Soil Cutting and Tillage

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Also in the 18th century, Jethro Tull promoted the use of horse-drawn cultivating hoes in wide crop row spacings. The purpose of this technique was to destroy ...

Developments in Agricultural Engineering 7

Soil Cutting and Tillage

O T H E R T I T L E S IN T H I S S E R I E S 1. Controlled Atmosphere Storage of Grains by J. Shejbal (Editor) 1980 viii + 6 0 8 pp. 2. Land and Stream Salinity by J.W. Holmes and T, Talsma (Editors) 1981 vi + 3 9 2 pp. 3. Vehicle Traction Mechanics by R.N. Y o n g , E.A. Fattah and N. Skiadas 1984 xi + 307 pp. 4. Grain Handling and Storage by G. Boumans 1984 xiii + 4 3 6 pp. 5. Controlled Atmosphere and Fumigation in Grain Storages by B.E. Ripp et al. (Editors) 1984 xiv + 798 pp. 6. Housing of Animals by A . Maton, J. Daelemans and J. Lambrecht 1985 xii + 4 5 8 pp.

Developments in Agricultural Engineering 7

Soi l Cuttin g an d Tillag e EDWAR D McKYE S Department Ste-Anne

of Agricultural



College of McGill


de Bellevue, Quebec, Canada

ELSEVIE R Amsterda m - Oxfor d - New Yor k - Toky o 198 5

E L S E V I E R S C I E N C E P U B L I S H E R S B.V. Sara Burgerhartstraa t 25 P.O. Bo x 211,100 0 A E Amsterdam , Th e Netherland s Distributors

for the United




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I S B N 0 4 4 4 4 2 5 4 8 - 9 (Vol . 7) I S B N 0-44441940- 3 (Series )

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CONTENT S 1. I N T R O D U C T I O N T O T I L L A G E A N D E A R T H M O V I N G 1.1 H I S T O R Y O F T I L L A G E 1.2 M A I N T Y P E S O F T I L L A G E T O O L S 1.2.1 Primary tillage 1.2.2 Secondary tillage tools 1.3 E A R T H M O V I N G E Q U I P M E N T 1.4 T H E A N A L Y S I S O F S O I L C U T T I N G A N D T I L L A G E 2. S O I L M E C H A N I C S 2.1 C O U L O M B ' S L A W O F F R I C T I O N A N D C O H E S I O N 2.2 T H E M E T H O D O F S T R E S S C H A R A C T E R I S T I C S 2.3 B O U N D A R Y C O N D I T I O N S 2.4 M E A S U R E M E N T O F S O I L S T R E N G T H P R O P E R T I E S 2.4.1 The direct shear box 2.4.2 The triaxial test 2.4.3 Field tests 2.5 S O I L B E H A V I O U R C O N S I D E R A T I O N S 2.5.1 Soil water pressure and movement 2.5.2 Critical state soil mechanics 2.5.3 Soil stress-strain behaviour 2.5.4 Shear rate effects 2.6 P R O B L E M S 3· SOI L C U T T I N G F O R C E S 3.1 T H E U N I V E R S A L E A R T H M O V I N G E Q U A T I O N 3.2 T W O D I M E N S I O N A L C A S E S : M E T H O D O F S T R E S S CHARACTERISTICS 3.2.1 Smooth, vertical blade 3.2.2 Smooth raked blade in a cohesive soil · · 3.2.3 Rough raked blade in a cohesive soil 3.2.4 Boundary conditions between very rough and smooth · · · 3.2.5 Unconstrained tool to soil adhesion 3.2.6 The shape of failure surfaces 3.2.7 Including soil weight - Hettiaratchi's calculations . . . . 3.2.8 Approximation for soil with weight 3.2.9 Validity of the weightless soil assumption 3.3 T H E M E T H O D O F T R I A L W E D G E S 3.4 S I M I L I T U D E M E T H O D S 3.5 T H R E E D I M E N S I O N A L C A S E S 3.5.1 Hettiaratchi and R e e c e (1967) 3.5.2 Godwin and Spoor (1977) 3.5.3 Three dimensional wedges ( M c K y e s and A l i , 1977) . . . . 3.5.4 Grisso et al. (1980) 3.5.5 Comparison of the methods

1 3 6 8 9 11 16 20 21 22 23 27 29 32 33 34 35 38 38 40 42 44 46 47 47 49 51 51 54 55 55 56 59 64 66


3.6 D Y N A M I C E F F E C T S 3.6.1 Inertial forces 3.6.2 Changes in soil strength 3.7 C R I T I C A L D E P T H 3.8 C O M P L E X T O O L S H A P E S 3.8.1 Curved tools 3.8.2 Shank and foot tools 3.8.3 The moldboard plow 3.8.4 Other tools 3.9 P R O B L E M S 4. S O I L L O O S E N I N G A N D M A N I P U L A T I O N 4.1 M E A S U R E M E N T S O F S O I L L O O S E N I N G 4.2 E F F I C I E N C Y O F S O I L L O O S E N I N G 4.2.1 D r a f t force efficiency 4.2.2 Loosening and pulverization efficiencies 4.3 S O I L M I X I N G A N D I N V E R S I O N 4.3.1 Soil properties 4.3.2 Tool shape and speed 4.4 T O O L S P A C I N G 4.4.1 Spacing and volume of soil disturbed 4.4.2 Tandem tool configurations 5. SOI L P R O P E R T I E S A N D P L A N T G R O W T H 5.1 S O I L C O M P A C T I O N 5.2 M E C H A N I C A L A N D H Y D R A U L I C P R O P E R T I E S O F C O M P A C T E D SOIL 5.2.1 Mechanical properties 5.2.2 Hydraulic properties 5.3 S O I L P H Y S I C A L P R O P E R T I E S A N D P L A N T G R O W T H 5.4 T I L L A G E O F C O M P A C T E D S O I L 5.5 P R O B L E M S 6. T R A C T I O N M A C H I N E S 6.1 F O R C E A N D E N E R G Y B A L A N C E S 6.1.1 Forces on a traction machine 6.1.2 Energy balance at the machine-soil interface 6.2 T R A C T I O N , S O I L D E F O R M A T I O N A N D S L I P 6.2.1 M a x i m u m traction force 6.2.2 Soil deformation and slip 6.2.3 Estimation of contact areas 6.3 S I N K A G E A N D R O L L I N G R E S I S T A N C E 6.3.1 Sinkage in soil 6.3.2 Rolling resistance 6.3.3 Bekker's formulae 6.3.4 M c K y e s (1978) 6.4 M A C H I N E I N T E R N A L L O S S E S 6.5 M A T C H I N G M A C H I N E S A N D T O O L S 6.6 P R O B L E M S

71 72 73 75 79 80 81 83 84 87 92 95 98 99 100 101 103 105 112 114 115 123 124

125 127 129 131 134 135 140 146 147 149 152 155




A P P E N D I X 1. Values of Í

factors, two dimensions


A P P E N D I X 2. Values of Í

factors, three dimensions


A P P E N D I X 3. Values of failure wedge angles


A P P E N D I X 4. Selected values of soil mechanical properties






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Chapter 1 I N T R O D U C T I O N TO T I L L A G E A N D


1.1 H I S T O R Y O E T I L L A G E E o r thousands of years of recorded history, groups of human beings have been tilling the soil in order to increase the production of food. Early evidence indicates that simple lightweight wooden plows, for instance, were employed extensively in the valleys of the Euphrates and Nile R i v e r s by the year 3000 B.C. Animals in the form of oxen provided the traction necessary to" pull the plows, preparing the soil for the seeding of barley, wheat and flax crops, (Encyclopedia Brittanica, 1979). The plows used during that period had no wheels or moldboards with which to invert the soil and prepare a true plow furrow. Nevertheless, they served to perform an initial breakup of the soil to a shallow depth and subsequently to cover the seeds of the crop. A n example of an early Egyptian wooden plow is depicted in E i g . 1.1.

E i g . 1.1. Early wooden plow, Thebes, Egypt, circa 3000 B.C. It was more than 2000 years ago that the first iron plows were fabricated in Northern Honan, China. A t first these were small hand-drawn tools having a f l a t V-shaped iron piece attached to a wooden blade and handles. During the first century B.C., water buffaloes were used to pull tillage implements. Subsequently, triple-shared plows, plow-and-sow instruments and harrows were developed. P l o w s have been used also in India for thousands of years. Early implements had no wheels or moldboard, being composed of wedge-shaped hardwood blocks, and pulled by bullocks. The soil was broken into clods but not turned over, and this primary tillage was followed by the passage of a rectangular wooden beam, also drawn by bullocks, for the breaking of clods and the levelling of the seedbed. I r o n plow shares appeared on R o m a n plows about 2000 years ago, as well as cutting coulter knives. Still no moldboard was used to turn soil over. These plows were pulled by teams of oxen, up to eight per team on a heavy soil with high strength. There were reports, but no solid evidence, that plows equipped with wheels appeared in Northern Italy around 100 A . D .




Draft pole Handles


Fig. 1.2. Two-wheeled plow with coulter and moldboard, 16th century Europe. W h e e l s , cutting coulters and moldboards all were included on plows in Europe by the year 1500 A . D . , such as in F i g . 1.2. These implements could invert the soil and make true furrows and a true seedbed. High ridges of soil were left in the fields, some of which remain in evidence today. Rather than chest yokes for animals to pull tools, padded horsecollars, apparently invented in China, were attached to horses. This innovation significantly improved the animals' ability to provide draft force. Teams of two, four or eight and more horses or oxen were often used in primary soil cultivation, depending on the strength of the soil to be tilled. Tillage implements very similar to those now in use began to appear with the introduction of the Rotherham plow in the Netherlands, England and Scotland by the early 1700's. The principal design features of this instrument remain virtually unchanged today. Also in the 18th century, Jethro Tull promoted the use of horse-drawn cultivating hoes in wide crop row spacings. The purpose of this technique was to destroy weeds competing with crops and to keep the soil in between the rows in a good crumbly and friable condition for water infiltration. M o r e than 100 years later, Robert R a n s o m e patented a cast-iron plow s h a r e in 1785, and a self-sharpening share in 1803. Later he introduced s t a n d a r d parts for tillage implements which could be replaced in the field, and a double-shared plow. Around the same time, the practice of "mole 11 plowing began in the United Kingdom to provide subsurface drainage channels in wet fields. This technique is accomplished with a deep soil cutting " l e g " trailing a bullet-shaped mole at the base, which leaves continuous tube-like cavities in certain plastic soils, and greatly improves the internal drainage of wetlands. O n the A m e r i c a n organic prairie soils, problems of satisfactory tillage induced John Deere, an Illinois blacksmith, to develop a steel plow with a one piece share and moldboard in the 1830 fs. Animal power for traction began to w a n e with the introduction of the steam powered tractor in the 1860's, beginning with the larger farm operations. The first gasoline powered tractor a p p e a r e d in the U n i t e d States in 1892, and many manufacturers were producing these machines in Europe and A m e r i c a within a few years. The steel wheels on tractors began to be replaced by rubber tires in 1932, and by 1968 it was estimated that there were over 15 million tractors in the world.



1.2 M A I N T Y P E S O E T I L L A G E T O O L S 1.2.1 Primary tillage Primary tillage of soil is mainly for the cutting and loosening of soil to a depth of 15 to 90 c m . The moldboard plow is the most common primary tillage tool in the world, and has the capacity to break up many types of soil. It has the further ability to turn over and cover sod, crop residues and weeds. E o r s p e c i a l applications there are a great many types of plow, including stubble, general purpose, clay soil, stiff-sod, blackland, chilled general purpose and slatted plows. They may be used singly or in groups of from two to a l a r g e number of bottoms (shares), with the width of each plowshare being between 25 and 45 cm or more.

Eig. 1.3. A five bottom moldboard plow in operation (Courtesy of John Deere & Company, Moline, Illinois). A n o t h e r common primary tillage tool is the disk plow, comprising a hardened steel round concave disk of 50 to 95 cm in diameter. The disks have s h a r p e n e d and s o m e t i m e s s e r r a t e d e d g e s , and are often fitted with s e l f - c l e a n i n g s c r a p e r s . The draft force needed to pull a disk plow is approximately the same as one moldboard share in similar soil conditions. But the disk plow performs better in sticky, non-scouring (poor plow cleaning)





1.4. A five bottom moldboard plow (Courtesy John Deere & Company, Moline, Illinois).

soils, in hard dry ground, in many organic peat soils and where it is necessary to break hardened plow soles or to accomplish deep plowing. Chisel plows also find use in many areas. They are tools with long shanks and double-ended chisel points, usually about 6.4 cm in width. Chisels are usually mounted on a frame in gangs of 5, 10 or more at s p a c i n g of 30 cm or so. Chisels can cut, loosen and stir the soil, but do very little turning over. They are well adapted to loosening hard dry soils, shattering hard pans and soles and conserving the mulch of crop residues on the field surface, which is useful for soil and water conservation in some areas. T h e s u b s o i l e r is similar in principle to the chisel, but it is more heavily built and rigid for operation at depths of 40 to 90 cm to loosen deep soil layers for the promotion of water movement and root growth. Sometimes a t o r p e d o or bullet-shaped " m o l e " drain former is attached to the rear of the subsoiler, and wings can be attached to the sides of the subsoiler leg to i n c r e a s e the effective working width up to 90 cm on very heavy models. A tractor of 40 to 60 kW power is needed to pull one subsoiler shank at a depth of 45 cm in heavy soil, while a large track-laying tractor in the order of 50t mass is needed for three winged subsoilers operating at 90 cm depth. T h e r o t a r y plow is a n o t h e r primary tillage instrument, requiring a mechanical power source, usually provided by an auxiliary drive on the towing t r a c t o r . It consists of a set of knives, tines or rods which are rotated on a horizontal shaft and covered by a sheet metal hood. Soil is chopped up by the k n i v e s and thrown against the inside of the hood, resulting in a fine, loose soil structure, depending on the forward speed of the implement. A good level seedbed of very loose soil is created, but at the expense of increased capital, m a i n t e n a n c e and energy costs compared to other primary tillage tools. The r o t a r y plow is used extensively in intensive vegetable production, but is not recommended in areas where severe soil erosion due to wind or water flow are prevalent, especially in light or organic soils.



Eig. 1.5. A chisel plow folded for transport (Courtesy John Deere & Company, Moline, Illinois).

Eig. 1.6. A rigid deep chisel plow with rolling cutting coulters (Courtesy John Deere & Company, Moline, Illinois).




1.2.2 Secondary tillage tools S e c o n d a r y t i l l a g e o p e r a t i o n s usually are performed after a primary treatment on a field for one or more purposes. These aims are improved seedbed levelness and structure, increased soil pulverization, conservation of moisture, destruction of weeds, chopping of crop residues and the like. The harrow is a common term for many of the secondary tillage inplements, being a frame fitted with one of a number of types of soil moving tools. The names of some of these are disk, spike-tooth, spring-tooth or rotary cross harrow, according to the instruments which are mounted thereon.

F i g . 1.7. A heavy disk harrow being used to break up corn stubble (Courtesy John Deere & Company, Moline, Illinois). Sometimes harrows are combined with rollers comprised of V-shaped wheels or hollow cage wheels for improved fineness and smoothness of the soil s u r f a c e . Other popular secondary tillage tools are cultivators with V-shaped sweeps or blades for the removal of weeds at shallow depths between crop rows, or rod weeders which have vertical rods that turn and gather up weeds. There are many other special tools used in agricultural operations, some of which closely resemble earthwork construction implements in form and function. Tractor backblades for levelling and shallow excavation, long-beam land levellers, bulldozer backblades for ditching and land smoothing, front end loaders for material transport and ridging blades which form ridges and furrows for specialized cropping systems are just some of these examples.


E i g . 1.8. A heavy disk harrow (Courtesy John Deere Illinois).


& Company, Moline,

F i g . 1.9. A spring-tooth cultivator with V-sweeps attached for weed cultivation (Courtesy John Deere & C o m p a n y , Moline, Illinois).




1.3 E A R T H M O V I N G E Q U I P M E N T C o n s t r u c t i o n equipment for earthmoving is highly diverse in shape and f u n c t i o n , but most of the soil cutting machines can be categorized into one of three principal classes, namely (1) blade, (2) ripper or (3) shovel. Tools w h i c h resemble blades include bulldozer front and back blades, road graders, h a u l i n g s c r a p e r s , s n o w p l o w s and other all straight-edged blades. These instruments cut and push soil or other granular material at a depth which is generally less than their width. Ripper types of tool, on the other hand, are m o r e n a r r o w compared to their working depth, and are often attached to graders and bulldozers when it is necessary to cut and loosen hard soil, pavement or even soft rock layers. Shovels are blades equipped with sides which form a space in which soil or other materials can be cut and lifted up. T h e action of shovels usually causes a vertical-sided trench to be excavated owing to the cutting action of the shovel sides in conjunction with the bottom cutting edge (Fig. 1.10).





1.10. A comparison of the aspect ratios of three different classes of earthmoving implements.

The basic shapes of earthmoving tools have not changed a great deal since a n t i q u i t y , although most are operated today by mechanical power sources in large operations, and their construction benefits from modern metallurgical engineering. However, in small scale earthmoving work, hand power is still used to operate many tools, such as shovels, hoes, picks and the like. In addition, there remain many areas of the world where animals are the principal source of traction for all construction and agricultural soil cutting. Whatever the source of operating power, a cutting tool requires a certain force for movement in a particular soil, and it will have a certain effect on the soil structure depending on the initial soil state, and on the geometry of the t o o l . It is to be able to find these required forces and resultant soil changes that comprises the principal purpose of this book.





A h y d r a u l i c excavator machine for construction and general earthmoving (Courtesy Caterpillar Tractor Co., Peoria, Illinois).

1.4 T H E A N A L Y S I S O F S O I L C U T T I N G A N D T I L L A G E Basically all soil cutting, moving and tillage instruments transfer soil from its original location. Thus the mechanical failure of the soil material is i n v o l v e d , in the sense that the mass of soil being moved does not retain its o r i g i n a l geometric shape. The design of effective and efficient cutting tools begins with the analysis of this soil failure, in order to predict the forces and energy required by the implements. The design process proceeds subsequently to the description of soil manipulation and structural changes which result f r o m cutting tool action, depending upon the special applications of interest. The subsequent chapters in this book treat the basics of soil mechanics necessary for these calculations, and the analytical techniques which are available for the prediction of forces acting on moving implements. Following that, the loosening and manipulation of soils are discussed, as well as some of their consequences to the growth of crops. Lastly, some basics of the design of machines which power soil cutting implements are presented.




Fig. 1.12. A tracked tractor fitted with a wide bulldozing blade on the front, and a narrow ripping blade on the rear (Fiat-Allis Company).


1.13. A root plow Forida).


2.3 m

width (Fleco






Chapter 2 SOIL


2.1 C O U L O M B ' S L A W O F S O I L F R I C T I O N A N D C O H E S I O N In Chapter 1 it was noted that the cutting of soil involves the material failure of soil. This mechanical failure usually occurs in the shear mode along internal rupture surfaces in the soil, and often at the boundary between soil and cutting tool surface.

Cutting tool

F i g . 2.1. Internal and boundary soil failure during cutting. In order to analyze the mechanism of soil failure, it is necessary to know under what regimes of forces or pressures soil failure does occur. The basis of soil mechanical strength knowledge is ascribed to Coulomb (1776). Working w i t h masonry and soils, Coulomb noted that there appeared to be two mechanical processes in action which determine the ultimate shearing strength of both these materials. One process he called friction, and the other cohesion. In the first case, a portion of the shear strength is proportional to the pressure acting perpendicularly on the shearing surface. In the second, part of the strength resisting shear movement is a constant, regardless of the normal pressure acting. The total material shear strength is the sum of these two components as follows.

s = c + óçtano^


where s = shear strength (ultimate shear force per unit area), c = cohesion ( f o r c e per unit area), á ð = the normal pressure acting on the internal shear s u r f a c e in question and ßÂçö = the coefficient of internal sliding friction. P a r a m e t e r ö is called the angle of internal friction and is sometimes a d i r e c t l y visible quantity, such as the angle of repose of a pile of granular material. If the internal plane of sliding is not known in a material beforehand, it is necessary to find the relationship among stresses on planes at different slopes in order to determine the direction of the potential rupture surface.





2.2. Some of the diagrams from Coulomb's work on masonry and soil mechanics (1776). F i g . 1 , Coulomb's tensile strength test on white quarry rock; F i g . 2 and 3, shear and bending tests on rock beams; F i g . 4, addition of force vectors; F i g . 5, compression test on a masonry pillar (from which E q n . 2.4 was subsequently developed); F i g . 6, beam bending stresses; F i g . 7 and 8, active soil failure behind a retaining wall and the calculation of the curved failure line shape.



In 1914, Mohr provided equations and a graphical method for finding s t r e s s e s on different planes in a material at equilibrium. P i g . 2.3 shows the s t r e s s conventions commonly adopted for soils, and the graphical results of M o h r ! s equations.



4 0\


Sign convention

Physical space



Fig. 2.3. Stresses at a point. Principal stresses and normal and shear stresses on an inclined plane, and the graphical form of normal and shear stress combinations on planes at different angles. In Fig. 2.3, the principal stresses, and G3, are shown acting on mutually perpendicular planes bounding a small element of material. The principal planes are, in general, the only two upon which there is no shear stress plane is another plane, a c t i n g . A t a counterclockwise angle, 0 f, from the ab, w h i c h has normal stress, ó ç · and shear stress, 7, acting on it. The convenient convention for soils is that normal stresses are positive when c o m p r e s s i v e , and shear stress is positive when acting in a counterclockwise sense on an element of material, as shown in F i g . 2.3. I f the material is in equilibrium, that is the acceleration of all points is equal to zero, then the forces on the triangular element abc must balance to z e r o . Taking a unit depth of material perpendicular to the planes shown in the Figure, the forces can be added up in directions perpendicular and parallel to plane ab as follows. a Rd x / cos0 ' - a^dx t a n s i n 0' - ó-^÷

cos0 ' = 0

T d x / cos0 ' + a^dx tan0 ' cos 0' - ó-^dx sin0 ' = 0 ï

óç = a 1 c o s > 7=

(ó ÷


+ a 3s i n z0 ' =

- a 3) s i n 0 fc o s 0 '


ó º 1 + á-, 2

(ó Ë1 - ó-\

^ + ß



^jcos 2ff

(2.2) (2.3)




By the use of these two equations, the combination of stresses, ó ç and r, can be found on a plane at any angle 0 ' from the G\ plane. Eqn. 2.2 and 2.3 happen to describe a circle on a plot of Ô versus a n if the same scale is used for each axis, as shown in E i g . 2.3. A l l of the points on the circle represent s t r e s s combinations, ó ç and T, on planes in the material, and each stress combination point on the graph can be located by a radius of the circle rotated at a counterclockwise angle, 20', from the major principal stress point, on the graph. There is another condition which relates stresses at a point in a material when failure occurs, or is impending, namely the material strength criterion. In m a t t e r which follows the friction and cohesion law of Coulomb, Eqn. 2.1 gives this relation on the shear failure plane. This law can also be represented on a plot of shear versus normal stresses by a line as shown in E i g . 2.4 labelled "strength limit".






F i g . 2.4. Shear failure of a soil under uniform stresses, and a plot of the strength limit law, together with M o h r ' s circle. The strength limit forms two straight lines in the graph making angles of ± ö with the a n axis, and intersecting the Ô axis at ± c . It is possible for a soil to obey both the C o u l o m b f r i c t i o n - c o h e s i o n strength law, and M o h r fs d e s c r i p t i o n of equilibrium simultaneously. A l l that is required is that the m a t e r i a l be in a failure condition on some plane, without being subjected to a c c e l e r a t i o n s , and this can take place during the process of soil failure at constant speed. When these requirements are met, then all stress combinations on planes at different angles in the material lie on the Mohr circle, and two planes in particular are subjected to the combination of shear and normal stresses which accompanies shear failure. In F i g . 2.4 these two failure planes are located on the stress graph where Mohr's circle touches the strength limit l i n e s , and they are labelled with their stresses, tff and ±T f . For all other points on Mohr's circle of stresses, the shear stress acting on the



corresponding material planes is less than the strength, and slip failure should not occur. In F i g . 2.4 the radius to each point of failure stresses is perpendicular to the tangent strength limit line, and thus the angle of the rupture plane plane can be calculated as: relative to the 0 f = ±(7Ã/4 +ö /2)

in radians


This angle was also found by Coulomb (1776) without the use of the stress circle, but following the same basic mechanical principles. It m u s t be noted that equilibrium can be maintained only if the Mohr circle touches each strength limit line at one point, and is not intersected at two places by each line. Otherwise shear stresses would exceed strength in some directions in the material and accelerations would be the result. P r o b l e m 2 . 1 : In F i g . 2.5 an element of a drained cohesionless sandy soil is shown in the state of failure. Find the angle of internal friction of the soil, and the angle of the rupture plane shown in the Figure as 0^.


2.5. The physical diagram, and corresponding Problem 2 . 1 .



circle for

Solution: F o r an aid in visualizing the stresses and angles involved, Mohr's c i r c l e is constructed, as in F i g . 2.5, by the stress combinations (30,5) k P a , normal and shear stress on the horizontal Ç plane, and (10,-5) k P a on the v e r t i c a l V plane. The principal stresses for this point in the soil are found from the geometry of the circle as follows. ó

3É "



20 + ^ 1 2 5 k P a =

In the right-angled triangle, O C F ;

EC _ oc -






( ^ - a 3) / 2

(óé +ó 3 ) / 2

11.2 20.0

0.56 = sin 34. Ã



^ | kPa




Using Eqn. 2.4, the angle from the

plane to the rupture plane;

In F i g . 2.5, from the Mohr circle;

Note that the above expression also tells us that the major principal plane on which ó•

F i g . 2.12. A plot of the failure shear force, F m, versus vertical force, N, values yields the soil angle of internal friction, ö , and c A . The direct shear test can be performed on either dry or wet soil samples, whether they be restructured and packed into the shear box, or cut from field s a m p l e blocks to fit closely into the box with a minimum of structural disturbance. If slow tests are to be conducted on wet soil, the box may be immersed in a water bath which does not alter the magnitudes of vertical and s h e a r f o r c e s applied, but prevents evaporation f r o m , and drying out of soil samples. The effects of water pressure within the soil during confinement and s h e a r actions, and the recommended speeds of shearing will be treated later in the Sections concerning pore water pressure and dynamic effects on soil strength. 2.4.2 The triaxial test In the traxial soil test, a sample of material, usually cylindrical in shape, is c o n f i n e d by a fluid, generally water, in a pressure-tight cell, as shown in Fig. 2.13. The pressure in the cell fluid is provided by an external source and is labelled a 3 in the figure. A flexible rubber membrane surrounds the soil sample to prevent the cell fluid from penetrating the sample and altering its constitution. By means of a movable piston rod in the top cover of the cell, a force, P 1 ? can also be applied to the soil cylinder. A porous stone is often p l a c e d in contact with the bottom of the sample, and sometimes above as w e l l , in order to allow access to the internal pore water of the soil, thus p e r m i t t i n g either drainage of the pore water, or the measurement of its hydrostatic pressure, u. This testing device provides the best approximation of a state of uniform s t r e s s e s acting on a soil body in three directions (triaxial) and it gives considerable flexibility in the control of these stresses, as well as of the pore




water within the soil. The principal stress on the sides of the sample is equal to G^, and the average stress acting vertically has a value; °l




] /



where A = the cross-sectional area of the soil cylinder at any time.

F i g ; 2.13. Elements of the triaxial soil testing apparatus. T o determine the strength parameters of a given soil, a series of shear tests is performed on samples as in the case of the direct shear box test. A s a n d y sample could be reused for successive tests, if it is kept at the same density for each of the tests. However a sensitive clay soil must be tested u s i n g multiple samples which are as similar to each other as possible, since the shear testing is destructive of the soil fabric. D u r i n g each shear test, the vertical deformation of the sample can be measured via the movement of the top piston rod, and volume changes can be monitored for saturated soils through the pore water access tube. It is d i f f i c u l t to determine the volume changes in unsaturated soil, because air w i t h i n the sample can change volume without being observed through pore water extrusion or infiltration. T h e v e r t i c a l force, P ^ , is increased continuously or in steps until a m a x i m u m value is reached, or until the soil fails with accompanying large v e r t i c a l deformation and obvious visible signs of rupture. After a series of t e s t s has been performed on a soil at different levels of cell fluid confining pressure, the major and minor principal stresses are known for the soil failure condition. These failure stresses can be used to construct a Mohr's circle for e a c h of the tests, as shown in F i g . 2.14. The strength parameters, c and ö, are then determined by drawing the best fit straight line envelope to these circles.








N^est 3




Fig. 2.14. Construction of M o h r ' s circles using the levels of principal stresses at failure of a series of soil samples, and the determination of strength parameters, c and ö . During slow triaxial shear testing, if water is allowed to drain freely from the s o i l s a m p l e , then the test is known as a drained triaxial test, and it is a s s u m e d that the water pressure within the soil is always close to zero. If the s o i l water is not allowed to flow out at all, the definition of undrained triaxial test is applied. In this latter case, the option exists of measuring the pore water pressure during the test by means of a gauge which requires very little volume change for readings (such as an electronic pressure transducer). This pore water pressure, u, can then be subtracted from the total stresses a p p l i e d to the soil in order to estimate the effective, intergranular stresses a c t i n g between soil particles. M o h r ' s circles constructed from such effective or intergranular stress values will give so-called effective strength parameters of cohesion and friction angle. The applications of pore pressure and effective stresses will receive further attention in a later Section of this Chapter. For further details on the construction and procedures of the triaxial test, the reader is referred to L a m b e (1951) or Bishop and Henkel (1957)· Problem 2.2: A triaxial test series is performed on four undrained samples of a c l a y taken from the same depth in a field. The measurements are listed below for the points when rupture occurred in each test. Find c, ö , c 1, ö 1. Initial sample diameter = 35.6 m m Initial sample height = 76.2 m m

ó í kPa 70 140 210 280

A h , mm

Ѻ, Í

15.2 14.9 15.1 15.0

310 415 500 590

u, k P a 4 34 68 98




A s s u m i n g that there is practically no air in the samples, and that no v o l u m e c h a n g e o c c u r r e d d u r i n g the u n d r a i n e d t r i a x i a l t e s t i n g , the cross-sectional area of the soil cylinders on which P i acts can be calculated by using the measured vertical deformations of samples. Volume V = h D 2 7Ã/4 = 76.2x35.6 2 7Ã/4 m m 3 = 75,848 m m 3

= (76.2 m m - Ah)D ð/4 = (76.2 m m - Ah) A = ó3 +

P 1/ A

a 3, k P a 70 140 210 280




A, mm 1243 1237 1241 1239

P 1( 7 6 . 2 m m - Ah)/75,848 m m 3 2

P X/ A , k P a 249 335 403 476

a^kPa 319 475 613 756

ó\ =

a-^-UjkPa a ! 3 , k P a 315 441 545 658

66 106 142 182

The strength parameters, c and ö , can be measured from the M o h r ' s circles constructed in E i g . 2.15 from the failure stress combinations. Two sets of c i r c l e s are shown, one for the total applied stresses, and the other for e f f e c t i v e , intergranular stresses, depending on which strength quantities are desired. Alternatively, the strength parameters can be calculated from each c o m b i n a t i o n of circle pairs using Eqn. 2.5, as shown below, and the average values for all the tests can be found. For the four tests in this example, six d i f f e r e n t pairs of stress combinations, (J-j, and (7g, exist, and six values of c and ö are available for finding average values. For each pair of test results;

sin0 =

[(ó ë - a 3 ) a - (ó 1 - = c cot ä




If the above assumption is accepted, then angle â can be found from the Mohr circle geometry to be +





T A B L E 3.3 Approximate and more exact Í ö

a° 30 60 90

1.73 1.43 2.66



^\ /

JL 2(ce - u)tan r ^ cos0 e since(l -sin0 )

= 7 g d zN 7 + c d N c + qdN^


t a n




30° Í

1.55 1.80 3.60

H e t t i a r a t c h i (1969).


3.5 3.7 5.8

factors for soil with weight. þ



3.5 3.6 5.8

2.7 4.4 15.8






3.0 5.6 21.0

45° Í


6.3 10.3 25.5



6.3 10.5 26.0




Calculations using Eqn. 3.37 in Table 3.3 show that the approximation for N q is nearly perfect, because soil weight is not directly involved in the effect of surcharge q on the tool force. The weight factors Íã , are generally about 2 0 % lower than Hettiaratchi's calculated values. This is understandable since the approximate model has not included any of the weight of region A B C of F i g . 3.10, and thus underestimates the frictional strength generated along the slip line section B C . 3.2.9 Validity of the weightless soil assumption T h e s h a p e s of soil failure lines in front of cutting tools may not be exactly logarithmic spirals because the unit weight of the soil invalidates E q n . 3.32 and 3.33. Nonetheless, a check can be attempted on the accuracy of predictions of cohesion and surcharge pressure factors calculated by the weightless assumption in E q n . 3.15 or 3.20. In Table 3.4, the results are shown and compared to those of Hettiaratchi (1969) who had included soil weight in more exact calculations. TABLE


N c and N q factors calculated assuming a weightless soil compared to more exact calculations.






Na c

þ = 30°

0 Í q






N c


Í q




þ Í

= 45° Na Í c c q

Í 3" * q


30 60 90

-.19 1.10 2.00

-b 1.10 2.00

2.00 1.15 1.00

1.96 1.15 1.00

-.36 1.28 3.46

-b 1.35 3.48

1.79 1.89 3.00

2.30 1.92 3.00

-.56 1.21 4.83


30 60 90

2.00 1.15 1.00

2.20 2.00 2.10 1.15 2.60 1.00

2.00 1.15 1.00

2.00 3.60 4.03 4.70 8.05 8.60

3.46 3.66 5.80

3.50 3.60 5.80

3.44 5.10 6.27 6.10 8.69 9.50 10.3 10.4 24.1 24.0 25.8 26.0


- b 1.44 1.55 2.36 4.95 5.83

2.52 2.52 5.80

H e t t i a r a t c h i (1969). N o t determined.


3.3 T H E M E T H O D O F T R I A L W E D G E S Terzaghi (1941) examined the possibility of approximating logarithmic spiral slip l i n e s by straight lines, for ease of resolution of forces, (an idea indeed proposed by C o u l o m b in 1772). In F i g 3.11, the exact shape of the lowest soil slip line, caused by a tool c u t t i n g the soil, is approximated by a straight line at an angle to the horizontal, â, of which the magnitude is not yet determined. The pressures on each surface of the resultant moving soil wedge are shown integrated to total f o r c e s on the various sides and over the volume. Assuming soil slip on the t o o l s u r f a c e and within the soil itself, the frictional components of shear s t r e n g t h on the two slip lines have been combined with perpendicular forces to form resultant forces Ñ and R as indicated, and cohesional resistance




F i g . 3.11. The wedge theory of passive soil failure. forces are shown separately as c a L and cL^. The basic model in F i g 3.11 can be expanded fairly easily to incorporate layered soils, sloped soil surfaces, curved tools, surface point loads and other complications which are difficult to include in the more exact methods of stress calculation. Leaving wedge angle â undetermined as yet, the net forces in the horizontal and vertical directions are assumed to be zero because of equilibrium, and Ñ solved as follows, for a unit tool width: Ó F x = P s i n ( a + 5 ) + c a L c o s a - Rsin(/3+(/>) - c L ^ o s /3 = 0 ÓÃ


- - P c o s ( a+ < 5 ) + c L s i n c e - Rcos(/J + 0Ë mg/q


R o o t d e n s i t y < 0.1 m g / g




F i g , 5.11· Root density distributions in a plots were subjected to different t r a f f i c , at an average contact al., 1979). By permission of the


clay field of silage corn wherein levels of compaction by machinery pressure of 61.7 k P a (Raghavan et Publisher.




18 16 14

Y = 16-90(7 -0.99) (1976)

r 2 = 0.76




10 D r y matter yield , Y




*Õ = 1 2 . 3 - 2 8 2 ( 7 - 1 . 1 3 ) '


r 2 = 0.75

0-1977 Ä -1980





Soil dry density , 7 d t / m

1.2 3


( 0 - 2 0 cm)

F i g . 5.12. D r y matter yield of silage corn as a function of soil dry density between 0 and 20 cm depth in a clay ( M c K y e s et al., 1979 plus unpublished data).

The results of F i g . 5.12 were obtained in successive plantings of the same hybrid of silage corn on a field of clay in the years 1976, 1977 and 1980. The various densities of topsoil were obtained by applying different levels of m a c h i n e r y traffic to plots following an initial rotary cultivation of the field in the spring of each year to a depth of approximately 25 c m . Between the years 1977 and 1980, the yields in plots of different density were close together. In 1976, the yields were higher at lower soil density, around 16 t/ha at one t / m 3. The curves in F i g . 5.12 fitted to the data points illustrate two important aspects of the effects of soil density upon crop growth, namely the phenomenon of optimum density, and the role of local seasonal precipitation. The occurrence of an optimum density for soil was observed by Vomocil (1955) and reported by Rosenberg (1964). Vomocil noted the yields of field corn, sweet corn and potatoes on some New Jersey soils, at different degrees of compaction, and concluded that a soil which is either looser or more dense



t h a n a particular density value will incur yield losses compared to that optimum structure. H e proposed an equation to describe this phenomenon with a parabolic shape, such that the loss in crop yield increases as the square of the difference in soil dry density from the optimum, as shown below. Y

* -



C (



d r

) 2


where Y * = the maximum obtainable crop yield Y

= the actual crop yield


= a constant

^dry Y*

H rv




a c t u a s o




density (averaged from 10-40 cm depth)

= the optimum soil dry density for maximum yield

Vomocil (1955) suggested that the constant C , which can be referred to as a f a c t o r of sensitivity to soil compaction, depends upon both the type and variety of crop, and the weather. The results of E i g . 5.12 confirm the concept of a parabolic change in crop yields. However, there appears to be a distinct d i f f e r e n c e in yield behaviour between the 1976 and the other two years' seasons. The rainfall pattern was also markedly different in these periods, and may well provide a sufficient explanation for the different behaviour. In 1976, one of the wettest summers on record in the Montreal area, 330 m m of rain fell in the combined months of June, July and A u g u s t . In the years 1977 and 1 9 8 0 , 215 and 220 m m , respectively, were experienced in the same period. These latter two amounts of precipitation are essentially the same, as are the silage corn yields at each soil density in E i g . 5.12 for those years. W i t h about 5 0 % more rainfall in 1976, not only has the sensitivity factor C of E q n . 5.5 diminished from 282 to 90 t/ha/(t/m 3)^ but the apparent optimum soil dry density has been reduced as well from about 1.13 to 0.99 t / m 3, and the maximum possible crop yield, all other factors being equal, has increased from 12.3 to 16 t/ha. It would appear then that it is not only the compaction s e n s i t i v i t y constant, C , which is dependent on annual weather patterns, but also the optimum soil density, and the maximum obtainable crop yield. S u c h behaviour is reasonable in view of the preceding comments on soil h y d r a u l i c properties as a function of compaction. A loose soil with large m a c r o p o r e s , such as the clay of E i g . 5.12 at a dry density of 1 t / m 3 and porosity 6 2 % , might normally be expected to drain too quickly, and to retain an insufficient quantity of moisture during dry periods of the growing season f o r optimum crop growth. However, during a year such as 1976, when the s u m m e r rainfall exceeded the normal average by some 5 0 % , there were no extended dry periods at any time in the growing season. Therefore, the loose s o i l f a b r i c , w i t h its r e d u c e d impedance to root growth and increased conductivity for moisture and nutrients, allowed the soil-plant system to take advantage of the plentiful rainfall and to produce higher than normal yields. The years 1977 and 1980 represented average weather conditions from June to September, and were thus more typical of what is to be expected in most




y e a r s . In such a case, the soil density of 1 t / m 3 was too low for water retention in periods between rainfalls, which extended up to ten days in those years. The available water in the soil fabric was used to an extent that there w a s considerable water stress in the plants at some times, and over the g r o w i n g season the total crop yields were reduced by some 3 0 % below the optimum quantity which occurred at a somewhat higher soil density (1.13 t/m 3). In all of the years of the study illustrated in F i g . 5.12, and in all studies r e p o r t e d on the effects of compaction on crop growth, an excessive amount of compaction above the optimum dry density also results in reduced crop g r o w t h . I n the case of the clay soil of F i g . 5.12, the dropping off of yields w a s quite rapid, with nearly half of the yield being lost for a dry density i n c r e a s e of only 0.13 or so t / m 3 above the optimum. Other soil types, such as the s a n d y loam of F i g . 5.13 below, appear to be less sensitive to the absolute amount of density change. The difference here is logical when one c o n s i d e r s that there is a much larger discrepancy between the sizes of m a c r o p o r e s and micropores in a clay soil, and only small overall density changes may be needed to close off the macropores among clay structural units. In a sandy soil, however, there is a more gradual distribution of macro and m i c r o p o r e volumes among soil particles, and larger overall density increases are required in order to have an equivalent effect on soil structure as it influences the movement of roots and moisture in the growing season.

Fig. 5.13. Relationship between silage corn crop yields and the dry density of the 0-20 cm layer of a sandy loam soil (Negi et al., 1981). By permission of the Publisher.



In a detailed study of root growth and water movement in field plots of silage corn on clay soil, Douglas and M c K y e s (1983) outlined the stresses which can arise in soils compacted or tilled to different structures. F i g . 5.14 shows an example of the rate of water extraction from corn roots at varying depth in the soil, along with the dry density patterns in the corresponding p l o t s of different cultural treatments. The roots in the plots tilled by a moldboard plow and a subsoiler were not active as deep as the roots in other treatments, most likely owing to a dense " p a n " layer of soil occurring at the 20-25 cm depth. This phenomenon shows up also in F i g . 5.15, which contains plots of crop growth stress factors as a function of time during the growing season. The two crop stress factors shown, one related to water deficits and the other to root growth impedance, were calculated as the difference b e t w e e n the initial and the later rates of crop growth divided by the initial and subsequent actual water transpiration rate, and effective rooting depth, respectively.

Root extraction rate, m/m/day

Soil dry density, t / m 3 ( 0 - 2 0 c m )

Fig. 5.14. Patterns of corn root extraction rate of water at depths in a clay soil in plots subjected to different tillage treatments, after 57 days f r o m seeding, and the dry density profiles of the various plots (Douglas and M c K y e s , 1983). By permission of the Publisher. T h e curves in F i g . 5.15 indicate that both the loose " c o n t r o l " clay plots and the compacted untitled plots suffered more water stress for crops than the others. However, it was the moldboard plow and subsoiler treatments which resulted in greater stresses for root growth. In this case, the root impedance stresses in the moldboard plow and subsoiler treatments were more severe than water availability stresses, because as the results of F i g . 5.16 s h o w , the final crop yield was lower in these treatments than in the loose control soil. The compacted untitled treatment caused the highest soil packing of all, and the combination of root and water stresses led to low corn yields. The pattern of crop yield versus soil dry density in F i g . 5.16 is similar to that described by the Vomocil (1955) E q n . 5.5, and those of F i g . 5.12 and 5.13 above.







5.15. Calculated growth stress factors for water availability and root g r o w t h in silage corn on a clay soil subjected to the same t r e a t m e n t s as in F i g . 5.13 (Douglas and M c K y e s , 1983). By permission of the Publisher.


é ·









11 D r y matter yield , t/ha



Moldboard plow ^ Í , ·

1.1 Fig.


1.2 S o i l dry d e n s i t y , t/m



5.16. D r y matter harvest yield of silage corn on plots of clay soil subjected to different treatments of compaction and tillage as a f u n c t i o n of soil dry density (Douglas and M c K y e s , 1983). By permission of the Publisher.



The f o l l o w i n g measures for optimizing crop growth by avoiding excessive s o i l compaction can be derived from the results shown previously in this chapter. (a) A v o i d high machinery contact pressures, especially during repeated passes on fields. For a cultivation program which requires between five and ten passes of machines on the field per year, it is recommended that the tire contact pressure of the vehicles involved be limited to less than 70 k P a . (b) A v o i d , if possible, travelling on fields with machines when the topsoil is m o i s t , c l o s e to the " o p t i m u m " m o i s t u r e content for compaction. D e n s i f i c a t i o n of soil can be up to five times as severe at the optimum water content under a given compacting pressure as when the soil is quite dry. (c) A v o i d excessive slipping could double soil density wheel slip also lead to A maximum slip rate of

of tractor tires during field operations, which changes under the same weight. Undue rates of premature wear and costly replacement of tires. 1 6 % is recommended.

(d) Attempt to manage cultural programs such that a healthy system of strong roots, and sufficient organic matter remain in the topsoil. Compaction s t u d i e s on a vigorous cereal stubble have shown that an extensive root s y s t e m near the soil surface can reduce compaction damage under m a c h i n e r y loads by about two thirds compared to bare soil with low organic matter content (Chasse et al., 1975). 5.4 T I L L A G E O F C O M P A C T E D S O I L A s T a b l e 5.2 a n d P r o b l e m 5.3 previously demonstrated, compaction increases the strength of a soil, and the energy which is required in order to cut or till it. Even when sufficient extra energy is expended to cut and loosen a compacted soil structure, the resulting structure will probably not be the s a m e as that of the original uncompacted state. Studies of possible methods by w h i c h to a l l e v i a t e the e f f e c t s on soil structure caused by heavy construction machinery, for instance, have indicated that the cutting and lifting of a severely compressed clayey or silty soil results in a rather blocky structure. L a r g e clods of 10 to 20 cm sizes are separated by the tillage action, but these structural units are relatively compact, hard and impervious in t h e m s e l v e s . Furthermore, it is very difficult to refine the topsoil texture subsequently, by means of conventional secondary tillage tools such as discs, cultivators and the like. T h e a b o v e findings do not necessarily infer that it is a useless excercise to a t t e m p t the loosening of a severely compacted soil. O n the contrary, c u t t i n g and loosening is about the only practice which can begin to improve a compacted and damaged topsoil structure. With the years, the open spaces b e t w e e n clods will allow the passage of roots, moisture and air. A n d their action will eventually enter into the compacted soil blocks, to gradually open larger pores through root penetration, wetting and drying, and the overall soil t i l t h will experience improvement. What cannot be expected is that a single l o o s e n i n g tillage action will immediately restore a heavily compacted soil profile to its former uncompacted structural quality.

124 5.5


5.4 What would be the different maximum compacted dry densities of a clay soil at 2 0 % moisture content under a tractor rear tire having an average contact pressure of 100 k P a (a) without and (b) with a rate of wheelslip of 2 0 % ? The clay has an initial dry density of 1.0 t / m 3 , equivalent pressure 7 k P a , A ^ O . 2 and B ^ O . 3 t / m 3 . =1.62 t / m 3 , (b) Y d

A n s w e r : (a) Y r f

=1.89 t / m 3 .

5.5 A c l a y loam field has been found to have different constants for silage c o r n y i e l d at a particular fertilization rate in Vomocil's E q n . 5.5, depending on the seasonal precipitation as follows. Rainfall from M a y to August, m m : Y * , t/ha: C , t/ha: 7 * d ry , t / m 3 :

280 12 250 1.15

400 16 100 1.00

If the rainfall in a certain region is 280 m m from M a y to August in three y e a r s out of four, and 400 m m one in four, (a) what would be the optimum dry density in the topsoil of this field for silage corn yields over the long term? (b) What would be the average yield? A n s w e r : (a) ã .

=1.13 t / m 3 , (b) Y = 12.5 t/ha.

5.6 A farm uses a 60 kW tractor having a mass of 5.3 t and an average tire contact pressure of 55 k P a . The farm fields are clay loam and have constants in the compaction E q n . 5.3 of ã ï = 0 . 8 5 t / m 3, p 0=7 k P a , Á^=0.2 and B-^0.2 t / m 3 . Grain corn is being grown on much of this f a r m , for which the average annual constants in Vomocil's yield E q n . 5.5 are Y * = 9 . 5 t/ha (at 1 5 % humidity), C = 1 0 0 t/ha a n d Y * d r =y 1 . 2 t / m 3. Assume that the tractor covers much of the field each year, that the average topsoil moisture content is 1 0 % , that wheelslip averages 5 % and that preseeding cultivation renders the topsoil close to its optimum density each year. What will be the e x p e c t e d change in grain yields in the wheel tracks if a larger 13 t tractor is acquired with an average contact pressure of 110 k P a ? Answer: Y i e l d change = 6.2-3.6 = 2.6 t/ha. 5.7 A Ste. Rosalie clay soil has the saturated hydraulic conductivity values at different dry densities shown in Table 5.3, and the constants in the compaction E q n . 5.3 as given in Table 5.1. Assuming an initially loose soil at ã 0 , what would be the limiting vehicle contact pressure in order to maintain a soil hydraulic conductivity of 1 0 " 3 cm/s (a) at 1 0 % and (b) at 2 0 % soil moisture content? A s s u m e also small wheelslip. Answer: (a) p=61 k P a , (b) p=7.2 k P a .



Chapter 6



In mechanized soil cutting, moving and tillage, motorized vehicles are used to provide the energy for the movement of tools in the soil. The net traction force required from a vehicle is the draft needed to perform a particular soil manipulation operation in question, and the useful output energy of the machine is the draft multiplied by the tool horizontal speed. The useful energy is invariably less than the net production of the vehicle's engine owing to losses in the transmission of power within the vehicle, and inefficiencies in t r a c t i o n between the machine and the ground surface upon which it travels. T h i s chapter aims to provide an introduction to the technical questions concerning vehicle traction forces, efficiency and losses, for the purpose of m a t c h i n g traction machines to soil cutting instruments, and ensuring that a desired operation in the soil can be performed at a reasonable rate of energy efficiency. 6.1 E O R C E A N D E N E R G Y B A L A N C E S 6.1.1 Eorces on a traction machine

E i g . 6 . 1 . A traction machine and soil cutting tool combination, and the free body force diagram of the machine alone. Eig. 6.1 schematically depicts a typical combination of a soil cutting tool and p o w e r e d wheeled machine. The free body force diagram can be used to represent most wheeled machines, with modifications for six or more wheels.




Assuming that the machine remains on the soil surface, vertical can be calculated as follows for the four wheeled machine. Í


= (WX


Nf = W + V where


+ HY)/X. L


(6.1) (6.2)

= total upward force from the ground to the rear wheels,


+ VX


= total upward force from the ground to the front wheels,


W = total machine weight, excluding the tool weight, X

distance from the front axle to the machine center of gravity,


vertical force transferred downward from tool to machine,



distance from front axle to effective tool attachment point,


total horizontal force required to pull the cutting implement,


effective height of the tool attachment point, machine wheelbase.

The summation of the horizontal forces to zero yields the following result. Ç. = Ç + R. + R t f r where Ç Rf, R


= total traction force of all driven wheels on the ground, = the rolling resistance forces on the pair of front or rear wheels, respectively, due to soil and tire body deformations.

The concept of rolling resistance will be discussed in Section 6.3.2. A track l a y i n g machine has a similar force diagram, as shown in F i g . 6.2, for which the equilibrium equations follow.

F i g . 6.2. Forces acting on a tracked machine pulling an instrument.




= W + V


Ht = Ç + R




= (WX




+ VX


+ HY)/N


= total upward force from the ground to the two tracks,

R = total rolling resistance of the tracks on the ground, Hj. = the total traction thrust from the tracks on the ground, X Ñ

= the distance from the front sprocket wheel center to the center of vertical pressure distribution on the tracks.

S i m i l a r equilibrium relations can be developed easily for other vehicles which might have, for instance, one wide track, four tracks, a mixture of wheels and tracks and the like. T h e f o r c e , H, needed to move the soil cutting tool at a desired speed is the useful output force of the machine, and it is often termed the "drawbar pull 11 of a vehicle. The total thrust or traction force, H t , can be considered as an input force from wheels or tracks to the ground surface, and the sum of rolling resistance forces is a loss of effort caused by soil and/or tire deformations. For a certain required level of drawbar pull, H, the force balances above are practically independent of vehicle speed, if the resistance to m o t i o n through the air is neglected. However, in order that the machine and tool combination be able to move at all on a particular surface, the vehicle engine and power transmission system must be capable of meeting the h o r i z o n t a l thrust requirement of E q n . 6.3 or 6.5, and the configuration and s t r e n g t h of the machine-soil interface must also be able to sustain the necessary level of total traction force. 6.1.2 Energy balance at the machine-soil interface

F i g . 6.3. Horizontal forces acting on a wheeled or tracked machine, and the speeds of the machines and driving wheels. The flow of energy from a traction machine to the soil depends partly on the s p e e d of operation of the vehicle, and also upon the stiffness of the g r o u n d beneath the wheels or tracks. In F i g . 6.3, the vehicle speed with respect to the ground, v, is usually less than the speed, í , at which the




wheel bottom or tracks are moving with repect to the vehicle. These two s p e e d s would be equal only if there were no soil horizontal deformation and no r e l a t i v e movement between the surface of the tires or tracks and the g r o u n d surface. Such a case is never achieved perfectly in practice for such machines. E v e n on hard travel surfaces, there is some relative movement between the traction device and the ground, and any horizontal soil deformation serves to reduce the vehicle speed further for a given wheel speed. It is practically convenient to combine the speed of slip motion of tires or tracks with respect to the machine-soil contact surface together with the horizontal deformation velocity of the soil. The sum of these relative speeds is a total slip velocity, v g, which is the difference between wheel and vehicle speeds. í


= í


- í


A parameter called slip, S, is commonly defined as the loss in vehicle speed divided by the wheel or track speed as follows. S = í /v = (v - v)/v s w w w


í = í



(1 - S)

E q n . 6.9 shows that when wheelslip is zero, the vehicle and wheel speeds are identical. When the slip is one, or 1 0 0 % , the vehicle speed is zero and the machine does not move even though the wheels or tracks are turning. Wheelslip generally increases with both drawbar pull, which requires horizontal f o r c e s on the soil surface, and soil softness with its attendant larger deformations. The total energy loss between the machine input to the soil and the output to the towed tool load is combination of speed and force losses arising from soil deformations as follows. Energy loss = Input power - Output power = H t v w - H v = H.v - ( H - 2 R ) v = H.v S + Z R v t w t t w


where Z R = total of rolling resistance forces on all tires or tracks. T h e tractive efficiency, T.E., of a machine is defined as the output, or d r a w b a r power divided by the input power from the wheel or track to the soil. T.E. = (Hv)/(H.v ) = (1 - S)(H. - Z R ) / H , = (1 - S)(l t W t t

hi t


It c a n be seen in E q n . 6.11 that decreases in tractive efficiency are p r o p o r t i o n a l to both rate of slip and the level of rolling resistance forces. F i g . 6.4 shows some typical curves of tractive efficiency as a function of wheelslip, for two wheel drive tractors with lugged agricultural tires on surfaces ranging from concrete to soft sand. The tractive efficiency can a c h i e v e levels of over 90 percent on very firm surfaces, but is reduced by slip and rolling resistance to near 50 percent on softer terrain.



1.0 0.9 /

o n c cr e t

0.8 Firn é



0.6 Tractive Efficiency T.E. 0.5







rilled s U N

S o f t or sane

10 20 Wheelslip, S %


Eig. 6.4. Typical curves of tractive efficiency versus wheelslip for two wheel d r i v e tractors with lugged agricultural tires on different surfaces (A.S.A.E., 1984). 6.2 T R A C T I O N , S O I L D E F O R M A T I O N A N D 6.2.1 M a x i m u m traction force


F i g . 6.5. Contact of lugged driving tires with the soil surface. The horizontal traction force transmitted between a machine and the soil has a limiting value which depends on both the soil strength and machine design parameters. The contact area of a driving tire is approximately flat as shown in F i g . 6.5, within certain limits of soil stiffness and tire inflation




pressure which will be shown later. The magnitude of the area of all driving wheels or tracks can be estimated as: (6.12)


T h e l u g s on tires and tracks penetrate the soil surface, unless it is very hard, in order to hold soil among them and to cause the development of soilt o - s o i l strength on the horizontal tractive surface. Thus it is approximately correct to use the strength of the soil in shear over the tire or track contact area in order to estimate the maximum possible traction force. This maximum force value, H m 7 is the summation of the soil shear strength over the total contact area of all driving tires or tracks. Ó Á/· H m = l ( c + ptan)dA = c Ó A + 2Ntan(/> (6.13) 0^ in which Ó A = the sum of contact areas of all driving wheels or tracks, ÓÍ

= the sum of vertical forces on all driving wheels or tracks.

T h e machine design specifications which can be changed to alter the m a x i m u m traction force are the driver contact areas and the weight on the d r i v e r s . On a coarse granular material which is cohesionless, it is only the increase in vehicle weight which can improve the traction effort. When, h o w e v e r , the soil is a wet fine-grained material with a small total angle of internal friction, it is the contact area which should be enlarged for better traction, and this is the reason for the popularity of wide tires or tracks, large diameter tires, long tracks and dual or even triple tire sets on tractors which operate on clayey terrain. Problem 6 . 1 :

E i g . 6.6. A n agricultural four wheel drive tractor with triple tires. T h e t r a c t o r in E i g . 6.6 has a mass of 15,400 kg and a total of twelve 5 8 . 7 - 8 6 . 4 cm tires, each of which has a width of 58.7 cm and outside diameter 180.8 c m . What is the maximum gross traction thrust of this machine on a soil of c=10 kPa and ö= 25 degrees? Estimate the contact length of the tires on the soil surface to be one quarter of their outside diameter.




Using Eqn. 6.13;

= c ÓÁ + Z N t a n O ^ = 10 k P a x l 2 x 0 . 5 8 7 x l . 8 0 8 m /4 + 15.4x9.8tan25° k N = 31.8 + 70.4 k N = 102.2 k N

6.2.2 Soil deformation and slip Because soil is not a perfectly rigid material, it deforms under the action of stresses. The horizontal deformation of the soil surface in reaction to a traction force of a track or low pressure tire is very similar to that observed in the rectangular shear plate soil strength test shown in F i g . 2.15 and Table 2.1. Bekker (1956, 1960) suggested that the horizontal deformation versus shear stress response of a plastic soil, such as sand or wet clay, is observed to be close to an exponential curve of the following f o r m . 7 = s(l - e ~ X ' / K) where ô s Xf Ê


= horizontal shear stress = soil shear strength = c A + ñ tan ö = horizontal deformation of the soil surface = a soil stiffness constant.

Fig. 6.7. Horizontal deformation of soil as a flexible tire rolls and slips on it with a traction force.




S u c h a curve cannot be derived from the results of a soil stress-strain t e s t , such as measured in the triaxial test, since the distribution of shear strain with depth in the soil is not defined in the ground under a machine, as shown in F i g . 6.7. The final horizontal deformation of the soil surface is shown in F i g . 6.7 as X f m, which can be related to wheelslip and contact length of a tire or track as s h o w n below, assuming that the tire or track maintains a relatively rigid length. X·


= SL


F i g . 6.7 indicates that the soil deformation, X ! , varies along the contact length, and the shear stress between soil and machine will also vary according to Eqn. 6.14 for a plastic soil. The total traction thrust can be calculated by integrating the shear stress distribution over the contact area. H


= b / r d X = b s / ( l - e ~ X ' / K) d X

= H l { i - f


= bs f(l

a - e - ^ , }



X / K




Bekker (1960) and Wong (1978) have shown that one can develop some more complicated stress-deformation relationships for soils which suffer some loss of strength with extensive shearing. They also pointed out the importance of the length of a tire or track contact area in the behaviour described by Eqn. 6.16, as shown in the following example. Problem 6.2: A tractor designer has the choice of two lugged rear tire sizes f o r a machine having a vertical load of 20 k N per rear wheel. One tire has a diameter of 200 c m , a ground contact length of 100 cm and width 40 c m , while the other has diameter 100 c m , contact length 50 cm and width 80 c m . Both tires have a contact area of 0.4 square meters, but which one will p r o d u c e the higher traction thrust level for wheel slips between 0 and 30 percent, on a soil surface having c=20 k P a , ö = 3 0 ° and K = 7 c m ? Solution: The maximum possible shear stress under both tires is: s = c +

p t a n ö = 20 + (20/0.4)tan30° k P a = 48.9 k P a

Eqn. 6.16, giving total thrust as a function of slip, becomes:



(1 -

e - S L -/ 0

7 m

)}l 0.05d) sinkage




Fig. 6.10. Estimation of lengths of contact of tires on hard and soft surfaces. B e k k e r (1960) proposed that the tire contact length, L, can be estimated approximately also as follows.



bCp .^l p c ) •

( Ú Ã Ã Ã)



where p. = tire inflation pressure P c = contact pressure component of tire body flexural stiffness. In addition, Wong (1978) and Bekker (1985) gave an approximate method for calculating length L from the maximum tire deflection, $ t , as in F i g . 6.10. L = 2(d5 t - 5



) l i2


O n e or another of these methods for estimating tire contact lengths is needed to predict the traction thrust and slip of wheels, as well as to predict sinkage, which is described in the next section. 6.3 S I N K A G E A N D R O L L I N G R E S I S T A N C E 6.3.1 Sinkage in soil T h e stress distribution in a soil under an object having a vertical load is more complex than the horizontal thrust strength relation given in E q n . 6.13. A s F i g . 6.11(b) indicates, penetration into the soil begins at small loads with a c c o m p a n y i n g shear strains, which are less than those required to develop f a i l u r e . After local shear failure has occurred (Fig. 6.11c), a wedge of soil b e n e a t h the s i n k i n g s t r u c t u r e continues to descend, but an increasing penetration load is needed in order to lift larger depths of "surcharge" soil above and to the sides of the object. N a t u r a l soil exhibits a gradual change in load required for the transition between states (b) and (c) in F i g . 6.11, and a smooth variation in sinkage with increasing penetration force Í is observed. Early descriptions of such




c. L a r g e


Eig. 6.11. A plate on the soil surface, (a) under no load, (b) under a small load which does not fail the soil and (c) developing local shear failure. behaviour were reported by Bernstein (1913) and Goriatchkin (1937), (cited by Bekker, 1956), and the following equation was proposed to describe it. ñ = kzn where ñ k æ ç

(6.20) = = = =

vertical average contact pressure, a soil stiffness constant for sinkage, sinkage distance into the soil, a soil constant.

The principal deficiency of E q n . 6.20 for predictions of machine sinkage and performance was found to be the variability of the soil stiffness constant, k, with the size of the object on the soil. In civil engineering technology, it was known that the sinkage of a rectangular plate, at a given average vertical pressure on a particular soil, depends also on the width of the rectangle ( T a y l o r , 1 9 4 8 ) . B e k k e r (1956) combined the two concepts, namely the exponential pressure-sinkage relationship of E q n . 6.20, and the plate size dependence of the soil stiffness constant as follows. ñ = N/(bL) = (k lb + k , ) z n = k z n ö c w h e r e k c and k^ = soil stiffness constants, which independent of plate width or diameter.

(6.21) are

presumed to be

Eqn. 6.21 was tested against the results of laboratory and field experiments and w a s found to express the measured pressure-sinkage curves with a reasonable degree of accuracy, as well as matching the results reported by other researchers. Bekker (1960) did find that when circular disks are used in penetration tests, rather than rectangular plates, the radius of the disks is the m o r e appropriate measurement to use as b, whereas b is the smaller overall dimension of a rectangular area.



In order to evaluate the soil constants in E q n . 6.21, it is necessary conduct at least two soil penetration tests using plates of different width r a d i u s , b. The measured sets of pressure and sinkage values must then a n a l y z e d graphically or analytically on a log-log basis to find the best exponential constants, as in E q n . 6.22 and F i g . 6.12. log(p) = log(k) + nlog(z)

to or be fit


From the best fit logarithmic curves, constants k and ç can be determined for each plate of the tests. The average value of ç is used together with the k values from the two plates to obtain k c and k0 as shown below. k




b b


( kb

l 2

( k

l "

2 2 '

k ) / ( b


lk b l )

2 "

/ ( b

2 '

b )



2 3)

b )



where subscripts 1 and 2 refer to the values measured for plates 1 and 2.

F i g . 6.12. Plotting order to best fit squares

of measured pressure and sinkage values on log scales in determine constants k and ç for each plate tested. The lines can be found either graphically, or by using a least analysis of the logarithms of measured values.




However, it may be risky to attempt the measurement of soil stiffness c o n t a n t s with tests which use only two plates, especially if they are small plates. A large variability exists in soils, even in carefully prepared laboratory samples, let alone at different locations in a field. Large plates, of the order of 30 cm or more in width, can reduce the variation in experimental results, but they require large loads to approach practical sinkage pressure levels and are thus inconvenient and costly to perform. Smaller rectangular or circular plates in the range of five to ten cm are handy for testing by one person, but E a n (1985) has shown that the variation in k c and \

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