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An Improved Empirical Equation for Uniaxial Soil Compression for a Wide Range of Applied Stresses D. D. Fritton* ABSTRACT 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 offour soils, and from soil mixed with four different amounts of sand, were subjected to applied stresses ranging fro contents.

Data froml 8 resulting curve shape Swipe to view nentp empirical equations. urve shapes better t four initial water senting the three existing and new nd that fit all three quations. 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 tress, compression index, and elastic rebound/recompression parameter) and the parameters of to page of the new equation. 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. Balley et al. (1986) introduced an equation, In( ) In( o) (a b )(1 3 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 suficient representation of soil compaction for agricultural activities.

This work started with the intent of evaluating the relationship between the compression index, a parameter denved from uniaxial soll-compression data, and the clay content reported by Gupta and Larson (1982). The extraction f a compression index from the data, however, required arbitrary decisions due to deviations from linearity between bulk density and applied stress at both the Iow- and highstress ends of each data set. A better description of uniaxial soil-compression data is needed.

It is the pur ose of this article to present a new empirical equation for uni ression that is capable of 28 article to present a new empincal equation for uniaxial soll compression that is capable offitting soil bulk-density data for the entire range of applied stresses for both disturbed and undisturbed soil at any fixed initial water content. ree 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. here is soil bulk density (Mg m ) 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 ultiple 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, max Assouline et al. 1997) introduced an equation, where max is an empirical parameter (Mg m ) 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. Objective and Proposed Equation 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 niaxial soil-compression curve shapes. The equation chosen is analogous to the equation extensively used to fit waterretentlon data (van Genuchten and Nielsen, 1985).

With soil bulk density replacing volumetric water content and applied stress plus one replacing soil water press quation becomes 4 28 soil bulk density, the particle Abbreviations: ci, compression index; K, elastic rebound/ recompression parameter; pC, preconsolidation stress. 678 FRITTON: IMPROVED EQUATION FOR UNIAXIAL SOIC COMPRESSION 679 density (Mg m 3 ), is an empirical parameter (kPa 1 ), n and m are nitless empirical parameters, and other symbols retain their previous definitions. MATERIALS AND METHODS 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. 5 to 1 m) soil in dry mass (sand/soil) ratios of 1:4, 2:3, 3:2, and to extend the range of particle-size distributions studied. Particle-size distribution was determined according to Kilmer and Alexander (1949). Organic C was determned 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. Fig. 1. Bulk densi roperties of the two components, assuming the white quartz sand had no organic C.

Fig. 1. Bulk density of three soils plotted as a function ofthe applied stress plus one. Points represent experimental data. The smooth lines are best-fit nonlinear regression curves based on the threeparameter version of Eq. [3]. 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. n addition, approximately six 63. 5-mm-diam. undisturbed-soil cores were taken to a depth of 1 m with a hydraulically driven soil ampler 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 seve. Air-dried rock fragments were weighed and used to calculate rock-fragment content as a fraction ofthe whole sample The sieved soil was poured into two 63. 5-mm-diam. rings, which were 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 in 6 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 hich the bulk samples were collected, contained no obvious distortions from sampling and had no displaced rock fragments. A 63. 5-mm-dlam. 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 contenta Water content was determined on the sample after compression, but could not be used in most ases 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 soil was then trimmed until the sample was level with the top of the compression cylinder. The excess soil was weighed to etermine 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 pplied to the top ofthe sample through a brass plate placed over the top porous stone using a triplebeam 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 leve’ of stress for at least 30 min. Applied stress leveis of O, 31, 62, 93, 186, 371, 557, 743, 11 14, 485, and 2971 kPa were used for all soils except the Hagerstown and Hagerstown/sand mixtures where 0, 31, 62, 186, 557, 1 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 of25. 4 mm, the volume of the soil and the soil bulk density were then calculated. Following the highest level of stress, the soll 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-mmdiam. 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. 680 SOIC SCI. SOC. AM. J. , VOL. 65, MAY-JUNE 2001 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. The various points and lines are discussed in the text to illustrate calculations. 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. [2], and [3] density as a function ofthe logarithm of stress were used to evaluate Eq. [2], and [3] using the NonlinearFit package in Mathematica (Wolfram Research, Champaign, 11_) (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 O to 2971 kPa, with a program written using Mathematica (Wolfram, 1991). The preconsolldation stress (pc ) was calculated using the Casagrande procedure described by Wu (1976). Th’s 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 ofthe Rayne soil. The Casagrande procedure starts by determining the point of maximum curvature for the ata Table 1.

Soil characteristics. 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 cume. Since the compression data were not always linear from the point of maxmum curvature to the point at the highest level of applied stress, this line (FG) wa 0 DF 28