According to the principle of equilibrium of internal
forces, we get
[푁
푄
]=[cos휃−sin휃
sin휃 cos휃][푁
푄
]. (31)
And the bending moment remains unchanged.
According to the principle of vector analysis, we get
[푢
V
]=[cos휃−sin휃
sin휃 cos휃][푢
V
]. (32)
And the rotation displacement remains unchanged.
Combining the above equations, the corresponding
transformation matrix for the local coordinate system rotat-
ing clockwise by훽degree can be expressed as follows:
[
[
[[
[
[
[[
[
[
푁푛
푄푛
푀푛
푢푛
V푛
휑푛
]
]
]]
]
]
]]
]
]
=
[
[[
[
[
[[
[
cos훽−sin훽0 0 0 0
sin훽 cos훽0 0 0 0
001000
000 cos훽−sin훽0
000 sin훽 cos훽0
000001]
]]
]
]
]]
]
×
[
[[
[
[
[
[[
[
푁푛
푄푛
푀푛
푢푛
V푛
휑푛
]
]]
]
]
]
]]
]
,
푋푛=퐶푛⋅푋푛.
(33)
As we can see, the above connection matrix is the so-
called orthogonal matrix whose inverse is its transposed
matrix.
As shown inFigure 3, the node of difference (no.푛)isthe
connection node. First we need to distinguish퐾and푝in the
recursion formula for the two connected segments and then
insert the connection matrix for transformation from푋푛to
푋푛when dealing with the퐷푚,퐹푚,and푋푛,recursively.The
detailed procedure is as follows.
(1) For the calculation of푋푛,푋푖+1=퐺푖푋푖+퐻푖훿, (푖=1,2,...,푛−1) (34)푋푛=퐶푛푋푛=퐶푛(퐺푛−1푋푛−1+퐻푛−1훿),where퐶푛is
the transformation matrix for point푛.Thus,푋푛+1=
퐺푛푋푛+퐻푛훿. Next,푋푖+1=퐺푖푋푖+퐻푖훿, (푖=푛+1,푛+2,...푚), (35)where푚indicates the end point of calculation.
(2) For the calculation of퐷푚:because푋푛=퐷(푛)푋 0 +퐹(푛),
and
푋푛+1=퐺푛푋푛+퐻푛훿=퐺푛(퐶푛푋푛)+퐻푛훿=퐺푛(퐶푛(퐷(푛)푋 0 +퐹(푛))) + 퐻푛훿=퐺푛퐶푛퐷(푛)푋 0 +퐺푛퐶푛퐹(푛)+퐻푛훿
=퐷(푛+1)푋 0 +퐹(푛+1)
㨐⇒ {
퐷(푛+1)=퐺푛퐶푛퐷(푛)
퐹(푛+1)=퐺푛퐶푛퐹(푛)+퐻푛훿,
(36)
So,theaboveprocedurealsoappliestocalculationof
퐷(푚)as follows:퐷(푖+1)=퐺푖퐷(푖), (푖=1,2,...,푛−1),퐷(푛+1)=퐺푛퐶푛퐷(푛),퐷(푖+1)=퐺푖퐷(푖), (푖=푛+1,푛+2,...푚).(37)
(3) For the calculation of퐹푚,퐹(푖+1)=퐺푖퐹(푖)+퐻푖훿, (푖=1,2,...,푛−1),
퐹(푛+1)=퐺푛퐶푛퐹(푛)+퐻푛훿,
퐹(푖+1)=퐺푖퐹(푖)+퐻푖훿, (푖=푛+1,푛+2,...푚),
(38)
where푚denotes the end point of calculation.5.2.4. The Solution Flow Process.In short, the proposed
solution procedure involves four main steps:(1) calculating the value of퐺푛and퐻푛using the given
Equation ( 25 );
(2) calculating퐷(푚)and퐹(푚)using the given recursion
formula of ( 29 );
(3) calculating the vector푋 0 by solving linear algebraic
Equations ( 30 );
(4) calculating푋푛using the given recursion formula of
( 23 ).Because the equations and solution formula are all given
in form of matrices, a simple computer program has been
writtenontheplatformofMATLABtorunthisprocedure.
At last, we can get the shear, bending-moment, and deflection
diagram along the pile.6. Verification
The practical examples of portal double-row piles used to
stabilizeanpotentiallandslide(showninFigure 4)arecon-
sidered herein to verify the developed numerical calculation
techniques. Soil strength parameters used in the stability
analysis are from laboratory shear testing on the undisturbed
soil samples. The resisting (shear) force required to achieve
the desired safety factor and transferred by the pile is
estimated to be 2147 kN/m. Before the pile is installed, the
slope is approaching limit state and the safety factor can be
assumedtobeone.Thevaluesof푐and휑can be determined
based on experiment data and satisfying this limit state
condition. Then, the required resisting force 2147 kN/m can
be obtained using back analysis. When this additional force is
applied to the specified place of the landslide, the safety factor
calculated using the limit equilibrium method can achieve
the desired value 1.3. The manual digging discrete reinforced
concrete piles were designed to be installed at a spacing of
4mtoincreasethefactorofsafetyofthewholeslopetothe
required value of 1.3.