foundation and the sand-gravel shell are used, which can be
defined as
휕
휕푟[
푟⋅푘푟(푟)
훾푤
휕푢푠
휕푟
]=−푟(
휕휀V
휕푡
+
푘V
훾푤
휕^2 푢푠
휕푧^2
),푟푤≤푟≤푟푒
(8)
휕
휕푟
(
푟⋅푘ℎ푤
훾푤
휕푢푤
휕푟
)=−푟(
휕휀V
휕푡
+
푘V푤
훾푤
휕^2 푢푤
휕푧^2
), 푟푐≤푟≤푟푤,
(9)
where훾푤istheunitweightofwater.
2.3.3. Solving Conditions.Consider the following
A푟=푟푒,휕푢푠/휕푟 = 0;
B푟=푟푤,푢푠=푢푤;
C푟=푟푐,휕푢푤/휕푟 = 0; (considering the concrete-cored
pile as an impervious pile)
D푟=푟푤,푘푟(푟푤)(휕푢푠/휕푟) = 푘ℎ푤(휕푢푤/휕푟); (the radial
velocity of pile- soil interface is equal).The vertical boundary conditions can be written as
E푧=0,푢푤=0,푢=0;
F푧=퐻,휕푢푤/휕푧 = 0,휕푢/휕푧 = 0; (in single-drainage
condition)
G푧=퐻,푢=0,푢=0; (in double-drainage condition).Assuming that there is no deformation of the pile and the
soil at initial time and the external load is bore all by pore
water, so the initial condition can be written as푡=0,푢(푧,0) =
휎 0.
2.4. The Establishment of the Governing Equations.Equation
( 10 ) can be obtained by integrating both sides of ( 8 )about푟
and using solving conditionA:
휕푢푠
휕푟=
훾푤
2푘ℎ
(
휕휀V
휕푡
+
푘V
훾푤
휕^2 푢푠
휕푧^2
)[
푟푒^2
푟푓(푟)
−
푟
푓(푟)
]. (10)
Integratingbothsidesof ( 10 )about푟again and using solving
conditionB, the following can be obtained:
푢푠(푟)=푢푤儨儨儨儨푟=푟푤+
훾푤
2푘ℎ
(
휕휀V
휕푡
+
푘V
훾푤
휕^2 푢푠
휕푧^2
)
×[푟푒^2 퐴 0 (푟)−퐵 0 (푟)],
(11)
퐴 0 (푟)=∫
푟푟푤푑휉
휉푓(휉)
,퐵 0 (푟)=∫
푟푟푤휉푑휉
푓(휉)
. (12)
Equation ( 11 )issubstitutedinto( 2 ):푢푠=푢푤儨儨儨儨푟=푟푤+
푟^2 푒훾푤퐹푐
2푘ℎ
(
휕휀V
휕푡
+
푘V
훾푤
휕^2 푢푠
휕푧^2
), (13)
where퐹푐=2(퐴 1 푟^2 푒−퐵 1 )/푟푒^2 푟푤^2 (푛^2 −1),퐴 1 =∫
푟푒
푟푤푟퐴^0 (푟) 푑푟,and퐵 1 =∫
푟푒
푟푤푟퐵^0 (푟)푑푟.By integrating both sides of ( 9 )about푟and using solving
conditionC,thefollowingisobtained:휕푢푤
휕푟=
훾푤
2푘ℎ푤
(
휕휀V
휕푡
+
푘V푤
훾푤
휕^2 푢푤
휕푧^2
)(
푟푐^2
푟
−푟). (14)
Equation ( 14 )isintegratedabout푟both sides to get푢푤(푟)=푢푤儨儨儨儨푟=푟푤+
훾푤
2푘ℎ푤
(
휕휀V
휕푡
+
푘V푤
훾푤
휕^2 푢푤
휕푧^2
)
⋅[푟^2 푤(
1
2
+푎^2 ln푟
푟푤
)−
1
2
푟^2 ].
(15)
Equation ( 15 )issubstitutedinto( 3 ):푢푤=푢푤儨儨儨儨푟=푟푤+
훾푤푅
8푘ℎ푤
(
휕휀V
휕푡
+
푘V푤
훾푤
휕^2 푢푤
휕푧^2
), (16)
where푅=푟^2 푤[(4푎^2 /(1−푎^2 ))(ln푟푤−푎^2 ln푎푟푤)−2푎^2 −4푎^2 ln푟푤−
(1 − 푎^4 )/(1 − 푎^2 )+2].
Equation ( 13 )minus( 16 ) meanwhile combining the
expression of푢in ( 4 )and( 5 )deduces푢−푢푤=−
(푛^2 −1)
퐸(훼+푛^2 −1+푌)
(
푟^2 푒퐹푐훾푤
2푘ℎ
−
훾푤푅
8푘ℎ푤
)
휕푢
휕푡
+
푟^2 푒퐹푐푘V
2푘ℎ
휕^2 푢
휕푧^2
−[
(1 − 푎^2 )
(푛^2 −푎^2 )
푟푒^2 퐹푐푘V
2푘ℎ
+
(푛^2 −1)
(푛^2 −푎^2 )
푅푘V푤
8푘ℎ푤
]
휕^2 푢푤
휕푧^2
.
(17)
Substituting ( 10 )and( 14 )intosolvingconditionD,
meanwhile combining ( 5 ), the equation can be deduced as−(푛^2 −푎^2 )
2퐸(훼+푛^2 −1+푌)
휕푢
휕푡
=(푎^2 −1)
푘V푤
훾푤
휕^2 푢푤
휕푧^2
−(푛^2 −1)
푘V
훾푤
휕^2 푢푠
휕푧^2
.
(18)
Substituting the expression of푢in ( 4 )into( 18 )weget
equation휕^2 푢푤
휕푧^2=퐴
휕푢
휕푡
−퐵
휕^2 푢
휕푧^2
, (19)
where퐴=(푛^2 −푎^2 )훾푤/(1 − 푎^2 )(푘V푤−푘V)퐸(훼 + 푛^2 −1+푌),
퐵=(푛^2 −푎^2 )푘V/(1 − 푎^2 )(푘V푤−푘V).
Equation ( 19 )issubstitutedinto( 17 )andweget푢푤=푢+퐶
휕푢
휕푡
−퐷
휕^2 푢
휕푧^2
, (20)
where 퐶 = (훾푤[(푛^2 −1)푘V푤 +푘V]/퐸(훼 + 푛^2 −1+
푌)(푘V푤−푘V))[푟푒^2 퐹푐/2푘ℎ+ 푅(푛^2 −1)/8푘ℎ푤]퐷=(푘V푤푘V/(푘V푤−
푘V))[푟푒^2 퐹푐/2푘ℎ+(푛^2 − 1)푅/(1 − 푎^2 )8푘ℎ푤].