Establishment of Bekker's Model for Predicting the Pressure-Sinkage Behaviour of a Loamy Sand Soil By 1*
Ajewole P. O. and 2Manuwa S. I. 1 Department of Agricultural and Bio-Environmental Engineering, The Federal Polytechnic, Ado-Ekiti, P.M.B. 5351, Ado-Ekiti, Nigeria 2 Department of Agricultural Engineering, Federal University of Technology, P.M.B. 704, Akure, Nigeria *Corresponding author, E-mail:
[email protected], Mobile No: 07068675164 Abstract Prediction of pressure-sinkage behaviour of soil is of great interest to the agricultural engineer in the evaluation of soil compaction caused by farm vehicles and to the construction equipment engineer in the assessment of the effectiveness of soil compactors and the like. To predict the pressure-sinkage behaviour of a loamy sand soil, multiple plate penetration tests were conducted using three small rectangular plates. From the pressure versus sinkage relationship of the soil, soil stiffness constants kc , k and n corresponding to Bekker’s model of soil pressure-sinkage were determined. At moisture content of 15.86 % dry basis and different cone indexes of 293, 422, 724 and 11017 kPa, the average values of 331.56 kN/mn-1, 4769.80 kN/mn and 1.2397 obtained for kc , k and n respectively were fitted into Bekker’s equation in order to establish the model for pressure-sinkage relationship for the soil under investigation and also to predict the sinkage-behaviour of the soil under higher loads and larger plates. Keywords: Soil, pressure-sinkage, plates, stiffness constants, Bekker’s model Introduction Soil compaction do cause considerable damage to the structure of the tilled soil and the subsoil and consequently to crop production, and the environment. The prediction of soil sinkage under wheels and tracks is of great importance for determining the off-road vehicle performance and the level of compaction in the agricultural soils. Soil stiffness constants govern the soil sinkage and the behaviour of soil under load (Rashidi et al, 2006). When a load is applied to the surface of a soil a reduction of soil pore volume occurs. Also soil shear at edges of the loading plate takes place. The loaded area sinks into the soil to a certain depth until the soil’s resistive force is in equilibrium with the applied force; therefore, sinkage of soil occurs. Resistance of soil to applied pressure can be characterized in terms of two parameters: cohesiveness, the bonding of the soil particles, and the angle of internal friction which is the resistance of movement between soil particles (Abou-Zeid, 2004). Many different methods and devices have been used for quantifying soil reactions to applied forces, but, penetration resistance and soil density measurement are the most techniques (Ronai and Shmulevich, 1995).
If a track or wheel is pulled a distance L in the horizontal direction, the work done by the towing force, which is equal to the magnitude of the motion resistance due to terrain compaction, can be equated to the vertical work done in making a rut of length L. For the measurement of the vertical soil strength, the plate-sinkage test has become widely accepted. The principle is depicted in Figure 1 and it illustrates the influence of the plates of different sizes used for penetration tests on soil. The result of the measurement is the pressure versus the sinkage (AESCO, 2005). These measured curves can be approximated by a simple exponential equation for homogenous terrain, which was proposed by Bekker (1956):
k p c k z n b
…..(1)
Where p is the contact pressure (kgcm-2), b is the width of a rectangular sinkage plate (cm), z is the sinkage (cm), n is the exponent of deformation, kc is cohesive modulus of deformation (kgcm –(n+1)) and is frictional modulus of deformation (kg cm–(n+2)). The values of p and z are measured while the parameters kc and k and n are derived by fitting experimental data to the equation (1). The value of kc , k and n can be derived from the results of the test with two sizes of plates having different widths as shown in equations (2) and (3): k p1 c k z n ….(2) b1
k p2 c k z n ….(3) b2 On the logarithmic scale, the equations (2) and (3) can be rewritten as follow: k log p1 log c k n log z ….(4) b1 k log p2 log c k n log z b2
….(5)
Figure 1: Principle of the measurement of the vertical soil strength (pressure-sinkage)
Materials and Methods Pressure-Sinkage Test Equipment An equipment for measuring pressure-sinkage behaviour of soils was designed, developed and mounted on an indoor soil bin of Soil Tillage Dynamics Laboratory of The Federal University of Technology, Akure, Nigeria. The equipment consists of a hydraulic loading system for sinking three different plates of different aspect ratios into the soil loaded in the soil bin. The aspect ratios of the plates ranged from 1.5 to 3.0 and are similar to the ones expected for contact area of pneumatic tyres. Table 1 shows the dimensions of the sinkage plates and their aspect ratios. Table 1: Dimensions and aspect ratios of the sinkage plates Sinkage Plate Width (mm) Length (mm) Aspect Ratio Number 1 25 37.5 1.5 2 30 75 2.5 3 50 150 3.0 Experimental Soil A loamy sand soil was used for the experiment. The soil was obtained from Igokoda in the South Western part of Nigeria. The physical and mechanical properties of the soil were determined in the laboratory following the procedures reported by Manuwa (2009) and are shown in Table 2 and 3 respectively. The soil samples collected were spread out on polythene sheets and air-dried for 3 days. The soil was turned at interval to ensure uniform drying. After they have been sufficiently dried, they were then packed and loaded into the soil bin. Table 2: Physical properties of experimental soil Classification Loamy Sand Sand (%) 85.00 Silt (%) 3.00 Clay (%) 12.00 Bulk Desity (kg/m3) 1560 3 Particle Density (kg/m ) 1725 Moisture Content (% dry basis) 7.05 Table 3: Mechanical properties of experimental soil Classification Loamy Sand Moisture content (%) 11.5 3 Bulk Density (kg/m ) 1435 Cohesion (kPa) 4.50 Angle of Internal Friction (degree) 40.6 Adhesion (kPa) 0.16 Cone index at 75mm depth (kPa) 345
Plate Penetration Test Procedure and Collection of Pressure-Sinkage Data The soil bin was filled with dry sample of the experimental soil in seven layers of 100 mm thickness. The soil was then compacted to a cone index of about 293 kPa at 75 mm depth according to the procedure reported in Manuwa and Ademosun (2007). The moisture content of the sample were then determined at this cone index using gravimetric method. Each of the three plates used was then driven into the soil at four different pressure intervals and at a particular cone index. At each pressure interval, the depth of sinkage was read on the slide gauge and recorded. After each cone index test, the soil was removed from the soil bin and loosened by hand shovel. The soil bin was then filled again with the loosened soil and compacted to higher cone index and the value of the cone index was recored. Plate penetration tests were carried out at four different cone indexes. The pressure intervals and depth of sinkage obtained at each cone index were also recorded. This procedure was replicated three times. Determination of soil stiffness constants kc , k and n The graphs of log of pressure, p against log of sinkage z obtained at different cone indexes for the three plates were plotted. From each graph, the slope which equals log k k was obtained where k = c k and b is the width of the plate as shown in equation b (4). Graph of K was then plotted against 1/b. The slope obtained gave the value of kc while the intercept gave the value of k . Result and Discussion Fig 2-5 shows the graphs of log p plotted against log z for each plate at different cone indexes while Table 4 gives the summary of the values of kc and k and n obtained at different cone indexes.
2.4 2.3 2.2 Plate 1 Plate 2 Plate 3
Log P
2.1 2.0 1.9 1.8 1.7 1.6 -1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
Log z
Fig.2: Graph of Log P against Log z for moist loamy sand at moisture content of 15.86% dry basis and cone index of 293 kN/m2
2.4 2.3 2.2
Log P
2.1
Plate 1 Plate 2 Plate 3
2.0 1.9 1.8 1.7 1.6 -1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
Log z
Fig. 3: Graph of Log P against Log z for moist loamy sand at moisture content of 15.86% dry basis and cone index of 422 kN/m2
2.4 2.3 2.2
Plate 1 Plate 2 Plate 3
Log P
2.1 2.0 1.9 1.8 1.7 1.6 -1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
Log z
Fig. 4: Graph of Log P against Log z for moist loamy sand at moisture content of 15.86% dry basis and cone index of 724 kN/m2
2.4 2.3 2.2 Plate 1 Plate 2 Plate 3
Log P
2.1 2.0 1.9 1.8 1.7 1.6 -1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
Log z
Fig. 5: Graph of Log P against Log z for moist loamy sand at moisture content of 15.86% dry basis and cone index of 1017 kN/m2 For the rectangular plates, the values of kc , k and n obtained were in the range of 16.88 to 513.51 kN/mn-1, 3660.60.13 to 5896.00 kN/mn and 1.1685 to 1.3284 respectively at different cone indexes. It was observed that the value of k are significantly higher than the values of kc . This may be due to the frictional nature of the soil. This also agrees with the results of researchers who have worked on similar soil as reported by Rashidi et al (2006 and 2012) and Nang et al (2007). The soil stiffness constants kc and k also increase as cone index increases. It was also discovered that the loamy sand soil sample has high values of exponent of deformation n when compared to values obtained in previous work done on other soil types. This may be due to the loose nature of the soil as it has little resistance to deformation. Table 4: Values of kc and k and n obtained for loamy sand soil at different cone indexes Moisture Content (%) dry basis 15.86 (moist)
Average
Cone Index (kN/m2) 293 422 724 1017 614
kc (kN/mn-1)
k (kN/mn)
16.88 355.35 440.52 513.51 331.56
3660.60 4693.70 5252.40 5472.50 4769.80
n 1.2338 1.2280 1.1685 1.3284 1.2397
The following relationship was obtained for predicting pressure-sinkage behaviour of loamy sand soil when the average values of kc and k and n were fitted into Bekker’s equation:
331.56 p 4769.80 z 1.2397 b The relationship obtained can be used to predict the soil sinkage values under heavier load p and plate width b.
Conclusion A pressure-sinkage equipment was used to carry out plate penetration tests and establish Bekker’s model for predicting pressure-sinkage behaviour of a loamy sand soil . Average values of 331.56 kN/mn-1, 4769.80 kN/mn and 1.2397 were obtained for soil stiffness constants kc and k and n respectively. These values were fitted into Bekker’s model of predicting soil pressure-sinkage behaviour. The relationship obtained can be used to predict sinkage of soils under heavier loads and different plate widths. The data obtained in this study are relevant in the design of tractive devices, evaluation of performance of off-road vehicles and mobility studies of natural terrains. References Abou-Zeid, A. (2004): Distributed soil displacement and pressure associated with surface loading. MSc. Thesis, University of Saskatchewan, Canada. AESCO (2005): Matlab/Simulink Module. AS2TM User’s Guide. Hamburg. Bekker, M. G. (1956): Theory of land locomotion-the mechanics of vehicle mobility. University of Michigan Press, Ann Arbor, MI: 522pp. Manuwa, S. I. (2009): Performance evaluation of tillage tines operating under different depths in a sandy clay loam soil. Soil and Tillage Research 103 (2009) 399-405. Manuwa, S. I. and Ademosun, O. C. (2007): Draught and soil disturbance of model tillage tines under varying soil parameters. Agricultural Engineering International: the CIGR Ejournal. Manuscript PM 06 016. Vol. IX. March, 2007. Nang, N. V., Takaaki, M., Tatsuya, K. and Shikegi, I. (2007): Experimental device for measuring sandy soil sinkage parameters. Bulletin of Faculty of Agriculture Rashidi, M., Keyhani, A. and Tabatabaeefar, A. (2006): Multiplate penetration tests to predict soil pressure-sinkage behaviour under rectangular region. International Journal of Agriculture and Biology. http://www.fspublishers.org. Rashidi, M, Fakhri, M., Sheikhi, M., Azadeh, S and Razavi, S. (2012): Evaluation of Bekker Model in Predicting Soil Pressure-Sinkage Behaviour under Field Conditions. Middle-East Journal of Scientific Research 12 (10): 1364-1369, 2012. Ronai, D. and Shmulevich, I. (1995): Comparative analysis of some soil compaction measurement techniques. International Journal of Agrophysics, 9: 227-233.
