Deduce from ( 20 )as휕푢푤
휕푧^2
=
휕^2 푢
휕푧^2
+퐶
휕^3 푢
휕푡휕푧^2
−퐷
휕^4 푢
휕푧^4
. (21)
Substituting ( 19 )into( 21 ) is deduced as퐷
휕^4 푢
휕푧^4
−퐶
휕^3 푢
휕푡휕푧^2
−(퐵+1)
휕^2 푢
휕푧^2
+퐴
휕푢
휕푡
=0. (22)
So far, the governing equations as ( 20 )and( 22 )are
obtained.
Thesolutioncanbeobtainedbyusingthemethodof
separation of variables for ( 22 ), which can be expressed as
푢(푧,푡)=휎 0
∞
∑
푚=12
푀
sin(푀
퐻
푧)푒−훽푚푡, (23)
푢푤(푧,푡)=휎 0
∞
∑
푚=12
푀
[1 − 퐶훽푚+퐷(
푀
퐻
)
2
]⋅sin(푀
퐻
푧)푒−훽푚푡,
(24)
푈=1−
∞
∑
푚=12
푀^2
푒−훽푚푡, (25)
where푀 = ((2푚 + 1)/2)휋,(푚 = 0,1,2,...),
훽푚=퐸(훼+푛^2 −1+푌){푘V푤푘V(
푀
퐻
)
2×[
푟^2 푒퐹푐
2푘ℎ
+
(푛^2 −1)푅
(1 − 푎^2 )8푘ℎ푤
]
+[(푛^2 −1)푘V+푘V푤]}
×(훾푤
{
{
{
(푛^2 −푎^2 )
21−푎^2
⋅(
퐻
푀
)
2
+[(푛^2 −1)푘V푤+푘V]×[
푟^2 푒퐹푐
2푘ℎ
+
(푛^2 −1)푅
8푘ℎ푤
]
}
}
}
)
−1.(26)
In order to verify the rationality of the assumptions and
the methods of the consolidation in this paper, the consolida-
tion solution can be degraded.
kwkhksokr(r)rw rs re rFigure 3: Five variation patterns of horizontal permeability coeffi-
cient in smear zone.When푋=1and푎=0,훽푚changes into훽푚=퐸(푛^2 −1+푌){푘V푤푘V(
푀
퐻
)
2×[
푟푒^2 퐹푐
2푘ℎ
+
(푛^2 −1)푟^2 푤
8푘ℎ푤
]
+[(푛^2 −1)푘V+푘V푤]}
×(훾푤{푛^4 (
퐻
푀
)
2
+[(푛^2 −1)푘V푤+푘V]×[
푟푒^2 퐹푐
2푘ℎ
+
(푛^2 −1)푟^2 푤
8푘ℎ푤
]})
−1
.(27)
This is the consolidation solution of common composite
foundation provided by Lu et al. [ 30 ] that considered the
radial flow within the pile.
When푌=1,푘V=푘V푤,and푘ℎ=푘ℎ푤,훽푚and푈change
into훽푚=
푘V퐸
훾푤
(
푀
퐻
)
2
=푐V(푀
퐻
)
2
,푈=1−
∞
∑
푚=12
푀^2
푒−푀
(^2) 푇V
,
(28)
where푐V=푘V퐸/훾푤,푇V=푐V푡/퐻^2.
This is the one-dimensional consolidation solution of
Terzaghi’s theory. The rationality of the consolidation solu-
tion in this paper can be reflected through the above answer
to degradation.
From ( 13 ), ( 23 ), and ( 25 ), it can be seen that the influence
of horizontal permeability coefficient in the influenced zone
to the consolidation solution is reflected mainly by the
parameter퐹푐, which is related to the changing pattern of
horizontal permeability coefficient.Figure 3displays a typical