it can be obtained that
퐴=−
푎^3 푞
4퐺
. (38)
Substituting the above expression of퐴back into ( 33 ), one
has
푓=−
푎^3 푞
4퐺
(
1
푅3푡
+
1
푅4푡
), (39)
andthenthestressanddisplacementcomponentscanbe
obtained by substituting ( 39 )into( 27 )to( 32 ).
For the second part (ii), there is only shear stress on the
horizontal ground surface. In the same way, the stress har-
monic function푔(푟,푧)isgiventoeliminatetheshearonthe
boundary, and the solutions of the stress and displacement
are as follows:
휎(2)푟 =2퐺( 2
휕^2 푔
휕푟^2
−
2 ]
푟
휕푔
휕푟
+푧
휕^3 푔
휕푟^2 휕푧
), (40)
휎(2)휃 =2퐺(
1
푟
휕푔
휕푟
+2]
휕^2 푔
휕푟^2
+
푧
푟
휕^2 푔
휕푟휕푧
), (41)
휎(2)푧 =2퐺푧
휕^3 푔
휕푧^3
, (42)
휏푟푧(2)=2퐺(
휕^2 푔
휕푟휕푧
+푧
휕^3 푔
휕푟휕푧^2
), (43)
푢(2)푟 =2(1−])
휕푔
휕푟
+푧
휕^2 푔
휕푟휕푧
, (44)
푢(2)푧 =−(1−2])
휕푔
휕푟
+푧
휕^2 푔
휕푧^2
. (45)
Similarly, the expression of stress function can be written
as follows:
푔=퐵(
1
푅3푡
+
1
푅4푡
), (46)
where퐵is a constant. The stress function푔satisfies the equi-
librium∇^2 푔=0, and the corresponding stress components
on the ground surface푧=0are
휎푧(2)
儨儨儨
儨儨푧=0=0,
휏푟푧(2)
儨儨儨
儨儨푧=0=2퐺퐵[
3ℎ(푡+푟)
푅3푡耠
+
3 (ℎ−푛)(푡+푟−푚)
푅耠4푡
].
(47)
Accordingly,itcanbeobtainedthat퐵=−푎^3 푞/4퐺.
The final expression for the stress function is then as
follows:
푔=−
푎^3 푞
4퐺
(
1
푅3푡
+
1
푅4푡
). (48)
Substituting ( 48 )into( 40 )to( 45 ), the stress and displace-
mentcomponentscanbeobtained.
ozrr
o 2o 1R 4R 3
p(r,z)zzFigure 5: The stress distribution of the slope surface.3.4. The Correction of the Stresses on the Slope Surface.
Considering a virtual source, positive mirror image of the
actual cavity with respect to a slope surface will produce the
same normal stresses and opposite shear stresses as the actual
cavity, the shear stress is eliminated, and the normal stress
increases doubly, as shown inFigure 5.
Using the method of coordinate transformation, the
normal and shear stresses can be obtained as follows:휎푧耠儨儨儨儨푧耠=0=3푎^3 푞[
[
(푡 + 푟耠sin훽)2
+푐^2
푅^50−
1
푅^30
]
]
×cos^2 훽−3푎^3 푞⋅[(푟耠cos훽−ℎ)(푡+푟耠sin훽) + 푐푑
2푅^50]
×sin2훽 +푎^3 푞
2
[
[
4
푅^30
−
3(푡 + 푟耠sin훽)2
+3푐^2
푅^50]
]
휏푟耠푧耠儨儨儨儨푧耠=0=0,
(49)
where푐=푡+푟耠sin훽−푚,푑=푟耠cos훽−ℎ+푛,and푅 0 =
√(푡 + 푟耠sin훽)^2 +(푟耠cos훽−ℎ)^2.
In order to satisfy the free surface boundary condition
as much as possible, the Boussinesq solution has been
introduced to correct the normal stress. Stress푞耠is applied
on the surface of the slope, which is equal to the normal stress
휎푧耠|푧耠=0in value but opposite in direction.
As shown inFigure 6,표푙is the intersection line of the
slope and the horizontal plane. A small element with an area
of휌푑휃푑휌is taken out of the slopelor耠(i.e.,or耠inFigure 5)
for analysis. Further, the force exerted on the small element is
equal toq耠휌d휃d휌. Using the Boussinesq solutions, the stress
and displacement components of soil under the action of the
force (q耠휌d휃d휌) can be derived as follows: