Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be...

Limit Equlibrium Method

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

• 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


Weak Clay layer( cu , u ) Short term

Long term(c´, ´ )


12

x

Weak Clay layer( cu , u ) Short term

Long term

Sandy Fill(c´, ´)

(c´, ´ )

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


12

v1

v2

v2 - v1


12

v1

v2

v2 - v1 v1 - v2

v1

v2


W1

P

X

C12

C1

R1

´

u


L

W2

W1

P

X

XC12

C12

R2

C2

C1

R1

´

´

´

u


´

L + W2

X

R2


´

L + W2

X

R2W1

X

C1

P

R1

u


Weak clay layer

Failure planes

• 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

• 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

Factor of Safety

mobcF F

tan

Factor of Safety

mobcF F

tan

mob m mc tan

This is usually written as

Factor of Safety

mobcF F

tan

mob m mc tan


where cm (=cF ) is known as the mobilised cohesion

Factor of Safety

mobcF F

tan

mob m mc tan


where cm (=cF ) is known as the mobilised cohesion

m (= tantan

1 F ) is known as the mobilised friction angle

Failure plane between wedges

Factor of Safety

Consider the 2 wedge mechanism


Factor of Safety


Between the wedges it is often assumed that the mobilised cohesion, c* and mobilised friction angle, * are given by

ccm m* * 2 2


Factor of Safety


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

Factor of Safetyv1

v2v1 - v2

C12

W1

R1

C1

mc

X 1

Factor of Safetyv1

v2v1 - v2

C12

C12

X 2

W2

C2

R 2

W1

R1

C1

mc

X 1

Factor of Safetyv1

v2v1 - v2

Factor of Safety

W1

C1C12

X1

R1

Factor of Safety

W1

C1C12

X1

R1

W2

C2

C12

R2

X2

• 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

• 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

• 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

• 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

Example

15 m

20 m

12

60o50o

1 2

Example

15 m

20 m

12

60o50o

1 2

1. Calculate areas:

A1 = 86.6 m2 A2 = 118.8 m2

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

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 =

Example

4. Calculate known forces

Example


60o

X1

W1

C1

R1

16.1

Example


60o50o60o

X1

X2

W1

W2

C1

R1

R2

16.1

16.1

16.1

Example

60o

X 1

W1

C1R1

16.1

Example

60o

X 1

W1

C1R1

16.1

60o50o

X2

W2

R216.1

16.1

Example

Example

X2 - X1

F1.0 1.5 2.0

610

238

F = 1.18

Limit Equlibrium Method. Limit Equilibrium Method Failure mechanisms are often complex and cannot be...

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