Soil stress distribution under lugged tires

Soil & T_illage . ELS EV I ER

Soil & Tillage Research 32 (1994) 163-181

KeSearcn

Soil stress distribution under lugged tires K. H a m m e l Institute of Soil Science, University of Hohenheim, Emil- 14blff-Strasse, D- 70599 Stuttgart, Germany Accepted 16 June 1994

Abstract The spatial distribution of stress under a lugged tire was investigated by model calculations and a field experiment. The stress model used is based on the analytical solution of the linear elastic case and was in satisfactory agreement with the field experiment. The spatial distribution of stress and resulting deformation is considered from several viewpoints. As expected, a tire resting only on its lugs introduces a large heterogeneity in the stress state near the soil surface, leading to deviations of the vertical stress at 5 cm depth of up to 100%. Additional deviations in the model are an artefact of the coarse resolution of the surface load. When calculated stress was related to Mohr-Coulomb theory, failure zones under the lugs occurred, unlike with a smooth tire. A driving force, which acts horizontally on the surface, will enhance shearing deformation and failure by introducing tensile stresses in front of the tire. Generally the heterogeneity introduced at the surface decreased with depth. To relate stress occurring in the field to laboratory measurements, the principal stresses and their time dependence were calculated for a commonly-used sample size. Since stress and strain states that can be obtained in laboratory tests are restricted, the model may help the choice of the proper test and loading paths for studying deformation behavior of a soil under a moving tire. Keywords: Soil stress distribution; Deformation behavior; Lugged tire; Stress model

1. Introduction As a result o f continuing m e c h a n i z a t i o n in agriculture, wheel loads have steadily increased. F o r instance the average t r a c t o r m a s s in G e r m a n y m o r e than doubled between 1958 a n d 1981 (Sch6n a n d Olfe, 1986). This has led to greater m e c h a n i c a l forces acting on the soil with a growing risk o f irreversible d e f o r m a tion. Soil d e f o r m a t i o n generally changes hydraulic processes like water m o v e m e n t , infiltration a n d surface r u n o f f ( R e i c o s k y et al., 1981; Walker a n d Chong, SSDI0167-1987(94)00416-C

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1986). Therefore, direct effects on plant growth and on neighboring ecosystems must be expected. To study the effect of deformation on soil properties, it is helpful to distinguish between four main types of deformation behavior. As long as the soil remains continuous in a macroscopic sense, deformation can be described by shear, compression and plastic flow. In this case, deformation may be defined by the strain tensor for small deformations. If the displacement of small volume elements becomes discontinuous, failure occurs and the deformation can no longer be represented by the strain tensor. Depending on loading conditions under a tire and on soil strength properties, a combination of these two types of deformation can happen, where the dominant type of deformation may vary spatially. Different types of deformation have different effects on soil properties. Compression reduces soil volume, mainly of large pores, and increases soil bulk density, penetration resistance and bearing capacity. Shear will change the shape of a volume element and thus the geometry of the pore space, while the volume remains constant. Wheeling of wet soil can cause remoulding and plastic flow near the soil surface which may increase the soil volume but reduce the volume of large pores, their continuity, and shear strength (Bradford, 1981 ). The processes described often result in a lower root density (Tardieu, 1988 ). The consequent decreasing in root uptake may in turn lead to an increasing risk of ground water pollution by nitrogen and pesticides. The process of soil deformation is determined by the external loading conditions, which create the stress distribution in the soil. These spatially and temporally varying stresses cause a deformation the extent of which is determined by the mechanical properties of the soil. In terms of continuum mechanics, the state of stress is completely determined by the stress tensor, which is a second rank, symmetric tensor with six independent components. The first approach to modelling stress distribution in soils has been to assume soil to be a perfectly elastic body. Formulation of the force balance for a differential volume element at rest, leads to a system of differential equations, which were solved about 100 years ago by Boussinesq for a vertical point load (Jumikis, 1984), and by Cerruti for a horizontal point load (Smoltczyk, 1988). Soil was idealized as a homogenous elastic semi-infinite medium. To obtain these solutions it was necessary to invoke a stress-strain-relationship, which was Hook's law describing isotropic and elastic behavior. However, description of stress state by elasticity theory was not satisfactory for soils which naturally vary in water content and density. Therefore FrShlich (1934) introduced the so-called concentration factor to account for certain soil conditions. I am not aware of any attempt to relate this concentration factor to any theoretical stress-strain concept and will thus consider it as an empirical quantity. Since the governing equations are linear, the stresses under a loaded area may be calculated by the principle of superposition from the stress caused by a point load. This was mostly done for simple forms of the contact area like circles, rectangles or ellipses. The surface load distribution was often chosen to be uniform or to vary according to a power law (SShne, 1958; Johnson and Burt, 1990). Although soil is treated as a homogenous continuous medium, these

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165

types of calculations of stress distribution were successful in predicting vertical soil stress distribution (Van den Akker, 1988 ). Moreover, the concentration factor was used to characterize stress distribution behavior of soils with respect to texture, type of aggregation and soil matric potential (Horn, 1988; Lebert et al., 1988). Model calculations of soil stress state can only be verified if the quality of measurements is known. Measurement uncertainties may occur due to the stiffness and size of the pressure cell (Bailey et al., 1988 ), due to lack of contact between pressure cell and the soil, and due to spatial variability. The measurement accuracy may be checked if several pressure cells are placed in an horizontal plane at different distances from and orthogonal to the track line. Integration of the measured vertical stress distribution over the area must be equal to the wheel load. The possibility of estimating the accuracy of stress measurements in this way was pointed out by Van den Akker and Carsjen ( 1989 ). As most tires are lugged, the design and density of these lugs may play an important role in influencing soil stresses near the surface, especially on firm soils where tire sinkage is small. Koolen et al. ( 1992 ) reported significant lug effects on the measured vertical stress at 15 cm depth. The objective of this study is to use Frrhlich's concept of soil stress distribution for the situation of a lugged tire and to relate the model calculation to measurements on a field site under permanent grassland. The model calculations yield the spatial and temporal distribution of the stress field and may be used to plan laboratory experiments or to calculate zones of shear failure, when the shear parameters are given. 2. Theory In the following analysis, soil is assumed to be a homogenous elastic semi-infinite medium. The force balance of a differential volume element at rest leads to a set of differential equations. To arrive at a complete description, a constitutive relation between stress and strain must be provided. I will assume that Hook's law of linear and isotropic elasticity is valid. Then there are two elastic constants, Young's modulus E and Poisson's ratio #, which is the ratio between axial and radial deformation. Only the latter has an influence on stress state (Eq. 1 ). If E is given, p determines volume change. This may vary from p = 0 , where volume change is largest, t o / t = 0.5, where volume remains constant. Consider a point force f a c t i n g on the surface at the origin, let z denote the vertical axis pointing downward, and let x and y denote the horizontal axes. Denote its vertical and horizontal components byf~ and fx, respectively, and let the tensor S with components tro be the resulting stress in the soil. Then (Jumikis, 1984; Smoltczyk, 1988 )

3ij (1-2#) ao=2rtRs(Zfz+Xfx ) - 2r~R2 Cij ,

i,j6{x,y,z}

(1)

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where

Cx~ ~I'R(xE-y2) Y2Z'~ , I x (3R+z)x 3 3xR "~ =]z~ r2(R+z) "l'~r2)+Jx~-~'~ ( R . I . z ) 3 R (R+z)U C

t'(R(y2-yZ)

xZz'~ . [ x

(3R+z)xy 2 +

[

xyz'~ - zR Rr2)+Jx~~

2Rxy

Cxy=f~r2--~-z)

(R+z)3R [

( 3R + z )x2"~ 1 ~ ]

Cxz=Cyz=czz=o and

R=x/x +y2+z 2, The cartesian stress tensor S is symmetric and may thus be diagonalized by an orthogonal transformation, producing the principal stress tensor

Sp =N'SN

(2)

where the columns of the matrix N are orthonormal vectors. The off-diagonal components of Sv vanish and the diagonal components ai, are called principal stresses. They are usually ordered with decreasing magnitude such that 0.1> o'2> 0"3 (the second index is dropped as it always equals the first). The columns of N point in the directions of these principal stresses. N and ai may be readily obtained from S by calculating its eigenvectors and eigenvalues. The principal stress tensor Su also provides information about the type of deformations if it is interpreted in terms of elasticity. Deformation then consists of compression and shear, where compression is the change of volume at constant shape and shear is the change of shape at constant volume. Compression may be associated with the hydrostatic stress SH and shear with the deviatoric stress So. These are defined as

Sp=S.+S,,

(3)

with components 3

aH=am=~ ~ ai and aD=ai--am i=l

(4)

where 0.,. is the mean principal stress. Since all off-diagonal components are zero, SH and SD may be represented as vectors and characterized by their lengths

I~H=N/c30.mand

0.0=

i~= 0.i-a21-1

(5)

Since the vertical stress 0"zzin soils tends to concentrate around the loading center due to increasing stiffness with depth and increasing plasticity with greater water contents, FrShlich (1934) introduced an empirical concentration factor v.

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To keep the stress state physically consistent, he had to assume deformation at constant volume, consequently/~ = 0.5. One can show that stress state is then onedimensional in a spherical coordinate system. Thus the spherical stress tensor is a principal stress tensor, where only the component in the radial direction is different from zero. Moreover, tensile stresses do not occur. This simplifies the stress tensor (Eq. l ) considerably but does not change any stress components, for which at least one index refers to the z-direction (axz, ayz and trzz). If the surface load exerts mainly a vertical force, the dominant stress component will be a~. Deformation behavior, which leads to that stress state, differs from linear isotropic elasticity however. Loads are often distributed over some area. Since the stress distribution depends linearly on the applied force, point loads may be superimposed in order to approximate a loaded area. Linearity is satisfied as long as Eq. ( l ) is a valid description (i.e. as long as the deformation of a volume element remains small). For a set o f n point loadsfik,fx~ at (Xk, Yk, Zk=0) and with FrShlich's modifications, the stress component trijat point (z, y, z) becomes

ao= ~ u(i--ik(J--fi)(Z~-2fi~+(X--XklX--xkl"-3f~) k=l

2nR~ +2

i,je{x,y,z}

(6)

where

Rk = ~/(x--xk)2 + (Y--yg)2+zZ One can show that for u = 3 and/t = 0.5 Eq. (6) represents linear elastic behavior. The stress state in a soil is not only introduced by external loads like vehicles, but the weight of the soil itself creates a geostatic stress, which is indicated by the superscript g. If a soil is undisturbed in a geological sense and level, there are no shear stresses (i.e. a,~ = 0, i ~ j and tr~ are principal stresses ). It follows from symmetry that the horizontal stresses are equal (i.e. tr~ = axx g - ayy g ). The geostatic stress state can be formulated as a~ = a~zKo z

agzz=g~pb(z' )dz'

(7)

0

where Pb is the dry bulk density, K0 the coefficient of earth pressure at rest, and g is the acceleration due to gravity. While the influence of external loads on the soil surface decreases with soil depth, geostatic stress increases. The geostatic stress was not included in the model calculations, as only the difference between the natural stress state and that with an external load was considered. Only the estimation of failure zones would require geostatic stress, but since all calculations were done for relatively small soil depths, the influence of geostatic stress was assumed to be neglible. However, there would be no difficulty in including geostatic stress if necessary.

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3. Material and methods

The pressure cells used in the field experiment were cylindrical with a diameter of l0 cm and a height of 2.8 cm. They consisted of a rigid body with an oil-filled cylindrical volume which was enclosed at the top by a strengthened rubber membrane. The rigid upper circular edge was narrow to prevent bridging. Soil stress was measured perpendicular to the membrane over an area of 80 c m 2. The oil pressure was measured with a piezoresistive pressure sensor, which was mounted inside the cell. The oil volume was connected to a 50-cm long tube to introduce or to retract oil. The pressure cells were installed horizontally from a trench, to keep the soil above them undisturbed. When the cells were installed a little oil was added to maintain sufficient contact. Three pressure cells were installed at 25 cm depth, one in the center of the track line, the second at a distance of 15 cm and the third at a distance of 30 cm perpendicular to the driving direction. The soil (Stagnic Cambisol) was a silt loam used as permanent grassland at the experimental farm Ihinger H o f near Stuttgart. Porosity was 0.53 at l0 cm depth, 0.58 at 25 cm and 0.54 at 45 cm. There was no plough pan and the well aggregated top layer had a greater content of organic matter and a greater root density than comparable arable soils. The soil surface had uniform turf cover. The volumetric water content at the time of the experiment was 0.34 at 10 cm depth, 0.33 at 25 cm and 0.38 at 45 cm. A 4 wheel-drive tractor (Mercedes-Benz Trac 8001 ) with a total weight of 56.9 kN was used for the experiment. The tractor moved in first gear at low revs and its speed, 0.21 m s -a, was calculated from the time differences of the peak values and the spatial distance of the cells. The experiment was run with Trelleborg 500/60-26.5 tires inflated to 80 kPa, and a 19 kN-wheel load on the rear axle. The signals from the pressure cells were recorded on-line with a digital data acquisition system. The driving force was assumed to be neglible, since the tractor had only to overcome the rolling resistance. In the following only one rear wheel is considered. To determine the size and the shape of the contact area, prints of the lug pattern were made on a rigid surface. Since no marked sinkage of the tire was observed, lug contact area on the rigid surface was chosen to represent the contact area in the field. Thetotal contact area was approximated by gathering several lug prints, between each of which the wheel was rotated slightly. The shape of that print was simplified to a geometric form, which consists of a rectangle in the center and two half circles (Fig. 1 ). The force distribution over the contact area was assumed to be uniform, where not indicated otherwise. As already mentioned, the measurement of soil stresses is often associated with large uncertainties. To validate measured normal stresses, the balance condition Use of a company or product name in this paper does not imply recommendation of the product or company to the exclusion of others which may also be suitable.

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169

Distance (cm) -30

-20

-10

0

10

20

30

30 2O tO 0 -10

-30 -

e

!i

Fig. 1. Lug pattern (grey) of the tire on a rigid surface. The areal resolution is 0.25 cm 2. The total contact area of an equivalent tire without lugs (black) is approximated by a rectangle and two half circles enclosing the lug print.

n

ajxdy=F+= Y,Lk --oo

--oo

(8)

k=l

was used, where Fz is the total wheel load. In this experiment, the integral along the driving direction x could be estimated from a large number of measurements. To evaluate this integral, time was related to length by the vehicle velocity. However, perpendicular to this direction, only three positions were measured. To obtain a reliable estimate, the three values were used to determine the parameters Po, Pt and a in the approximation function fizz =poexp ( _ p 2 lyl a)

(9)

where y is the distance perpendicular to the center of the wheel track and ~ is the approximation of the measured vertical stress. This was then used to evaluate the integral in Eq. ( 8 ). The parameters Po and Pl were fitted for every time step with the Levenberg-Marquardt algorithm (Press et al., 1992). The exponent a was varied to get the minimal deviation from the whole measured data set. The standard deviation per time step was minimal for a = 2.5 and had the value 3.3 kPa. The integral in y-direction was then calculated as +or

f O=dy=0.8pot(0.4) pO.8

(10)

--(2O

w h e r e / ' i s the gamma function (Abramowitz and Stegun, 1970). All calculations of stress state are based on the stress model SOCOMO developed by Van den Akker and Van Wijk ( 1987 ), which was expanded to determine the directions of the principal axes according to Eq. (2). All calculations were carried out with v = 5 and neglible driving force, unless otherwise stated.

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K. Hammel /Soil & Tillage Research 32 (1994) 163-181

4. Results and discussion

4.1. Application of model tofieM experiment In order to relate model calculations to physical reality, one needs to consider the averaging effect caused by the finite size of the pressure cell, the influence of the spatial resolution of the surface load on the calculated stress distribution, and the influence of the spatial distribution of the surface load on the measured stress distribution. At a depth of 5 cm, the stress heterogeneity introduced by the lugs would not have been resolved by the pressure cells and the measured stresses would have given a blurred picture of reality (Fig. 2 ). Since the fine details of the stress distribution disappear rapidly with increasing depth, the difference between measured and actual stresses also decreased and fell below 5% at 25 cm depth. Two resolutions of the lug contact area were compared with respect to the smooth tire. The differences in vertical stress distribution for a resolution of 0.25 cm 2, corresponding to 3044 point loads (see Fig. 5), and a resolution of 20.25 cm 2, corresponding to 38 point loads (Fig. 3 ), were large at 5 cm depth but practically neglible at 25 cm. The lower resolution also increased the range of vertical stress and thus added artificial heterogeneity at 5 cm depth. This is visible in Fig. 3, where deviations greater than 100% occurred, unlike for high resolution (see Fig. 5 ). Greater resolutions of the surface load would have led to a better representation of the actual load, but the required computing time increases in proportion to the square of the increase of the linear resolution. The measured vertical stresses at 25 cm depth were used to evaluate Eq. (8). It became clear that the measured values had to be corrected by a factor of 0.7 to satisfy the force balance condition. Then the stress state was modelled with high resolution of the surface load. The result was then averaged over a region with a radius of 5 cm corresponding to the size of the pressure cell. The concentration factor v was varied in steps of 0.5 and chosen by eye to fit best at v= 5.5. The corrected measured values and the calculated values are shown in Fig. 4. D i s t a n c e (cm) -4O 30.

-20 ~L , , , . . . . I . . . . . . . . .

0 I .........

20 I .........

40

20-~

2OO

10-

110 t~

0-10

60

33 ~ 18

-30

Fig. 2. Calculated vertical stress distribution in the xy-plane at 5 cm depth (contour lines) and averaged over a radius of 5 cm (grey levels) corresponding to the size of the pressure cell.

K. Hammel / Soil & Tillage Research 32 (1994) 163-181 (a) -40

171

Distance (cm) -20

0

20

40

40 30

4:

20 10 0 r~

-10 -20 -30 -40 (b)

~5 -I z

Fig. 3. Differences between the calculated vertical stress distribution in the xy-plane at 5 c m depth ( a ) a n d 2 5 cm depth (b) for a l u g g e d and an unlugged tire. The areal resolution of the surface load

was 20.25 cm 2 (38 point loads). The contour values are normalized to the peak vertical stress value for the unlugged case, which was 108 kPa at 5 c m a n d 85 kPa at 2 5 c m . T h e white contour line in ( a ) corresponds to the contour value 1. The dashed lines indicate the lug print. (Note the different scale of the grey levels).

Assuming a uniform load distribution led to a marked underestimation of vertical stress in the track center compared with the calculated vertical stress. Therefore load distribution was changed to be trapezoid, which greatly improved the agreement between theory and measurement (Fig. 4). This type of surface load distribution was also observed by Plackett (1984) and applied by Van den Akker (1988). In order to compare measured and calculated vertical stress, the measured stress-time function of the moving tire was converted to a stress-space function. This implies that the surface load is shifted with the vehicle velocity which is true if the contact area is independent of tire rotation. This is not the case for the lug print, which is periodically changing. Thus, the center of gravity becomes a function of tire rotation, and oscillates around the center of the track

K. Hammel / Soil & Tillage Research 32 (1994) 163-181

172

120 100

~

~

(a)

-

80~

60-

:~

40200 -40

-20

0

20

40

Distance(cm) Fig. 4. Measured (solid lines) and calculated (symbols) vertical stress in the track center (a) at a distance of 15 cm (b) and 30 cm (c). The contour plot in the upper left corner indicates the surface load distribution. The black color refers to a weighting factor of 1, which decreased linearly towards the boundary of the contact area. The weighting factor at the boundary was 0.2.

line due to the asymmetry of the lug print. Although the effect of the single lugs has disappeared, the asymmetry in the vertical stress distribution is still visible at 25 cm depth (Fig. 5 ). The slight irregularities in the measured curves are probably due to this phenomenon. However, this effect was neglected in the calculation, as it is symmetric to the center of the track line and its amplitude is relatively low, where the latter is additionally damped by the averaging property of the pressure cell. Additional deviations between measured and calculated curves may be attributed to: (i) the spatial variability of mechanical properties within the profile; (ii) to the simplifying assumptions about the surface load distribution in the field and (iii) to differing correction factors among the pressure cells. However, it is not feasible to quantify the influence of these effects based on the available data.

4.2. Application of model to hypothetical loading situations The tire resting entirely on its lugs or on its total potential contact area may be considered as the two possible extremes in a field situation. The latter may happen as a consequence of tire construction (when it is smooth) or with a certain amount of sinkage. While the unlugged tire creates a smooth vertical stress distribution, the lugged tire introduces a highly variable one, where large stress under the lugs is close to low stress between them. This characteristic was lost with increasing soil depth. At 5 cm depth, vertical peak stress under the lugged tire was about twice as high as under the smooth tire. At 25 cm depth, that difference

K. Hammel ~Soil & Tillage Research 32 (1994) 163-181 (a) -40

173

Distance(cm) -20

0

20

40

(b)

E ~2

Z }

Fig. 5. Differences between the calculated vertical stress distribution in the xy-plane for a lugged and unlugged tire 5 cm depth (a) and 25 cm depth (b). The areal resolution of the surface load was 0.25 cm ~ (3044 point loads). The contour values are normalized to the peak vertical stress, which was 108 kPa in 5 cm depth and 85 kPa in 25 cm depth, for the unlugged case. The dashed lines indicate the lug print. (Note the different scale of the grey levels).

decreased to only about 10% (Fig. 5 ). The asymmetric stress distribution, which was clear at 25 cm depth, is caused by the asymmetric contact area distribution of the lug print, whereas the unlugged tire is symmetrical. To estimate the quality of deformation, it is necessary to consider the complete stress state, not only vertical stress. As long as the soil is continuous and its deformation is a differentiable function of space, deformation may be interpreted as compression and shear, associated with the characteristic stresses aH and ao (Eq. 5). Thereby aD is a measure of the compressive impact and aD of the shear impact on soil. To illustrate the behavior of these two variables as a function of time, a volume element with a radius of 2.8 cm in the center of the track line at 5 cm depth is considered (Fig. 6). The symbol in the upper left comer of Fig. 6 and the following ones indicates the direction of motion of the tire. If the lugged tire is moving with

174

K. Hammel / Soil & Tillage Research 32 (1994) 163-181 (a)

180

_ _ tTll --~---- t7D

,

160 140 120 100 80

40 0 -20 -40

"1 0

140 7

....

' ....

1

~

I ....

' ....

2

I ....

' ....

3

I .... 4

" or.

(b)

120 100 80 6O 4O 20 0 I .........

0

I ....

I

' ....

I ....

' ....

2 Time

F ....

3

' ....

I

....

4

(s)

Fig. 6. The compressive (on) and shear component (aD) of the principal stress Sp along the center of the wheel track in the direction of movement at a depth of 5 cm as function of time, calculated from the state of stress as a function of space. In (a) a driving force of 0.4 relative to wheel load was added, while in (b) the driving force was 0.

neglible driving force, the mechanical impact at a shallow depth will be generally both compressive and shearing. In this case the component an dominates and is always positive, and the time dependence of an and ao are similar. When a driving force was added, the deformation behavior may be expected to differ significantly. In front of the tire the compressive component was negative, thus introducing tensile stress. Additionally the maxima of an and aD were greater and occurred at different times (Fig. 6). Whether or not the stress will develop in this way depends on the soil's resistance to failure. The stress state may also be related to the Mohr-Coulomb failure criterion, which divides the phase space of the principal stresses a~ and a3 into two parts. Depending on the shear parameters angle of internal friction 0 and cohesion c for a given a~, failure will occur if

K. Hammel ~Soil& Tillage Research 32 (1994) 163-181

175

a3 < a~ tan2 (45 ° - ~ / 2 ) - 2ctan (45 ° - ~ / 2 ) =tr¢3

(11)

where trY3is the minor principal stress at failure. When this limit is reached, a failure plane occurs and irreversible deformation will result. Failure is often but not necessarily preceded by compression. The concentration of forces under the lugs and the driving force have also a marked effect on the magnitude of the failure zones created by a lugged tire. To illustrate the intensity of failure the difference tr¢3-tr3 was evaluated in the calculations, where a~ and a3 were obtained from Eq. (2). For shear parameters ~ = 31 ° and c = 10 kPa the zones of failure at 5 cm depth were mainly influenced by the lug pattern (Fig. 7 ). Failure was reached much earlier when a driving force was added (Fig. 8) as a consequence of the tensile stress in front of the wheel which reduced ira, while cr~ remained nearly unchanged (see Figs. 10 and I 1 ). The deformation may then be characterized as follows. The soil breaks in front of the wheel, which then rolls over this disturbed structure, remoulding it completely. The same calculations carried out for the unlugged tire with zero driving force showed failure only in a small area. If, for the unlugged tire, a driving force is added, the shape and structure of failure zones are similar to the lugged tire, but the intensity was much smaller (Fig. 9). It is thus clear that a driving force increases the shear stress and enhances plastic flow and macroscopic failure. Volume change behavior is not determined by the Mohr-Coulomb failure criterion. However, its effect on soil physical properties is important. Therefore, it is necessary to know the actual deformation of soil samples as a function of stress state and time. The observed deformation may then be related to the change of physical properties. Model calculations may serve to estimate the loading path Distance (cm) -40

40 30 20 E

-20

0

20

40

Q 5O 40

10

30 ~ 0 20

'~

-10 -20

-3o

10 0

- 4 0 -~

Fig. 7. Failure zones in the xy-plane at 5 cm depth. The contour values are the difference a ~ - a3, where al and tr3 were obtained from Eq. (2) and a~ from Eq. ( 11 ).

K. Hammel ~Soil & Tillage Research 32 (1994) 163-181

176

Distance (cm) -40

-20

0

20

40

40 30

50

20

I

10

30 "-~ t~

ea 0

20 -10

~5 10

-20

r~ 0

-30 -40

Fig. 8. Failure zones in the xy-plane at 5 cm depth. The contour values are the difference t r ~ - a3, where tr~ and tr3 were obtained from Eq. ( 2 ) and trf3 from Eq. ( 11 ). A driving force with a ratio of 0.4 to wheel load was added. Distance (cm) -40

-20

0

20

40

40 30

50

20

4o~

10 30 ~ , 0

~5

20

-10 -20 -30

D

-40

Fig. 9. Failure zones in the xy-plane for an unlugged tire at 5 cm depth. The contour values are the difference trf3-tr3, where al and aa were obtained from Eq. ( 2 ) and af3 from Eq. ( 11 ). A driving force with a ratio of 0.4 to wheel load was added.

which should be chosen to study a certain field situation. Common instruments for this purpose are an oedometer and the triaxial apparatus. Stress state in these tests is axially symmetric and, consequently, determined by two different principal stresses tr~ and a3 = a2. During the test, the directions of the principal stresses did not change. Even if a vehicle moves relatively slowly, as in this experiment,

K. Hammel /Soil & Tillage Research 32 (1994) 163-181

177

stress duration at a small volume element was of the order of seconds. The influence of time is neglected by the Mohr-Coulomb failure theory. However, it may be an essential factor that determines the amount of deformation. To arrive at half of the total volume change may require loading times of the order of 0.1 to 1005 depending on the size of the stress step (Dexter and Tanner, 1974). This is the time range we also have to face in the case of moving vehicles. It is worth mentioning the effect of soil depth and driving force on al and a3 as functions of time. These were calculated for the vehicle velocity 0.21 m s-~ corresponding to the experiment, and averaged over a region with a radius of 2.8 cm to get the actual values for a sample of that size (Figs. 10 and 11 ). At 5 cm depth, tr~ rose rapidly to a large value with marked fluctuations from which it decreased rapidly again. At a depth of 25 cm, the peak stress was less as a result of broadening of the input force pulse and tr~ was a much smoother function of time. With no driving force, the two principal stresses behaved similarly in time and were symmetrical with respect to loading and unloading. The directional pattern of al was also influenced by depth (Fig. 10). At a depth of 5 cm, the structure of the

~

180

(a)

"

120 100

E

40?

/.-

204

/,--

0

I

.... --.

........

120 =

~--

, . . . . . . .

,

2

3

~

6040-

4

(b)

100 -_

~

k --.,

a ~

2

2o ~ ........ 0

I .... ' .... I .... 1 2

' .... I .... ' .... I .... 3 4

Time (s)

Fig. 10. Major (a~) and minor (a3) principal stresses along the center of the wheel track in the direction of motion at 5 cm depth (a) and 2 5 cm depth (b) as a function of time, calculated from the state of stress as a function of space. The arrows show the direction of a~, where the 3-dimensiona! direction vector is projected on the xz-plane. The length of each arrow is proportional to a~.

178

K. Hammel / Soil & Tillage Research 32 (1994) 163-181 200 (a)

180

.

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Fig. 1 I. Major (a~) and minor (or3) principal stresses along the center of the wheel track in the direction of motion at 5 cm depth (a) and 25 cm depth (b) as a function of time, calculated from the state of stress as a function of space. The arrows show the direction ofcq, where the 3-dimensional direction vector is projected on the xz-plane. The length of each arrow is proportional to at.

contact area and surface load dominates stress and also the direction of the principal stresses. That applied in the case of trl almost being vertical for greater stress. When that direction turned to almost horizontal, stresses were low. At 25 cm depth, the directional pattern was less influenced by the contact area, and the volume element considered underwent continuously varying directions of a~. When a driving force was added, the directional pattern of trl rotated for a certain value depending on the ratio between wheel load and driving force. The stress function became asymmetric with respect to loading and unloading. In front of the driving wheel, typically negative tr3 (tensile stress) occurred (Fig. I l ). This state of stress is therefore characterized by larger shear components, as shown before (Fig. 6 ), and an inclined directional pattern of the principal axes. Due to the general property that linear dissipative systems smooth out structures with

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high frequencies, the highly variable structure of surface load and contact area disappears with depth. With increasing depth only wheel load and driving force dominate the state of stress. To meet the modelled stress-time functions in a loading experiment exactly would take an extraordinary experimental effort. Even if stress state could be reproduced satisfactorily, it would be difficult to change the direction of the principal axes during the test. Conversely, the calculated stress may serve to design experiments, which will meet the essentials of the external loading condition considered. The time frame and the relative proportions of shear and compressive stress as a function of depth especially may be identified.

5. Conclusions Measured vertical stress in a field experiment is well described by FrShlich's stress model. Although the concentration factor is empirical and consequently needs to be determined by measurement or assumption, modelled stress state cannot be arbitrary. Therefore, if several reliable measurements are present, the model may be validated considering the resolution area of the pressure cell and a sufficient resolution of surface load distribution. Model calculations showed that tire lugs introduce a large heterogeneity in the stress state, which decreased with increasing soil depth. Shear and compressive stress near the soil surface was increased and focussed by the lugs in comparison with an equivalent unlugged tire. Shear and compressive stress were generally increased by a driving force. The distribution of stress in this case was characterized by a large shear component combined with a relatively low compressive component in front of the wheel. This feature was amplified by the lugs. From application of Mohr-Coulomb failure theory it follows that the lugs generally increased soil failure. As Fr/Shlich's motivation to introduce a concentration factor was to decrease the obvious deviation of measured stresses from elasticity theory, he derived a stress state, which is physically possible and includes the linear elastic case but is not based on a specified tensorial stress-strain relation. If the latter, which requires a set of parameters such as effective stress, time and stress history, were known, the finite element method (FEM) would be the preferable tool to calculate the state of stress. However, the experimental effort required to define the parameters with sufficient precision is enormous. Recent work pointed out that the main difficulty in FEM (and also in our understanding of soil behavior) is to get consistent and comprehensive stress-strain relations (Raper and Erbach, 1990; Schafer et al., 1991 ). Consequently, the stress model according to Frrhlich expanded by an arbitrary surface load distribution represents a relatively simple procedure that is useful for examining: (i) the effect of external loading conditions on stress state; (ii) the quality of expected deformation and (iii) to design loading conditions of deformation experiments.

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Acknowledgement I would like to thank the German Research Foundation (DFG) for financial support. I am grateful to J.J.H. van den Akker from the DLO Winand Staring Centre in Wageningen for providing the stress model SOCOMO. The manuscript has been considerably improved by the editorial suggestions of the Editor-in-Chief and anonymous reviewers. References Abramowitz, M. and Stegun, I.A., 1970. Handbook of Mathematical Functions. Dover Publishing Co., New York, USA, 1046 pp. Bailey, A.C., Timothy, A.N. and Johnson, C.E., 1988. Soil stress state determination under wheel loads. Trans. ASAE, 51: 1309-1314. Bradford, J.M., 1981. The shear strength of a moderately well-structured soil in its natural and remoulded states. Soil Sci. Soc. Am. J., 45: 9-12. Dexter, A.R. and Tanner, D.W., 1974. Time dependance of compressibility for remoulded and undisturbed soils. J. Soil Sci., 25:153-164. FriShlich, O.K., 1934. Druckverteilung im Baugrunde. Verlag Julius Springer, Wien, Austria, 184 pp. (in German). Horn, R., 1988. Compressibility of arable land. In: J. Drescher, R. Horn and M. de Boodt (Editors), Impact of Water and External Forces on Soil Structure. Catena Suppl., 1l: 53-73. Johnson, C.E. and Burr, E.C., 1990. A method of predicting soil stress state under tires. Trans. ASAE, 33: 713-717. Jumikis, R.A., 1984. Soil Mechanics. R.E. K_riegerPubl. Company, Inc., Malabar, Florida, USA, 576 PP. Koolen, A.J., Lerink, P., Kurstjens, D.A.G., van den Akker, J.J.H. and Arts, W.B.M., 1992. Prediction of aspects of soil-wheel systems. Soil Tillage Res., 24:381-396. Lebert, M., Burger, N. and Horn, R., 1988. The effect of soil structure and stress duration on the pressure transmission in tilled soils. Proc. 11th Int. Conf. ISTRO, Edinburgh, UK, Vol. 1, pp. 281 287. Plackett, C.W., 1984. The ground pressure of some agricultural tires at low load and with zero sinkage. J. Agric. Eng. Res., 29: 159-166. Press, W.H., Flannery, B.P., Teucholsky, S.A. and Vetterling, W.T., 1992. Numerical Recipes. The Art of Scientific Computing. Cambridge University Press, Cambridge, USA, 994 pp. Raper, R.L. and Erbach, D.C., 1990. Prediction of soil stress using the finite element method. Trans. ASAE, 33: 725-730. Reicosky, D.C., Voorhees, W.B., and Radke, J.K., 1981. Unsaturated water flow through a simulated wheel track. Soil Sci. Soc. Am. J., 45: 3-8. Schafer, R.L., Bailey, A.C., Johnson, C.E. and Raper, R.L., 1991. A rationale for modeling soil compaction behavior: An engineeringmechanics approach. Trans. ASAE, 34:1609-1617. Smoltczyk, U. (Editor), 1988. Grundbautaschenbuch. Part I, Verlag Ernst und Sohn, Berlin, Germany, 191 pp. (in German). S~Shne,W., 1958. Fundamentals of pressure distribution and soil compaction under tractor tires. Agric. Eng., 39: 276-281. Sch6n, H. and Olfe, G., 1986. Entwicklung und Stand des Schiepper- und Maschineneinsatzes. In: M. Brennd~rfer (Editor), Bodenverdichtungen. KTBL-Schrift 308, Landwirtschaftsverlag GmbH, Miinster-Hiltrup, Germany, pp. 33-42 (in German ). Tardieu, F., 1988. Effect of the structure of the ploughed layer on the spatial distribution of root density. Proc. 1ltb Int. Conf. ISTRO, Edinburgh, UK, Vol. l, pp. 153-158. Van den Akker, J.J.H. and Van Wijk, A.L.M., 1987. A model to predict subsoil compaction due to

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field traffic. In: G. Monier and M.J. Goss (Editors), Soil Compaction and Regeneration. Proc. of a Workshop on Soil Compaction: Consequences und Structural Regeneration Processes, Avignon. 17-18 September 1985, A.A. Balkema Rotterdam, Netherlands, pp. 69-84. Van den Akker, J.J.H., 1988. Model computation of subsoil stress distribution and compaction due to field traffic. Proc. 1lth Int. Conf. ISTRO, Edinburgh, UK, Vol. 1, pp. 403-409. Van den Akker, J.J.H. and Carsjen, G.J., 1989. Reliability of pressure cells to measure traffic-induced stress in the topsoil-subsoil interface. Proc. 4th Eur. Conf. ISTVS, Wageningen, Netherlands, Vol. 1+ pp. 1-7. Walker, J. and Chong, S.K., 1986. Characterization of compacted soil using sorptivity measurements. Soil Sci. Soc. Am. J., 50: 288-291.