SHEAR STRENGTH OF GEOMATERIALS - A BRIEF HISTORICAL PERSPECTIVE

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SHEAR STRENGTH OF GEOMATERIALS - A BRIEF
HISTORICAL PERSPECTIVE


Richard H.G. Parry
1




ABSTRACT

Although the shear strength expression proposed by Coulomb (1776) for masonry, brick and earth, made
up of cohesion and friction components, is still in general use today, it was not until Terzaghi (1936)
presented the principle of effective stress that it could be used with confidence in most soils and soft or
jointed rocks. The Griffith (1921, 1924) crack theory, while inadequate in itself, has provided a starting
point for developing empirical expressions for the shear strength of hard intact rocks. Many factors
influence the shear strength of soils and rocks, such as drainage conditions, rate of strain or shear
displacement, degree of saturation, stress path, anisotropy, fissures and joints. A brief review is attempted
here of the more salient advances which have been made in understanding and quantifying the influence of
these various factors.



1.0 INTRODUCTION

Throughout the five thousand years or more that humans have been concerned with works of construction
some assessment of rock and soil strengths have had to be made, at least by responsible builders, to ensure
that they were adequate to serve the purpose, either in their natural state or as constructional materials in
their own right. Until the 20th Century such assessments were necessarily qualitative, based mostly on
experience or rule of thumb. The problem is that soil and rock come in an infinite number of guises, and
while experience and rule of thumb may often have sufficed in the past, there were no doubt many instances,
mostly long forgotten, when they proved inadequate and even misleading, leading to failures. The advances
in understanding the strength of engineering materials which occurred in the 19th Century did not extend to
soils and rock masses. A fundamental understanding of the strength of soils and soft rocks had to await an
understanding of effective stress, first expressed formally by Terzaghi in 1936 and for which he is justifiably
regarded as the Father of Soil Mechanics. A different, more empirical approach has been required in the
case of hard rocks for which it is recognised that their strength is dominated by the presence of
imperfections.

2.0 COULOMB 1776

In 1773 Coulomb read his paper to the French Academy of Sciences which, after being refereed by his
peers, was published by the Academy in 1776. He proposed an expression for the shear resistance of
masonry, brick and earth of the form:

s = ca + (1/n) N (1)

in which he visualised the frictional component to conform to Amonton's Law with N as the normal force
and 1/n the coefficient of friction, and the cohesion c to be the resistance of solid bodies to the simple
separation of their parts, giving a total cohesive or adhesive force of ca where a is the area of rupture. He
related cohesion to tensile strength and showed by experiment for a "white stone" that the shear fracture
force directed along the plane of rupture was slightly greater than the tensile fracture force perpendicular to
the rupture plane, but the difference was very small and he chose to neglect it.
___________________


1
Cambridge, UK.

In modern geotechnical terms the Coulomb equation is written in the form:

τ
f
= c + σ
n
tanφ (2)

where τ
f
is the shear strength per unit area, c is the unit cohesion, σ
n
is the normal compressive stress on the
shear plane and φ is the angle of shearing resistance. The values of c and φ for a soil depend on many factors
including stress path, stress level, drainage conditions and strain rate. Although widely used for soils, the
Coulomb equation is less used in rock mechanics as strength envelopes often show strong curvature.
Coulomb applied the shear strength parameters to studying the strength of an axially loaded vertical
masonry pier and to earth pressures on retaining walls. He examined first a masonry pier with cohesive
strength only and showed correctly that the least strength was given by a rupture plane at 45° to the pier axis.
He repeated his analysis for a masonry pier having both cohesive and frictional strength and showed that the
minimum strength was given by a rupture plane with angle x to the horizontal, where x is given by:

tan x = [(1 + n
-2
)
0..5
- n
-1
)]
-1
(3)

It is now usual to use φ rather than n, giving the much simpler expression:

x = (45° + φ /2) (4)

In analysing earth pressure on a vertical retaining wall Coulomb first considered the stability of a triangle
of earth behind the wall, deducing correctly that the geometry of the triangle giving the maximum pressure
was determined by the friction alone and was not influenced by the cohesion. Assuming the soil to have
both cohesion and friction, and assuming zero friction between the wall and the soil, he obtained the
expression for the maximum horizontal force on unit length of the retaining wall:

P
H
= mh
2
- clh (5)

where h equalled the height of the earth behind wall and, in modern terms, putting γ equal to the unit weight
of earth:


m = 0.5γ tan
2
(45 - φ /2) (6)

l = 2tan(45 - φ /2) (7)

In an example illustrating the use of his expression Coulomb assumed a coefficient of friction of unity,
which he considered to be the angle of repose and he took cohesion to be zero for "newly-turned soils
(Heyman 1972 translation). This presumably included any reworked soil.

3.0 CULMANN 1866, MOHR 1882.

Clear graphical representation of stress and shear strength has played an important role in their
understanding in relation to soils and rocks. Culmann (1866) first used the stress circle in considering
longitudinal and vertical stresses in horizontal beams during bending. His exploitation of the stress circle
included the establishment of a point on the circle, now known as the pole point, allowing the stresses on a
plane at any specified inclination to be found by drawing a line through this point parallel to this plane. Such
a line met the circle again at the required stress point. This gave him a simple means for plotting principal
stress trajectories for a beam.
Aware of the fact that the currently favoured failure criterion, based on the maximum strain criterion of
Saint-Venant, did not give good agreement with experiments on steel specimens, Mohr (1882) promoted the
use of a failure criterion based on limiting shear resistance, and proposed that stress circles should be drawn
to give a full understanding of stress conditions at failure. As an illustration of the shear stress criterion,
Mohr used the example of cast iron tested to failure in compression, in tension and in pure shear. Drawing
failure envelopes just touching the compression and tension circles, he showed that the failure stress in shear
was given by the radius of a circle with its centre at the origin of stresses and just touching the two
envelopes. These values agreed satisfactorily with experimental observations.
Although the motivations of Coulomb and Mohr towards developing a failure criterion were very
different and on different materials, the end point was much the same: a stress dependent criterion based on
shear resistance, familiar to geotechnical engineers as the Mohr-Coulomb criterion.

4.0 ANGLE OF REPOSE

Heyman (1972) cites a number of workers in the 18
th
and 19
th
centuries who appear to have been
primarily concerned with setting up classical solutions for the thrust of soils as an intellectual exercise, rather
than with any practical application. Although some included cohesion in their analyses, the bulk of the work
avoided the difficulties this introduced, assuming cohesionless behaviour and a friction angle equal to the
angle of repose. This clearly led to difficulties in interpreting the behaviour of clays which were known to
be able to stand on steep, or even vertical, faces.
In his memorandum on Landslides in Clays, Collin (1846) recognised that soils may have both frictional
and cohesive (sometimes referred to as adhesive) components of strength as evidenced in the passage:

"The force of cohesion of clay plays an important role in the equilibrium of soils although so far the
geometricians who have established the proper formulas to determine the conditions of the equilibrium of
clayey soils have neglected this factor, which one might just as well neglect as gravity or friction".

Collin himself apparently considered that friction played little part in the strength of clays, with remarks such
as "... the mathematical formula for stability must include two forces, one of constant gravity and the other of
cohesion...". He warned that this cohesive strength could be greatly diminished by the infiltration of water,
by freezing and thawing and other factors, or could simply diminish with time.
Collin carried out tests on long 40 mm square-section specimens of clay compacted to different
consistencies (presumably different moisture contents, although these are not given) in which a central
section was pulled out at right angles to the axis of the specimen thus shearing on two parallel surfaces 50
mm apart. Collin's memorandum was concerned with the reality and shape of slip surfaces and he made no
attempt to use measured shear strengths to analyse the slips. His purpose in carrying out the tests was to
show the influence of soil consistency (ie moisture content) and time or loading rate effects on the cohesive
strength.
Rankine (1876) coped with the problem of cohesion in clays by stating that "...the adhesion of earth is
gradually destroyed by the action of air and moisture, and of changes in the weather, especially by alternate
frost and thaw; sothat friction is the only force that can be relied upon to produce permanent stability". He
considered the adhesion, however, to be "...useful in providing temporary stability in the execution of
earthworks, enabling the side of a cutting to stand for a time with a vertical face for a certain depth below its
upper edge". This still led him to incorrect assumptions, such as stating the angle of repose for a "damp
clay" to be 45°.
In discussing the stability of rock cuttings, Rankine stated that the permanence of cohesion in firm, sound
rock, such as igneous and metamorphic rocks without fissures, may be depended upon and cuttings made
vertical or nearly so. He warned, however, that sedimentary rocks containing clay, such as shale, even if
hard when cut may soften with time; and that sandstones and limestones could have variable strengths. He
also observed that sedimentary rocks in the side of a cutting were more stable when the beds were horizontal
or dipping away from the cutting, than when they dipped into the cutting.
The lack of understanding of clay strength and the anomalous, and often incorrect, results obtained by
applying formulae based on an angle of repose led practising engineers to rely on their judgement and
experience when dealing with this material. This led to frequent failures in construction works. In an
attempt to clarify the relationship between normal pressure and shear strength in clays, Bell (1915)
constructed an apparatus consisting of a brass cylinder 3 inches (76 mm) internal diameter, which could be
filled to a depth of about 5 inches (127 mm) with clay and loaded axially by means of a plunger. The mid-
portion of the cylinder consisted of a brass ring which could be translated laterally by a measured force, thus
shearing the cylinder of soil on two parallel faces 3 inches in diameter. Tests were made under different
axial stresses and, as the tests were essentially undrained (and for reasons now clearly understood), his plots
of shear resistance against normal pressure showed substantial zero intercepts and low angles of inclination,
generally less than 6.5°. Higher values were obtained with stiff sandy clay and dry very stiff boulder clays,
probably as a result of unsaturation or cavitation in the clay. While these tests confirmed that it was not
appropriate to apply an angle of repose to clay, they did not clarify the true behaviour of clay and the role of
pore water pressure in the soil.

5.0 DILATANCY AND THE SHEAR STRENGTH OF SAND

Citing earlier experiments by Osborne Reynolds (1885), Casagrande (1936) observed that during shear
deformation dense sand expanded and very loose sand reduced in volume. Under large shear deformation a
unique critical voids ratio was reached, and in the case of dense sand the applied shear stress reached a peak
before dropping to a lower ultimate value at the critical voids ratio. Casagrande also realised that the critical
voids ratio depended upon the static pressure applied to the soil. Taylor (1948) introduced the concept that
the strength of dense sand consisted in part of internal rolling and sliding friction between grains and in part
of interlocking. In the shear box he proposed the interlocking portion s
e
of the strength to be given by:

s
e
.A.Δh = σ.A.Δt (8)

where A is the area of the test specimen, σ is the applied vertical pressure, Δh is an incremental horizontal
shear displacement at failure and Δt the corresponding increase in thickness of the specimen. For a sample
of dense Ottawa sand tested in the shear box he found the interlocking portion to make up 26 per cent of the
total shear strength.
Rowe (1962,1969) proposed a dilatancy rate D for granular materials such as sand given by:

D = ( 1 - dv/dε
a
) (9)

where v is volume decrease per unit volume and ε
a
is the major principal strain in a compression test and
minor principal strain in an extension test. Applying minimum energy principles Rowe derived an expression
for principal stress ratio σ′
1
/σ′
3
at failure:



σ′
1
/σ′
3
=

( 1 - dv/dε
a
) tan
2
(45 + φ′
f
/2) (10)

The friction parameter φ′
f
depends on relative density, pressure and stress path and has a magnitude in the
range

φ′
u
< φ′
f
< φ′
cs
, where φ′
u
is individual particle slip friction and φ′
cs
is the frictional strength at critical
state. At the densest state up to peak φ′
f
=

φ′
u
and at its loosest state (critical state where dv/dε
a
= 0),
φ′
f
= φ′
cs
.
Rowe introduced the theoretical finding by Horne (1965) that the maximum value of ( 1 - dv/dε
a
) is 2 (ie
dv = -dε
a
and thus dv/dε
a
= -1), giving 1≤ ( 1 - dv/dε
a
) ≤ 2. The upper limit for sands in triaxial compression
becomes:

σ′
1
/σ′
3
= 2 tan
2
(45 +φ′
u
/2) (11a)

and the lower limit (critical state value) for sands in triaxial compression is:

σ′
1
/σ′
3
= tan
2
(45 + φ′
cs
/2) (11b)

In plane strain, φ
f
= φ
cs
for any packing, so the upper limit for sands in plane strain becomes:

σ′
1
/σ′
3
= 2 tan
2
(45 + φ′
cs
/2) (12a)

and the lower limit for sands in plane strain becomes:

σ′
1
/σ′
3
= tan
2
(45 + φ′
cs
/2) (12b)

Rowe observed that the limiting dilatancy rate of 2 was not necessarily reached by dense packings.

In order to illustrate the influence of dilatancy rates Rowe considered the case of Mersey River quartz
sand for which φ
u
= 26° and φ
cs
= 32°. Applying Equations 11a and 11b for triaxial compression:

(σ′
1
/σ′
3
)
max
= 5.1, thus φ′
max
= 42°

and (φ′
max
- φ′
min
) = (φ′
max
- φ′
cs
) = 10°.

Applying Equations 12a and 12b for plane strain φ′
max
= 47°, (φ′
max
- φ′
cs
) = 15° for Mersey River sand.
After analysing results from 17 different sands tested in 12 different laboratories Bolton (1986) proposed
the use of a dilatancy index I
R
, given by the best fit relation ship:

I
R
= I
D
(10 - ln p′) - 1 (13)

The experimental data yielded the following best fit relationships (Figure 1):

Triaxial compression: (φ′ - φ′
cs
) = 3I
R
(in degrees) (14a)

Plane strain: (φ′ - φ′
cs
) = 5I
R
(in degrees) (14b)

All conditions: -(dε
v
/dε
1
)
max
= 0.3I
R
(14c)

It can be seen in Figure 1 that (φ′
max
- φ′
cs
) for triaxial compression (given by I
D
= 1) is 10° and 16° for plane
strain, which correspond closely to the values derived by Rowe for Mersey quartz sand.

6.0 EFFECTIVE STRESS

After studying many case records of earth dam and foundation failures in Europe and the USA, Terzaghi
recognised the inadequacies of existing soil classifications and put in hand an experimental programme
lasting 7 years. This culminated in 1925 in his book Erdbaumechanik and a series of articles in Engineering
News Record. His strength tests consisted of compressing 20 mm and 40 mm cubes of soil at various
moisture contents. His realisation in these articles of the importance of pore water pressure in determining
soil behaviour led to his formal statement in 1936 of the principle of effective stress, in which he states that
the effective principal stress is the difference between the total principal stress and the neutral stress (now
called the pore water pressure).
In his 1925 articles Terzaghi stated that frictional resistance in soils was produced by the difference
between the externally applied stress and the hydrostatic excess. He also stated that when load was applied
to a clay a large part was taken by the pore water and hence there was no large additional frictional
resistance, and furthermore if a clay was not in hydrostatic equilibrium it could demonstrate an angle of
friction anywhere between zero and the normal value.
He appears to have been much
less clear about cohesion in clays
and some of his comments are
confusing. For the most part he
refered to "cohesion" or "apparent
cohesion", as the internal
resistance arising from pressure
exerted by the surface tension of
the capillary water. In modern
terms this is the unconfined
(undrained) strength of a
saturated clay. In a rather
obscure comment he contrasts
"apparent cohesion" with "true
cohesion" produced by the
internal friction.
In an unpublished MIT Report
(1930) Casagrande and Albert
concluded from a series of direct
shear tests on Boston blue clay
that consolidation can take place
during a shear test and the
measured strength depended upon the rate of load application. In the first systematic investigation of shear
strength using the triaxial test, Jurgenson (1934) showed shear strength for remoulded Boston blue clay to be
independent of total normal stress if no consolidation was allowed; and he also concluded that the higher
resistances obtained in slow tests were caused by partial consolidation of the soil.. He concluded too, as did
Casagrande and Albert, that previous history influenced soil strength. In a discussion on Jurgenson's paper,
Casagrande (1934) distinguished, perhaps for the first time, the relationship between the Mohr-Coulomb
strength envelopes for quick (undrained) tests and slow (drained) tests (Fig.2).

7.0 COHESION AND FRICTION IN CLAYS

With the principle of effective stress firmly established (Terzaghi 1936) researchers directed their
attention to separating the true cohesion and true friction components of the shear strength of clays. A
proposal by Tiedeman (1937) that the true cohesion was proportional to the maximum consolidation pressure
was criticised by Hvorslev (1937) on the rather flimsy basis that it gave friction angles greater than indicated
by the inclination of observed failure planes, and on the stronger evidence that all samples under the same
normal stress, and having the same maximum consolidation stress history, would have equal strengths.
Observations did not support this. Hvorslev also pointed out that this criterion also ignored volume changes
during shear.
On the basis of shear box tests on Wiener Tegel and Klein Belt clays, Hvorslev (ibid) proposed a true
cohesion component of strength c
e
depending only on water content at failure, and he related this cohesion to
the "equivalent pressure p
e
", that is the value of pressure for that water content given by the unique
relationship between water content and consolidation pressure for the soil when normally consolidated,
giving the shear strength expression:
τ
f
= c
e
+ (σ
n
– u) tanφ
e
= κ
_
p
e
+ σ′
n
tanφ
e
(15)

where κ is a constant, u is pore pressure at failure and φ
e

is the true friction angle. Dividing Equation 15
throughout by p
e
gives the advantage of a non-dimensional plot of τ
f
/ p
e
vs σ′
n
/ p
e
with abscissa κ and slope
φ
e
. Terzaghi (1938) supported these findings in a paper on the Coulomb equation.
An outstanding contribution to the
understanding and usage of the shear
strength parameters for clays was made
by Skempton in his study of a slip in
the soft clays of the west bank of the
Eau Brink Cut near King's Lynn in
Norfolk (Skempton 1943). Skempton
performed both total stress and
effective stress slip circle stability
calculations and showed that, although
both gave factors of safety close to
unity, the critical slip circles differed in
location as shown in Figure 3, with the
effective stress circle approximating
more closely to the estimated location
of the actual slip surface than the total
stress circle. Referring to the undrained
shear strength as the "existing shear
strength", Skempton measured this on
sealed specimens in a triaxial cell
(together with some confirmatory tests
in direct shear) and showed it to be
essentially independent of applied cell
pressure as already demonstrated by
other workers. He measured the
effective stress cohesion and angle of
friction in drained direct shear box
tests, but pointed out that while it was
appropriate to use these in analysis they
were not fundamental strength
parameters. The fundamental Hvorslev
parameters were difficult to determine
in the laboratory and unsuitable to use
in the field with existing analytical
methods. Skempton also showed that
the measured inclination of failure
planes in undrained triaxial tests was
determined by the true angle of internal
friction and not by the (zero) angle of
shearing resistance measured in the undrained tests, which explained why the critical slip circle given by the
total stress analysis of the field slips differed considerably in location from the critical circle given by the
effective stress analysis and from the actual slip surface.
In a supplement to a report on the influence of strain rates Taylor (1944) discussed various other factors
which would modify the shear strength expression for clays, the basic form of which he gave for a normally
consolidated clay as:

s = σ′ tanφ (16)

where σ ′ is the intergranular pressure normal to the failure surface and φ is the true friction angle. For
overconsolidated clays (Taylor used the term precompressed) he introduced the concept of intrinsic stress p
i

arising from the observation that overconsolidated specimens in the shear box at failure under drained
conditions had a voids ratio less than normally consolidated specimens of the same clay under the same
applied pressure. The intrinsic pressure was taken to be the pressure increment along the failure line for the
normally consolidated clay on an e - σ ′

plot corresponding to the voids ratio difference. This gave the shear
strength expression:

s = (p
i
+ σ′ )tanφ (17)

Taylor further suggested the possibility of including a term for soil density in the strength equation which,
together with p
i
tanφ, comprised the Hvorslev true cohesion term c
e
.
For consolidated undrained tests on normally consolidated clays Taylor gave the expression for shear
strength:

s = σ
c
tanφ
a
(18)


where σ
c
is the consolidation pressure and φ
a

is an apparent friction angle. He illustrated
the differences between drained and
consolidated undrained strength envelopes by
means of the diagram in Figure 4, the
portions of the envelopes deviating from a
straight line through the origin giving the
strengths for overconsolidated specimens.
Taylor drew attention to a number of points
with respect to this diagram, notably that for
normally consolidated specimens the
consolidated undrained strengths were about
half the drained strengths for the same
consolidation pressure, the envelopes
crossing over where the drained test
underwent no volume change, and at lower
pressures than this the consolidated
undrained strengths could be much higher than drained strengths because of expansion of the drained
specimens during shear (and the development of high negative pore pressures in the undrained tests,
although Taylor does not state this). Taylor also commented on the relative flatness of the consolidated
undrained envelope for overconsolidated specimens (again reflecting the development of high negative pore
pressures during shear).
The Triaxial Shear Report prepared by Rutledge (1947) contained the results of a large number of triaxial
tests carried out between 1940 and 1944 in three American laboratories. Observations from the tests on
undisturbed, saturated clays led to the American Working Hypothesis, that for normally consolidated clay
the strength at failure depended only on water content and that water content itself was a function solely of
major principal effective stress for clays of the type tested (Chicago and Massena clays). Similar findings
were reported by Bjerrum (1950).
By relating strains to effective stress changes during shear and assuming zero volume change, Skempton
(1948) derived for undrained triaxial compression tests on saturated clays the expression:

Δu = A(Δσ
1
- Δσ
3
) + Δσ
3
(19a)

where Δu is the pore pressure increment caused by the applied major and minor total principal stress
increments Δσ
1
and Δσ
3
respectively, and A is a constant. In a subsequent paper Skempton (1953) modified
this expression to include the constant B for soils which were not fully saturated, thus:

Δu = B[Δσ
3
+ A(Δσ
1
- Δσ
3
)] (19b)

where B is the relationship between pore pressure change and an applied isotropic total stress change Δσ
3
.
Skempton showed that A = 1/3 for a soil behaving as an isotropic elastic material. He also listed values of A
at failure ranging from -1/2 to 0 for heavily overconsolidated clays up to +1/2 to +1 for normally
consolidated clays (up to +1.5 for clays of high sensitivity).
The earlier work of Taylor on energy expended at failure was taken up for clays by Gibson (1953), who
postulated that the Hvorslev "true friction angle" comprised the sum of the resistance arising from frictional
interaction between grains and a strength related to the rate of volume change of the test specimen. He
proposed a modified form of the Hvorslev expression for drained shear box tests:

τ
f
= c
R
+ σ′
n
tanφ′
R
+ σ′
n
g
f
(20)

whereσ′
n
g
f
is the work expended at failure to expand the specimen against the applied stress σ′
n
. He found
φ′
R
to have a value ranging from 3° to 26° for a number of clays tested in the shear box. Cylindrical
specimens of each soil, 76 mm in height and 38 mm diameter submitted to unconfined axial compression at
constant strain rate, showed the angles of failure planes as they developed to reflect good agreement with the
true angle of friction φ′
R
(identical to φ′
e
for undrained conditions).
A basic weakness in the attempts by Hvorslev and others in the early studies was the fact that the peak
shear strength, while important to engineers, was a transient point on the stress strain curve which varied
with a number of factors such as applied stress path and boundary conditions. It was not a fundamental state
of the soil. A full understanding of soil strength needed the establishment of a fundamental "shear state" in
the soil as a platform from which to view any other state during shear including, and most importantly, the
condition of failure, usually taken to be the point of maximum deviator stress. With interest concentrated on
peak strength, little attention was given to post-peak behaviour and to the fact that under sustained shearing a
specimen might be expected to achieve an ultimate "steady state" condition. One problem discouraging such
studies was the often considerable distortion of the test specimen in the post-peak phase or even in reaching
peak strength.
In order to understand the influence of factors such as overconsolidation ratio and stress path on the
behaviour of clays a basic reference state in shear was required and this was provided by the critical state
concept (Roscoe, Schofield and Wroth 1958, Parry 1958), although not known by this name when first
proposed. The basis of the critical state concept is that under sustained uniform shearing, a unique condition
of e vs p′
cs
and q′
cs
vs p′
cs
is achieved

as shown in Figure 5, regardless of the test type and initial stress history
of the soil, where e is voids ratio, p′
cs
is mean effective stress and q′
cs
is deviator stress at critical state. In
Figure 5 the paths AB, FB are undrained tests, ACD, FGH are consolidated undrained tests and ACE, FGJ
are consolidated drained compression tests.
The critical state strength expression is usually written:

q
cs
= Mp′
cs

(21)



where M is a constant for a particular soil.

Combining equation 21 with the Mohr-Coulomb failure
expression (assuming c′ =0) gives for:

Triaxial compression: M = (6sinφ′
cs
)/(3 - sinφ′
cs
) (22a)
Triaxial extension: M = (6sinφ′
cs
)/(3 + sinφ′
cs
) (22b)

The ratio of critical state deviator stress in extension (E) to compression (C) is thus given by:

q
cs
(E)/q
cs
(C) = (3 - sinφ′
cs
)/(3 + sinφ′
cs
) (23)

Equation (23) gives undrained shear strength ratios reducing from 0.8 for φ′
cs
= 20° to 0.68 for φ′
cs
= 35°.
These ratios accord well with peak strength ratios observed by many experimentalists, commonly in the
range 0.6 to 0.9, confirming the strong influence exerted by the basic critical state condition even where a
peak strength is experienced before the soil reaches critical state. Theoretical plane strain strengths lie
intermediate between triaxial compression and extension values which is also in accord with observations.
Most natural soils and even laboratory reconstituted soils do not deform uniformly during shearing owing
to a lack of uniformity in the soil structure and the manner in which the boundary forces are applied to the
test specimen. This obviates the possibility of the soil reaching a true critical state. In the field, too, failure
and the mode of failure when it occurs is likely to be governed by specific weaknesses in the soil structure.



Thus, the adoption of strength parameters for design or analysis becomes a matter of judgement, based on
experience, a sound knowledge of the site conditions and selection of the most relevant laboratory or field
tests to determine strength parameters. A particular case is residual strength (Skempton 1964) where large
displacements can occur in the field along a well defined failure surface, which, in some cases, may be a pre-
existing failure surface. It is considered to have particular relevance in stiff heavily overconsolidated clays.
Residual strength parameters cannot be determined in the triaxial test and require the use of a ring shear or
several reversals of a shear box.
The influence of soil structure in natural sedimented clays was examined by Burland (1990) in the light of
the soil in a reconstituted state, the strength and compressibility for which he refers to as the "intrinsic"
properties. In order to compare natural and reconstituted states of a soil, he introduces a normalising
parameter I
v
, termed the "void index", such that:

I
v
= (e - e∗
100
)/ (e∗
100
- e∗
1000
) (24a)

= (e - e∗
100
)/C
c
*
(24b)

where e is voids ratio, e∗
100
and e∗
1000
are voids ratios for the reconstituted soil under consolidation pressures
of 100 kPa and 1000 kPa respectively.


In Figure 6 values of I
vo
for a number of natural sedimented clays (e = e
0
) are plotted against effective
overburden pressure σ′
vo
and the best fit regression line through these points termed the sedimentation
compression line (SCL). The separation of the SCL from the intrinsic compression line (ICL) is a measure of
the enhanced resistance of a naturally deposited clay over that in its reconstituted state. Over the region
shown in Figure 6 the effective overburden carried by the reconstituted clay is approximately five times that
carried by the equivalent reconstituted clay and would be much greater for highly sensitive and quick clays.
Plotting shear strength against normal effective stress for two heavily overconsolidated natural clays, low
plasticity Todi clay and high plasticity London clay, Burland shows the intact strength envelope to lie above
the intrinsic strength envelope as expected. Normalising the applied stresses by dividing by the Hvorslev
equivalent stress he shows the intact drained and undrained surfaces to again lie above the intrinsic Hvorslev
surface. He also noted for these two clays that their strengths dropped quickly after peak to achieve a
reasonably stable value after only a few millimetres further displacement, and he terms this the "post-rupture
strength", as opposed to the residual strength which will only be achieved after much greater displacements.
The post-rupture envelope lies close to the intrinsic critical state envelope at low stresses and below it at high
stresses.
Despite the great attention which has been given to the effective strength shear strength parameters c′ and
φ′ for clays, and the many papers written on the subject, there are still uncertainties about the basic nature of
these sources of strength and the appropriate values to adopt for design and analysis. This applies
particularly to c′, as a measured φ′ value, although it may be stress path and stress magnitude dependent, is
usually regarded as reliable, providing the appropriate test is performed to evaluate it. It is much more
difficult to decide on the right value of c′ to adopt. It is embodied in Critical State theory that c′ = 0 and
although in practice most natural clays, particularly if initially moderately to heavily overconsolidated (i.e.
on the "dry" side of critical state), do not achieve a true critical state, it is common for c′ = 0 to apply at large
strain or shear displacement (e. g. at the residual state).
Effective stress strength envelopes for peak strength conditions, particularly for drained tests on
moderately to heavily overconsolidated clays, are likely to exhibit a zero stress cohesive intercept. This may
simply arise from the linear extension of a portion of a curved envelope, or may have a specific physical
meaning. In most cases the overriding contribution will come from interlocking of the densely packed
particles leading to dilation, but some contribution to the cohesion intercept may come from interparticle
bonding, at least in natural undisturbed clays.
True cohesion has been attributed by various writers to interparticle electrical forces (e.g. Lambe 1953,
Rosenquist 1955) and interparticle bonding through the coalesence of modified "rigid" water layers
surrounding the particles (Hvorslev 1937, Terzaghi 1941, Grim 1948, Haefeli 1951). The possible existence
of some form of interparticle attraction or bonding is supported by the fact that effective stress tensile
strengths have been measured in natural heavily overconsolidated London Clay (Bishop and Garga 1969)
and in a natural soft lightly overconsolidated clay (Parry and Nadarajah 1974). In the latter case effective
stress tensile strengths up to 8 kPa were measured in undrained triaxial compression and extension tests,
while effective stress cohesion intercepts ranged from 3 kPa to 11kPa for specimens at different orientations.
These relative magnitudes are of interest in view of Coulomb's assumption of tensile and cohesive strengths
having essentially the same magnitude. Whatever the origin of the effective stress cohesion intercept, it
remains a major uncertainty what value should be used in design in any particular circumstance. While
putting c′ = 0 is common, and no doubt often justified, it will be overconservative in some instances.

8.0 INFLUENCE OF STRAIN RATE, ANISOTROPY AND STRUCTURE ON SOIL STRENGTH


A study by Taylor (1944, 1948) of the
influence of strain rate on measured
undrained shear strengths showed that
for Boston blue clay tested in the triaxial
cell the strength increased by about 6 per
cent for every tenfold increase in axial
compression rate from about 0.0003%
per minute up to 1.0% per minute.
Some early studies of strain rate effects
on clays were made by Casagrande and
Shannon (1948, 1949) and Casagrande
and Wilson (1951). Unconfined and
undrained triaxial tests were performed
with failure times ranging from 0.015
seconds to 30 days. The reports by
Casagrande and Shannon describe, in
general, tests performed at rates faster than normal laboratory rates, while the report by Casagrande and
Wilson describes creep tests at rates slower than normal laboratory rates. Tests were also performed at
normal laboratory rates (2 to 10 minutes to failure) for comparison purposes. Quick undrained tests on a
wide range of natural clays and on reconstituted kaolin mostly showed linear increases in strength with
reducing log-time to failure, with a few showing increases at an increasing rate. The increases ranged from
1.07 to 1.44 for one (order of magnitude) time cycle and 1.38 to 2.20 for four time cycles. There appeared to
be no correlation between these rates and the plasticity of the soil. In the slow tests the strength decreases
with increasing time to failure were fairly constant, ranging from 4 percent to 10 percent per time-cycle over
four time cycles.
A series of fast tests on a number of clays, mostly on unconfined specimens, were performed by Whitman
(1957) with failure times ranging from 5 minutes to 0.005 seconds. All clays showed linear increases in
strength, ranging from 3% up to 15% per time-cycle up to three reducing time-cycles, with sharper increases
in the fourth and fifth time-cycles. Similar behaviour as shown in Figure 7 was reported by Richardson and
Whitman (1963) for undrained triaxial compression tests on reconstituted, overconsolidated Buckshot clay
(w
L
= 62, w
P
= 24).
Slow triaxial compression tests performed on undisturbed specimens of Fornebu clay (w
L
= 36-59, w
P
=
15-34) by Bjerrum, Simons and Torblaa (1958), consolidated under isotropic pressures up to 400 kPa,
showed a marked drop of 40% in strength over the first time-cycle from 10 minutes to 100 minutes to failure,
with subsequent drops of only about 6% per time-cycle up to 500 hours to failure as shown in Figure 8.
Although it has long been recognised that all soils are anisotropic in their behaviour either as a result of
their stress history and applied stress during shear, or fabric anisotropy built in during their formation, little
or no account is usually taken of this in design and analysis and basic textbooks on soil mechanics give it
scant coverage. This, despite the fact that much has been written on the subject. In an early paper on this
subject Casagrande and Carrillo (1944) distinguished between induced anisotropy due exclusively to strains
arising from the applied stresses, and inherent fabric anisotropy termed inherent anisotropy. They derived
graphical and analytical solutions for shear strength in the case of induced anisotropy for (a) purely cohesive
material and (b) purely frictional material, assuming in both cases the principal strengths (maximum and
minimum) occurred on principal stress planes and varied elliptically between these extremes. For a cohesive
soil with inherent anisotropy they considered the case where the principal strengths did not occur on the
planes of the principal stresses. Assuming isotropic Hvorslev strength parameters for a one-dimensionally
normally consolidated clay, Hansen and Gibson (1949) derived an expression for undrained shear strength
for any orientation of the failure plane. They concluded from this analysis that in the field the greatest
undrained strength would be mobilised in the case of active earth pressure and the least in the case of
passive pressure, with other types of failures, such as circular slides, lying between these two extremes.
Duncan and Seed (1966) concluded that, as a result of initial stress anisotropy and principal stress rotation in
the field, the effective stress strength parameters
and pore pressure parameters would vary with
orientation of the failure plane and account should
be taken of this in applying the otherwise valid
Hansen and Gibson analysis.
Assuming a cross-anisotropic elastic material
Henkel (1971) derived expressions for effective
stress paths for three types of undrained tests: (a)
triaxial tests on vertical specimens (b) triaxial tests
on horizontal specimens and (c) plane strain tests
on vertical specimens with zero strain in one of the
horizontal directions. He showed that these
coincided well with the initial linear portion of
experimental stress paths for heavily over-
consolidated London clay assuming a horizontal
stiffness equal to 1.6 times the vertical stiffness.
The influence of stress path on the peak shear
strength of an anisotropic soft clay can be seen in
Figure 9. Consolidation tests by Wesley (1975) on
vertical and horizontal specimens of Mucking
Flats clay (w
L
= 56, w
P
= 25) showed a vertical
stiffness about twice the lateral stiffness.
Theoretical elastic effective stress paths are shown
in Figure 9 (broken lines) for undrained triaxial
compression and extension tests on vertical and
horizontal specimens of a soil with vertical to
horizontal elastic moduli in the ratio of 2:1. The
stress paths for Mucking Flats clay observed by
Wesley and shown in Figure 9 correspond closely
to these and it can be seen that directions of the
stress paths determine the points at which the
specimens reach the failure envelope and thus
determine the peak undrained strengths. The actual
values of peak shear strength c
u
are given below:

Vertical specimens: Compression 16.3 kPa,
Extension 9.2 kPa.

Horizontal specimens: Compression 11.7 kPa,
Extension 11.8 kPa.

As noted previously undrained extension tests on
isotropic laboratory prepared specimens usually give
shear strengths of about 0.7 to 0.8 times the
compression values. The influence of anisotropy in
the Mucking Flats specimens is to exaggerate this
difference in vertical specimens and suppress it
altogether in horizontal specimens
An example of the effect of fabric anisotropy on
drained shear strength can be seen for heavily
overconsolidated Oxford Clay (w
L
= 70%,
w
P
= 25%) in Figure 10. Oxford Clay is an upper
Jurassic deposit outcropping in South-East England
and estimated to have been under 900 metres of overburden in the past. It is strongly laminated in the
horizontal direction to the extent that pieces of clay defoliate on drying like the leaves of a book. In order to
determine design parameters for reservoir excavations, drained direct shear box tests were performed to
measure peak and residual strengths of specimens with the shear plane vertical (V), horizontal (H) and at 30°
to the horizontal (I). The results for tests with a normal applied stress of 350 kPa are shown in Figure 10, and
it can be seen that the H and I tests gave comparable results, but the strength was much higher for the V test.
In a study of short-term failures of deep excavations in heavily overconsolidated London Clay (w
L
= 95,
w
P
=30) an Eocene deposit thought to have been under some 150 metres of sediments in the past, Skempton
and LaRochelle (1965) found back-calculated undrained shear strengths from the slips to be only 55% of the
values measured on 38 mm dia by 76 mm long triaxial specimens in the laboratory. They attributed about
20% of the reduction to time effects associated with pore water migration and the remainder to the presence
of fissures, which would have greatly influenced the mass behaviour of the clay, but would have had little
influence on small test specimens consisting essentially of intact clay. It was also found that 100 mm dia
triaxial test specimens gave undrained shear strengths about 10% lower than 38 mm dia specimens.
Marsland (1972) showed that the undrained laboratory measured strength of London clay was influenced by
the shapes, roughnesses and inclinations of fissures in the test specimen (cut from blocks). Noting also the
importance of the ratio of fissure spacing to specimen diameter in determining laboratory undrained
strengths, he plotted non-dimensionally against this ratio the ratio of laboratory strengths to those deduced
from various large scale field tests, using both his own data and data from Imperial College (Figure 11). He
showed that undrained shear strengths measured on specimens with a small diameter relative to fissure
spacing could be almost five times those from specimens with a large diameter relative to fissure spacing.


9.0 UNSATURATED CLAYS

In three-phase unsaturated clays, with both air and water in the voids, the shear strength may be
determined very largely by the high suction in the water phase rather than external pressures. Total suction
is made up of osmotic or solute suction and matric suction. Solute suction is the negative gauge pressure to
which a pool of pure water must be subjected to achieve equilibrium through a semi-impermeable membrane
with a pool containing a solution identical to the soil water. Matric suction is the negative gauge pressure
relative to external gas pressure on soil water to which a solution identical to the soil water must be subjected
to achieve equilibrium with the soil water through a permeable barrier. The effective stress equation is
usually expressed in terms of matric suction only (Bishop 1959, Aitchison 1960, Bishop etal 1960):

σ′ = (σ - u
a
) + χ (u
a

-

u
w

) (25)
leading to the expression for shear strength τ
f :

τ
f
= c′ + [σ
n
- u
a
+ χ (u
a

-

u
w
)] tanφ′ (26)

where u
a
is pore air pressure in
excess of atmospheric pressure and
(u
a
- u
w
) is matric suction. In
practice this expression has been
little used owing to the problem of
assigning values to the coefficient
χ and to the extreme difficulty of
measuring directly the large
negative values of pore water
pressure, greatly exceeding
atmospheric pressure in magnitude,
which often exist in unsaturated
clays. Recent work, however, offers
hope that the direct measurement of
large matric suction may become
possible ( Ridley 1993, Gan and
Fredlund 1997).
Fredlund and Morgenstern
(1977) showed that the shear
strength of unsaturated soils could
be described by any two of three
state variables (σ - u
a
), (σ - u
w
),
and (u
a
-

u
w
), based on which a
linear shear strength equation was
proposed by Fredlund etal (1978):

τ
f
= c′ + (σ
n
- u
a
)tanφ′ + (u
a
-

u
w
)tanφ
b
(27)

Using a curve fitting technique on results from a residual unsaturated soil with low suctions, Abramento and
Carvalho (1989) concluded that tanφ
b
should be treated as a variable with respect to suction.
The problems of measuring in the laboratory the shear strength parameters for unsaturated soils has
limited the use of equations such as (26) and (27) and has led to the use of indirect methods of assessing their
strength using saturated shear strength parameters and the soil-water characteristic curve, defined as the
relationship between the soil suction and the volumetric water content θ
w
(the ratio of the volume of water to
the total volume of soil). An expression of this form was proposed by Lamborn (1986) based on a
micromechanics/thermodynamic model and in a more general non-linear form by Vanapalli etal (1996):

τ
f
= c′ + (σ
n
- u
a
)tanφ′ + (u
a
-

u
w
) (Θ
κ
tanφ′)

(28)

in which Θ is the normalised water content of the soil θ
w

s
,

θ
s
is the saturated volumetric water content and
κ is a fitting parameter used for obtaining a best fit between the measured and predicted values. A
comparison between the measured and predicted shear strength function for a glacial till based on the soil-
water characteristic curve is shown in Figure 12 (Fredlund 1998). An alternative equation not using the
fitting parameter κ has been proposed by Vanapalli etal (ibid):

τ
f
= c′ + (σ
n
- u
a
)tanφ′ + (u
a -
u
w
)tanφ′ [(θ
w

r
)/(θ
s

r
)]

(29)

where θ
r
is residual volumetric water content estimated from the soil-water characteristic curve. Equations
28 and 29 are designated University of Saskatchewan Procedures 1 and 2 respectively.
Direct shear box tests performed on three compacted, unsaturated soils by Escario and Juca′ (1989), at
constant vertical stress, gave a plot of shear strength versus suction with an initial slope of tanφ′, then
gradually flattening until a maximum was reached. Adopting an empirical approach, the authors found that
the variation of shear strength with suction could be represented by an ellipse of 2.5 degrees up to the
maximum.
It should logically be possible to fit the stress-strain behaviour of unsaturated soils within a critical state
framework and suggested approaches for doing this have been published (eg Alonso etal 1990. Wheeler and
Sivakumar 1995). The basic framework is made up of four state variables namely mean net stress
p = [(σ
1

2

3
)/3 - u
a
], deviator stress q, suction s = (u
a
-

u
w
) and specific volume v. Limited experimental
data on compacted kaolin has given encouragement to continue this line of research, but as pointed out by
Wheeler and Sivakumar their elastic-plastic approach requires the determination of three elastic constants
and six parameters which vary with suction, and a realistic simplified model will be needed for the method to
have practical applications.

10.0 SOFT ROCK

Unconsolidated undrained triaxial tests performed in 1958 on cores of Melbourne mudstone for the design
of the Kings Bridge foundations established, apparently for the first time, the importance of moisture content
in determining the strength and deformation characteristics of a soft rock (Parry 1963,1972). A Silurian
deposit, Melbourne mudstone consists predominantly of siltstone with interbedded claystones and
sandstones. All the test specimens were initially tested under a confining pressure of 690 kPa (roughly
equal to the total overburden pressure) and in some cases the specimens were subjected to multi-stage testing
at confining pressures of 1725
kPa, 3450 kpa and occasionally
6900 kPa. Fourteen of the
twenty multi-stage tests showed
undrained angles of shearing
resistance close to zero, while
the other six showed angles of
shearing resistance up to 23°,
probably reflecting structural
features in the rock. It was
concluded that the mudstone
could be regarded as a φ
u
= 0
material. A site specific
correlation was found between
the shear strength of the
mudstone and its moisture
content (Figure 13), which was
subsequently confirmed for
other sites in Melbourne (eg
Parry 1972, McDonald and
Jellie 1991). Full use was made
of this correlation in fixing the
founding level for the
foundation caissons in the
highly variable, and almost
vertically bedded, mudstone, lying under about 30 metres of soft recent sediments. The φ
u
= 0 behaviour of
soft rock has been observed by other workers (eg Akai etal 1981).
Subsequent research on Melbourne mudstone at Monash University has confirmed the primary role of
moisture content, and consequently effective stress, in determining its strength and deformation
characteristics (Chiu and Johnston 1980, 1984). While confirming the strong similarity in its behaviour to
that of soils, particularly heavily overconsolidated clays, these workers have drawn attention to differences in
behaviour, such as the curvature of the effective stress Mohr-Coulomb envelope for both drained and
consolidated undrained tests, exhibiting a slope of 34° up to a normal effective stress of 10 MPa, reducing to
22° at 60 MPa, although this may be a consequence of the very high pressure range rather than inherent
differences between soil and soft rock behaviour.

11.0 GENERALISED STRENGTH CRITERIA FOR ROCK

The Griffith Crack Theory for brittle materials (Griffith1921,1924) provided for a number of workers a
basic conceptual model for the strength of rocks, but although giving a parabolic plot of shear strength
against normal pressure, which corresponds well with many experimental observations, it has not proved to
be a good quantitative model. Attempts to modify the theory
by introducing friction on the cracks (McClintock and
Walsh 1962) and extending it from a plane compression
model to three dimensions (Murrell 1963, 1965) have had
only limited success. Consequently Hoek and Brown
(1980), recognising the parabolic nature of the strength
envelope, proposed an empirical strength criterion for hard
rock masses given in the normalised form:

σ
1n
= σ
3n
+ (m σ
3n
+ S)
1/2
(30)

where σ
1n,
σ
3n
are the major and minor principal stresses
divided by the unconfined uniaxial compressive strength σ
c
.
For intact rocks the empirical factor m ranges from about 7
for carbonate rocks to 25 for coarse grained polyminerallic
igneous rocks, but may have values as low as 0.001 for
highly disturbed rock masses. The value of S ranges from 0
for jointed rock masses to 1.0 for intact rock. The special
cases of unconfined compressive strength σ
cm
and uniaxial
tensile strength σ
tm
for a rock mass can be found by putting
σ′
3
= 0 and σ′
1
= 0 respectively into Equation 30. The
general case and special cases are depicted in Figure 14. A
more convenient form of the general equation for slope
stability calculations, expressed in terms of shear strength,
was derived by Bray (Hoek 1983).
Widespread use of Equation 30, sometimes in relation to
unsuitable materials such as very poor quality rocks, has led
to modifications by the authors (Hoek 1994, Hoek etal 1995,
Hoek and Brown 1997) in which the square root power is replaced by a more general parameter a, the value
of which, together with the value of S, is fixed according to the Geological Strength Index GSI, introduced
by the above authors as a system for estimating the reduction in rock mass strength for different geological
conditions. For better quality rock masses GSI can be estimated from the Rock Mass Rating (Bieniawski
1976), but for poorer quality rocks physical appearance alone has to be relied upon.
Johnston and Chiu (1984) and Johnston (1985) presented an alternative expression for intact rock:

σ′
1N
= (Mσ′
3N
/B + 1)
B
(31)

where M and B are constants which, for a wide range of clay soils and rocks can be reasonably represented
by:

B = 1 - 0.0172(log σ
c
)
2
(32a)

M = 2.065 + 0.276(log σ
c
)
2
(32b)

where σ
c
is in kPa. Putting σ′
3N
= 0 in Equation 31 gives the uniaxial compressive strength σ
c
= σ
1
and the
ratio of uniaxial compressive strength to uniaxial tensile strength is given by putting σ′
1
= 0, σ′
3
= σ
t
,
yielding:

σ
c
/ σ
t
= - M / B (33)

Equation 31 is capable of representing a wide range of geomaterials. As σ
c
→ 0, B→ 1 and M→ 2.065,
implying a virtually linear envelope with φ′ = 20° for soft normally consolidated clay. In stiff heavily
overconsolidated clays, typically σ
c
= 200 kPa, giving B = 0.9 and thus a slightly curved envelope, while in
hard rocks, typically σ
c
= 250 Mpa and B = 0.5, which is the familiar parabolic form.
Based on published data, Hoek and Brown (1980) found the measured strengths of rock test specimens to
be influenced by specimen size and suggested the expression:

σ
cd
= σ
c50
(50/d)
0.18
(34)

where σ
cd

c50
are uniaxial compressive strengths for test specimens with diameter d mm and 50 mm
respectively.
Representation of a range of geomaterials by an expression such as Equation 31 poses the question
whether soils and rocks can be encapsulated within a common critical state framework. Attempts to do this
inevitably encounter the problem that rock behaviour, particularly strength, is strongly influenced by
structural defects which militate against the uniform deformation in shear required to reach a true critical
state. Nevertheless some qualified success has been reported for soft rocks (Chiu and Johnston 1984,
Johnston and Novello 1985) and for hard rocks (Gerogiannopoulos and Brown 1978, Michelis and Brown
1986). Soft sedimented rocks such as Melbourne mudstone and Ohya tuff (Adachi etal 1981), in which the
water content has a dominating influence, exhibit behaviour identical to very heavily overconsolidated soils,
and even the non-linear critical state line on a plot of deviator stress against mean effective stress ( Novello
and Johnston 1995), which is a reflection of a friction angle reducing with increasing applied stress, can also
occur with soils taken to the same high test pressures normally applied in rock testing. Novello and Johnston
also found that a form of critical state behaviour could be identified in hard rocks, assuming the brittle to
ductile transition with increasing applied stress in the rock to be equivalent to the change from
overconsolidated to normally consolidated shear behaviour in soils, and the strength in the ductile plastic
state to be of a cohesive, rather than a frictional, nature. Gerogiannopoulos and Brown introduced work
absorbed by brittle fracture mechanism in addition to friction, together with normality at peak strength, to
formulate an equation able to represent peak and residual strengths for a wide range of brittle strain softening
rocks (Brown and Michelis 1978). By exploiting the experimental observation that the plastic strain vector
had an approximately constant inclination for both work softening and work hardening materials Michelis
and Brown (1986) developed a general expression for intact or granular materials in which the relative scale
of their size to the spacing of discontinuities was such that the mass could be regarded as isotropic and
homogeneous.

12.0 ROCK JOINTS

If a rock joint is infilled with granular or cohesive material weaker than the rock itself this material will
strongly influence the strength, whereas the strength along a clean joint will be governed by the basic smooth
rock to rock friction φ′
b
, plus dilatancy depending on the roughnesses of the surfaces and usually represented
by angle i. A simple expression embodying this concept was suggested by Patton (1966):

τ
jf
= σ′
n
tan (φ′
b
+ i) (35)

As the effective normal stress σ′
n
increases the failure occurs increasingly through the asperities, giving a
curved relationship between τ
jf
and σ′
n
as observed by a number of workers (eg see Jaeger 1971).
An empirical expression for saturated joints based on Equation 35 was presented by Barton (1973):

τ
jf
= σ′
n
tan [(JRC)log
10
(JCS/σ′
n
) + φ′
r
] (36)

where JRC is the joint roughness coefficient, JCS is the joint wall compressive strength (saturated) and φ′
r
is
the residual friction angle (wet, drained). Simple empirical methods were given for evaluating φ′
r
from φ′
b

using the Schmidt hammer test and for
evaluating JRC from tilt tests using core
sticks. Magnitudes of JRC range from zero
for smooth planar surfaces to about 15 or 20
for very rough surfaces. Refinements to
Equation 36 have concentrated mainly on
attempts to evaluate the effects of scale and
normal stress magnitude on the dilatancy
portion of the equation.
Barton and Choubey (1977) deduced
from a study of 136 joint samples that the
total friction angle (arctan τ
jf
/σ′
n
) could be
estimated by these methods to within about
1° in the absence of scale effects. However,
they concluded that scale effects were
important in determining values of JCS and
JRC, both values decreasing with increasing
length of joint. Citing tests by Pratt etal
(1974) on quartz diorite which showed a
drop in peak shear strength of nearly 40%
for rough joint surfaces with areas ranging
from 0.006 m
2
to 0.5 m
2
, and adopting
JRC = 20, they concluded that these tests
indicated a fourfold reduction in JCR for a
fivefold increase in joint length from 140
mm to 710 mm. Tilt tests by Barton and
Choubey (ibid) on rough planar joints of
Drammen granite showed JRC values to
decrease from 8.7 for 100mm long joints to
5.5 for 450 mm long joints. ISRM (1977)
recommendations for assessing joint
roughness are based largely on the methods
proposed by Barton and Choubey (ibid).
Ten "typical" roughness profiles ranging
from JRC = 0-2 to 18-20 shown in Figure
15 are presented in this ISRM report.
Based on limited laboratory tests Barton
and Bandis (1982) proposed the scale correction for JRC given by:

JRC

n
= JRC
o
(L
n

/L
o

)
-0.02JRC
o

(37)
Where L
o,
L
n
refer to laboratory scale (100 mm) and in-situ joint lengths respectively.
In an attempt to reduce the degree of empiricism
in quantifying joint roughness a number of workers
have introduced fractal geometry (Mandelbrot
1983), seeing an analogy between coastal
irregularities and rock joint roughnesses. A straight
line linking the end points of a joint with length L
can be divided into a number N
s
of equal chords
each of length r (=L/N
s
). If this chord length is
stepped out along the joint profile (eg by means of
dividers) the number N of segments it measures will
be greater than N
s
by an amount exceeding unity
according to the roughness of the joint. The fractal
dimension D, which is given by the expression:

D = - log N / log r (38)

can be evaluated from the slope of a log-log plot of
N against r. For practical usage it is necessary to
convert this value into a quantity which can be used
in design or analysis and one approach has been to try and establish a partly or wholly empirical relationship
between JRC and fractal dimension (Turk etal 1987, Lee etal 1990). Lee etal established an empirical
expression relating the two giving the plot shown in Figure 16. It will be noted that fractal dimension values
for rock joints are not greatly in excess of unity and it is necessary determine the value to at least four
decimal places. Nevertheless the value is operator dependent, as shown by published values of 1.0045 (Turk
etal 1987), 1.005641 (Lee etal 1990) and 1.0040 (Seidel and Haberfield 1995) for the "standard" ISRM joint
JRC = 10-12.
Adopting a more fundamental approach,
and assuming gaussian distribution of chord
angles θ, Seidel and Haberfield (1995)
derived a relationship between fractal
dimension and the standard deviation of
asperity angle s
θ
:

s
θ
≈ cos
-1
(

N
(1 -
D
) /
D
) (39)

which plots as shown in Figure 17. It can
be seen in Figure 17 that s
θ



increases with
D, which expresses the divergence from a
straight line and with N, the number of
opportunities for divergence. As asperity
height is a function of chord length and
angle, an expression for the standard
deviation of asperity height can be derived
from Equation 39:

s
h
≈ ( N

-2/
D
- N
-2
)
0.5
(40)

By generating random profiles for s
θ
values and comparing these to the ISRM profiles for different ranges
of JRC in Figure 15, Seidel and Haberfield established a subjective basis for predicting JRC from fractal
geometry and showed, in addition, the possibility of the fractal approach providing conceptual models for the
effects of normal stress on shear behaviour of joints and the scale-dependence of joints
Krahn and Morgenstern (1979) drew attention to the difference in residual strength between soil and a
rock joint. The residual strength is independent of the original state in a soil whereas the residual strength
(preferred term ultimate frictional resistance) of a rock joint, as shown by their tests with limestone, is related
to the initial surface roughness and surface alterations which take place during shearing.
.Owing to their complex nature and the impossibility of submitting identical samples to a systematic
series of tests, natural filled rock joints present unique problems in assessing their shear strength,
compounded by the fact that not only are there the normal problems of determining the strengths of both the
rock and the filling, but also the interaction of the two. Consequently systematic research has been confined
to model studies, often using materials such as plaster or concrete in place of rock and a variety of artificial
or reconstituted soils as fillers. Although some patterns of behaviour have emerged, there is also much
conflicting evidence (Toledo and de Freitas 1993) probably arising in part from differences in model joint
preparation, but more particularly from the behaviour under shear of the infill material. If it is a silt or clay
the degrees of dissipation of pore pressures generated during joint displacement, which profoundly influence
its behaviour, are governed by the rate of displacement and boundary conditions such as rock type and
confining conditions.

Much of the research on the shear strength of filled rock joints has concentrated on the influence of joint
wall roughness, expressed as asperity height a, compared to the thickness t of the fill material. An example
of the influence of the relative thickness t/a for a series of tests on clay infilled sandstone joints is shown in
Figure 18 (Toledo and de Freitas ibid ), and Figure 19 shows a strength model presented by the same authors
embodying in a general way their own observations and those of a number of other researchers, embracing
mica filler and plaster as rock (Goodman 1970), kaolin clay and concrete blocks (Ladanyi and Archambault
1977), kaolin and hard gypsum rock (Lama 1978), oven dried bentonite and gypsum rock (Phien-Wej etal
1990) and fillers of kaolin, marble dust and fuel ash with plaster and cement as rock simulating joint
materials.
The inset diagram in Figure 18 shows the infill soil providing the initial resistance to joint displacement
and reaching a peak τ
Soil
at small displacement, followed by a decline in strength depending on the
brittleness of the infill soil, then an increase in strength with increasing displacement brought about initially
by a flow of infill material from highly stressed to low stressed zones leading to contact of the rock
asperities and a second, higher, peak strength τ
Rock
depending on the rock properties. The decline in joint
strength with increasing t/a is seen in Figure 18 up to about t/a = 1, after which the strength remains
essentially constant. It is noticeable that the peak rock strength still exceeds that of the infill at t/a = 1.
These features are incorporated into Figure 19, but it is noticeable that in this generalised model the critical
thickness at which the soil peak reaches its minimum may be less than the asperity height for clayey infill
and greater than asperity height for granular infill. A further feature of Figure 19 is the division of the
different thicknesses into three ranges as proposed by Nieto (1974), namely interlocking in which the rock
surfaces come into contact, interfering when there is no rock contact but the joint strength exceeds that of the
infilling and non-interfering when the joint strength equals that of the filler.

13.0 CONCLUSIONS

The formulation of the principle of effective stress by Terzaghi made possible a fundamental
understanding of the shear strength of soils and soft rocks. This was insufficient in itself, however, as the
strength of these materials in the field may be influenced by many other factors such as degree of saturation,
rate of loading, loading history and loading stress path, physical anisotropy, and the presence of fissures and
joints, and other imperfections, in a soil or rock mass. Much of the research work and writings of the 20th
Century have concentrated on understanding and quantifying these influences, as well as furthering the
understanding of fundamental strength behaviour and introducing improved laboratory and field methods of
measuring shear strength. Mohr stress circles and the Mohr-Coulomb envelope have proved invaluable in
providing an instantly comprehensible representation of stress and strength conditions. The critical state
concept has proved useful in understanding soil behaviour, although most soil test specimens do not reach a
true critical state because of imperfections in the specimens and in the test methods. There is evidence that
critical state may also apply to some limited extent to the behaviour of rocks. In the case of hard rocks the
Griffith crack theory, while inadequate in itself, has provided a conceptual starting point for developing
realistic empirical expressions for the commonly observed curved shear strength envelope. Rock joints, even
if clean, present a problem in assessing their frictional properties, but the introduction of fractal geometry
may allow quantitative assessments of joint strength to be made, including scale effects. If infilled, the
strength may be influenced by the strength of both the rock and the joint infill and also by the interaction
between them.

14.0 ACKNOWLEDGEMENTS

I am indebted to Professors Del Fredlund and S K Vanapalli for helping me to comprehend the mysteries of
unsaturated soils

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