rrrC
ddRRBzolrA(r,z)Figure 6: The stress analysis of calculation point by the force on the
small element.
푑휏푟耠耠푧耠=
3푞耠푟耠耠푧耠2
2휋푅耠5
휌푑휃푑휌,
푑푢푧耠=
(1+])푞耠
2휋퐸푅耠
[2(1−])+
푧耠2
푅耠2
]휌푑휃푑휌,
푑푢푟耠耠=
(1+])푞耠
2휋퐸푅耠
[
푟耠耠푧耠
푅耠2
−
(1−2])푟耠耠
푅耠+푧耠
]휌푑휃푑휌,
(50)
in which
푟耠耠2=휌^2 +푟耠2−2휌푟耠cos(휃 −휋
2
),
푅耠2=푟耠耠2+푧耠2=휌^2 +푟耠2−2휌푟耠cos(휃 −휋
2
)+푧耠2.
(51)
The directions of stress and displacement induced by the
force on the small element are not always in accordance
with coordinate axes (Figure 6). By means of coordinate
transformation, the solutions of stresses and displacements in
the coordinate systemr耠oz耠(Figure 5)canbeobtained.After
that, integrating results with respect to the whole sloping
ground surface can lead to the following:
휎푧耠=∫
휋0∫
∞0푑휎푧耠,
휎휃耠=∫
휋0∫
∞0(푑휎푟耠耠sin^2 휃耠+푑휎휃耠耠cos^2 휃耠),휎푟耠=∫
휋0∫
∞0(푑휎푟耠耠cos^2 휃耠+푑휎휃耠耠sin^2 휃耠),휏푟耠푧耠=∫
휋0∫
∞0푑휏푟耠耠푧耠cos휃耠,푢푧耠=∫
휋0∫
∞0푑푢푧耠,
푢푟耠=∫
휋0∫
∞0푑푢푟耠耠cos휃耠.(52)
Hence, the stress and displacement components in the
coordinate systemrozcan be derived as follows:휎푟(3)=휎푟耠sin^2 훽+휎푧耠cos^2 훽−휏푟耠푧耠sin2훽,휎푧(3)=휎푟耠cos^2 훽+휎푧耠sin^2 훽+휏푟耠푧耠sin2훽,휏푟푧(3)=
1
2
(휎푟耠−휎푧耠)sin2훽 − 휏푟耠푧耠cos2훽,휎(3)휃 =휎휃耠,
푢(3)푟 =푢푟耠sin훽−푢푧耠cos훽,푢(3)푧 =푢푟耠cos훽+푢푧耠sin훽,(53)
where휃耠=arctan(휌sin(휃 − (휋/2))/(푟耠−휌cos(휃 − (휋/2)))),
푟耠=푟sin훽+푧cos훽,and푧耠=−푟cos훽+푧sin훽.
With ( 53 ), the stress and displacement fields induced by
the stressesq耠on the slope can be derived. Using a virtual
source of the actual cavity at the image point, the lateral
deformations of soil around spherical cavity were predicted
by Rao et al. [ 22 ]. Meanwhile, the Cerruti solutions are used
to eliminate the shear stresses produced by the expansion
of both the real and the imaginary spherical cavities in
an infinite space. However, their results are inappropriate
because of the incorrect using of the Cerruti solutions. It is
knownthatadistributedstressisclearlynotapointforcein
the elementary sense. Hence, the stress should be integrated
with respect to the slope surface when it is substituted into
the Cerruti solutions.
Accordingly, when considering the effects of both hor-
izontal and sloping free boundaries, the final results of the
expansion of a single spherical cavity near a slope (Figure 2)
can be obtained by superposition of all the parts stresses and
displacements:휎푟=휎푟(0)+휎(1)푟 +휎(2)푟 +휎푟(3),
휎푧=휎푧(0)+휎푧(1)+휎(2)푧 +휎(3)푧,
휎휃=휎휃(0)+휎휃(1)+휎(2)휃 +휎(3)휃,
휏푟푧=휏푟푧(0)+휏푟푧(1)+휏푟푧(2)+휏푟푧(3),
푢푟=푢(0)푟 +푢(1)푟 +푢(2)푟 +푢(3)푟 ,
푢푧=푢(0)푧 +푢(1)푧 +푢(2)푧 +푢(3)푧.
(54)
4. Discussion of the Solutions
The presence of the horizontal and sloping free surfaces is
considered in this paper. Consequently, the present solutions
have more extensive applications compared with solutions of
Keer et al. [ 16 ]. According toSection 3, the solutions for this
problem can be derived in four steps (cavity expansion in an
infinite medium, cavity and its image expansion in an infinite
medium, and the corrections of stresses on horizontal surface