﻿ Craig's Soil Mechanics 7th Edition - Mecânica dos Solos 2 - 34
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# Craig's Soil Mechanics 7th Edition

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and the prin-
cipal stress difference at failure 170 kN/m2. Assuming that the above parameters
are appropriate to the failure stress state of the test, what would be the expected
value of pore water pressure in the specimen at failure?

Problems 133

4.5 The results below were obtained at failure in a series of consolidated–undrained
triaxial tests, with porewater pressuremeasurement, on specimens of a fully saturated
clay. Determine the values of the shear strength parameters c0 and �0. If a specimen of
the same soil were consolidated under an all-round pressure of 250kN/m2 and the
principal stress difference applied with the all-round pressure changed to 350kN/m2,
what would be the expected value of principal stress difference at failure?

�3 (kN/m
2) 150 300 450 600

�1 � �3 (kN/m2) 103 202 305 410
u (kN/m2) 82 169 252 331

4.6 The following results were obtained at failure in a series of drained triaxial tests
on fully saturated clay specimens originally 38mm in diameter by 76mm long.
Determine the secant parameter �0 for each test and the values of tangent
parameters c0 and �0 for the stress range 300–500 kN/m2.

All-round pressure (kN/m2) 200 400 600
Axial compression (mm) 7.22 8.36 9.41
Axial load (N) 565 1015 1321
Volume change (ml) 5.25 7.40 9.30

4.7 Derive Equation 4.12.
In an in-situ vane test on a saturated clay a torque of 35Nm is required to shear
the soil. The vane is 50mm wide by 100mm long. What is the undrained strength
of the clay?

4.8 A consolidated–undrained triaxial test on a specimen of saturated clay was
carried out under an all-round pressure of 600 kN/m2. Consolidation took place
against a back pressure of 200 kN/m2. The following results were recorded
during the test.

�1 � �3 (kN/m2) 0 80 158 214 279 319
u (kN/m2 ) 200 229 277 318 388 433

Draw the stress paths and give the value of the pore pressure coefficient A at
failure.

4.9 In a triaxial test a soil specimen is allowed to consolidate fully under an all-round
pressure of 200 kN/m2. Under undrained conditions the all-round pressure is
increased to 350 kN/m2, the pore water pressure then beingmeasured as 144 kN/m2.
Axial load is then applied under undrained conditions until failure takes place,
the following results being obtained.

Axial strain (%) 0 2 4 6 8 10
Principal stress difference (kN/m2) 0 201 252 275 282 283
Pore water pressure (kN/m2) 144 244 240 222 212 209

Determine the value of the pore pressure coefficient B and plot the variation of
coefficient A with axial strain, stating the value at failure.

134 Shear strength

REFERENCES

1 Bishop, A.W. (1966) The strength of soils as engineering materials, Geotechnique, 16,

91–128.

2 Bishop, A.W., Alpan, I., Blight, G.E. and Donald, I.B. (1960) Factors controlling the

strength of partly saturated cohesive soils, in Proceedings of the ASCE Conference on

Shear Strength of Cohesive Soils, Boulder, CO, USA, ASCE, New York, pp. 503–32.

3 Bishop, A.W., Green, G.E., Garga, V.K., Andresen, A. and Brown, J.D. (1971) A new ring

shear apparatus and its application to the measurement of residual strength, Geotechnique,

21, 273–328.

4 Bishop, A.W. and Wesley, L.D. (1975) A hydraulic triaxial apparatus for controlled stress

path testing, Geotechnique, 25, 657–70.

5 Bjerrum, L. (1973) Problems of soil mechanics and construction on soft clays, in Proceed-

ings of the 8th International Conference of SMFE, Moscow, Vol. 3, pp. 111–59.

6 Bolton, M.D. (1986) The strength and dilatancy of sands, Geotechnique, 36, 65–78.

7 British Standard 1377 (1990) Methods of Test for Soils for Civil Engineering Purposes,

British Standards Institution, London.

8 Bromhead, E.N. (1979) A simple ring shear apparatus, Ground Engineering, 12 (5), 40–4.

9 Cornforth, D.H. (1964) Some experiments on the influence of strain conditions on the

strength of sand, Geotechnique, 14, 143–67.

10 Duncan, J.M. and Seed, H.B. (1966) Strength variation along failure surfaces in clay,

Journal of the ASCE, 92 (SM6), 81–104.

11 Head, K.H. (1986) Manual of Soil Laboratory Testing, three volumes, Pentech, London.

12 Hight, D.W., Gens, A. and Symes, M.J. (1983) The development of a new hollow cylinder

apparatus for investigating the effects of principal stress rotation in soils, Geotechnique, 33,

355–83.

13 Lupini, J.F., Skinner, A.E. and Vaughan, P.R. (1981) The drained residual strength of

cohesive soils, Geotechnique, 31, 181–213.

14 Roscoe, K.H., Schofield, A.N. and Wroth, C.P. (1958) On the yielding of soils, Geotech-

nique, 8, 22–53.

15 Rowe, P.W. (1962) The stress–dilatancy relation for static equilibrium of an assembly of

particles in contact, Proceedings of the Royal Society A, 269, 500–27.

16 Skempton, A.W. (1954) The pore pressure coefficients A and B, Geotechnique, 4, 143–7.

17 Skempton, A.W. (1985) Residual strength of clays in landslides, folded strata and the

laboratory, Geotechnique, 35, 1–18.

18 Skempton, A.W. and Sowa, V.A. (1963) The behaviour of saturated clays during sampling

and testing, Geotechnique, 13, 269–90.

References 135

Chapter 5

Stresses and displacements

5.1 ELASTICITY AND PLASTICITY

The stresses and displacements in a soil mass due to applied loading are considered in
this chapter. Many problems can be treated by analysis in two dimensions, i.e. only the
stresses and displacements in a single plane need to be considered. The total normal
stresses and shear stresses in the x and z directions on an element of soil are shown in
Figure 5.1, the stresses being positive as shown; the stresses vary across the element. The
rates of change of the normal stresses in the respective directions are @�x/@x and @�z/@z;
the rates of change of the shear stresses are @�xz/@x and @�zx/@z. Every such element in
a soil mass must be in static equilibrium. By equating moments about the centre point
of the element, and neglecting higher-order differentials, it is apparent that �xz ¼ �zx.
By equating forces in the x and z directions the following equations are obtained:

@�x
@x

þ @�zx
@z

� X ¼ 0 ð5:1aÞ

@�xz
@x

þ @�z
@z

� Z ¼ 0 ð5:1bÞ

where X and Z are the respective body forces per unit volume. These are the equations
of equilibrium in two dimensions; they can also be written in terms of effective stress.
In terms of total stress the body forces are X ¼ 0 and Z ¼ � (or �sat). In terms of
effective stress the body forces are X 0 ¼ 0 and Z0 ¼ �0; however, if seepage is taking
place these become X 0 ¼ ix�w and Z0 ¼ �0 þ iz�w where ix and iz are the hydraulic
gradients in the x and z directions, respectively.
Due to the applied loading, points within the soil mass will be displaced relative to

the axes and to one another. If the components of displacement in the x and z
directions are denoted by u and w, respectively, then the normal strains are given by

"x ¼ @u
@x

; "z ¼ @w
@z

and the shear strain by

�xz ¼ @u
@z

þ @w
@x

However, these strains are not independent; they must be compatible with each other
if the soil mass as a whole is to remain continuous. This requirement leads to the
following relationship, known as the equation of compatibility in two dimensions:

@2"x
@z2

þ @
2"z
@x2

� @ �xz
@x@z

¼ 0 ð5:2Þ

Equations 5.1 and 5.2, being independent of material properties, can be applied to
both elastic and plastic behaviour.
The rigorous solution of a particular problem requires that the equations of equilib-

rium and compatibility are satisfied for the given boundary conditions; an appro-
priate stress–strain relationship is also required. In the theory of elasticity [18] a linear
stress–strain relationship is combined with the above equations. In general, however,
soils are non-homogeneous, exhibit anisotropy and have non-linear stress–strain
relationships which are dependent on stress history and the particular stress path