




































 

 
42 

† Corresponding author 
© 2014 Conscientia Beam. All Rights Reserved. 

 

DEVELOPMENT OF A MATHEMATICAL MODEL FOR ANGLE OF SOIL 

FAILURE PLANE IN CASE OF 3-DIMENSSIONAL CUTTING 

 

Moayad B. Zaied1 --- Mohammed H. Dahab2 --- Ahmed M. El Naim3 
1Department of Agric. Eng., Faculty of Natural Resources, University of Kordofan, Elobied, Sudan 

2Department of Agric. Eng., Faculty of Agriculture, University of Khartoum, Sudan 

3Department of crop Sciences, Faculty of Natural Resources, University of Kordofan, Elobied, Sudan 

 

ABSTRACT 

The interaction between tillage tools and soil is of a primary interest to the design and use of these tools for 

soil manipulation. A new explicit mathematical model to calculate the angle of soil failure plane when soil is 

cut with narrow tine was developed. Equations of Soil cohesion and soil adhesion cutting factors were 

partially differentiated with respect to angle of soil external friction, were maximized; Values of angle of 

soil failure plane were calculated by the model for tine rake angle range from 00 to 900 operated at 0.3 m 

depth and 0.15 m width in soils with different mechanical properties. It was found that angle of soil failure 

plane is acute and its values were used for calculating soil frictional cutting factor, soil overburden cutting 

factor, soil cohesion cutting factor, soil adhesion cutting factor, rupture distance, width of side crescent and 

soil resistance force. It was found that the force values were realistic. Therefore, the model is valid. 

Keywords: Mathematical modeling, Soil failure plane, 3-D soil cutting. 

 

Contribution/ Originality 

The study originates new explicit formula for angle of soil failure plane (β) with respect to 

tool rake angle (α), external frictional angle at a soil–tool interface (δ), soil internal angle of 

friction (ϕ), tool operating depth (d) and tool operating width (w) in case of 3-D soil cutting 

 

1. INTRODUCTION 

The dynamic response of soil to farm implements is a main factor in determining their 

performance. The interaction between tillage tools and soil is of a primary interest to the design 

and use of these tools for soil manipulation. The definition of notations is as a follow: 

 

C Soil cohesion, kPa 
Ca Adhesion force at a soil–metal interface, kPa 
d Tool working depth, m 
Fs Soil resistance force, kN 

 

Current Research in Agricultural Sciences 
2014 Vol. 1, No. 2, pp. 42-52 
ISSN(e): 2312-6418 
ISSN(p): 2313-3716 
© 2014 Conscientia Beam. All Rights Reserved. 
 

 
 
 
 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
43 

© 2014 Conscientia Beam. All Rights Reserved. 

g Acceleration due togravity, m / s2 
Kp Soil pressure cutting factor, dimensionless 
Kc Soil cohesion cutting factor, dimensionless 
Kca Soil adhesion cutting factor, dimensionless 

Nγ Soil frictional cutting coefficient, dimensionless 

Nq Soil overburden cutting coefficient, dimensionless 
Nc Soil cohesion cutting coefficient, dimensionless 
Nca Soil adhesion cutting coefficient, dimensionless 
r Rupture distance, m 
w Tool working width, m 
Ws Width of side crescent, m 

α  Tool rake angle, deg. 

β Angle of soil failure plane, deg. 

δ External frictional angle at a soil–tool interface, deg. 

ϕ  Internal frictional angle, deg. 

ρd Soil bulk density, g / cm3 

σn Normal stress, kPa 

τ Shear strength, kPa 

 

The force acting on a failure surface in the soil body can be determined by Mohr–Coulomb 

equation as follows 

)1...(....................................................................................................tan nC 
 

 The forces acting at a metal-soil interface are determined by the following equation 

)2.........(..........................................................................................tan naC   

 Limit equilibrium is one of the most important approaches used to analyze soil–tool systems. 

Two most important factors in the approach are the shape of soil failure surface, and equilibrium 

equations, which are two or three dimensional cases. Grisso and Perumpral [1] reviewed four 

narrow tillage tool models, discussed assumptions, capabilities and limitations associated with 

each model, predicted tillage tool performance under two different soil conditions using the four 

models and compared simulated results with the experimental results. 

Many models for prediction of soil pressure coefficients, cutting factors, soil forces and draft 

were developed. Terzaghi [2] and Hettiaratchi and Reece [3] established a two-dimensional 

model and evaluated soil loads and soil resistance to tillage tools. They reported that the soil in 

front of tool and above the failure surface is assumed to consist of Rankine passive zone and a 

complex shear zone bounded by part of logarithmic spiral as shown in Fig. 1.  

Payne [4] developed a three-dimensional soil failure model depending on the upward 

movement of soil in front of the tool during tillage. In this model a failure zone includes a 

triangular center wedge, a center crescent and two side blocks called wings of the crescent (Fig. 

2). It is proposed that failure wedge ahead of a cutting blade and the failure wedge consists of 

center wedge, two side crescents and straight rupture plane at the bottom [5], [6] as shown in 

Fig. 3. Zeng and Yao [7] developed a dynamic soil cutting model included the acceleration and 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
44 

© 2014 Conscientia Beam. All Rights Reserved. 

strain-rate effects. Kuczewsk and Piotrowska [8] introduced a new model for forces on narrow 

soil cutting tines taking into account variability of the inclination angle of bottom failure surface 

in the side segment and inertial forces for different side segments. Draft force and power 

requirement for tillage implements were considerably affected by implement design and 

conditions of soil. In terms of effects on draft force and soil disturbance, Kheiralla, et al. [9] 

formulated a draft force model for ploughs based on traveling speed and tillage depth. Abo-Elnor, 

et al. [10]concluded that the blade cutting width had a significant effect on cutting forces so that 

the cutting forces increased but not in linear proportion as the cutting width increased. McKyes 

and Maswaure [11] demonstrated that designing a tillage tool for minimum draft requirement 

and high soil cutting efficiency called for a shallow operating depth and rake angle of 300. Chung 

and Sudduth [12] developed a model for soil failure caused by a vertically operating conical tool 

and concluded that the angle of soil failure plane and angle of internal soil friction are negatively 

correlated. Kasisira and DuPlessis [13] developed mathematical force models employing limit 

equilibrium analysis based on the soil-volume tilled to predict the draft requirements of tillage 

tool and reported that such models require a preliminary assumption of the soil failure pattern 

ahead of the tool. 

Two-dimensional soil cutting model can be valid if the tool working width is larger than its 

depth. McKyes [14] mentioned two-dimensional models to calculate soil cutting factors and 

angle of soil failure plane as follow: 

)3........(..................................................
)sin(2

)sin()cot(cot








PK

 

)4.........(..................................................
)sin(sin

cos






CK

 

)5.........(..................................................
)sin(sin

)cos(








CaK

 

)6.......(....................
)sin(

)cos(
sinsin

)sin()sin(

cot 1


























 










  

Three-dimensional soil cutting model can be applied if the tool working width is smaller than 

its depth. Zhang and Kushwaha [15] mentioned three-dimensional models to predict soil pressure 

factors and cutting resistance force as follow 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
45 

© 2014 Conscientia Beam. All Rights Reserved. 

)7....(....................
)sin(

)sin()cotcot2cot
3

2
1)(cot(cot5.0 2









 w

d

N

 

)8..(..............................
)sin(

)sin()cotcot2cot1)(cot(cot 2








 w

d

Nq

)9......(..................................................
)sin(

).......cotcot2cot1(cos 2








 w

d

NC

)10.........(......................................................................
)sin(sin

)cos(








CaN

)11.......(................................................................................).........cot(cot   dr

)12.........(......................................................................cotcot2cot2   dWS

)13.......(............................................................)( 2 wdNCCdNNgdF CaaCdS  

 

2. MODEL DEVELOPMENT METHOD 

2.1. Problem  

In case of two-dimensional soil cutting, the angle of soil failure plane β is given explicitly as a 

function of α, δ and ϕ (equation 6). In three- dimensional soil cutting case, in equations [8-11] all 

parameters except angle of soil failure plane (β) are known. The soil failure plane angle can be 

identified by solving any of the following equations: 

)11......(............................................................0






N
 

)12.......(............................................................0






qN
 

)13.......(............................................................0





cN

 

)14.......(............................................................0





caN  



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
46 

© 2014 Conscientia Beam. All Rights Reserved. 

Since differentiations of these equations are rather complex and result formidable terms, thus, 

there is no explicit function for β with respect to α, δ, ϕ, d and w, therefore a numerical procedure 

was used to determine the failure plane angle (β). To this regard, the present work aims to look 

for a simple explicit model to calculate β whenever α, δ, ϕ, working depth (d) and working width 

(w) are known. 

 

2.2. Solution for Model Development 

The developed model will be in the form of ),,,,( wdf   . To get that, some 

selections should be considered. Equation (7) contains constant values, so, it is excluded while 

Equation (8) and Equation (9) have no constant values in their terms, therefore any of them with 

Equation (10) can be chosen to conduct the partial derivative. Now Equation (9) is selected:  

)sin(

)cotcot2cot1(cos 2








 w

d

NC

 

The Partial derivative of Nc with respect to δ and maximization will be as follow 

)15......(....................0
)(sin

)cos()cotcot2cot1(cos

2

2
















w

d
NC

)16.........(......................................................................0cotcot2cot1 2  
w

d

)17....(......................................................................01cotcot2cot
2

2
2

2

2

 
w

d

w

d

 

The resulting function is in the form of a quadratic where cot β is in the positive root of equation: 

)18....(................................................................................
2

4
cot

2

a

acbb 
  

Where, 

2

2

w

d
a   

cot2
2

2

w

d
b   

1c  



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
47 

© 2014 Conscientia Beam. All Rights Reserved. 

Solving with respect to β will result the following relationship: 

 

)19.(......................................................................cot1cotcot

2

 









w

d

d

w

 

From equation (10):  

)sin(sin

)cos(








CaN

 

Partial derivative of Nca with respect to δ and maximization will be as follow: 

)20.(..................................................0
)(sinsin

)cos(sin)cos(
22
















CaN

Let, 

0)cos(  
 

Then,  

)21.(............................................................0)sin(sin)cos(cos  

 

Dividing by )cos(sin    

)22...(................................................................................0)tan(cot  

 

Solving with respect to β will result the following relationship: 

)23.(................................................................................).........tan(cot    

If   )(  or   090)( , where


 is very small number   then, 

)()tan(    

Then β as function of  ,   and   was given as follow: 

)24......(................................................................................).........(cot    



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
48 

© 2014 Conscientia Beam. All Rights Reserved. 

Adding equation (18) to equation (24): 

  )25.(........................................cot1cot)(cot2

2

 









w

d

d

w

 

Then, 

)26......(........................................
2

cot1cot

cot

2

















































w

d

d

w

Accordingly, 

)27........(........................................
2

cot1cot

cot

2

1































 




w

d

d

w

Therefore, equation (27) is the developed model for calculating the angle of soil failure plane in 

case of three-dimensional soil cutting.  

 

2.3. Model Validation Test  

Values of angle of soil failure plane (β) are calculated using the developed model for different 

tine rake angles and varied soil mechanical properties for narrow tine operated at 0.3 m depth and 

0.15 m width. The validity of the model depends on realistic of the results, that, (β) should be 

acute angle and real positive number. The resultant values of β were used to calculate soil 

frictional coefficient (N), soil overburden coefficient (Nq), soil cohesion coefficient (NC), soil 

adhesion coefficient (NCa) as well as rupture distance, width of side crescent and soil resistance 

force.  

 

3. RESULTS AND DISCUSSION 

Table 1, Table 2 and Table 3 demonstrated the values of angle of soil failure plane, resultant 

soil coefficients, rupture distance, width of side crescent and soil resistance force at different soil 

mechanical properties and rake angles. 

  It was found that angle of soil failure plane (β) is acute and its values are real positive 

number. Moreover, it was shown that as the angle of internal soil friction (ϕ) increased, the value 

of β decreased and vice versa which agreed with Chung and Sudduth [12]. 

When values of β were used for calculating rupture distance, width of side crescent, soil 

frictional coefficient (Nγ), soil overburden coefficient (Nq), soil cohesion coefficient (NC) and soil 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
49 

© 2014 Conscientia Beam. All Rights Reserved. 

adhesion coefficient (NCa), the resultant values of these variables were found to be defined and 

determined. The values of coefficients were applied to calculate soil resistance force, it was found 

that the force values were realistic.  

 

4. CONCLUSIONS 

In case of three dimensional soil cutting, there is no explicit function for angle of soil failure 

plane (β) with respect to rake angle (α), external frictional angle at a soil–tool interface (δ), angle 

of soil internal friction (ϕ), tool operating depth (d) and tool operating width (w), therefore, a 

numerical procedures were used to determine the value of angle of soil failure plane. 

A new explicit mathematical model to calculate angle of soil failure in case of three-

dimensional cutting was developed. The model was developed by partially differentiating of soil 

cohesion and soil adhesion cutting coefficients with respect external frictional angle at a soil–tool 

interface and the two resultant equations were maximized and were solved simultaneously for 

angle of soil failure plane with respect to rake angle, external frictional angle at a soil–tool 

interface, angle of soil internal friction, tool depth and tool width,.  

The resultant values of the angle of soil failure plane, correspondent soil cutting coefficients, 

rupture distances, width of side crescent and soil resistance forces obtained by the developed 

model at different tool rake angles were found to be realistic. Therefore, the model is valid. 

 

REFERENCES 

[1] R. D. Grisso and J. V. Perumpral, "Review of models for predicting performance of narrow tillage 

tool," Transactions of the ASAE, vol. 28, pp. 1062-1067, 1985. 

[2] K. Terzaghi, Theoretical soil mechanics. New York: Wiley, 1959. 

[3] D. R. P. Hettiaratchi and A. R. Reece, "Symmetrical three-dimensional soil failure," J. Terramech, 

vol. 4, pp. 45–67, 1967. 

[4] P. C. J. Payne, "The relationship between mechanical properties of soil and the performance of 

simple cultivation implements," J. Agric. Eng. Res., vol. 1, pp. 23–50, 1956. 

[5] E. McKyes and O. S. Ali, "The cutting of soil by narrow blades," J. Terramech, vol. 14, pp. 43–58, 

1977. 

[6] W. C. Swick and J. V. Perumpral, "A model for predicting soil–tool interaction," J. Terramech, vol. 

25, pp. 43–56, 1988. 

[7] D. Zeng and Y. Yao, "A dynamic model for soil cutting by blade and tine," J. Terramech, vol. 29, 

pp. 317–327, 1992. 

[8] J. Kuczewsk and E. Piotrowska, "An improved model for forces on narrow soil cutting tines," Soil 

Till. Res., vol. 46, pp. 231–239, 1998. 

[9] F. A. Kheiralla, A. Yahia, M. Zohadie, and W. Ishak, "Modelling of power and energy 

requirements for tillage implements operating on serdang sandy clay loam, Malaysia," Soil Till. 

Res., vol. 78, pp. 21–34, 2004. 

[10] M. Abo-Elnor, R. Hamilton, and J. T. Boyle, "Simulation of soil–blade interaction for sandy soil 

using advanced 3-D finite element analysis," Soil Till. Res., vol. 75, pp. 61–73, 2004. 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
50 

© 2014 Conscientia Beam. All Rights Reserved. 

[11] E. McKyes and J. Maswaure, "Effect of design parameters of flat tillage tools on loosening of clay 

soil," Soil Till. Res., vol. 43, pp. 195–204, 1997. 

[12] S. O. Chung and K. A. Sudduth, "Soil failure models for vertically operating and horizontally 

operating strength sensor," Transactions of the ASABE, vol. 49, p. 851−863, 2006. 

[13] L. L. Kasisira and H. L. M. DuPlessis, "Width Prediction of a side circular crescent failed by a 

tillage tool in a sandy clay loam soil," Agricultural Engineering International: The CIGR E Journal. 

Manuscript PM 1334, vol. 11, pp. 1-8, 2009. 

[14] E. McKyes, Soil cutting and tillage. Developments in agricultural engineering 10. Agricultural engineering 

soil mechanics. Amsterdam: Elsevier Science Publishing Company Inc., 1989. 

[15] J. Zhang and R. L. Kushwaha, "A modified model to predict soil cutting resistance," Soil Till. Res., 

vol. 34, pp. 157–168, 1995. 

 

Fig-1. Logarithmic spiral failure zone, [3]. 

 
Source: [3]. 

 

Fig-2. Failure zone of the Payne model [4]. 

 

Source: [4]. 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
51 

© 2014 Conscientia Beam. All Rights Reserved. 

Fig-3.Double-wedge failure zone of the McKyes-Ali model (a-rake angle; b-angle of soil failure 

plane; r-rupture distance).[5]. 

 

Source: [5]. 

 

Table-1. First Soil Sample Mechanical Properties, ϕ = 200, δ = 100, ρd = 1.77 g / cm3, C = 3 kPa, 

Ca = 1.62 kPa 

α0 β0 Nγ Nq Nc Nca r (m) Ws (m) Fs (kN) 

5 72.8 29.55 23.73 6.28 1.62 3.52 1.29 7.89 

10 70.2 12.88 11.11 5.11 1.09 1.81 0.94 3.79 

15 67.8 8.27 7.60 4.68 0.92 1.24 0.78 2.64 

20 65.4 6.20 5.99 4.48 0.86 0.96 0.69 2.12 

25 63.1 5.05 5.06 4.39 0.83 0.80 0.62 1.84 

30 60.8 4.34 4.45 4.37 0.83 0.69 0.57 1.67 

35 58.7 3.87 4.01 4.41 0.84 0.61 0.53 1.56 

40 56.6 3.55 3.66 4.49 0.86 0.56 0.50 1.50 

45 54.5 3.33 3.37 4.60 0.90 0.51 0.47 1.47 

50 52.6 3.18 3.11 4.76 0.95 0.48 0.44 1.46 

55 50.6 3.09 2.87 4.97 1.01 0.46 0.41 1.47 

60 48.7 3.05 2.63 5.23 1.09 0.44 0.38 1.50 

65 46.9 3.04 2.37 5.55 1.19 0.42 0.35 1.55 

70 45.0 3.08 2.09 5.94 1.31 0.41 0.31 1.62 

75 43.2 3.15 1.75 6.43 1.46 0.40 0.28 1.71 

80 41.4 3.26 1.32 7.05 1.65 0.39 0.23 1.84 

85 39.5 3.42 0.77 7.84 1.89 0.39 0.16 2.00 

90 37.7 3.62 0.01 8.86 2.21 0.39 0.02 2.21 

 

 

 

 



Current Research in Agricultural Sciences, 2014, 1(2): 42-52 

 

 
52 

© 2014 Conscientia Beam. All Rights Reserved. 

Table-2. Second Soil Sample Mechanical Properties, ϕ = 300, δ = 200, ρd = 1.51 g / cm3, C = 2.5 

kPa, Ca = 1.4 kPa 

α0 β0 Nγ Nq Nc Nca r (m) Ws (m) Fs (kN) 

5 64.2 38.37 48.97 7.66 2.08 3.58 1.42 8.67 

10 61.9 16.60 21.55 6.16 1.39 1.86 1.03 4.10 

15 59.8 10.73 14.14 5.62 1.19 1.30 0.85 2.85 

20 57.7 8.16 10.85 5.40 1.12 1.01 0.75 2.31 

25 55.7 6.78 9.02 5.35 1.10 0.85 0.68 2.03 

30 53.9 5.96 7.88 5.40 1.11 0.74 0.62 1.87 

35 51.9 5.46 7.10 5.55 1.15 0.66 0.57 1.79 

40 50.2 5.17 6.55 5.78 1.22 0.61 0.54 1.76 

45 48.4 5.03 6.13 6.10 1.30 0.57 0.50 1.77 

50 46.8 5.00 5.80 6.53 1.42 0.53 0.47 1.82 

55 45.1 5.09 5.53 7.10 1.58 0.51 0.43 1.92 

60 43.6 5.30 5.29 7.85 1.78 0.49 0.40 2.06 

65 42.0 5.66 5.06 8.87 2.06 0.47 0.37 2.26 

70 40.5 6.21 4.79 10.28 2.44 0.46 0.33 2.55 

75 38.9 7.05 4.43 12.34 3.00 0.45 0.29 2.99 

80 37.4 8.41 3.84 15.55 3.88 0.45 0.24 3.68 

85 35.8 10.78 2.73 21.19 5.43 0.44 0.17 4.88 

90 34.2 15.87 0.04 33.46 8.82 0.44 0.02 7.49 

 

Table-3. Third Soil Sample Mechanical Properties, ϕ = 320, δ = 230, ρd = 1.42 g / cm3, C = 2.4 

kPa, Ca = 1.14 kPa 

α0 β0 Nγ Nq Nc Nca r (m) Ws (m) Fs (kN) 

5 62.2 41.05 57.21 8.05 2.15 3.59 1.46 8.70 

10 60.0 17.78 24.99 6.47 1.46 1.88 1.05 4.12 

15 57.9 11.54 16.34 5.92 1.26 1.31 0.87 2.87 

20 55.9 8.82 12.51 5.71 1.19 1.03 0.76 2.34 

25 54.0 7.38 10.41 5.68 1.18 0.86 0.69 2.06 

30 52.2 6.54 9.11 5.77 1.20 0.75 0.63 1.92 

35 50.4 6.05 8.25 5.97 1.26 0.68 0.58 1.85 

40 48.7 5.79 7.65 6.28 1.34 0.62 0.54 1.84 

45 47.1 5.70 7.23 6.70 1.45 0.58 0.51 1.87 

50 45.5 5.76 6.93 7.28 1.61 0.55 0.47 1.95 

55 43.9 5.97 6.71 8.05 1.81 0.52 0.44 2.09 

60 42.4 6.37 6.55 9.11 2.09 0.50 0.41 2.29 

65 40.9 7.00 6.44 10.60 2.49 0.49 0.37 2.59 

70 39.4 8.02 6.36 12.82 3.08 0.47 0.34 3.05 

75 37.9 9.70 6.25 16.36 4.03 0.47 0.29 3.80 

80 36.5 12.81 6.01 22.86 5.77 0.46 0.24 5.17 

85 35.0 20.19 5.25 38.32 9.92 0.46 0.18 8.44 

90 33.5 58.81 0.15 119.86 31.87 0.45 0.02 25.64 

 

 

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