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Design equations for undrained stability of opening in underground walls
Tunnelling and Underground Space Technology 70 (2017) 214–220
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Technical note
Design equations for undrained stability of opening in underground walls ⁎
MARK
Boonchai Ukritchon , Suraparb Keawsawasvong Geotechnical Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
A R T I C L E I N F O
A B S T R A C T
Keywords: Underground walls Opening Limit analysis Stability
Undrained stability of an opening in underground walls in clays with constant and linearly increasing shear strengths with depth was investigated by two-dimensional finite element limit analysis under plane strain condition. Three parametric studies were performed to study the effects of the cover depth ratio, overburden stress factor and linear strength gradient ratio on the load factor of the opening in underground walls. Predicted failure mechanisms associated with these parameters were discussed and examined. New design equations for the undrained stability of an opening in underground walls in clays with an arbitrary linear increase in strength with depth were developed for a practical application in the field of trenchless constructions.
1. Introduction In recent years, a concrete diaphragm wall is commonly employed as a permanent retaining structure for deep excavations in soft soils in various constructions including basements, underpasses, shaft stations of tunnels, cut-and-cover tunnels, and among others (e.g., Ou, 2006; Puller, 2003). This paper concerns with the study of undrained stability of an opening in underground walls in clays, as shown in Fig. 1. The problem statement consists of a rigid underground wall with an opening in cohesive soils in which the wall is completely supported by lateral bracings. The active collapse of this problem is caused by the driving force generated from the soil self-weight (γ) and a uniform surcharge on the ground surface (σs), and resisted by the mobilized soil shear resistance (su), a pressure at the opening (σt) and a side friction along the underground wall. The study of the problem is important for the stability evaluation of an opening in underground walls for many practical trenchless constructions such as the opening for a launch of a tunnel at a working shaft (Guglielmetti et al., 2008; Li et al., 2009), ventilating shaft (Wu et al., 2017), microtunnelling and horizontal drilling (Barla et al., 2006; Chapman and Ichioka, 1999; Cui et al., 2015), underpass constructions (Zhang et al., 2016), etc. In addition, it can be practically applied for the opening that may occur due to a failure of some segmental linings of underground walls and other circumstances. The soil collapse at the opening can induce the failure zone that propagates up to the ground surface resulting in severe damages to adjacent structures and facilities. The stability of this problem was firstly studied by Broms and Bennermark (1967) who performed experiments of extruding clay under pressure through vertical circular openings and proposed the
⁎
stability ratio (N) as:
N = (σs + γH −σt )/ su
where γ = soil unit weight, su = undrained shear strength of soil, H = C + D/2 is depth measured from the ground surface to the center of opening, C = cover depth and D = opening height (see Fig. 1). This equation has become widely referred by practicing engineers and researchers for empirically estimate expected deformations of being “collapse” or “unstable” for N ≥ 6 and: “elasto-plastic” for N = 2–4. As shown in Fig. 2, a problem closely related to the opening stability in underground wall corresponds to the tunnel face stability that were firstly studied by Mair (1979) using a centrifugal model test. Later, the analytical upper and lower bound limit analyses were employed by Davis et al. (1980) to derive classical plasticity solutions of the problem in a purely cohesive material, considering the two-dimensional (2D) plane strain heading of tunnel in a longitudinal direction (Fig. 2a) and the 2D plane strain unlined circular tunnel in a transverse direction (Fig. 2b). Since then, a large number of researches using experimental tests and numerical simulations were carried out to investigate the stability of tunnel face, including centrifugal model tests (Kimura and Mair, 1981; Chambon and Corté, 1994), 1g model tests (Kirsch, 2010; Berthoz et al., 2012; Chen et al., 2013), limit equilibrium method (Anagnostou and Kovári, 1994; Anagnostou and Kovári, 1996; Jancsecz and Steiner, 1994; Broere, 2001), finite difference method (Li et al., 2009; Chen et al., 2013; Senent et al., 2013; Senent and Jimenez, 2015), finite element analysis (Vermeer et al., 2002; Lu et al., 2014; Ibrahim et al., 2015; Ukritchon et al., 2017), discrete element method (Funatsu et al., 2008; Zhang et al., 2011; Chen et al., 2011), kinematic approach of
Corresponding author. E-mail addresses: [email protected] (B. Ukritchon), [email protected] (S. Keawsawasvong).
http://dx.doi.org/10.1016/j.tust.2017.08.004 Received 13 February 2017; Received in revised form 9 April 2017; Accepted 2 August 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
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Tunnelling and Underground Space Technology 70 (2017) 214–220
B. Ukritchon, S. Keawsawasvong
Fig. 1. Problem definition of an opening in underground wall in clay.
2011) are not applicable to the proposed study since their boundary conditions are completely different. This is because the face pressure of the 2D unlined circular tunnel is applied normal to the circumference of the circular tunnel, but that of the proposed study is applied normal to the front face of the opening. So far, there are very few studies of the undrained stability problem of the opening in underground walls. In practice, the conventional calculation of safety analysis for this problem still relies on the empirical Eq. (1) based on Broms and Bennermark (1967) since there is no reliable and accurate formula of hand calculation currently available in the literature. In this paper, the computational limit analysis, known as the finite element limit analysis (FELA) (e.g., Sloan, 2013), was employed to investigate the undrained stability of an opening in underground walls in clays with linearly increasing shear strength with depth. The results of this study can be applied to the stability evaluation of a vertical opening with relatively long length as compared to its vertical height. Because of its plane strain condition, the studied solutions provide a conservative estimate for a certain length of opening due to the three-dimensional effect of opening shape, as compared to the in-plane opening. Three parametric studies were performed to examine the effects of the cover depth ratio of opening, overburden stress factor and linear strength gradient ratio on the load factor of the problem. Predicted failure mechanisms associated with these parameters were discussed and examined. Based on the computed numerical lower bound solutions, new design equations were developed for the reliable and accurate predictions of the required opening pressure and safety factor (FS) of an opening in underground walls in clays with an arbitrary linear increase in strength profile with depth in practice.
upper bound limit analysis (Leca and Dormieux, 1990; Li et al., 2009; Mollon et al., 2009, 2010, 2011, 2012; Mollon, 2012; Han et al., 2016a, 2016b), finite element limit analysis (Sloan, 2013). Note that most of those works were involved with three-dimensional (3D) stability of tunnel face. However, Sloan and Assadi (1994) and Augarde et al. (2003) employed finite element limit analysis to assess the stability of 2D plane strain heading in clay layers with profiles of homogeneous and linearly increasing strengths, respectively. In addition, the upper bound solutions of the 2D plane strain heading were obtained by Huang and Song (2013) using the multi-rigid-block mechanisms. Recently, the 3D effects in the different geometry between the 3D tunnel face stability and 2D plane strain heading and unlined circular tunnel were investigated by Ukritchon et al. (2017), whereas the upper bound limit analysis with rigid and deformed elements was employed by Yang et al. (2016) to compute the drained stability of 2D plane strain heading in a cohesive-frictional soil. A similarity of the stability problem can be observed between the 2D plane strain heading of tunnel face (Fig. 2a) and the opening in underground walls (Fig. 1), in which an opening appears in these two problems. However, existing solutions for the plane strain heading (e.g., Davis et al., 1980; Sloan and Assadi, 1994; Augarde et al., 2003; Huang and Song, 2013) may not be directly applied to the proposed study since there are some differences in their boundary conditions. In particular, there is no rigid horizontal lining of tunnel in the proposed study, while the effect of rigid underground wall is not considered in the stability of 2D plane strain heading. In other words, the stability of tunnel face can be considered of a later stage of the opening in underground wall that is relatively located far away from the tunnel heading. Consequently, the application of existing solutions of the plane strain heading of tunnel face to the proposed study is still questionable. In addition, it is clear that the solutions of unlined circular tunnel (Fig. 2b) (Davis et al., 1980; Sloan and Assadi, 1994; Wilson et al.,
2. Method of analysis FELA is the numerical computational method of limit analysis that Fig. 2. Two-dimensional stability of tunnel face by Davis et al. (1980).
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vertical and horizontal directions, while the left boundary of the problem was allowed to move only in the vertical direction. In addition, the rigid wall is achieved by constraining zero movement in both directions. The size of the problem domains were chosen to be large enough that the plastic yielding zone was contained within the domain and did not intersect the left and bottom boundaries. Therefore, the UB and LB limit loads are not altered by an extension of the domain size, and there is no influence of the domain size on the computed solutions. In OptumG2, the UB and LB solutions of the limiting pressure σs are solved by employing the second-order cone programming (SOCP) (e.g., Krabbenhoft et al., 2007). An automatically adaptive mesh refinement, a powerful feature in OptumG2, was employed in both the UB and LB simulations to compute the tight UB and LB solutions. Five iterations of adaptive meshing with the number of elements increasing from 5000 to 10,000 were used for all analyses, since initial calibration tests revealed that this setting was adequate enough to obtain a suitable accurate solution. Note that the plane strain FELA using SOCP in OptumG2 is comparable to that of previous formulations proposed by Krabbenhoft et al. (2005), Makrodimopoulos and Martin (2006, 2007). Full details of the numerical FELA formulation in OptumG2 can be found in Krabbenhoft et al. (2015).
employs the classical plasticity theorems (Drucker et al. 1952) with the concept of finite element and mathematical programming. At present, it has become a powerful approach for analyzing various complex stability problems in geotechnical engineering, as demonstrated by Sloan (2013). The technique utilizes the powerful capabilities of finite element discretization for handling complicated soil stratifications, loadings and boundary conditions with the plastic bound theorem (Drucker et al., 1952) to bracket the exact limit load by upper bound (UB) and lower bound (LB) solutions, while the computational efficiency is achieved by the technique of convex optimization (e.g., Krabbenhoft et al., 2007) in mathematical programming. The underlying bound theorems assume a rigid-perfectly plastic material with an associated flow rule. Details of limit analysis and FELA are omitted here but may be found in Chen and Liu (1990) and Sloan (2013), respectively. In this study, the FELA software, OptumG2 (Krabbenhoft et al., 2015), under the plane strain condition was employed to investigate the 2D undrained stability of an opening in underground walls in cohesive soils. Recently, OptumG2 was successfully applied to solve a variety of undrained stability problems in geotechnical engineering, including suction caissons (Keawsawasvong and Ukritchon 2016; Ukritchon and Keawsawasvong 2016), conical slopes (Keawsawasvong and Ukritchon, 2017a), active failure of trapdoors (Keawsawasvong and Ukritchon, 2017b), and contiguous pile walls (Keawsawasvong and Ukritchon, 2017c). Consequently, OptumG2 was chosen in this study to simulate the undrained soil collapse at an opening in underground walls and compute its numerical plasticity solutions. The problem definition of the proposed study is shown in Fig. 1. An opening with a height (D) is made in the underground wall at the distance (C) measured from the ground surface. It is assumed that the wall is rigid and fully rough and there is a complete lateral bracing of the wall. The soil profile is a cohesive soil with unit weight (γ) and undrained shear strength at the ground surface (su0) that increases linearly with depth with a gradient (ρ). Because the geometry of the problem is a continuous wall, a plane strain condition was used in FELA, where an example of numerical model is shown in Fig. 3. In both upper and lower bound calculations, the soil mass was discretized as triangular elements and modelled as Tresca material with the associated flow rule. The continuous wall was modelled as the rigid plate element while the soil-structure interface element with a fully rough condition was employed along the length of wall in order to simulate the shear sliding of soil along the wall. The opening is resisted by a uniform pressure (σt), which is considered to be a general case of an opening with free surface. A uniform surcharge (σs) is applied all over the ground surface and is optimized in both upper and lower bound simulations to compute the bound solution of the limiting pressure σs. The boundary condition of the problem was defined such that the bottom boundary of the model was fixed in both
3. Results and discussions For the undrained stability of an opening in underground walls, there are seven input parameters, namely γ, su0, ρ, σs , σt, C and D. Based on the dimensional analysis, it can be shown that four dimensionless parameters can be created from these input parameters, namely: (i) cover depth ratio (C/D), (ii) overburden stress factor (γD/su0), (iii) linear strength gradient ratio (ρD/su0) and (iv) load factor ((σs − σt)/ su0). In this study, the latter was taken as the dependent parameter that is a function of the first three terms considered as the independent input parameters. Note that Augarde et al. (2003) also employed the same set of four dimensionless parameters to investigate the undrained stability of 2D plane strain heading in clays with linearly increasing strength with depth. Parametric studies were performed on the three dimensionless input parameters, covering their practical ranges of C/D = 1–10, γD/ su0 = 0–5 and ρD/su0 = 0–2. Fig. 4 shows the relationship between (σs − σt)/su0 as a function of γD/su0, C/D and ρD/su0. For all values of these input parameters, the exact load factor can be accurately bracketed by computed UB and LB solutions within 5%. In general, a linearly decreasing relationship between (σs − σt)/su0 and γD/su0 is observed. The gradient of this linear relationship is significantly affected by C/D and ρD/su0, where these larger ratios lead to the large gradient. Closer inspections of the results indicate that a nonlinear increasing relationship between (σs − σt)/su0 and C/D is found, while a linearly increasing relationship between (σs − σt)/su0 and ρD/su0 is observed. A positive sign of (σs − σt)/su0 implies that the surcharge required to produce a soil collapse at the opening must be greater than the applied pressure at the opening. On the contrary, a negative sign of (σs − σt)/su0 implies that the applied pressure at the opening must be larger than the applied surcharge in order to maintain the soil stability at the opening. A comparison of the stability ratio (N) of Broms and Bennermark (1967) defined in Eq. (1) is shown in Fig. 5, corresponding to the case of homogenous clay with weightless material (i.e., γ = 0). Note that the N ratios of the present study correspond to the opening in underground walls, while existing solutions correspond to those of 2D plane strain heading using analytical (Davis et al., 1980) and numerical (Augarde et al., 2003) methods of lower bound limit analysis. A good agreement between the stability ratio of opening and that of plane strain heading by Davis et al. (1980) using the mechanism of LB thick cylinder and Augarde et al. (2003) is observed, where the former is higher than the latter about 7–9%. Such result is mainly attributed from the influence of
Fig. 3. Numerical model of an opening in underground wall in clay (initial non-adaptive mesh).
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Fig. 4. Relationship between (σs − σt)/su0 and γD/su0 for ρD/su0 values of (a) 0, (b) 0.5, (c) 1 and (e) 2.
corners of opening and extends to outcrop at the ground surface. In addition, a shear band is developed along the upper length of underground wall, which is not taken into account for the failure mechanism of the 2D plane strain heading. Note that there is a difference in the size of plastic yielding zone between two ratios of ρD/su0, where a higher ρD/su0 ratio results in a smaller size of plastic zone. This result suggests that the size of plastic yielding zone depends on this parameter as well as the other two dimensionless parameters (i.e., C/D and γD/su0). Table 1 shows a closer comparison in the load factor (σs − σt)/su0 between the present study of 2D opening in underground wall and the existing LB solutions of 2D plane strain heading. The latter include the numerical LB solutions by Augarde et al. (2003) and the analytical LB solutions by Davis et al. (1980) considering a stress field in a large cylindrical volume of soil (thick cylinder). Note that the solutions by Davis et al. (1980) are available only to the case of homogeneous strength. In the comparison, the dimensionless parameters were selected to represent most practical cases at their limiting ratios as: (1) shallow vs. deep openings (i.e., C/D = 1 and 10); (2) small vs. large opening heights and/or overburden stresses (i.e., γD/su0 = 0 and 3); and (3) homogenous vs. non-homogenous strengths (i.e., ρD/su0 = 0 and 1). In general, it can be observed that the solutions of the present study are higher than those of 2D plane strain heading by 2–40% because of the kinematic constraint of underground wall above the opening. The large differences in (σs − σt)/su0 seem to be associated with large ratios of C/D and γD/su0 and smaller ratios of ρD/su0. Under these conditions, the influence of fully rough rigid wall becomes more prominent on (σs − σt)/su0. Consequently, the comparison results confirm the difference in the boundary conditions between the two problems and justify the need of the proposed study. Another closer comparison between the present study of 2D undrained stability of opening in underground wall and 3D undrained stability of tunnel face is also shown in Table 1. In this comparison, the
Fig. 5. Comparison of stability ratios (N) between the present study and existing solutions for homogeneous clay.
rough rigid underground wall on the stability of opening, which is not considered in the 2D plane strain heading. The comparison also indicates that the stress field in a large spherical volume of soil (thick sphere) proposed by Davis et al. (1980) furnishes significantly unsafe LB solutions for the stability of opening in underground walls, and hence they should not be used in practice. Two examples of predicted failure mechanisms deduced from the final adaptive meshes and velocity fields are shown in Fig. 6, where C/ D = 4, γD/su0 = 1, ρD/su0 = 0 and 1. The mode of failure is caused by the soil squeezed out of the opening due to the actions of the soil selfweight and the applied surcharge, which are considered as the driving force of the problem. The plastic yielding zone initiates from the
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Fig. 6. Example of adaptive mesh and velocity field, where C/D = 4 and γD/su0 = 1 for ρD/su0 values of (a) 0 and (b) 1.
(a) ρD/su0 = 0
(b) ρD/su0 = 1 parameters, the solutions of the present study are significantly smaller than those of the 3D tunnel face stability by 50–100%, confirming the assumption of conservativeness in applying the results of the present study to the 3D problems.
solutions of the latter were obtained by employing Tunnel Face Stability Software (TFSS) (Mollon, 2012) that is based on the kinematic approach of upper bound limit analysis, which represents the most upto-date of previous studies by Mollon et al. (2009, 2010, 2011, 2012). Similarly, comparisons were made between the limiting cases of each dimensionless parameter covering most practical cases of only homogenous strengths, namely (1) C/D = 1 and 10 (i.e., shallow vs. deep openings) and (2) γD/su0 = 0 and 3 (small vs. large opening heights and/or overburden stresses). Note that there is no result of TFSS in the case of non-homogeneous strength since it is limited to the stability analysis of only a homogeneous strength profile (i.e., ρD/su0 = 0). As expected, it is very evident that there are substantial differences in the load factor (σs − σt)/su0 between the present study of 2D opening in underground wall and the 3D tunnel face stability (TFSS), which are mainly attributed from the 3D effect in the different geometry between the two problems. For most practical ranges of the dimensionless
3.1. New design equations A technique of nonlinear regression analysis was employed to develop the approximate solution for the undrained stability of the opening in underground walls. Since the lower bound theorem provides a safe estimate of the limit load for a stability problem, the computed LB solutions were chosen in the regression analysis. It was found that a linear relationship between (σs − σt)/su0 and (γD/su0, ρD/su0) exists, while a power function yields an accurate curve-fitting between (σs − σt)/su0 and C/D. Therefore, the accurate approximate LB solution for the undrained stability of the opening in underground wall is
Table 1 Comparison of the load factor (σs − σt)/su0 between the present study and other solutions. C/D
γD/su0
ρD/su0
Present study Numerical LB 2D opening in underground wall
Augarde et al. (2003) Numerical LB 2D plane strain heading
Davis et al. (1980) Analytical LB (thick cylinder) 2D plane strain heading
Mollon (2012) Numerical UB 3D tunnel face stability
1 10 1 10 1 10 1 10
0 0 3 3 0 0 3 3
0 0 0 0 1 1 1 1
4.23 8.67 −0.44 −22.97 8.86 82.00 4.36 50.49
4.00 7.70 −0.74 −24.39 8.73 80.29 4.19 48.82
4.197 8.089 −0.303 −23.411 – – – –
7.89 21.24 3.10 −10.98 – – – –
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Fig. 8. Relationship between the stability factors (Nc, Nρ and Nγ) of an opening in underground walls and cover depth ratio (C/D).
Fig. 7. Comparison of the load factor (σs − σt)/su0 between the proposed Eq. (2) and lower bound solutions for the undrained stability of an opening in underground walls.
compute the opening pressure that is required to maintain the stability for a certain level of safety factor (FS). The conventional FS used in a slope stability problem is adopted here such that it is defined as the ratio of the full shear strength to the mobilized shear strength. Thus, su0 and ρ in the right hand side of Eq. (3) are treated as the “mobilized” strength parameter. By replacing su0 by su0/FS and ρ by ρ/FS in Eq. (3) and solving for σt, the design equation for the required opening pressure maintaining the stability can be obtained:
proposed, as shown in Eq. (2).
σs−σt C a2 C a4 ρD ⎞ C γD = a1 ⎛ ⎞ + a3 ⎛ ⎞ ⎛ −⎛0.5 + ⎞ ⎛ ⎞ su0 D ⎠ ⎝ su0 ⎠ ⎝D⎠ ⎝ D ⎠ ⎝ su0 ⎠ ⎝ ⎜
⎟
⎜
⎟
(2)
where a1 = 4.5690, a2 = 0.2786, a3 = 4.0725 and a4 = 1.2540. The comparison of (σs − σt)/su0 between the predictions in Eq. (2) and the lower bound solutions is shown in Fig. 7, where a good agreement between these two solutions can be achieved, with a coefficient of determination (R2) of 99.99%. In practice, Eq. (2) is not convenient for a stability evaluation of the problem. Instead, a “nicer” expression can be obtained from Eq. (2) by multiplying su0 to both sides in the equation. Hence, the new design equation for a safe estimate of the limiting uniform surcharge (σs) of an opening in underground walls in clays with linearly increasing strength profiles is proposed, as shown in Eq. (3).
σs = su0 Nc + ρDNρ + σt −γDNγ
σt = σs + γDNγ −
Nρ =
(3)
FS =
C a2 a1 D C a4 a3 D
() ()
Nγ = 0.5 +
C D
(5)
where FS = required safety factor. Another practical application is to assess the safety level of an opening in underground walls when the problem geometry (C, D), soil properties (γ, su0, ρ), loadings (σs, σt) are given. Therefore, the design equation for the factor of safety against the soil collapse at the opening can be determined straightforwardly from Eq. (5), as shown below.
where
Nc =
ρ su0 Nc− DNρ FS FS
su0 Nc + ρDNρ σs + γDNγ −σt
(6)
The physical meaning of Eq. (6) can be interpreted as follows. The FS factor is defined as the ratio of the total shear resistance of soil to the net driving pressure of the problem. The former is contributed from the effects of the constant soil cohesion (su0Nc) and linear strength gradient (ρDNρ), while the latter is calculated from the sum of the uniform surcharge on the ground surface (σs) and the effect of the overburden pressure (γDNγ) less the trapdoor pressure (σt) that is the resistance of the problem. Consequently, Eqs. (5) and (6) represent the design stability equations for the opening in underground walls in clays with an arbitrary linear increase in strength with depth, including constant strength (ρ = 0) and linearly increasing shear strengths (su0 ≠ 0, ρ ≠ 0). The latter also includes a special case of zero strength at the ground surface (su0 = 0, ρ ≠ 0), where it corresponds to the case of a normally consolidated clay.
(4)
The new design equation in Eq. (3) and associated factors in Eq. (4) employ the concept that is similar to the Terzaghi’s classical bearing capacity equation. The terms Nc, Nρ and Nγ represent the stability factors of the opening in underground walls for a constant soil cohesion (su0), linear strength gradient (ρ) and soil unit weight (γ), respectively. The negative sign of the last term in the right hand side indicates that the unit weight of soil (γ) represents the driving force of the problem that produces the same effect as the uniform surcharge (σs). Consequently, the physical meaning of Eq. (3) can be interpreted such that the limiting surcharge on the ground surface causing the soil collapse at the opening can be calculated from the sum of three effects of problem resistance, including the cohesion part, the linear strength gradient part, and the opening pressure less the effect of soil self-weight that is the driving force of the problem. Fig. 8 shows the relationship of new three stability factors, where they are a function of only the cover depth ratio. Nonlinear increasing functions between Nc, Nρ and C/D were observed, whereas a linearly increasing relationship between Nγ and C/D was found. It should be noted that the same expression Nγ = 0.5 + C/D was also proposed by Davis et al. (1980) using analytical LB and UB limit analysis of 2D plane strain heading. Therefore, the proposed expression of Nγ in this study seems to be reasonable. For a practical application, the safety evaluation of the problem is to
4. Conclusions In this paper, the undrained stability problem of an opening in underground walls in clays with linearly increasing shear strength with depth was investigated by the computational limit analysis, FELA. Three parametric studies were performed to study the effects of the cover depth ratio (C/D), overburden stress factor (γD/su0) and linear strength gradient ratio (ρD/su0) on the load factor ((σs − σt)/su0) of the opening in underground walls. In all cases, the exact load factors were accurately bracketed by the computed bound solutions within 5%. It was found that (σs − σt)/su0 had a nonlinear increasing function with C/D, a linearly decreasing function with γD/su0 and a linearly increasing function with ρD/su0. By employing the nonlinear regression 219
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non-homogeneous clays. Comput. Geotech. 81, 125–136. Keawsawasvong, S., Ukritchon, B., 2017b. Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth. Comput. Geotech. 81, 284–293. Keawsawasvong, S., Ukritchon, B., 2017c. Undrained limiting pressure behind soil gaps in contiguous pile walls. Comput. Geotech. 83, 152–158. Kimura, T., Mair, R.J., 1981. Centrifugal testing of model tunnels in soft clay. Proc. Tenth Int. Conf. Soil Mech. Found. Eng. 1, 19–322. Kirsch, A., 2010. Experimental investigation of the face stability of shallow tunnels in sand. Acta Geotech. 5 (1), 43–62. Krabbenhoft, K., Lyamin, A.V., Hjiaj, M., Sloan, S.W., 2005. A new discontinuous upper bound limit analysis formulation. Int. J. Numer. Meth. Eng. 63, 1069–1088. Krabbenhoft, K., Lyamin, A.V., Sloan, S.W., 2007. Formulation and solution of some plasticity problems as conic programs. Int. J. Solids Struct. 44, 1533–1549. Krabbenhoft, K., Lyamin, A., Krabbenhof, T.J., 2015. Optum Computational Engineering, (OptumG2), Available on, < www.optumce.com > . Leca, E., Dormieux, L., 1990. Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Géotechnique 40 (4), 581–606. Li, Y., Emeriault, F., Kastner, R., Zhang, Z.X., 2009. Stability analysis of large slurry shield-driven tunnel in soft clay. Tunn. Undergr. Space Technol. 24 (4), 472–481. Lu, X., Wang, H., Huang, M., 2014. Upper bound solution for the face stability of shield tunnel below the water table. Math. Probl. Eng, 727964. Mair, R.J., 1979. Centrifugal Modelling of Tunnel Construction In Soft Clay. (PhD Thesis) Cambridge University. Makrodimopoulos, A., Martin, C.M., 2006. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Meth. Eng. 66, 604–634. Makrodimopoulos, A., Martin, C.M., 2007. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int. J. Geomech. 31, 835–865. Mollon, G., 2012. Tunnel Face Stability Software (TFSS): Ver. 1.0; July, 2012. Available at: < http://guilhem.mollon.free.fr/Accueil_Eng.html > (accessed October 2015). Mollon, G., Dias, D., Soubra, A., 2009. Probabilistic analysis and design of circular tunnels against face stability. Int. J. Geomech. 9 (6), 237–249. Mollon, G., Dias, D., Soubra, A., 2010. Face stability analysis of circular tunnels driven by a pressurized shield. J. Geotech. Geoenviron. Eng. 136 (1), 215–229. Mollon, G., Dias, D., Soubra, A., 2011. Rotational failure mechanisms for the face stability analysis of tunnels driven by a pressurized shield. Int. J. Numer. Anal. Meth. Geomech. 35 (12), 1363–1388. Mollon, G., Dias, D., Soubra, A., 2012. Continuous velocity fields for collapse and blowout of a pressurized tunnel face in purely cohesive soil. Int. J. Numer. Anal. Meth. Geomech. 37 (13), 2061–2083. Ou, C.Y., 2006. Deep Excavation, Theory and Practice. Taylor & Francis Group, London. Puller, M., 2003. Deep Excavations, A Practical Manual, second ed. Thomas Telford, London. Senent, S., Jimenez, R., 2015. A tunnel face failure mechanism for layered ground, considering the possibility of partial collapse. Tunn. Undergr. Space Technol. 47, 182–192. Senent, S., Mollon, G., Jimenez, R., 2013. Tunnel face stability in heavily fractured rock masses that follow the Hoek-Brown failure criterion. Int. J. Rock Mech. Min. Sci. 60, 440–451. Sloan, S.W., 2013. Geotechnical stability analysis. Géotechnique 63 (7), 531–572. Sloan, S.W., Assadi, A., 1994. Undrained stability of a plane strain heading. Can. Geotech. J. 31, 443–450. Ukritchon, B., Keawsawasvong, S., 2016. Undrained pullout capacity of cylindrical suction caissons by finite element limit analysis. Comput. Geotech. 80, 301–311. Ukritchon, B., Keawsawasvong, S., Yingchaloenkitkhajorn, K., 2017. Undrained face stability of tunnels in Bangkok subsoils. Int. J. Geotech. Eng. 11 (3), 262–277. Vermeer, P., Ruse, N., Marcher, T., 2002. Tunnel heading stability in drained ground. Felsbau 20 (6), 8–24. Wilson, D.W., Abbo, A.J., Sloan, S.W., Lyamin, A.V., 2011. Undrained stability of a circular tunnel where the shear strength increases linearly with depth. Can. Geotech. J. 48 (9), 1328–1342. Wu, G., Chen, W., Bian, H., Yuan, J., 2017. Structure optimisation of a diaphragm wall with special modelling methods in a large-scale circular ventilating shaft considering shield crossing. Tunn. Undergr. Space Technol. 65, 35–41. Yang, F., Zhang, J., Zhao, L., Yang, J., 2016. Upper-bound finite element analysis of stability of tunnel face subjected to surcharge loading in cohesive-frictional soil. KSCE J. Civil Eng. 20 (6), 2270–2279. Zhang, Z.X., Hu, X.Y., Scott, K.D., 2011. A discrete numerical approach for modeling face stability in slurry shield tunnelling in soft soils. Comput. Geotech. 38 (1), 94–104. Zhang, D., Liu, B., Qin, Y., 2016. Construction of a large-section long pedestrian underpass using pipe jacking in muddy silty clay, a case study. Tunn. Undergr. Space Technol. 60, 151–164.
analysis to the computed lower bound solutions, the new design stability equations of the required opening pressure (σt) and the safety factor (FS) for the opening in underground walls were developed. In addition, the new stability factors of the opening in underground walls, with respect to constant soil cohesion, linear shear strength gradient and soil unit weight were proposed for the use of these design equations that can be applied for an arbitrary linear increase in undrained shear strength of clay with depth. Therefore, the new design stability equations provide a simple, accurate and reliable hand calculation of the undrained stability problem of an opening in underground walls in practice. References Anagnostou, G., Kovári, K., 1994. The face stability of slurry shield-driven tunnels. Tunn. Undergr. Space Technol. 9 (2), 165–174. Anagnostou, G., Kovári, K., 1996. Face stability conditions with earth-pressure-balanced shield. Tunn. Undergr. Space Technol. 11 (2), 165–173. Augarde, C.E., Lyamin, A.V., Sloan, S.W., 2003. Stability of an undrained plane strain heading revisited. Comput. Geotech. 30 (5), 419–430. Barla, M., Camusso, M., Aiassa, S., 2006. Analysis of jacking forces during microtunnelling in limestone. Tunn. Undergr. Space Technol. 21 (6), 668–683. Berthoz, N., Branque, D., Subrin, D., Wong, H., Humbert, E., 2012. Face failure in homogeneous and stratified soft ground, theoretical and experimental approaches on 1g EPBS reduced scale model. Tunn. Undergr. Space Technol. 30, 25–37. Broere, W., 2001. Tunnel Face Stability and New CPT Applications. (PhD Thesis) Delft University of Technology, The Netherlands. Broms, B.B., Bennermark, H., 1967. Stability of clay in vertical openings. J. Geotech. Eng. Divis. 193, 71–94. Chambon, P., Corté , J.F., 1994. Shallow tunnels in cohesionless soil, stability of tunnel face. J. Geotech. Eng. 120 (7), 1148–1165. Chapman, D.N., Ichioka, Y., 1999. Prediction of jacking forces for microtunnelling operations. Tunn. Undergr. Space Technol. 14 (1), 31–41. Chen, W.F., Liu, X.L., 1990. Limit Analysis in Soil Mechanics. Elsevier, Amsterdam. Chen, R.P., Tang, L.J., Ling, D.S., Chen, Y.M., 2011. Face stability analysis of shallow shield tunnels in dry sandy ground using the discrete element method. Comput. Geotech. 38 (2), 187–195. Chen, R.P., Li, J., Kong, L.G., Tang, L.J., 2013. Experimental study on face instability of shield tunnel in sand. Tunn. Undergr. Space Technol. 33, 12–21. Cui, Q.L., Xu, Y.S., Shen, S.L., Yin, Z.Y., Horpibulsuk, S., 2015. Field performance of concrete pipes during jacking in cemented sandy silt. Tunn. Undergr. Space Technol. 49, 336–344. Davis, E.H., Gunn, M.J., Mair, R.J., Seneviratn, H.N., 1980. The stability of shallow tunnels and underground openings in cohesive material. Geotechnique 30 (4), 397–416. Drucker, D.C., Prager, W., Greenberg, H.J., 1952. Extended limit design theorems for continuous media. Q. Appl. Math. 9, 381–389. Funatsu, T., Hoshino, T., Sawae, H., Shimizu, N., 2008. Numerical analysis to better understand the mechanism of the effects of ground supports and reinforcements on the stability of tunnels using the distinct element method. Tunn. Undergr. Space Technol. 23 (5), 561–573. Guglielmetti, V., Grasso, P., Mahtab, A., Xu, S., 2008. Mechanized Tunnelling in Urban Areas – Design Methodology and Construction Control. Taylor & Francis Group, London. Han, K., Zhang, C., Li, W., Guo, C., 2016a. Face stability analysis of shield tunnels in homogeneous soil overlaid by multilayered cohesive-frictional soils. Math. Probl. Eng. 18. Han, K., Zhang, C., Zhang, D., 2016b. Upper-bound solutions for the face stability of a shield tunnel in multilayered cohesive–frictional soils. Comput. Geotech. 79, 1–9. Huang, M., Song, C., 2013. Upper-bound stability analysis of a plane strain heading in non-homogeneous clay. Tunn. Undergr. Space Technol. 38, 213–223. Ibrahim, E., Soubra, A.H., Mollon, G., Raphael, W., Dias, D., Reda, A., 2015. Three-dimensional face stability analysis of pressurized tunnels driven in a multilayered purely frictional medium. Tunn. Undergr. Space Technol. 49, 18–34. Jancsecz, S., Steiner, W., 1994. Face support for a large mix-shield in heterogeneous ground conditions. Tunnelling 94, 531–550. Keawsawasvong, S., Ukritchon, B., 2016. Finite element limit analysis of pullout capacity of planar caissons in clay. Comput. Geotech. 75, 12–17. Keawsawasvong, S., Ukritchon, B., 2017a. Stability of unsupported conical Excavations in
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Technical note
Design equations for undrained stability of opening in underground walls ⁎
MARK
Boonchai Ukritchon , Suraparb Keawsawasvong Geotechnical Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
A R T I C L E I N F O
A B S T R A C T
Keywords: Underground walls Opening Limit analysis Stability
Undrained stability of an opening in underground walls in clays with constant and linearly increasing shear strengths with depth was investigated by two-dimensional finite element limit analysis under plane strain condition. Three parametric studies were performed to study the effects of the cover depth ratio, overburden stress factor and linear strength gradient ratio on the load factor of the opening in underground walls. Predicted failure mechanisms associated with these parameters were discussed and examined. New design equations for the undrained stability of an opening in underground walls in clays with an arbitrary linear increase in strength with depth were developed for a practical application in the field of trenchless constructions.
1. Introduction In recent years, a concrete diaphragm wall is commonly employed as a permanent retaining structure for deep excavations in soft soils in various constructions including basements, underpasses, shaft stations of tunnels, cut-and-cover tunnels, and among others (e.g., Ou, 2006; Puller, 2003). This paper concerns with the study of undrained stability of an opening in underground walls in clays, as shown in Fig. 1. The problem statement consists of a rigid underground wall with an opening in cohesive soils in which the wall is completely supported by lateral bracings. The active collapse of this problem is caused by the driving force generated from the soil self-weight (γ) and a uniform surcharge on the ground surface (σs), and resisted by the mobilized soil shear resistance (su), a pressure at the opening (σt) and a side friction along the underground wall. The study of the problem is important for the stability evaluation of an opening in underground walls for many practical trenchless constructions such as the opening for a launch of a tunnel at a working shaft (Guglielmetti et al., 2008; Li et al., 2009), ventilating shaft (Wu et al., 2017), microtunnelling and horizontal drilling (Barla et al., 2006; Chapman and Ichioka, 1999; Cui et al., 2015), underpass constructions (Zhang et al., 2016), etc. In addition, it can be practically applied for the opening that may occur due to a failure of some segmental linings of underground walls and other circumstances. The soil collapse at the opening can induce the failure zone that propagates up to the ground surface resulting in severe damages to adjacent structures and facilities. The stability of this problem was firstly studied by Broms and Bennermark (1967) who performed experiments of extruding clay under pressure through vertical circular openings and proposed the
⁎
stability ratio (N) as:
N = (σs + γH −σt )/ su
where γ = soil unit weight, su = undrained shear strength of soil, H = C + D/2 is depth measured from the ground surface to the center of opening, C = cover depth and D = opening height (see Fig. 1). This equation has become widely referred by practicing engineers and researchers for empirically estimate expected deformations of being “collapse” or “unstable” for N ≥ 6 and: “elasto-plastic” for N = 2–4. As shown in Fig. 2, a problem closely related to the opening stability in underground wall corresponds to the tunnel face stability that were firstly studied by Mair (1979) using a centrifugal model test. Later, the analytical upper and lower bound limit analyses were employed by Davis et al. (1980) to derive classical plasticity solutions of the problem in a purely cohesive material, considering the two-dimensional (2D) plane strain heading of tunnel in a longitudinal direction (Fig. 2a) and the 2D plane strain unlined circular tunnel in a transverse direction (Fig. 2b). Since then, a large number of researches using experimental tests and numerical simulations were carried out to investigate the stability of tunnel face, including centrifugal model tests (Kimura and Mair, 1981; Chambon and Corté, 1994), 1g model tests (Kirsch, 2010; Berthoz et al., 2012; Chen et al., 2013), limit equilibrium method (Anagnostou and Kovári, 1994; Anagnostou and Kovári, 1996; Jancsecz and Steiner, 1994; Broere, 2001), finite difference method (Li et al., 2009; Chen et al., 2013; Senent et al., 2013; Senent and Jimenez, 2015), finite element analysis (Vermeer et al., 2002; Lu et al., 2014; Ibrahim et al., 2015; Ukritchon et al., 2017), discrete element method (Funatsu et al., 2008; Zhang et al., 2011; Chen et al., 2011), kinematic approach of
Corresponding author. E-mail addresses: [email protected] (B. Ukritchon), [email protected] (S. Keawsawasvong).
http://dx.doi.org/10.1016/j.tust.2017.08.004 Received 13 February 2017; Received in revised form 9 April 2017; Accepted 2 August 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
(1)
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Fig. 1. Problem definition of an opening in underground wall in clay.
2011) are not applicable to the proposed study since their boundary conditions are completely different. This is because the face pressure of the 2D unlined circular tunnel is applied normal to the circumference of the circular tunnel, but that of the proposed study is applied normal to the front face of the opening. So far, there are very few studies of the undrained stability problem of the opening in underground walls. In practice, the conventional calculation of safety analysis for this problem still relies on the empirical Eq. (1) based on Broms and Bennermark (1967) since there is no reliable and accurate formula of hand calculation currently available in the literature. In this paper, the computational limit analysis, known as the finite element limit analysis (FELA) (e.g., Sloan, 2013), was employed to investigate the undrained stability of an opening in underground walls in clays with linearly increasing shear strength with depth. The results of this study can be applied to the stability evaluation of a vertical opening with relatively long length as compared to its vertical height. Because of its plane strain condition, the studied solutions provide a conservative estimate for a certain length of opening due to the three-dimensional effect of opening shape, as compared to the in-plane opening. Three parametric studies were performed to examine the effects of the cover depth ratio of opening, overburden stress factor and linear strength gradient ratio on the load factor of the problem. Predicted failure mechanisms associated with these parameters were discussed and examined. Based on the computed numerical lower bound solutions, new design equations were developed for the reliable and accurate predictions of the required opening pressure and safety factor (FS) of an opening in underground walls in clays with an arbitrary linear increase in strength profile with depth in practice.
upper bound limit analysis (Leca and Dormieux, 1990; Li et al., 2009; Mollon et al., 2009, 2010, 2011, 2012; Mollon, 2012; Han et al., 2016a, 2016b), finite element limit analysis (Sloan, 2013). Note that most of those works were involved with three-dimensional (3D) stability of tunnel face. However, Sloan and Assadi (1994) and Augarde et al. (2003) employed finite element limit analysis to assess the stability of 2D plane strain heading in clay layers with profiles of homogeneous and linearly increasing strengths, respectively. In addition, the upper bound solutions of the 2D plane strain heading were obtained by Huang and Song (2013) using the multi-rigid-block mechanisms. Recently, the 3D effects in the different geometry between the 3D tunnel face stability and 2D plane strain heading and unlined circular tunnel were investigated by Ukritchon et al. (2017), whereas the upper bound limit analysis with rigid and deformed elements was employed by Yang et al. (2016) to compute the drained stability of 2D plane strain heading in a cohesive-frictional soil. A similarity of the stability problem can be observed between the 2D plane strain heading of tunnel face (Fig. 2a) and the opening in underground walls (Fig. 1), in which an opening appears in these two problems. However, existing solutions for the plane strain heading (e.g., Davis et al., 1980; Sloan and Assadi, 1994; Augarde et al., 2003; Huang and Song, 2013) may not be directly applied to the proposed study since there are some differences in their boundary conditions. In particular, there is no rigid horizontal lining of tunnel in the proposed study, while the effect of rigid underground wall is not considered in the stability of 2D plane strain heading. In other words, the stability of tunnel face can be considered of a later stage of the opening in underground wall that is relatively located far away from the tunnel heading. Consequently, the application of existing solutions of the plane strain heading of tunnel face to the proposed study is still questionable. In addition, it is clear that the solutions of unlined circular tunnel (Fig. 2b) (Davis et al., 1980; Sloan and Assadi, 1994; Wilson et al.,
2. Method of analysis FELA is the numerical computational method of limit analysis that Fig. 2. Two-dimensional stability of tunnel face by Davis et al. (1980).
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vertical and horizontal directions, while the left boundary of the problem was allowed to move only in the vertical direction. In addition, the rigid wall is achieved by constraining zero movement in both directions. The size of the problem domains were chosen to be large enough that the plastic yielding zone was contained within the domain and did not intersect the left and bottom boundaries. Therefore, the UB and LB limit loads are not altered by an extension of the domain size, and there is no influence of the domain size on the computed solutions. In OptumG2, the UB and LB solutions of the limiting pressure σs are solved by employing the second-order cone programming (SOCP) (e.g., Krabbenhoft et al., 2007). An automatically adaptive mesh refinement, a powerful feature in OptumG2, was employed in both the UB and LB simulations to compute the tight UB and LB solutions. Five iterations of adaptive meshing with the number of elements increasing from 5000 to 10,000 were used for all analyses, since initial calibration tests revealed that this setting was adequate enough to obtain a suitable accurate solution. Note that the plane strain FELA using SOCP in OptumG2 is comparable to that of previous formulations proposed by Krabbenhoft et al. (2005), Makrodimopoulos and Martin (2006, 2007). Full details of the numerical FELA formulation in OptumG2 can be found in Krabbenhoft et al. (2015).
employs the classical plasticity theorems (Drucker et al. 1952) with the concept of finite element and mathematical programming. At present, it has become a powerful approach for analyzing various complex stability problems in geotechnical engineering, as demonstrated by Sloan (2013). The technique utilizes the powerful capabilities of finite element discretization for handling complicated soil stratifications, loadings and boundary conditions with the plastic bound theorem (Drucker et al., 1952) to bracket the exact limit load by upper bound (UB) and lower bound (LB) solutions, while the computational efficiency is achieved by the technique of convex optimization (e.g., Krabbenhoft et al., 2007) in mathematical programming. The underlying bound theorems assume a rigid-perfectly plastic material with an associated flow rule. Details of limit analysis and FELA are omitted here but may be found in Chen and Liu (1990) and Sloan (2013), respectively. In this study, the FELA software, OptumG2 (Krabbenhoft et al., 2015), under the plane strain condition was employed to investigate the 2D undrained stability of an opening in underground walls in cohesive soils. Recently, OptumG2 was successfully applied to solve a variety of undrained stability problems in geotechnical engineering, including suction caissons (Keawsawasvong and Ukritchon 2016; Ukritchon and Keawsawasvong 2016), conical slopes (Keawsawasvong and Ukritchon, 2017a), active failure of trapdoors (Keawsawasvong and Ukritchon, 2017b), and contiguous pile walls (Keawsawasvong and Ukritchon, 2017c). Consequently, OptumG2 was chosen in this study to simulate the undrained soil collapse at an opening in underground walls and compute its numerical plasticity solutions. The problem definition of the proposed study is shown in Fig. 1. An opening with a height (D) is made in the underground wall at the distance (C) measured from the ground surface. It is assumed that the wall is rigid and fully rough and there is a complete lateral bracing of the wall. The soil profile is a cohesive soil with unit weight (γ) and undrained shear strength at the ground surface (su0) that increases linearly with depth with a gradient (ρ). Because the geometry of the problem is a continuous wall, a plane strain condition was used in FELA, where an example of numerical model is shown in Fig. 3. In both upper and lower bound calculations, the soil mass was discretized as triangular elements and modelled as Tresca material with the associated flow rule. The continuous wall was modelled as the rigid plate element while the soil-structure interface element with a fully rough condition was employed along the length of wall in order to simulate the shear sliding of soil along the wall. The opening is resisted by a uniform pressure (σt), which is considered to be a general case of an opening with free surface. A uniform surcharge (σs) is applied all over the ground surface and is optimized in both upper and lower bound simulations to compute the bound solution of the limiting pressure σs. The boundary condition of the problem was defined such that the bottom boundary of the model was fixed in both
3. Results and discussions For the undrained stability of an opening in underground walls, there are seven input parameters, namely γ, su0, ρ, σs , σt, C and D. Based on the dimensional analysis, it can be shown that four dimensionless parameters can be created from these input parameters, namely: (i) cover depth ratio (C/D), (ii) overburden stress factor (γD/su0), (iii) linear strength gradient ratio (ρD/su0) and (iv) load factor ((σs − σt)/ su0). In this study, the latter was taken as the dependent parameter that is a function of the first three terms considered as the independent input parameters. Note that Augarde et al. (2003) also employed the same set of four dimensionless parameters to investigate the undrained stability of 2D plane strain heading in clays with linearly increasing strength with depth. Parametric studies were performed on the three dimensionless input parameters, covering their practical ranges of C/D = 1–10, γD/ su0 = 0–5 and ρD/su0 = 0–2. Fig. 4 shows the relationship between (σs − σt)/su0 as a function of γD/su0, C/D and ρD/su0. For all values of these input parameters, the exact load factor can be accurately bracketed by computed UB and LB solutions within 5%. In general, a linearly decreasing relationship between (σs − σt)/su0 and γD/su0 is observed. The gradient of this linear relationship is significantly affected by C/D and ρD/su0, where these larger ratios lead to the large gradient. Closer inspections of the results indicate that a nonlinear increasing relationship between (σs − σt)/su0 and C/D is found, while a linearly increasing relationship between (σs − σt)/su0 and ρD/su0 is observed. A positive sign of (σs − σt)/su0 implies that the surcharge required to produce a soil collapse at the opening must be greater than the applied pressure at the opening. On the contrary, a negative sign of (σs − σt)/su0 implies that the applied pressure at the opening must be larger than the applied surcharge in order to maintain the soil stability at the opening. A comparison of the stability ratio (N) of Broms and Bennermark (1967) defined in Eq. (1) is shown in Fig. 5, corresponding to the case of homogenous clay with weightless material (i.e., γ = 0). Note that the N ratios of the present study correspond to the opening in underground walls, while existing solutions correspond to those of 2D plane strain heading using analytical (Davis et al., 1980) and numerical (Augarde et al., 2003) methods of lower bound limit analysis. A good agreement between the stability ratio of opening and that of plane strain heading by Davis et al. (1980) using the mechanism of LB thick cylinder and Augarde et al. (2003) is observed, where the former is higher than the latter about 7–9%. Such result is mainly attributed from the influence of
Fig. 3. Numerical model of an opening in underground wall in clay (initial non-adaptive mesh).
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Fig. 4. Relationship between (σs − σt)/su0 and γD/su0 for ρD/su0 values of (a) 0, (b) 0.5, (c) 1 and (e) 2.
corners of opening and extends to outcrop at the ground surface. In addition, a shear band is developed along the upper length of underground wall, which is not taken into account for the failure mechanism of the 2D plane strain heading. Note that there is a difference in the size of plastic yielding zone between two ratios of ρD/su0, where a higher ρD/su0 ratio results in a smaller size of plastic zone. This result suggests that the size of plastic yielding zone depends on this parameter as well as the other two dimensionless parameters (i.e., C/D and γD/su0). Table 1 shows a closer comparison in the load factor (σs − σt)/su0 between the present study of 2D opening in underground wall and the existing LB solutions of 2D plane strain heading. The latter include the numerical LB solutions by Augarde et al. (2003) and the analytical LB solutions by Davis et al. (1980) considering a stress field in a large cylindrical volume of soil (thick cylinder). Note that the solutions by Davis et al. (1980) are available only to the case of homogeneous strength. In the comparison, the dimensionless parameters were selected to represent most practical cases at their limiting ratios as: (1) shallow vs. deep openings (i.e., C/D = 1 and 10); (2) small vs. large opening heights and/or overburden stresses (i.e., γD/su0 = 0 and 3); and (3) homogenous vs. non-homogenous strengths (i.e., ρD/su0 = 0 and 1). In general, it can be observed that the solutions of the present study are higher than those of 2D plane strain heading by 2–40% because of the kinematic constraint of underground wall above the opening. The large differences in (σs − σt)/su0 seem to be associated with large ratios of C/D and γD/su0 and smaller ratios of ρD/su0. Under these conditions, the influence of fully rough rigid wall becomes more prominent on (σs − σt)/su0. Consequently, the comparison results confirm the difference in the boundary conditions between the two problems and justify the need of the proposed study. Another closer comparison between the present study of 2D undrained stability of opening in underground wall and 3D undrained stability of tunnel face is also shown in Table 1. In this comparison, the
Fig. 5. Comparison of stability ratios (N) between the present study and existing solutions for homogeneous clay.
rough rigid underground wall on the stability of opening, which is not considered in the 2D plane strain heading. The comparison also indicates that the stress field in a large spherical volume of soil (thick sphere) proposed by Davis et al. (1980) furnishes significantly unsafe LB solutions for the stability of opening in underground walls, and hence they should not be used in practice. Two examples of predicted failure mechanisms deduced from the final adaptive meshes and velocity fields are shown in Fig. 6, where C/ D = 4, γD/su0 = 1, ρD/su0 = 0 and 1. The mode of failure is caused by the soil squeezed out of the opening due to the actions of the soil selfweight and the applied surcharge, which are considered as the driving force of the problem. The plastic yielding zone initiates from the
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Fig. 6. Example of adaptive mesh and velocity field, where C/D = 4 and γD/su0 = 1 for ρD/su0 values of (a) 0 and (b) 1.
(a) ρD/su0 = 0
(b) ρD/su0 = 1 parameters, the solutions of the present study are significantly smaller than those of the 3D tunnel face stability by 50–100%, confirming the assumption of conservativeness in applying the results of the present study to the 3D problems.
solutions of the latter were obtained by employing Tunnel Face Stability Software (TFSS) (Mollon, 2012) that is based on the kinematic approach of upper bound limit analysis, which represents the most upto-date of previous studies by Mollon et al. (2009, 2010, 2011, 2012). Similarly, comparisons were made between the limiting cases of each dimensionless parameter covering most practical cases of only homogenous strengths, namely (1) C/D = 1 and 10 (i.e., shallow vs. deep openings) and (2) γD/su0 = 0 and 3 (small vs. large opening heights and/or overburden stresses). Note that there is no result of TFSS in the case of non-homogeneous strength since it is limited to the stability analysis of only a homogeneous strength profile (i.e., ρD/su0 = 0). As expected, it is very evident that there are substantial differences in the load factor (σs − σt)/su0 between the present study of 2D opening in underground wall and the 3D tunnel face stability (TFSS), which are mainly attributed from the 3D effect in the different geometry between the two problems. For most practical ranges of the dimensionless
3.1. New design equations A technique of nonlinear regression analysis was employed to develop the approximate solution for the undrained stability of the opening in underground walls. Since the lower bound theorem provides a safe estimate of the limit load for a stability problem, the computed LB solutions were chosen in the regression analysis. It was found that a linear relationship between (σs − σt)/su0 and (γD/su0, ρD/su0) exists, while a power function yields an accurate curve-fitting between (σs − σt)/su0 and C/D. Therefore, the accurate approximate LB solution for the undrained stability of the opening in underground wall is
Table 1 Comparison of the load factor (σs − σt)/su0 between the present study and other solutions. C/D
γD/su0
ρD/su0
Present study Numerical LB 2D opening in underground wall
Augarde et al. (2003) Numerical LB 2D plane strain heading
Davis et al. (1980) Analytical LB (thick cylinder) 2D plane strain heading
Mollon (2012) Numerical UB 3D tunnel face stability
1 10 1 10 1 10 1 10
0 0 3 3 0 0 3 3
0 0 0 0 1 1 1 1
4.23 8.67 −0.44 −22.97 8.86 82.00 4.36 50.49
4.00 7.70 −0.74 −24.39 8.73 80.29 4.19 48.82
4.197 8.089 −0.303 −23.411 – – – –
7.89 21.24 3.10 −10.98 – – – –
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Fig. 8. Relationship between the stability factors (Nc, Nρ and Nγ) of an opening in underground walls and cover depth ratio (C/D).
Fig. 7. Comparison of the load factor (σs − σt)/su0 between the proposed Eq. (2) and lower bound solutions for the undrained stability of an opening in underground walls.
compute the opening pressure that is required to maintain the stability for a certain level of safety factor (FS). The conventional FS used in a slope stability problem is adopted here such that it is defined as the ratio of the full shear strength to the mobilized shear strength. Thus, su0 and ρ in the right hand side of Eq. (3) are treated as the “mobilized” strength parameter. By replacing su0 by su0/FS and ρ by ρ/FS in Eq. (3) and solving for σt, the design equation for the required opening pressure maintaining the stability can be obtained:
proposed, as shown in Eq. (2).
σs−σt C a2 C a4 ρD ⎞ C γD = a1 ⎛ ⎞ + a3 ⎛ ⎞ ⎛ −⎛0.5 + ⎞ ⎛ ⎞ su0 D ⎠ ⎝ su0 ⎠ ⎝D⎠ ⎝ D ⎠ ⎝ su0 ⎠ ⎝ ⎜
⎟
⎜
⎟
(2)
where a1 = 4.5690, a2 = 0.2786, a3 = 4.0725 and a4 = 1.2540. The comparison of (σs − σt)/su0 between the predictions in Eq. (2) and the lower bound solutions is shown in Fig. 7, where a good agreement between these two solutions can be achieved, with a coefficient of determination (R2) of 99.99%. In practice, Eq. (2) is not convenient for a stability evaluation of the problem. Instead, a “nicer” expression can be obtained from Eq. (2) by multiplying su0 to both sides in the equation. Hence, the new design equation for a safe estimate of the limiting uniform surcharge (σs) of an opening in underground walls in clays with linearly increasing strength profiles is proposed, as shown in Eq. (3).
σs = su0 Nc + ρDNρ + σt −γDNγ
σt = σs + γDNγ −
Nρ =
(3)
FS =
C a2 a1 D C a4 a3 D
() ()
Nγ = 0.5 +
C D
(5)
where FS = required safety factor. Another practical application is to assess the safety level of an opening in underground walls when the problem geometry (C, D), soil properties (γ, su0, ρ), loadings (σs, σt) are given. Therefore, the design equation for the factor of safety against the soil collapse at the opening can be determined straightforwardly from Eq. (5), as shown below.
where
Nc =
ρ su0 Nc− DNρ FS FS
su0 Nc + ρDNρ σs + γDNγ −σt
(6)
The physical meaning of Eq. (6) can be interpreted as follows. The FS factor is defined as the ratio of the total shear resistance of soil to the net driving pressure of the problem. The former is contributed from the effects of the constant soil cohesion (su0Nc) and linear strength gradient (ρDNρ), while the latter is calculated from the sum of the uniform surcharge on the ground surface (σs) and the effect of the overburden pressure (γDNγ) less the trapdoor pressure (σt) that is the resistance of the problem. Consequently, Eqs. (5) and (6) represent the design stability equations for the opening in underground walls in clays with an arbitrary linear increase in strength with depth, including constant strength (ρ = 0) and linearly increasing shear strengths (su0 ≠ 0, ρ ≠ 0). The latter also includes a special case of zero strength at the ground surface (su0 = 0, ρ ≠ 0), where it corresponds to the case of a normally consolidated clay.
(4)
The new design equation in Eq. (3) and associated factors in Eq. (4) employ the concept that is similar to the Terzaghi’s classical bearing capacity equation. The terms Nc, Nρ and Nγ represent the stability factors of the opening in underground walls for a constant soil cohesion (su0), linear strength gradient (ρ) and soil unit weight (γ), respectively. The negative sign of the last term in the right hand side indicates that the unit weight of soil (γ) represents the driving force of the problem that produces the same effect as the uniform surcharge (σs). Consequently, the physical meaning of Eq. (3) can be interpreted such that the limiting surcharge on the ground surface causing the soil collapse at the opening can be calculated from the sum of three effects of problem resistance, including the cohesion part, the linear strength gradient part, and the opening pressure less the effect of soil self-weight that is the driving force of the problem. Fig. 8 shows the relationship of new three stability factors, where they are a function of only the cover depth ratio. Nonlinear increasing functions between Nc, Nρ and C/D were observed, whereas a linearly increasing relationship between Nγ and C/D was found. It should be noted that the same expression Nγ = 0.5 + C/D was also proposed by Davis et al. (1980) using analytical LB and UB limit analysis of 2D plane strain heading. Therefore, the proposed expression of Nγ in this study seems to be reasonable. For a practical application, the safety evaluation of the problem is to
4. Conclusions In this paper, the undrained stability problem of an opening in underground walls in clays with linearly increasing shear strength with depth was investigated by the computational limit analysis, FELA. Three parametric studies were performed to study the effects of the cover depth ratio (C/D), overburden stress factor (γD/su0) and linear strength gradient ratio (ρD/su0) on the load factor ((σs − σt)/su0) of the opening in underground walls. In all cases, the exact load factors were accurately bracketed by the computed bound solutions within 5%. It was found that (σs − σt)/su0 had a nonlinear increasing function with C/D, a linearly decreasing function with γD/su0 and a linearly increasing function with ρD/su0. By employing the nonlinear regression 219
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non-homogeneous clays. Comput. Geotech. 81, 125–136. Keawsawasvong, S., Ukritchon, B., 2017b. Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth. Comput. Geotech. 81, 284–293. Keawsawasvong, S., Ukritchon, B., 2017c. Undrained limiting pressure behind soil gaps in contiguous pile walls. Comput. Geotech. 83, 152–158. Kimura, T., Mair, R.J., 1981. Centrifugal testing of model tunnels in soft clay. Proc. Tenth Int. Conf. Soil Mech. Found. Eng. 1, 19–322. Kirsch, A., 2010. Experimental investigation of the face stability of shallow tunnels in sand. Acta Geotech. 5 (1), 43–62. Krabbenhoft, K., Lyamin, A.V., Hjiaj, M., Sloan, S.W., 2005. A new discontinuous upper bound limit analysis formulation. Int. J. Numer. Meth. Eng. 63, 1069–1088. Krabbenhoft, K., Lyamin, A.V., Sloan, S.W., 2007. Formulation and solution of some plasticity problems as conic programs. Int. J. Solids Struct. 44, 1533–1549. Krabbenhoft, K., Lyamin, A., Krabbenhof, T.J., 2015. Optum Computational Engineering, (OptumG2), Available on, < www.optumce.com > . Leca, E., Dormieux, L., 1990. Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Géotechnique 40 (4), 581–606. Li, Y., Emeriault, F., Kastner, R., Zhang, Z.X., 2009. Stability analysis of large slurry shield-driven tunnel in soft clay. Tunn. Undergr. Space Technol. 24 (4), 472–481. Lu, X., Wang, H., Huang, M., 2014. Upper bound solution for the face stability of shield tunnel below the water table. Math. Probl. Eng, 727964. Mair, R.J., 1979. Centrifugal Modelling of Tunnel Construction In Soft Clay. (PhD Thesis) Cambridge University. Makrodimopoulos, A., Martin, C.M., 2006. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Meth. Eng. 66, 604–634. Makrodimopoulos, A., Martin, C.M., 2007. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int. J. Geomech. 31, 835–865. Mollon, G., 2012. Tunnel Face Stability Software (TFSS): Ver. 1.0; July, 2012. Available at: < http://guilhem.mollon.free.fr/Accueil_Eng.html > (accessed October 2015). Mollon, G., Dias, D., Soubra, A., 2009. Probabilistic analysis and design of circular tunnels against face stability. Int. J. Geomech. 9 (6), 237–249. Mollon, G., Dias, D., Soubra, A., 2010. Face stability analysis of circular tunnels driven by a pressurized shield. J. Geotech. Geoenviron. Eng. 136 (1), 215–229. Mollon, G., Dias, D., Soubra, A., 2011. Rotational failure mechanisms for the face stability analysis of tunnels driven by a pressurized shield. Int. J. Numer. Anal. Meth. Geomech. 35 (12), 1363–1388. Mollon, G., Dias, D., Soubra, A., 2012. Continuous velocity fields for collapse and blowout of a pressurized tunnel face in purely cohesive soil. Int. J. Numer. Anal. Meth. Geomech. 37 (13), 2061–2083. Ou, C.Y., 2006. Deep Excavation, Theory and Practice. Taylor & Francis Group, London. Puller, M., 2003. Deep Excavations, A Practical Manual, second ed. Thomas Telford, London. Senent, S., Jimenez, R., 2015. A tunnel face failure mechanism for layered ground, considering the possibility of partial collapse. Tunn. Undergr. Space Technol. 47, 182–192. Senent, S., Mollon, G., Jimenez, R., 2013. Tunnel face stability in heavily fractured rock masses that follow the Hoek-Brown failure criterion. Int. J. Rock Mech. Min. Sci. 60, 440–451. Sloan, S.W., 2013. Geotechnical stability analysis. Géotechnique 63 (7), 531–572. Sloan, S.W., Assadi, A., 1994. Undrained stability of a plane strain heading. Can. Geotech. J. 31, 443–450. Ukritchon, B., Keawsawasvong, S., 2016. Undrained pullout capacity of cylindrical suction caissons by finite element limit analysis. Comput. Geotech. 80, 301–311. Ukritchon, B., Keawsawasvong, S., Yingchaloenkitkhajorn, K., 2017. Undrained face stability of tunnels in Bangkok subsoils. Int. J. Geotech. Eng. 11 (3), 262–277. Vermeer, P., Ruse, N., Marcher, T., 2002. Tunnel heading stability in drained ground. Felsbau 20 (6), 8–24. Wilson, D.W., Abbo, A.J., Sloan, S.W., Lyamin, A.V., 2011. Undrained stability of a circular tunnel where the shear strength increases linearly with depth. Can. Geotech. J. 48 (9), 1328–1342. Wu, G., Chen, W., Bian, H., Yuan, J., 2017. Structure optimisation of a diaphragm wall with special modelling methods in a large-scale circular ventilating shaft considering shield crossing. Tunn. Undergr. Space Technol. 65, 35–41. Yang, F., Zhang, J., Zhao, L., Yang, J., 2016. Upper-bound finite element analysis of stability of tunnel face subjected to surcharge loading in cohesive-frictional soil. KSCE J. Civil Eng. 20 (6), 2270–2279. Zhang, Z.X., Hu, X.Y., Scott, K.D., 2011. A discrete numerical approach for modeling face stability in slurry shield tunnelling in soft soils. Comput. Geotech. 38 (1), 94–104. Zhang, D., Liu, B., Qin, Y., 2016. Construction of a large-section long pedestrian underpass using pipe jacking in muddy silty clay, a case study. Tunn. Undergr. Space Technol. 60, 151–164.
analysis to the computed lower bound solutions, the new design stability equations of the required opening pressure (σt) and the safety factor (FS) for the opening in underground walls were developed. In addition, the new stability factors of the opening in underground walls, with respect to constant soil cohesion, linear shear strength gradient and soil unit weight were proposed for the use of these design equations that can be applied for an arbitrary linear increase in undrained shear strength of clay with depth. Therefore, the new design stability equations provide a simple, accurate and reliable hand calculation of the undrained stability problem of an opening in underground walls in practice. References Anagnostou, G., Kovári, K., 1994. The face stability of slurry shield-driven tunnels. Tunn. Undergr. Space Technol. 9 (2), 165–174. Anagnostou, G., Kovári, K., 1996. Face stability conditions with earth-pressure-balanced shield. Tunn. Undergr. Space Technol. 11 (2), 165–173. Augarde, C.E., Lyamin, A.V., Sloan, S.W., 2003. Stability of an undrained plane strain heading revisited. Comput. Geotech. 30 (5), 419–430. Barla, M., Camusso, M., Aiassa, S., 2006. Analysis of jacking forces during microtunnelling in limestone. Tunn. Undergr. Space Technol. 21 (6), 668–683. Berthoz, N., Branque, D., Subrin, D., Wong, H., Humbert, E., 2012. Face failure in homogeneous and stratified soft ground, theoretical and experimental approaches on 1g EPBS reduced scale model. Tunn. Undergr. Space Technol. 30, 25–37. Broere, W., 2001. Tunnel Face Stability and New CPT Applications. (PhD Thesis) Delft University of Technology, The Netherlands. Broms, B.B., Bennermark, H., 1967. Stability of clay in vertical openings. J. Geotech. Eng. Divis. 193, 71–94. Chambon, P., Corté , J.F., 1994. Shallow tunnels in cohesionless soil, stability of tunnel face. J. Geotech. Eng. 120 (7), 1148–1165. Chapman, D.N., Ichioka, Y., 1999. Prediction of jacking forces for microtunnelling operations. Tunn. Undergr. Space Technol. 14 (1), 31–41. Chen, W.F., Liu, X.L., 1990. Limit Analysis in Soil Mechanics. Elsevier, Amsterdam. Chen, R.P., Tang, L.J., Ling, D.S., Chen, Y.M., 2011. Face stability analysis of shallow shield tunnels in dry sandy ground using the discrete element method. Comput. Geotech. 38 (2), 187–195. Chen, R.P., Li, J., Kong, L.G., Tang, L.J., 2013. Experimental study on face instability of shield tunnel in sand. Tunn. Undergr. Space Technol. 33, 12–21. Cui, Q.L., Xu, Y.S., Shen, S.L., Yin, Z.Y., Horpibulsuk, S., 2015. Field performance of concrete pipes during jacking in cemented sandy silt. Tunn. Undergr. Space Technol. 49, 336–344. Davis, E.H., Gunn, M.J., Mair, R.J., Seneviratn, H.N., 1980. The stability of shallow tunnels and underground openings in cohesive material. Geotechnique 30 (4), 397–416. Drucker, D.C., Prager, W., Greenberg, H.J., 1952. Extended limit design theorems for continuous media. Q. Appl. Math. 9, 381–389. Funatsu, T., Hoshino, T., Sawae, H., Shimizu, N., 2008. Numerical analysis to better understand the mechanism of the effects of ground supports and reinforcements on the stability of tunnels using the distinct element method. Tunn. Undergr. Space Technol. 23 (5), 561–573. Guglielmetti, V., Grasso, P., Mahtab, A., Xu, S., 2008. Mechanized Tunnelling in Urban Areas – Design Methodology and Construction Control. Taylor & Francis Group, London. Han, K., Zhang, C., Li, W., Guo, C., 2016a. Face stability analysis of shield tunnels in homogeneous soil overlaid by multilayered cohesive-frictional soils. Math. Probl. Eng. 18. Han, K., Zhang, C., Zhang, D., 2016b. Upper-bound solutions for the face stability of a shield tunnel in multilayered cohesive–frictional soils. Comput. Geotech. 79, 1–9. Huang, M., Song, C., 2013. Upper-bound stability analysis of a plane strain heading in non-homogeneous clay. Tunn. Undergr. Space Technol. 38, 213–223. Ibrahim, E., Soubra, A.H., Mollon, G., Raphael, W., Dias, D., Reda, A., 2015. Three-dimensional face stability analysis of pressurized tunnels driven in a multilayered purely frictional medium. Tunn. Undergr. Space Technol. 49, 18–34. Jancsecz, S., Steiner, W., 1994. Face support for a large mix-shield in heterogeneous ground conditions. Tunnelling 94, 531–550. Keawsawasvong, S., Ukritchon, B., 2016. Finite element limit analysis of pullout capacity of planar caissons in clay. Comput. Geotech. 75, 12–17. Keawsawasvong, S., Ukritchon, B., 2017a. Stability of unsupported conical Excavations in
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