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Research ArticleStability of Three-Dimensional Slurry Trenches
withInclined Ground Surface: A Theoretical Study
Xiao-Fei Jin,1 Shu-Ting Liang,1 and Xiao-Jun Zhu2
1School of Civil Engineering, Southeast University, Nanjing
210096, China2Architectural Design and Research Institute,
Southeast University, Nanjing 210096, China
Correspondence should be addressed to Shu-Ting Liang;
[email protected]
Received 29 April 2015; Accepted 24 May 2015
Academic Editor: João M. P. Q. Delgado
Copyright © 2015 Xiao-Fei Jin et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
Stability of slurry trenches is an important issue during the
construction of the groundwater cutoff walls and diaphragm walls,
andthus gradually draws attention. In this paper, a theoretical
method for a three-dimensional trench model with an inclined
groundwas proposed. Based on the Coulomb-type force equilibrium, a
safety factor assessing the stability was derived.The results
showedthat the existing two-dimensional model was conservative
compared to the present three-dimensional model; concretely, a
greaterinclined angle of the inclined ground and trench length
decreased the safety factor. This work could be used to assess the
trenchstability for both 2D and 3D cases with inclined ground
surfaces.
1. Introduction
Slurry trenches are oftenused as a hydraulic barrier to
preventthe groundwater from flowing into the trenches during
theconstruction of groundwater cutoff walls and diaphragmwalls, and
the slurry in the excavated trenches plays animportant role in
providing a lateral supporting force tothe trench walls before
backfilling. Therefore, slurry trenchstability is a major concern.
However, most of the existingstudies focused on the ground
movements and stress reliefinduced by diaphragm wall construction:
Poh and Wongconcluded that evaluated aspects of performance
includelateral and vertical soil movements during the
constructionof the test wall panel [1]; Ng and Yan confirmed the
hor-izontal arching and downward load transfer mechanismsduring
diaphragm wall installation using the finite differencemethod [2];
Gourvenec and Powrie investigated the impactof three-dimensional
effects and panel length on horizontalground movements and changes
in lateral stress during thesequential installation of a number of
diaphragm wall panels[3]; Ng and Lei derived an analytical solution
for calculatinghorizontal stress changes and displacements caused
by theexcavation for a diaphragm wall panel [4]; Schäfer
andTriantafyllidis investigated the influence of a diaphragm
wall
construction on the stress field in a soft clayey soil [5];Arai
et al. conducted to examine ground movement andstress after the
installation of circular diaphragm walls andsoil excavation within
the walls [6]; Conti et al. studiedthe mechanisms of load transfer
and the deformations ofthe ground during slurry trenching and
concreting in drysand [7]; Comodromos et al. proposed a new
approach forsimulating the excavation and construction of
subsequentpanels to investigate the effects from the installation
ofdiaphragm walls on the surrounding and adjacent buildings[8]; Lei
et al. proposed an approximate analytical solutionto predict ground
surface settlements along the centre-lineperpendicular to a
slurry-supported diaphragm wall panel[9]. Until now, the problem of
the slurry trench stability grad-ually receivesmuch attention in
underground engineering. Infact, in the early stage, a
two-dimensional limit equilibriummethod for trench stability based
on a simpleCoulombwedgefor dry soil conditions was developed [10];
later, the methodwas extended to account for influences of
different levels ofslurry in the trench and groundwater in the
cohesionlesssoil [11]. Moreover, based on the Rankin theorem, the
trenchstability was assessed by considering the pressures
fromsoils, hydrostatic slurry, and groundwater [12]. However,these
mentioned methods treated the trench stability as
Hindawi Publishing CorporationAdvances in Materials Science and
EngineeringVolume 2015, Article ID 362160, 7
pageshttp://dx.doi.org/10.1155/2015/362160
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2 Advances in Materials Science and Engineering
a two-dimensional problem and neglected stabilizing forcesor
shear forces acting on both ends of the failure mass, whichmay
produce conservative results. Indeed, considering thereal case
including contribution of the shear forces on theboth ends of the
failure wedge, three-dimensional methodderived from the
two-dimensional limit equilibrium theorywas already used to analyze
the trench stability; Prater [13]and Washbourne [14] proposed
planar sides to enhancedexisting two-dimensional models. Nonplanar
sides also wereproposed [15, 16] andmay yield a closer
approximation to thecurved geometry observed for trench failure
surfaces [14, 17];Fox presented Coulomb-type force equilibrium
analyses forgeneral three-dimensional stability of a
slurry-supported [18].
Stability analyses using the methods from the abovemen-tioned
works have already taken into account the influencesof several
primary design parameters, including trench lengthand depth, slurry
depth and density, groundwater depth,and tension crack. However, if
cutoff walls locate below aninclined ground, it is necessary to
consider the effect ofthe inclined ground surface on the trench
stability. In thisregard, Li et al. investigated the influence of
inclination angleon trench stability by a two-dimensional model,
and theyshowed that it was unconservative to neglect ground
surfaceinclination when analyzing trench stability [19].
Here, extending the two-dimensional model of Li et al.[19] to
three-dimensional case, we presented an analyticalsolution of the
safety factor and critical failure angle for aslurry-supported
trench with an inclined ground, and anexample was finally discussed
to illustrate the variation of thesafety factor and critical
failure angle with different inclinedangle, trench length, and
groundwater depth.
2. Theory
A three-dimensional model of a failure wedge with length𝑙 and
force analysis of the model are shown in Figure 1.In this model,
the top surface of slurry is assumed to behigher than that of the
groundwater surface (i.e., ℎ
𝑠> ℎ𝑤),
and the trench walls were considered to be impermeableafter the
excavation, because there existed a layer of grad-ually thickened
filter cake-like bentonite, which results inindependency between
the pore pressure in the soil andslurry pressure in the trench. It
is worth mentioning that theshear strength due to soil suctions
above the groundwatersurface can be included by specifying an
appropriate valuefor 𝑐1, which is called effective stress cohesion
intercept abovegroundwater surface, according to the total cohesion
method[18, 20]. Besides, the wedge was assumed to be rigid,
andtemporary loads were considered uniformly distributed, andthe
geometric parameters 𝑙1, 𝑙2, 𝑎, 𝑏, and 𝑐 in Figure 1 satisfy𝑙1 =
(cos𝛽/ sin(𝜃 − 𝛽))ℎ, 𝑙2 = ℎ𝑤csc𝜃, 𝑎 = 𝑙1cos𝜃 − ℎcot𝜃,𝑏 = 𝑙1cos𝜃,
and 𝑐 = ℎcot𝜃, in which 𝜃 and 𝛽 are angles madeby the failure plane
and inclined ground with respect to thehorizontal plane,
respectively, and ℎ is the depth of the trench.
Considering the three-dimensional case different fromthe
two-dimensional case by Li et al. [19], the shear resistanceforce 𝑆
acting on end planes of the wedge is treated to beparallel to the
failure plane (see Figures 1(b) and 1(c)). The
shear resistance force is assumed to be uniformly
distributedalong the failure plane, and its magnitude
proportionallyvaries; thus, the progressive failure effect can be
neglected andthe safety factor with respect to shear failure of
each end planeis equal to that of the failure plane [18]. The total
shear forceon each end plane of the wedge is calculated as
𝑆 = 𝑆1 + 𝑆2, (1)
where 𝑆1 and 𝑆2 are shear resistant forces acting on twoportions
of each end plane of the wedge, respectively, thatis, above (I) and
below (II and III) the horizontal plane; seeFigure 1(c).
For the portion (I) above the horizontal plane of thewedge,
through the force analysis, 𝑆1 can be expressed as
𝑆1 =1𝐹(∫
𝑏
0∫
0
− tan𝛽𝑥(𝑐
1 +𝜎
ℎ1,1 tan𝜙
) 𝑑𝑧 𝑑𝑥
−∫
𝑏
𝑐
∫
0
(−𝑏 tan𝛽/𝑎)(𝑥−𝑐)(𝑐
1 + 𝜎
ℎ2,1tan𝜙
) 𝑑𝑧 𝑑𝑥)
=ℎ2
2𝐹(𝑐
1 tan𝛽cot 𝜃
tan 𝜃 − tan𝛽+14𝐾𝛾1 tan𝜙
(tan𝛽)2
⋅ ℎcot 𝜃
(tan 𝜃 − tan𝛽)2+𝐾𝑞 tan𝜙 ( 1
tan 𝜃 − tan𝛽
− cot 𝜃)) ,
(2)
where 𝑞 is the uniformly distributed temporary load on
theinclined ground surface,𝐹 is the safety factor, 𝛾1 and 𝑐
1 are theunit weight and effective stress cohesion intercept of
the soilabove the groundwater surface, respectively, 𝜙 is the
effectivestress friction angle of the entire soil profile, 𝐾 taken
as theat-rest lateral earth pressure coefficient 𝐾0 = 1 − sin𝜙
is theaverage lateral earth pressure coefficient for the end
planes ofthe failure wedge, and the horizontal effective stresses
are
𝜎
ℎ1,1 = 𝐾(𝑞+ 𝛾1𝑥
𝑏(𝑧 + 𝑥 tan𝛽)) ,
𝑥 ∈ [0, 𝑏] , 𝑧 ∈ [− tan𝛽𝑥, 0] ,
𝜎
ℎ2,1 = 𝐾(𝑞+ 𝛾1𝑥 − 𝑐
𝑎(𝑧+
𝑏 tan𝛽𝑎
(𝑥 − 𝑐))) ,
𝑥 ∈ [𝑐, 𝑏] , 𝑧 ∈ [−𝑏 tan𝛽𝑎
(𝑥 − 𝑐) , 0] .
(3)
For the shear resistance force 𝑆2 acting on the portion (IIand
III) below the horizontal plane, it can be calculated as
𝑆2 =1𝐹(∫
𝑧𝑤
0∫
−𝑐𝑧/ℎ+𝑐
0(𝑐
1 +𝜎
ℎ1,2 tan𝜙
) 𝑑𝑥 𝑑𝑧
+∫
ℎ
𝑧𝑤
∫
−𝑐𝑧/ℎ+𝑐
0(𝑐
2 +𝜎
ℎ2,2 tan𝜙
) 𝑑𝑥 𝑑𝑧)
=cot 𝜃2𝐹
{𝐾 tan𝜙
3[3𝑞ℎ2 + 𝛾2ℎ
3
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Advances in Materials Science and Engineering 3
x
yz
𝜃
𝛽
𝛾1
𝛾2
c1
c2
End p
lane
Inclined ground surface
l
(a)
𝜃
𝛽
Q q
zs
hhs
hw
l1
l2
zw
Trench with filled slurry
Failu
re pla
ne
Inclined g
round sur
face
Horizontal plane
Top surface of slurry
Groundwater surface
(b)
S1
S2W
I
I
II
IIIPSU
Q
T
N
xa
bcdx
𝜃
𝛽
𝛽
(c)
Figure 1: Three-dimensional failure wedge. (a) Schematic model
of failure wedge, (b) definitions of geometric parameters, and (c)
forceanalysis.
+ 𝑧𝑤(𝛾1 − 𝛾2) (𝑧
2𝑤− 3ℎ𝑧𝑤+ 3ℎ2)] + 𝑧
𝑤(𝑐
1 − 𝑐
2)
⋅ (2ℎ − 𝑧𝑤) + 𝑐
2ℎ2} ,
(4)
where 𝛾2 and 𝑐
2 are the unit weight and effective stresscohesion intercept of
the soil below the groundwater surface,respectively, 𝑧
𝑤is the distance between the horizontal plane
and groundwater surface, and the horizontal effective
stressesare
𝜎
ℎ1,2 = 𝐾(𝑞+ 𝛾1𝑥2 tan𝛽𝑏
+ 𝛾1𝑧) , 𝑧 ∈ [0, 𝑧𝑤] ,
𝜎
ℎ2,2 = 𝐾(𝑞+ 𝛾1𝑥2 tan𝛽𝑏
+ 𝛾1𝑧𝑤 + 𝛾2 (𝑧 − 𝑧𝑤)) ,
𝑧 ∈ [𝑧𝑤, ℎ] .
(5)
Substituting (2) and (4) into (1), the total shear
resistanceforce applied on each end plane is
𝑆 =Φ
𝐹, (6)
where
Φ =12tan𝛽ℎ2 cot 𝜃
(tan 𝜃 − tan𝛽)2[𝑐
1 (tan 𝜃 − tan𝛽)
+14𝐾𝛾1 tan𝜙
tan𝛽ℎ] +𝐾 tan𝜙
6[
3𝑞ℎ2
tan 𝜃 − tan𝛽+ 𝛾2ℎ
3 cot 𝜃
+ 𝑧𝑤(𝛾1 − 𝛾2) (𝑧
2𝑤− 3ℎ𝑧𝑤+ 3ℎ2) cot 𝜃]
+cot 𝜃2
[𝑧𝑤(𝑐
1 − 𝑐
2) (2ℎ − 𝑧𝑤) + 𝑐
2ℎ2] .
(7)
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4 Advances in Materials Science and Engineering
For the entire wedge, the force equilibrium in the direc-tions
normal and tangential to the failure plane yields
∑𝐹𝑁= 𝑁
+𝑈− (𝑊+𝑄) cos 𝜃 −𝑃𝑆sin 𝜃 = 0,
∑𝐹𝑇= 2𝑆 +𝑇− (𝑊+𝑄) sin 𝜃 +𝑃
𝑆cos 𝜃 = 0,
(8)
where𝑊 is the soil weight, 𝑄 is the equivalent concentratedforce
by 𝑞, 𝑃
𝑆is the lateral force by filled slurry, 𝑈 is the
hydrostatic groundwater force, 𝑁 is the effective normalforce,
and 𝑇 is the shear force on the failure plane. Throughthe
calculations of the forces in (8), they are expressed
𝑊+𝑄
𝑙= Γ =
ℎ
tan 𝜃 − tan𝛽(𝑞+
ℎ𝛾12)
+(𝛾2 − 𝛾1) ℎ
2𝑤
2 tan 𝜃,
𝑇
𝑙=
1𝐹(Ψ+
𝑁 tan𝜙
𝑙) =
1𝐹(𝑐
1ℎcos𝛽
sin (𝜃 − 𝛽)
+ (𝑐
2 − 𝑐
1) ℎ𝑤 csc 𝜃 +𝑁 tan𝜙
𝑙) ,
𝑈
𝑙= Λ =
𝛾𝑤csc 𝜃ℎ2
𝑤
2,
𝑃𝑆
𝑙= Ω =
𝛾𝑠ℎ2𝑠
2
(9)
in the expression of 𝑃𝑆/𝑙, and 𝛾
𝑠and ℎ
𝑠are unit weight and
the height of the filled slurry, respectively.Solving (8) for
the safety factor of thewedge, the following
is obtained:
𝐹 =2Φ
(Γ sin 𝜃 − Ω cos 𝜃) 𝑙+
Ψ
Γ sin 𝜃 − Ω cos 𝜃
+ (Γ
Γ tan 𝜃 − Ω+
Ω
Γ − Ω cot 𝜃
−Λ
Γ sin 𝜃 − Ω cos 𝜃) tan𝜙.
(10)
The critical angle 𝜃cr of the failure plane that corresponds
tothe minimum safety factor 𝐹
𝑆for the failure wedge is found
by taking 𝑑𝐹/𝑑𝜃 = 0, and the equation can be solved by
aniterative method. It is noted that the solution of 𝜃cr
shouldlocate in the range 45∘ ≤ 𝜃cr ≤ 90
∘; otherwise 𝐹𝑆will be less
than zero if 𝜃cr < 45∘ [18].
Considering a peculiar case of cohesionless soils (i.e., 𝑐1
=𝑐
2 = 0), the expression of 𝐹 reduces to
𝐹 = 𝑓 (𝜃) tan𝜙, (11)
where
𝑓 (𝜃) =2Π
(Γ sin 𝜃 − Ω cos 𝜃) 𝑙+
Γ
Γ tan 𝜃 − Ω
+Ω
Γ − Ω cot 𝜃−
Λ
Γ sin 𝜃 − Ω cos 𝜃,
Π = Φ cot𝜙,
Π =18𝐾𝛾1 (tan𝛽)
2ℎ3 cot 𝜃(tan 𝜃 − tan𝛽)2
+16
⋅ 𝐾 [3𝑞ℎ2
tan 𝜃 − tan𝛽
+ 𝛾2ℎ3 cot 𝜃
+ 𝑧𝑤(𝛾1 − 𝛾2) (𝑧
2𝑤− 3ℎ𝑧𝑤+ 3ℎ2) cot 𝜃] .
(12)
Equation (11) indicates that 𝜃cr is independent of 𝜙, which
is
consistent with the analytical result for the horizontal
groundby Fox [18].The proposed analyticalmethod is also
applicablewhen the ground is sloping away from the trench (i.e., 𝛽
< 0).Under this condition, the intersection point of the
inclinedground surface and the failure plane should be above
thegroundwater surface (i.e., 𝑙
1> 𝑙2).
3. Example
Here, we investigate the influences of different
groundinclinations and trench lengths on the trench stability.
Thegeometric and physical parameters of the trench, soil, andslurry
are from Fox [21], namely, ℎ = 20m, ℎ
𝑠= 20m,
𝑧𝑤= 3m, 𝛾
𝑠= 11.8 kN/m3, 𝛾1 = 19.0 kN/m
3, 𝑐2 = 0 kPa,𝛾2 = 20.0 kN/m
3, 𝜙 = 37∘, and 𝑞 = 0 kN/m2. Moreover,𝑐
1 = 10 kPa is used for the soil above the groundwater surfaceto
consider the soil suction effect.
Figure 2 shows the relationships of the minimum safetyfactor
𝐹
𝑆or critical angle 𝜃cr versus inclined angle 𝛽 for the
two-dimensional case. In Figure 2(a), we can see that 𝐹𝑆is
negatively relevant with 𝛽, and 𝐹𝑆decreases from 1.48 to
1.19
as 𝛽 varies from 0∘ to 15∘. This indicates that the assumptionof
a horizontal ground (i.e., 0∘) for the inclined ground resultsin an
overestimation of the trench stability, which may resultin sliding
failure of the excavated panel. When the inclinedangle 𝛽 comes down
into negative value for a specific case𝛽 = −10
∘, 𝐹𝑆equals 1.74, which is greater than 1.48 for
𝛽 = 0∘. And more, the present prediction is comparable
to that based on Rankin’s earth pressure theory (the dashedline
in Figure 2(a), Morgenstern and Amir-Tahmasseb, 1965[11]), which
validates our derived theory, and is also equal tothe calculated
values with the method proposed by Li et al.[19]. With the given
𝜃cr, engineers can calculate the size ofthe potential failure mass,
and reinforcement design can bemade if the trench stability is not
satisfactory; for example,when 𝛽 = 0∘ and 𝜃cr = 58.56
∘, the area of the end cross-section is 22.6% greater than that
for 𝜃cr = 45
∘
+ 𝜙 (= 63.5∘).
In Figure 2(b), it is readily seen that 𝜃cr decreases and
then
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Advances in Materials Science and Engineering 5
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25 30 35−10 −5
Fact
or o
f saf
ety,FS
𝛽 (∘)
FS for 𝜃 = 45∘ + 𝜙/2
FS with 𝛽
(a)
55
56
57
58
59
60
61
62
Criti
cal f
ailu
re an
gle,𝜃
cr
c1 = 0
c1 = 10
0 5 10 15 20 25 30 35−10 −5𝛽 (∘)
(b)
Figure 2: Influences of the inclined angle 𝛽 on (a) 𝐹𝑆and (b)
𝜃cr.
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 20 40 60 80 100
Fact
or o
f saf
ety,FS
Trench length, L (m)
c1 = 0, 𝛽 = 0 c1 = 0, 𝛽 = 5c1 = 10kPa, 𝛽 = 5c
1 = 10kPa, 𝛽 = 0
(a)
57
58
59
60
61
62
63
64
0 20 40 60 80 100Trench length, L (m)
Criti
cal f
ailu
re an
gle,𝜃
cr
c1 = 0, 𝛽 = 0c1 = 0, 𝛽 = 5c1 = 10kPa, 𝛽 = 5 c
1 = 10kPa, 𝛽 = 0
(b)
Figure 3: Influences of the trench length 𝐿 on (a) 𝐹𝑆and (b)
𝜃cr.
increases as 𝛽 changes from −10∘ to 35∘, and the minimumvalue
𝜃cr = 56
∘ is obtained when 𝛽 = 25∘. This again showsthat it is necessary
to determine 𝜃cr instead of using 𝜃cr =45∘ + 𝜙. Plus, Figure 2(b)
also shows that increasing 𝑐1 tendsto a reduced value of 𝜃cr.
The influences of the trench length 𝐿 on 𝐹𝑆and 𝜃cr are
plotted in Figure 3. It shows that 𝐹𝑆is obviously affected
by
the trench length 𝐿, and the results of the present
three-dimensional case move toward the two-dimensional case as𝐿
tends to infinity. Figure 3(a) shows that increasing 𝑐1 and𝛽
results in increasing and decreasing 𝐹
𝑆, respectively. For
this example, values of the trench length 𝐿 which correspondto
𝐹𝑆equalling 1.8 are 39.3m, 34.8m, 29.3m, and 22.6m for
𝑐
1 = 10 kPa (𝛽 = 0), 𝑐
1 = 0 kPa (𝛽 = 0), 𝑐
1 = 10 kPa (𝛽 = 5),and 𝑐1 = 0 kPa (𝛽 = 5), respectively,
whereas, correspondingto 𝑐1 and 𝛽, 𝐹𝑆 is calculated as 1.48, 1.44,
1.37, and 1.32 for thetwo-dimensional case, respectively. Figure
3(b) shows that 𝜃crdecreases with increasing trench length 𝐿, and
increasingboth 𝑐1 and 𝛽 results in decreasing 𝜃cr, and 𝜃cr is
apparentlyaffected by 𝛽. For this example, values of 𝜃cr that
correspondto 𝐿 = 40 are 59.8m, 58.4m, 60.1m, and 58.5m for 𝑐1 =10
kPa (𝛽 = 0), 𝑐1 = 10 kPa (𝛽 = 5), 𝑐
1 = 0 kPa (𝛽 = 0),and 𝑐1 = 0 kPa (𝛽 = 5), respectively.
Finally, based on the parameters 𝐿 = 40m and 𝛽 = 5∘,the
influence of the rising groundwater on the safety factor isshown in
Figure 4. It is noticed that larger 𝑧
𝑤represents lower
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6 Advances in Materials Science and Engineering
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5
Fact
or o
f saf
ety,FS
Depth to groundwater, zw (m)
c1 = 10
c1 = 0
L = 40m, 𝛽 = 5∘c1 = 10, FS with zw
c1 = 0, FS with zw
Figure 4: Influence of the depth of groundwater 𝑧𝑤on 𝐹𝑆.
groundwater surface. Then, Figure 4 shows that increasingboth
𝑧
𝑤and 𝑐1 produces an increasing safety, and this can be
easily understood.
4. Conclusions
In this paper, we have presented an analytical solutionof the
safety factor for the three-dimensional model of aslurry trench
with an inclined ground surface. The solutionincludes several
parameters, such as the inclined angle ofthe inclined ground
surface, trench length and depth, slurrydepth, temporary load, and
groundwater surface elevation.The results showed that increasing
inclined angle 𝛽 andtrench length 𝐿 results in decreasing the
safety factor, butincreasing the depth to groundwater 𝑧
𝑤and suction effect
𝑐
1 produces an increase in the safety factor; meanwhile,
theexisting two-dimensional model was conservative comparedto the
present three-dimensional model. The study could beuseful to assess
the trench stability for both 2D and 3D caseswith inclined ground
surface.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgments
Thefinancial support received fromNational Natural
ScienceFoundation of China (NSFC), Grant no. 51208181, and
KeyProjects in the National Science & Technology Pillar
Pro-gram during the Twelfth Five-Year Plan Period, Grant
no.2011BAJ10B08, is gratefully acknowledged. The authors alsothank
ProfessorDr. QiangChen, Southeast University, for theEnglish
correction.
References
[1] T. Y. Poh and I. H.Wong, “Effects of construction of
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