. The excess soil was weighed to determine the mass of soil remaining in the compression cylinder. The mass of moist soil in the compression cylinder was then corrected for the water content determined from the second sample to estimate the dry mass of solids contained in the compression cylinder.
Compression Apparatus and Procedure
The compression apparatus (Model C-320, ELE International, Pelham, AL) consisted of a sample cylinder placed so that the soil sample rested upon a porous stone and was then covered by a second porous stone. The bottom porous stone was drained so that this was a uniaxial drained compression test. Force was applied to the top of the sample through a brass plate placed over the top porous stone using a triple-beam arrangement with a 10:1 beam ratio. Weight added to the beam hanger was converted to the equivalent pressure applied to the soil surface in kPa. A dial gauge (Model LC-3M, ELE International) with a measurement precision of 0.025 mm was initially calibrated to read zero with the porous stone and the brass plate set on a solid spacer of 25.4-mm length, and then read after the soil sample had been loaded at each level of stress for at least 30 min. Applied stress levels of 0, 31, 62, 93, 186, 371, 557, 743, 1114, 1485, and 2971 kPa were used for all soils except the Hagerstown and Hagerstown/sand mixtures where 0, 31, 62, 186, 557, 1114, and 2971 kPa were used. Using an initial sample length of 25.4 mm, the volume of the soil and the soil bulk density were then calculated.
Following the highest level of stress, the soil sample was removed from the compression apparatus and oven-dried at 105°C to determine the final water content and provide a second estimate of the mass of solids contained in the compression cylinder. In the case of undisturbed samples, the sample was broken into enough pieces to make sure that >11.5-mm-diam. rock fragments had not been included. This size ensured that the initial sample size was at least 100 times the volume of the largest rock fragment and that rock fragments did not interfere with the compression process by bridging across the two porous stones. In no case were rock fragments found large enough or numerous enough to interfere with the compression process.
Mathematical and Statistical Calculations
Data (see Fig. 1) selected to represent the variety of curve shapes of bulk density as a function of the logarithm of stress were used to evaluate Eq. [1], [2], and [3] using the NonlinearFit package in Mathematica (Wolfram Research, Champaign, IL) (Boyland et al., 1992) with the Levenberg-Marquardt method to minimize the error sum of squares. The selected mathematical equation (Eq. [3]) was then fit to each data set. The resulting parameters were used to generate the slope and curvature for each data set for stresses ranging from 0 to 2971 kPa, with a program written using Mathematica (Wolfram, 1991).
The preconsolidation stress (pc) was calculated using the Casagrande procedure described by Wu (1976). This procedure is illustrated in Fig. 2
. The dashed curve represents part of the data (the complete data set is shown in Fig. 1) from a sample of the Rayne soil. The Casagrande procedure starts by determining the point of maximum curvature for the data set. This Rayne sample had a maximum curvature at an applied stress of 97 kPa and a bulk density of 1.24 Mg m-3. The tangent line to the curve (line AB in Fig. 2) is then drawn at this point (labeled C in Fig. 2) and a line (CD) is drawn parallel to the x axis through point C. The angle DCB is then bisected giving the line CE. A line (FG) is then extended from the steep linear portion of the curve. Since the compression data were not always linear from the point of maximum curvature to the point at the highest level of applied stress, this line (FG) was drawn through a point on the curve represented by the geometric mean of the stresses at the maximum (97 kPa) and minimum curvature (590 kPa). The slope of FG was set equal to the slope of the curve at the geometric mean stress (239 kPa). The highest stress (2971 kPa) was used for the minimum curvature when the curve had no s-shape. The intersection of lines CE and FG (point H in Fig. 2) determines the preconsolidation stress (115 kPa at a bulk density of 1.25 Mg m-3 for the sample in Fig. 2).
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Fig. 2. A partial set of data represented by the best-fit dashed line from Eq. [3] for a sample of the Rayne soil. The complete data set is shown in Fig. 1. The various points and lines are discussed in the text to illustrate calculations
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The compression index (Ci) was set equal to the slope of the bulk density–log stress curve at the midpoint (the geometric mean) between the maximum and minimum curvature. In the example shown in Fig. 2, the compression index is the slope of the line FG (the virgin compression curve, the estimated response of a soil that has not experienced previous stress) and is equal to 0.397. This procedure is equivalent to rejecting points at either end of the curve that cause nonlinearity without arbitrary decisions.
The elastic rebound/recompression parameter (K) is the slope of the compression curve from zero stress to the preconsolidation stress. It was calculated, as illustrated by Kirby (1994), by finding the slope of the line (IH in Fig. 2) that connects the preconsolidation stress point on the virgin compression curve (H) with the initial point on the curve (zero applied stress with the initial bulk density, o). For the case illustrated, the difference between the bulk density on the virgin compression curve at the preconsolidation stress, 1.25 Mg m-3, and the initial bulk density, 1.21 Mg m-3, was divided by the logarithm of the preconsolidation stress giving the elastic rebound/recompaction parameter, 0.0198. All calculations were done within the Mathematica package.
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RESULTS
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Shown in Table 1 are the depth interval, the particle-size distribution, the organic-C content, and the rock fragment content for each soil horizon sampled and material used. For tested materials, the organic-C content is somewhat higher in surface horizons than the underlying horizons, but is generally low. The clay (<0.002 mm) content varies from 91 to 486 g kg-1; the silt (0.05 to 0.002 mm) content is relatively high in most samples and varies from 87 to 594 g kg-1; and the sand (2 to 0.05 mm) content is relatively evenly distributed across the various sand fractions in most samples and varies from 110 to 822 g kg-1. The rock-fragment content varies from 0 to 347 g kg-1 but is applicable only to undisturbed samples since all disturbed samples contained only <2.0-mm material.
Data in Fig. 1 represent the variety of response curves obtained. The points represent the experimental data and the smooth lines represent the nonlinear optimized fit to the data using Eq. [3] as a three-parameter model. The Bucks sample (disturbed) was sieved from the surface horizon and run at an initial water content of 0.247 kg kg-1. This sample is representative of only 3 other samples of the 120 samples run. The response is nearly linear when plotted against the logarithm of one plus the applied stress (kPa) over the whole range of applied stresses. The Rayne sample (disturbed) was also sieved from the surface horizon and was run at an initial water content of 0.219 kg kg-1. This sample had a nearly constant bulk density at low stress, followed by a sharp change in slope, and ended with a decreasing rate of change at higher levels of stress. This somewhat s-shaped response is representative of 63 of the 120 samples run. After the preconsolidation stress was exceeded, the remaining samples responded in a linear fashion instead of a decreasing rate of change at higher levels of stress when plotted against the logarithm of one plus the applied stress (kPa). The undisturbed Glenelg sample taken from the 0.99- to 1.22-m depth and run at an initial water content of 0.134 kg kg-1 is a typical example of this third group.
Table 2 shows the parameters used in Eq. [3] for the three data sets shown in Fig. 1 with additional data for the Hagerstown sample that had the lowest R2, and for all four sets when Eq. [3] was used as a five-parameter model. For the whole data set, the value of was related to the point of maximum curvature and varied from 0.0000845 to 0.230 kPa-1. The value of n was sensitive to the steepness of the curve after the preconsolidation stress was exceeded and varied from 0.689 to 19.4. The value of m was related to the curvature after the preconsolidation stress was exceeded and varied from 0.00576 to 1.11. All R2 values for Eq. [3] fit to each of the 120 sets of data exceeded 0.97.
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Table 2. Parameters used in Eq. [3] as a three-parameter model to generate the smooth curves in Fig. 1 and for the Hagerstown sample that had the poorest fit compared with using Eq. [3] as a five-parameter model. R2 values are shown to facilitate comparisons
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Estimating Preconsolidation Stress
Figure 3
shows a plot of the calculated preconsolidation stress (pc) as a function of the value obtained by fitting Eq. [3] to each data set. The points are the individual data pairs and the line is calculated from the equation
 | (4) |
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Fig. 3. Preconsolidation stress plotted as a function of the parameter derived from fitting Eq. [3] to 119 sets of soil-compression data. The smooth line is the nonlinear regression curve based on Eq. [4]
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One point at = 0.0000845 kPa and pc = 1620 is not shown on the left side of Fig. 3 to improve clarity and because eliminating this point improved the consistency of the statistical comparisons among equations [pc= a e-b/( + c); pc = 1/(a + ); and pc = a/ where a, b, and c are best-fit constants] fit to the data of Fig. 3. Equation [4] was selected because it fit the data nearly as well (R2 = 0.95) as other equations attempted, and provided a direct connection between Eq. [3] and the preconsolidation stress, a traditional parameter used to describe compression data.
Estimating Compression Index
To relate the compression index to the parameters of Eq. [3], the derivative of Eq. [3] was examined. By assuming that >> 1, ()n >> 1, and mn << 1, the combination formed by multiplying (m - o), m, and n resulted. The functional relationship between the compression index and (m - o)mn was examined. The same sample that produced an outlier in Eq. [4] was also an outlier in this case and was dropped from the data set. A linear relationship gave an R2 of 0.67, while a rectangular hyperbolic relationship improved the R2 to 0.72. Since both functional forms resulted in unequal variances, square root and inverse transformations of the compression index were tried. A square root transformation improved the R2 to 0.77 and the inverse transformation resulted in an R2 of 0.88. A plot of the inverse of the compression index and (m - o)mn is shown in Fig. 4
. The equation used for the solid line in Fig. 4 was
 | (5) |
Estimating the Elastic Rebound/Recompression Parameter
In a similar manner, the procedure for calculating K was examined to find a functional connection between K and the parameters of Eq. [3]. By substituting pc into Eq. [3] for , 1/pc for , and assuming pc >> 1, this combination resulted:
 | (6) |
Figure 5
is a plot of K as a function of (m - o) (1 - 1/2m)/(-log ). The straight line
 | (7) |
fits the data points shown (R2 = 0.97). There is increasing variance with increasing K, and this effect can be removed by using an inverse transform and fitting the resulting data with the inverse of a rectangular hyperbola similar to the procedure used for Eq. [5]. In this case, there is no improvement in R2, and the resulting equation is much more complex. It was felt that the linear representation was better. The sample that had been an outlier earlier was also an outlier in this case and has been dropped from the dataset for this figure.
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DISCUSSION
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Experimental Procedures
The data collected in this study was obtained after the applied stress had been in place for at least 30 min. Preliminary data were taken every 5 min for 120 min at each level of applied stress on a disturbed Bucks sample from the soil surface at an intermediate water content. These data showed that bulk density increased rapidly with time after the applied stress was in place, but approached equilibrium slowly. The error in using 30 min rather than 120 min was estimated to be <0.01 Mg m-3 and probably <0.02 Mg m-3 relative to a true equilibrium. As a result, bulk-density values were rounded to the nearest 0.01 Mg m-3 before being used in the analyses reported. Equation [3] fit most of the data with an average difference <0.01 Mg m-3 (see Table 3) indicating that greater accuracy would have been justified. It seems unlikely that the decision to take data after 30 min would affect the conclusions reported in this article.
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Table 3. A comparison of the fit for three equations as measured by R2 and the average difference (absolute values) between the measured and calculated bulk density for 7 (Hagerstown) or 11 (Bucks, Rayne, and Glenelg) values of applied pressure for four soil samples from the 120 sets of compression data
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A second uncertainty in the data arose from the two estimates made of the dry soil mass contained in the compression cylinder. The first estimate relied on the accuracy of the initial water content determined on a soil sample treated the same way as the one being compressed. This initial water content is subject to variation especially in the case of undisturbed samples due to the natural soil heterogeneity. The second estimate relied on the ability of the operator to recover all the soil at the end of the last compression step. Differences caused by these two estimates were usually <0.02 Mg m-3. The data collected for this study involved two different operators. In the early data collection with the first operator, it was concluded that the first estimate was the best. This involved 83 of the 120 samples run. During later data collection with the second operator, procedures were refined and it was concluded that the second estimate was better. This involved 37 of the 120 samples run. These decisions would affect all bulk densities for a given sample. All would increase or decrease by the same amount. This does not change the shape of the curve and would not change the conclusions of this article. A change in o in Eq. [3], however, affects the empirical parameters , m, and n and thus might slightly change the regression coefficients reported in Eq. [5] and [7].
Statistical Comparisons
Empirical equations used in the literature to fit soil-compression data have deficiencies when used across the whole range of curve shapes. The experience from fitting data for soils representing each of the three curve shapes as shown in Fig. 1 will be used to highlight the problems encountered.
The equation developed in Assouline et al. (1997) missed bulk-density data points by an average difference (see Eq. [2] in Table 3) of 0.060 Mg m-3 for data that is nearly linear on a semi-log plot such as the Bucks data in Fig. 1. When the Assouline et al. (1997) equation was used to fit curves similar to the Glenelg soil, the statistical fit was good (R2 = 0.9869), but the fitted curve missed the shape of the data in the region where the applied stress is near (93 to 743 kPa) the preconsolidation stress (395 kPa). In addition, the equation cannot fit a constant level of bulk density at applied-stress levels below the preconsolidation stress. A similar result occurred when the equation was fit to the Rayne soil.
The empirical equation developed by Bailey et al. (1986) fit bulk-density points within 0.033 Mg m-3 (see Eq. [1] in Table 3) for data that was nearly linear on a semi-log plot (Bucks soil from Fig. 1), but the predicted points oscillated along the measured data first predicting low then high then low again in a repeated pattern. The compression index then varies with the level of applied stress making it difficult to calculate from the equation. When the Bailey et al. (1986) equation was used to fit curves similar to the Glenelg soil, the statistical fit was good (R2 = 0.9911). However, as with the Assouline et al. (1997) equation, the fitted curve missed the shape of the data in the region of the preconsolidation stress and was unable to fit the constant level of bulk density at applied-stress levels below the preconsolidation stress. When the Bailey et al. (1986) equation is used to fit data of the type shown by the Rayne soil, the predicted points again smooth out the curve in the preconsolidation region and oscillate about the curve at the higher stresses. While the statistical fit was good (R2 = 0.9902), the shape of the fitted curve was not satisfactory.
In contrast to the previous two empirical equations, Eq. [3] fit all three curve shapes with an average difference of <0.009 Mg m-3 (Table 3). For the data that is nearly linear on a semi-log plot, Eq. [3] fit the data points with an average difference between measured and predicted data of 0.009 Mg m-3, compared with an average difference of 0.033 Mg m-3 for the Bailey et al. (1986) equation and 0.060 Mg m-3 for the Assouline et al. (1997) equation. For the Glenelg soil, Eq. [3] fit the data with an average difference of 0.002 Mg m-3; the Bailey et al. (1986) equation fit the data with an average difference of 0.011 Mg m-3; and the Assouline et al. (1997) equation fit the data with an average difference of 0.014 Mg m-3. For the Rayne soil, Eq. [3] fit the data with an average difference of 0.003 Mg m-3; the Bailey et al. (1986) equation fit the data with an average difference of 0.015 Mg m-3; and the Assouline et al. (1997) equation fit the data with an average difference of 0.017 Mg m-3. Thus Eq. [3] fits all three compression curve shapes better than the best models reported in the literature.
One additional comparison is shown in Table 3 for a Hagerstown sample. This sample represented the poorest fit for Eq. [3]. This data set represents a small number of samples where the bulk density starts to level out and then jumps. It seems probable that this is an experimental artifact, but in all other aspects, these data sets were consistent with the rest of the data sets and were not eliminated from the study. In this case, two values deviate from an otherwise smooth relationship. The Bailey et al. (1986) and Assouline et al. (1997) equations fit this Hagerstown data better than Eq. [3].
In the above comparisons, Eq. [3] was used as a three-parameter model. The goodness of fit increased when the number of parameters in Eq. [3] was increased to five, except in the case of the Glenelg sample (see Table 2 for comparisons). Equation [3] is very sensitive to the initial values of the fitting parameters when it is used as a five-parameter model. Sometimes the parameters met the convergence criteria after iteration, but did not produce a good fit to the data. An acceptable fit, however, was always found using a trial-and-error approach to selecting initial values. The reduction of the number of parameters to three eliminated this problem as long as a reasonable initial value of was used.
The fact that not all curves showed an s-shape, indicating an approach to the particle density as a limiting value, probably resulted from a need to take data at even higher applied stresses to reach levels where the s-shape would be evident. The time response problem alluded to earlier seems more likely to have caused an s-shape when one might not have existed at that level of applied stress than to have kept a curve from showing an s-shape. It is also not clear whether the value of 2.65 Mg m-3 is the right value for m. In a previous evaluation of the data using Eq. [3] as a five-parameter model, m was treated as an empirical parameter and 49 of the 120 data sets had better fits with m < 2.65 Mg m-3. Using m as a constant resulted in better correlations between traditional measures of compression and the coefficients of Eq. [3], as shown in Fig. 4 and 5, than occurred when it was allowed to vary as an empirical parameter. Most of this effect was probably due to data sets where m was larger than 2.65 Mg m-3, in some cases as high as 77.8 Mg m-3.
The sample that produced the outlier in Fig. 3, 4, and 5 was an undisturbed Bucks sample (0.76-1.22 m) run at a water content of 0.173 kg kg-1. This soil produced a compression curve with a very slowly changing curvature over the whole range of applied stress, resulting in a poorly defined preconsolidation stress. Equation [3] fit the data better than the Bucks sample used in Table 3 although not as well as the Rayne or Glenelg samples. This compression curve deviated widely from the traditional concept. Two other replications run on undisturbed samples from this same depth at slightly different water contents resulted in a response similar to the Glenelg in Fig. 1.
During the various evaluations, the problem of using a zero applied stress in equations using a logarithm of applied stress arose and was solved by adding a constant (1) to the applied stress. This practice allowed the various equations to be compared even at zero applied stress using semi-log representations like Fig. 2. In the case of Eq. [3], an error frequently occurred in the nonlinear regression procedure if the applied stress was used without adding the constant (1) but did not occur otherwise.
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SUMMARY AND CONCLUSIONS
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A new empirical equation (Eq. [3]) is proposed for uniaxial soil compression that is capable of fitting bulk-density data for the entire range of applied stresses for data from both disturbed and undisturbed soil at any fixed initial water content. The equation statistically fits compression data better than either the Bailey et al. (1986) or Assouline et al. (1997) equations, previously the most used models for soil compression. This equation satisfies the requirement set by Assouline et al. (1997) that the bulk density has an upper limit at high values of applied stress. The equation does not meet the boundary condition proposed by Bailey et al. (1986) that the slope of the stress–strain curve approaches a constant as the applied stress becomes very large. The equation does, however, slowly approach the upper limit, meeting to some extent the intent of the Bailey et al. (1986) boundary condition. The equation satisfies the other Bailey et al. (1986) condition that the strain should be zero when the soil has had no stress. The equation has not been extended to the variable initial-density and water-content situations studied by McNabb and Boersma (1993)(1996). Instead, the approach has been to fit each situation as a separate entity.
Equation [3] also allows the generation of three material properties: the preconsolidation stress (Eq. [4]), the compression index (Eq. [5]), and the elastic rebound/recompression parameter (Eq. [7]), as shown in the results. Since these properties cannot be generated exactly from the parameters of Eq. [3] and since not all samples match the traditional response, it might be desirable to instead use either the three or five parameters from Eq. [3] with Eq. [3] to represent soil behavior in computer-simulation equations such as those being used by Kirby (1994).
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ACKNOWLEDGMENTS
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The Pennsylvania Department of Agriculture, the Pennsylvania Agricultural Experiment Station Project 3488, and the Penn State Fund for Research funded this work. I thank Joel Hunter and Andrea Long for their dedication and patience in making the experimental measurements. I also thank the anonymous reviewers who provided the incentive to investigate the three-parameter version of Eq. [3].
Received for publication April 24, 2000.
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REFERENCES
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- Assouline, S., J. Tavares-Filho, and D. Tessier. 1997. Effect of compaction on soil physical and hydraulic properties: Experimental results and modeling. Soil Sci. Soc. Am. J. 61:390–398.[Abstract/Free Full Text]
- Bailey, A.C., C.E. Johnson, and R.L. Schafer. 1986. A model for agricultural soil compaction. J. Agric. Eng. Res. 33:257–262.
- Boyland, P., J. Keiper, E. Martin, J. Novak, M. Petkovsek, S. Skiena, I. Vardi, A. Wenzlow, T. Wickham-Jones, and D. Withoff. 1992. Guide to standard Mathematica packages. Ver. 2.1. Wolfrom Research, Champaign, IL.
- Gupta, S.C., and R.R. Allmaras. 1987. Models to assess the susceptibility of soils to excessive compaction. Adv. Soil Sci. Springer-Verlag, New York.
- Gupta, S.C., and W.E. Larson. 1982. Modeling soil mechanical behavior during tillage. p. 151–178. In van Doren et al. (ed.) Predicting tillage effects on soil physical properties and processes. ASA Spec. Publ. 44. ASA, Madison, WI.
- Kilmer, V.J., and L.T. Alexander. 1949. Methods of making mechanical analyses of soils. Soil Sci. 68:15–24.
- Kirby, J.M. 1994. Simulating soil deformation using a critical-state model: I. Laboratory tests. Eur. J. Soil Sci. 45:239–248.
- Koolen, A.J., and H. Kuipers. 1983. Agricultural soil mechanics. p.27–28. Springer-Verlag, New York.
- McNabb, D.H., and L. Boersma. 1993. Evaluation of the relationship between compressibility and shear strength of Andisols. Soil Sci. Soc. Am. J. 57:923–929.[Abstract/Free Full Text]
- McNabb, D.H., and L. Boersma. 1996. Nonlinear model for compressibility of partly saturated soils. Soil Sci. Soc. Am. J. 60:333–341.[Abstract/Free Full Text]
- van Genuchten, M.Th., and D.R. Nielsen. 1985. On describing and predicting the hydraulic properties of unsaturated soils. Ann. Geophys. 3:615–628.
- Wolfram, S. 1991. Mathematica: A system for doing mathematics by computer. 2nd ed. Addison-Wesley Publishing, Redwood City, CA.
- Wu, T.H. 1976. Soil mechanics. 2nd ed. p. 113. Allyn and Bacon, Boston, MA.
- Young, J.L., and M.R. Lindbeck. 1964. Carbon determination in soils and organic materials with a high frequency induction furnace. Soil Sci. Soc. Am. Proc. 28:377–381.
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D. D. Fritton
Fitting Uniaxial Soil Compression Using Initial Bulk Density, Water Content, and Matric Potential
Soil Sci. Soc. Am. J.,
June 21, 2006;
70(4):
1262 - 1271.
[Abstract]
[Full Text]
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ABSTRACT
INTRODUCTION
