The Delft Sand, Clay & Rock Cutting Model, 2019a

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The Delft Sand, Clay & Rock Cutting Model. 8.3.1. The Model of Evans. For brittle rock the cutting theory of Evans (1964) and (1966) can be used to calculate cutting forces (Figure 8-21). The forces are derived from the geometry of the chisel (width, cutting angle and cutting depth) and the tensile strength (BTS) of the rock. Evans suggested a model on basis of observations on coal breakage by wedges. In this theory it is assumed that: 1. A force R is acting under an angle δ (external friction angle) with the normal to the surface A-C of the wedge. 2. A resultant force T of the tensile stresses acting at the center of the arc C-D, the line C-D is under an angle β (the shear angle) with the horizontal. 3. A third force S is required to maintain equilibrium in the buttock, but does not play a role in the derivation. 4. The penetration of the wedge is small compared to the layer thickness hi. The action of the wedge tends to split the rock and does rotate it about point D. It is therefore assumed that the force S acts through point D. Along the fracture line, it is assumed that a state of plain strain is working and the equilibrium is considered per unit of width w of the wedge. The force due to the tensile strength σT of the rock is: T T (8-41) T r cos d w 2 r sin w Where r·d is an element of the arc C-D making an angle with the symmetry axis of the arc. Let hi be the depth of the cut and assume that the penetration of the edge may be neglected in comparison with hi. This means that the force R is acting near point C. Taking moments on the chip cut about point D gives: hi R cos T r sin 2 T r sin w r sin sin (8-42) From the geometric relation it follows: h i r sin (8-43) 2sin Hence: T hi w R (8-44) 2sin cos The horizontal component of R is R·sin(α+δ) and due to the symmetry of the forces acting on the wedge the total cutting force is: c T i sin F 2 Rsin h w sin cos (8-45) The normal force ( on cutting force) is per side: Fn Rcos Thi w 2sin cos cos (8-46) The angle β can be determined by using the principle of minimum energy: dF c d 0 (8-47) Giving: Page 262 of 454 TOC Copyright © Dr.ir. S.A. Miedema
Rock Cutting: Atmospheric Conditions. cos cos sin sin 0 cos 2 0 (8-48) Resulting in: 1 2 2 4 2 (8-49) With: 1 sin sin cos (8-50) 2 This gives for the horizontal cutting force: 2sin F h w h w 1 sin c T i T i HT (8-51) For each side of the wedge the normal force is now (the total normal/vertical force is zero): cos F h w h w 1 sin n T i T i VT (8-52) Evans Brittle Horizontal Force Coefficient λ HT vs. Blade Angle α φ=00° Brittle Horizontal Force Coefficient λ HT (-) 100 10 φ=05° φ=10° φ=15° φ=20° φ=25° φ=30° φ=35° φ=40° © S.A.M. 1 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Blade Angle α (Degrees) Figure 8-20: The brittle-tear horizontal force coefficient λHT (Evans). φ=45° Copyright © Dr.ir. S.A. Miedema TOC Page 263 of 454
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The Delft Sand, Clay & Rock Cutting
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Chapter 1: Introduction. 1.1. Appro
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Introduction. In dry sand cutting t
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Chapter 2: Basic Soil Mechanics. 2.
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Basic Soil Mechanics. 2.2.2. Soil C
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Basic Soil Mechanics. soil characte
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Basic Soil Mechanics. 2.3. Soils. 2
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Basic Soil Mechanics. 2.3.2. Clay.
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Basic Soil Mechanics. 2.3.3. Rock.
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Basic Soil Mechanics. Figure 2-12:
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Basic Soil Mechanics. Figure 2-15 B
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2.4. Soil Mechanical Parameters. Ba
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Basic Soil Mechanics. 2.4.2.4. Impo
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Basic Soil Mechanics. Table 2-3: Em
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Basic Soil Mechanics. 2.4.3.4. Poro
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Basic Soil Mechanics. 3 3 2 l e 3 g
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Basic Soil Mechanics. 46 Friction a
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Basic Soil Mechanics. c tan (2-
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2.4.9. Unconfined Tensile Strength.
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Basic Soil Mechanics. 2.5. Criteria
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Basic Soil Mechanics. increase, to
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Basic Soil Mechanics. Table 2-14: T
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2.6. Soil Mechanical Tests. 2.6.1.
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Basic Soil Mechanics. be regarded a
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2.6.5.1. Consolidated Drained (CD).
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Basic Soil Mechanics. 2.8. The Mohr
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Basic Soil Mechanics. Squaring equa
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Basic Soil Mechanics. 2.9. Active S
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Basic Soil Mechanics.
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Basic Soil Mechanics. 2.10. Passive
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cos sin cos sin 1 1 sin
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Basic Soil Mechanics. 2.11. Summary
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2.12. Shear Strength versus Frictio
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Basic Soil Mechanics. 2.13. Nomencl
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The General Cutting Process. Chapte
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The General Cutting Process. 10. β
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The General Cutting Process. The fo
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The General Cutting Process. W2 si
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The General Cutting Process.
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The General Cutting Process.
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3.6. The Snow Plough Effect. The Ge
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The General Cutting Process. The ve
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3.6.5. The Resulting Cutting Forces
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The General Cutting Process. Q c Pc
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Which Cutting Mechanism for Which K
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4.6. Summary. Which Cutting Mechani
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Chapter 5: Dry Sand Cutting. 5.1. I
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Dry Sand Cutting. The force K1 on t
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Dry Sand Cutting. Horizontal Cuttin
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Dry Sand Cutting. 2 v s i VD F
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Dry Sand Cutting. Horizontal Cuttin
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Dry Sand Cutting. Shear Angle β vs
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Dry Sand Cutting. 5.6. Specific Ene
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5.8. Experiments in Dry Sand. 5.8.1
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Dry Sand Cutting. 5.8.2. Wismer & L
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Chapter 6: Saturated Sand Cutting.
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Saturated Sand Cutting. 2 ci c i F
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Saturated Sand Cutting. the Waterlo
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Saturated Sand Cutting. The normal
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Saturated Sand Cutting. Figure 6-8:
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Saturated Sand Cutting. Figure 6-10
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6.7. The Blade Tip Problem. Saturat
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Saturated Sand Cutting. s2 0.8 L
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Saturated Sand Cutting. However Fig
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Saturated Sand Cutting. S1=L1*(1-I/
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Saturated Sand Cutting. 6.9. Determ
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' h Saturated Sand Cutting.
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Saturated Sand Cutting. α=45° and
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Saturated Sand Cutting. 6.12.1. Spe
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Saturated Sand Cutting. 100.0 SPT v
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Saturated Sand Cutting. 6.12.2. The
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Saturated Sand Cutting. 6.13. Exper
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Saturated Sand Cutting. The tests a
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Saturated Sand Cutting. old laborat
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Saturated Sand Cutting. 6.13.2. Tes
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Saturated Sand Cutting. side effect
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Saturated Sand Cutting. 6.13.7. Com
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Saturated Sand Cutting. which the t
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Saturated Sand Cutting. The dimensi
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Saturated Sand Cutting. 10 Partial
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Saturated Sand Cutting. The equatio
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Saturated Sand Cutting. 0.30 No Cav
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Saturated Sand Cutting. P4 (bar) P3
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Saturated Sand Cutting. 10.0 8.0 Fh
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Saturated Sand Cutting. Preal Real
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Clay Cutting. Chapter 7: Clay Cutti
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Clay Cutting. 7.3. The Influence of
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Clay Cutting. From this equation, s
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Clay Cutting. a material on which a
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Clay Cutting. 7.3.4. The Proposed T
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Clay Cutting. 100 90 Shear Strength
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Clay Cutting. 7.3.6. Abelev & Valen
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Clay Cutting. 2500 The Strain Rate
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Clay Cutting. 7.4. The Flow Type. 7
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Clay Cutting. F s ch i w s ah b
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Rock Cutting: Hyperbaric Conditions
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9.8. Specific Energy Graphs. Rock C
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The Occurrence of a Wedge. Chapter
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The Occurrence of a Wedge. 19. A fo
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The Occurrence of a Wedge. h 4 4
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10.3. The Equilibrium of Moments. T
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A Wedge in Dry Sand Cutting. Chapte
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A Wedge in Dry Sand Cutting. The no
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A Wedge in Dry Sand Cutting. Figure
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11.4. Results of some Calculations.
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A Wedge in Dry Sand Cutting. 11.5.
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A Wedge in Dry Sand Cutting. 11.6.
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A Wedge in Saturated Sand Cutting.
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12.4. The Equilibrium of Moments. A
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12.8. Experiments. A Wedge in Satur
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12.10. Nomenclature. A Wedge in Sat
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A Wedge in Clay Cutting. Chapter 13
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A Wedge in Clay Cutting. on the equ
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A Wedge in Clay Cutting. Figure 13-
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A Wedge in Clay Cutting. L3 L7
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A Wedge in Atmospheric Rock Cutting
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14.4. Nomenclature. A Wedge in Atmo
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A Wedge in Hyperbaric Rock Cutting.
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15.4. Nomenclature. A Wedge in Hype
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Exercises. Chapter 16: Exercises. 1
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Exercises. 16.2.8. Calc.: Bulldozer
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Exercises. 16.3. Chapter 3: The Gen
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Exercises. 16.4.5. MC: Hyperbaric R
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Exercises. 16.6. Chapter 6: Water S
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Exercises. 2 2 1 w c i c g v h
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Exercises. F: Determine the pore pr
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Exercises. E sp Fh vc Fh Fh 10
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Exercises. 1,000 Pore Pressures on
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Exercises. A tensile strength of -2
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Exercises. Shear Stress vs. Normal
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Exercises. 70 60 50 40 Shear Stress
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Exercises. 16.8.4. Calc.: Cutting F
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Exercises. 16.8.5. Calc.: Cutting F
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Exercises. 16.9. Chapter 9: Hyperba
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Bibliography. Chapter 17: Bibliogra
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Bibliography. Miedema, S. (1989, De
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Figures & Tables. Chapter 18: Figur
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Figures & Tables. Figure 5-20: The
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Figures & Tables. Figure 8-5: Const
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Figures & Tables. Figure 12-11: The
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18.2. List of Figures in Appendices
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Figures & Tables. Figure U-2: Speci
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Figures & Tables. Figure Y-27: A la
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Figures & Tables. 18.3. List of Tab
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18.4. List of Tables in Appendices.
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Appendices. Chapter 19: Appendices.
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Active & Passive Soil Failure Coeff
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Dry Sand Cutting Coefficients. Appe
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Dry Sand Cutting Coefficients. B.1.
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Dry Sand Cutting Coefficients. B.2
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Dry Sand Cutting Coefficients. B.3
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Dry Sand Cutting Coefficients. Perc
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Dimensionless Pore Pressures p1m &
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The Shear Angle β Non-Cavitating.
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The Shear Angle β Non-Cavitating.
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Appendix E: The Coefficient c1. The
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The Coefficient c1. Table E-3: c1 f
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Appendix F: The Coefficient c2. The
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The Coefficient c2. Table F-3: c2 f
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Appendix G: The Coefficient a1. The
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The Coefficient a1. Table G-3: a1 f
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The Shear Angle β Cavitating. Appe
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The Shear Angle β Cavitating. Tabl
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Appendix I: The Coefficient d1. The
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The Coefficient d1. Table I-3: d1 f
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Appendix J: The Coefficient d2. The
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The Coefficient d2. Table J-3: d2 f
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The Properties of the 200 μm Sand.
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Experiments in Water Saturated Sand
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The Snow Plough Effect. Appendix N:
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The Snow Plough Effect. 10.0 8.0 Fh
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The Snow Plough Effect. 20.0 16.0 F
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Specific Energy in Sand. Appendix O
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Specific Energy in Sand. 2500 Speci
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Occurrence of a Wedge, Non-Cavitati
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Occurrence of a Wedge, Non-Cavitati
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Occurrence of a Wedge, Cavitating.
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Occurrence of a Wedge, Cavitating.
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Pore Pressures with Wedge. Appendix
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Pore Pressures with Wedge. Table R-
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Pore Pressures with Wedge. Table R-
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FEM Calculations with Wedge. Append
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FEM Calculations with Wedge. Figure
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S.3 The 75 Degree Blade. FEM Calcul
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Appendix T: Force Triangles. Force
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Force Triangles. Figure T-3: The fo
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Force Triangles. Figure T-5: The fo
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Specific Energy in Clay. Appendix U
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Specific Energy in Clay. 6000 Speci
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Clay Cutting Charts. Appendix V: Cl
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Clay Cutting Charts. The Vertical C
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Clay Cutting Charts. Shear Angle β
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Clay Cutting Charts. V.3 The Curlin
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Rock Cutting Charts. Appendix W: Ro
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Rock Cutting Charts. W.2 The Transi
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Rock Cutting Charts. A & B: Tensile
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Rock Cutting Charts. W.5 Brittle Te
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Rock Cutting Charts. W.6 Brittle Te
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Hyperbaric Rock Cutting Charts. App
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Hyperbaric Rock Cutting Charts. 100
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Hyperbaric Rock Cutting Charts. X.2
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Applications & Equipment. Appendix
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Y.2 Bucket Ladder Dredges. Applicat
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Y.3 Cutter Suction Dredges. Applica
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Applications & Equipment. Figure Y-
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Applications & Equipment. Y.4 Trail
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Y.5 Backhoe Dredges. Applications &
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Y.6 Clamshell Dredges. Applications
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Y.7 Bucket Wheel Dredges. Applicati
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Y.8 Braun Kohle Bergbau. Applicatio
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Y.9 Deep Sea Mining. Applications &
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Y.10 Cable Trenching. Applications
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Y.11 Offshore Pipeline Trenching. A
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Y.12 Dry Trenching. Applications &
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Applications & Equipment. Y.13 PDC
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Applications & Equipment. Y.14 Bull
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Applications & Equipment. Y.15 Dry
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Y.16 Tunnel Boring Machines. Applic
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Publications. Appendix Z: Publicati
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Publications. 52. Miedema, S.A.,
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