퐵=퐵푇. At last, we develop the following geometric and
constitutive equations:
푑푢
푑푠=
푁
퐸퐴
,
푑V
푑푠
=
훼푄
퐺퐴
+휑,
푑휑
푑푠
=
푀
퐸퐼
.
(9)
2.5. Matrix Form of the Governing Equations.For the sake
of convenience of problem solving, combining the three
equilibrium differential equations ( 3 )andthethreegeometric
and constitutive equations ( 9 ) leads to a system of six
equations:
푑푁
푑푠=푘푠퐻퐷푢 + 푞휏,
푑푄
푑푠
=푘푛퐻푏V+푞푛,
푑푀
푑푠
=−푄,
푑푢
푑푠
=
푁
퐸퐴
,
푑V
푑푠
=
훼푄
퐺퐴
+휑,
푑휑
푑푠
=
푀
퐸퐽
.
(10)
Let퐾=[[
[
[
[[
[
[
[
[[
[
[
[[
[
[
[
[
000퐻푘푠퐷00
000 0퐻푘푛푏0
0−10 0 00
1
퐸퐴
00 0 00
0
훼
퐺퐴
00 01
00
1
퐸퐼
000
]]
]
]
]]
]
]
]
]]
]
]
]]
]
]
]
]
,
푋={푁푄푀푢V 휑}
푇
,푝={푞휏 푞푛 0000 }
푇
.(11)
Then, we can use matrix notation to present the above
equations ( 10 )intheform:
푑푋
푑푠=퐾푋+푝=퐹(푋, 푠), (12)
where퐾isthecoefficientssquarematrixofsixorderand푋
and푝represent two column matrices.
This system of six independent differential equations can
be solved for six unknown functions (three independent
forces and three independent displacements).
3. Boundary Conditions
Asmentionedabove,therearetotalsixunknownstobedeter-
mined (푁,푄,푀,푢,V,휑). Therefore, six boundary conditions
are needed for the problem solving.
The boundary conditions for ( 12 ) are determined accord-
ingtothewayinwhichthepile’sheadandbasearesupported
or restrained. There are three conditions at the base point and
three conditions at the head point. We use matrix notation to
present these boundary conditions in the following form:퐶푋
儨儨儨
儨儨푆=0=표,
퐷푋
儨儨儨
儨儨푆=퐿=표,
(13)
where푆=0indicates the beginning point of calculation
(the pile’s base point),푆=퐿indicates the end point of
calculation(thepile’sheadpoint),퐶isthematrixofboundary
condition on the beginning point, and퐷is the matrix of
boundary condition on the end point. They are3×6matrices.
In this study, the following possible pile end conditions were
considered.(1) Free head (allows both displacement and rotation):
푁=푄=0,푀=0. The corresponding matrix of
boundary condition is[
[
100000
010000
001000
]
]
×
{{
{{
{{
{
{{
{{
{{
{
{
푁 푄 푀 푢 V 휑
}}
}}
}}
}
}}
}}
}}
}
}
=표. (14)
(2) In case of bottom end hinged (allows rotation without
displacement):푢=V=0,푀=0. The corresponding
matrix of boundary condition is[
[
001000
000100
000010
]
]
×
{{
{{
{{
{
{{
{{
{{{
{
푁 푄 푀 푢 V 휑
}}
}}
}}
}
}}
}}
}}}
}
=표. (15)
(3) In case of bottom end fixed (allows neither displace-
ment nor rotation):푢=V =0,휑=0.The
corresponding matrix of boundary condition is[
[
000100
000010
000001
]
]
×
{{
{{
{{{
{{
{{{
{{
{
푁 푄 푀 푢 V 휑
}}
}}
}}}
}}
}}}
}}
}
=표. (16)
(4) In case of bottom end partially hinged (allows rota-
tion without vertical displacement):푢=0,푄=0,