Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be...
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Transcript of Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be...
![Page 1: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/1.jpg)
Limit Equlibrium Method
![Page 2: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/2.jpg)
Limit Equilibrium Method
• Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces.
• Analysis is straightforward if mechanism consists of
• Multiple wedges• Circular failures
• Most situations can be approximated by one of these mechanisms
![Page 3: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/3.jpg)
• Mechanism appropriate when soil stratigraphy contains weak layers.
These can occur due to
• Thin clay layers in sedimentary deposits
• Pre-existing slips in clayey soils
• Fissures and joints in stiff clays
• Joints in rocks and other cemented soil materials
Multiple wedge mechanisms
![Page 4: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/4.jpg)
Multiple wedge mechanisms
Weak Clay layer( cu , u ) Short term
Long term(c´, ´ )
![Page 5: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/5.jpg)
Multiple wedge mechanisms
12
x
Weak Clay layer( cu , u ) Short term
Long term
Sandy Fill(c´, ´)
(c´, ´ )
![Page 6: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/6.jpg)
Multiple wedge mechanismsOnce the mechanism has been specified, the directions of the forces must be determined. These can be determined from common sense or by drawing a velocity diagram to get the relative velocities.
![Page 7: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/7.jpg)
Multiple wedge mechanismsOnce the mechanism has been specified, the directions of the forces must be determined. These can be determined from common sense or by drawing a velocity diagram to get the relative velocities.
12
![Page 8: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/8.jpg)
Multiple wedge mechanismsOnce the mechanism has been specified, the directions of the forces must be determined. These can be determined from common sense or by drawing a velocity diagram to get the relative velocities.
12
v1
v2
v2 - v1
![Page 9: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/9.jpg)
Multiple wedge mechanismsOnce the mechanism has been specified, the directions of the forces must be determined. These can be determined from common sense or by drawing a velocity diagram to get the relative velocities.
12
v1
v2
v2 - v1 v1 - v2
v1
v2
![Page 10: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/10.jpg)
Multiple wedge mechanisms
W1
P
X
C12
C1
R1
´
u
![Page 11: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/11.jpg)
Multiple wedge mechanisms
L
W2
W1
P
X
XC12
C12
R2
C2
C1
R1
´
´
´
u
![Page 12: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/12.jpg)
Multiple wedge mechanisms
´
L + W2
X
R2
![Page 13: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/13.jpg)
Multiple wedge mechanisms
´
L + W2
X
R2W1
X
C1
P
R1
u
![Page 14: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/14.jpg)
Multiple wedge mechanisms
Weak clay layer
Failure planes
![Page 15: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/15.jpg)
Multiple wedge mechanisms
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• When performing stability analyses we are often interested in determining a factor of safety
• The factor of safety can be determined from a limit equilibrium analysis using factored strength parameters
• At failure the stresses are related by the Mohr-Coulomb criterion
c + tan
• At states remote from failure the stresses are assumed to be given by
Factor of Safety
![Page 17: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/17.jpg)
• When performing stability analyses we are often interested in determining a factor of safety
• The factor of safety can be determined from a limit equilibrium analysis using factored strength parameters
• At failure the stresses are related by the Mohr-Coulomb criterion
c + tan
• At states remote from failure the stresses are assumed to be given by
Factor of Safety
mobcF F
tan
![Page 18: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/18.jpg)
Factor of Safety
mobcF F
tan
![Page 19: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/19.jpg)
Factor of Safety
mobcF F
tan
mob m mc tan
This is usually written as
![Page 20: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/20.jpg)
Factor of Safety
mobcF F
tan
mob m mc tan
This is usually written as
where cm (=cF ) is known as the mobilised cohesion
![Page 21: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/21.jpg)
Factor of Safety
mobcF F
tan
mob m mc tan
This is usually written as
where cm (=cF ) is known as the mobilised cohesion
m (= tantan
1 F ) is known as the mobilised friction angle
![Page 22: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/22.jpg)
Failure plane between wedges
Factor of Safety
Consider the 2 wedge mechanism
![Page 23: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/23.jpg)
Failure plane between wedges
Factor of Safety
Consider the 2 wedge mechanism
Between the wedges it is often assumed that the mobilised cohesion, c* and mobilised friction angle, * are given by
ccm m* * 2 2
![Page 24: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/24.jpg)
Failure plane between wedges
Factor of Safety
Consider the 2 wedge mechanism
Between the wedges it is often assumed that the mobilised cohesion, c* and mobilised friction angle, * are given by
ccm m* * 2 2
It is more convienient to take c* = cm and * = m as this must be the case when F=1
![Page 25: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/25.jpg)
Factor of Safetyv1
v2v1 - v2
![Page 26: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/26.jpg)
C12
W1
R1
C1
mc
X 1
Factor of Safetyv1
v2v1 - v2
![Page 27: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/27.jpg)
C12
C12
X 2
W2
C2
R 2
W1
R1
C1
mc
X 1
Factor of Safetyv1
v2v1 - v2
![Page 28: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/28.jpg)
Factor of Safety
W1
C1C12
X1
R1
![Page 29: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/29.jpg)
Factor of Safety
W1
C1C12
X1
R1
W2
C2
C12
R2
X2
![Page 30: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/30.jpg)
• Equilibrium requires that the forces between the two wedges are equal and opposite.
• In the analysis we have assumed a factor of safety which enables X1 and X2 to be determined.
• The values of X1 and X2 depend on F and will not in general be equal
• We need to determine the value of F that gives X1 = X2
• This can be determined by a trial and error process, followed by interpolation
Factor of Safety
![Page 31: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/31.jpg)
• the solution is not necessarily the factor of safety of the slope.
• To determine the actual factor of safety all the possible mechanisms must be considered to determine the mechanism giving the lowest factor of safety.
Factor of SafetyX1 - X2
F
Required solution
![Page 32: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/32.jpg)
• For the slope analysis a unique factor of safety can be determined.
• In the analysis of the retaining wall considered above the force on the wall is related to the factor of safety.
• When analysing retaining walls we are often concerned with the limiting stability, that is when the factor of safety is 1.
• For an active failure mechanism increasing the factor of safety results in a greater horizontal force being required
• For a gravity retaining wall, the wall can be analysed as a single block or wedge
Factor of Safety
![Page 33: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/33.jpg)
• In any analysis the appropriate parameters must be used for c and . In an undrained analysis (short term in clays) the parameters are cu, u with total stresses, and in an effective stress analysis (valid any time if pore pressures known) the parameters are c’, ’ used with the effective stresses.
• In an effective stress analysis if pore pressures are present the forces due to the water must be considered, and if necessary included in the inter-wedge forces.
Factor of Safety
![Page 34: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/34.jpg)
Example
15 m
20 m
12
60o50o
1 2
![Page 35: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/35.jpg)
Example
15 m
20 m
12
60o50o
1 2
1. Calculate areas:
A1 = 86.6 m2 A2 = 118.8 m2
![Page 36: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/36.jpg)
Example
15 m
20 m
12
60o50o
1 2
1. Calculate areas:
A1 = 86.6 m2 A2 = 118.8 m2
2. Assume Factor of Safety
F = 2
![Page 37: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/37.jpg)
Example
15 m
20 m
12
60o50o
1 2
1. Calculate areas:
A1 = 86.6 m2 A2 = 118.8 m2
2. Assume Factor of Safety
F = 2
3. Calculate c, parameters
Weak layer cm = cu/F = 40/2 = 20 kPa, m = 0
Clayey sand cm = 0, ’m =
![Page 38: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/38.jpg)
Example
4. Calculate known forces
![Page 39: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/39.jpg)
Example
4. Calculate known forces
60o
X1
W1
C1
R1
16.1
![Page 40: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/40.jpg)
Example
4. Calculate known forces
60o50o60o
X1
X2
W1
W2
C1
R1
R2
16.1
16.1
16.1
![Page 41: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/41.jpg)
Example
60o
X 1
W1
C1R1
16.1
![Page 42: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/42.jpg)
Example
60o
X 1
W1
C1R1
16.1
60o50o
X2
W2
R216.1
16.1
![Page 43: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/43.jpg)
Example
60o
X 1
W1
C1R1
16.1
60o50o
X2
W2
R216.1
16.1
![Page 44: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/44.jpg)
Example
![Page 45: Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be modelled by single wedges with plane surfaces. Analysis.](https://reader036.fdocuments.in/reader036/viewer/2022062819/5706386a1a28abb8239045ff/html5/thumbnails/45.jpg)
Example
X2 - X1
F1.0 1.5 2.0
610
238
F = 1.18