soil mechanics ii 土の力学ii · 2015-03-30 · soil mechanics • geotechnical engineering •...
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Soil Mechanics II 土の力学II
Hiroyuki Tanaka
田中洋行
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.
Shear strength
Evaluation of strength
Compression Small tensile strength
Concrete
Bending Moment
Steel
P 119
Page number of the text book
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
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
Mohr’s Stress Circle
Principal Stress
σ1
σ1
σ1
σ3
σ3 σ3
σ
σ
σ τ
τ
τ
P122-123
τ
σ
P125
300
100 100
σ
τ
50
50
300
50
50
Positive σ: compression τ: anti-clockwise
How to draw the Mohr’s circle
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
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
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
σ1
σ3
α
σ1
σ3
σ1
σ3
Ⅰ Ⅰ
Plane acting the maximum principal stress
Pole
α α
Failure point
P127
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
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
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
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.
Young Modulus, E50 and Sensitivity
Sensitivity = qu/qr
P140-141
Performance of CU Test
σ’
τ φ’
C’
After consolidation, σ3 = σ’3
Pore pressure generated by shearing
Undrained shear strength
su
P136-137
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
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
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
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)
Su measured by in situ test Vane test
P142
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
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
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
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
Drained and Undrained strength
σ’
τ φ’
C’
CD CU: Positive Dilatancy
CU: Negative Dilatancy
Liquefaction Liquid
σ3
σ1
σ’3=σ3-u
increase constant
If σ’3=0, fully liquefied
P147-149
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
Earth Pressure
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
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
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
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
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
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
Rankine method φ=0
σ
τ
C=(qu/2)
σa: Active
σp: Passive
γz
2c
2c
σa=γz-2c σp=γz+2c
For Clay
With surcharge
Change in properties or existence of water table
Hydraulic pressure Earth pressure (Effective stress)
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
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
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
Stability of slope
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
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
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
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
Search for the smallest SF
Critical Circle
P193
Tailor’s Chart
P194
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
Failure pattern
Base Failure Toe Failure Slope Failure
P192
Bearing Capacity
Shallow Foundation
Deep Foundation
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
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
Calculation of Q
B
Q=qB=?
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
Deep Foundation (Piles)
Types of Pile classified by support system
Skin
Fri
ctio
n
Point Resistance
Pointed Pile Friction Pile
P217
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
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
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
Negative Skin Friction
Conventional case Reclaimed Ground
Positive Friction
Negative Friction
P223
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