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Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.

Volume 9
Issue 4
. , EJPAU 9(4), #35.
Available Online:



Results of measurements of a vertical deformation of soil of four forest site types, i.e. fresh mixed coniferous forest (fMCF), boggy mixed pine forest (bMPF), moist mixed broadleaved forest (mMBF), and moist broadleaved forest (mBF), using a plate bevameter and taking into account different moisture variants, are presented in this paper. The moduli kc, kφ, and n of Bekker’s equation were determined. The soils were grouped according to their resistance to vertical deformation through agglomeration using the Ward’s method. The Bekker’s equation moduli were determined once again for separated clusters.

Key words: .


Properties of the terrain have a crucial importance for traction possibilities of vehicles, such as developed driving forces, slips, and rolling resistance [1, 9, 22].

The rolling resistance is one of significant factors deciding about power losses of terrain vehicles. Since they result from formation of a wheel track, a method of their determination is based on a mathematical representation of vertical pressures caused by penetration of a plate (a model element) and measured in the field. A basic equation, developed at the beginning of the last century (eq. 1) shows that penetration of the element pressed into the soil z depends on its pressure per unit area p, the modulus of soil rigidity k, and the soil state (moisture, cohesion, density) expressed by a dimensionless exponent n [5, 7, 10, 14, 15, 17, 20, 21].


However, investigations of Bekker [2, 3] showed that the modulus of rigidity k significantly depends on dimensions of the element acting upon soil, especially on its width. Therefore, he modified the equation 1 introducing width of the acting element b, and two moduli of deformation (kc and kφ), thus separating the effect of cohesion c from the effect of the soil internal friction angle φ. This way, the equations 2 was developed.


In laboratory as well as at field tests a high agreement between measured values and those calculated by equation 2 was obtained. Moduli kc and kφwithin the range of occurring values turned out to be independent of the loading surface [6, 8, 13, 16, 19, 23].

The moduli of rigidity and exponent of soil state are being determined on the basis of fitting of equation 2 to experimentally obtained vertical pressures in relation to penetration depth of the measuring element. A knowledge of them for a given soil permits to calculate the rolling resistances for low-pressure tires (eq. 3), taking into account characteristics of both the soil and the driving wheel [20].


The measuring devices used in determination of soil vertical pressure curves, called plate bevameters, measure the force with which plates are driven into the soil, registering at the same time the depth of their penetration [11].

Within the domain of soil traction characteristics concerning agricultural soils great amount of research have been undertaken worldwide. As far as the forest soils concerns little attention was paid to these problems which are considered important in effective use of vehicles. Undertaken research contributes to better knowledge on forest soils reaction to vertical pressures, thus making possible the assessment of their tractability.


The objectives of this study were:



The measurements were carried out with a plate bevameter designed and built at the Department of Forest Works Mechanization, Agricultural University of Cracow [11]. They were conducted in the Niepołomice forest in four types of forest site, i.e. fresh mixed coniferous forest (fMCF), boggy mixed pine forest (bMPF), moist mixed broadleaved forest (mMBF), and moist broadleaved forest (mBF), which together with different moisture states gave 8 measuring variants. On each site, 4 blocks, corresponding to 4 replications, were established. Penetrations of the soil with three measuring plates of different size were conducted along each block, making one passage for each plate size.

The measurements of volumetric bulk density and moisture content were made according to generally accepted methods [12] in order to determine physical properties of the investigated soils.

Data processing and calculation of Bekker’s equation parameters

Values of the force used to press the plates into the soil were registered by the computer every 1 mm of plate penetration, till the depth of 0.1 m. A text format of created files permitted to process them in the program MS Excel, where forces exerted on surface of measuring plates were calculated thus obtaining for each plate the pattern of pressure-sinkage relationship.

Such factors as the number of replications, the number of measuring plates of different dimensions, their shape and velocity of their driving into soil, affect the accuracy of determination of traction parameters kc, kφ, and n [2, 3, 20]. In this study three circular plates of radii 15 mm, 20 mm, and 25 mm were used. They were pressed into the soil with velocity of 0.01 m/sec. In each measuring variant there were 30 replications made for each of 4 blocks, i.e. 120 for each site type.

In equation 2, assumed to represent the pattern of pressure-sinkage, a plate width b was replaced by plate radii r1, r2, and r3 (eq. 4).


Fig. 1. Pressure – sinkage curves for three penetration plates (based on the Authors’ research)

The field measurements yielded curves of vertical pressures in function of plate penetration depth p = f (z) (FIGURE 1). They were plotted in a coordinate system of a double logarithmic scale, thus obtaining lines close to straight lines, to which regression lines were fitted (FIGURE 2).

Fig. 2. Pressure-sinkage curves presented in double logarithmic scale (based on the Authors’ research)

The direction coefficients of regression lines corresponded to dimensionless coefficients of soil state n. In the case when they were different for each of measuring plates the resultant n was equal to their arithmetic mean [3, 14, 15].

To determine constant values of k called moduli of soil rigidity, logarithm of equation 4 was found for each measuring plate, obtaining

log (p)= log (k) + n log (z)            (5)

from which for depth z = 1 will be

n log (z) = 0                                (6)

and thus:

log (p) = log (k)                             (7)

Constant values of soil rigidity k1, k2, and k3 corresponded to measuring plates r1, r2, and r3, differing in dimensions.

In the case of using more than 2 measuring plates a diagram of calculated values of k in function of ratio l/r is made. The slope of a straight line fitted to marked points is equal to kc, while the intersection of the fitted line with y-axis points out the value of kφ [3, 14].

Fig. 3. Method of determination of cohesive kc and frictional k φ moduli using more than two penetration plates (based on the Authors’ research)

For a complete measuring range, i.e. to depth of 0.1 m, parameters kc, kφ, and n were calculated according to methods assumed earlier. Then, they were introduced into equation 4, together with radii of measuring plates on the basis of which they were determined, and vertical pressures were determined for the depth range from 0 to 0.1 m, with a 1 mm increment. Curves obtained from the model along with experimental ones were plotted in the system of logarithm coordinates, and also in the system of uniform coordinates, and then the obtained parameters were evaluated. When estimating model (calculated) and actual (from measurements) vertical pressures it was assumed that situation when calculated curves were in the area limited by curves from measurements, or in their vicinity, is the correct one. If this was not fulfilled the parameters kc, kφ, and n were determined once again, but this time using plate pairs: 15 and 20 mm, 20 and 25 mm, and 15 and 25 mm.

If the calculated curves were much different from the measured ones, the analyzed depth range of plate penetration was narrowed. This was in agreement with methods used in studies of this kind [3], when in the case of heterogeneous soils, characterized by a laminar structure of different strength, the division of a curve obtained from measurements into sub-ranges is advised.

In this study the investigated variants were grouped in respect of their resistance to vertical pressure. Using the Ward’s method of agglomeration the similarities between pressure-sinkage curves were searched for. For separated clusters the curves representing mean data values obtained in variants belonging to a given cluster were determined, and the Bekker’s parameters were calculated according to assumed methods.


Physical and mechanical characteristics of investigated substrata

The rule followed in studying the described soil parameters was to conduct measurements for two different moisture states of soil of a given forest site type in a single place and in time when the change in moisture was perceptible.

The denotations w1, w2, and w3 placed at the name of sites identify them in respect of soil moisture, and most often a higher index number mean a higher moisture content. TABLE 1 contains data on volumetric bulk density and moisture of soil during measurements on all measuring sites.

Table 1. Soil moisture and bulk density on the investigated sites



(by weight)

Dry bulk density

















































mMBF _291_w2











mMBF _244_w3






















mBF _w2











VC – variation coefficient

Bevameter measurements on site of fMCF were carried out for two moisture states (TABLE 1). Podzolic soils on this site represented the soil textural group of a coarse sand over a loose one [4]. Measurement results are shown in FIGURE 4.

The shape of curves (FIGURE 4) and lower values of vertical pressures for plates of a larger radius permitted to conclude that to depth of 50 mm the soil was being deformed as a homogeneous formation [3]. At higher depths diagrams of vertical pressures began to differ.

According to assumed methods, curve fragments and the number of measuring plates were chosen for calculations on the basis of an initial estimation of parameters of a vertical soil sinkage determined for a variable depth range and a variable number of plates.

Fig. 4. Pressure-sinkage curves of soil measured on fresh mixed coniferous forest (fMCF)

In the case of the first moisture variant (fMCF_w1) the depth range to 50 mm, and the measurements results obtained on the basis of two measuring plates: r=15 mm and r=20 mm, were accepted as optimum.

For the second moisture variant of this site (fMCF_w2) calculations were carried out on the basis of a curve fragment to the depth of 70 mm for all three plates.

Calculated traction parameters for this site, determined for the model equation 4, are presented in TABLE 2.

Table 2. Traction parameters of soil on fresh mixed coniferous forest (fMCF)

Site and the level of moisture




fMCF _w1




fMCF _w2




The soil of the fresh mixed coniferous forest (fMCF), evaluated according to value n, should be accepted as a resistant one. Moisture diversification (TABLE 1) turned out to have a small effect on the assigned parameters.

In the case of this site (fMCF) curves in their initial phase have a characteristic bend towards higher values (FIGURE 4). This fragment corresponds to compacting of the surface organic layer. During action of the plate upon this layer it was not pierced, but within the area of the plate effect soil particles came closer to one another. This slow consolidation (FIGURE 4) took place till the moment when there was a rapid increase of vertical pressure values, with values of the soil state exponent n much higher than in the initial part of curves. On diagrams shown in FIGURE 4 this takes place at pressure of about 0.1 MPa with corresponding penetration depth of about 15 mm. Further increase of vertical pressures as the plate was getting deeper lasted till about 1.0 MPa, and then a characteristic bending took place, manifesting piercing of the top more resistant layer, and the plate reaching a less resistant layer (sand) situated below.

On soils of a fresh mixed coniferous forest (fMCF) the organic layer successfully prevented formation of deep wheel tracks. This was true till breaking of the surface layer by action of elements of high pressures per unit area, or through break during action of tangential stresses [12].

Fig. 5. Pressure-sinkage curves of soil measured on boggy mixed pine forest (bMPF)

A muck soil of a boggy mixed pine forest site (bMPF) represented a textural group of a marshy muck over a coarse sand [4], and was characterized by a low bulk density (TABLE 1). This was also confirmed by the relationship between vertical deformation of soil and exerted pressures (FIGURE 5). The shape of curves indicates that it was a homogeneous medium being in a state of low density. The maximum specific pressures did not exceed 0.7 MPa.

The depth range of 0–100 mm was accepted as the optimum for which on this substrate the pressure-sinkage model parameters should be determined. Calculations were based on results from all three measuring plates.

Table 3. Traction parameters of soil on boggy mixed pine forest (bMPF)









It may be concluded on the basis of results that the soil associated with this site had a low resistance to vertical stresses, and the formation of deep wheel tracks may be expected, even at low specific pressures. In the case of this site the vegetation layer only slightly protected soil against vertical stresses.

Soils tested in two compartments associated with a moist mixed broadleaved forest site (mMBF) were not distinctly diversified in respect of consolidation determined by bulk density measurement (TABLE 1). There were differences in moisture, and this was an experimental intention. In the case of mMBF in compartment 244 the soil was described as a pseudogley soil belonging to a textural group of a medium sand deposited over a coarse sand, while in compartment 291 it was sand loosely interbedded with a skeletonless coarse sand [4].

In this point it should be noticed that planning and realization of an experiment of this kind under natural conditions are very complicated. Apart from a correct laying out of the measurement areas it is necessary during the entire study period to react to changing conditions, especially under the effect of weather variability.

The pressure-sinkage pattern of soils of a moist mixed broadleaved forest (mMBF) in three moisture variants (FIGURE 6) showed that these were substrata composed of two layers. A layer of a rather low strength was present under a surface layer of a considerable resistance. In the case of smaller measuring plates the maximum values, manifesting the breaking of the surface layer, were distinctly marked. In the case of these soils, similarly as in the case of soil of a fresh mixed coniferous forest (FIGURE 4), the shape of curves in the depth range to 15 mm reflected a gradual crushing of the surface organic layer.

Fig. 6. Pressure-sinkage curves of soil measured on moist mixed broadleaved forest (mMBF)

Shape of experimental curves (FIGURE 6) shows a distinct effect of moisture on vertical pressures depending on plate sinkage. In the case of the first two soil moisture variants on this site (mMBF_w1 and mMBF_w2) a laminar soil structure was quite evident, while in the third variant (mMBF_w3) this diversification was small due to a high moisture.

The depth ranges used to determine parameters of a vertical soil deformation were as follows: 0–70 mm for mMBF_w1, 0–40 mm for mMBF_w2, and 0–60 mm for mMBF w3. The traction parameters (TABLE 4) for the variant of the lowest moisture (mMBF_w1) were calculated on the basis of two, 15 and 20 mm, plates, while in the case of the remaining variants all three plates were taken into consideration.

The determined parameters (TABLE 4) showed that soil of a moist mixed broadleaved forest (mMBF) was characterized, similarly as soil of a fresh mixed coniferous forest (fMCF) by a considerable strength. Its damage was also prevented by an organic layer overgrown with rich vegetation. This prevented formation of deep wheel tracks. The lower layer in the investigated soils of a moist mixed broadleaved forest (mMBF) was distinctly less resistant, which turned out in measurements of the first two moisture variants (mMBF_w1 and mMBF_w2). On this site, out of all analyzed substrata, a drop in soil resistance to vertical pressures in the moment of punch breaking through a more resistant upper layer was most evident.

Table 4. Traction parameters of soil on moist mixed broadleaved forest (mMBF)





mMBF _w1




mMBF _w2




mMBF _w3




Bevameter measurements on a moist broadleaved forest site (mBF) were carried out for two moisture variants. Practically there were no differences in volume density (TABLE 1). The soil of this site consisted of a skeletonless brown soil representing a textural group of a heavy loam underlain by clay [4].

Fig. 7. Pressure-sinkage curves of soil measured on moist broadleaved forest (mBF)

The pattern of curves derived from field measurements (FIGURE 7) indicates that this soil was homogeneous in the entire range of measurement depths to 0.1 m. The same graph also shows that even small differences in moisture (TABLE 1) may cause in heavy soils a distinctly different resistance to deformations.

The shape of curves reflects a limited occurrence of the organic layer over the soil of the investigated site of a moist broadleaved forest (mBF), because in the initial phase of the chart there is no section corresponding to an elastic compaction, characteristic for soils of the remaining investigated sites.

When selecting fragments of experimental curves (FIGURE 7) for determination of coefficients the depth range of 0–80 mm was assumed for the variant mBF_w1, and the range 0–100 mm for the variant mBF_w2. In both cases calculations of the traction parameters for this soil (TABLE 5) were based on measurements carried out using all 3 plates.

Table 5. Traction parameters of soil on moist broadleaved forest (mBF)





mBF _w1




mBF _w2




Values of Bekker’s parameters (TABLE 5) show a considerable resistance of the soil associated with a moist broadleaved forest of a lower moisture variant (mBF_v1).

Agglomeration of investigated substrata on the basis of load capacity

The agglomeration yielded four clusters (FIGURE 8) listed in order of decreasing resistance to vertical pressures:

Cluster 1: mBF_w1
Cluster 2: mMBF_w1, mMBF_w2, mMBF_w3, mBF_w2
Cluster 3: fMCF_w1, fMCF_w2
Cluster 4: bMPF.

Fig. 8. Clusters of the investigated sites made according to pressure –sinkage curves

To determine the traction parameters of respective clusters (TABLE 6) the fragments of pressure-sinkage curves corresponding to the following depths were chosen: 0–80 mm for the first cluster, 0–60 for the second, 0–50 for the third, and 0–100 for the fourth. Data of all three measuring plates were used in calculations, with the exception of cluster 2 where data of two larger plates were used.

Table 6. Terrain properties for predicting vertical deformation of soil for obtained clusters

Emerged clusters








Cluster 2




Cluster 3




Cluster 4





  1. The soils of the fresh mixed coniferous forest (fMCF) and moist mixed broadleaved forest (mMBF) are characterized by considerable strength with small effect of moisture on the calculated parameters.

  2. The soil associated with boggy mixed pine forest (bMPF) had a low resistance to vertical pressures and deep wheel tracks may be expected even at low specific pressures.

  3. Results obtained for moist broadleaved forest (mBF) indicate that even small differences in moisture may cause here a distinctly different resistance to deformations.

  4. The agglomeration of the investigated forest soils according to their strength showed that a forest site type determines the resistance of a substratum to a vertical deformation.

  5. Due to a stratified structure of forest soils, in the determination of Bekker’s parameters it is required to make a division of curves obtained from measurements into subranges which are analyzed separately.

  6. In distinguished subranges the Bekker’s pressure-sinkage model obtained a good compatibility with curves derived from measurements.


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b – smaller dimension of the penetrating plate (mm, m)
c – soil cohesion (kPa)
f – coefficient of rolling resistance (-)
Ff – rolling resistance (kN)
k – modulus of soil rigidity (MPa)
kc – soil cohesive modulus (MPa/m n)
kφ – soil frictional modulus (MPa/m n+1)
n – exponent of soil deformation
p – vertical pressure (MPa)
pi – tyre inflation pressure (MPa)
pc – tyre rigidity (MPa)
r –radius of the penetrating plates (mm)
z –sinkage (m)


Accepted for print: 24.11.2006

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.

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