Mechanics of Tillage and Traction 3(2+1)

LESSON 18. Traction

18.2. Traction in soils

Soils, like metals, can behave both elastically and plastically. Elastic deformation refers to the ability of the deformed material to return to its original dimensions. Plastic deformation refers to a condition of permanent deformation. For a soil in the elastic condition, a given applied force causes a known deformation. On removal of the force, recovery takes place.

If, however, the force is continually increased, a loading condition will occur that will cause the soil to deform permanently, i.e., it will behave plastically. The onset of this plastic condition is generally con­sidered to be induced by shear failure, i.e., the sliding of one particle over another. In this case, the ability of a particular soil to support a given load before a permanent change to the soil structure occurs is called the shear strength of the soil.

18.3. Shear strength

Granular materials such as soils exhibit cohesive and frictional prop­erties. Cohesion is the bonding together of soil elements irrespective of the type of applied load. Pure clay fits this category; dry sand, on the other hand, exhibits a frictional resistance to shear loads in that the resistance to shear increases with applied load. Most soils exhibit a combination of cohesive and frictional proper­ties. The water content in clay soil has a strong influence on shear strength: the higher the water content, the lower the shear strength. An applied load will result in a wedge-shaped cone below the tire, which will cause deformation and displacement of adjacent soil par­ticles. If the resultant shear stresses are greater then the soil can sus­tain, sinkage will occur, increasing the surface area. This will continue until the soil is able to support the tire and load.

Two forms of shear strength may be considered:

 i. Bulk shear strength: the resistance offered to movement by a relatively large volume of

    soil aggregates.

ii. Clod shear strength: the resistance offered by the individual clod or aggregate.

The main factors that influence shear strength are:

                                                              i.      moisture content

                                                            ii.      packing density and particle size

                                                          iii.      organic matter content.

Figure 1.1 shows the changes in bulk shear strength with changes in moisture content. The moisture content is also related to the upper and lower plastic limits.

 

 

 

 

 

 

 

 

18.4. Plastic limit

As the drying process continues, the plastic state reaches a consis­tency at which the soil ceases to behave as a plastic and begins to break apart and crumble. The increase in shear strength with decreasing moisture content from the upper to the lower plastic limit is clearly seen. The lower plastic limit represents the maximum moisture content where a farmer can break clods during seedbed preparations without causing structural damage. It is a condition frequently accepted as the upper moisture limit for working soils in agriculture. At high moisture contents, clods are very weak, and susceptible to deformation. The higher the organic matter content, the stronger the aggregates. Aggregates produced on fine sand and silt soils tend to be very weak.

18.5. Coulomb and Micklethwaite equations

The failure of an agricultural soil can be described by Coulomb's equation:

                                  ------------------------------------------------- (1.1)

where t is the shear stress of the material, c the cohesive property of the material, s the normal stress on the sheared surface, and  the angle of internal shearing resistance of the material.

For saturated clay, the cohesion is independent of the applied normal load, and so

t = c                ----------------------------------------------- (1.2)

Sandy soils, however, have little cohesion but have larger values for the angle of internal shearing resistance. Thus for a sandy soil,

                                                        ----------------------------------------------- (1.3)

An agricultural soil is composed of both sand and clays, and therefore has properties intermediate between those for sand and clay alone. The typical agricultural soil has therefore both cohesive and frictional prop­erties. Figures 1.2 and 1.3 show the shear versus normal stress char­acteristics for clay and sand soils, and for agricultural soils, respec­tively.

 

 

 

 

 

 

 

 

Micklethwaite (1944), as cited by Reece (1966), proposed the fol­lowing modification to the Coulomb equation.

Let A be the contact area; then

                                           ----------------------------------------------- (1.4)

If t= t_max, then ta is the maximum thrust, or H_max.

The normal stress  s = W/A, and so

H_max  = cA + W tanf ----------------------------------------------- (1.5)

This expression is usually referred to as the Micklethwaite equation. The pressure under a rigid wheel on frictionless soils at small sinkages was given by Reece (1966) in citing Uffelmann (1961) as

                                    p/c = 5.7                      --------------------------------------------- (1.6)

where p is the pressure. The sinkage z necessary to support the wheel for radial pressures given by the above equation is

                                    z = W²/(5.7c)²b²d        ----------------------------------------------- (1.7)

Where,

W - vertical load on the wheel,

 b - width of the wheel, and

 d -  wheel diameter.

The rolling resistance can be considered as principally due to the work done in forming the wheel rut.

 

 

 

 

 

 

 

 

 

 

                       

The distance moved is the sinkage z, and the work done is given by

E_R = 5.7cbz       

where E_R is the energy required to form the rut. Thus the rolling resistance R (assuming that it is due entirely to rut formation) is given by

  R = W²/5.7cbd

 

 

 

 

 

 

 

 

 

 

For a driven wheel, the maximum (drawbar) pull can be obtained by considering the thrust that can be developed given a value for the max­imum shearing stress along the contact patch:

F_D = H - R

Thus, for frictionless soils that have a maximum shear stress equal to c, the drawbar pull is

F_D = cbr sin  - W²/5.7cbd

where  is the angle of sinkage given in figure 1.4.

BEKKER THEORY

Fundamental to the Bekker approach to the theory of land traffic is the relationship between the sinkage z and the normal pressure p. The relationship developed by Bekker is a modification of an assumed linear relationship between pressure and sinkage that is used in civil engi­neering soil mechanics for small sinkages. Bekker (1960) gives the fol­lowing equation:

                                          ----------------------------------------------- (1.8)

where,

k_c - the cohesive modulus of sinkage,

k_f - frictional modulus of sinkage,and

 n - exponent reflecting the hyperbolic shape of the load sinkage curve.

The values of k_c,, and n can be determined for any given soil by conducting load sinkage studies on two plates with different areas. Log-log plots of pressure against sinkage will give straight-line rela­tionships of slope n. Two equations for p at z = 1 enable values for k_c and , to be obtained.

The horizontal shear stress is given by a modification of the Cou-lomb-Micklethwaite equation. For plastic soils, the following rela­tionship is given by Bekker (1969):

                                     ----------------------------------- (1.9)

Where,

            K - slip coefficient and

 j  - the amount of soil deformation that produces stress t

The Coulomb constants c and  can be determined for a given soil by plotting maximum shear stress against normal pres­sure to give the straight-line equation. The slip coefficient (also termed the deformation constant) can be obtained from stress-deformation curves that are obtained with a bevameter (A bevameter is a device used to determine, in situ, the pressure-sinkage relationship for a given supporting surface. It is usually mounted to a vehicle subframe, which acts to provide force reaction during the penetration test).   

           

 

 

 

 

 

 

 

 

 

 

 

Figure 1.5 shows shear stress-deformation curves for plastic and brittle soils (Bekker, 1969). If an annular shear ring is used, then the shear strength is measured in terms of the shear torque, and the displacement is measured in terms of the angular deformation. These tests are usually re­peated at various levels of normal pressures. Idealized shearing stress-deformation graphs for different normal pressures are shown in figure 1.6. The yield points I, II, III, IV mark the end of the quasi-elastic deformation and the beginning of plastic flow. This yield point, for a given soil, will occur at the same deformation value and represents the constant K.

 

 

 

 

 

 

 

 

 

 

 

The idealized shear stress-deformation curves may not be obtained in practice. On soft ground, and at high loads that are greater than the bearing capacity of the soil, the shear stress may continue to rise with deformation without giving any exact yield point. This situation requires correction of the previous equation for .

Slip i can be expressed in terms of the deformation j and the distance x measured from the start of the ground contact area and some location along the ground contact area:

j = ix

The thrust H at a particular slip can be found by using the above re­lationship and the equation for horizontal shear stress. Integration over the track length l gives

             ----------- (1.10)

where H_i is the thrust at slip i.

The rolling resistance R₁ is determined by considering the work done in making a rut of length l and of depth z:

                                               ----------------------------------------------- (1.11)

Substituting for the radial pressure p gives

                                   ----------------------------------------------- (1.12)

For a wheel, Bekker provides mathematical approximations for the sinkage z and the rolling resistance R, such that

                                           ------------------------------- (1.13)

 

where D is the wheel diameter, and

                            ---------- (1.14)

                       

The drawbar pull f_d can then be determined from

                                    F_D = H_i – R_wheel             ------------------------------------------------ (1.15)

A slightly modified version of the equation for the tractive force developed by a tire was given by Bekker (1956) in terms of the slip i, where

                                            --------------------------------------------- (1.16)

and

                                                        ------------------------- (1.17)

where l is the length of the contact area (= 2[¶(d-¶)]^½), ¶  the tire deflection, d the tire diameter, and K the tangent modulus of defor­mation from the ring shear test.