- Email: [email protected]
Stability analysis of stone column-supported and geosynthetic-reinforced embankments on soft ground
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem
Stability analysis of stone column-supported and geosynthetic-reinforced embankments on soft ground Gang Zhenga,b,c, Xiaoxuan Yua,b, Haizuo Zhoua,b,c,∗, Shun Wangd, Jiapeng Zhaoa,b, Xiaopei Hea,b, Xinyu Yanga,b a
School of Civil Engineering, Tianjin University, Tianjin, 300072, China Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin, 300072, China c State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, 300072, China d Engineering, Department of Civil Engineering and Natural Hazards, University of Natural Resources and Life Sciences, Vienna, Feistmantelstraße 4, 1180, Wien, Austria b
A R T I C LE I N FO
A B S T R A C T
Keywords: Geosynthetics Stone columns Failure Stability Embankment
This study focuses on the stability of stone column-supported and geosynthetic-reinforced embankments on soft soil. An upper-bound limit state plasticity failure discretization scheme (known as discontinuity layout optimization (DLO)), which determines the embankment stability without pre-assuming a slip surface, is used. The relationships between the stability of stone column-supported and geosynthetic-reinforced embankments and various influencing parameters, including the soil strength, geometric configuration, reinforcement strength, and area replacement ratio, are analysed. It is found that geosynthetics provide a significant contribution to embankment stability. Two failure mechanisms of geosynthetics (i.e., rupture failure and bond failure) are revealed and the effect of geosynthetics on embankment stability is governed by the failure mode. The application of stone columns mitigates the risk of geosynthetic failure. To provide an analytical solution for primary design in engineering practice, an approach based on the limit equilibrium method is proposed. Validations are performed with the DLO solution to demonstrate the accuracy and reliability of the developed analytical approach.
1. Introduction The stability of embankments over soft soil has attracted considerable attentions (e.g., Mohapatra and Rajagopal, 2017). The application of stone columns serve as an economically and environmentally friendly solution for increasing both the slope stability and the bearing capacity when stabilizing an embankment (e.g., Abusharar and Han, 2011; Zhou et al., 2017, 2018). The stability of column-supported embankments is governed by the failure mechanism of columns (Zhou et al., 2019). Kitazume and Maruyama (2006, 2007) categorized the shear and bending failure modes as internal failure modes, whereas the sliding and rotational failure modes were categorized as external failure modes. For stone columns, shear failure is the critical failure mode, and the strength of stone columns depends on the friction angle of the granular material and the confining stress from the surrounding soils (Abusharar and Han, 2011; Ambily and Gandhi, 2007; Fattah et al., 2010). The influences of various reinforcement strength parameters have been investigated in prior numerical and experimental investigations (Christoulas et al., 1997; Chen et al., 2015).
Geosynthetic reinforcement is widely used to improve embankment performance on soft soil (Liu et al., 2017; Zhang et al., 2019). A basal geosynthetic reinforcement can serve to resist the earth pressure within an embankment and to prevent the lateral deformation of its foundation, thereby increasing the bearing capacity and stability (Chai et al., 2016; Leshchinsky, 1987; Rowe and Li, 2005; Rowe and Liu, 2015 ). The failure mechanisms of geosynthetics can be divided into two categories: rupture failure and bond failure (Rowe and Soderman, 1987; Hird and Kwok, 1989). For rupture failure, the stability of an embankment depends on the strength of the geosynthetic. When bond failure occurs, the embankment stability is governed by the bond strength between the geosynthetic and the surrounding soil in which the tensile capacity of the geosynthetic is not fully developed (Gilbert et al., 2010; Ochiai et al., 1996; Rowe and Soderman, 1985). These two mechanisms were described by Tandjiria et al. (2002), who numerically investigated the effects of the reinforcement force distribution. An alternative method for ensuring the embankment stability for rapid construction on soft ground is the use of geosynthetic reinforcement above the tops of stone columns (Deb et al., 2008; Hosseinpour
∗
Corresponding author. School of Civil Engineering, Tianjin University, Tianjin, 300072, China. E-mail addresses: [email protected] (G. Zheng), [email protected] (X. Yu), [email protected] (H. Zhou), [email protected] (S. Wang), [email protected] (J. Zhao), [email protected] (X. He), [email protected] (X. Yang). https://doi.org/10.1016/j.geotexmem.2019.12.006
0266-1144/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Gang Zheng, et al., Geotextiles and Geomembranes, https://doi.org/10.1016/j.geotexmem.2019.12.006
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
List of symbols H D n S d as ϕc ϕf ϕeq γc γf γs
γeq cf cu ceq η α
Embankment height Clay thickness Slope gradient Axis-to-axis spacing of stone column walls Width of stone column walls Area replacement ratio of stone column walls Friction angle of stone column walls Friction angle of embankments Equivalent friction angle of reinforced zone Unit weight of stone column walls Unit weight of embankments Unit weight of clay
R q Q σh x K aeq
Equivalent unit weight of reinforced zone Cohesion of embankments Undrained shear strength of the clay Equivalent cohesion of reinforced zone Stress concentration ratio Friction coefficient at the interface between geosynthetics and clay Rupture strength of geosynthetic Surcharge on the ground Surcharge on the top of ground Horizontal stress below loads of embankment Distance from the edge of loading of embankment Coefficient of active earth pressure of the equivalent model
based on plasticity theorems. It serves as a highly effective tool to evaluate the critical plastic collapse mechanism and the limit state of soils, where an associated flow rule is used. The DLO technique has been successfully used in a variety of geotechnical engineering endeavours (Leshchinsky and Ambauen, 2015; Smith and Tatari, 2016; Leshchinsky, 2015; Zhou et al., 2018a Zhou et al., 2019; Zheng et al., 2019). In this study, the LimitState: GEO v3.4 software (LimitState, 2013) adopting the DLO is used for stability analysis of stone columnsupported and geosynthetic-reinforced embankments, and an associated flow rule is adopted. A shear strength reduction technique was selected to calculate the value of factor of safety (FS), which is applied to the strength parameters of the soil and columns (Dawson et al., 1999; Abusharar and Han, 2011). The plastic analysis presented applies only to ductile geosynthetic reinforcement, which is modelled using the engineered element. A nodal spacing of H/5 is selected based on an accuracy of 1–2% in terms of the factor of safety (FS) (Smith and Tatari, 2016). A friction coefficient of 0.8 was assumed at the interface between the geosynthetics and soil (Huang and Han, 2009; Smith and Tatari, 2016). For stone column-supported and geosynthetic-reinforced embankments, two modes of the slip surface, namely, the lateral sliding failure and deep-seated failure modes, are shown in Fig. 2. The following parametric studies are focused on the latter failure mode.
et al., 2015). The application of geosynthetic reinforcement helps to transfer stresses from the soft soil to the columns (King et al., 2017; Rui et al., 2018). Prior studies have presented insights into the effects of a single geosynthetic or single stone columns on the stability of reinforced embankments. However, limited insight has been provided into the interactions among the soil, stone column, and geosynthetics in the embankment stability. In this study, stability analyses of stone column-supported and geosynthetic-reinforced embankments on soft ground are investigated using the upper-bound limit state plasticity failure discretization scheme, known as discontinuity layout optimization (DLO). The influences of the soil strength, geometric parameters, reinforcement strength, and area replacement ratio on the embankment stability are discussed. An analytical solution that considers both the shear failure behaviours of the stone column and the frictional properties along the interface between the geosynthetics and the surrounding soil is proposed. 2. Problem definition and verification 2.1. Numerical model A reinforced embankment is constructed on undrained soil, as shown in Fig. 1. Geosynthetics are placed at the embankment base, and stone columns are installed in the soft soil. Due to the complexity and the demand for significant time of a three-dimensional (3D) arrangement of multiple columns, a 3D problem of individual column model has been commonly converted into a two-dimensional (2D) columnwall model to investigate the embankment stability (Abusharar and Han, 2011; Zhang et al., 2014; Marandi et al., 2016); A column-wall model is thus adopted in this study. The parameters depicted in Fig. 1 are as follows: H is the embankment height; n is the slope gradient (n (horizontal):1 (vertical)); D is the thickness of the clay; S, d and as (i.e., d ) are the axis-to-axis spacing, the width and the area replacement ratio S of the stone column walls, respectively; γf , γc and γs are the unit weights of the embankment, stone column walls and clay, respectively; ϕf and ϕc are the friction angles of the embankment and stone column walls, respectively; c f and c u are the cohesion of the embankment and the undrained shear strength of the clay, respectively; α is the friction coefficient at the interface between the geosynthetic and the soil; R is the rupture strength of the geosynthetic; and Q is the surcharge applied at the top of the embankment. Commonly used approaches for stability analysis of embankments include the method of slices (Jewell, 1988), finite element method (Ambily and Gandhi, 2007; Chen et al., 2015) and limit analysis (LA) (Smith and Tatari, 2016). LA, which acts as a rigorous analytical solution, can provide lower and upper bounds to a real limit load. The discontinuity layout optimization (DLO) (Gilbert et al., 2010; Smith and Gilbert, 2007) is an upper bound LA procedure, which is rigorously
2.2. Verification 2.2.1. Behaviour of geosynthetic-reinforced embankments To verify the accuracy of the established model, the ultimate height of geosynthetic-reinforced embankments obtained by DLO is compared with the results presented in the literature. The established model is compared with the solution of Rowe and Li (1999) utilizing the finite element method and the solution of Smith and Tatari (2016) using DLO. An embankment with a crest width of 27 m and a 2 (horizontal):1 (vertical) side slope is constructed atop soft soil with a thickness of 15 m. The parameters of the soft soil and embankment fill are provided in Table 1. Fig. 3 shows the comparison of the results calculated by DLO
Fig. 1. Schematic of the model. 2
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 2. Failure mechanisms of stone column-supported and geosynthetic-reinforced embankments: (a) Lateral sliding failure and (b) Deep-seated failure mode.
Table 1 Parameters of the reinforced embankment analysis for calibration (Rowe and Li, 1999). Parameter
Clay
Embankment
Undrained shear strength (kPa), z=0m Undrained shear strength (kPa), z = 15 m Friction angle (°) Unit weight (kN/m³)
5
N/A
27.5
N/A
0 Varied (from 14.7 to 15.6)
37 20
Note: the undrained shear strength varies linearly with the depth z below the clay surface; N/A represents the phrase not available.
Fig. 4. Validation of the behaviour of stone column-supported embankments: (a) comparison of the FS versus the friction angle of the stone columns between the results of the DLO solution and those of Abusharar and Han (2011) (unit: m), (b) comparison of the ultimate loads versus the spacing of the stone columns between the results of the DLO solution and those of Fattah et al. (2016) (unit: mm). Table 3 Parameters of the numerical analysis for calibration (Fattah et al., 2016).
Fig. 3. Comparison of the maximum height Hm of an embankment versus the geosynthetic rupture strength (i.e., tensile force of the geosynthetic) between the results of the present DLO solution and the results of previous studies.
Thickness (m)
Unit weight (kN/m³)
Undrained shear strength (kPa)
Friction angle (°)
Embankment Clay Sand Stone column
5 10 2 10
20.5 18.5 20.5 19.5
0 20 0 0
32 0 30 38
Unit weight (kN/ m³)
Undrained shear strength (kPa)
Friction angle (°)
Embankment Clay Stone column
21:84 18.5 15.7
0 10 0
40 0 41:5
(using a non-associated flow rule and a non-associated flow rule, respectively) with the results in prior literatures (Rowe and Li, 1999; Smith and Tatari, 2016). Notably, the results using an associated flow rule in this study accurately fit the results of Smith and Tatari (2016), which are slight overestimations relative to the result of Rowe and Li (1999). Considering a non-associated flow rule (i.e., modified soil cohesion and friction angles are used (Drescher and Detournay, 1993; Bolton, 1986)), the results show a slight decrease, which demonstrate an agreement with those of Rowe and Li (1999).
Table 2 Parameters of the numerical analysis for calibration (Abusharar and Han, 2011). Material
Material
3
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
approximately linear increase in the FS. The DLO approach based on the upper bound limit theorem yields a slightly higher solution than that based on the finite difference method (approximately 2.5%, 1.7% and 3.8% higher for friction angles of 34°, 38°, and 42°, respectively). To further elaborate the effect of individual stone columns on embankment stability, a comparison of DLO numerical modelling was made with laboratory tests from Fattah et al. (2016), who investigated the relationship between the axis-to-axis spacing of individual stone columns and ultimate surcharge. The parameters of the clay, stone columns and embankment are provided in Table 3. The scale of the numerical model is shown in Fig. 4(b). The results demonstrate reasonable agreement for various stone column spacings (approximately 3.8%, 5.1% and 6 0.1% higher for spacing between the stone columns of 175, 210, and 280 mm, respectively). The presented comparisons demonstrate that the implementations of DLO accurately capture the contributions of the stone columns and geosynthetic to the stability of the embankments.
Table 4 Parameters of stone column-supported and geosynthetic-reinforced embankments. Influencing factor
Range of values
H D
0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2
ϕc as
38, 42, 46 0.1, 0.2, 0.3 0, 0.5, 1, 1.5, 2, ∞
R γf D2
Note: The value of
R γf D2
= 10 are used to simulate
R γf D2
= ∞, and
R γf D2
=0
represents that there is no geosynthetic reinforcement. Geosynthetics are placed at the interface between the embankment and clay (Jewell, 1988; Smith and Tatar, 2016).
2.2.2. Behaviour of stone column-supported embankments To further validate the modelling of stone column walls, solutions of Abusharar and Han (2011) are selected. The geometric parameters of the column walls and embankment are provided as follows: the width of the columns is 0.8 m, the length of the columns is 10 m, the spacing between adjacent columns is 4 m, the crest width of the embankment is 20 m, the height of the embankment is 5 m, and the angle of the side slopes is 2 (horizontal):1 (vertical). The material parameters of the numerical model are shown in Table 2. Fig. 4(a) shows a comparison between the FS from the DLO and that from Abusharar and Han (2011). An increase in the friction angle of the column walls results in an
3. Parametric analysis To investigate the influences of various factors on the stability of stone column-supported and geosynthetic-reinforced embankments, parametric studies on the FS are performed. The solution is an inherent function of the following series of influencing factors:
FS = f (H , D , R, n, α, c f , c u, γf , γc, γs, ϕf , ϕc , as)
(1)
The corresponding parameters are mentioned in the numerical
Fig. 5. Parametric analysis for the various values of 4
cu γf D
and. H
D
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 6. Cross sections of the calculation models (modified from Abusharar and Han, 2011): (a) stone column walls, (b) equivalent area model.
practice. Following the specifications for the design of highway subgrades (Ministry of Transport (MOT) 2015), a FS of 1.35 is adopted to illustrate the effect of influencing factors on the embankment stability. A set of parametric analyses containing the evaluation results with the required FS is illustrated in Fig. 5, which shows the relationships between cu and H (when the FS is always equal to 1.35) for a stone γf D
D
column-supported embankment both with and without geosynthetics. Each chart in Fig. 5 is divided into three parts by two boundaries. The first boundary (i.e., the black solid line) is plotted for an embankment without geosynthetics, illustrating a notable increase in cu with inγf D
creasing H . This behaviour represents a significant contribution of the D soil strength to the stability of embankments. Obviously, the FS values corresponding to the region above this boundary, denoted as the stable zone, are always larger than 1.35. The other boundary (i.e., the red solid line) is depicted for the embankment with sufficient geosynthetics, denoted as R 2 = ∞, representing that the further increase in R 2 has γf D
γf D
no influence on the stability of the embankment. This boundary shows an initial convexly increasing trend with the increasing H , eventually D
levelling off as H exceeds a threshold value. The rate of increase for the D red line is lower than that for the black solid line. This is because the contribution of soil strength to the stability of an embankment is less than the contribution of the presence of high-strength geosynthetics. For the region below this boundary corresponding to the unstable zone, an instability of the embankment is always observed despite a high geosynthetic strength. The region between these two boundaries represents the conditions for which the geosynthetic strength effectively improves the embankment stability; this region is denoted as the geosynthetic-required zone. Commonly, there are two failure modes for geosynthetics. Rupture failure often occurs when the geosynthetic tensile strength is relatively small, whereas bond failure can be observed when tensile strength is high. The lower boundary (i.e., the red solid line) merely represents the bond failure. The dotted lines (with FS = 1.35), in the geosyntheticrequired zone, are depicted here for geosynthetics with a certain rupture strength (e.g., R 2 = 0.5, 1, 1.5, and 2) to show both bond and
Fig. 7. Mechanism of the deep-seated failure of a reinforced embankment over an equivalent area: (a) failure mechanism, (b) block analysis.
model, as depicted in Fig. 1. The above parameters can be nondimensionalized and given as follows:
γf D
H c R FS = f ⎜⎛ , u , , ϕf , as ⎞⎟ 2 D γ D γ f fD ⎠ ⎝
rupture failure modes. Correspondingly, a piecewise function of (2)
H D
versus is observed for geosynthetics with a certain rupture strength. Specifically, as shown in Fig. 5(a), a consistency between geosynthetics with a certain rupture strength (i.e., the dotted line) and geosynthetics with R = ∞ (i.e., the red solid line) is found for the first segment (i.e., curve AB), in which a bond failure mode occurs. As H increases, a deD viation is found (i.e., curve BC), representing a rupture failure, and a cu nearly linear relationship is observed between and H . Compared
in which the dimensionless parameters include the nondimensionalized height of the embankment H , the strength of the clay cu , and the strength of the geosynthetic
D R γf D2
cu γf D
γf D
(Smith and Tatar, 2016). The thickness
of the mat under the embankment and the location of the geosynthetic have a minor effect on the stability of the embankment (Jewell, 1988). Additionally, the embankment strength is not considered as an influencing factor because its effect on the performance of a reinforced embankment is negligible compared with that of the clay strength (Smith and Tatar, 2016). The effects of the influencing factors in Eq. (2) are studied, and the ranges of the parameters are listed in Table 4. The constant parameters are as follows: α = 0.8, q = 0, n = 2, c f = 0, ϕf = 32°, γf = 20 kN/m³, γs = 18.5 kN/m³, and γc = 19.5 kN/m³. The meaning of each parameter can be found in the list of symbols. Theoretically, the slope is stable when the FS is equal to 1.0. However, a FS between 1.3 and 1.5 is usually required in engineering
γf D
D
with curve AB, the slope of curve BC increases, becoming similar to the black solid line (i.e., R = 0). Point B, the inflection point of a dotted curve, represents a transition from bond failure to rupture failure in the geosynthetic. The H corresponding to point B is denoted as a critical D
( ) H
nondimensionalized embankment height D , in which the bond cri strength of the interface between the geosynthetic and the surrounding soil is equal to the rupture strength of the geosynthetic. An increase in the area of the stable zone and a reduction in the area of the unstable zone are observed with a larger friction angle ϕc and a 5
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 8. Plot of the FS against the friction angle of the stone columns for Eq. (7) and the DLO (red markers) using c u = 15 kPa; d = 0.8 m; S = 4 m; D = 10 m; H = 5 m; γf = 17 kN/m3; and γeq = 16 kN/m3: (a) influence of the friction angle of the stone column walls, (b) influence of the undrained shear strength of the clay, (c) influence of the width of the stone column walls, (d) influence of the axis-to-axis spacing of stone column walls.
Fig. 6. The properties of the equivalent area model can be calculated based on the weighted average between the soils and columns as follows (Cooper and Rose, 1999):
larger area replacement ratio as of the stone column walls. When the friction angle ϕc is large, a significant decrease in the area of the geosynthetic-required zone is found as as increases, indicating a relatively apparent contribution of the geosynthetic to the embankment stability for a low area replacement ratio of the stone column walls. Meanwhile, for a certain R 2 value, an increase in the area replacement ratio and γf D
friction angle results in a great
( )
H . D cri
For a designed embankment
constructed on soft soil, an applicable solution via both stone column and geosynthetics technologies can be directly determined from the presented parametric analysis in engineering practice.
ceq = cc as + c u (1 − as)
(3)
ϕeq = tan−1 (μ tan ϕc + (1 − μ)tan ϕs)
(4)
γeq = γc as + γs (1 − as)
(5)
where
μ= 4. Analytical solutions
as η 1 + as (η − 1)
(6)
where ϕs and cc are the friction angles of the soft soil and the cohesions of the column, respectively; The meaning of the other parameters can be found in the list of symbols. The stress concentration ratio η is a critical coefficient, which is considered in the analysis of Christoulas et al. (1997) with usual values of 2–6. The value of η is influenced by a number of factors, such as the elastic modulus, Poisson's ratio, the location of the columns and the area replacement ratio (Fattah et al., 2016; Hanna et al., 2013). The stress concentration ratio versus the strain shows an initial increase and
To develop an explicit equation for an evaluation of reinforced embankments and provide solutions with any FS, a simplified method based on the limit equilibrium method considering the influences of geosynthetics and stone columns is proposed. To derive an analytical solution, it is common to convert the column walls into an equivalent area model for simplification (Christoulas et al., 1997; Abusharar and Han, 2011; Zhang et al., 2014), as shown in 6
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
zone. 3. A simplified analytical solution for stone column-supported and geosynthetic-reinforced embankment stability is proposed based on the limit equilibrium method. Validations are performed to demonstrate the accuracy of the FS values predicted with the proposed model. This simplified method can be adopted for primary design in geotechnical engineering projects involving similar construction and soil conditions.
a subsequent decrease afterwards, eventually exhibiting a degradation to approximately 1.0 when the failure occurs (Juran and Guermazi, 1988; Zhang et al., 2014). The values of η with 1.0–1.3 is commonly adopted in the stability analysis (Marandi et al., 2016; Zhou et al., 2017), and a stress concentration ratio of 1.0 is used in this study. For the calculation of the FS of stone column-supported embankment, reduction factors of 0.9–1.0 were recommended to convert the columnwall model into the equivalent area model (Abusharar and Han, 2011; Zhang et al., 2014). In this study, a reduction factor of 1.0 is selected. Jewell (1988) presents an analytical solution to determine safety factors and reinforcement forces for geosynthetic-reinforced embankments on soft soils based on the translational failure mechanism. Based on this mechanism proposed by Jewell (1988), the geosynthetic is modelled as a single representative force applied to the surrounding soil. Considering the equivalent area model, the corresponding distribution of the soil reactions in this study is shown in Fig. 7(a). Specifically, a vertical surface load q that increases with a gain in the distance x from the edge is shown in Fig. 7(b). The solution FS via the integration of these relationships is as follows:
FS =
Acknowledgements This research was funded by the National Key R&D Program of China (Grant No. 2017YFC0805407), the National Natural Science Foundation of China (No. 51708405and No. 41630641) and the Project of Tianjin Science and Technology Plan (No. 16YDLJSF00040). The authors appreciate the financial support. References Abusharar, S.W., Han, J., 2011. Two-dimensional deep-seated slope stability analysis of embankments over stone column-improved soft clay. Eng. Geol. 120 (1–4), 103–110. Ambily, A.P., Gandhi, S.R., 2007. Behavior of stone columns based on experimental and FEM analysis. J. Geotech. Geoenviron. Eng. 133 (4), 405–415. Bolton, M.D., 1986. The strength and dilatancy of sands. Geotechnique 36 (1), 65–78. Chai, J.C., Sari, K., Shen, S.L., Cai, Y., 2016. Predicting self-healing ratio of GCL with a damage hole. Geotext. Geomembranes 44 (5), 761–769. Chen, J.F., Li, L.Y., Xue, J.F., Feng, S.Z., 2015. Failure mechanism of geosynthetic-encased stone columns in soft soils under embankment. Geotext. Geomembranes 43 (5), 424–431. Christoulas, S., Giannaros, C., Tsiambaos, G., 1997. Stabilization of embankment foundations by using stone columns. Geotech. Geol. Eng. 15 (3), 247–258. Cooper, M.R., Rose, A.N., 1999. Stone column support for an embankment on deep alluvial soils. Proc. Inst. Civ. Eng. Geotech. Eng. 37 (1), 15–25. Dawson, E.M., Roth, W.H., Drescher, A., 1999. Slope stability analysis by strength reduction. Geotechnique 49 (6), 835–840. Deb, K., Chandra, S., Basudhar, P.K., 2008. Response of multilayer geosynthetic-reinforced bed resting on soft soil with stone columns. Comput. Geotech. 35 (3), 323–330. Drescher, A., Detournay, E., 1993. Limit load in translational failure mechanisms for associative and non-associative materials. Geotechnique 43 (3), 443–456. Fattah, M.Y., Shlash, K.T., Al-Waily, M.J.M., 2010. Stress concentration ratio of model stone columns in soft clays. Geotech. Test J. 34 (1), 50–60. Fattah, M.Y., Zabar, B.S., Hassan, H.A., 2016. Experimental analysis of embankment on ordinary and encased stone columns. Int. J. Geomech. 16 (4) 04015102. Gilbert, M., Smith, C., Haslam, I., Pritchard, T., 2010. Application of discontinuity layout optimization to geotechnical engineering. In: Proceedings of the 7th European Conference on Numerical Methods in Geotechnical Engineering, pp. 169–174. Griffiths, D.V., 2015. Observations on load and strength factors in bearing capacity analysis. J. Geotech. Geoenviron. Eng. 141 (7) 06015004. Hanna, A.M., Etezad, M., Ayadat, T., 2013. Mode of failure of a group of stone columns in soft soil. Int. J. Geomech. 13 (1), 87–96. Hird, C.C., Kwok, C.M., 1989. Finite element studies of interface behaviour in reinforced embankments of soft ground. Comput. Geotech. 8 (2), 111–131. Huang, J., Han, J., 2009. 3D coupled mechanical and hydraulic modeling of a geosynthetic-reinforced deep mixed column-supported embankment. Geotext. Geomembranes 27 (4), 272–280. Jewell, R.A., 1988. The mechanics of reinforced embankments on soft soils. Geotext. Geomembranes 7 (4), 237–273. Juran, I., Guermazi, A., 1988. Settlement response of soft soils reinforced by compacted sand columns. J. Geotech. Geoenviron. Eng. 114 (8), 930–943. King, D.J., Bouazza, A., Gniel, J.R., Rowe, R.K., Bui, H.H., 2017. Serviceability design for geosynthetic reinforced column supported embankments. Geotext. Geomembranes 45 (4), 261–279. Leshchinsky, D., 1987. Short-term stability of reinforced embankment over clayey foundation. Soils Found. 29, 105–114. Leshchinsky, B., 2015. Bearing capacity of footings placed adjacent to c′-φ′ slopes. J. Geotech. Geoenviron. Eng. 141 (6) 04015022. Leshchinsky, B., Ambauen, S., 2015. Limit equilibrium and limit analysis: comparison of benchmark slope stability problems. J. Geotech. Geoenviron. Eng. 141 (10) 04015043. LimitState, 2013. LimitState: Geo Manual V 3.0 (Sheffield, U.K). Liu, K.W., Rowe, R.K., Su, Q., Liu, B., Yang, Z., 2017. Long–term reinforcement strains for column supported embankments with viscous reinforcement by FEM. Geotext. Geomembranes 45 (4), 307–319. Marandi, S.M., Anvar, M., Bahrami, M., 2016. Uncertainty analysis of safety factor of embankment built on stone column improved soft soil using fuzzy logic α-cut technique. Comput. Geotech. 75, 135–144. Mohapatra, S.R., Rajagopal, K., 2017. Undrained stability analysis of embankments supported on geosynthetic encased granular columns. Geosynth. Int. 24 (5), 465–479. Ministry of Transport of the People's Republic of China (MOT), 2015. Specifications for
γH 1 ⎛ nH ⎛ (1 + α ) ⎛ceq + f tan ϕeq ⎞ + γeq D tan ϕeq ⎞ + 4c u ⎞⎟ ⎜ γf H ⎝ DK aeq ⎝ 2n ⎝ ⎠ ⎠ ⎠ ⎜
⎟
(7) Where K aeq can be expressed as follows according to Griffiths (2015):
K aeq = −tan ϕeq +
1 + tan2 ϕeq
(8)
The geosynthetic rupture strength can be determined by the bond strength. Therefore, the maximum possible restraint (i.e., maximum available tension force Tmax) on the surrounding soil is exerted by the geosynthetic. The minimum required rupture strength Rc (i.e., a critical value representing the transition from rupture failure to bond failure), which equals to Tmax, can be given:
Rc =
αnH 1 (ceq + γf tan ϕeq ) + K a f γf H 2 2 FS
(9)
To verify the effectiveness of Eq. (7), a comparison of the stone column-supported and geosynthetic-reinforced embankment results between the limit equilibrium and DLO methods is conducted, as is shown in Fig. 8. The results from Eq. (7) provide a generally good fit to the DLO results. 5. Conclusions In this study, the stability of stone column-supported and geosynthetic-reinforced embankments is analysed. Parametric analysis that can be used to determine the strength of the required geosynthetics and stone column walls for various geometric conditions and soil properties are developed. Additionally, a limit equilibrium solution is developed to predict the FS. The following conclusions are drawn: 1. Geosynthetics have a noticeable effect on the embankment stability, which is influenced by the nondimensionalized height of the embankment H . Two failure modes (i.e., rupture failure and bond D failure) in the geosynthetics are observed. With a certain geosynthetic rupture strength, there is a transition from bond failure to rupture failure. The value of H corresponding to this transition is denoted as the critical value
D H . D cri
( )
Once
H D
exceeds
( )
H , D cri
the effect
of the geosynthetic on the embankment stability is limited due to its insufficient strength. 2. The application of stone columns and geosynthetic provides a costeffective solution for the stability of embankments over soft soil. The stone column reduces the risk of these two failure modes of geosynthetics. With an increase in the friction angle ϕc and the area replacement ratio as of the stone column walls, there is a gain in the area of the stable zone and a reduction in the area of the unstable 7
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
stability of embankments. Geotext. Geomembranes 20 (6), 423–443. Zhang, Z., Han, J., Ye, G., 2014. Numerical investigation on factors for deep-seated slope stability of stone column-supported embankments over soft clay. Eng. Geol. 168, 104–113. Zhang, Z., Wang, M., Ye, G., Han, J., 2019. A novel 2D-3D conversion method for calculating maximum strain of geosynthetic reinforcement in pile-supported embankments. Geotext. Geomembranes 47 (3), 336–351. Zheng, G., Zhao, J., Zhou, H., Zhang, T., 2019. Ultimate bearing capacity of strip footings on sand overlying clay under inclined loading. Comput. Geotech. 106, 266–273. Zhou, H., Diao, Y., Zheng, G., Han, J., Jia, R., 2017. Failure modes and bearing capacity of strip footings on soft ground reinforced by floating stone columns. Acta Geotech 12 (5), 1089–1103. Zhou, H., Zheng, G., He, X., Xu, X., Zhang, T., Yang, X., 2018. Bearing capacity of strip footings on c - φ soils with square voids. Acta Geotech. 13 (3), 747–755. Zhou, H., Zheng, G., Liu, J., Yu, X., Yang, X., Zhang, T., 2019. Performance of embankments with rigid columns embedded in an inclined underlying stratum: centrifuge and numerical modelling. Acta Geotechnica 14 (5), 1571–1584. Zhou, H., Zheng, G., Yang, X., Li, T., Yang, P., 2019. Ultimate seismic bearing capacities and failure mechanisms for strip footings placed adjacent to slopes. Can. Geotech. J. 56 (11), 1729–1735. Zhou, H., Zheng, G., Yin, X., Jia, R., Yang, X., 2018a. The bearing capacity and failure mechanism of a vertically loaded strip footing placed on the top of slopes. Comput. Geotech. 94, 12–21.
Design of Highway Subgrades. (JTG D30-2015), MOT [in Chinese]. Ochiai, H., Otani, J., Hayashic, S., Hirai, T., 1996. The pull-out resistance of geogrids in reinforced soil. Geotext. Geomembranes 14 (1), 19–42. Rowe, R.K., Li, A.L., 1999. Reinforced embankments over soft foundations under undrained and partially drained conditions. Geotext. Geomembranes 17 (3), 129–146. Rowe, R.K., Li, A.L., 2005. Geosynthetic-reinforced embankments over soft foundations. Geosynth. Int. 12 (1), 50–85. Rowe, R.K., Liu, K.W., 2015. Three-dimensional finite element modelling of a full-scale geosynthetic-reinforced, pile-supported embankment. Can. Geotech. J. 52 (12), 2041–2054. Rowe, R.K., Soderman, K.L., 1985. An approximate method for estimating the stability of geotextile-reinforced embankments. Can. Geotech. J. 22 (3), 392–398. Rowe, R.K., Soderman, K.L., 1987. Stabilization of very soft soils using high strength geosynthetics: the role of finite element analyses. Geotext. Geomembranes 6 (1–3), 53–80. Rui, R., Han, J., van Eekelen, S.J.M., Wan, Y., 2018. Experimental investigation of soilarching development in unreinforced and geosynthetic-reinforced pile-supported embankments. J. Geotech. Geoenviron. Eng. 145 (1) 04018103. Smith, C.C., Gilbert, M., 2007. Application of discontinuity layout optimization to plane plasticity problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2086), 2461–2484. Smith, C.C., Tatari, A., 2016. Limit analysis of reinforced embankments on soft soil. Geotext. Geomembranes 44 (4), 504–514. Tandjiria, V., Low, B.K., Teh, C.I., 2002. Effect of reinforcement force distribution on
8
Contents lists available at ScienceDirect
Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem
Stability analysis of stone column-supported and geosynthetic-reinforced embankments on soft ground Gang Zhenga,b,c, Xiaoxuan Yua,b, Haizuo Zhoua,b,c,∗, Shun Wangd, Jiapeng Zhaoa,b, Xiaopei Hea,b, Xinyu Yanga,b a
School of Civil Engineering, Tianjin University, Tianjin, 300072, China Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin, 300072, China c State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, 300072, China d Engineering, Department of Civil Engineering and Natural Hazards, University of Natural Resources and Life Sciences, Vienna, Feistmantelstraße 4, 1180, Wien, Austria b
A R T I C LE I N FO
A B S T R A C T
Keywords: Geosynthetics Stone columns Failure Stability Embankment
This study focuses on the stability of stone column-supported and geosynthetic-reinforced embankments on soft soil. An upper-bound limit state plasticity failure discretization scheme (known as discontinuity layout optimization (DLO)), which determines the embankment stability without pre-assuming a slip surface, is used. The relationships between the stability of stone column-supported and geosynthetic-reinforced embankments and various influencing parameters, including the soil strength, geometric configuration, reinforcement strength, and area replacement ratio, are analysed. It is found that geosynthetics provide a significant contribution to embankment stability. Two failure mechanisms of geosynthetics (i.e., rupture failure and bond failure) are revealed and the effect of geosynthetics on embankment stability is governed by the failure mode. The application of stone columns mitigates the risk of geosynthetic failure. To provide an analytical solution for primary design in engineering practice, an approach based on the limit equilibrium method is proposed. Validations are performed with the DLO solution to demonstrate the accuracy and reliability of the developed analytical approach.
1. Introduction The stability of embankments over soft soil has attracted considerable attentions (e.g., Mohapatra and Rajagopal, 2017). The application of stone columns serve as an economically and environmentally friendly solution for increasing both the slope stability and the bearing capacity when stabilizing an embankment (e.g., Abusharar and Han, 2011; Zhou et al., 2017, 2018). The stability of column-supported embankments is governed by the failure mechanism of columns (Zhou et al., 2019). Kitazume and Maruyama (2006, 2007) categorized the shear and bending failure modes as internal failure modes, whereas the sliding and rotational failure modes were categorized as external failure modes. For stone columns, shear failure is the critical failure mode, and the strength of stone columns depends on the friction angle of the granular material and the confining stress from the surrounding soils (Abusharar and Han, 2011; Ambily and Gandhi, 2007; Fattah et al., 2010). The influences of various reinforcement strength parameters have been investigated in prior numerical and experimental investigations (Christoulas et al., 1997; Chen et al., 2015).
Geosynthetic reinforcement is widely used to improve embankment performance on soft soil (Liu et al., 2017; Zhang et al., 2019). A basal geosynthetic reinforcement can serve to resist the earth pressure within an embankment and to prevent the lateral deformation of its foundation, thereby increasing the bearing capacity and stability (Chai et al., 2016; Leshchinsky, 1987; Rowe and Li, 2005; Rowe and Liu, 2015 ). The failure mechanisms of geosynthetics can be divided into two categories: rupture failure and bond failure (Rowe and Soderman, 1987; Hird and Kwok, 1989). For rupture failure, the stability of an embankment depends on the strength of the geosynthetic. When bond failure occurs, the embankment stability is governed by the bond strength between the geosynthetic and the surrounding soil in which the tensile capacity of the geosynthetic is not fully developed (Gilbert et al., 2010; Ochiai et al., 1996; Rowe and Soderman, 1985). These two mechanisms were described by Tandjiria et al. (2002), who numerically investigated the effects of the reinforcement force distribution. An alternative method for ensuring the embankment stability for rapid construction on soft ground is the use of geosynthetic reinforcement above the tops of stone columns (Deb et al., 2008; Hosseinpour
∗
Corresponding author. School of Civil Engineering, Tianjin University, Tianjin, 300072, China. E-mail addresses: [email protected] (G. Zheng), [email protected] (X. Yu), [email protected] (H. Zhou), [email protected] (S. Wang), [email protected] (J. Zhao), [email protected] (X. He), [email protected] (X. Yang). https://doi.org/10.1016/j.geotexmem.2019.12.006
0266-1144/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Gang Zheng, et al., Geotextiles and Geomembranes, https://doi.org/10.1016/j.geotexmem.2019.12.006
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
List of symbols H D n S d as ϕc ϕf ϕeq γc γf γs
γeq cf cu ceq η α
Embankment height Clay thickness Slope gradient Axis-to-axis spacing of stone column walls Width of stone column walls Area replacement ratio of stone column walls Friction angle of stone column walls Friction angle of embankments Equivalent friction angle of reinforced zone Unit weight of stone column walls Unit weight of embankments Unit weight of clay
R q Q σh x K aeq
Equivalent unit weight of reinforced zone Cohesion of embankments Undrained shear strength of the clay Equivalent cohesion of reinforced zone Stress concentration ratio Friction coefficient at the interface between geosynthetics and clay Rupture strength of geosynthetic Surcharge on the ground Surcharge on the top of ground Horizontal stress below loads of embankment Distance from the edge of loading of embankment Coefficient of active earth pressure of the equivalent model
based on plasticity theorems. It serves as a highly effective tool to evaluate the critical plastic collapse mechanism and the limit state of soils, where an associated flow rule is used. The DLO technique has been successfully used in a variety of geotechnical engineering endeavours (Leshchinsky and Ambauen, 2015; Smith and Tatari, 2016; Leshchinsky, 2015; Zhou et al., 2018a Zhou et al., 2019; Zheng et al., 2019). In this study, the LimitState: GEO v3.4 software (LimitState, 2013) adopting the DLO is used for stability analysis of stone columnsupported and geosynthetic-reinforced embankments, and an associated flow rule is adopted. A shear strength reduction technique was selected to calculate the value of factor of safety (FS), which is applied to the strength parameters of the soil and columns (Dawson et al., 1999; Abusharar and Han, 2011). The plastic analysis presented applies only to ductile geosynthetic reinforcement, which is modelled using the engineered element. A nodal spacing of H/5 is selected based on an accuracy of 1–2% in terms of the factor of safety (FS) (Smith and Tatari, 2016). A friction coefficient of 0.8 was assumed at the interface between the geosynthetics and soil (Huang and Han, 2009; Smith and Tatari, 2016). For stone column-supported and geosynthetic-reinforced embankments, two modes of the slip surface, namely, the lateral sliding failure and deep-seated failure modes, are shown in Fig. 2. The following parametric studies are focused on the latter failure mode.
et al., 2015). The application of geosynthetic reinforcement helps to transfer stresses from the soft soil to the columns (King et al., 2017; Rui et al., 2018). Prior studies have presented insights into the effects of a single geosynthetic or single stone columns on the stability of reinforced embankments. However, limited insight has been provided into the interactions among the soil, stone column, and geosynthetics in the embankment stability. In this study, stability analyses of stone column-supported and geosynthetic-reinforced embankments on soft ground are investigated using the upper-bound limit state plasticity failure discretization scheme, known as discontinuity layout optimization (DLO). The influences of the soil strength, geometric parameters, reinforcement strength, and area replacement ratio on the embankment stability are discussed. An analytical solution that considers both the shear failure behaviours of the stone column and the frictional properties along the interface between the geosynthetics and the surrounding soil is proposed. 2. Problem definition and verification 2.1. Numerical model A reinforced embankment is constructed on undrained soil, as shown in Fig. 1. Geosynthetics are placed at the embankment base, and stone columns are installed in the soft soil. Due to the complexity and the demand for significant time of a three-dimensional (3D) arrangement of multiple columns, a 3D problem of individual column model has been commonly converted into a two-dimensional (2D) columnwall model to investigate the embankment stability (Abusharar and Han, 2011; Zhang et al., 2014; Marandi et al., 2016); A column-wall model is thus adopted in this study. The parameters depicted in Fig. 1 are as follows: H is the embankment height; n is the slope gradient (n (horizontal):1 (vertical)); D is the thickness of the clay; S, d and as (i.e., d ) are the axis-to-axis spacing, the width and the area replacement ratio S of the stone column walls, respectively; γf , γc and γs are the unit weights of the embankment, stone column walls and clay, respectively; ϕf and ϕc are the friction angles of the embankment and stone column walls, respectively; c f and c u are the cohesion of the embankment and the undrained shear strength of the clay, respectively; α is the friction coefficient at the interface between the geosynthetic and the soil; R is the rupture strength of the geosynthetic; and Q is the surcharge applied at the top of the embankment. Commonly used approaches for stability analysis of embankments include the method of slices (Jewell, 1988), finite element method (Ambily and Gandhi, 2007; Chen et al., 2015) and limit analysis (LA) (Smith and Tatari, 2016). LA, which acts as a rigorous analytical solution, can provide lower and upper bounds to a real limit load. The discontinuity layout optimization (DLO) (Gilbert et al., 2010; Smith and Gilbert, 2007) is an upper bound LA procedure, which is rigorously
2.2. Verification 2.2.1. Behaviour of geosynthetic-reinforced embankments To verify the accuracy of the established model, the ultimate height of geosynthetic-reinforced embankments obtained by DLO is compared with the results presented in the literature. The established model is compared with the solution of Rowe and Li (1999) utilizing the finite element method and the solution of Smith and Tatari (2016) using DLO. An embankment with a crest width of 27 m and a 2 (horizontal):1 (vertical) side slope is constructed atop soft soil with a thickness of 15 m. The parameters of the soft soil and embankment fill are provided in Table 1. Fig. 3 shows the comparison of the results calculated by DLO
Fig. 1. Schematic of the model. 2
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 2. Failure mechanisms of stone column-supported and geosynthetic-reinforced embankments: (a) Lateral sliding failure and (b) Deep-seated failure mode.
Table 1 Parameters of the reinforced embankment analysis for calibration (Rowe and Li, 1999). Parameter
Clay
Embankment
Undrained shear strength (kPa), z=0m Undrained shear strength (kPa), z = 15 m Friction angle (°) Unit weight (kN/m³)
5
N/A
27.5
N/A
0 Varied (from 14.7 to 15.6)
37 20
Note: the undrained shear strength varies linearly with the depth z below the clay surface; N/A represents the phrase not available.
Fig. 4. Validation of the behaviour of stone column-supported embankments: (a) comparison of the FS versus the friction angle of the stone columns between the results of the DLO solution and those of Abusharar and Han (2011) (unit: m), (b) comparison of the ultimate loads versus the spacing of the stone columns between the results of the DLO solution and those of Fattah et al. (2016) (unit: mm). Table 3 Parameters of the numerical analysis for calibration (Fattah et al., 2016).
Fig. 3. Comparison of the maximum height Hm of an embankment versus the geosynthetic rupture strength (i.e., tensile force of the geosynthetic) between the results of the present DLO solution and the results of previous studies.
Thickness (m)
Unit weight (kN/m³)
Undrained shear strength (kPa)
Friction angle (°)
Embankment Clay Sand Stone column
5 10 2 10
20.5 18.5 20.5 19.5
0 20 0 0
32 0 30 38
Unit weight (kN/ m³)
Undrained shear strength (kPa)
Friction angle (°)
Embankment Clay Stone column
21:84 18.5 15.7
0 10 0
40 0 41:5
(using a non-associated flow rule and a non-associated flow rule, respectively) with the results in prior literatures (Rowe and Li, 1999; Smith and Tatari, 2016). Notably, the results using an associated flow rule in this study accurately fit the results of Smith and Tatari (2016), which are slight overestimations relative to the result of Rowe and Li (1999). Considering a non-associated flow rule (i.e., modified soil cohesion and friction angles are used (Drescher and Detournay, 1993; Bolton, 1986)), the results show a slight decrease, which demonstrate an agreement with those of Rowe and Li (1999).
Table 2 Parameters of the numerical analysis for calibration (Abusharar and Han, 2011). Material
Material
3
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
approximately linear increase in the FS. The DLO approach based on the upper bound limit theorem yields a slightly higher solution than that based on the finite difference method (approximately 2.5%, 1.7% and 3.8% higher for friction angles of 34°, 38°, and 42°, respectively). To further elaborate the effect of individual stone columns on embankment stability, a comparison of DLO numerical modelling was made with laboratory tests from Fattah et al. (2016), who investigated the relationship between the axis-to-axis spacing of individual stone columns and ultimate surcharge. The parameters of the clay, stone columns and embankment are provided in Table 3. The scale of the numerical model is shown in Fig. 4(b). The results demonstrate reasonable agreement for various stone column spacings (approximately 3.8%, 5.1% and 6 0.1% higher for spacing between the stone columns of 175, 210, and 280 mm, respectively). The presented comparisons demonstrate that the implementations of DLO accurately capture the contributions of the stone columns and geosynthetic to the stability of the embankments.
Table 4 Parameters of stone column-supported and geosynthetic-reinforced embankments. Influencing factor
Range of values
H D
0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2
ϕc as
38, 42, 46 0.1, 0.2, 0.3 0, 0.5, 1, 1.5, 2, ∞
R γf D2
Note: The value of
R γf D2
= 10 are used to simulate
R γf D2
= ∞, and
R γf D2
=0
represents that there is no geosynthetic reinforcement. Geosynthetics are placed at the interface between the embankment and clay (Jewell, 1988; Smith and Tatar, 2016).
2.2.2. Behaviour of stone column-supported embankments To further validate the modelling of stone column walls, solutions of Abusharar and Han (2011) are selected. The geometric parameters of the column walls and embankment are provided as follows: the width of the columns is 0.8 m, the length of the columns is 10 m, the spacing between adjacent columns is 4 m, the crest width of the embankment is 20 m, the height of the embankment is 5 m, and the angle of the side slopes is 2 (horizontal):1 (vertical). The material parameters of the numerical model are shown in Table 2. Fig. 4(a) shows a comparison between the FS from the DLO and that from Abusharar and Han (2011). An increase in the friction angle of the column walls results in an
3. Parametric analysis To investigate the influences of various factors on the stability of stone column-supported and geosynthetic-reinforced embankments, parametric studies on the FS are performed. The solution is an inherent function of the following series of influencing factors:
FS = f (H , D , R, n, α, c f , c u, γf , γc, γs, ϕf , ϕc , as)
(1)
The corresponding parameters are mentioned in the numerical
Fig. 5. Parametric analysis for the various values of 4
cu γf D
and. H
D
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 6. Cross sections of the calculation models (modified from Abusharar and Han, 2011): (a) stone column walls, (b) equivalent area model.
practice. Following the specifications for the design of highway subgrades (Ministry of Transport (MOT) 2015), a FS of 1.35 is adopted to illustrate the effect of influencing factors on the embankment stability. A set of parametric analyses containing the evaluation results with the required FS is illustrated in Fig. 5, which shows the relationships between cu and H (when the FS is always equal to 1.35) for a stone γf D
D
column-supported embankment both with and without geosynthetics. Each chart in Fig. 5 is divided into three parts by two boundaries. The first boundary (i.e., the black solid line) is plotted for an embankment without geosynthetics, illustrating a notable increase in cu with inγf D
creasing H . This behaviour represents a significant contribution of the D soil strength to the stability of embankments. Obviously, the FS values corresponding to the region above this boundary, denoted as the stable zone, are always larger than 1.35. The other boundary (i.e., the red solid line) is depicted for the embankment with sufficient geosynthetics, denoted as R 2 = ∞, representing that the further increase in R 2 has γf D
γf D
no influence on the stability of the embankment. This boundary shows an initial convexly increasing trend with the increasing H , eventually D
levelling off as H exceeds a threshold value. The rate of increase for the D red line is lower than that for the black solid line. This is because the contribution of soil strength to the stability of an embankment is less than the contribution of the presence of high-strength geosynthetics. For the region below this boundary corresponding to the unstable zone, an instability of the embankment is always observed despite a high geosynthetic strength. The region between these two boundaries represents the conditions for which the geosynthetic strength effectively improves the embankment stability; this region is denoted as the geosynthetic-required zone. Commonly, there are two failure modes for geosynthetics. Rupture failure often occurs when the geosynthetic tensile strength is relatively small, whereas bond failure can be observed when tensile strength is high. The lower boundary (i.e., the red solid line) merely represents the bond failure. The dotted lines (with FS = 1.35), in the geosyntheticrequired zone, are depicted here for geosynthetics with a certain rupture strength (e.g., R 2 = 0.5, 1, 1.5, and 2) to show both bond and
Fig. 7. Mechanism of the deep-seated failure of a reinforced embankment over an equivalent area: (a) failure mechanism, (b) block analysis.
model, as depicted in Fig. 1. The above parameters can be nondimensionalized and given as follows:
γf D
H c R FS = f ⎜⎛ , u , , ϕf , as ⎞⎟ 2 D γ D γ f fD ⎠ ⎝
rupture failure modes. Correspondingly, a piecewise function of (2)
H D
versus is observed for geosynthetics with a certain rupture strength. Specifically, as shown in Fig. 5(a), a consistency between geosynthetics with a certain rupture strength (i.e., the dotted line) and geosynthetics with R = ∞ (i.e., the red solid line) is found for the first segment (i.e., curve AB), in which a bond failure mode occurs. As H increases, a deD viation is found (i.e., curve BC), representing a rupture failure, and a cu nearly linear relationship is observed between and H . Compared
in which the dimensionless parameters include the nondimensionalized height of the embankment H , the strength of the clay cu , and the strength of the geosynthetic
D R γf D2
cu γf D
γf D
(Smith and Tatar, 2016). The thickness
of the mat under the embankment and the location of the geosynthetic have a minor effect on the stability of the embankment (Jewell, 1988). Additionally, the embankment strength is not considered as an influencing factor because its effect on the performance of a reinforced embankment is negligible compared with that of the clay strength (Smith and Tatar, 2016). The effects of the influencing factors in Eq. (2) are studied, and the ranges of the parameters are listed in Table 4. The constant parameters are as follows: α = 0.8, q = 0, n = 2, c f = 0, ϕf = 32°, γf = 20 kN/m³, γs = 18.5 kN/m³, and γc = 19.5 kN/m³. The meaning of each parameter can be found in the list of symbols. Theoretically, the slope is stable when the FS is equal to 1.0. However, a FS between 1.3 and 1.5 is usually required in engineering
γf D
D
with curve AB, the slope of curve BC increases, becoming similar to the black solid line (i.e., R = 0). Point B, the inflection point of a dotted curve, represents a transition from bond failure to rupture failure in the geosynthetic. The H corresponding to point B is denoted as a critical D
( ) H
nondimensionalized embankment height D , in which the bond cri strength of the interface between the geosynthetic and the surrounding soil is equal to the rupture strength of the geosynthetic. An increase in the area of the stable zone and a reduction in the area of the unstable zone are observed with a larger friction angle ϕc and a 5
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
Fig. 8. Plot of the FS against the friction angle of the stone columns for Eq. (7) and the DLO (red markers) using c u = 15 kPa; d = 0.8 m; S = 4 m; D = 10 m; H = 5 m; γf = 17 kN/m3; and γeq = 16 kN/m3: (a) influence of the friction angle of the stone column walls, (b) influence of the undrained shear strength of the clay, (c) influence of the width of the stone column walls, (d) influence of the axis-to-axis spacing of stone column walls.
Fig. 6. The properties of the equivalent area model can be calculated based on the weighted average between the soils and columns as follows (Cooper and Rose, 1999):
larger area replacement ratio as of the stone column walls. When the friction angle ϕc is large, a significant decrease in the area of the geosynthetic-required zone is found as as increases, indicating a relatively apparent contribution of the geosynthetic to the embankment stability for a low area replacement ratio of the stone column walls. Meanwhile, for a certain R 2 value, an increase in the area replacement ratio and γf D
friction angle results in a great
( )
H . D cri
For a designed embankment
constructed on soft soil, an applicable solution via both stone column and geosynthetics technologies can be directly determined from the presented parametric analysis in engineering practice.
ceq = cc as + c u (1 − as)
(3)
ϕeq = tan−1 (μ tan ϕc + (1 − μ)tan ϕs)
(4)
γeq = γc as + γs (1 − as)
(5)
where
μ= 4. Analytical solutions
as η 1 + as (η − 1)
(6)
where ϕs and cc are the friction angles of the soft soil and the cohesions of the column, respectively; The meaning of the other parameters can be found in the list of symbols. The stress concentration ratio η is a critical coefficient, which is considered in the analysis of Christoulas et al. (1997) with usual values of 2–6. The value of η is influenced by a number of factors, such as the elastic modulus, Poisson's ratio, the location of the columns and the area replacement ratio (Fattah et al., 2016; Hanna et al., 2013). The stress concentration ratio versus the strain shows an initial increase and
To develop an explicit equation for an evaluation of reinforced embankments and provide solutions with any FS, a simplified method based on the limit equilibrium method considering the influences of geosynthetics and stone columns is proposed. To derive an analytical solution, it is common to convert the column walls into an equivalent area model for simplification (Christoulas et al., 1997; Abusharar and Han, 2011; Zhang et al., 2014), as shown in 6
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
zone. 3. A simplified analytical solution for stone column-supported and geosynthetic-reinforced embankment stability is proposed based on the limit equilibrium method. Validations are performed to demonstrate the accuracy of the FS values predicted with the proposed model. This simplified method can be adopted for primary design in geotechnical engineering projects involving similar construction and soil conditions.
a subsequent decrease afterwards, eventually exhibiting a degradation to approximately 1.0 when the failure occurs (Juran and Guermazi, 1988; Zhang et al., 2014). The values of η with 1.0–1.3 is commonly adopted in the stability analysis (Marandi et al., 2016; Zhou et al., 2017), and a stress concentration ratio of 1.0 is used in this study. For the calculation of the FS of stone column-supported embankment, reduction factors of 0.9–1.0 were recommended to convert the columnwall model into the equivalent area model (Abusharar and Han, 2011; Zhang et al., 2014). In this study, a reduction factor of 1.0 is selected. Jewell (1988) presents an analytical solution to determine safety factors and reinforcement forces for geosynthetic-reinforced embankments on soft soils based on the translational failure mechanism. Based on this mechanism proposed by Jewell (1988), the geosynthetic is modelled as a single representative force applied to the surrounding soil. Considering the equivalent area model, the corresponding distribution of the soil reactions in this study is shown in Fig. 7(a). Specifically, a vertical surface load q that increases with a gain in the distance x from the edge is shown in Fig. 7(b). The solution FS via the integration of these relationships is as follows:
FS =
Acknowledgements This research was funded by the National Key R&D Program of China (Grant No. 2017YFC0805407), the National Natural Science Foundation of China (No. 51708405and No. 41630641) and the Project of Tianjin Science and Technology Plan (No. 16YDLJSF00040). The authors appreciate the financial support. References Abusharar, S.W., Han, J., 2011. Two-dimensional deep-seated slope stability analysis of embankments over stone column-improved soft clay. Eng. Geol. 120 (1–4), 103–110. Ambily, A.P., Gandhi, S.R., 2007. Behavior of stone columns based on experimental and FEM analysis. J. Geotech. Geoenviron. Eng. 133 (4), 405–415. Bolton, M.D., 1986. The strength and dilatancy of sands. Geotechnique 36 (1), 65–78. Chai, J.C., Sari, K., Shen, S.L., Cai, Y., 2016. Predicting self-healing ratio of GCL with a damage hole. Geotext. Geomembranes 44 (5), 761–769. Chen, J.F., Li, L.Y., Xue, J.F., Feng, S.Z., 2015. Failure mechanism of geosynthetic-encased stone columns in soft soils under embankment. Geotext. Geomembranes 43 (5), 424–431. Christoulas, S., Giannaros, C., Tsiambaos, G., 1997. Stabilization of embankment foundations by using stone columns. Geotech. Geol. Eng. 15 (3), 247–258. Cooper, M.R., Rose, A.N., 1999. Stone column support for an embankment on deep alluvial soils. Proc. Inst. Civ. Eng. Geotech. Eng. 37 (1), 15–25. Dawson, E.M., Roth, W.H., Drescher, A., 1999. Slope stability analysis by strength reduction. Geotechnique 49 (6), 835–840. Deb, K., Chandra, S., Basudhar, P.K., 2008. Response of multilayer geosynthetic-reinforced bed resting on soft soil with stone columns. Comput. Geotech. 35 (3), 323–330. Drescher, A., Detournay, E., 1993. Limit load in translational failure mechanisms for associative and non-associative materials. Geotechnique 43 (3), 443–456. Fattah, M.Y., Shlash, K.T., Al-Waily, M.J.M., 2010. Stress concentration ratio of model stone columns in soft clays. Geotech. Test J. 34 (1), 50–60. Fattah, M.Y., Zabar, B.S., Hassan, H.A., 2016. Experimental analysis of embankment on ordinary and encased stone columns. Int. J. Geomech. 16 (4) 04015102. Gilbert, M., Smith, C., Haslam, I., Pritchard, T., 2010. Application of discontinuity layout optimization to geotechnical engineering. In: Proceedings of the 7th European Conference on Numerical Methods in Geotechnical Engineering, pp. 169–174. Griffiths, D.V., 2015. Observations on load and strength factors in bearing capacity analysis. J. Geotech. Geoenviron. Eng. 141 (7) 06015004. Hanna, A.M., Etezad, M., Ayadat, T., 2013. Mode of failure of a group of stone columns in soft soil. Int. J. Geomech. 13 (1), 87–96. Hird, C.C., Kwok, C.M., 1989. Finite element studies of interface behaviour in reinforced embankments of soft ground. Comput. Geotech. 8 (2), 111–131. Huang, J., Han, J., 2009. 3D coupled mechanical and hydraulic modeling of a geosynthetic-reinforced deep mixed column-supported embankment. Geotext. Geomembranes 27 (4), 272–280. Jewell, R.A., 1988. The mechanics of reinforced embankments on soft soils. Geotext. Geomembranes 7 (4), 237–273. Juran, I., Guermazi, A., 1988. Settlement response of soft soils reinforced by compacted sand columns. J. Geotech. Geoenviron. Eng. 114 (8), 930–943. King, D.J., Bouazza, A., Gniel, J.R., Rowe, R.K., Bui, H.H., 2017. Serviceability design for geosynthetic reinforced column supported embankments. Geotext. Geomembranes 45 (4), 261–279. Leshchinsky, D., 1987. Short-term stability of reinforced embankment over clayey foundation. Soils Found. 29, 105–114. Leshchinsky, B., 2015. Bearing capacity of footings placed adjacent to c′-φ′ slopes. J. Geotech. Geoenviron. Eng. 141 (6) 04015022. Leshchinsky, B., Ambauen, S., 2015. Limit equilibrium and limit analysis: comparison of benchmark slope stability problems. J. Geotech. Geoenviron. Eng. 141 (10) 04015043. LimitState, 2013. LimitState: Geo Manual V 3.0 (Sheffield, U.K). Liu, K.W., Rowe, R.K., Su, Q., Liu, B., Yang, Z., 2017. Long–term reinforcement strains for column supported embankments with viscous reinforcement by FEM. Geotext. Geomembranes 45 (4), 307–319. Marandi, S.M., Anvar, M., Bahrami, M., 2016. Uncertainty analysis of safety factor of embankment built on stone column improved soft soil using fuzzy logic α-cut technique. Comput. Geotech. 75, 135–144. Mohapatra, S.R., Rajagopal, K., 2017. Undrained stability analysis of embankments supported on geosynthetic encased granular columns. Geosynth. Int. 24 (5), 465–479. Ministry of Transport of the People's Republic of China (MOT), 2015. Specifications for
γH 1 ⎛ nH ⎛ (1 + α ) ⎛ceq + f tan ϕeq ⎞ + γeq D tan ϕeq ⎞ + 4c u ⎞⎟ ⎜ γf H ⎝ DK aeq ⎝ 2n ⎝ ⎠ ⎠ ⎠ ⎜
⎟
(7) Where K aeq can be expressed as follows according to Griffiths (2015):
K aeq = −tan ϕeq +
1 + tan2 ϕeq
(8)
The geosynthetic rupture strength can be determined by the bond strength. Therefore, the maximum possible restraint (i.e., maximum available tension force Tmax) on the surrounding soil is exerted by the geosynthetic. The minimum required rupture strength Rc (i.e., a critical value representing the transition from rupture failure to bond failure), which equals to Tmax, can be given:
Rc =
αnH 1 (ceq + γf tan ϕeq ) + K a f γf H 2 2 FS
(9)
To verify the effectiveness of Eq. (7), a comparison of the stone column-supported and geosynthetic-reinforced embankment results between the limit equilibrium and DLO methods is conducted, as is shown in Fig. 8. The results from Eq. (7) provide a generally good fit to the DLO results. 5. Conclusions In this study, the stability of stone column-supported and geosynthetic-reinforced embankments is analysed. Parametric analysis that can be used to determine the strength of the required geosynthetics and stone column walls for various geometric conditions and soil properties are developed. Additionally, a limit equilibrium solution is developed to predict the FS. The following conclusions are drawn: 1. Geosynthetics have a noticeable effect on the embankment stability, which is influenced by the nondimensionalized height of the embankment H . Two failure modes (i.e., rupture failure and bond D failure) in the geosynthetics are observed. With a certain geosynthetic rupture strength, there is a transition from bond failure to rupture failure. The value of H corresponding to this transition is denoted as the critical value
D H . D cri
( )
Once
H D
exceeds
( )
H , D cri
the effect
of the geosynthetic on the embankment stability is limited due to its insufficient strength. 2. The application of stone columns and geosynthetic provides a costeffective solution for the stability of embankments over soft soil. The stone column reduces the risk of these two failure modes of geosynthetics. With an increase in the friction angle ϕc and the area replacement ratio as of the stone column walls, there is a gain in the area of the stable zone and a reduction in the area of the unstable 7
Geotextiles and Geomembranes xxx (xxxx) xxx–xxx
G. Zheng, et al.
stability of embankments. Geotext. Geomembranes 20 (6), 423–443. Zhang, Z., Han, J., Ye, G., 2014. Numerical investigation on factors for deep-seated slope stability of stone column-supported embankments over soft clay. Eng. Geol. 168, 104–113. Zhang, Z., Wang, M., Ye, G., Han, J., 2019. A novel 2D-3D conversion method for calculating maximum strain of geosynthetic reinforcement in pile-supported embankments. Geotext. Geomembranes 47 (3), 336–351. Zheng, G., Zhao, J., Zhou, H., Zhang, T., 2019. Ultimate bearing capacity of strip footings on sand overlying clay under inclined loading. Comput. Geotech. 106, 266–273. Zhou, H., Diao, Y., Zheng, G., Han, J., Jia, R., 2017. Failure modes and bearing capacity of strip footings on soft ground reinforced by floating stone columns. Acta Geotech 12 (5), 1089–1103. Zhou, H., Zheng, G., He, X., Xu, X., Zhang, T., Yang, X., 2018. Bearing capacity of strip footings on c - φ soils with square voids. Acta Geotech. 13 (3), 747–755. Zhou, H., Zheng, G., Liu, J., Yu, X., Yang, X., Zhang, T., 2019. Performance of embankments with rigid columns embedded in an inclined underlying stratum: centrifuge and numerical modelling. Acta Geotechnica 14 (5), 1571–1584. Zhou, H., Zheng, G., Yang, X., Li, T., Yang, P., 2019. Ultimate seismic bearing capacities and failure mechanisms for strip footings placed adjacent to slopes. Can. Geotech. J. 56 (11), 1729–1735. Zhou, H., Zheng, G., Yin, X., Jia, R., Yang, X., 2018a. The bearing capacity and failure mechanism of a vertically loaded strip footing placed on the top of slopes. Comput. Geotech. 94, 12–21.
Design of Highway Subgrades. (JTG D30-2015), MOT [in Chinese]. Ochiai, H., Otani, J., Hayashic, S., Hirai, T., 1996. The pull-out resistance of geogrids in reinforced soil. Geotext. Geomembranes 14 (1), 19–42. Rowe, R.K., Li, A.L., 1999. Reinforced embankments over soft foundations under undrained and partially drained conditions. Geotext. Geomembranes 17 (3), 129–146. Rowe, R.K., Li, A.L., 2005. Geosynthetic-reinforced embankments over soft foundations. Geosynth. Int. 12 (1), 50–85. Rowe, R.K., Liu, K.W., 2015. Three-dimensional finite element modelling of a full-scale geosynthetic-reinforced, pile-supported embankment. Can. Geotech. J. 52 (12), 2041–2054. Rowe, R.K., Soderman, K.L., 1985. An approximate method for estimating the stability of geotextile-reinforced embankments. Can. Geotech. J. 22 (3), 392–398. Rowe, R.K., Soderman, K.L., 1987. Stabilization of very soft soils using high strength geosynthetics: the role of finite element analyses. Geotext. Geomembranes 6 (1–3), 53–80. Rui, R., Han, J., van Eekelen, S.J.M., Wan, Y., 2018. Experimental investigation of soilarching development in unreinforced and geosynthetic-reinforced pile-supported embankments. J. Geotech. Geoenviron. Eng. 145 (1) 04018103. Smith, C.C., Gilbert, M., 2007. Application of discontinuity layout optimization to plane plasticity problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2086), 2461–2484. Smith, C.C., Tatari, A., 2016. Limit analysis of reinforced embankments on soft soil. Geotext. Geomembranes 44 (4), 504–514. Tandjiria, V., Low, B.K., Teh, C.I., 2002. Effect of reinforcement force distribution on
8