Stability of rock blocks - UNICAEN · Influence of water on slope stability EPFL -LMR Sliding...

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ÉCOLE POLYTECHNIQUEFÉDÉRALE DE LAUSANNE

L M RRock mechanics

LABORATOIRE DEMÉCANIQUE DES ROCHES

V. Labiouse, J. Abbruzzese

Stability of rock blocks

EPFL - LMR

Stability of rock blocks

1. Stability of one block

2. Stability of a column

3. Stability of two blocks

4. Stability of several blocks (fauchage)

5. Influence of water on slope stability

EPFL - LMR

Sliding stability of a block

ββ W cos β

W sin β

W σ = N/A = W cos β /A

b h

τres = σ tan ϕ + c*

τsol = T/A = W sin β /A

EPFL - LMR

ββ W cos β

W sin β

W

b h

W sin β /A < (W cos β /A) tan ϕ + c*

τsol < τres = σ tan ϕ + c*Stable if


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EPFL - LMR

Safety Factor with respect to sliding :

solsol

ress

ctanFτ

+ϕ⋅σ=

ττ

=∗

β W

b h

β⋅

+βϕ

=∗

sinWAc

tantanFs

β

+ϕ⋅β=

∗

sinAW

ctancosAW

Fs


EPFL - LMR

ϕϕ β

β

β > ϕ

β < ϕ

β = ϕ

Sliding

Stable

Stability limit

ϕ β

Sliding of a block on a smooth plane (c* = 0)

EPFL - LMR

SlidingStable

tan β

b/h

β > ϕβ < ϕ

tan ϕ

Sliding of a block on a smooth plane (c* = 0)

EPFL - LMR

β

β

W cos β

W sin β

W

Mdestabilizing,0 = W sin β . h/2 b

h

Toppling stability of a block

Toppling instability occurs if the direction line of the weight vector Wintersects the slope surface beyond the base of the column.

Μdestab,0 < Μstab,0

Stable if

0

Mstabilizing,0 = W cos β . b/2

tan β < b/h

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EPFL - LMR

β

β

W cos β

W sin β

W

b

h

Safety Factor with respect to toppling :

β==

tanhb

MM

F0,déstab

0,stabs

Toppling stability of a blockEPFL - LMR

b/h < tan β

b/h > tan βb/h = tan β

Toppling

Stable

Stability limit

ββ

b

h

β

Toppling stability of a block

EPFL - LMR

Toppling

Stable

tan β

b/h

b/h < tan β

b/h > tan β

b / h= ta

n β

Toppling stability of a blockEPFL - LMR

Sliding and toppling stability of a block on a smooth plane (c* = 0)

Sliding

Stable

tan ϕ

tan β

b/h

b / h= ta

n β

Slidingand

Toppling

Toppling

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EPFL - LMR

Safety Factor with respect to sliding :

β

A

β⋅

+βϕ

=∗

sinWAc

tantanFs

τres = σ tan ϕ + c*

WCG

Sliding stability of a columnEPFL - LMR

β

β==

tanhb

MMF

CG

CG

déstab

stabsCG

bCG

hCG

Toppling instability occurs if the direction line of the weight vector Wintersects the slope surface beyond the base of the column.

Toppling stability of a column

Safety Factor with respect to toppling :

EPFL - LMR

Sliding and toppling stability ofrock columns

EPFL - LMR

Stability of two blocks

Toppling of the two blocks

Toppling and sliding

Stability of the two blocks

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EPFL - LMR

The « fauchage » phenomenon

Toppling in layered or

fractured rocks

characterized by a

system steeply dipping

into the slope EPFL - LMR

β

b

huu = γw h cos β

Pressure distributions for allowed seepage

0

Hypotheses:1. Hydrostatic along the

rear fracture2. Flow with constant

gradient in the basaldiscontinuity

u0 = 0

V

UV = ½ γw h2 cos β

U = ½ γw h b cos β

EPFL - LMR

βW cos β

W

b

h

τres = σ´ tan ϕ + c*V

U

W sin βσ´ = N/Aτsol = T/A

Sliding stability for allowed seepage

0N = W cos β - UT = W sin β + V

Stable if τsol < τres

Stability highly endangered

EPFL - LMR

βW cos β

W

b

h

V

U

W sin β

Toppling stability for allowed seepage

0

Mdestabilizing,0 = W sin β . h/2 + V. h/3 + U. 2b/3

Μdestab,0 < Μstab,0

Stable if

Mstabilizing,0 = W cos β . b/2

Stability highly endangered

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EPFL - LMR

Pressure distributions if no outflow possible

β

b

h

V

U

u0

u = γw h cos β

Hypotheses:1. Hydrostatic along the

rear fracture2. No outflow at the toe of

the basal discontinuity

u0 = γw (h cos β + b sin β)

V = ½ γw h2 cos β

U = ½ γw b (2h cos β + b sin β)


L M RRock mechanics



Plane slide

EPFL - LMR

Stability of a plane slide

1. Kinematical conditions

2. Sliding along a plane

3. Sliding along a plane, with a rear tension crack

4. Stabilising measures

Control of water

Pre-tense anchors (active)

Grouted bar bolts (passive)

EPFL - LMR

Plane slide on the D526 roadconnecting Mens and Clelles (Isère – France)

http://www.irma-grenoble.com/

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EPFL - LMR

Plane slide in layered rocksin Sylans (France)

EPFL - LMR

When is a sliding mechanism possible ?

Yes No

NoNoautomatic detection of potential sliding planes, based on the use

of DTM25.

examplefor plane sliding

EPFL - LMR

Slide on a single plane joint: dry slope

Factor of safety:max. shear strengthapplied shear stressFS =

- applied shear stress:τsol = (T/A) = W/A sin β

A = L x 1 (m)σ´= (N/A) = W/A cos βτ = (T/A) = W/A sin βc*= 0β

ϕ=

βϕ⋅β

=tantan

sinWtancosWFS

- max. shear strength:τres = σ´ tan ϕ + c*

= (N/A) tanϕ ββ

W cos β

W

W sin β

L

EPFL - LMR

U

Hw

Slide on a single plane joint: role of water

U = resultant of the pore water pressure distribution, as a function of the hydraulic conditions

( )β

ϕ⋅−β=

sinWtanUcosWFS

τres = (σ - u) tan ϕτres =[ (Wcosβ-U)/Α ] tanϕ

The maximum shear strength on the failure surface is reduced

β W cos β

W

W sin β

L

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EPFL - LMR

Slide on a single joint (with tension crack)

Seepage allowedActions due to water presence:

1. Hydrostatic pressure in the tension crack;

2. Seepage through the joint at the base.

at the toe of the tension crack:u = γw hw

β

L

V

UW cos β

W

W sin βu

hw

2ww h

21V γ= Lh

21U wwγ=

EPFL - LMR


Seepage allowedstability against sliding

↑ : Ν = W cosβ – U – V sinβ

← : Τ = W sinβ + V cosβ

( )β+β

⋅+ϕβ−−β=

cosVsinWA*ctansinVUcosWFS

Shear strength reduced and applied forces increasedstability highly endangered.

βL

V

UW cos β

W

W sin βu

hw

EPFL - LMR


No outflow possible (at 0)Actions due to water presence:

1. Hydrostatic pressure inthe tension crack;

2. No water flow; Hydrostatic pressure in the failure plane.

at the toe 0 of the basal plane:u = γw (hw+ L sinβ)

βL

V

UW cos β

W

W sin βu

hw

0

2ww h

21V γ=

( )2

sinLh2LU ww β+γ=

EPFL - LMR

1. Surface drainage(cut-off ditch)

2. Pumping from wells3. Gravity drainage of

the rear tension crack4. Gravity drainage of

the basal plane5. Drainage gallery and

radial drains

Methods to control water in jointed rock slopes

1.2.

3.

4.

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EPFL - LMR

Support methods: active measures

Pre-tension active anchors

Free length

Spiral winding cables or rodsGrouted zone

Fixed anchor length

β

EPFL - LMR

↑ : Ν = Wcosβ - U - Vsinβ + Pa sin(β + γ)← : Τ = Wsinβ + Vcosβ - Pa cos(β + γ)

β

V

UW cos β

W

W sin β

Pa

Pa cos(β + γ)

Pa sin(β + γ)

βγ


EPFL - LMR

β

V

UW cos β

W

W sin β

Pa

Pa cos(β + γ)

Pa sin(β + γ)

βγ


( )( )( )γ+β−β+β

ϕγ+β+β−−β=

cosPcosVsinWtansinPsinVUcosWFS

a

a

The pre-tension of anchors both increases N and

decreases T, thus improving the stability of the slope.

EPFL - LMR

Support methods: passive measures

Grouted boltsGrouting all along the bar’s length

Thread bar Grouted zone

Passive anchors increasethe rock mass cohesion(Bjurström, 1974):

f

ca

aa S

SCσσ

τ=

Sa = area of bar’s transversal sectionS = area of the surface reinforced by boltsτa = shear strength of the bar (τa ≈ 0.6 σf)σc = rock’s compressive strengthσf = yielding stress of the bar’s material (steel)

β

! Take care that there is also tension in the bolts in addition to bending and shear stresses…

10

EPFL - LMR


↑ : Ν = Wcosβ - U - Vsinβ

← : Τ = Wsinβ + Vcosβ

β

V

UW cos β

W

W sin βT

ScS*ctanNFS a ⋅+⋅+ϕ=

! either c*·S or ca·S(do not add contributions)

EPFL - LMR


TS*ctanNFS ⋅+ϕ

=

( )β+β

⋅+ϕβ−−β=

cosVsinWSCtansinVUcosWFS a

β

V

UW cos β

W

W sin β


L M RRock mechanics



Wedge slide

EPFL - LMR

Wedge slide above the road of Cogne (Italy)

http://www.crealp.ch/

Wedge sliding on

two intersecting

discontinuities

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EPFL - LMR

Wedge sliding stability

Vertical plane Transverse section to i direction

Sliding failure of a rock wedge on the S1 and S2planes, which define an intersection line in i direction.

βW

i

nW cos β

W sin βS1

S2

W cos β

N1 N2θ1

n

hθ2

i

S1

h

S2

n

EPFL - LMR

Wedge sliding stability

β⋅+⋅+ϕ+ϕ

=sinW

ScSctanNtanNFS 2*21

*12211

Solving the equilibrium equations in the h, i and n directions:

( )21

21 sin

sincosWNθ+θ

θ⋅β=

( )21

12 sin

sincosWNθ+θ

θ⋅β=

βW

i

nW cos β

W sin βS1

S2

W cos β

N1 N2θ1

n

hθ2

EPFL - LMR

Wedge stability higher than plane failure

For example: if c1*=c2*=0 (smooth joints)and ϕ1= ϕ2= ϕ

βϕ

=tantanKFS

( )21

21sin

sinsinKθ+θ

θ+θ=

with K : wedge factor

Wed

ge fa

ctor

K

Aperture angle of the wedge ω

Stability of rock blocks - UNICAEN · Influence of water on slope stability EPFL -LMR Sliding...

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Transcript of Stability of rock blocks - UNICAEN · Influence of water on slope stability EPFL -LMR Sliding...