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# Craig's Soil Mechanics 7th Edition

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```of plane strain in which the strain in the direction of the intermediate principal stress is
zero due to restraint imposed by virtue of the length of the structure in question. In the
triaxial test, consolidation proceeds under equal all-round pressure (i.e. isotropic
consolidation), whereas in-situ consolidation takes place under anisotropic stress
conditions.
Tests of a more complex nature, generally employing adaptions of triaxial equip-

ment, have been devised to simulate the more complex states of stress encountered in
practice but these are used principally in research. The plane strain test uses a
prismatic specimen in which strain in one direction (that of the intermediate principal
stress) is maintained at zero throughout the test by means of two rigid side plates tied
together. The all-round pressure is the minor principal stress and the sum of the
applied axial stress and the all-round pressure the major principal stress. A more
sophisticated test, also using a prismatic specimen, enables the values of all three
principal stresses to be controlled independently, two side pressure bags or jacks being
used to apply the intermediate principal stress. Independent control of the three
principal stresses can also be achieved by means of tests on soil specimens in the form
of hollow cylinders in which different values of external and internal fluid pressure can
be applied in addition to axial stress. Torsion applied to the hollow cylinders results in
the rotation of the principal stress directions.
Because of its relative simplicity it seems likely that the triaxial test will continue to

be the main test for the determination of shear strength characteristics. If considered
necessary, corrections can be applied to the results of triaxial tests to obtain the
characteristics under more complex states of stress.

4.3 SHEAR STRENGTH OF SANDS

The shear strength characteristics of a sand can be determined from the results of
either direct shear tests or drained triaxial tests, only the drained strength of a sand
normally being relevant in practice. The characteristics of dry and saturated sands are
the same, provided there is zero excess pore water pressure in the case of saturated
sands. Typical curves relating shear stress and shear strain for initially dense and loose
sand specimens in direct shear tests are shown in Figure 4.8(a). Similar curves are
obtained relating principal stress difference and axial strain in drained triaxial com-
pression tests.
In a dense sand there is a considerable degree of interlocking between particles.

Before shear failure can take place, this interlocking must be overcome in addition
to the frictional resistance at the points of contact. In general, the degree of
interlocking is greatest in the case of very dense, well-graded sands consisting of
angular particles. The characteristic stress–strain curve for an initially dense sand
shows a peak stress at a relatively low strain and thereafter, as interlocking is

102 Shear strength

progressively overcome, the stress decreases with increasing strain. The reduction in
the degree of interlocking produces an increase in the volume of the specimen
during shearing as characterized by the relationship, shown in Figure 4.8(c),
between volumetric strain and shear strain in the direct shear test. In the drained

Dense

Dense

Dense

Loose

Loose

Loose

Same σ′

Same e0

φcv σ′

Different σ′

φmax

φcv

ecv

(φmax)

(φcv)
(φµ)

σ′
(a) (b)

(secant)

(c)

(e)

(g)

(d)

Vo
lu

m
e

in
cr

ea
se

Vo
lu

m
e

de
cr

ea
se

dεvεv

dγ

τ

τ

τ/σ′

τ

γ

γ

γ γ

γ

e

e

e0

(f )

A

B

C

A

B

C

A

B

C

′

′

′

′

′

′

Figure 4.8 Shear strength characteristics of sand.

Shear strength of sands 103

triaxial test a similar relationship would be obtained between volumetric strain and
axial strain. The change in volume is also shown in terms of void ratio (e) in Figure
4.8(d). Eventually the specimen would become loose enough to allow particles to
move over and around their neighbours without any further net volume change and
the shear stress would reach an ultimate value. However, in the triaxial test non-
uniform deformation of the specimen becomes excessive as strain is progressively
increased and it is unlikely that the ultimate value of principal stress difference can
be reached.
The term dilatancy is used to describe the increase in volume of a dense sand during

shearing and the rate of dilation can be represented by the gradient d"v/d�, the
maximum rate corresponding to the peak stress. The angle of dilation ( ) is tan�1

(d"v/d�). The concept of dilatancy can be illustrated in the context of the direct shear
test. During shearing of a dense sand the macroscopic shear plane is horizontal but
sliding between individual particles takes place on numerous microscopic planes
inclined at various angles above the horizontal, as the particles move up and over
their neighbours. The angle of dilation represents an average value of this angle for the
specimen as a whole. The loading plate of the apparatus is thus forced upwards, work
being done against the normal stress. For a dense sand the maximum angle of shearing
resistance (�0max) determined from peak stresses (Figure 4.8(b)) is significantly greater
than the true angle of friction (�m) between the surfaces of individual particles, the
difference representing the work required to overcome interlocking and rearrange the
particles.
In the case of initially loose sand there is no significant particle interlocking to be

overcome and the shear stress increases gradually to an ultimate value without a prior
peak, accompanied by a decrease in volume. The ultimate values of stress and void
ratio for dense and loose specimens under the same values of normal stress in the direct
shear test are essentially equal as indicated in Figures 4.8(a) and (d). Thus at the
ultimate (or critical) state, shearing takes place at constant volume, the corresponding
angle of shearing resistance being denoted �0cv (or �

0
crit). The difference between �

0
m and

�0cv represents the work required to rearrange the particles.
It may be difficult to determine the value of the parameter �0cv because of the

relatively high strain required to reach the critical state. In general, the critical state
is identified by extrapolation of the stress–strain curve to the point of constant stress,
which should also correspond to the point of zero rate of dilation on the volumetric
strain–shear strain curve. Stresses at the critical state define a straight line failure
envelope intersecting the origin, the slope of which is �0cv.
In practice the parameter �0max, which is a transient value, should only be used for

situations in which it can be assumed that strain will remain significantly less than that
corresponding to peak stress. If, however, strain is likely to exceed that corresponding
to peak stress, a situation that may lead to progressive failure, then the critical-state
parameter �0cv should be used.
An alternative method of representing the results from direct shear tests is to plot

the stress ratio �/�0 against shear strain. Plots of stress ratio against shear strain
representing tests on three specimens of sand, each having the same initial void
ratio, are shown in Figure 4.8(e), the values of effective normal stress (�0) being
different in each test. The plots are labelled A, B and C, the effective normal stress
being lowest in test A and highest in test C. Corresponding plots of void ratio

104 Shear strength

against shear strain are shown in Figure 4.8(f ). Such results indicate that both the
maximum stress ratio and the ultimate (or critical) void ratio decrease with increas-
ing effective normal stress. The ultimate values of stress ratio,```