Research Article Stability of Three-Dimensional Slurry ...and depth, slurry depth and density, groundwater depth, and tension crack. However, if cuto walls locate below an inclined

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

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

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)


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


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


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.

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Research Article Stability of Three-Dimensional Slurry ...and depth, slurry depth and density, groundwater depth, and tension crack. However, if cuto walls locate below an inclined