﻿ Craig's Soil Mechanics 7th Edition - Mecânica dos Solos 2 - 37
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# Craig's Soil Mechanics 7th Edition

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```was shown that at small strains the shear stress in the soil at
the wall of the expanding cavity is given by

� ¼ "c dp
d"c

ð5:8Þ

and at larger strains by

� ¼ dp
d½lnð�V=VÞ� ð5:9Þ

both the circumferential and volumetric strains being defined with respect to the
reference state. Equations 5.8 and 5.9 can be used to derive the entire stress–strain
curve for the soil.
An analysis for the interpretation of pressuremeter tests in sands has been given by

Hughes et al. [9]. The analysis enables values for the angle of shearing resistance (�0)
and the angle of dilation ( ) to be determined. A comprehensive review of the use of
pressuremeters, including examples of test results and their application in design, has
been given by Mair and Wood [10].

5.2 STRESSES FROM ELASTIC THEORY

The stresses within a semi-infinite, homogeneous, isotropic mass, with a linear stress–
strain relationship, due to a point load on the surface, were determined by Boussinesq
in 1885. The vertical, radial, circumferential and shear stresses at a depth z and a
horizontal distance r from the point of application of the load were given. The stresses
due to surface loads distributed over a particular area can be obtained by integration
from the point load solutions. The stresses at a point due to more than one surface
load are obtained by superposition. In practice, loads are not usually applied directly
on the surface but the results for surface loading can be applied conservatively in
problems concerning loads at a shallow depth.
A range of solutions, suitable for determining the stresses below foundations, is

given in the following sections. Negative values of loading can be used if the stresses
due to excavation are required or in problems in which the principle of superposition is
used. The stresses due to surface loading act in addition to the in-situ stresses due to the
self-weight of the soil.

Referring to Figure 5.5(a), the stresses at X due to a point load Q on the surface are as
follows:

�z ¼ 3Q
2
z2

1

1þ ðr=zÞ2
( )5=2

ð5:10Þ

144 Stresses and displacements

Figure 5.5 (a) Stresses due to point load and (b) variation of vertical stress due to point load.

�r ¼ Q
2

3r2z

ðr2 þ z2Þ5=2
� 1� 2v
r2 þ z2 þ zðr2 þ z2Þ1=2

( )
ð5:11Þ

�� ¼ � Q
2

ð1� 2vÞ z
ðr2 þ z2Þ3=2

� 1
r2 þ z2 þ zðr2 þ z2Þ1=2

( )
ð5:12Þ

�rz ¼ 3Q
2

rz2

ðr2 þ z2Þ5=2
( )

ð5:13Þ

It should be noted that when v ¼ 0:5 the second term in Equation 5.11 vanishes and
Equation 5.12 gives �� ¼ 0.
Equation 5.10 is used most frequently in practice and can be written in terms of an

influence factor Ip, where

Ip ¼ 3
2

1

1þ ðr=zÞ2
( )5=2

Then,

�z ¼ Q
z2
Ip

Values of Ip in terms of r/z are given in Table 5.1. The form of the variation of �z with z
and r is illustrated in Figure 5.5(b). The left-hand side of the figure shows the variation
of �z with z on the vertical through the point of application of the load Q (i.e. for
r ¼ 0); the right-hand side of the figure shows the variation of �z with r for three
different values of z.
It should be noted that the expression for �z (Equation 5.10) is independent of

elastic modulus (E ) and Poisson’s ratio (�).

Table 5.1 Influence factors for vertical stress due to point load

r/z Ip r/z Ip r/z Ip

0.00 0.478 0.80 0.139 1.60 0.020
0.10 0.466 0.90 0.108 1.70 0.016
0.20 0.433 1.00 0.084 1.80 0.013
0.30 0.385 1.10 0.066 1.90 0.011
0.40 0.329 1.20 0.051 2.00 0.009
0.50 0.273 1.30 0.040 2.20 0.006
0.60 0.221 1.40 0.032 2.40 0.004
0.70 0.176 1.50 0.025 2.60 0.003

146 Stresses and displacements

Referring to Figure 5.6(a), the stresses at point X due to a line load ofQ per unit length
on the surface are as follows:

�z ¼ 2Q

z3

ðx2 þ z2Þ2 ð5:14Þ

�x ¼ 2Q

x2z

ðx2 þ z2Þ2 ð5:15Þ

�xz ¼ 2Q

xz2

ðx2 þ z2Þ2 ð5:16Þ

Figure 5.6 (a) Stresses due to line load and (b) lateral pressure due to line load.

Stresses from elastic theory 147

Equation 5.15 can be used to estimate the lateral pressure on an earth-retaining
structure due to a line load on the surface of the backfill. In terms of the dimensions
given in Figure 5.6(b), Equation 5.15 becomes

�x ¼ 2Q

h

m2n

ðm2 þ n2Þ2

However, the structure will tend to interfere with the lateral strain due to the load Q
and to obtain the lateral pressure on a relatively rigid structure a second load Q must
be imagined at an equal distance on the other side of the structure. Then, the lateral
pressure is given by

px ¼ 4Q

h

m2n

ðm2 þ n2Þ2 ð5:17Þ

The total thrust on the structure is given by

Px ¼
Z 1
0

pxh dn ¼ 2Q

1

m2 þ 1 ð5:18Þ

Strip area carrying uniform pressure

The stresses at point X due to a uniform pressure q on a strip area of width B and
infinite length are given in terms of the angles
and � defined in Figure 5.7(a).

�z ¼ q

f
þ sin
cosð
þ 2�Þg ð5:19Þ

�x ¼ q

f
� sin
cosð
þ 2�Þg ð5:20Þ

�xz ¼ q

fsin
sinð
þ 2�Þg ð5:21Þ

Contours of equal vertical stress in the vicinity of a strip area carrying a uniform
pressure are plotted in Figure 5.8(a). The zone lying inside the vertical stress contour of
value 0.2q is described as the bulb of pressure.

Strip area carrying linearly increasing pressure

The stresses at point X due to pressure increasing linearly from zero to q on a strip area
of width B are given in terms of the angles
and � and the lengths R1 and R2, as
defined in Figure 5.7(b).

148 Stresses and displacements

Figure 5.7 Stresses due to (a) uniform pressure and (b) linearly increasing pressure, on strip area.

Figure 5.8 Contours of equal vertical stress: (a) under strip area and (b) under square area.

�z ¼ q

x

B

� 1

2
sin 2�

� �
ð5:22Þ

�x ¼ q

x

B

� z

B
ln
R21
R22

þ 1
2
sin 2�

� �
ð5:23Þ

�xz ¼ q
2

1þ cos 2� � 2 z
B

�
ð5:24Þ

Circular area carrying uniform pressure

The vertical stress at depth z under the centre of a circular area of diameter D ¼ 2R
carrying a uniform pressure q is given by

�z ¼ q 1� 1
1þ ðR=zÞ2

( )3=224
3
5 ¼ qIc ð5:25Þ

Values of the influence factor Ic in terms of D/z are given in Figure 5.9.
The radial and circumferential stresses under the centre are equal and are given by

�r ¼ �� ¼ q
2

ð1þ 2vÞ � 2ð1þ vÞ
f1þ ðR=zÞ2g1=2

þ 1
f1þ ðR=zÞ2g3=2

" #
ð5:26Þ

Figure 5.9 Vertical stress under the centre of a circular area carrying a uniform pressure.

150 Stresses and displacements

Rectangular area carrying uniform pressure

A solution has been obtained for the vertical stress at depth z under a corner of a
rectangular area of dimensions mz and nz (Figure 5.10) carrying a uniform pressure q.
The solution can be written in the form

�z ¼ qIr

Values of the influence factor Ir in terms of m and n are given in the chart, due to
Fadum [5], shown in Figure 5.10. The factors m and n are interchangeable. The chart
can also be used for a strip area, considered as a rectangular area of infinite length.
Superposition enables any area based on rectangles to be dealt with and enables the
vertical stress under any point within or outside the area to be obtained.

Figure 5.10 Vertical stress under a corner of a rectangular area carrying a uniform pressure.
(Reproduced from R.E. Fadum (1948) Proceedings of the 2nd International Conference
of SMFE, Rotterdam, Vol. 3, by permission of Professor Fadum.)

Stresses from elastic theory 151

Contours of equal vertical stress in the vicinity of a square area carrying a uniform
pressure are plotted in Figure 5.8(b). Influence factors for �x and �y (which depend on v)
are given in Ref. [16].

Influence chart for vertical stress

Newmark [12] constructed an influence chart, based on the Boussinesq solution,
enabling the vertical stress to be determined```