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Soil Mechanics II 土の力学II
Hiroyuki Tanaka
田中洋行
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Soil Mechanics
• Geotechnical Engineering
• Meaning of “Geo” The earth, or Ground
• Geology
• Geo-sciences, chemistry, graphy, so on.
• This lecture is proceeding based on “土質力学入門”, written by Prof. 三田地利之.
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Shear strength
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Evaluation of strength
Compression Small tensile strength
Concrete
Bending Moment
Steel
P 119
Page number of the text book
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Criteria for Soil Vertical Force: N
Shear Force: S
Normal Stress: σn = N/A A: Cross Area Shear Stress: τ=S/A
τ=c+σntanφ Coulomb’s Criteria
c
φ
Stable
Unstable
Impossible
Boundary Criteria
τ
σ
C: cohesion, φ: friction angle
P129
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Shear and Normal Stresses
σ and τ are changed according to an angle
We can find α for τ = 0 When τ = 0, we call this plane “Principal” Plane. Principal stresses: The maximum and minimum principal stresses: σ1 and σ3
P121-122
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Mohr’s Stress Circle
Principal Stress
σ1
σ1
σ1
σ3
σ3 σ3
σ
σ
σ τ
τ
τ
P122-123
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τ
σ
P125
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300
100 100
σ
τ
50
50
300
50
50
Positive σ: compression τ: anti-clockwise
How to draw the Mohr’s circle
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Measuring parameters c, φ
• Laboratory Test – Sampling from a borehole
– Direct Shear Test
– Triaxial (Unconfined compression) Test
• In situ Test – No sample
– Vane Test
– Standard Penetration Test (N value)
P129, 8-12
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Direct Shear Test
Failure envelopment τ=c+tanφ
σ τ
Merit: Easily understand Demerit: stress and strain are not uniform Control of drainage is difficult
P132
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Triaxial Test
Tri: Three Axial: Axis Piston
Cell
Specimen
Rubber Membrane
Pressure Gauge
bure
tte
No shear stress because of water Principal plane Lateral Pressure, Cell pressure
Deviator stress (σ1-σ3)
P134
σ2=σ3
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σ1
σ3
α
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σ1
σ3
σ1
σ3
Ⅰ Ⅰ
Plane acting the maximum principal stress
Pole
α α
Failure point
P127
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Failure criterion of Mohr and Coulomb In stead of using σ and τ, the failure criterion is presented by σ1 and σ3
τ
σ σ1 σ3
φ c
(σ1-σ3)/2
(σ1+σ3)/2 c cotφ
σ1-σ3=(σ1+σ3)sinφ+2c・cosφ P99 Mohr and Coulomb’s criteria
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Three conditions by Drainage
• Unconsolidated Undrained (UU) Consolidation Shear
• Consolidated Undrained (CU)
• Consolidated Drained (CD)
Principle of Effective Stress
The behavior including the strength is governed by the effective stress. σ’=σ-u σ’: Effective stress, σ: Total stress
P130-132
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Performance of UU Test
σ’3
τ
σ’1
σ’: Effective stress
σ: Total stress
τ Failure envelop : φ=0
Cu or Su Apparent cohesion, or undrained shear strength
P135
Fully Saturated
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Unconfined Compression Test Sometimes called Uniaxial Test
σ: Total stress
τ Failure envelop : φ=0
σ3=0
σ1: At failure, we call this strength qu unconfined compression strength
σ3 qu
cu
Cu=qu/2 P140-142 In practice, cu is called “cohesion”, or apparent cohesion.
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Young Modulus, E50 and Sensitivity
Sensitivity = qu/qr
P140-141
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Performance of CU Test
σ’
τ φ’
C’
After consolidation, σ3 = σ’3
Pore pressure generated by shearing
Undrained shear strength
su
P136-137
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Incremental Undrained Shear Strength
σ’, p
su
After consolidation, σ3 = σ’3
If Normally consolidated Su/p: constant
For Japanese clays, su/p=0.3~0.35
P136-137
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Performance of CD Test
σ’
τ φ’
C’
σ’3 does not change during shearing
Total and effective stresses are always the same because of no excess pore water pressure
P138-139
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Effective stress and Total stress analysis
• The effective stress analysis (ESA) seems more reasonable.
• Permeability is high (sandy soil), the ESA is applicable. Sand
• For clayey soil, effective stress or pore water pressure is unknown. Total stress analysis, in an other word, φ=0 method Clay
P137
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Shear strength for Total stress analysis
• Undrained shear strength
• For low OCR, i.e., negative dilatancy, undrained shear strength (UC or UU test) always is smaller than the drained shear strength.
• For long consolidation, the increase in the undrained shear strength can be expected (CU test)
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Su measured by in situ test Vane test
P142
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Dilatancy
• Volume change during shearing • Performance is changed under undrained or
drained conditions – Undrained:
• No volume change ∆V=0 • Pore water pressure change ∆u
– Positive: Negative Negative: Positive
– Drained: • No Pore water pressure u=0 • Volume change ∆V
– Positive: Expand Negative: Compressive
P144
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Critical Void Ratio Void ratio, e
loose,γ :small
large small
dense,γ: unit weight large
Dilatancy Positive Negative
Drained:Volume change
Undrained:Pore water pressure
Positive
Compression
Negative
Expand
ecrit
P 144-147
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Pore water Pressure
σ3
σ1
∆u=B{∆σ3+A(∆σ1-∆σ3)}
Excess pore water pressure: Deviator Stress: shear
Skempton’s A and B Coefficient: B=1 for saturated soil, A: dependent on Dilatancy
P151
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Dilatancy for sand and clay
e: large, loose e: small, dense
Dilatancy Positive Negative
Drained:Volume change
Undrained:Pore water pressure
Positive
Compression
Negative
Expand
ecrit
Sand
Clay OCR: Over-Consolidation Ratio OCR=pc/pvo: pc=preconsolidation pressure pvo= the current pressure
OCR: High Heavily OverConsolidated
OCR:1~2 Normally Slightly
Skempton’s A: Low or negative A: Hight
P 144-147, 150-151
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Drained and Undrained strength
σ’
τ φ’
C’
CD CU: Positive Dilatancy
CU: Negative Dilatancy
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Liquefaction Liquid
σ3
σ1
σ’3=σ3-u
increase constant
If σ’3=0, fully liquefied
P147-149
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Counter measurements for liquefaction
• Densification – Positive dilatancy. No positive water pressure
– Vibration
• Lowering the ground water table – No water
• Stabilized with cement – Cohesion τ=c’+(σ-u)tanφ
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Earth Pressure
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Earth Pressure Relating the movement of a wall
Passive Earth Pressure
Active Earth Pressure
Earth pressure at rest
Eart
h Pr
essu
re
Retaining wall P160, 169
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How to calculate the earth pressure?
• Rankine’s Method – Plastic equilibrium
– Theoretically, limitation of its application
• Coulomb’s Method – Stability of soil mass
– Trial calculation, more extensive application
P161, 170
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Rankine method
σ
τ φ’
C’
σv=γz: γ=unit weight of soil, z=depth
Passive State
Active State
σh
σv
σa: Active σp: Passive
P161
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Rankine method
σ
τ φ’
C’=0
σa: Active σp: Passive
γz
(σp-γz)/2
(σp+γz)/2
21 sin tan (45 )1 sin 2
ppK
zσ φ φγ φ
−= = = +
+sin p
p
zz
σ γφσ γ
−=
+
2tan (45 )2
aaK
zσ φγ
= = −
Using half angle formulae
Important Value: φ=30°Ka=1/3, Kp=3.0
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Earth Pressure at Rest
σh
σv For conventional ground
σv>σh
ho
vK σ
σ= Ko: Coefficient of earth pressure at rest
Depending on OCR. For NC (OCR=1) Ko=1-sinφ
P169
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Earth Pressure acting on the wall
Total Pressure
Center of Gravity
2 2 21 1tan (45 ) tan (45 )2 2 2 2
a t tP H H Hφ φγ γ= ° − = ° −
P166
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Rankine method φ=0
σ
τ
C=(qu/2)
σa: Active
σp: Passive
γz
2c
2c
σa=γz-2c σp=γz+2c
For Clay
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With surcharge
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Change in properties or existence of water table
Hydraulic pressure Earth pressure (Effective stress)
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Coulomb’s Method
Force Polygon
δ: Friction between the ground and the wall in Rankine’s theory, cannot take account.
β
Pa
Max. Pa
P170-174
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0 m
φ=30゚
Water Level
9m
3m
12m
φ=30゚
γ=20kN/m3
γ’(γsub)=10kNm3
Sheet pile
Tie Rod Force
Embedded Depth
Active Earth Force
Passive Earth Force
①
②
③
la
lp
② x la<③ x lp
Check for Embedded Depth
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0 m
φ=30゚
Water Level
9m
3m
12m
Active Earth Pressure
Passive Earth Pressure
The embedded depth is enough?
①
② ③
Pa1=(20 x 3)/2=30kN/m
Pa2=(20 x 9)=180kN/m Pa3=(30 x 9)/2=135kN/m
Pp=(90 x 3)/2=135kN/m
Center of Gravity
la3=3 + (2/3)x9=9m
lp=9 + (2/3)x3=11m
la3=3 + 9/2=7.5m
la1=(2/3)x3=2m
Pp x lp=135 x 11 = 1485 kN < Pa x la= 30x2+180x7.5+135x9=2625
Unstable
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Stability of slope
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Evaluation of the slope
Slip Slip surface
SF or Fs : Safety Factor = Resistance Drive Force
Resistance: Shear Strength
Drive Force: Gravity, Seismic
Safe or Danger?
Resistance
Drive
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Stability of infinitive slope 1
h
W=1 x cosi x h x γt N=W x cosi S=W x sini Driving Force R=N x tanφ Coulomb’s criteria: Resistance Force
N
S
FS= R S
= cos2ihγttanφ cosihγtsini
= tanφ tani FS=1, i=φ φ: angle of repose
P188
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Circle Failure method φ=0: Short term
SF= Moment for Resistance Moment for Drive
Resistance moment = (FE x su or cu)xR
Drive Moment=W x x P192
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Circle Failure method Slice method
SF= ΣMoment for Resistance ΣMoment for Drive
Resistance moment = (FE x su or cu)xR
Drive Moment=W x x P196
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Search for the smallest SF
Critical Circle
P193
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Tailor’s Chart
P194
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Stability Factor Ns
t cs
HNc
γ= No dimension
N/m3 x m
N/m2
P194
Toe Failure Base Failure Slope Failure
Stability Number Slope angle
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Failure pattern
Base Failure Toe Failure Slope Failure
P192
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Bearing Capacity
Shallow Foundation
Deep Foundation
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Pattern of Failure Load
Sett
lem
ent
General Failure
Local Failure
Bearing Capacity: Ultimate: Qu Allowable : Qal=Qu/SF SF: Settlement, uncertainty for soil parameters
P209
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Theoretical Value of Prandtl
III Passive Zone
I Active Zone
II Transition Zone
α:dependent on the roughness of the footing. Smooth:
452φα = ° +
Important value: φ=0, Qu=2bqu, qu=(2+π)c=5.14c
P210
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Calculation of Q
B
Q=qB=?
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Terzaghi’s equation
Shape Factor
Factors General Failure Local Failure
Continuous Square Rectangular Circle
1 2u c f qq acN BN D Nγβγ γ= + +Cohesion (Cu, Su)
Friction Surcharge
B
Df
Qu=Bqu
γ2
γ1
P211, 212
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Deep Foundation (Piles)
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Types of Pile classified by support system
Skin
Fri
ctio
n
Point Resistance
Pointed Pile Friction Pile
P217
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Design of Pile
• Using N value (Standard Penetration Test).
• Empirical equation
1 1(40 )x9.85 2
u p s s c cR NA N A N A= + +
Ap: cross sectional area As: surface area of the sand layer Ac: surface area of the clay layer
P218
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Standard Penetration Test (SPT)
Raymond sampler
Hammer (Mass = 63.5 kg)
Height = 76 cm
Knocking Head
Definition of N value How many blows for penetration of 30 cm
P10
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Pile Group
Pressure Bulb
RT = E・n・Ru
RT: Bearing capacity of the pile group Ru: Bearing capacity of the single pile n: Number of the Pile
E: Efficiency of the pile group <1.0 P221
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Negative Skin Friction
Conventional case Reclaimed Ground
Positive Friction
Negative Friction
P223