and푀=0. The corresponding matrix of boundary
condition is[
[
010000
001000
000100
]
]
×
{{
{{{
{{
{{
{{{
{{
{
푁 푄 푀 푢 V 휑
}}
}}
}}}
}}
}}}
}}
}
=표. (17)
(5) Elastic vertical support at the bottom end:푁=퐾V⋅푢⋅
Area,푄=0,and푀=0. The corresponding matrix
of boundary condition is[
[
100−퐾V⋅Area 00
010 0 00
001 0 00]
]
×
{{
{{{
{{
{{
{{{
{{
{
푁 푄 푀 푢 V 휑
}}
}}}
}}
}}
}}}
}}
}
=표, (18)
where퐾Vis the modulus of vertical compressibility.Areais
the cross-sectional area of the pile.
4. Definition of Boundary Value Problem
Next, we impose the boundary conditions ( 13 )atthepile
head and base upon the derived new governing differential
equations ( 12 )todefineaboundaryvalueproblemofthe
following equations:
푑푋
푑푠=퐾푋+푝,
퐶푋|푆=0=표,
퐷푋|푆=퐿=표.
(19)
To this end, the response of double-row portal stabilizing pile
is mathematically idealised as the boundary value problem of
( 19 ).
Thus, many numerical methods to solve the ordinary
differential equations can be adopted to solve the boundary
value problem of ( 19 ).
It should be noted that the existence and uniqueness
of solution for the boundary value problem of ( 19 )should
be mathematically proved. This matter is however outside
thescopeofthewriter’smajor.Accordingtothephysical
character of the problem, we can imagine that the solution
exists and is unique. The solution can be validated through
comparative studies.
5. Method of Solution
5.1. Uniformity Preprocessing.The orders of magnitude of the
section internal forces (푁,푄,푀)aresomuchhigherthan
those of the displacements (푢,V,휑) that numerical solving of
the equations may meet singularity difficulty. So for reasons
of numerical stability, it is necessary to perform uniformity
preprocessing for the order of magnitude of the element in the
coefficient matrix퐾. We multiply the displacements (푢,V,휑)
by퐸and substitute the original displacement variables by
the expressions (퐸푢,퐸V,퐸휑). So, we redefine two variables as
follows:푋=̃
{{
{{
{{{
{{
{{
{{{
{
푁 푄 푀 푢 V 휑
}}
}}
}}}
}}
}}
}}}
}
=
{{
{{
{{{
{{
{{
{{{
{
푁
푄
푀
퐸푢
퐸V
퐸휑
}}
}}
}}}
}}
}}
}}}
}
,
퐾=̃
[
[[
[
[
[[
[
[
[
[[
[
[
[[
[
[
[
[
[
000
퐻푘푠퐷
퐸
00
000 0
퐻푘푛푏
퐸
0
0−10 0 0 0
1
퐴
00 0 0 0
0
퐸훼
퐺퐴
00 01
00
1
퐼
000
]
]]
]
]
]]
]
]
]
]]
]
]
]]
]
]
]
]
]
.
(20)
Finally, we obtain the following system of ordinary
differential equations:푑푋̃
푑푠=퐾̃푋+푝,̃
퐶⋅푋̃
儨儨儨
儨儨푆=0=표,
퐷⋅푋̃
儨儨
儨儨儨푆=퐿=표.
(21)
This system of six independent differential equations
subjected to boundary conditions can be numerically solved
for six unknown functions (three forces and three displace-
ments).5.2. The Runge-Kutta Finite Difference Algorithm.The Runge-
Kutta algorithm is commonly used for the solution of the
ordinary differential equation of the form푑푋/푑푠 = 퐹(푋,푠).
So,itischosentosolve( 21 ).5.2.1. Derivation of the Recursion Formula.The following
finite difference formula ( 22 )isoneformatoftheRunge-
Kutta methods:푋푛+1=푋푛+