Shear Strength of Rocks: Principles, Testing Methods, and Engineering Applications

Rock Slope Engineering
Main purposes of rock slope engineering:
 To determine rock slope stability conditions
 To stabilize unstable natural slopes
 To design (while maintaining safety conditions) the rock
excavation slopes by obtaining optimal conditions from
the reliability and the economical point of view.

Scope of Rock Slope Engineering (RSE)
Piteau & Peckover (1978):
 RSE is not concerned with large landslides
 RSE analyze individual rock block falls, translation of small rock
masses, occasional slides of accumulated debris from gullies,
talus* slopes and postglacial slide area
* Talus: fallen disintegrated material formed a slope at the foot of a steeper declivity
 Large scale (km) instability phenomena are examined in the
engineering geology field
 Small scale instability analysis pertains to the soil and rock
mechanics field  both geological and engineering background
are required to study small or large landslide problems
Giani (1992):

Problem Definition
 “Rock” : a compact semi-hard mass of a variety of mineral
 “Defects” : all features (ultra microscopic to macroscopic) in which
influence the strength and the deformation characteristics of rocks
 Defects  decrease the load carrying capacity of rocks and cause
a concentration of stresses in certain directions around excavation
(Lama & Vutukuri, 1978)
 Fabric defects : part of rock arranged in a regular or irregular
order relatively to each other
 Structural defects : Folds, Faults and Joints  due to
tectonic stresses. Quantitative description of structural defects =
discontinuity  main problems of RSE
Defects in rocks:

Problem Definition
 Rock slope stability depends on (1) strength features of the
rocks, (2) geometrical and strength features of discontinuities;
and (3) presence of weathering action and rock defects.
 Rock weathered by deferent causes : mechanical processes and
chemical dissolutions
 Rock weathering is a process which cause alteration of rock due
the action of water, CO2, O2  decrease competency of the rock
 Classified on the basis of relative importance of defects and
alteration to slope stability
 Divided into continuous, pseudo-continuous and
discontinuous mass
Rock slope excavation

Continuous
Pseudo-
Continuous
Discontinuous

 Shear strength and unit weight of intact rock determine
stability conditions of a homogenous slope
 Intact rock shear failure envelope is nonlinear  strength
features depends on applied normal stress
 Uniaxial compressive strength most important features for
mechanical characterization of intact rock

Uniaxial compressive strength of the rocks

Principal Factors Affecting Rock Slope Stability Analysis
 Rock slope stability is conditioned by the presence of weakness
or discontinuity planes. Strength and deformation of the rock
depends on continuity, spacing, orientation and mechanical
features of these planes.
1. Find a connection between discontinuity sets and potential
instability kinematics
2. Determine shear strength on discontinuity planes
3. Assess water flow conditions
4. Determine safety factor or the motion of the block
5. The efficiency of stabilizing techniques applications
(excavation, drainage, rock bolts etc.)
Steps of analysis

Shear Strength
Intact rock strength envelope


 Isotropic material specimens
sheared in direct shearing
device
 fi = internal shearing resistance
angle
 fi = 35 – 65  decreasing with
normal load increment
 Shear failure surface is pre-
determined in a direct shearing
device  triaxial tests

Shear Strength
Typical shear stress-shear
displacement diagram

Maximum (a) and residual (b)
shear resistance diagram



Types of strength criterion
 Peak strength criterion expresses a relationship between
stress components  peak strength developed under various
stress combinations can be predicted
 Residual strength criterion is defined using a relationship able
to predict residual strength under various stress conditions Brady
& Brown (1985)
 Strength criterion is best written in effective stress terms. As
pore pressures are usually low, total and effective pressures are
almost the same.
 General form of peak strength criterion express maximum
principal stress as a function of minimum and intermediate
principal stresses.
 
3
1 
 f


Coulomb Shear Strength Criterion
Coulomb (1776)  soil and rock shear strengths:
f

 tan
n
c 
  = shear strength (kN/m2)
c = cohesion (kN/m2)
f = internal friction angle ( 0)
 = normal stress at the failure surface
(kN/m2)
1
1
3 3
n

b
Stress transformation
    b




 2
cos
2
1
2
1
3
1
3
1 



  b


 2
sin
2
1
3
1 


Coulomb Shear Strength Criterion
Stress limit conditions on any planes defined by the angle of b :
1
1
3 3
n

b
Critical plane : the plane on which all available shear strength is
firstly reached, for increments of 1.
Orientation of the critical plane:
 
 
 
b
f
b
b
f
b


2
cos
1
tan
2
sin
2
cos
1
tan
2
sin
2 3






c
2
4
f

b 


Coulomb criterion in  – n plane Coulomb criterion in 1 – 3 plane

n
f
c
3 1
2b
Considering f
b cos
2
sin 
 
f
f

f

sin
1
sin
1
cos
2 3
1




c
f
f

sin
1
sin
1
tan



Linear relationship of 3
and peak value of 1
3

c
1

f
f

sin
1
cos
2


c
c
Theoretical value for Uniaxial
compressive strength
By extrapolating, shear strength Coulomb envelope, for 1 = 0
Apparent value T of uniaxial tensile strength
3

c
1
f
f

sin
1
cos
2


c
T
 Coulomb criterion is not suitable for prediction of shear strength
conditions when tension stress applied perpendicular to a shear failure
plane
 Strength behavior in a tensile field is different from the extrapolated
Coulomb envelope, the uniaxial tensile strength are usually lower

 It implies that a major shear fracture exists at peak strength.
Observations show that this is not always the case.
 It implies a direction of shear failure which does not always
agree with experimental observations
 Experimental peak strength envelopes are generally non-linear.
They can be considered only over a limited range of n or 3
Coulomb criterion is not a particularly satisfactory peak strength
criterion for rock material. The reasons are:
 Other peak strength criteria are preferred for intact rock, even
though, in a slope stability problem, the value of n are generally
low enough to justify a linear strength envelope assumption.
 The coulomb criterion may instead be applied to shear strength
behavior in residual conditions and particularly for rock
discontinuity residual conditions

Rock Discontinuity Shear Strength
 Planar discontinuity surfaces
 Inclined discontinuity surfaces
 Multiple inclined discontinuity surfaces
 Ladanyi & Archambault criterion
 Rough discontinuity surfaces
 Barton criterion

Planar discontinuity surfaces
 Surface shape of natural rock discontinuities : planar, undulated, stepped
 At lower scale : rough, smooth, slickenslide
 With large displacement, initially polished rock surfaces become
scratched and gouged  fb increases to f  similar to sawn planar but
not polished
 If the original surface is quite rough  it becomes progressively smoother
with increasing displacements  fb progressively decreases to fr
Friction angle of a discontinuity :
1. Peak friction angle fp  evaluated on natural discontinuities 
maximum shear strength determined by roughness failure or overstep
2. Basic friction angle fb  on an artificially planar slickenslide surface
3. Residual friction angle fr  shear strength is stabilized on a minimum
value  altered and smooth surfaces

Friction angle of a discontinuity :
 Ultimate friction angle fu (first residual)  down to residual stress fr
 Barton & Choubey (1977) : residual friction angle fr of a joint is a function
of the relative strengths of the joint wall material and the stronger
unweathered material in the interior of each block:
R = Schmidt hammer rebound on sawn surfaces (unweathered)
r = Schmidt hammer rebound on wet joint surfaces (weathered)
When wall material is unweathered fr = fb
   
R
r
b
r


20
20 

 f
f

A Schmidt hammer  to measure the elastic
properties or strength of concrete or rock.
 Measures the rebound of a spring loaded mass
impacting against the surface of sample
 Reading affected by the orientation of hammer
 Schmidt hammer is an arbitrary scale ranging
from 10 to 100. Schmidt hammers are available
in several different energy ranges.
Schmidt Hammer
The test is also sensitive to other factors:
- Local variation in the sample. To minimize  take a selection of readings. Average
of 10 readings should be obtained.
- Water content of the sample, a saturated material give different results from dry one.
Classed as indirect test as it does not give a direct measurement of the strength of
the material. It simply gives an indication based on surface properties, it is only
suitable for making comparisons between samples.

Typical shear strength envelope from direct shear test on a series
of rock specimens with a relatively flat surface, for a stress range of
normal stresses of 0 – 1.5 MPa.
Residual friction fr for most rocks is usually between 25 – 35.

Range of ultimate friction angles for rocks
using Hoek shear box

Inclined discontinuity surfaces
 If the shearing surfaces are inclined at an angle i to the direction of the
shearing stress, then the shearing resistance for displacements along the
inclined surface is given by:
 
i
b 
 f

 tan
Negative inclination Positive inclination

 The maximum value of the inclination of a surface for which there is still
a possibility of the upper half sliding under the action of a shear force (S):
For this reason, when rough surfaces have asperities inclined so that
the failure movements of the surfaces will occur together
with the failure of the asperities and not with the sliding along the surface.
When the inclination is negative, the upper half will slide when
0
tan
sin
cos 
 b
i
S
i
S f

90
and
1
tan
tan 


 i
i b f
f

90

i
b
f
b
i f


 Planar discontinuity surfaces
 Inclined discontinuity surfaces
 Multiple inclined discontinuity surfaces
 Ladanyi & Archambault criterion
 Rough discontinuity surfaces
 Barton criterion
 Scale effects
 Joint Roughness Coefficient measurements from
large scale index tests

Multiple inclined discontinuity surfaces
 Patton (1966) and Deeree et.al. (1967) : to closely study the influence of
the asperities and phenomenon of interlocking on the strength envelopes
 Horizontal surface containing a number of regular “teeth” with the same
size and shape.
 Each teeth having surface inclined at an angle i to the direction of applied
shearing force. The teeth had a constant internal strength identical to the
rock mass itself.
N
S

Shear strength envelopes for specimens
with different teeth inclination
Kaoline and rough
plaster surface (1:1)
Line A  i = 25
Line B  i = 35
Line C  i = 45
Line D  residual
strength of all three
series ( 1)
fb
frfb
frfb
fbi
Max shear strengths are
related to the frictional
resistance due to teeth
inclination

Shear strength envelopes for specimens
with different number of teeth
Kaoline and rough
plaster surface (1:1)
fbi
b
N
K
S f
tan


K : cohesive strength
K

Bi-linear relationship  2 modes of failure
 First Linear Tract :
- At low normal loads  maximum shearing strength related to frictional
resistance along the inclined surface = internal shearing resistance of the
teeth at failure point
- Displacements perpendicular to the shearing force direction (dilatants
behavior)
 Second Linear Tract :
- At high normal loads  maximum shearing strength unrelated to sliding
along the inclined surface
- Horizontal displacements occurred when the teeth sheared at their base.
Displacements perpendicular to the shearing force direction: very small
Bi-linear relationship is not obtained in natural joint shear tests  different
teeth superimposition types and complicated nature of asperity failure

Ladanyi & Archambault Criterion
 Two failure modes occur during shearing along an irregular surface:
shearing and sliding
 is the rough surface projected area portion where the
asperities are sheared off
 is the remaining portion of the projected area where sliding occurs
dx
dy
V 

A
As



Dilatation rate
Shear area ratio
 

 s
s
s A
A
A
s
A
A

Components of shear force mobilized for sliding:
 S1 : due to external work carried out in order to dilate against the external
normal force N
 S2 : due to additional friction dilatancy internal work
 S3 : due to the work of internal friction, if the specimen does not change
in volume during shearing



 V
N
i
N
dx
dy
N
S tan
1
b
b V
S
i
S
S f
f tan
tan
tan
2



b
N
S f
tan
3 
S
N
i
S
i
N
S
S
S b
b 




 f
f tan
tan
tan
tan
3
2
1
 
i
i
N
S
i
N
S b
b
b 



 f
f
f tan
tan
tan
tan
tan
 
i
b 
 f

 tan
The same as the result obtained by Patton (1966)

Components of shear force which occurs as a result of teeth shearing:
 S4 (assuming portion As of the teeth sheared off at the base):
K and fo : Coulomb parameters related to the strength of rock substance
 By adding all 4 components :
 When (flat surfaces and persistence lower than 100%):
o
s N
K
A
S f
tan
4 

b
s
s
b
n
b
s
n
V
a
a
K
V
a
A
S
f
f

f


tan
)
1
(
1
)
tan
(
)
tan
)(
1
(










0


V
)
tan
(
tan
)
1
( K
a
a o
n
s
b
s
n 


 f

f


Difficult to determine K and fo and taking into account that Mohr envelope
is an initially curve shape as results of different multiplies of asperity
heights and inclination  shears of at different stages

Use parabolic law (Fairhurst, 1964)  shear resistance of the material
„adjacent‟ to the discontinuity surfaces  :
5
.
0
1
1
)
1
(












j
n
c n
n
n




j : uniaxial compressive strength of rock material adjacent to the
discontinuity < uniaxial compressive strength of intact rock
n : ratio of uniaxial compressive c and uniaxial tensile t of intact rock
Hoek (1968)  hard rock  n = 10
Two extreme situations of strength envelopes:
1. Extremely low normal stress and no shearing of the asperities
2. Normal stress high enough to completely shear off asperities.

1. Extremely low normal stress and no shearing of the asperities
i
V
A
A
A
a s
s
s tan
0
0 




2. Normal stress high enough to completely shear off asperities.
0
and
1 


V
As
Approximate values of as and for extreme condition 0 < n < t :
i
a
L
j
n
s tan
1
1













i
V
K
j
n
tan
1














V
For rough surfaces, the empirical values found by Ladanyi & Archambault
based on large number of shear tests are K = 4 ; L = 1.5

Shear strength envelopes for the cases f = 30 and i = 10
(a) Fairhurst equation for rock material failure
(b) Ladanyi & Archambault criterion equation
(c) Patton equation
(d) Residual strength for slickenslide and planar surface equation
Bi-linear
envelope
Adherent to reality of physical
phenomenon  transition
zone due to progressive
shearing of asperities and
superimposition of teeth of
discontinuity

 Mechanical measurement
of roughness on
discontinuity surfaces
 Roughness contour
diagram
Rough discontinuity surfaces
In nature, discontinuity surface shape
is not regular but is almost random

Natural discontinuity shear strength as a function of several parameters:
 Applied normal stress or state of stress in general terms
 Wall roughness characteristics
 Strength and deformability of the asperities and of the wall
 Thickness type and physical properties of any filling material
 Initial contact area and distribution of apertures and contacts between
the walls
 Orientation of the shearing plane and direction of shear forces
 Discontinuity dimension with respect to shear direction and cross
direction
 In nature, discontinuity surface shape is not regular but is almost
random
1. Difficult to (1) evaluate these parameters and (2) to analytically formulate
a strength criterion equation which takes all parameters into account
2. Empirical approaches relate shear behavior observation to a limited
number of parameters which mainly govern the phenomenon.

Barton Criterion
 Barton criterion is empirical and able to predict and describe the peak
shear strength of rock discontinuities
 Advantage: the relative facility of determining the parameters which
governs the criterion equation.
 
 
r
n
n f


 
 /
JCS
log
JRC
tan 10
JRC: Joint Roughness Coefficient  a scale roughness factor and varies
within the range 0 and 20 increasing with wall surface roughness.
JCS: joint compressive strength  using Schmidt hammer
fr : residual friction angle
Components of shear strength of natural discontinuities:
1. Basic frictional component (fr)
2. Geometrical component controlled by surface roughness (JRC)
3. Asperity failure component controlled by the ratio (JCS/n)

Roughness classification and shear failure envelope for non planar joints
fr  constant
fr = 30
JRC = 20 JRC = 10

Roughness classification and shear failure envelope for non planar joints
JRC = 5
• Taking FS  not considering arc tan  /n > 70
or every possible intercept cohesion.
• Uniaxial compressive strength of joint wall JCS
strongly influences the shear strength of rough
joints

Peak shear resistance envelopes for
natural discontinuities (experimental data)
• Peak shear strength is less influenced
by JCS with smoother wall surface, as
the asperity failure is of an importance
which decreases with JRC value
• Joint strength depends on rock
mineralogy for smooth and slikenslide
planar surfaces

Scale Effects
 JRC and JCS/n are not independent of scale effect  important to
determine shear strength parameter measurements free of scale
effect and scale correction factor for scale-dependent parameters
 Bandis et al. (1981) examined the scale effects of shear behavior of
discontinuities by experimental  scale effect on peak displacement,
dilatancy value, JRC value, asperity failure, size and distribution of
contact area, limit size of specimen, ultimate shear resistance,
strongly jointed mass for different value of normal stress
 Scale dependence of the laboratory specimen size on the three
components of shear strength of natural discontinuities

Gradual increase of
peak displacement
Dependence of specimen size on 3 components of shear strength
A: Component due to asperity failure
B: Dilatancy component
C: Residual frictional component
D = A+B  contribution to shear resistance given by wall discontinuities roughness
E = A+B+C  peak resistance angle fp = fr + i
Brittle to plastic
Decrease of peak friction angle

Scale Effects
 Small blocks in a densely jointed mass may mobilize higher JRC
values than larger blocks in a mass with wider-space joints
 The scale effect on peak shear strength implies that there should be
a minimum size concerning the test specimen considered as
technically acceptable.
 Barton & Choubey (1977)  as a first approximation  natural block
size of the rock mass or more specially, the spacing of cross-joints

Shear Strength of Rocks: Principles, Testing Methods, and Engineering Applications

JRC measurement from large scale index tests
 Tilt, pull and push test  very cheap method of assessing JRC 
Large scale index test (Bandis, et al., 1981)
Insitu tilt test  In tilt test, JRC :
ns
r

f

JCS
log
JRC
10


 : tilt angle : upper half slides on lower half
ns : normal stress when sliding
Tilt test on a block: base of
block A-A parallel to mean
reference line M-M

Insitu pull test
 In pull test, JRC :





 




N
A
N
T
T
r
JCS
log
tan
JRC
10
2
1
1
f
T1 : tangential component of weight of
overlying block
T2 : external puling force
N : normal component of block weight (W)
A : joint area
In push test T2 = pushing force applied by
means of a flat jack inserted between the
walls of two adjacent blocks opened with a
drilled line
JCS can be estimated by Schmidt Hammer

 Roughness surface slope angle dependence on joint length L is shown in
tilt test by means of a modification of Patton (1966) law:
 By referring to Barton criterion:
 The reduced tilt angle  may be attributed to an effective reduction in i
and to a joint roughness reduction with an increase in length
 Size of this scale effect for a tilt test  empirical formulas (next figures):
 Patton law
 When L = L0
))
(
tan( L
i
r
n 
 f

 )
/
JCS
(
log
JRC
)
( 10 n
L
i 

  0
JRC
02
.
0
0
0 /
JRC
JRC

 L
Ln
  0
JRC
03
.
0
0
0 /
JCS
JCS

 L
Ln
Ln and L0 : length referring to in situ
scale to laboratory scale,
respectively
   
0
0 JRC
03
.
0
0
0
10
JRC
02
.
0
0
0 )
/
log(
)
/
JCS
(
log
/
JRC 


 L
L
L
L
i n

)
/
JCS
(
log
JRC 0
10
0
0 n
i 

 
 
r
n
n f


 
 /
JCS
log
JRC
tan 10

Laboratory tilt test
Different sizes of rock specimens used
to assess discontinuity shear strength
scale effect by means tilt tests

 In tilt test  normal stress acting on joint when sliding occurs:
 In the hypothesis: a block slides at 66 tilt angle; unit weight of 25 kN/m3
and height 0.1 m
 For JCS = 100 MPa 
 When L = L0 
 The surface slope of angle i decreases with an increase of L, as fr is not
considered scale dependent
A
W /
cos
i
r 
f

    5
/
)
/
(
log
5
/
/ 0
0 JRC
03
.
0
0
10
JRC
02
.
0
0
0



 L
L
L
L
i
i
0
0 JRC
5

i
MPa
001
.
0
cos
cos

 


h
A
W
0
0 i
r 
f

  0
0
0
0 /
/
1
/
/ i
i
i
i
r 

 
f


)
/
JCS
(
log
JRC 0
10
0
0 n
i 


 For all lengths it has been assumed that:
1. The value of i, for a given normal load, is a single value
2. Reference line M-M defining shear plane always remains parallel to
measured surface A-A
 Prediction of tilt angle is based on triangular roughness representation
and the scale effect is evaluated by assuming a roughness median line,
constant in inclination (horizontal) for each joint length.
Influence of JRC on the slope stability
L0 = 10 cm
JCS0 = 50 MPa
0 = 61
fb = 30 ; r/R = 0.75
ns = 0.00126 MPa
    


25
20
20 


 R
r
b
r f
f

Influence of JRC on the slope stability
8
.
7
JCS
log
JRC
0
10
0
0 


ns
r

f
 The scale effect on the length of the joint:
(L = 2 m)
  88
.
4
/
JRC
JRC 0
JRC
02
.
0
0
0 


L
Ln
  MPa
80
.
24
/
JCS
JCS 0
JRC
03
.
0
0
0 


L
Ln
 
  36
.
1
Wsin
)
/
JCS
log
JRC
tan(
cos 10




f

 r
n
W
F
Safety factor of block sliding (n = 0.052 MPa) :
F without taking into account the scale factor and by using JRC = 7.8 and
JCS = 50 MPa  F = 1.94, which is 42% difference with respect to scale
corrected safety factor
As joint profiles are rougher, the scale effect increases. If L/L0 = 20 and
JRC0 = 20  JRC = 6  three times lower than 20
))
(
tan( L
i
r
n 
 f



 Statistical methods for JRC determination and shear
behavior prediction
 Fractal characterization of joint surface roughness for
estimating shear strength
 Geostatistical operators applied to the rock joint shear
strength prediction
 Influence of the wall discontinuity interlock level on the
shear resistance
 Filled discontinuities
 Discontinuity shear behavior under dynamic conditions

Shear Strength of Rocks: Principles, Testing Methods, and Engineering Applications

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