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AGEN 304: SOIL MECHANICS AND CULTIVATIONS
LECTURE 11: SOIL CUTTING AND TILLAGE
This component of the course deals with the forces and mechanisms in soil
failure during cutting and tillage operations
Soil tillage
What is tillage?
a) Lal 1979)
Physical, chemical and biological soil manipulation to optimize conditions
for seed germination, emergence and seedling establishment
b) Ahn and Hintze (1990)
Restrict the definition to physical loosening of the soil which is carried
out in a range of cultivation operations.
Tillage objectives with respect to the crop environment
 To produce a suitable seedbed for crop germination through primary and
secondary tillage operations
 To prepare a favourable plant environment by achieving the following
a) Aeration
b) Water movement
c) Nutrient movement
d) Chemical incorporation
e) Weed and pest control
f) Residue management
g) Optimize soil temperature and moisture conditions for crop growth.
 Enhance soil and water conservation, soil structural development, soil
aggregation
Soil temperature: -is critical for germination and growth of plants and for
microorganism development. Microorganisms enhance nutrient release,
stabilization of soil structure and water movement.
Tillage/soil loosening, soil inversion and compaction - all influence soil
temperature eg volumetric heat capacity of soil depends on soil bulk density
among other things
Aeration air is vital for root development and overall plant growth. Tillage
enhances pore space and therefore aeration
Water. - water is required for overall plant establishment i.e. germination,
growth, nutrient transport, evapotranspiration needs.
2
Important characteristics are infiltration, storage, evaporation and drainage -
which are functions of pore size, pore size distribution and pore continuity as well
as surface configuration, texture, slope etc. .
-most of these are affected by tillage
-clod size and pore size control infiltration rate
Storage Capacity (Available Water Capacity) is affected by pore sizes in the soil
The finer the pores, the greater the storage
Drainage - is a function of aggregate sizes and pore size distribution and pore
continuity.
Through cultivation, we can vary tilth thus directly influencing infiltration, storage
and drainage
Tillage also breaks surface crusts, thus enhancing infiltration
Crop Support
Soils provide a support media for crop growth. More compact soils provide firm
support for crops (Soil strength)
Soil Conservation: Large aggregates vs small aggregates
Seedbed: Tillage for seedbed preparation should aim to achieve good seed to
soil contact.
-Also need a well aerated seedbed with good water storage properties, good
drainage and good infiltration capacity.
Mechanisation
Different equipment have different soil requirements eg Precision seed drills
require a fine and smooth seedbed/surface for accurate seed placement
Potato harvester requires a clod and stone free surface
3
TILLAGE AND EARTHMOVING
There are two main tillage classes
 Primary tillage
 Secondary tillage
Primary Tillage
The first stage of land preparation.The main purposes of primary tillage are to
cut and loosen soil to a depth of 15-90 cm
Tools used include:
 Mouldboard plough, which has the capacity to break up most soils and
also invert soil, weeds and crop residues. Ploughs can be used singly or
in groups of 2 or more
 Disc plough (dp)
Concave disk 50-90 cm diameter
approximately same draught requirements as the mouldboard plough
in similar conditions, but the dp performs better in sticky non-scouring
soils, hard dry ground; in many organic peat soils and where it is
necessary to break hardened plough soles or to accomplish deep
ploughing.
Can also handle stony/ rocky conditions
 Special application tools
e.g Chiesel ploughs (CPs)
-cps have long shanks and double ended chisel points about 6.4
cm in width.
-they cut, loosen and stir the soil
-adapted to loosen hard dry soils and shattering hard pans
 Subsoilers
-are similar in principle to chisel ploughs but are more heavily built and rigid
for operating at depths of 40-90 cm
-used for loosening deep layers of soil in order to promote water movement
and root development.
-very useful for breaking plough-pans developed at depths as a result of
compaction due to the trafficking of heavy machinery and heavy traction
equipment
power requirements -To pull one subsoiler shank at a depth of 45 cm in the
soil, a 40-60 kW tractor is needed.
Rotary plough (rp)
consists of a set of knives or rods that are rotated on a horizontal shaft and
covered by a sheet metal hoad. Soil is chopped up by the knives and
thrown against the inside of the hoad, resulting in a fine, loose soil structure
4
< The fineness of the soil depends on the implement forward speed.
Rp is used extensively in vegetable production where fine seedbeds are the
norm.
Rp is not recommended where severe soil erosion is prevalent
Secondary tillage
Secondary tillage is usually performed after primary tillage operations for:
 Improving seedbed levelness and at times to improve tilth.
 Weed control
 Chopping crop residues
Harrows such as:
 disc harrows
 spike tooth harrows
 spring tooth narrows
 spring tooth harrows
 rotary cross harrows
are popularly used for this purpose.
Earthmoving equipment
Usually used in the construction industry, e.g. soil moving in dam
construction, buildings, roads, channel and canal construction
There are three principal classes:
 Blade
Commonly used as road graders and bulldozer front end loaders etc.
 Ripper
The ripper is very narrow, and therefore has large d/w ratios Often attached
5
to graders or bulldozers when its necessary to cut and loosen hard soil.
 Shovel
Shavels are blades equipped with sides to form a bucket. They cut and lift
up the soil.
 All tools have some effect on soil structure which depends on the initial
state of the soil and tool geometry
 Soil cutting, moving and tillage implements displace soil from its original
location.
 These operations involve mechanical soil failure
 The design of effective cutting tools begins with the analysis of soil
failure, which enables the prediction of force and energy requirements by
the tools.
 The design procedure also involves a description of soil manipulation
and structural changes which result from the cutting tool action.
Soil Physical properties and plant growth
Ultimate objective of tillage operations
-To optimize crop growth using the most cost effective ways to achieve
this
Measures to optimize crop growth
-Reduce compaction/ avoid compaction (and enhance all the
other good properties) by
- use of low draft/weight machinery/ tyre contact pressures
- work below optimal soil moisture content (which results in
maximum compaction for that soil)
-always work under no slip conditions (maximum allowable
slip =< 15%)
6
Mechanical properties of compacted soils
Cohesion - increases logarithmically with soil density
Angle of internal friction - increases linearly with density
Therefore soil strength increases with density
- Use appropriate tillage tools
- always match tool with objective
two questions that need to be answered
(1) what is the objective of this tillage operation?
(2)- How best (machinery wise) can objective be achieved?
7
Lecture 12: AGEN 304: Soil Mechanics and Cultivations
SOIL FAILURE WITH TILLAGE TOOLS AND THE UNIVERSAL
EARTH MOVING EQUATION
Analogy between a retaining wall and a soil cutting blade in the passive case
The inclination of the failure plane is 45-/2 as compared to 45 + /2 in the active
case.
The limitations of the wedge theory in dealing with passive cases were discussed
earlier.
Reece (1965) proposed the universal earth moving equation (general soil
mechanics equation ) for describing the force necessary to cut a soil with a tool.
wqdNcdNNgdP qc )( 2
 
Where P= total tool force
= total soil (mass) density
g= acceleration due to gravity
d= tool working depth
c= soil cohesion
q= surcharge pressure vertically acting on the soil surface
w= tool weight
N , Nc, Nq = dimensionless factors, similar to Terzaghi’s dimensionless
coefficients
The magnitude of the dimensionless factors N , Nc, Nq depends on
 Soil frictional strength
 Tool geometry
 Tool to soil strength properties
8
Tool geometry factors affecting the N values are:
 Angle of tool blade from the horizontal (rake angle )
 Possible curvature of tool shape
 Depth to width ratio of a narrow tool (d/w)
This equation is of value and is applied in the design and evaluation of soil
cutting tools for acceptable performance and minimising required force or draft
inputs thereby reducing the need for extensive field work.
SOIL CUTTING
In soil cutting we are interested in:
 The force required to move the soil in upward (passive) failure.
 Changes to the soil properties due to the tillage process.
There are two main types of cutting tools from a mechanical point of view
classified on the basis of their depth to width ratio (d/w):
 Wide tools with d/w <5
 Narrow tools with d/w>5-10
Wide tools ( eg a bulldozer blade) cause soil failure in two dimensions i.e. soil
moves forwards and upwards.
Narrow tools (eg a ripper tine) cause failure in 3 dimensions i.e. soil moves
forwards, upwards and sideways.
Soil cutting tools are also classified according to the angle of attack from the
horizontal which is termed the rake angle, . (see diagram above).
We can have either small or large rake angles.
The principles involved in analysing soil failure during soil cutting are similar to
the passive case of a retaining wall. Using this approach the solutions obtained
by Reece resulting in the Universal Earth Moving Equation is also obtainable by
use of Coulomb’s wedge theory. Thus the principles applying to earth retaining
walls also apply to soil cutting tools
2 dimensional soil failure occurs in wide tools while 3 dimensional failure occurs
in narrow tools. This necessitates different analysis procedures for wide and
narrow tools.
9
2-D Cases (Wide Tools)
The same forces as in a wall exist (see diagrams above)
The force required to move the blade can be resolved into horizontal H and
vertical directions V.
The weight of the blade W is included in the calculation of V if significant
compared to V
For a blade of finite width w, this width should be included in the calculation of
the soil force, P.
wqhKhKcchKKhP qcaacp )( 2
 
Kp (=N), Kc(=Nc), Kca(=Nca), Kq(= Nq) are passive earth pressure coefficients h or
d is the height or depth of blade, w is its width.
The values N , Nc, Nca, Nq are obtainable from Appendix 5 in Mckyes, appendix
1 and 2 in Mckyes (Soil cutting and Tillage) or they can be calculated (easier to
look them up in the graphs).
The horizontal draft force H is then given by
 cot)sin( dwcPH a
and
WdwcPV a  )cos( 
however the log spiral method developed by Hettiaratchi (1969) is more accurate
than the straight wedge model.
The horizontal force is the draft force required in soil cutting tools and has to be
provided by the draft power source eg a tractor, a pair of draft animals.
The wedge theory of passive soil failure
10
Example (refer to Mckyes pp193)
A 317 kW tracked tractor shown above with a mass of 41.8 tonnes is shown with
a flat bulldozer blade having a total width of 4.80 m. the tractor is beginning to
cut the soil to a depth of 40 cm as indicated. The blade itself excluding its arms
has a mass of 4 tonnes. Soil properties = 17.6 kN/m3, =30o
, = 20o
,
c=10kPa, ca=4kPa.
Find the horizontal (draft) force and vertical uplift forces required to move the
blade through the soil.
Solution A
The N coefficients can be determined from appendix 5 (Mckyes) or by calculation
using the straight line wedge model
11
Then using the universal equation plug in the appropriate values of N or K to
obtain the total tool force, the draft force and the vertical force.
P= 70.1 kN;
H= 72.8 kN and
V= 52.0 kN
SOLUTION B
The calculation of the N factors using the above equations which can be tedious,
can be avoided by use of graphs prepared from these equations for known
conditions.
Typical such graphs are available from Appendix 5 (Mckyes, Agricultural
Engineering Soil Mechanics or Appendices 1 and 2 (Mckyes, Soil cutting and
Tillage) proceed as follows:
d/w =0.40/4.80=0.083 i.e narrow or wide tool?
N (=20o,=30o, =53o ) or Kp= approx 1.45
Nc (=20o,=30o, =53o ), or Kc=approx 2.6
Nca===20o,=30o, =53o, Kca =0.8
q=0, therefore we have no surcharge and the surcharge term falls out.
whKcchKKhP caacp )( 2
 
Interpolations for N values
The N values can also be obtained from approximations Hettiaratchi and
Reece,1974) and use of Appendix 1 and 2 in Soil Cutting and Tillage
(Mckyes1985) as follows using
(i) Hettiaratchi and Reece (1974)




 ][
)0(
)(
)0(



N
N
NN
Where N is the required N factor
N(=0) is the N factor for a perfectly smooth blade
N (=) is the N factor for a perfectly rough blade
alternatively this may also be achieved as follows
12
(ii) Mckyes (1985) linear extrapolation


 )( )0()()0(   NNNN
Using these approximations
N=0.94 +(1.7-0.94)*20/30 =1.446
Nc using the first aproximation = 1.3(3.4/1.3)0.666
Nc = 2.46
Nca (appendix 2) = 0.8
therefore
P= (17.6*0.42 1.446 +10*0.4*2.46+4*0.4*0.8)*4.8
P=72.92 kN
and as before
 cot)sin( dwcPH a
H = 72.92 sin(53+20)+4*0.4*4.8*cot 53
H= (69.73+5.787) kN = 75.52 kN
WdwcPV a  )cos( 
V= 72.92 cos (53+20)-4*0.4*4.8 +(4000*9.8/1000)
V = 21.32-7.68+39.24
V= 52.88 kN
NB.
 The difference between the results from two approaches.
 There is need for caution when using the graphs so as to ensure you use the
correct graph for the right purpose.
 The difference between the wedge theory result and this result should
normally not exceed 10 %.
13
14
15
16

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11 ag304 soil_cutting_and_tillage_+_worked_example

  • 1. 1 AGEN 304: SOIL MECHANICS AND CULTIVATIONS LECTURE 11: SOIL CUTTING AND TILLAGE This component of the course deals with the forces and mechanisms in soil failure during cutting and tillage operations Soil tillage What is tillage? a) Lal 1979) Physical, chemical and biological soil manipulation to optimize conditions for seed germination, emergence and seedling establishment b) Ahn and Hintze (1990) Restrict the definition to physical loosening of the soil which is carried out in a range of cultivation operations. Tillage objectives with respect to the crop environment  To produce a suitable seedbed for crop germination through primary and secondary tillage operations  To prepare a favourable plant environment by achieving the following a) Aeration b) Water movement c) Nutrient movement d) Chemical incorporation e) Weed and pest control f) Residue management g) Optimize soil temperature and moisture conditions for crop growth.  Enhance soil and water conservation, soil structural development, soil aggregation Soil temperature: -is critical for germination and growth of plants and for microorganism development. Microorganisms enhance nutrient release, stabilization of soil structure and water movement. Tillage/soil loosening, soil inversion and compaction - all influence soil temperature eg volumetric heat capacity of soil depends on soil bulk density among other things Aeration air is vital for root development and overall plant growth. Tillage enhances pore space and therefore aeration Water. - water is required for overall plant establishment i.e. germination, growth, nutrient transport, evapotranspiration needs.
  • 2. 2 Important characteristics are infiltration, storage, evaporation and drainage - which are functions of pore size, pore size distribution and pore continuity as well as surface configuration, texture, slope etc. . -most of these are affected by tillage -clod size and pore size control infiltration rate Storage Capacity (Available Water Capacity) is affected by pore sizes in the soil The finer the pores, the greater the storage Drainage - is a function of aggregate sizes and pore size distribution and pore continuity. Through cultivation, we can vary tilth thus directly influencing infiltration, storage and drainage Tillage also breaks surface crusts, thus enhancing infiltration Crop Support Soils provide a support media for crop growth. More compact soils provide firm support for crops (Soil strength) Soil Conservation: Large aggregates vs small aggregates Seedbed: Tillage for seedbed preparation should aim to achieve good seed to soil contact. -Also need a well aerated seedbed with good water storage properties, good drainage and good infiltration capacity. Mechanisation Different equipment have different soil requirements eg Precision seed drills require a fine and smooth seedbed/surface for accurate seed placement Potato harvester requires a clod and stone free surface
  • 3. 3 TILLAGE AND EARTHMOVING There are two main tillage classes  Primary tillage  Secondary tillage Primary Tillage The first stage of land preparation.The main purposes of primary tillage are to cut and loosen soil to a depth of 15-90 cm Tools used include:  Mouldboard plough, which has the capacity to break up most soils and also invert soil, weeds and crop residues. Ploughs can be used singly or in groups of 2 or more  Disc plough (dp) Concave disk 50-90 cm diameter approximately same draught requirements as the mouldboard plough in similar conditions, but the dp performs better in sticky non-scouring soils, hard dry ground; in many organic peat soils and where it is necessary to break hardened plough soles or to accomplish deep ploughing. Can also handle stony/ rocky conditions  Special application tools e.g Chiesel ploughs (CPs) -cps have long shanks and double ended chisel points about 6.4 cm in width. -they cut, loosen and stir the soil -adapted to loosen hard dry soils and shattering hard pans  Subsoilers -are similar in principle to chisel ploughs but are more heavily built and rigid for operating at depths of 40-90 cm -used for loosening deep layers of soil in order to promote water movement and root development. -very useful for breaking plough-pans developed at depths as a result of compaction due to the trafficking of heavy machinery and heavy traction equipment power requirements -To pull one subsoiler shank at a depth of 45 cm in the soil, a 40-60 kW tractor is needed. Rotary plough (rp) consists of a set of knives or rods that are rotated on a horizontal shaft and covered by a sheet metal hoad. Soil is chopped up by the knives and thrown against the inside of the hoad, resulting in a fine, loose soil structure
  • 4. 4 < The fineness of the soil depends on the implement forward speed. Rp is used extensively in vegetable production where fine seedbeds are the norm. Rp is not recommended where severe soil erosion is prevalent Secondary tillage Secondary tillage is usually performed after primary tillage operations for:  Improving seedbed levelness and at times to improve tilth.  Weed control  Chopping crop residues Harrows such as:  disc harrows  spike tooth harrows  spring tooth narrows  spring tooth harrows  rotary cross harrows are popularly used for this purpose. Earthmoving equipment Usually used in the construction industry, e.g. soil moving in dam construction, buildings, roads, channel and canal construction There are three principal classes:  Blade Commonly used as road graders and bulldozer front end loaders etc.  Ripper The ripper is very narrow, and therefore has large d/w ratios Often attached
  • 5. 5 to graders or bulldozers when its necessary to cut and loosen hard soil.  Shovel Shavels are blades equipped with sides to form a bucket. They cut and lift up the soil.  All tools have some effect on soil structure which depends on the initial state of the soil and tool geometry  Soil cutting, moving and tillage implements displace soil from its original location.  These operations involve mechanical soil failure  The design of effective cutting tools begins with the analysis of soil failure, which enables the prediction of force and energy requirements by the tools.  The design procedure also involves a description of soil manipulation and structural changes which result from the cutting tool action. Soil Physical properties and plant growth Ultimate objective of tillage operations -To optimize crop growth using the most cost effective ways to achieve this Measures to optimize crop growth -Reduce compaction/ avoid compaction (and enhance all the other good properties) by - use of low draft/weight machinery/ tyre contact pressures - work below optimal soil moisture content (which results in maximum compaction for that soil) -always work under no slip conditions (maximum allowable slip =< 15%)
  • 6. 6 Mechanical properties of compacted soils Cohesion - increases logarithmically with soil density Angle of internal friction - increases linearly with density Therefore soil strength increases with density - Use appropriate tillage tools - always match tool with objective two questions that need to be answered (1) what is the objective of this tillage operation? (2)- How best (machinery wise) can objective be achieved?
  • 7. 7 Lecture 12: AGEN 304: Soil Mechanics and Cultivations SOIL FAILURE WITH TILLAGE TOOLS AND THE UNIVERSAL EARTH MOVING EQUATION Analogy between a retaining wall and a soil cutting blade in the passive case The inclination of the failure plane is 45-/2 as compared to 45 + /2 in the active case. The limitations of the wedge theory in dealing with passive cases were discussed earlier. Reece (1965) proposed the universal earth moving equation (general soil mechanics equation ) for describing the force necessary to cut a soil with a tool. wqdNcdNNgdP qc )( 2   Where P= total tool force = total soil (mass) density g= acceleration due to gravity d= tool working depth c= soil cohesion q= surcharge pressure vertically acting on the soil surface w= tool weight N , Nc, Nq = dimensionless factors, similar to Terzaghi’s dimensionless coefficients The magnitude of the dimensionless factors N , Nc, Nq depends on  Soil frictional strength  Tool geometry  Tool to soil strength properties
  • 8. 8 Tool geometry factors affecting the N values are:  Angle of tool blade from the horizontal (rake angle )  Possible curvature of tool shape  Depth to width ratio of a narrow tool (d/w) This equation is of value and is applied in the design and evaluation of soil cutting tools for acceptable performance and minimising required force or draft inputs thereby reducing the need for extensive field work. SOIL CUTTING In soil cutting we are interested in:  The force required to move the soil in upward (passive) failure.  Changes to the soil properties due to the tillage process. There are two main types of cutting tools from a mechanical point of view classified on the basis of their depth to width ratio (d/w):  Wide tools with d/w <5  Narrow tools with d/w>5-10 Wide tools ( eg a bulldozer blade) cause soil failure in two dimensions i.e. soil moves forwards and upwards. Narrow tools (eg a ripper tine) cause failure in 3 dimensions i.e. soil moves forwards, upwards and sideways. Soil cutting tools are also classified according to the angle of attack from the horizontal which is termed the rake angle, . (see diagram above). We can have either small or large rake angles. The principles involved in analysing soil failure during soil cutting are similar to the passive case of a retaining wall. Using this approach the solutions obtained by Reece resulting in the Universal Earth Moving Equation is also obtainable by use of Coulomb’s wedge theory. Thus the principles applying to earth retaining walls also apply to soil cutting tools 2 dimensional soil failure occurs in wide tools while 3 dimensional failure occurs in narrow tools. This necessitates different analysis procedures for wide and narrow tools.
  • 9. 9 2-D Cases (Wide Tools) The same forces as in a wall exist (see diagrams above) The force required to move the blade can be resolved into horizontal H and vertical directions V. The weight of the blade W is included in the calculation of V if significant compared to V For a blade of finite width w, this width should be included in the calculation of the soil force, P. wqhKhKcchKKhP qcaacp )( 2   Kp (=N), Kc(=Nc), Kca(=Nca), Kq(= Nq) are passive earth pressure coefficients h or d is the height or depth of blade, w is its width. The values N , Nc, Nca, Nq are obtainable from Appendix 5 in Mckyes, appendix 1 and 2 in Mckyes (Soil cutting and Tillage) or they can be calculated (easier to look them up in the graphs). The horizontal draft force H is then given by  cot)sin( dwcPH a and WdwcPV a  )cos(  however the log spiral method developed by Hettiaratchi (1969) is more accurate than the straight wedge model. The horizontal force is the draft force required in soil cutting tools and has to be provided by the draft power source eg a tractor, a pair of draft animals. The wedge theory of passive soil failure
  • 10. 10 Example (refer to Mckyes pp193) A 317 kW tracked tractor shown above with a mass of 41.8 tonnes is shown with a flat bulldozer blade having a total width of 4.80 m. the tractor is beginning to cut the soil to a depth of 40 cm as indicated. The blade itself excluding its arms has a mass of 4 tonnes. Soil properties = 17.6 kN/m3, =30o , = 20o , c=10kPa, ca=4kPa. Find the horizontal (draft) force and vertical uplift forces required to move the blade through the soil. Solution A The N coefficients can be determined from appendix 5 (Mckyes) or by calculation using the straight line wedge model
  • 11. 11 Then using the universal equation plug in the appropriate values of N or K to obtain the total tool force, the draft force and the vertical force. P= 70.1 kN; H= 72.8 kN and V= 52.0 kN SOLUTION B The calculation of the N factors using the above equations which can be tedious, can be avoided by use of graphs prepared from these equations for known conditions. Typical such graphs are available from Appendix 5 (Mckyes, Agricultural Engineering Soil Mechanics or Appendices 1 and 2 (Mckyes, Soil cutting and Tillage) proceed as follows: d/w =0.40/4.80=0.083 i.e narrow or wide tool? N (=20o,=30o, =53o ) or Kp= approx 1.45 Nc (=20o,=30o, =53o ), or Kc=approx 2.6 Nca===20o,=30o, =53o, Kca =0.8 q=0, therefore we have no surcharge and the surcharge term falls out. whKcchKKhP caacp )( 2   Interpolations for N values The N values can also be obtained from approximations Hettiaratchi and Reece,1974) and use of Appendix 1 and 2 in Soil Cutting and Tillage (Mckyes1985) as follows using (i) Hettiaratchi and Reece (1974)      ][ )0( )( )0(    N N NN Where N is the required N factor N(=0) is the N factor for a perfectly smooth blade N (=) is the N factor for a perfectly rough blade alternatively this may also be achieved as follows
  • 12. 12 (ii) Mckyes (1985) linear extrapolation    )( )0()()0(   NNNN Using these approximations N=0.94 +(1.7-0.94)*20/30 =1.446 Nc using the first aproximation = 1.3(3.4/1.3)0.666 Nc = 2.46 Nca (appendix 2) = 0.8 therefore P= (17.6*0.42 1.446 +10*0.4*2.46+4*0.4*0.8)*4.8 P=72.92 kN and as before  cot)sin( dwcPH a H = 72.92 sin(53+20)+4*0.4*4.8*cot 53 H= (69.73+5.787) kN = 75.52 kN WdwcPV a  )cos(  V= 72.92 cos (53+20)-4*0.4*4.8 +(4000*9.8/1000) V = 21.32-7.68+39.24 V= 52.88 kN NB.  The difference between the results from two approaches.  There is need for caution when using the graphs so as to ensure you use the correct graph for the right purpose.  The difference between the wedge theory result and this result should normally not exceed 10 %.
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