a nonlinear 3d finite element analysis of the soil forces acting on a disk plow

Soil & Tillage Research 74 (2003) 115–124

A nonlinear 3D finite element analysis ofthe soil forces acting on a disk plow

Nidal H. Abu-Hamdeha,∗, Randall C. Reederb

a Postal Code 21110, P.O. Box 422, Irbid, Jordanb The Ohio State University, Agricultural Engineering Bldg., RM 228C,

590 Woody Hayes Drive, Columbus, OH 43210-1057, USA

Received 6 June 2002; received in revised form 7 May 2003; accepted 19 May 2003

Abstract

This study aimed to compare predicted soil forces on a disk plow with measured forces within the tillage depth of clay(90 g kg−1 sand, 210 g kg−1 silt, 700 g kg−1 clay) and sandy loam (770 g kg−1 sand, 40 g kg−1 silt, 190 g kg−1 clay) soils. Themodel assumed the effects of both tilt angle and plowing speed. Two plowing speeds (4 and 10 km/h) at three tilt angles (15◦,20◦ and 25◦) were compared and the draft, vertical, and side forces determined. A 3D nonlinear finite element model was usedto predict the soil forces while a dynamometer was used to measure them on a disk plow in the field. An incremental methodwas used to deal with material nonlinearity and the Trapezoidal rule method was used to analyze the dynamic response ofsoil during tillage. Field tillage experiments were conducted to verify the results of the finite element model. It was found thatincreasing the tilt angle of the plow increased the draft and vertical forces and decreased the side force. Increasing plowingspeed increased the draft and side forces and decreased the vertical force. Generally, the results from the finite element modelwere found to be compatible with the experimental results in clay soil, while in sandy loam the differences between predictedand measured data were probably due to problems of measuring soil mechanical characteristics in the triaxial test.© 2003 Elsevier B.V. All rights reserved.

Keywords:Tillage; Finite element model; Soil forces; Disk angle; Plowing speed

1. Introduction

Tillage, a process of applying energy to the soil tochange its soil physical condition or to disturb soilfor some other purpose, is one component in any sys-tem of soil management for crop production. Tillageprocesses are used in crop production for several pur-poses, such as loosening soil to create a seedbed or arootbed, moving soil to change the microtopography,

∗ Corresponding author. Tel.:+962-79-5614261;fax: +962-2-7095018.E-mail addresses:[email protected] (N.H. Abu-Hamdeh),[email protected] (R.C. Reeder).

or mixing soil to incorporate amendments. Soil tillagehas always been a major research area in agriculture.As a tillage operation is a procedure for breaking upsoil, soil failure depends mainly upon the soil prop-erties, tool geometry, and cutting speed. Almost allof the soil cutting tools used in agriculture have beendeveloped by field experiment and by trial and error(Kepner et al., 1978).

Experimental and theoretical analysis techniquesare essential to develop efficient tillage or soil cuttingtools which will require less energy and still pro-vide a satisfactory soil condition for crop emergenceand growth. The field experiment allows prototypeverification with specialize instrumented equipment

0167-1987/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0167-1987(03)00152-1

116 N.H. Abu-Hamdeh, R.C. Reeder / Soil & Tillage Research 74 (2003) 115–124

(McLaughlin et al., 1993). Theoretical analysis isgrowing in popularity to accelerate the productionprocess requirements and the desire to decrease pro-totype construction and verification. Analytical andempirical models are still used to solve soil–tool inter-action applications (McKyes, 1985), but many of thesemodels are 2D and are theoretically suitable only forvery wide tools (Osman, 1964; Siemens et al., 1965;Hettiaratchi et al., 1966). A few 3D models to pre-dict narrow tillage tool behavior in soils are available(Payne, 1956; Hettiaratchi and Reece, 1967; McKyes,1978; Perumpral et al., 1983). The majority of thesemodels, however, are for slow-moving tools and donot take into consideration the speed effects. Mosttillage operations, on the other hand, are performed atspeeds in the range 4–10 km/h, where the soil forceson the tillage tools are expected to vary with toolspeed. The latest advances in computer performanceare proving to be promising for numerical approachesto tool design such as the finite element method. Muchwork has been reported on the static analysis of tillageproblems using the finite element method (Konder andZelasko, 1963; Yong and Hanna, 1977; Liu and Hou,1985; Zeng and Fu, 1985; Xie and Zhang, 1985; Chiand Kushwaha, 1990; Wang and Gee-Clough, 1991;Plouffe et al., 1998a,b). This method was shown tobe capable of simulating different tool shapes andthe dynamic effect of travel speed. Simulations, how-ever, must always be performed in conjunction withexperimental tests to verify their validity.

The objectives of this study were to: (1) use a 3Dfinite element model for predicting the soil forces on adisk plow as affected by tilt angle and plowing speed;(2) measure soil forces on a disk plow in the fieldexperiments in clay and sandy loam soils at differentspeeds and tilt angles, and (3) compare and evaluatethe predicted soil forces against actual field measure-ments of these forces.

2. Materials and methods

The finite element model and the program devel-oped during this study have the capability to predictthe effect of plow tilt angle, plowing speed, and soiltype on the soil forces acting on a disk plow. The de-tails of these analyses are included in the followingsections.

2.1. Constitutive relationships

The hyperbolic model developed byDuncan andChang (1970)to represent a typical stress–strain rela-tionship was used in this study. The model representsthe nonlinear elastic behavior of soil and the tangentmodulus of elasticity in this model is expressed asa function of soil stress level and soil strength. Thismodel was selected for its generality as well as forthe convenience involved in the determination of themodel parameters using triaxial tests. The hyperbolicmodel is given by:

Et = KPa

[σ3

Pa

]n [1 − Rf (σ1 − σ3)

(σ1 − σ3)f

]2

(1)

whereEt is the tangent modulus of elasticity,Pa theatmospheric pressure,σ1 the major principal stressin soil, σ3 the minor principal stress in soil,(σ1 −σ3)f = (σ1 − σ3) at soil failure,Rf the failure ratiodefined as the ratio of ultimate deviatoric stress tothe soil strength, andK, n the dimensionless numbersdetermined from triaxial test results.

In Duncan’s equation, the tangent modulus,Et, ofsoil was expressed as a function of the major and minorprinciple stresses.Qun and Shen (1988)established arelation between soil strain rate and shear stress bymeans of statistical mechanics as follows:

ln ξ = α + β(12(σ1 − σ3)) (2)

where ξ is the actual strain rate in soil andα, β arethe equation coefficients.

The following expression for a modified tangentmodulus based onEqs. (1) and (2)was proposed byShen and Kushwaha (1993)to account for loading rateeffects:

Et =KPa

[σ3

Pa

]n [1 − Rf (σ1 − σ3)

(σ1 − σ3)fo[1 + Bt ln(ξ/ξo)]

]2

(3)

where ξo is the maximum strain rate at conventionaltriaxial apparatus,(σ1 − σ3)fo = (σ1 − σ3) at soilfailure from conventional triaxial apparatus andBt thecoefficient relating to the stain rate effect.

The draft and vertical forces of a tillage tool are af-fected by adhesion and friction between the soil andthe surface of the tillage tool and hence an interface

N.H. Abu-Hamdeh, R.C. Reeder / Soil & Tillage Research 74 (2003) 115–124 117

element needs to be introduced to study their resis-tance effect in the finite element method. The tangentmodulus used was derived fromClough and Duncan(1971)hyperbolic model as:

Et = KiPa

[σn

Pa

]ni[1 − Rif τ

τf

]2

(4)

whereEt is the tangent modulus of soil–tool interface,Pa the atmospheric pressure,σn the normal stress atsoil–tool interface,Rif the failure ratio of soil–toolinterface,Ki, ni the parameters obtained from tests,τ the actual shear stress at soil–tool interface,τf theshear stress at soil failure.

The shear stress at soil failure is given by:

τf = Ca + σn tanθ (5)

whereCa is the adhesion between soil and cutting tool,σn the normal stress, andθ the friction angle betweenthe soil and tool.

The dynamic interface model was obtained fromEqs. (2) and (4)as follows (Shen and Kushwaha,1993):

Et = KiPa

[σn

Pa

]ni[1 − Rif τ

τfo[1 + Bi ln(γ/γo)]

]2

(6)

whereτfo is the shear stress at soil failure obtainedfrom direct shear-box test,γ the actual shear strainrate at soil–tool interface,γo the maximum strain rateon direct shear-box apparatus, andBi the coefficientrelating to the sliding rate effect.

Laboratory tests were conducted in a modified shearbox to study the characteristics of the soil–tillage toolinterface (Chi and Kushwaha, 1990). The estimatedparameters for the interface element are listed inTable 1.

Duncan and Chang (1970)stated that two materialproperties, namely the tangent modulus and Poisson’sratio are necessary to completely describe the me-chanical behavior of any material under a general sys-tem of changing stresses. The tangent modulus can becalculated fromEqs. (3) or (4). The following equa-tion proposed byChi et al. (1993)calculates values ofPoisson’s ratio,ν, as a linear function of stress ratio;it is written in terms of stress level and soil strength:

ν = a + b

[(1 − sinφ)(σ1 − σ3)

2c cosφ + 2σ3 sinφ

](7)

Table 1Soil parameters used for the finite element analysis

Terms Soil type

Clay Sandyloam

Parameters of initial modulus of soilAngle of internal friction (φ), ◦ 17.30 31.80Cohesion (c), kPa 59.00 10.10K 41.87 21.58n 0.00 0.55Failure ratio (Rf ) 0.77 0.76a 0.15 0.10b 0.32 0.34

Parameters of interface of soil–toolSoil–tool friction (θ) 22.6 22.6Adhesion (Ca), kPa 27.01 5.05Ki, kPa/cm 4.71 4.71ni 0.81 0.81Failure ratio (Rif ) 0.88 0.87

whereν is the Poisson’s ratio, anda, b are the soilparameters obtained from the regression analysis.

2.2. Boundary conditions applied

The soil-cutting model considered in the analysiswas idealized with tetrahedral constant strain elementsbecause of their simplicity and convenience for nonlin-ear material.Fig. 1 illustrates the finite element meshfor a disk plow. The region of influence consideredin the analysis had a length in the longitudinal direc-tion of seven times the tool operating depth, a depthof three times the tool operating depth and a width inthe lateral direction of twice the tool diameter. A totalof 1500 nodes and 6000 elements were generated foreach meshing. The boundary conditions applied wereas follows:

1. The nodes on bottom surface were fixed vertically(z-direction).

2. The nodes on front and rear surface were fixedlongitudinally (x-direction).

3. The nodes on right and left side surface were fixedlaterally (y-direction).

4. The nodes on blade–soil interface had a specifiedlongitudinal (x-direction) displacement during eachincrement and were constrained from movementsand rotations in all other directions.


Soil surface

Z

X

Disk blade

Y

Fig. 1. Finite element mesh of disk plow.

5. The nodes on side wall of furrow were fixed later-ally (y-direction).

6. All other nodes were free in three directions.

2.3. Finite element formulation

The general matrix differential equation for time de-pendent problems can be expressed as follows (Cooket al., 1988):

Ma + Ca + Ka + f = 0 (8)

wheref is the external load vector,a, a, a are the nodalacceleration, velocity, and displacement vectors, re-spectively, andM, C, K the mass, damping and stiff-ness matrix, respectively.

For tillage problems, the soil forces are determinedby the f vector which can be considered to compriseof three components: acceleration, damping, and staticequilibrium.

Since soil is a nonlinear elastic material, the incre-mental method was implemented into the finite ele-ment program to solve the nonlinear behavior of soiland the interaction between the soil and tool surfacein order to obtain information at every stage of thesoil-cutting process. The total load was applied in sev-

eral increments. This allowed the linear elastic theoryto be used because the calculated strains and stressesfor each increment were small. During each incrementthe soil tangent modulus at each Gauss point was up-dated according to the current stress status. The loadwas continued until the soil structure collapsed. Closerapproximations to the exact solution are obtained asthe load increment is decreased and the number ofload steps is increased.

The Trapezoidal rule integration method was usedto simulate the dynamic response of soil (Cook et al.,1988). The time interval corresponded exactly to theload interval. During each time step the accelerationat each node and the velocity at each Gauss pointwere calculated, and the soil tangent modulus at eachGauss point was updated continuously according tothe present speed value and stress status.

It was assumed that only shear and tensile stressescause failure in agricultural soils. Shear failure oc-curred at one Gauss point when the difference betweenthe major and minor principal stresses at this point ex-ceeded the maximum shear strength, and the tangentmodulus was modified to a small value (10−6 timesthe initial modulus). Tensile failure occurred at oneGauss point if a tensile stress at this point was larger


than five times the soil cohesion determined from lab-oratory triaxial tests, and the tangent modulus wasmodified to a very small value (10−6 times the initialmodulus). This tensile stress value was used becausethe value of cohesion prevailing in the field is usuallyseveral times greater than that determined from labo-ratory tests, thus, the fivefold cohesion was selectedas a threshold (Shen and Kushwaha, 1993).

2.4. Disk plow and field measurementsof soil forces

Disk plowing has always been the most popularprimary tillage practice in Jordan. Its ability to cutthrough crop residues, roll over roots and other ob-structions, and control weeds mechanically, even inwet soil conditions, is probably the major reason forits popularity. The disk is tilted backward at an an-gle of 15–25◦ from the vertical (Fig. 2). Blades ondisk plows are concave, usually representing sectionsof hollow spheres. The action of a concave disk bladeis such that the soil is lifted, pulverized, partially in-verted, and displaced to one side.

The net effect of all the soil forces acting on a diskblade resulting from its cutting, pulverizing, elevating,and inverting action, plus any parasitic forces actingon the disk, can be expressed in terms of three forces,namely longitudinal (draft) force, lateral (side) force,and vertical upward force, L, S, and V (Fig. 3).

Tilt angle

Vertical

Ground line

Fig. 2. Identification of tilt angle for a plow disk.

Fig. 3. Soil forces acting upon a disk blade; longitudinal or draftforce (L), lateral or side force (S), and vertical upward force (V).

Field experiments were conducted to compare thepredicted force values with the measured force values.The influence of tilt angle and plowing speed upon soilreactions were investigated experimentally in a seriesof field tests. Two typical soils in Jordan were used,namely a sandy loam soil (770 g kg−1 sand, 40 g kg−1

silt, 190 g kg−1 clay) at an average moisture contentof 13.6% and a clay soil (90 g kg−1 sand, 210 g kg−1

silt, 700 g kg−1 clay) at an average moisture content of10.9%. Several core samples were taken from differentlocations in the test fields for soil density and moisturecontent measurement. The average soil density andmoisture content were used in laboratory tests to deter-mine the soil parameters for the constitutive equations.The average soil parameters obtained from the labora-tory triaxial tests on field samples are listed inTable 1for the sandy loam and clay soils. Details of the labora-tory triaxial tests can be found inShen and Kushwaha(1994). In these tests, the size of the soil specimens


was 52 mm in diameter and 106 mm in length, test rateof soil samples was set to be 0.1 mm/s, and stress levelswere controlled around four values: 20, 40, 60, 80 kPa.

The field tests were with a 610 mm diameter plowdisk having a 55.8 radius of curvature working at adepth of 18 cm with a width of cut of 21 cm. A pre-liminary run with the disk plow to leave an open fur-row for the actual test run was performed before eachtest run. The analysis was conducted for four differentcases to observe the effects of tilt angle and plowingspeed. They were:

1. Plowing speed of 4 km/h at three tilt angles(15◦, 20◦, and 25◦) in clay.

2. Plowing speed of 10 km/h at three tilt angles(15◦, 20◦, and 25◦) in clay.

3. Plowing speed of 4 km/h at three tilt angles(15◦, 20◦, and 25◦) in sandy loam.

4. Plowing speed of 10 km/h at three tilt angles(15◦, 20◦, and 25◦) in sandy loam.

Tillage forces were measured using a force trans-ducer mounted on the plow frame and consisting ofsix load cells to measure draft force, side force, andvertical force. Signal conditioners with second-orderlow pass filters adjusted at 100 Hz were connected oneach load cells. A custom application created underthe LabView v5.0 environment (National Instruments)monitored input signals, displayed, and saved the re-sults in data files. A transformation matrix was used toconvert the six load cell forces into forces along all or-thogonal axes. The operating speed was measured by awheel with a magnetic signal and an ultrasonic sensorwas used to measure the operating depth. Data werecollected with a data logger (Model CR7X, CampbellScientific Inc., Logan, UT). The results reported forthese tests include values of L, S, and V at the differ-ent tilt angles and plowing speeds. For each case, fourtest replicates were used and the average force valueswere used for comparison with the force predictions.

2.5. Finite element analysis

A finite element program, written in FORTRAN,was developed using all the techniques and equationspreviously discussed.

For the finite element analysis, appropriate boun-dary-condition information and nodal and elementaldata were inputted as required. The initial modulus of

elasticity was computed for each element usingEq. (3)for soil elements andEq. (6)for soil–tool interface el-ements. The total load was applied in increments. Foreach incremental load, the displacement of each nodalpoint and the stresses and strains within each elementwere computed. Soil reaction forces were calculatedfrom a small displacement assigned to the nodes on theinterface at each increment. The modulus of elasticityand Poisson’s ratio values for each element were thencomputed and updated based on the current state ofstress. This process was continued until the total loadwas applied. Average predicted values of draft, side,and vertical forces were calculated from the summa-tion of the nodal forces on the soil-interface elementsin the longitudinal, lateral, and upward directions, re-spectively. All four cases studied in this research wereanalyzed using the same procedure; tilt angle, plow-ing speed, and soil parameter values being changedfor the case under consideration.

3. Results and discussion

Results of the finite element analysis included thecalculation of the reaction forces from the summa-tion of the node forces on the interference at eachdisplacement increment. Average measured values ob-tained from the force transducer and predicted valuesof draft, side, and vertical upward forces at 4 km/hplowing speed are shown inFigs. 4 and 6for the clayand sandy loam soils, respectively. The same reactionforces on the disk blade at 10 km/h plowing speed areshown inFigs. 5 and 7for the clay and sandy loamsoils, respectively.

Increasing the tilt angle of the disk blade in claysoil, within the 15–25◦ range, increased the measureddraft (L) and the measured vertical upward force (V)but decreased the measured side force (S) as shown inFigs. 4 and 5. Thus, penetration is better at the smallertilt angles.

When in the clay soil the speed was increased from4 to 10 km/h,Figs. 4 and 5show the draft force (L) in-creased 40%, the side force (S) increased because thesoil was thrown farther to the side and the vertical up-ward force (V) decreased. Thus, with the blade tilted,increasing the speed would improve soil penetrationunder these soil conditions. The forces calculated atthe soil–tool interface do increase with speed due to


0

1

2

3

4 S

oil R

eact

ions

, kN

10 15 20 25 30 Tilt Angle, degrees

Draft measured

Draft predicted

Vertical measured

Vertical predicted

Side measured

Side predicted

Fig. 4. Measured and predicted soil reactions versus tilt angle for a 61 cm disk having a 55.8 cm spherical radius of curvature at 4 km/hin clay soil.

greater soil acceleration. The dynamic algorithm inthe analysis considered soil acceleration and translatethat into higher soil stresses.

The relationships between both the predicted andmeasured reaction forces in the clay soil and the tilt

0

1

2

3

4

Soi

l Rea

ctio

ns, k

N


Draft measured

Draft predicted

Vertical measured

Vertical predicted

Side measured

Side predicted

Fig. 5. Measured and predicted soil reactions versus tilt angle for a 61 cm disk having a 55.8 cm spherical radius of curvature at 10 km/hin clay soil.

angle and speed are similar, seeFigs. 4 and 5. A com-parison of the reaction forces shows that the finiteelement model predicted the forces relatively accu-rately; in general, the predicted values tending to belower than the measured. The relative error between


0

1

2

3

4 S

oil R

eact

ions

, kN


Draft measured

Draft predicted

Vertical measured

Vertical predicted

Side measured

Side predicted

Fig. 6. Measured and predicted soil reactions versus tilt angle for a 61 cm disk having a 55.8 cm spherical radius of curvature at 4 km/hin sandy loam soil.

the finite element model and field tests values rangedfrom 0.9 to 9% for draft force, from 1.5 to 8% for ver-tical upward force, and from 2 to 8.5% for side force.Chi and Kushwaha (1991)found a relative error in thedraft force between their finite element model and soil

0

1

2

3

4

Soi

l Rea

ctio

ns, k

N


Draft measured

Draft predicted

Vertical measured

Vertical predicted

Side measured

Side predicted

Fig. 7. Measured and predicted soil reactions versus tilt angle for a 61 cm disk having a 55.8 cm spherical radius of curvature at 10 km/hin sandy loam soil.

bin test ranging from 0.8 to 10.5% for a simple tillagetool.

Figs. 6 and 7show both the test data and finiteelement simulation for the disk plow in the sandyloam soil at speeds 4 and 10 km/h, respectively. The


variations in measured soil forces with tilt angle weresimilar to those obtained in the clay soil. Increasingthe tilt angle increased the draft and vertical upwardforces but decreased the side force. When the speedwas increased from 4 to 10 km/h, the draft and sideforces increased with speed while the vertical forcedecreased as the speed was increased. In the soil–toolinteraction model, the terms containing tool speed arethe accelerational force terms. Large variations in re-action forces were observed to occur over the speedrange used, yet the prediction error (difference be-tween predicted and measured values) did not vary asa function of plow speed. This would appear to be dueto the accelerational force terms accounting for a largeportion of the variation in plow forces.

The predicted reaction forces were not as close tothe experimentally measured values in the sandy loamas in the clay. In sandy loam soil, the finite elementmodel under-predicted the draft force by an averageof 21%, the vertical upward force by an average of19%, and the side force by an average of 17%. Thisgreater prediction error is probably caused by the factthat the sandy loam soils have a low cohesion and itis not easy to collect undisturbed samples from thetest fields for laboratory triaxial compression tests.This certainly introduced some errors into determin-ing the soil constitutive parameters to simulate fieldconditions.

The results indicated that soil type and tilt anglehave the most pronounced effect on soil reactions, asevidenced by the comparative results for the two soilsin Figs. 4–7. It should be kept in mind that these resultswere obtained in soils that had not been subjected tothe effects of plant growth and other field environmentconditions.

4. Conclusions

The effects of soil type, tilt angle, and plowing speedon soil reaction forces were investigated through the-oretical predictions and field measurements. Based onthe results from this study, the following conclusionscan be drawn:

• Increasing the tilt angle of the disk plow increasedthe draft and vertical upward forces and decreasedthe side force.

• Increasing plowing speed increased the draft andside forces and decreased the vertical upward force.The amount of increase and decrease was affectedby soil type.

• The finite element analysis gave a relatively accu-rate prediction of the reaction forces on the diskplow blade in the clay soil. The relative predictionerror of the finite element model ranged from 0.9to 9%.

• The finite element analysis under-predicted the re-action forces in the sandy loam soil. This was dueto some errors in determining the soil constitutiveparameters to simulate field conditions.

• The accelerational force terms in the model can ac-count for a large portion of the variation in reactionforces observed to occur with an increase in plowspeed.

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a nonlinear 3d finite element analysis of the soil forces acting on a disk plow

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