Limit Analysis for Local and Overall Stability of a Slurry Trench in Cohesive Soil

Han, Chang-Yu; Wang, Jian-Hua; Xia, Xiao-He; Chen, Jin-Jian

doi:10.1061/(ASCE)GM.1943-5622.0000268

Open access

Technical Notes

Nov 3, 2012

Limit Analysis for Local and Overall Stability of a Slurry Trench in Cohesive Soil

Authors: Chang-Yu Han [email protected], Jian-Hua Wang, Xiao-He Xia, and Jin-Jian ChenAuthor Affiliations

Publication: International Journal of Geomechanics

Volume 15, Issue 5

https://doi.org/10.1061/(ASCE)GM.1943-5622.0000268

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Abstract

This paper uses limit analysis to develop a two-dimensional (2D) and a three-dimensional (3D) analysis of slurry trench local and overall stability for cohesive soil. Formulas for the slurry trench stability analysis are obtained through theoretical derivation based on limit analysis theory, and rotational mechanisms are then presented for slurry trench stability. For 2D slurry trench local and overall stability, the failure surface has the shape of a circular arc, whereas it has the shape of spherical cap for 3D local stability, and it has the shape of a torus with an outline defined by a circle for 3D overall stability. Examples are provided to illustrate the safety factor influenced by the slurry and soil bulk density ratio, slurry level depth and trench depth ratio, thickness of the weak soil layer, cohesion, and trench width and depth ratio. The safety factor for the 3D solutions is approximately 1.1 times greater than the safety factor for the 2D solutions for local stability but 1.2 times greater (

B / h = 1

) for overall stability.

Introduction

Slurry trenches are used in the construction of groundwater cutoff walls and subsurface structural diaphragm walls. The stability of slurry trenches has attracted great attention among geotechnical researchers and the industry itself because instability of the slurry trench is common in underground engineering. The safety factor is an important index in the design of the slurry trench, but little research has been carried out on the subject (Choy et al. 2007; Fox 2004; Morgenstern and Amir-Tahmasseb 1965; Wong 1984).

With a slurry trench, the slope angle is 90° and the slope face has slurry pressure action. To date, the existing methods for determining slope stability have been classified into the following three types.

The limit equilibrium method has been commonly used in practice because of its relative simplicity. This method is based on discretization into slices, in which global static equilibrium conditions are satisfied. Basically, they are based on assumptions of the interslice forces to make the problem statically (Ahmed et al. 2012; Alejano et al. 2011; Mendjel and Messast 2012; Tinti and Manucci 2008).

Finite-element modeling, based on continuum mechanics, can be used to determine deformations under loading or the safety factor by numerical iterations. An appropriate constitutive model for the soil mass in the slope is needed with these models. With finite-element modeling, both soil movement and progressive failure can be estimated (Srivastava et al. 2010; Stolle and Guo 2008; Xu 2011).

Limit analysis, based on plasticity limit theorems, has an advantage in that the lower- and upper-bound theorems bracket the true solution. This approach is rigorous so that the stress field with a lower-bound solution is in equilibrium and yield criterion at every point in the soil; the velocity field with an upper-bound solution is compatible with flow rule and the imposed displacements. In recent years, great efforts have been devoted to the application of the plasticity limit theorems to slope stability (Kumar and Sahoo 2012; Li et al. 2010; Loukidis et al. 2003; Michalowski 2010; Yang and Yang 2010).

The overall stability of trenches has been addressed numerous times; however, a limit analysis for local stability of slurry trenches has not been reported. This paper aimed to develop a two-dimensional (2D) and a three-dimensional (3D) analysis of slurry trench local and overall stability for cohesive soil. Formulas for slurry trench stability analysis are obtained through theoretical derivation based on the limit analysis theory, and rotational mechanisms are then presented for slurry trench stability. Numerical examples are provided to illustrate the safety factor influenced by the trench width and depth ratio (

B / h

), thickness of the weak soil layer (

h_{n}^{'}

), cohesion (

c

), slurry and soil bulk density ratio (

γ_{sr} / γ

), distance from the slurry level to the trench top, and trench width ratio (

h_{sr} / h

).

Limit Analysis for Slurry Trench Stability

The application of limit analysis to earth slopes started with a paper by Drucker and Prager (1952), who applied the kinematic approach of limit analysis to the stability of slopes undergoing plane-strain failure. Limit analysis aims to evaluate bounds on the limit load inducing or resisting failure in structures built of perfectly plastic materials. In the application to slurry trenches, the limit load can be identified by the weight of the soil. An upper bound can be obtained from the kinematic method, in which the kinematically admissible velocity field defines the possible mechanism of failure. The strain rates resulting from the velocity field must satisfy the flow rule that is associated with the yield condition of the material, and the velocities must satisfy the boundary conditions. The most common yield condition used for undrained, purely cohesive soils is the Tresca hexagonal yield criterion. The flow rule requires the following relationship among the principal strain rates for problems of plane plastic flow:

{\dot{ε}}_{1} + {\dot{ε}}_{3} = 0

(1)

where

{\dot{ε}}_{1}

and

{\dot{ε}}_{3}

= principal strain rates. The mode of deformation in the transition layer is a combination of the shear flow parallel to the layer with the extension normal to it. The dissipation rate can be written as

δ D = c δ [v_{t}]

(2)

where

D

= dissipation of energy;

δ

= variational symbol;

c

= cohesion stress of the soil; and

[v_{t}]

= tangential velocity change.

The work rate of the soil weight (

W_{s}

) can be calculated as the dot product of the total weight of a block

W_{i}

and the velocity of the block centroid

v_{i}^{c}

W_{s} = W_{i} v_{i}^{c}

(3)

The work rate of the slurry pressure (

W_{sr}

) can be calculated as the dot product of the pressure (

P_{i}

) of the slurry and the velocity of the interface of the soil and the slurry (

v_{i}^{p}

)

W_{sr} = \int P_{i} v_{i}^{p} {dS}_{p}

(4)

For a slurry trench of a given geometry, it is possible to evaluate the safety factor, defined as

F = \frac{D + W_{sr}}{W_{s}}

(5)

Equating the external rate of work to the rate of internal energy dissipation, a least upper bound for the safety factor can be obtained.

Limit Analysis for Local Stability Of Slurry Trench

Slurry-supported trenches are excavated in soft soil or sand. When the liquid level fluctuations are too large or the surface decreases sharply, a local instability and overexcavation phenomenon often occur (Fig. 1), which leads to an increase in the filling factor of the concrete and impermeable material. Therefore, the amount of construction materials and the difficulty are increased. Fig. 2 shows an example of concrete filling caused by instability. Redundant concrete must be drilled and removed in subsequent construction, which is wasteful, increases the difficulty, and increases the noise and damage to adjacent buildings.

Fig. 2. Concrete overflow caused by local sloughing of slurry trench

2D Local Stability

Soils are described as purely cohesive when their shear strength is independent of the level of stress. This is typical of clays subjected to undrained conditions. Constructing rotational mechanisms for such soils is relatively simple, because their deformation occurs without volume change. Fig. 3 shows a kinematically admissible velocity field. The rotating mechanism has the shape of a cylinder. This mechanism was considered earlier by Baligh and Azzouz (1975) and Gens et al. (1988).

Fig. 3. Two-dimensional rotational mechanism for slurry trench local instability

The bow-shaped CFD (a surface of velocity discontinuity shown in Fig. 3) rotates about the center of rotation

O

with the materials below the surface CFD remaining at rest. Thus, the surface CFD is a surface of velocity discontinuity.

It is very complicated to determine the direct integration of the external work rate due to the soil weight in the region CFD. An easier alternative is to first find the area centroid and the velocity vector at the centroid. The rate of external work for the region CFD is then found by the dot product of the soil weight concentrated at its volume centroid and the velocity vector at the centroid.

The weight of CFD, which is instable and rotational, can be calculated by the following equation:

G = \frac{1}{2} γ R^{2} (π - 2 δ - \sin 2 δ)

(6)

where

R

and

δ

are shown in Fig. 3.

The work rate of the soil weight can be written as

W_{s} = \frac{2}{3} γ ω R^{3} \cos^{3} δ

(7)

where

γ

= unit weight of soil; and

ω

= angular velocity of the region CFD for rotation.

The internal dissipation of energy occurs along the discontinuity surface CFD. The total internal dissipation of energy is found by integration over the whole surface.

The internal dissipation of energy can be computed as

D = 2 c ω R^{2} (\frac{π}{2} - δ)

(8)

The rate of external work due to the slurry pressure occurs along the surfaces

CE

and

ED

. The rate of external work due to the slurry pressure is found by integration over the whole surface.

The rate of external work due to the slurry pressure on

CE

is

W_{CE} = \int_{0}^{(π / 2) - δ} ω γ_{sr} R^{2} \sin^{2} δ \tan θ \sec^{2} θ (h_{1} + R \cos δ - R \sin δ \tan θ) d θ

(9)

The rate of external work due to the slurry pressure on

ED

is

W_{ED} = - \int_{0}^{(π / 2) - δ} ω γ_{sr} R^{2} \sin^{2} δ \tan β \sec^{2} β (h_{1} + R \cos δ + R \sin δ \tan β) d β

(10)

where

γ_{sr}

= unit weight of the slurry in the trench; and

h_{1}

= distance from the slurry level to the weak layer soil.

The total rate of external work due to the slurry pressure is

W_{sr} = \frac{2}{3} ω γ_{sr} R^{3} \cos^{2} δ \cot δ

(11)

The safety factor (

F

) in Eq. (5) can then be obtained by substituting Eqs. (7), (8), and (11) into Eq. (5) as follows:

F = \frac{3 c}{γ h_{n}^{'}} \frac{(π - 2 δ)}{\cos^{2} δ} + \frac{γ_{sr}}{γ} \frac{1}{\sin δ}

(12)

where

h_{n}^{'}

= thickness of the weak soil layer; and

δ

= medivariation. The function

F (δ)

has a minimum value when

δ

satisfies the condition

\partial F / \partial δ = 0

.

3D Local Stability

The actual instability of a slurry trench is a 3D problem, so one must consider the 3D space effects when the length of the slurry trench is short. Fig. 4 shows a 3D rotational mechanism in cohesive soils (undrained behavior), with a spherical failure surface confined to the vertical portion of the slurry trench. This mechanism is often used to analyze slope instability problems (Griffiths and Marquez 2008; Hungr et al. 1989; Silvestri 2006); a special case of mechanisms was considered earlier by Baligh and Azzouz (1975).

Fig. 4. Spherical cap failure surface: (a) cross section of mechanism; (b) coordinate system

The weight of spherical cap CFD is

G = \frac{π}{3} γ R^{3} {(1 - \sin δ)}^{2} (2 + \sin δ)

(13)

The work rate of the soil weight

W_{s}

was obtained as the dot product of the soil weight concentrated at its volume centroid and the centroid velocity vector, yielding

W_{s} = \frac{1}{4} ω γ π R^{4} \cos^{4} δ

(14)

The work dissipation rate is

D = 4 ω c R^{3} \int_{δ}^{π / 2} \int_{0}^{π / 2} \cos θ \sqrt{1 - \sin^{2} α \cos^{2} θ} d α d θ

(15)

The rate of external work due to the slurry pressure occurs along the surface

CED

, but the rate is found by integration over the whole surface.

The rate of external work due to the slurry pressure on

CE

is

W_{CE} = \int_{0}^{α} 2 ω γ_{sr} R^{3} \sin^{2} δ \tan β \sec^{2} β (h_{1} + R \cos δ - R \sin δ \tan β) \sqrt{\cos^{2} δ - \sin^{2} δ \tan^{2} β} d β

(16)

The rate of external work due to the slurry pressure on

ED

is

W_{ED} = \int_{0}^{α} 2 ω γ_{sr} R^{3} \sin^{2} δ \tan β \sec^{2} β (h_{1} + R \cos δ + R \sin δ \tan β) \sqrt{\cos^{2} δ - \sin^{2} δ \tan^{2} β} d β

(17)

The total rate of external work due to the slurry pressure is

W_{sr} = 4 ω γ_{sr} R^{4} \sin^{2} δ \int_{0}^{(π / 2) - δ} \frac{\tan^{2} θ}{\cos^{2} θ} \sqrt{\cos^{2} δ - \sin^{2} δ \tan^{2} θ} d θ

(18)

The safety factor (

F

) in Eq. (5) can then be obtained by substituting Eqs. (14), (15), and (18) into Eq. (5) as follows:

F = \frac{32 c}{π γ h_{n}^{'} \cos^{3} δ} f_{1} + \frac{16 γ_{sr} \sin^{2} δ}{π γ \cos^{4} δ} f_{2}

(19)

in which the functions

f_{1}

and

f_{2}

are defined as

f_{1} = \int_{δ}^{π / 2} \int_{0}^{π / 2} \cos θ \sqrt{1 - \sin^{2} α \cos^{2} θ} d α d θ

f_{2} = \int_{0}^{(π / 2) - δ} \frac{\tan^{2} θ}{\cos^{2} θ} \sqrt{\cos^{2} δ - \sin^{2} δ \tan^{2} θ} d θ

The function

F (δ)

has a minimum value when

δ

satisfies the condition

\partial F / \partial δ = 0

.

Limit Analysis for Overall Stability of Slurry Trench

2D Overall Stability

In cohesive soil (

c > 0

,

φ = 0

), the slip surface is usually assumed to be circular (Baligh and Azzouz 1975; Hungr et al. 1989; Silvestri 2006). In this study, a circular slip surface is assumed for the overall stability analysis of a slurry trench in cohesive soil, as shown in Fig. 5.

Fig. 5. Two-dimensional rotational mechanism for slurry trench overall instability

The circular slip surface is described by

r = r_{0}

(21)

where

r_{0}

is shown in Fig. 5. Hence, the magnitude of the velocity vector

v

along this surface varies according to

v = v_{0}

(22)

where

v_{0}

= magnitude of the velocity at point

A

. From the geometrical relations, it may be shown that the values

h

and

L

can be expressed in terms of the angles

θ_{0}

and

θ_{h}

in the form

h = r_{0} (\sin θ_{h} - \sin θ_{0})

(23)

L = r_{0} \csc θ_{h} \sin (θ_{h} - θ_{0}) - r_{0} \cot θ_{h} (\sin θ_{h} - \sin θ_{0})

(24)

Following Ausilio et al. (2000), the work rate of the soil weight (

W_{s}

) can be written as

W_{s} = γ R^{3} ω [f_{11} - f_{12} - f_{13}]

(25)

where

ω

= angular velocity of the region

ABC

for rotation. The functions

f_{21}

,

f_{22}

, and

f_{23}

will be explained in Eqs. (34)–(36).

The internal dissipation of energy occurs along the discontinuity surface

AC

. The total internal dissipation of energy (

D

) is found by integration over the whole surface

D = c ω r^{2} (θ_{h} - θ_{0})

(26)

The slurry velocity (

v_{sr}

) during rotation about axis

O

is

v_{sr} = r_{x_{2}} ω \sin θ

(27)

The slurry pressure is

p_{sr} = - γ_{sr} r_{x_{2}} h^{″} s \sec θ

(28)

where the functions

r_{x_{1}}

,

r_{x_{2}}

, and

h^{″}

are defined by Eqs. (29)–(31)

r_{x_{1}} = \frac{\cos θ_{h}}{\cos θ_{sr}} r_{0}

(29)

r_{x_{2}} = \frac{\cos θ_{h}}{\cos θ} r_{0}

(30)

h^{″} = r_{x_{2}} \sin θ - r_{x_{1}} \sin θ_{w}

(31)

The total rate of external work (

W_{sr}

) due to the slurry pressure is

W_{sr} = ω γ_{sr} \int_{θ_{sr}}^{θ_{h}} r_{x_{2}}^{2} h^{″} \tan θ d θ = - \frac{1}{12} ω γ_{sr} R^{3} \sec^{3} θ_{sr} \sin^{2} (θ_{sr} - θ_{h}) [\sin (θ_{sr} - θ_{h}) - 3 \sin (θ_{sr} + θ_{h})]

(32)

The safety factor (

F

) in Eq. (5) can then be obtained by substituting Eqs. (25), (26), and (32) into Eq. (5) as follows:

F = \frac{c}{γ h} \frac{(θ_{h} - θ_{0}) [\sin θ_{h} - \sin θ_{0}]}{(f_{11} - f_{12} - f_{13})} + \frac{1}{12} \frac{γ_{sr}}{γ} \frac{f_{14}}{f_{11} - f_{12} - f_{13}}

(33)

in which the functions

f_{11}

,

f_{12}

,

f_{13}

,

f_{14}

,

f_{15}

,

θ_{sr}

, and

θ_{B}

are defined as

f_{11} = \frac{1}{3} (\sin θ_{h} - \sin θ_{0})

(34)

f_{12} = \frac{1}{6} f_{02} (2 \cos θ_{0} - f_{02})

(35)

f_{13} = \frac{1}{6} [\sin (θ_{h} - θ_{0}) - f_{02} \sin θ_{h}] [\cos θ_{0} - f_{02} + \cos θ_{h}]

(36)

f_{14} = \sec^{3} θ_{sr} \sin^{2} (θ_{sr} - θ_{h}) [\sin (θ_{h} - θ_{sr}) + 3 \sin (θ_{sr} + θ_{h})]

(37)

θ_{sr} = \arccos [\frac{f_{01} \cot θ_{B} \sin θ_{0}}{\sqrt{1 + f_{01} \sin θ_{0} (2 + f_{01} \csc^{2} θ_{B} \sin θ_{0})}}]

(38)

θ_{B} = \arctan (\frac{\sin θ_{0}}{\cos θ_{h}})

(39)

where

f_{01} = \frac{h}{h_{sr}} \frac{1}{\sin θ_{h} - \sin θ_{0}}

(40)

f_{02} = \frac{\sin (θ_{h} - θ_{0})}{\sin θ_{h}} - \frac{\cos θ_{h}}{\sin θ_{h}} (\sin θ_{h} - \sin θ_{0})

(41)

The function

F

has a minimum value when

θ_{0}

and

θ_{h}

satisfy the conditions

\frac{\partial F}{\partial θ_{0}} = 0, \frac{\partial F}{\partial θ_{h}} = 0

(42)

Solving these equations and substituting the values of

θ_{0}

and

θ_{h}

thus obtained into Eq. (33) yields a least upper bound for the safety factor,

F

, of a slurry trench.

3D Overall Stability

Soils are described as purely cohesive (

c > 0

,

φ = 0

) when their shear strength is independent of the level of stress. This is typical of clays subjected to undrained conditions. Constructing rotational mechanisms for such soils is relatively simple, because their deformation occurs without volume change. Consequently, the surface of revolution provides a torus surface, as illustrated in Fig. 6. The similar shape of this mechanism was considered by Michalowski and Drescher (2009) and Xia et al. (2012).

Fig. 6. Three-dimensional rotational mechanism for slurry trench overall instability

The trace of the mechanism on the symmetry plane is described by two circular arcs,

AC

and

A^{'} C^{'}

. The equation for the torus surface is given by

r = r_{0}, r^{'} = r_{0}^{'}

(43)

With the circular arcs

r_{0}

and

r_{0}^{'}

defining the shape of the failure surface in Fig. 6 given in Eq. (43), the centerline of the conical volume

r_{m} (θ)

and the radius of the circular cross section

R

are found as

r_{m} = \frac{r + r^{'}}{2}, R = \frac{r - r^{'}}{2}

(44)

To calculate the work of the soil weight, a local coordinate system

x

,

y

was introduced, as shown in Fig. 6. The velocity during rotation about axis

O

is

v = (r_{m} + y) ω

(45)

where

ω

= angular velocity, and the infinitesimal volume element is

dV = dxdy (r_{m} + y) d θ

(46)

The work rate of the soil weight (

W_{s}

) can be written as

W_{s} = 2 ω γ [\int_{θ_{0}}^{θ_{B}} \int_{0}^{x_{1}} \int_{a}^{y_{1}} {(r_{m} + y)}^{2} \cos θ dydxd θ + \int_{θ_{B}}^{θ_{h}} \int_{0}^{x_{2}} \int_{d}^{y_{1}} {(r_{m} + y)}^{2} \cos θ dydxd θ]

(47)

where the functions

x_{1}

,

x_{2}

,

y_{1}

,

a

,

d

will be defined by Eqs. (53) and (54).

The internal dissipation of energy (

D

) can be computed by

D = 2 ω c R [\int_{θ_{0}}^{θ_{B}} \int_{a}^{R} \frac{{(r_{m} + y)}^{2}}{\sqrt{R^{2} - y^{2}}} dy d θ + \int_{θ_{B}}^{θ_{h}} \int_{d}^{R} \frac{{(r_{m} + y)}^{2}}{\sqrt{R^{2} - y^{2}}} dy d θ]

(48)

The slurry velocity during rotation about axis

O

is

v_{sr} = r_{x 2} ω \sin θ

(49)

The slurry pressure is

p_{sr} = - 2 γ_{sr} r_{x_{2}} h^{″} \sqrt{R^{2} - d^{2}} \sec θ

(50)

where the functions

r_{x_{1}}

,

r_{x_{2}}

, and

h^{″}

are defined by Eqs. (29)–(31).

The total rate of external work (

W_{sr}

) due to the slurry pressure is

W_{sr} = ω γ_{sr} r_{0}^{3} \cos^{3} θ_{h} \int_{θ_{sr}}^{θ_{h}} \frac{2 x_{2} \tan θ (\tan θ - \tan θ_{sr})}{\cos^{2} θ} d θ

(51)

The safety factor (

F

) in Eq. (5) can then be obtained by substituting Eqs. (47), (48), and (51) into Eq. (5) as follows:

F = \frac{2 c R f_{21} + γ_{w} r_{0}^{3} \cos^{3} θ_{h} f_{22}}{2 γ f_{23}}

(52)

In Eqs. (47), (48), (51), and (52),

x_{1}

,

x_{2}

,

y_{1}

,

a

,

d

,

f_{21}

,

f_{22}

, and

f_{23}

are functions of the soil strength parameters and the geometry of the slip surface, which can be defined as follows:

x_{1} = \sqrt{R^{2} - a^{2}}, x_{2} = \sqrt{R^{2} - d^{2}}, y_{1} = \sqrt{R^{2} - x^{2}}

(53)

a = \frac{\sin θ_{0}}{\sin θ} r_{0} - r_{m}, d = \frac{\cos θ_{h}}{\cos θ} r_{0} - r_{m}

(54)

f_{21} = \int_{θ_{0}}^{θ_{B}} \int_{a}^{R} \frac{{(r_{m} + y)}^{2}}{\sqrt{R^{2} - y^{2}}} dy d θ + \int_{θ_{B}}^{θ_{h}} \int_{d}^{R} \frac{{(r_{m} + y)}^{2}}{\sqrt{R^{2} - y^{2}}} dyd θ

(55)

f_{22} = \int_{θ_{sr}}^{θ_{h}} \frac{2 x_{2} \tan θ (\tan θ - \tan θ_{sr})}{\cos^{2} θ} d θ

(56)

f_{23} = \int_{θ_{0}}^{θ_{B}} \int_{0}^{x_{1}} \int_{a}^{y_{1}} {(r_{m} + y)}^{2} \cos θ dydxd θ + \int_{θ_{B}}^{θ_{h}} \int_{0}^{x_{2}} \int_{d}^{y_{1}} {(r_{m} + y)}^{2} \cos θ dydxd θ

(57)

The function

F

has a minimum value when

θ_{0}

,

θ_{h}

, and

r_{0}^{'} / r_{0}

satisfy the conditions

\frac{\partial F}{\partial θ_{0}} = 0, \frac{\partial F}{\partial θ_{h}} = 0, \frac{\partial F}{\partial (r_{0}^{'} / r_{0})} = 0

(58)

The safety factor of a slurry trench also can be calculated with Eq. (58). To avoid lengthy computations, these simultaneous equations may be solved by a numerical procedure. The minimum

F

is calculated with independent variable parameters

θ_{0}

,

θ_{h}

, and

r_{0}^{'} / r_{0}

.

Results and Discussion

It is commonly accepted that 3D analyses yield safety factors for slopes that are greater than those from 2D analyses (Li et al. 2010; Tutluoglu et al. 2011). This statement is supported by direct comparison of analytical results and by intuition, because 2D analysis is less restrictive. Michalowski (2010) offered a more formal justification for the statement that a 2D safety factor for a uniform slope cannot be greater than that from a 3D analysis. Based on the 2D and 3D mechanisms for slurry trench local and overall stability, examples are presented in this section to demonstrate the difference between 2D and 3D models.

Comparison between 2D and 3D Solutions of Local Stability

For slurry trench local stability, the estimates of safety factor

F

were obtained using a procedure for a given slurry and soil bulk density ratio, thickness of the weak soil layer, cohesion, and soil bulk density. The independent variable in the procedure was angle

δ

. These parameters were varied by a small increment in computational loops, and the process was repeated until the minimum of

F

was reached, with the increments of 0.01° used for angle

δ

. The results of these computations are represented graphically in Figs. 7–9 for 2D and 3D solutions.

Fig. 7. Comparison of results for 2D and 3D local stability with different $γ_{sr} / γ$ ( $h_{n}^{'} = 25 m$ , $γ = 20 kN / m^{3}$ )

Fig. 8. Comparison of results for 2D and 3D local stability with different $h_{n}^{'}$ ( $γ_{sr} / γ = 0.6$ , $γ = 20 kN / m^{3}$ )

Fig. 9. Comparison of results for 2D and 3D local stability with different $c$ ( $γ_{sr} / γ = 0.6$ , $γ = 20 kN / m^{3}$ )

Fig. 7 illustrates the distribution of the safety factor versus the slurry and soil bulk density ratio (

γ_{sr} / γ

). It can be seen that increasing the

γ_{sr} / γ

from

γ_{sr} / γ = 0.5

to

γ_{sr} / γ = 0.7

can increase the factor of safety by more than 23% (2D) and 21% (3D). The safety factors for the 3D solutions are approximately 1.08 and 1.1 times greater than the safety factors for the 2D solutions for

c = 10 kPa

and

c = 20 kPa

, respectively.

Fig. 8 shows the distribution of the safety factor versus the thickness of the weak soil layer (

h_{n}^{'}

). It can be seen that increasing the

h_{n}^{'}

from

h_{n}^{'} = 5 m

to

h_{n}^{'} = 30 m

can decrease the factor of safety by more than 49.6% (2D) and 53.4% (3D). The safety factors for the 3D solutions are approximately 1.1 and 1.13 times greater than the safety factors for the 2D solutions for

c = 10 kPa

and

c = 20 kPa

, respectively.

Fig. 9 presents the distribution of the safety factor versus cohesion (

c

). It can be seen that increasing the

c

from

c = 10 kPa

to

c = 20 kPa

can increase the factor of safety by more than 24.8% (2D) and 28.4% (3D). The safety factors for the 3D solutions are approximately 1.12 and 1.08 times greater than the safety factors for the 2D solutions for

h_{n}^{'} = 10 m

and

h_{n}^{'} = 25 m

, respectively.

The following special example of slurry trench local stability has the following parameters: slurry and soil bulk density ratio

γ_{sr} / γ = 0.6

, thickness of the weak soil layer

h_{n}^{'} = 25 m

, cohesion

c = 20 kPa

, and soil bulk density

γ = 20 kN / m^{3}

. Therefore, the calculated 2D and 3D safety factors

F

are 1.18 and 1.31, respectively. For the selected geometric parameters (no slurry), a value of

F = 0.33

was obtained by Michalowski and Drescher (2009) using the cylindrical surface failure mechanism, and a value of

F = 0.43

was obtained by Baligh and Azzouz (1975) using the spherical cap failure mechanism. Because slurry pressure prevents the slope from failing, as the slurry pressure increases, the safety factor becomes larger and larger.

Comparison between 2D and 3D Solutions of Overall Stability

The estimates of safety factor

F

were obtained using a procedure for a given slurry and soil bulk density ratio, slurry level depth and trench depth ratio, cohesion, trench width and depth ratio (in 3D), soil bulk density, and trench depth. Independent variables in the procedure were angles

θ_{0}

and

θ_{h}

and ratio

r_{0}^{'} / r_{0}

(in 3D). These parameters were varied by a small increment in computational loops, and the process was repeated until the minimum of

F

was reached, with the increments of 0.1° used for angles

θ_{0}

and

θ_{h}

and 0.01 for ratio

r_{0}^{'} / r_{0}

. The results of these computations are represented graphically in Figs. 5–9 for 2D and 3D solutions.

This section examines the difference between 2D and 3D models for various parameters, including

γ_{sr} / γ

,

h_{sr} / h

,

c

, and

B / h

. To facilitate comparison of 2D and 3D calculations,

B / h = 0.5 1, 5, and 10

in the 3D cases.

Fig. 10 illustrates the distribution of the safety factor versus the slurry and soil bulk density ratio (

γ_{sr} / γ

) for various trench width and depth ratios (

B / h

). As expected, the safety factor of the slurry trench increases as

γ_{sr} / γ

increases. From this figure, the slurry and soil bulk density ratio,

γ_{sr} / γ

, is found to have a great effect on the solutions. It can be seen that increasing the

γ_{sr} / γ

from

γ_{sr} / γ = 0.5

to

γ_{sr} / γ = 0.7

can increase the factor of safety by more than 19.6% (2D) and 15.6% (

B / h = 1

). The safety factors for the 3D solutions are approximately 1.38, 1.18, 1.05, and 1.04 times greater than the safety factors for the 2D solutions for

B / h = 0.5

,

B / h = 1

,

B / h = 5

, and

B / h = 10

, respectively.

Fig. 10. Comparison of results for 2D and 3D overall stability with different $γ_{sr} / γ$ ( $h_{sr} / h = 0.1$ , $c = 20 kPa$ , $γ = 18 kN / m^{3}$ , $h = 10 m$ )

Fig. 11 shows the distribution of the safety factor versus the slurry level depth and trench depth ratio (

h_{sr} / h

). The safety factor of the slurry trench decreases as

h_{sr} / h

increases. From this figure, the slurry level depth and trench depth ratio,

h_{sr} / h

, is found to have a great effect on the solutions. It can be seen that increasing the

h_{sr} / h

from

h_{sr} / h = 0.1

to

h_{sr} / h = 0.5

can decrease the factor of safety by more than 36.4% (2D) and 31.2% (

B / h = 1

). The safety factors for the 3D solutions are approximately 1.42, 1.21, 1.05, and 1.03 times greater than the safety factors for the 2D solutions for

B / h = 0.5

,

B / h = 1

,

B / h = 5

, and

B / h = 10

, respectively.

Fig. 11. Comparison of results for 2D and 3D overall stability with different $h_{sr} / h$ ( $γ_{sr} / γ = 0.6$ , $c = 20 kPa$ , $γ = 18 kN / m^{3}$ , $h = 10 m$ )

Fig. 12 presents the distribution of the safety factor versus cohesion (

c

). The safety factor of the slurry trench increases as

c

increases. From this figure, the cohesion,

c

, is found to have a great effect on the solutions. It can be seen that increasing the

c

from

c = 10 kPa

to

c = 20 kPa

can increase the factor of safety by more than 23.3% (2D) and 28.4% (

B / h = 1

).The safety factors for the 3D solutions are approximately 1.32, 1.16, 1.04, and 1.03 times greater than the safety factors for the 2D solutions for

B / h = 0.5

,

B / h = 1

,

B / h = 5

, and

B / h = 10

, respectively.

Fig. 12. Comparison of results for 2D and 3D overall stability with different $c$ ( $γ_{sr} / γ = 0.6$ , $h_{sr} / h = 0.1$ , $γ = 18 kN / m^{3}$ , $h = 10 m$ )

Fig. 13 illustrates the distribution of the safety factor versus the trench width and depth ratio (

B / h

) in the 3D cases. As expected, the safety factor of the slurry trench decreases with increasing trench width and depth ratio. From this figure, the trench width and depth ratio,

B / h

, is found to have a great effect on the chart solutions when

B / h < 5

. It can be seen that increasing the

B / h

from

B / h = 1

to

B / h = 5

can decrease the factor of safety by more than 13%. This means that the 3D boundary effect on slurry trench stability is very large when

B / h < 5

. As shown in Fig. 13, this difference changes by less than 1% when the ratio of

B / h

increases from 5 to 10. This implies that the 3D end boundary effect decreases more significantly with an increasing

B / h

ratio for the slurry trench. It also means that the 3D boundary end effect on slurry trench stability is small and is almost insignificant when

B / h \geq 10

.

Fig. 13. Variation of safety factor $F$ with $B / h$ ( $γ_{sr} / γ = 0.6$ , $h_{sr} / h = 0.1$ , $c = 20 kPa$ , $γ = 18 kN / m^{3}$ )

The example of slurry trench overall stability has the following parameters: slurry and soil bulk density ratio

γ_{sr} / γ = 0.6

, slurry level depth and trench depth ratio

h_{sr} / h = 0.1

, cohesion

c = 20 kPa

, soil bulk density

γ = 18 kN / m^{3}

, and trench depth

h = 10 m

. The calculated 2D and 3D (

B / h = 1

) safety factors

F

are 0.92 and 1.12, respectively. For the selected geometric parameters (no slurry), a value of

F = 0.43

was obtained using the cylindrical surface failure mechanism, and

F = 0.63

was obtained using the torus failure mechanism by Michalowski and Drescher (2009).

Conclusions

The upper-bound methods for the 2D and the 3D analysis of slurry trench local and overall stability for cohesive soil are presented in this paper. Formulas for the stability analysis of a slurry trench are obtained through theoretical derivation based on limit analysis theory. Rotational mechanisms are presented for slurry trench stability. For 2D slurry trench local and overall stability, the failure surface has the shape of a circular arc, whereas for 3D local stability, it has the shape of a spherical cap, and it has the shape of torus with an outline defined by a circle for 3D overall stability.

Examples are provided to illustrate variations in the safety factor with the following parameters: slurry and soil bulk density ratio, slurry level depth and trench depth ratio, thickness of weak soil layer, cohesion, trench width and depth ratio, soil bulk density, and trench depth. The stability factor increases with increasing slurry and soil bulk density ratio and cohesion, and with decreasing slurry level depth and trench depth ratio, thickness of the weak soil layer, trench width and depth ratio, soil bulk density, and trench depth.

Based on the comparisons between 2D and 3D solutions, the 3D effects tend to lose significance when

B / h \geq 10

. It can be concluded that when using limit analysis, 2D solutions can replace 3D solutions for preliminary slurry trench design where

B / h \geq 10

. Based on the results presented, the following conclusions can be made:

1.

Using the limit analysis theory, the 2D and 3D solutions for the local and overall stability analysis of slurry trench are obtained; for the application example presented, the safety factors for the 3D solutions are approximately 1.1 times greater than the safety factors for the 2D solutions for local stability but 1.2 times greater (

B / h = 1

) for overall stability.

2.

The slurry and soil bulk density ratio is found to have a great effect on the solutions; increasing the

γ_{sr} / γ

from

γ_{sr} / γ = 0.5

to

γ_{sr} / γ = 0.7

can increase the factor of safety by more than 23% (2D) and 21% (3D) for local stability, and by 19.6% (2D) and 15.6% (

B / h = 1

) for overall stability.

3.

The safety factor of a slurry trench decreases as the slurry level depth and trench depth ratio increases for overall stability; increasing the

h_{sr} / h

from

h_{sr} / h = 0.1

to

h_{sr} / h = 0.5

can decrease the factor of safety by more than 53.7% (2D) and 51.5% (

B / h = 1

).

4.

The boundary effect is found to reduce with increasing trench width and depth ratio for the 3D slurry trench; in addition, the safety factors for the 3D slurry trenches are almost unchanged when

B / h \geq 10

.

Acknowledgments

This research was supported financially by the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110073120012), the Shanghai Pujiang Talent plan (Grant No. 11PJ1405700), and the National Natural Science Foundation of China (Grant Nos. 41002095, 41172251, and 41272317).

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Information & Authors

Information

Published In

International Journal of Geomechanics

Volume 15 • Issue 5 • October 2015

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

History

Received: Feb 2, 2012

Accepted: Oct 31, 2012

Published online: Nov 3, 2012

Published in print: Oct 1, 2015

Authors

Affiliations

Chang-Yu Han [email protected]

Ph.D., Dept. of Civil Engineering, Shanghai Jiaotong Univ., Shanghai 200240, China; Associate Professor, School of Civil Engineering and Architecture of Henan Univ., Kaifeng 475004, China (corresponding author). E-mail: [email protected]

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Jian-Hua Wang

Professor, Dept. of Civil Engineering, Shanghai Jiaotong Univ., Shanghai 200240, China.

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Xiao-He Xia

Professor, Dept. of Civil Engineering, Shanghai Jiaotong Univ., Shanghai 200240, China.

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Jin-Jian Chen

Associate Professor, Dept. of Civil Engineering, Shanghai Jiaotong Univ., Shanghai 200240, China.

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Limit Analysis for Local and Overall Stability of a Slurry Trench in Cohesive Soil

Abstract

Introduction

Limit Analysis for Slurry Trench Stability

Limit Analysis for Local Stability Of Slurry Trench

2D Local Stability

3D Local Stability

Limit Analysis for Overall Stability of Slurry Trench

2D Overall Stability

3D Overall Stability

Results and Discussion

Comparison between 2D and 3D Solutions of Local Stability

Comparison between 2D and 3D Solutions of Overall Stability

Conclusions

Acknowledgments

References

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Abstract

Introduction

Limit Analysis for Slurry Trench Stability

Limit Analysis for Local Stability Of Slurry Trench

2D Local Stability

3D Local Stability

Limit Analysis for Overall Stability of Slurry Trench

2D Overall Stability

3D Overall Stability

Results and Discussion

Comparison between 2D and 3D Solutions of Local Stability

Comparison between 2D and 3D Solutions of Overall Stability

Conclusions

Acknowledgments

References

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