The Runge-Kutta algorithm of this type ( 22 )isanumeri-
cal method of fifth order, where훿is the stepsize of difference
and퐹(푋,푠) =퐾푋+푝. For convenience of computation, this
formulationmayberewritteninthefollowingform:
푋푛+1=퐺푛푋푛+퐻푛훿. (23)Aswecansee,thevalueoffunctionatpoint(푛+1)can
be determined from the value of function at point (푛), where
푛=0represents the beginning point of calculation and푛=푚
represents the end point of calculation.
In the above equation,퐺푛and퐻푛canbeobtainedfrom
the following recursion formula:
퐾푛(푗)=퐾(푠̃ 푛+훿푗), 푃푛(푗)=푝(푠푛+훿푗),훼 1 =훼 2 =훼 3 =
1
2
,훼 4 =1,
훽 1 =훽 4 =
1
6
,훽 2 =훽 3 =
1
3
,
훿푗=(
푗
∑
푘=1훾푘)훿, 훾 1 =훾 3 =0, 훾 2 =훾 4 =
1
2
,
㨐⇒ 훿 1 =0, 훿 2 =훿 3 =
1
2
훿, 훿 4 =훿,
(24)
퐺푛=퐼+
4
∑
푗=1훽푗퐺(푗),퐻푛=
4
∑
푗=1훽푗퐻(푗),
퐺(푗)=(훿퐾(푗))(퐼+훼푗퐺(푗−1)), 퐺(0)=0,
퐻(푗)=(훿퐾(푗))훼푗퐻(푗−1)+푃(푗),퐻(0)=0,
(25)
where퐼is the identity matrix.
5.2.2. Determination of the Initial Vector푋 0 .The initial value
is the start point of the recursion formula. Now, we discuss
inthefollowinghowtoobtaintheinitialvector푋 0 by using
the recursion formula of ( 23 ) and imposing the boundary
conditions at pile head and base.
Considering the recursion formula of ( 23 ),푋푛can be
expressed in terms of푋 0 as follows:
푋푛=퐷(푛)푋 0 +퐹(푛). (26)In the case of푛=0,wehave퐷(0) =퐼,퐹(0) =0and
substituteitintotherecursionformulaof ( 23 ). We get
푋푛+1=퐺푛(퐷(푛)푋 0 +퐹(푛))+퐻푛훿. (27)Itcanberewrittenasfollows:푋푛+1=(퐺푛퐷(푛))푋 0 +(퐺푛퐹(푛)+퐻푛훿),푋푛=퐷(푛)푋 0 +퐹(푛)㨐⇒ 푋푛+1=퐷(푛+1)푋 0 +퐹(푛+1),
푋푛+1=(퐺푛퐷(푛))푋 0 +(퐺푛퐹(푛)+퐻푛훿),
푋푛+1=퐷(푛+1)푋 0 +퐹(푛+1).
(28)
XnXn− 1Xn− 2Xn+1Xn+2uuuPilePileBeamXnFigure 3: Scheme of local coordinate system transformation
between pile and connection beam.Comparingtheabovetwoequations,therecursionfor-
mula for퐷(푛)and퐹(푛)is obtained as follows:퐷(0)=퐼, 퐷(푛+1)=퐺푛퐷(푛),
퐹(0)=0, 퐹(푛+1)=퐺푛퐹(푛)+퐻푛훿.
(29)
Now considering the case of boundary point:푋푚 =
퐷(푚)푋 0 +퐹(푚), we substitute the boundary conditions at end
point퐶푋 0 =표,퐷푋푚=표into the above equation. This leads
to the equation to solve for푋 0 :[
퐶
퐷퐷(푚)
]푋 0 =[
표
−퐷퐹(푚)
]. (30)
The above set of linear algebraic equations can be solved
for푋 0 by using the method of Gaussian elimination with
pivot selection. Once푋 0 is known,푋푛can be obtained in
sequence using the recursion formula of ( 23 ).5.2.3. Coordinate Transformation between Pile and Beam.
When solving the problem of double-row portal piles using
the recursion formula of ( 23 ), due to the direction change
of the axes of the pile and the beam at the connection point
(shown inFigure 3), first we need to distinguish퐾and푝in
the recursion formula for the two connected segments. And
then in order to satisfy the equilibrium of internal forces and
maintain the continuity of displacements at the connection
point, we need to introduce the so-called connection matrix
to the corresponding formula when dealing with퐷푚,퐹푚,and
푋푛,recursively.
The centroidal axes’ rotation from pile to beam at con-
nection point means mathematically that the local coordinate
system rotates clockwise by훽degree at the connection point
(shown inFigure 3).