1 Introduction

Slurry trenches are used in the construction of groundwater cutoff walls and subsurface structural diaphragm walls. They are also the first stage of the building process of barettes (they were used as deep foundations of the Petronas Towers in Kuala Lumpur). The stability of slurry trench has attracted great attention among geotechnical researchers and industry. The safety factor is an important index in the design of slurry trench. Theoretical analysis, numerical analysis and experiments have been undertaken to investigate slurry trench stability (Ng and Lings, 1995; Ng et al., 1995; Tsai and Chang, 1996; Ng and Yan, 1998; 1999; Tsai et al., 1998; 2000).

Tsai (1997) presented an analytical method to evaluate the stability of weak sublayers against lateral extrusion in a slurry supported trench. Oblozinsky et al. (2001) proposed suggestions for the practical design on the stability of the slurry trench based on the elasto-plastic finite-element method (FEM) analysis. Filz et al. (2004) presented a method for analyzing global stability, and pointed out that increasing the bentonite concentration of the slurry had beneficial impacts on stability. Fox (2004) employed the limit equilibrium method to estimate the stability factors for general 2D and 3D stability of a slurry-supported trench. Based on the diaphragm wall construction of two open-cut metro stations, Xu et al. (2011) studied the influence factors of stability of slurry trench sides during diaphragm wall building in soft soil by analyzing the fullness coefficient of slurry trenches corresponding to different controlling parameters. Based on the upper bound limit analysis theorem, Han et al. (2012a) developed a 2D and 3D analysis of slurry trench for local and overall stability for cohesive soil.

This paper is aimed at developing a 2D and 3D analysis of slurry trench stability for frictional/cohesive soil. Formulas for the slurry trench stability analysis are obtained through theoretical derivation with the basis of limit analysis theory, and then rotational mechanisms are presented for slurry trench stability. Examples are provided to illustrate the safety factor influence by the trench width and depth ratio (B/h), friction angle (φ), cohesion (c), slurry and soil bulk density ratio (γ sr/γ), and distance from slurry level to trench top and trench depth ratio (h sr/h).

2 Limit analysis theorems

The limit analysis method is based on the extensions of the maximum work principle derived by Hill (1948), and was given in the form of theorems by Drucker et al. (1952). Limit analysis aims at evaluating bounds on the limit load inducing or resisting failure in structures built of perfectly plastic materials. 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 satisfy the boundary conditions (Chen, 1975).

The most common yield condition used for soils is the Mohr-Coulomb function, which contains two material constants: the internal friction angle φ and the cohesion c. The perfectly plastic idealization of Coulomb yield criterion with associated flow rule implies that any plastic deformation must be accompanied by an increase in volume (for details please refer to (Michalowski, 1995)).

3 2D analysis of slurry trench stability

In this study, a 2D rotational mechanism with a log-spiral surface AC is shown in Fig. 1, in which the failure surface is assumed to pass through the toe of the slurry trench. The log-spiral slip surface is used for problems of slope stability (Chen et al., 2003; Loukidis et al., 2003; Zhu et al., 2003; Jiang et al., 2009) and foundation stability (Kumar and Ghosh, 2007; Han et al., 2012b). The region ABC rotates as a rigid body around the center of rotation O with the materials below the logarithmic surface AC remaining at rest. Thus, the surface AC is a surface of velocity discontinuity. In Fig.1, radii of r 0, r and r h, angles of θ 0, θ B , θ sr and θ h , and distances of h and h sr are presented; ω is the angular velocity of the region ABC for rotation; and φ is the internal friction angle.

Fig. 1
figure 1

2D rotational mechanism for frictional/cohesive soil

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Slurry trench can be considered as a special sit-uation of the slope, when the slope angle is 90°. Expressions of the rate of internal energy dissipation (D) and the rate of work done by soil weight (W s) for the 2D failure mechanism can be found in (Chen, 1975).

The work rate of the soil weight (W s) can be written as

$${W_{\rm{s}}} = \gamma r_0^3\omega [{f_{11}} - {f_{12}} - {f_{13}}],$$
(1)

where γ is the unit weight of soil, ω is the angular velocity of the region for rotation, and r 0 are shown in Fig. 1. The functions f 1, f 12 and f 13 will be explained later in Eqs. (11)–(13).

The internal dissipation of energy occurs along the discontinuity surface AC. The total internal dissipation rate of energy (D) is found by integration over the whole surface.

$$D = {{c\omega r_0^2} \over {2\tan \varphi }}[{{\rm{e}}^{2({\theta _h} - {\theta _0})\tan \varphi }} - 1].$$
(2)

The slurry velocity (v sr) during rotation about axis O is

$${v_{{\rm{sr}}}} = {r_{{x_2}}}\omega \sin \theta .$$
(3)

The slurry pressure is

$${p_{{\rm{sr}}}} = {\gamma _{{\rm{sr}}}}{r_{{x_2}}}h''\sec \theta ,$$
(4)

where γ sr is the unit weight of the slurry in the trench. The functions \(r_{x_1}\), \(r_{x_2}\) and h″ are defined by

$${r_{{x_1}}} = {{\cos {\theta _h}} \over {\cos {\theta _{{\rm{sr}}}}}}{r_0}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }},$$
(5)
$${r_{{x_2}}} = {{\cos {\theta _h}} \over {\cos \theta }}{r_0}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }},$$
(6)
$$h'' = {r_{{x_1}}}\sin \theta - {r_{{x_1}}}\sin {\theta _{\rm{w}}}.$$
(7)

The rate of external work (W sr) due to the slurry pressure on CD is

$$\matrix{ {{W_{{\rm{sr}}}} = \omega {\gamma _{{\rm{sr}}}}\int_{{\theta _{{\rm{sr}}}}}^{^{{\theta _h}}} {r_{{x_2}}^2} h''\tan \theta {\rm{d}}\theta}\cr { = - {1 \over {12}}\omega {\gamma _{{\rm{sr}}}}r_0^3{{\rm{e}}^{3({\theta _h} - {\theta _0})\tan \varphi }}{{\sec }^3}{\theta _{{\rm{sr}}}}{{\sin }^2}({\theta _{{\rm{sr}}}} - {\theta _h})} \cr { \times [\sin ({\theta _{{\rm{sr}}}} - {\theta _h}) - 3\sin ({\theta _{{\rm{sr}}}} + {\theta _h})].} \cr}$$
(8)

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

$$F = {{D + {W_{{\rm{sr}}}}} \over {{W_{\rm{s}}}}}.$$
(9)

The same expression of safety factor (F) for cohesive soil can be found in (Han et al., 2012a).

The safety factor (F) in Eq. (9) can then be obtained by substituting Eqs. (1), (2) and (8) into Eq. (9) as follows:

$$\matrix{ {F = {1 \over 2}{c \over {\gamma h}}{{{f_{15}}} \over {\tan \varphi ({f_{11}} - {f_{12}} - {f_{13}})}}} \cr { + {1 \over {12}}{{{\gamma _{{\rm{sr}}}}} \over \gamma }{{{f_{14}}} \over {{f_{11}} - {f_{12}} - {f_{13}}}}.} \cr }$$
(10)

In Eqs. (1) and (10), f 11, f 12, f 13, f 14, f 15, θ sr and θ B are functions of soil strength parameters and geometry of the slip surface, which can be defined as follows:

$$\matrix{ {{f_{11}} = {1 \over {3(1 + 9{{\tan }^2}\varphi )}}[(3\tan \varphi \cos {\theta _h} + \sin {\theta _h})} \hfill \cr { \times {{\rm{e}}^{3({\theta _h} - {\theta _0})\tan \varphi }} - (3\tan \varphi \cos {\theta _0} + \sin {\theta _0})],} \hfill \cr }$$
(11)
$${f_{12}} = {1 \over 6}{f_{02}}(2\cos \theta_{0} - {f_{02}}),$$
(12)
$$\matrix{ {{f_{13}} = {1 \over 6}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }}[\sin ({\theta _h} - {\theta _0}) - {f_{02}}\sin {\theta _h}]} \hfill \cr { \times [\cos {\theta _0} - {f_{02}} + \cos {\theta _h}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }}],} \hfill \cr }$$
(13)
$$\matrix{ {{f_{14}} = {{\rm{e}}^{3({\theta _h} - {\theta _0})\tan \varphi }}{{\sec }^3}{\theta _{{\rm{sr}}}}{{\sin }^2}({\theta _{{\rm{sr}}}} - {\theta _h})} \hfill \cr { \times [\sin ({\theta _h} - {\theta _{{\rm{sr}}}}) + 3\sin ({\theta _{{\rm{sr}}}} - {\theta _h})],} \hfill \cr }$$
(14)
$${f_{15}} = [{{\rm{e}}^{2({\theta _h} - {\theta _0})\tan \varphi }} - 1][\sin {\theta _h}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }} - \sin {\theta _0}],$$
(15)
$${\theta _{{\rm{sr}}}} = \arccos \left[ {{{{f_{01}}\cot {\theta _B}\sin {\theta _0}} \over {\sqrt {1 + {f_{01}}\sin {\theta _0}(2 + {f_{01}}{{\csc }^2}{\theta _B}\sin {\theta _0})} }}} \right],$$
(16)
$${\theta _{\rm{B}}} = \arctan \left[ {{{\sin {\theta _0}} \over {\cos {\theta _h}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }}}}} \right],$$
(17)

where

$${f_{01}} = {h \over {{h_{{\rm{sr}}}}}}{1 \over {\sin {\theta _h}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }} - \sin {\theta _0}}},$$
(18)
$$\matrix{ {{f_{02}} = {{\sin ({\theta _h} - {\theta _0})} \over {\sin {\theta _h}}} - {{\cos {\theta _h}} \over {\sin {\theta _h}}}} \cr { \times [\sin {\theta _h}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }} - \sin {\theta _0}],} \cr}$$
(19)

where h sr is the distance from slurry level to trench top. The upper-bound theorem of limit analysis gives an upper bound for the value of the safety factor, F. The function F has a minimum value when θ 0 and θ h satisfy the conditions:

$${{\partial F} \over {\partial {\theta _0}}} = 0,\;{{\partial F} \over {\partial {\theta _h}}} = 0.$$
(20)

4 3D analysis of slurry trench stability

A 3D rotational mechanism is shown in Fig. 2 for frictional/cohesive soil with a logarithmic helicoid surface, in which the failure surface is assumed to pass through the top and the toe of the slurry trench. The same shape of this mechanism is considered by Michalowski and Drescher (2009), Han et al. (2013), and Xia et al. (2012) for the evaluating safety factor of slope. Soil over the failure surface rotates about the center of rotation O. Failure surface AC is the velocity discontinuous surface. Assume the mechanism can be specified completely by three variables. For convenience, we select the angles θ 0, θ h and \(r_{0}^{'}/r_0\).

Fig. 2
figure 2

3D rotational mechanism for frictional/cohesive soil

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With the slope angle β=90°, expressions of the rate of internal energy dissipation (D) and the work rate of soil weight (W s) for the 3D failure mechanism can be found in (Michalowski and Drescher, 2009; Han et al., 2013; Xia et al., 2012).

The work rate of the soil weight (W s) can be written as

$$\matrix{ {{W_s} = 2\omega \gamma \left[ {\int_{{\theta _0}}^{{\theta _B}} {\int_0^{{x_1}} {\int_a^{{y_1}} {{{({r_m} + y)}^2}\cos \theta {\rm{d}}y{\rm{d}}x\theta}}}} \right.} \cr {\left. { + \int_{{\theta _B}}^{{\theta _h}} {\int_0^{{x_2}} {\int_d^{{y_1}} {{{({r_m} + y)}^2}\cos \theta {\rm{d}}y{\rm{d}}x\theta}}}} \right].} \cr}$$
(21)

The internal dissipation of energy (D) can be computed:

$$\matrix{ {D = 2\omega cr_0^2\cot \varphi \left[ {{{\cos }^2}{\theta _h}{{\rm{e}}^{2({\theta _h} - {\theta _0})\tan \varphi }}} \right.} \cr { \times \int_{{\theta _{\rm{B}}}}^{{\theta _h}} {{{\sin \theta}\over {{{\cos }^3}\theta }}\sqrt {{R^2} - {d^2}} {\rm{d}}\theta}} \cr {\left. { - {{\sin }^2}{\theta _0}\int_\theta ^{{\theta _{\rm{B}}}} {{{\cos \theta}\over {{{\sin }^3}\theta }}\sqrt {{R^2} - {a^2}} {\rm{d}}\theta}} \right],} \cr }$$
(22)

where the functions r m , R, x 1, x 2, y 1, a, d will be explained in Eqs. (27)–(29). The two integrals relate to the work rate in the upper portion of the slope, in the range (θ 0, θ B ), and in the remaining part of the slope (θ B , θ h ).

The slurry velocity during rotation about axis O is

$${v_{{\rm{sr}}}} = {r_{{x_2}}}\omega \sin \theta .$$
(23)

The slurry pressure is

$${p_{{\rm{sr}}}} = - 2{\gamma _{{\rm{sr}}}}{r_{{x_2}}}h''\sqrt {{R^2} - {d^2}} \sec \theta ,$$
(24)

where γ sr is the unit weight of the slurry in the trench.

The rate of external work (W sr) due to the slurry pressure is

$$\matrix{ {{W_{{\rm{sr}}}} = 2\omega {\gamma _{\rm{w}}}\int_{{\theta _{\rm{w}}}}^{{\theta _h}} {r_{{x_2}}^2h''\sqrt {{R^2} - d} \tan \theta {\rm{d}}\theta}} \cr { = \omega {\gamma _{{\rm{sr}}}}r_0^3{{\cos }^3}{\theta _h}{{\rm{e}}^{3({\theta _h} - {\theta _0})\tan \varphi }}} \cr { \times \int_{{\theta _{{\rm{sr}}}}}^{{\theta _h}} {{{2{x_2}\tan \theta (\tan \theta - \tan {\theta _{{\rm{sr}}}})} \over {{{\cos }^2}\theta }}{\rm{d}}\theta .}}\cr}$$
(25)

The safety factor in Eq. (9) can then be obtained by substituting Eqs. (21), (22), (25) into Eq. (9) as follows:

$$F = {{c\cot (\varphi )r_0^2{f_{22}} + {\gamma _{{\rm{sr}}}}{f_{21}}} \over {2\gamma ({f_{23}} + {f_{24}})}}.$$
(26)

In Eqs. (21), (22), (25) and (26), r m , R, x 1, x 2, y 1, a, d, f 21, f 22, f 23 and f 24 are functions of soil strength parameters and geometry of the slip surface which can be defined as follows:

$${r_m} = {{r + r'} \over 2},R = {{r + r'} \over 2},$$
(27)
$${x_1} = \sqrt {{R^2} - {a^2}} ,{x_2} = \sqrt {{R^2} - {d^2}} ,{y_1} = \sqrt {{R^2} - {x^2}} ,$$
(28)
$$a = {{\sin {\theta _0}} \over {\sin \theta }}{r_0},d = {{\cos {\theta _h}} \over {\cos \theta }}{r_0}{{\rm{e}}^{({\theta _h} - {\theta _0})\tan \varphi }} - {r_m},$$
(29)
$$\matrix{ {{f_{21}} = r_0^3{{\rm{e}}^{3({\theta _h} - {\theta _0})\tan \varphi }}{{\cos }^3}{\theta _h}} \cr { \times \int_{{\theta _{{\rm{sr}}}}}^{^{{\theta _h}}} {{{2{x_2}\tan \theta (\tan \theta - \tan {\theta _{{\rm{sr}}}})} \over {{{\cos }^2}\theta }}{\rm{d}}\theta ,}}\cr}$$
(30)
$$\matrix{ {{f_{22}} = {{\rm{e}}^{2({\theta _h} - {\theta _0})\tan \varphi }}{{\cos }^2}{\theta _h}\int_{{\theta _B}}^{{\theta _h}} {{{\sin \theta}\over {{{\cos }^3}\theta }}{x_2}{\rm{d}}\theta}} \cr { - {{\sin }^2}{\theta _0}\int_{{\theta _0}}^{{\theta _B}} {{{\cos \theta}\over {{{\sin }^3}\theta }}{x_1}{\rm{d}}\theta {\rm{,}}}}\cr}$$
(31)
$${f_{23}} = \int_{{\theta _0}}^{{\theta _B}} {\int_0^{{x_1}} {\int_a^{{y_1}} {{{({r_m} + y)}^2}\cos \theta {\rm{d}}y{\rm{d}}x{\rm{d}}\theta ,}}}$$
(32)
$${f_{24}} = \int_{{\theta _B}}^{{\theta _h}} {\int_0^{{x_2}} {\int_d^{{y_1}} {{{({r_m} + y)}^2}\cos \theta {\rm{d}}y{\rm{d}}x{\rm{d}}\theta ,}}}$$
(33)

where

$$r = {r_0}{{\rm{e}}^{(\theta - {\theta _0})\tan \varphi }},\;r' = {r'_0}{{\rm{e}}^{(\theta - {\theta _0})\tan \varphi }}.$$
(34)

The function F has a minimum value when θ 0, θ h and \(r_{0}^{'}/r_0\) satisfy the conditions:

$${{\partial F} \over {\partial {\theta _0}}} = 0,\;{{\partial F} \over {\partial {\theta _h}}} = 0,\;{{\partial F} \over {\partial ({{r'}_0}/{r_0})}} = 0.$$
(35)

5 Results and discussion

The estimates of safety factor F were obtained using a procedure for the given slurry and soil bulk density ratio, slurry level depth and trench depth ratio, cohesion, friction angle, 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 \(r_{0}^{'}/r_0\).

In this section, the solutions obtained by Han et al. (2012a) and Fox (2004) will be compared for various parameters, including γ sr/γ, h sr/h, c, φ and B/h. The results of these com-putations are graphically represented in Figs. 3–7. The differences in 3D and 2D factors of safety of slurry trenches can be measured by the vertical distance between the respective lines in these figures. It can be found that the factors of safety increase with increasing slurry and soil bulk density ratio, cohesion, friction angle, and with decreasing slurry level depth and trench depth ratio, trench width and depth ratio.

Fig. 3
figure 3

Comparison of results with different γ sr/γ for h sr/h =0.05, c =20 kPa, γ =18 kN/m 3 , and h =20 m

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Fig. 4
figure 4

Comparison of results with different h sr /h for γ sr /γ =0.6, c =20 kPa, γ =18 kN/m 3, and h =20 m

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Fig. 5
figure 5

Comparison of results with different c for γ sr /γ =0.6, h sr / h =0.05, γ =18 kN/m 3, and h =20 m

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Fig. 6
figure 6

Comparison of results with different φ for c =20 kPa, γ sr /γ =0.6, h sr /h =0.05, γ =18 kN/m 3, and h =20 m

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Fig. 7
figure 7

Variation of safety factor F with B/h for c =20 kPa, γ sr/γ =0.6, h sr/h =0.05, γ =18 kN/m 3 , and h=20 m

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In Figs. 3–5, and 7 the data points corresponding to φ=0° (cohesive soils), represent the solutions obtained by Han et al. (2012a). The analytical upper bound solutions of this study are higher than those presented by Han et al. (2012a). The friction angle φ>0° is the reason why the factors of safety obtained from this study are greater than those presented by Han et al. (2012a).

Figs. 6 and 7 also show the comparisons between the safety factor calculated from this study and from the limit equilibrium method by Fox (2004). It can be seen that the trends of the upper bound results are similar to those of Fox (2004), but the factors of safety obtained from this study are higher than those presented by Fox (2004). It means that, the factors of safety derived from the limit analysis method are higher than those calculated form the limit equilibrium method. The limit equilibrium analyses lead to a conservative estimate on the safety of slurry trenches. Using limit analysis can not only provide a useful way of analyzing the stability of slurry trenches, but can also avoid the shortcomings of the arbitrary assump-tions of the limit equilibrium method (Yu et al., 1998).

6 Conclusions

The upper bound method for the 2D and 3D stability analysis of slurry trenches in frictional/cohesive soil are presented in this paper. It is inves-tigated via examples for the effects of slurry and soil bulk density ratio, slurry level depth and trench depth ratio, cohesion, friction angle, trench width and depth ratio on slurry trench stability. The factor of safety increases with increasing slurry and soil bulk density ratio, cohesion, friction angle, and with decreasing slurry level depth and trench depth ratio, trench width and depth ratio. The factor of safety from a 3D analysis will approach the factor of safety from a 2D analysis gradually when the ratio of B/h increases. Using 2D solutions to evaluate the stability of 3D slurry trenches may underestimate factors of safety. Compared with the limit equilibrium method, the limit equilibrium method yields a conservative estimate on the safety factors.