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: JoaΜo M. P. Q. Delgado
Copyright Β© 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]; SchaΜfer 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
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 lengthπ 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., β
π > βπ€),
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 π1, 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 π1, π2, π, π, and π in Figure 1 satisfyπ1 = (cosπ½/ sin(π β π½))β, π2 = βπ€cscπ, π = π1cosπ β βcotπ,π = π1cosπ, and π = βcotπ, in which π and π½ are angles madeby the failure plane and inclined ground with respect to thehorizontal plane, respectively, and β is the depth of the trench.
Considering the three-dimensional case different fromthe two-dimensional case by Li et al. [19], the shear resistanceforce π 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
π = π1 + π2, (1)
where π1 and π2 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, π1 can be expressed as
π1 =1πΉ(β«
π
0β«
0
β tanπ½π₯(π
1 +π
β1,1 tanπ
) ππ§ ππ₯
ββ«
π
π
β«
0
(βπ tanπ½/π)(π₯βπ)(π
1 + π
β2,1tanπ
) ππ§ ππ₯)
=β2
2πΉ(π
1 tanπ½cot π
tan π β tanπ½+14πΎπΎ1 tanπ
(tanπ½)2
β βcot π
(tan π β tanπ½)2+πΎπ tanπ ( 1
tan π β tanπ½
β cot π)) ,
(2)
where π is the uniformly distributed temporary load on theinclined ground surface,πΉ is the safety factor, πΎ1 and π
1 are theunit weight and effective stress cohesion intercept of the soilabove the groundwater surface, respectively, π is the effectivestress friction angle of the entire soil profile, πΎ taken as theat-rest lateral earth pressure coefficient πΎ0 = 1 β sinπ
is theaverage lateral earth pressure coefficient for the end planes ofthe failure wedge, and the horizontal effective stresses are
π
β1,1 = πΎ(π+ πΎ1π₯
π(π§ + π₯ tanπ½)) ,
π₯ β [0, π] , π§ β [β tanπ½π₯, 0] ,
π
β2,1 = πΎ(π+ πΎ1π₯ β π
π(π§+
π tanπ½π
(π₯ β π))) ,
π₯ β [π, π] , π§ β [βπ tanπ½π
(π₯ β π) , 0] .
(3)
For the shear resistance force π2 acting on the portion (IIand III) below the horizontal plane, it can be calculated as
π2 =1πΉ(β«
π§π€
0β«
βππ§/β+π
0(π
1 +π
β1,2 tanπ
) ππ₯ ππ§
+β«
β
π§π€
β«
βππ§/β+π
0(π
2 +π
β2,2 tanπ
) ππ₯ ππ§)
=cot π2πΉ
{πΎ tanπ
3[3πβ2 + πΎ2β
3
Advances in Materials Science and Engineering 3
x
yz
π
π½
πΎ1
πΎ2
c1
c2
End p
lane
Inclined ground surface
l
(a)
π
π½
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
π
π½
π½
(c)
Figure 1: Three-dimensional failure wedge. (a) Schematic model of failure wedge, (b) definitions of geometric parameters, and (c) forceanalysis.
+ π§π€(πΎ1 β πΎ2) (π§
2π€β 3βπ§π€+ 3β2)] + π§
π€(π
1 β π
2)
β (2β β π§π€) + π
2β2} ,
(4)
where πΎ2 and π
2 are the unit weight and effective stresscohesion intercept of the soil below the groundwater surface,respectively, π§
π€is the distance between the horizontal plane
and groundwater surface, and the horizontal effective stressesare
π
β1,2 = πΎ(π+ πΎ1π₯2 tanπ½π
+ πΎ1π§) , π§ β [0, π§π€] ,
π
β2,2 = πΎ(π+ πΎ1π₯2 tanπ½π
+ πΎ1π§π€ + πΎ2 (π§ β π§π€)) ,
π§ β [π§π€, β] .
(5)
Substituting (2) and (4) into (1), the total shear resistanceforce applied on each end plane is
π =Ξ¦
πΉ, (6)
where
Ξ¦ =12tanπ½β2 cot π
(tan π β tanπ½)2[π
1 (tan π β tanπ½)
+14πΎπΎ1 tanπ
tanπ½β] +πΎ tanπ
6[
3πβ2
tan π β tanπ½+ πΎ2β
3 cot π
+ π§π€(πΎ1 β πΎ2) (π§
2π€β 3βπ§π€+ 3β2) cot π]
+cot π2
[π§π€(π
1 β π
2) (2β β π§π€) + π
2β2] .
(7)
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
βπΉπ= π
+πβ (π+π) cos π βππsin π = 0,
βπΉπ= 2π +πβ (π+π) sin π +π
πcos π = 0,
(8)
whereπ is the soil weight, π is the equivalent concentratedforce by π, π
πis the lateral force by filled slurry, π is the
hydrostatic groundwater force, π is the effective normalforce, and π is the shear force on the failure plane. Throughthe calculations of the forces in (8), they are expressed
π+π
π= Ξ =
β
tan π β tanπ½(π+
βπΎ12)
+(πΎ2 β πΎ1) β
2π€
2 tan π,
π
π=
1πΉ(Ξ¨+
π tanπ
π) =
1πΉ(π
1βcosπ½
sin (π β π½)
+ (π
2 β π
1) βπ€ csc π +π tanπ
π) ,
π
π= Ξ =
πΎπ€csc πβ2
π€
2,
ππ
π= Ξ© =
πΎπ β2π
2
(9)
in the expression of ππ/π, and πΎ
π and β
π are unit weight and
the height of the filled slurry, respectively.Solving (8) for the safety factor of thewedge, the following
is obtained:
πΉ =2Ξ¦
(Ξ sin π β Ξ© cos π) π+
Ξ¨
Ξ sin π β Ξ© cos π
+ (Ξ
Ξ tan π β Ξ©+
Ξ©
Ξ β Ξ© cot π
βΞ
Ξ sin π β Ξ© cos π) tanπ.
(10)
The critical angle πcr of the failure plane that corresponds tothe minimum safety factor πΉ
πfor the failure wedge is found
by taking ππΉ/ππ = 0, and the equation can be solved by aniterative method. It is noted that the solution of πcr shouldlocate in the range 45β β€ πcr β€ 90
β; otherwise πΉπwill be less
than zero if πcr < 45β [18].
Considering a peculiar case of cohesionless soils (i.e., π1 =π
2 = 0), the expression of πΉ reduces to
πΉ = π (π) tanπ, (11)
where
π (π) =2Ξ
(Ξ sin π β Ξ© cos π) π+
Ξ
Ξ tan π β Ξ©
+Ξ©
Ξ β Ξ© cot πβ
Ξ
Ξ sin π β Ξ© cos π,
Ξ = Ξ¦ cotπ,
Ξ =18πΎπΎ1 (tanπ½)
2β3 cot π(tan π β tanπ½)2
+16
β πΎ [3πβ2
tan π β tanπ½
+ πΎ2β3 cot π
+ π§π€(πΎ1 β πΎ2) (π§
2π€β 3βπ§π€+ 3β2) cot π] .
(12)
Equation (11) indicates that πcr is independent of π, 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., π½ < 0).Under this condition, the intersection point of the inclinedground surface and the failure plane should be above thegroundwater surface (i.e., π
1> π2).
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, β = 20m, β
π = 20m,
π§π€= 3m, πΎ
π = 11.8 kN/m3, πΎ1 = 19.0 kN/m
3, π2 = 0 kPa,πΎ2 = 20.0 kN/m
3, π = 37β, and π = 0 kN/m2. Moreover,π
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 πΉ
πor critical angle πcr versus inclined angle π½ for the
two-dimensional case. In Figure 2(a), we can see that πΉπis
negatively relevant with π½, and πΉπdecreases from 1.48 to 1.19
as π½ varies from 0β to 15β. This indicates that the assumptionof a horizontal ground (i.e., 0β) for the inclined ground resultsin an overestimation of the trench stability, which may resultin sliding failure of the excavated panel. When the inclinedangle π½ comes down into negative value for a specific caseπ½ = β10
β, πΉπequals 1.74, which is greater than 1.48 for
π½ = 0β. And more, the present prediction is comparable
to that based on Rankinβs 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 πcr, engineers can calculate the size ofthe potential failure mass, and reinforcement design can bemade if the trench stability is not satisfactory; for example,when π½ = 0β and πcr = 58.56
β, the area of the end cross-section is 22.6% greater than that for πcr = 45
β
+ π (= 63.5β).
In Figure 2(b), it is readily seen that πcr decreases and then
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 35β10 β5
Fact
or o
f saf
ety,FS
π½ (β)
FS for π = 45β + π/2
FS with π½
(a)
55
56
57
58
59
60
61
62
Criti
cal f
ailu
re an
gle,π
cr
c1 = 0
c1 = 10
0 5 10 15 20 25 30 35β10 β5π½ (β)
(b)
Figure 2: Influences of the inclined angle π½ on (a) πΉπand (b) πcr.
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, π½ = 0 c1 = 0, π½ = 5c1 = 10kPa, π½ = 5c
1 = 10kPa, π½ = 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,π
cr
c1 = 0, π½ = 0c1 = 0, π½ = 5c1 = 10kPa, π½ = 5 c
1 = 10kPa, π½ = 0
(b)
Figure 3: Influences of the trench length πΏ on (a) πΉπand (b) πcr.
increases as π½ changes from β10β to 35β, and the minimumvalue πcr = 56
β is obtained when π½ = 25β. This again showsthat it is necessary to determine πcr instead of using πcr =45β + π. Plus, Figure 2(b) also shows that increasing π1 tendsto a reduced value of πcr.
The influences of the trench length πΏ on πΉπand πcr are
plotted in Figure 3. It shows that πΉπis obviously affected by
the trench length πΏ, and the results of the present three-dimensional case move toward the two-dimensional case asπΏ tends to infinity. Figure 3(a) shows that increasing π1 andπ½ results in increasing and decreasing πΉ
π, respectively. For
this example, values of the trench length πΏ which correspondto πΉπequalling 1.8 are 39.3m, 34.8m, 29.3m, and 22.6m for
π
1 = 10 kPa (π½ = 0), π
1 = 0 kPa (π½ = 0), π
1 = 10 kPa (π½ = 5),and π1 = 0 kPa (π½ = 5), respectively, whereas, correspondingto π1 and π½, πΉπ is calculated as 1.48, 1.44, 1.37, and 1.32 for thetwo-dimensional case, respectively. Figure 3(b) shows that πcrdecreases with increasing trench length πΏ, and increasingboth π1 and π½ results in decreasing πcr, and πcr is apparentlyaffected by π½. For this example, values of πcr that correspondto πΏ = 40 are 59.8m, 58.4m, 60.1m, and 58.5m for π1 =10 kPa (π½ = 0), π1 = 10 kPa (π½ = 5), π
1 = 0 kPa (π½ = 0),and π1 = 0 kPa (π½ = 5), respectively.
Finally, based on the parameters πΏ = 40m and π½ = 5β,the influence of the rising groundwater on the safety factor isshown in Figure 4. It is noticed that larger π§
π€represents lower
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, π½ = 5βc1 = 10, FS with zw
c1 = 0, FS with zw
Figure 4: Influence of the depth of groundwater π§π€on πΉπ.
groundwater surface. Then, Figure 4 shows that increasingboth π§
π€and π1 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 π½ andtrench length πΏ results in decreasing the safety factor, butincreasing the depth to groundwater π§
π€and suction effect
π
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
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