Modelling of geosynthetic-reinforced column-supported embankments using 2D full-width model and modified unit cell approach

Geotextiles and Geomembranes xxx (2017) 1e18

Contents lists available at ScienceDirect

Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem

Modelling of geosynthetic-reinforced column-supported embankments using 2D full-width model and modified unit cell approach Yan Yu*, Richard J. Bathurst GeoEngineering Centre at Queen's-RMC, Department of Civil Engineering, Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 April 2016 Received in revised form 30 September 2016 Accepted 26 December 2016 Available online xxx

Soil-cement deep mixing (DM) columns combined with geosynthetic basal reinforcement are an accepted technique in geotechnical engineering to construct road and railway embankments over soft foundations. Both full-width and unit cell models have been used to numerically simulate the performance of geosynthetic-reinforced and column-supported (GRCS) embankments. However, the typical unit cell model with horizontally fixed side boundaries cannot simulate the lateral spreading of the embankment fill and foundation soil. As a result, the calculated reinforcement tensile loads using typical unit cell models are much less than those from matching full-width models. The paper first examines GRCS embankments using a full-width model with small- and large-strain modes in FLAC and then compares the calculated results from the full-width model with those using a typical unit cell model, a recently proposed modified unit cell model, and a closed-form solution. The paper also examines the influence of the soft foundation soil modulus, reinforcement tensile stiffness, and DM column modulus on the reinforcement tensile loads. Numerical analyses show that the reinforcement tensile loads from the modified unit cell model are in good agreement with those from the full-width model for zones under the embankment crest for all cases and conditions examined in the paper. Both the full-width model and modified unit cell model perform better than the typical unit cell model for the prediction of the reinforcement tensile load when compared to the closed-form solution. However, while the modified unit cell developed by the writers is shown to be more accurate than the typical unit cell when predictions are compared to results using full-width numerical simulations, the benefit of using this approach to reduce computation times may be limited in practice. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Geosynthetic reinforcement Column-supported embankments Soil-cement deep mixing Numerical modelling Unit cell model Full-width model

1. Introduction Geosynthetic-reinforced and column-supported (GRCS) embankments have proven to be an effective technique to support road and railway embankments over soft foundations and to widen existing road embankments. Compared to embankment construction techniques that use the preloading method with and without prefabricated vertical drains, the GRCS technique has advantages of: (a) accelerating embankment construction, and (b) protecting adjacent existing embankments from distress. The columns used to support GRCS embankment fills are generally spaced in a square or triangular grid pattern.

* Corresponding author. E-mail addresses: [email protected] (Y. Yu), [email protected] (R.J. Bathurst).

Concrete and timber piles with and without pile caps have been used for the supports (e.g., Liu et al., 2007; Briançon and Simon, 2012; Nunez et al., 2013; Zhang et al., 2013; Blanc et al., 2014; Xing et al., 2014; Bhasi and Rajagopal, 2015a, 2015b; Rowe and Liu, 2015; Xu et al., 2016; Zhang et al., 2016). Other support types for embankments over soft foundations are geosynthetic-encased stone columns (e.g., Yoo, 2010; Ali et al., 2014; Hosseinpour et al., 2015; Khabbazian et al., 2015; Gu et al., 2016). Soil-cement deep mixing (DM) columns in combination with geosynthetic reinforcement have also been used to improve the bearing capacity and to reduce the differential settlement of soft foundations (e.g., Forsman et al., 1999; Lai et al., 2006; Han et al., 2007; Huang and Han, 2009, 2010; Huang et al., 2009; Borges and Marques, 2011; Bruce et al., 2013; Yapage and Liyanapathirana, 2014; Liu and Rowe, 2015, 2016; Borges and Gonçalves, 2016). One or more layers of geosynthetic

http://dx.doi.org/10.1016/j.geotexmem.2017.01.002 0266-1144/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

2

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

reinforcement are included at the base of the embankment to improve the load transfer from the backfill self-weight and any surcharge loading to the stiffer DM columns and then to the deeper and stiffer soil stratum (Fig. 1). Most often the DM columns are cylindrical. One notable exception is a GRCS embankment which was constructed in Finland with a series of parallel DM walls with isolated columns located between the column walls (Forsman et al., 1999). 2. Analysis methods The fundamental load transfer mechanisms for GRCS embankments are soil arching and tensioned membrane effects. Design methods using closed-form solutions are available that take advantage of one or both load transfer mechanisms for these types of soil structures (e.g., Hewlett and Randolph, 1988; Low et al., 1994; Love and Milligan, 2003; Kempfert et al., 2004; Van Eekelen et al., 2011, 2013, 2015; Zhuang et al., 2014; Zhang et al., 2016). These methods are attractive to designers because they are relatively simple. Several of these approaches and variants can be found in design guidance documents (e.g., BS8006, 2010; EBGEO, 2011). However, the simplicity of these design methods means that their use is best suited to field cases with simple geometry and ground conditions. The next level of analysis complexity is the use of twodimensional (2D) numerical models of GRCS embankment structures using the finite element method (FEM) and finite difference method (FDM) (e.g., Han et al., 2007; Huang and Han, 2010; Huang et al., 2009; Yapage and Liyanapathirana, 2014). Almost all GRCS embankments are constructed with isolated columns having cylindrical geometry. One exception is the Finnish case history mentioned earlier. An obvious shortcoming of 2D full-width models for the typical case of an array of cylindrical column supports is that numerical predictions of embankment behaviour may diverge from predictions using three-dimensional (3D) models. A practical exception may be the case of rows of cylindrical columns placed at small centre-to-centre spacing. Full-width embankment simulations using 3D FEM and FDM simulations offer the possibility of including the actual embankment geometry, periodic plan arrangement of column locations and cylindrical column geometry. This general approach has the disadvantage of formidable computational overhead which is compounded when advanced constitutive models are used for the component materials. Nevertheless, valuable insights have been

20 m

10 m

gained from these modelling efforts, but the complexity and computational demands restrict their use to research-focused applications (e.g., Huang and Han, 2009; Bhasi and Rajagopal, 2015a, 2015b; Khabbazian et al., 2015; Liu and Rowe, 2015, 2016). A strategy to reduce computations to manageable times is to focus on axisymmetric models centred on one column (e.g., Han and Gabr, 2002; Smith and Filz, 2007; Borges and Marques, 2011; Bhasi and Rajagopal, 2015a; Hosseinpour et al., 2015; Khabbazian et al., 2015). However this method has the disadvantage that true 3D effects beyond the axisymmetric unit cell are not captured. Based on the advantages and disadvantages summarized above for different approaches to model cylindrical column-supported geosynthetic-reinforced embankments, it is likely that 2D models will remain the preferred choice for designers at least in the near term. Nevertheless, computation times can remain prohibitive particularly if parametric analyses are required to optimize column dimensions and geosynthetic reinforcement options. Therefore, parametric analyses using the unit cell approach may be attractive as long as the behaviour of the unit cell can be expected to capture the behaviour of the same unit volume in a larger 2D full-width model. 3. Prior related work on unit cells Unit cell models have been reported by Han and Gabr (2002), Smith and Filz (2007), Zhuang et al. (2010), Borges and Marques (2011), Bhasi and Rajagopal (2015a), Hosseinpour et al. (2015), and Khabbazian et al. (2015). However, results using the typical unit cell with fixed lateral boundary conditions (Fig. 2b) can be quite different from those using a full-width model for the same embankment structure (Fig. 2a). The differences in numerical outcomes between a full-width model and unit cell model have been demonstrated by Bhasi and Rajagopal (2015a) and Khabbazian et al. (2015) who showed that the reinforcement axial tensile loads using the unit cell model were much lower than those using the full-width model. The reason for this discrepancy is that the typical unit cell cannot capture the lateral spreading of the embankment fill and foundation soil (Bhasi and Rajagopal, 2015a; Khabbazian et al., 2015). Yu et al. (2016b) proposed a modified unit cell that is able to simulate the lateral spreading of the embankment fill and foundation soil for the case of 2D embankment models. They demonstrated that the reinforcement axial tensile loads using the modified unit cell (Fig. 2c) were much greater than those using the

16.8 m

10 m

20 m

Example unit cell

10 m

5m

Platform fill

1

2

Granular embankment fill

2

1

Reinforcement

DM column

Soft clay Bedrock

Plan view of DM walls 0.7 m

2.8 m

Fig. 1. Schematic showing a GRCS embankment with DM column walls.

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

8.4 m

10 m

3

20 m

10 m

5m

Embankment fill Soft clay 1 Soft clay 2 Soft clay 3 Soft clay 4 Soft clay 5

y x 0.70 m

0.35 m

0.35 m

2.8 m (a) Full-width model

0.35 m

0.35 m

10 m

5m

Embankment fill

Fx

kb

Soft clay 1

kf,1

Soft clay 2

kf,2

Soft clay 3

kf,3

Soft clay 4 y

0.35 m

Soft clay 5

kf,4 y

2.8 m (b) Typical unit cell model

kf,5 x

x

2.8 m (c) Modified unit cell model

Fig. 2. Boundary conditions for (a) full-width model, (b) typical unit cell model (fixed horizontal displacements at both left and right sides of the cell), and (c) modified unit cell model (fixed horizontal displacements at left side of the cell and springs at right side of the cell; adapted from Yu et al., 2016b).

typical unit cell approach (Fig. 2b). However, they concluded that more investigation was required to develop confidence that the modified unit cell model can be used in place of 2D full-width models for the same embankment structure.

4. Objectives of current study

The numerical analyses reported in this paper were carried out using the two-dimensional FDM program FLAC (Itasca, 2011). The choice of magnitude of the soft soil modulus, DM column modulus and reinforcement stiffness on numerical outcomes using both approaches is examined in the current study. The influence of choice of small-strain and large-strain mode options in program FLAC on numerical results and computational effort is also investigated.

The main objectives of this paper are as follows: 5. Problem definition and parameter values 1) Demonstrate the difference in results between a full-width model, a typical unit cell model and the modified unit cell approach proposed by Yu et al. (2016b) for the same GRCS embankment with DM column walls; 2) Introduce a methodology to calibrate the modified unit cell model to achieve a good match with results using full-width 2D numerical embankment models; 3) Identify the practical limitations of using the modified unit cell approach compared to the typical unit cell approach and fullwidth numerical modelling; and 4) Compare results using all three numerical models with those using the closed-form solution by Low et al. (1994).

5.1. General This paper is focused on an example GRCS embankment constructed on soil-cement DM column walls (Fig. 1). The soft foundation soil is 10 m thick and founded on bedrock. The DM column walls are 0.7 m thick with spacing of 2.8 m (e.g., Forsman et al., 1999; Han et al., 2007; Huang and Han, 2010). The installation of the DM column walls results in an area replacement ratio of 25%. A 0.5-m thick working platform with a layer of geosynthetic reinforcement is placed over the foundation. The geosynthetic layer is located 0.3 m above the foundation. The height of the embankment

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

4

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

is H ¼ 5 m corresponding to the maximum embankment fill thickness. Embankment symmetry was invoked so that only one half of the full-width plane strain embankment was modelled (Fig. 2a). The parameter values for the soft foundation soil, working platform/embankment fill, DM column and geosynthetic stiffness were taken from Yu et al. (2016b). They modelled the same GRCS embankment using only unit cell models. The soft foundation soil was modelled as a linear elastic model with Mohr-Coulomb failure criterion. The working platform and the embankment fill were modelled as linear elastic Mohr-Coulomb materials. A linear elastic model was used to model the DM columns. The soil and DM column parameter values were selected based on ranges reported in the literature (Budhu, 2010; Huang and Han, 2010; Huang et al., 2009). For the base case, a depth-dependent Young's modulus for the foundation soil was used as follows: Efs,1 ¼ 0.5 MPa for foundation soil depth between 0 and 2 m, Efs,2 ¼ 1 MPa between 2 and 4 m, Efs,3 ¼ 1.5 MPa between 4 and 6 m, Efs,4 ¼ 2 MPa between 6 and 8 m, and Efs,5 ¼ 2.5 MPa between 8 and 10 m. The DM column walls have a Young's modulus of 150 MPa and Poisson's ratio of 0.3. Table 1 summarizes the soil and DM column parameter values for the base case full-width model in this investigation. The secant stiffness values of the geosynthetic for the instrumented field case reported by Forsman et al. (1999) were 1790 and 2120 kN/m at strain levels of 2% and 6%, respectively. Thus the axial stiffness of the geosynthetic in this investigation was set to J ¼ 2000 kN/m (base case). The influence of a wider range of geosynthetic stiffness values (i.e., J ¼ 500e5000 kN/m) is examined later in the paper. Load transfer between the dissimilar materials comprising the embankment problem was modelled using interfaces with normal and shear stiffness (e.g., Yu et al., 2015a, 2015b, 2016a, 2016b, 2016c). The estimation of interface parameter values based on the adjacent soil is explained by Yu et al. (2015b). Examples of the calculation of the interface friction angle (fi,pf) and cohesion (ci,pf) between the platform fill and DM column are: fi,pf ¼ tan1[Ri  tan(fef)] ¼ tan1[2/3  tan(35 )] ¼ 25 and ci,pf ¼ Ri  cef ¼ 2/3  1.0 ¼ 0.67 kPa, where Ri is the reduction factor, and fef and cef are the friction angle and cohesion of the working platform fill, respectively. The shear stiffness and normal stiffness were set to be ks,pf ¼ 10 MPa/m and kn,pf ¼ 100 MPa/m (Yu et al., 2015a). The interface parameter values used in this paper are

Table 1 FLAC material property values (base case values from Yu et al., 2016b). Properties Embankment and working platform fill Unit weight, gef (kN/m3) Young's modulus, Eef (MPa) Poisson's ratio, nef () Friction angle, fef (degree) Dilation angle, jef (degree) Cohesion, cef (kPa) Soft foundation soil Unit weight, gfs (kN/m3) Young's modulus of soft clay 1, Efs,1 (MPa) Young's modulus of soft clay 2, Efs,2 (MPa) Young's modulus of soft clay 3, Efs,3 (MPa) Young's modulus of soft clay 4, Efs,4 (MPa) Young's modulus of soft clay 5, Efs,5 (MPa) Poisson's ratio, nfs () Friction angle, ffs (degree) Dilation angle, jfs (degree) Cohesion, cfs (kPa) DM column Unit weight, gc (kN/m3) Young's modulus, Ec (MPa) Poisson's ratio, nc ()

Base case value (range) 20 40 0.3 35 5 1 16 0.5 (0.5e2.5) 1.0 (0.5e2.5) 1.5 (0.5e2.5) 2.0 (0.5e2.5) 2.5 (0.5e2.5) 0.25 15 0 5 20 150 (50e250) 0.3

Table 2 FLAC interfaces and corresponding parameter values for base case (from Yu et al., 2016b). Interface location and parameters Soft clay-DM column and soft clay-platform fill Friction angle, fi,fs (degree) Dilation angle, ji,fs (degree) Adhesion, ci,fs (kPa) Normal stiffness, kn,fs (MPa/m) Shear stiffness, ks,fs (MPa/m) Platform fill-DM column Friction angle, fi,pf (degree) Dilation angle, ji,pf (degree) Cohesion, ci,pf (kPa) Normal stiffness, kn,pf (MPa/m) Shear stiffness, ks,pf (MPa/m)

Value 10 0 3.3 10 1 25 0 0.67 100 10

provided in Table 2. The interaction between the geosynthetic reinforcement and working platform is modelled using the FLAC cable element grout. The cable elements have a cross-section area of Ag ¼ 0.002 m2 per metre running length of DM column and perimeter of Pg ¼ 2 m (out-of-plane perimeter of 1 m). Each cable element has a Young's modulus Eg ¼ J/Ag ¼ 2000/0.002 ¼ 1000 MPa. The cable gout properties were calculated based on the parameter values of the interface between the platform fill and DM column with shear stiffness Ks,g ¼ Pg  ks,pf ¼ 2  10 ¼ 20 MN/m/m, adhesion Cg ¼ Pg  ci,pf ¼ 2  0.67 ¼ 1.33 kPa, and friction angle fg ¼ fi,pf ¼ 25 . A summary of the cable grout properties is given in Table 3. The parameter values given in Tables 1e3 are taken from Yu et al. (2016b). For the same GRCS embankment, three different FLAC models, identified hereafter as the full-width model, the typical unit cell model, and the modified unit cell model, were built and executed. The details for each of the FLAC models are given below: (a) Full-width model (Fig. 2a): The bottom boundary was fixed in both x- and y-directions. The left side boundary (i.e., along the centre line of the embankment) was fixed only in x-direction as was the right side boundary. A total of 67069 zones (elements) were used for the soft foundation soil, DM columns, and working platform/embankment fill. The geosynthetic reinforcement was modelled using 712 cable elements. The cable node at left side (e.g., centre of the embankment) was defined using grid number with the remaining cable nodes defined using x- and y-coordinates. (b) Typical unit cell model (Fig. 2b): The bottom of the unit cell was fixed in both x- and y-directions. The unit cell left and right side boundaries were fixed only in x-direction. The unit cell has a width of 2.8 m with 0.35-m thick DM column wall on both left and right sides. The typical unit cell was modelled using 8400 zones with a total of 112 cable elements

Table 3 FLAC parameter values for cable elements (base case values from Yu et al., 2016b). Cable element parameters a

Young's modulus , Eg (MPa) Cross-sectional area, Ag (m2/m) Exposed perimeter, Pg (m) Grout stiffness, Ks,g (MN/m/m) Grout cohesion, Cg (kN/m) Grout frictional resistance, fg (degree)

Base case value (range) 1000 (250e2500) 2  103 2 20 1.33 25

Note. a Eg ¼ J/Ag where J is geosynthetic axial stiffness ranging from 500 to 5000 kN/m in this investigation.

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

for the geosynthetic reinforcement. Two cable nodes at left and right sides of the unit cell were defined using grid number and the remaining cable nodes were defined using x- and y-coordinates. (c) Modified unit cell model (Fig. 2c): The only difference from the typical unit cell was at the right side boundary of the unit cell. To allow for the lateral spreading of the embankment fill and foundation soil, a column of horizontal springs with axial stiffness was applied at the right side boundary (e.g., kfs for the foundation soil and kef for the embankment fill). The reason for using only horizontal springs on the right side of the modified unit cell is explained later. The spring stiffness values are different for the embankment fill and soft foundation soil. A foundation soil with depth-dependent modulus has depth-dependent spring stiffness. For the soft foundation soil with the Young's modulus Efs, Poisson's ratio vfs, and foundation width B ¼ 1 m, the spring stiffness (kfs) can be calculated using an empirical equation from Barden (1962):

0:65Efs  kfs ¼
B 1  v2fs

(1)

It should be noted that there are other empirical equations in the literature to calculate the spring stiffness (summaries of empirical equations have been collected by Selvadurai, 1979 and Prendergast and Gavin, 2016). These empirical equations were generally derived by comparing the maximum bending moment or displacement of an infinite (or finite) beam from elastic continuum models (e.g., an isotropic elastic half space) with that from the Winkler model. Equation (1) is selected because it is based on a finite beam model which is a better match to the DM columns in this investigation. However, for the embankment fill, the use of available empirical equations for the spring stiffness is not possible because the embankment fill has limited width (e.g., width of 16.8 m at embankment crest elevation and 36.8 m at the embankment toe elevation in this study). Furthermore, the displacement of the embankment is affected by the embankment fill itself, foundation soil modulus (Efs) and reinforcement stiffness (J). Thus the following empirical equation is proposed to calculate the embankment fill spring stiffness (kef) by comparing the numerical results from the modified unit cell model with those from the full-width model:

kef ¼ k1

Efs gef H þ q

!b1

2

1b2 3 6  A 7 41 þ @
5 gef H þ q B 0

J

(2)

Here k1 ¼ 0.11 MPa/m and b1 ¼ 0.46 and b2 ¼ 0.43 are dimensionless coefficients. The other parameter values are the embankment fill unit weight gef ¼ 20 kN/m3, embankment height H ¼ 5 m, surcharge pressure q ¼ 0 and foundation width B ¼ 1 m in this investigation. Stiffness parameter k1 and coefficients b1 and b2 are determined by fitting to back-calculated spring stiffness values for the GRCS embankment cases examined in this paper. For example, the full-width model in this investigation is used to examine the influence of only three different geosynthetic stiffness values J ¼ 500, 2000 and 5000 kN/m. However, the modified unit cell model with the spring stiffness value calculated using Equation (2) is for any geosynthetic stiffness value in the range J ¼ 500e5000 kN/m. It should be noted that Equation (2) does well for all cases examined in this paper. Using the full-width model again to recalibrate the coefficients in Equation (2) is unavoidable when using the modified unit cell model for geometries other than that examined in this paper. However, only a limited number of

5

simulations using the full-width model are required (e.g., five simulations for the embankment geometry examined in this paper). In addition to the springs used by Yu et al. (2016b), a horizontal load (Fx) was also applied to the right end of the geosynthetic reinforcement to approximate the continuous geosynthetic tensile load in the full-width model (Fig. 2c). When using the fixed xdisplacement boundary at the left side of the unit cell, the geosynthetic reinforcement tensile load on the left side of the unit cell was continuous across the cell boundary (similar to the left side symmetric boundary in Fig. 2a to model the GRCS embankment in Fig. 1). The magnitude of the applied horizontal load Fx was taken as 90% of the tensile load of the last cable element near the right side of the unit cell. Further discussion regarding Fx appears later in the paper. The applied horizontal load Fx was updated when the new tensile load of the last cable element on the right side of the modified unit cell was calculated at each step (or specified steps). All three FLAC models followed the same simulation procedure described next. The numerical modelling began by activating the foundation soil zones (elements) and applying foundation soil material properties. The initial stresses for the soft foundation (without DM column walls) were set by using K0 ¼ 1sin(ffs) ¼ 0.741. After the zones with DM column walls were replaced by the true DM column material properties, the model was then solved to reach force equilibrium. The stress distribution within the soft foundation due to installing the DM column walls was not considered. The FLAC models were run with the smallstrain mode prior to placement of the fill. To predict the displacements induced by the embankment fill only, the displacements and velocities of the numerical grid were then set to zero. The influence of strain mode on numerical outcomes was examined by selecting large- or small-strain mode during subsequent embankment construction. The construction of the GRCS embankment was modelled by activating 0.2-m thick working platform/embankment fill lifts in sequence until reaching the final embankment height of 5 m. For each working platform/embankment fill lift, the model was solved to reach force equilibrium. When the second working platform lift was activated, the cable elements were added to the model. In the current study, only idealized (fully drained) conditions were considered for the foundation soil. The methodology proposed in this paper for GRCS embankments with DM column walls (2D cases) with lateral springs can be used with other constitutive models for the soft foundation and when mechanical-hydraulic coupled analyses are performed. However, the empirical equation for the spring stiffness (Equation (2)) will be different because the Young's modulus (Efs) in this formulation does not appear in most other constitutive models including the modified Cam-Clay model.

5.2. Large- and small-strain options in FLAC The geometric nonlinearity of soils and structure elements can be modelled using large-strain mode in FLAC which is based on the Lagrangian formulation (Itasca, 2011). The other option in FLAC is to use the small-strain mode based on the Eulerian formulation. The difference between large- and small-strain modes is that the grid incremental displacements are added to the numerical grid coordinates at the end of each calculation step (or specified steps) when using large-strain model, while the calculation of stresses and displacements is based on the fixed grid representing the original geometry and material zones when using small-strain mode. For more details regarding the large- and small-strain mode options in FLAC, readers can refer to the FLAC manual (Itasca, 2011).

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

6

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

5.3. Influence of strain mode and model type on computation time

0

Embankment base settlement (cm)

2.7 cm

A desktop computer with two Intel Xeon X5670 Six-Core 2.93 GHz central processor units was used to execute all numerical simulations reported in this paper. Execution time was about 48 h to run a full-width model using large-strain mode. However, for the same GRCS embankment structure using the modified unit cell model with large-strain mode, it took the same computer about 1.5 h. For the same FLAC full-width model, small-strain mode required about 44% less computer time than large-strain mode. The typical unit cell model also required less computer time (e.g., less than half an hour) than the modified unit cell model when other conditions remained the same.

5

13.1 cm

10

15 H=5m Large-strain mode Small-strain mode

17.2 cm

6. Results

20 0

6.1. Full-width model (base case)

5

The base case embankment base vertical stresses at end of construction corresponding to H ¼ 5 m are shown in Fig. 3 using both small- and large-strain modes. The vertical stresses on the DM columns were much higher than those on the soft foundation soil because of the soil arching effect that resulted in load transfer from the embankment fill to the DM columns. For example, with largestrain mode the maximum vertical stress was about 371 kPa between x ¼ 0 and 0.7 m on the DM column (average pressure over this distance is 330 kPa which includes 100 kPa due to soil selfweight), and 30 kPa between x ¼ 0.7 and 2.45 m on the soft foundation (average of 26 kPa which is 87% of the maximum value between x ¼ 0.7 and 2.45 m). The influence of strain mode on the embankment base vertical stress varied with location, but in general vertical stresses using large-strain mode were less than those using small-strain mode. Fig. 4 shows the base (reference) case embankment base settlements calculated using both small- and large-strain modes at embankment height H ¼ 5 m. Due to the higher elastic modulus of the DM columns compared to that of the soft foundation, the embankment base just above the DM columns settled much less than that over the soft foundation between the DM columns. For example, when using large-strain mode the embankment base settlement was about 2.7 cm at x ¼ 0 and increased to about 13.1 cm at x ¼ 1.4 m. The use of small-strain mode resulted in larger

60

56.2 kN/m

Reinforcement tensile load (kN/m)

600

Large-strain mode Small-strain mode

51.7 kN/m

H=5m

500

400 Soil self-weight

Embankment base normal stress (kPa)

20

embankment base settlements between DM columns when comparing with the large-strain mode (e.g., 17.2 cm at x ¼ 1.4 m for small-strain mode versus 13.1 cm at x ¼ 1.4 m for large-strain mode). The difference in calculated embankment base settlements between large- and small-strain modes was due to the effect of geometric nonlinearity of soil and geosynthetic reinforcement for this type of soil-structure interaction problem which is more accurately computed using large-strain mode than small-strain mode. GRCS embankments take advantage of both the DM columns and geosynthetic reinforcement to transfer load vertically and laterally. Fig. 5 shows the reinforcement tensile loads calculated using both small- and large-strain modes at embankment height H ¼ 5 m using the base case parameter values. The tensioned geosynthetic reinforcement was the result of the load transfer from the embankment fill to the reinforcement. The maximum tensile load was about 56.2 kN/m using large-strain mode and about 51.7 kN/m using small-strain mode. In general, the reinforcement tensile loads using small-strain mode were less than those using

Large-strain mode Small-strain mode

200

15

Fig. 4. Embankment base settlements calculated using small- and large-strain modes with embankment fill thickness H ¼ 5 m (base case).

700

300

10

Distance from centre of embankment, x (m)

100

50

H=5m

40

30

20

10

0

0 0

5

10

15

20

Distance from centre of embankment, x (m) Fig. 3. Embankment base normal stresses calculated using small- and large-strain modes with embankment fill thickness H ¼ 5 m (base case).

0

5

10

15

20

Distance from centre of embankment, x (m) Fig. 5. Reinforcement tensile loads calculated using small- and large-strain modes with embankment fill thickness H ¼ 5 m (base case).

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

large-strain mode when other conditions were equal. Again, the differences are due to geometric nonlinearity of soil and reinforcement as noted above. Numerical results reported above for the GRCS embankment with base case parameter values show that the influence of strain mode on numerical outcomes is important. An important mechanism for GRCS embankments is the geosynthetic membrane effect. However, this mechanism can only be numerically modelled using FLAC large-strain mode (or mesh updating option in FEM software packages) (Yu and Bathurst, 2016). Hence, only numerical results from computations using large-strain mode are reported hereafter. Fig. 6 shows the vertical stresses above and below the reinforcement at embankment height H ¼ 5 m. A uniform load distribution acting on the geosynthetic reinforcement is assumed in BS8006 (2010) and a triangular load distribution is assumed in EBGEO (2011). However, the numerical results for the full-width model in this paper and from the modified unit cell model used by Yu et al. (2016b) showed that the net vertical stresses (i.e., the differences between the vertical stresses above and below the reinforcement) on the reinforcement under the embankment crest between the DM column walls were closer for load distributions having an inverted triangular shape as proposed by Van Eekelen et al. (2013).

7

reinforcement vertical displacements when other conditions were the same. The maximum reinforcement vertical displacement was about 14.6 cm for the case of foundation soil modulus of 0.5 MPa and decreased to about 7.6 cm when the foundation modulus was increased to 2.5 MPa. 6.3. Influence of geosynthetic axial stiffness Fig. 8a shows the influence of reinforcement axial stiffness on reinforcement tensile loads at embankment height H ¼ 5 m. The numerical results indicate that higher reinforcement axial stiffness resulted in greater reinforcement tensile loads when all other conditions remain unchanged. Increasing the reinforcement axial stiffness 10 fold from 500 to 5000 kN/m increased the maximum reinforcement tensile load from 26.6 to 67.2 kN/m (i.e., about 2.5 times increase). The influence of the reinforcement axial stiffness on the reinforcement vertical displacement at embankment H ¼ 5 m using large-strain mode is shown in Fig. 8b. Increasing the reinforcement axial stiffness reduced the reinforcement vertical displacements. Furthermore, the use of higher reinforcement axial stiffness also reduced the differential settlement of the reinforcement. 6.4. Influence of DM column modulus

6.2. Influence of foundation soil modulus Fig. 7a shows the influence of the soft foundation modulus on reinforcement tensile loads at embankment height H ¼ 5 m. Increasing the soft foundation modulus reduced the reinforcement tensile loads when other conditions were equal. For a foundation soil with soil modulus of 0.5 MPa, the maximum reinforcement tensile load was about 69.1 kN/m. After increasing the foundation soil modulus from 0.5 to 2.5 MPa, the maximum reinforcement tensile decreased from 69.1 to 37.2 kN/m. The influence of the soft foundation modulus on reinforcement vertical displacements at embankment height H ¼ 5 m is shown in Fig. 7b. The increase in soft foundation modulus reduced the

Fig. 9 shows the influence of DM column modulus on reinforcement tensile loads and vertical displacements at embankment height H ¼ 5 m. DM column modulus values ranging from 50 to 250 MPa have minor influence on the reinforcement tensile loads (Fig. 9a). However, the case with lower DM column modulus resulted in higher reinforcement vertical displacements because of the greater compressibility of the DM columns associated with the lower DM column modulus value (Fig. 9b). 6.5. Lateral displacements of DM column centres The foundation of the GRCS embankment examined in this

500 Vertical stress below reinforcement Vertical stress above reinforcement H=5m Large-strain mode

400

Embankment side slope

Vertical stress (kPa)

Embankment crest

Net vertical stress on reinforcement under embankment crest between DM columns

300

200

100

0 0

5

10

15

20

Distance from centre of embankment, x (m) Fig. 6. Vertical stresses above and below the reinforcement using large-strain mode with embankment fill thickness H ¼ 5 m (base case).

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

8

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

100 Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 0.5 MPa

Reinforcement tensile load (kN/m)

Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 2.5 MPa

80

H=5m Large-strain mode

69.1 kN/m 56.2 kN/m 37.2 kN/m

60

40

20

0 0

5

10

15

20

Distance from centre of embankment, x (m) (a)

Reinforcement vertical displacement (cm)

0 7.6 cm

5

10

15

H=5m Large-strain mode

12.3 cm

Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 0.5 MPa

14.6 cm

Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 2.5 MPa

20 0

5

10

15

20

Distance from centre of embankment, x (m) (b) Fig. 7. Influence of foundation soft clay modulus on (a) reinforcement tensile loads and (b) reinforcement vertical displacements (using large-strain mode with embankment fill thickness H ¼ 5 m).

paper can be divided into six different zones (Fig. 10). It should be noted that only zones 1, 2 and 3 in Fig. 10 were fully loaded by the uniform 5-m thick embankment fill. The thickness of embankment fill above zones 4, 5 and 6 was less because they are located below the embankment side slope. Fig. 11 shows the horizontal displacements at the DM column centres for embankment height H ¼ 5 m (i.e., maximum embankment fill thickness) and

computations using base case parameter values. For the DM column centre at x ¼ 2.8 m, the maximum horizontal displacement was about 2.8 cm at the DM column top and gradually decreased to zero at the DM column base matching the fixed DM column base boundary condition. The maximum horizontal displacement increased to about 6.0 cm for the DM column centre at x ¼ 5.6 m and 9.6 cm for the DM column centre at x ¼ 8.4 m. For DM columns

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

9

Reinforcement tensile load (kN/m)

100 J = 500 kN/m J = 2000 kN/m J = 5000 kN/m 80

H=5m Large-strain mode

60

40

20

0 0

5

10

15

20

Distance from centre of embankment, x (m) (a)

Reinforcement vertical displacement (cm)

0

5

10

H=5m Large-strain mode

15

J = 500 kN/m J = 2000 kN/m J = 5000 kN/m 20 0

5

10

15

20

Distance from centre of embankment, x (m) (b) Fig. 8. Influence of geosynthetic reinforcement axial stiffness on (a) reinforcement tensile loads and (b) reinforcement vertical displacements (using large-strain mode with embankment fill thickness H ¼ 5 m).

below the embankment side slope, the maximum horizontal displacement at the DM column centre was about 12.5, 13.6 and 11.6 cm at x ¼ 11.2, 14 and 16.8 m, respectively.

6.6. Comparison of results using full-width and unit cell models As noted earlier, a uniform 5-m thick embankment fill exists only above foundation zones 1, 2 and 3. These locations will have the greatest reinforcement loads and thus are important when selecting the geosynthetic material for the GRCS embankment.

Hence, only calculated reinforcement loads from full-width model zones 1, 2 and 3 are compared with results using typical and modified unit cell models. It should be noted that to simplify interpretation of model results, possible bending failure of the DM columns was not considered in this paper by making the DM column material elastic with no tensile or compressive yield limit. Fig. 12 shows the calculation of net horizontal displacements of DM column centres from the full-width model. To model the GRCS embankments using the modified unit cell (e.g., Fig. 2c) with fixed left side boundary, the net (or relative) horizontal displacements of

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

10

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

100 Ec = 50 MPa

Reinforcement tensile load (kN/m)

Ec = 150 MPa Ec = 250 MPa

80

H=5m Large-strain mode 60

40

20

0 0

5

10

15

20

Distance from centre of embankment, x (m) (a)

Reinforcement vertical displacement (cm)

0

5

10

H=5m Large-strain mode

15

Ec = 50 MPa Ec = 150 MPa Ec = 250 MPa 20 0

5

10

15

20

Distance from centre of embankment, x (m) (b) Fig. 9. Influence of DM column modulus on (a) reinforcement tensile loads and (b) reinforcement vertical displacements (using large-strain mode with embankment fill thickness H ¼ 5 m).

the DM column centres from full-width model zones 1, 2 and 3 are needed. It is these net (or relative) horizontal displacements that influence the reinforcement tensile loads. Fig. 13 shows the net horizontal displacements of DM column centres from full-width model zones 1, 2 and 3 at embankment height H ¼ 5 m using base case parameter values. The maximum net horizontal displacement was about 2.8, 3.3 and 3.6 cm for zones 1, 2 and 3, respectively with an average value of 3.2 cm which is close to the net horizontal displacement of 3.3 from zone 2. The calculation of the spring stiffness value (kef) for the embankment fill is given

below. For example, when the geosynthetic reinforcement stiffness value is set to J ¼ 500 kN/m, the spring stiffness value (kef,J¼500 kN/m) for the modified unit cell model was selected by the trial-and-error method until the maximum horizontal displacement of the DM column centre on the right side of the modified unit cell agrees with the averaged maximum net horizontal displacement from the full-width model. Using the above procedure, the spring stiffness values kef,J¼500 kN/m, kef,J¼2000 kN/m and kef,J¼5000 kN/m were selected for the modified unit cell model with the reinforcement stiffness values J ¼ 500, 2000 and 5000 kN/m, respectively. Together with

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

11

5.6 m

x = 2.8 m

16.8 m

14 m

11.2 m

8.4 m

Full-width model zone 6

Full-width model zone 5

Full-width model zone 4

Full-width model zone 2

5

Full-width model zone 3

10

Full-width model zone 1

Height above DM column base (m)

15

0 0

5

10

15

20

Distance from centre of embankment, x (m)

Height above DM column base (m)

Fig. 10. Location of DM columns and six different foundation zones in the full-width model.

10

10 (a)

9

9 8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3 x = 2.8 m H=5m Large-strain mode

2 1 0

3 x = 5.6 m

2

10

15

1

20

10

1 0 0

5

10

15

20

10 (d)

9

0

(e)

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3 x = 11.2 m

5

10

15

20

x = 16.8 m

1

0 0

20

(f)

2

1

0

15

3 x = 14 m

2

1

10

9

8

2

5

10

9

3

x = 8.4 m

2

0 5

(c)

9

8

0

Height above DM column base (m)

10 (b)

0 0

5

10

15

20

0

5

10

15

20

DM column wall centre horizontal displacement, sh,x (cm) Fig. 11. Horizontal displacements of DM column centres using the full-width model with embankment fill thicknesses H ¼ 5 m and using large-strain mode (base case).

spring stiffness values for the modified unit cell model with different foundation soil moduli, the coefficients (k1 ¼ 0.11 MPa/m,

b1 ¼ 0.46 and b2 ¼ 0.43) of the empirical Equation (2) are obtained using nonlinear regression.

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

12

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

sh,r

δh = sh,r - sh,l

tensile loads using the full-width model were much greater than those using the typical unit cell, but were in practical good agreement with those using the modified unit cell model. Further comparisons between results using the full-width and unit cell models are shown in Fig. 15 for different soft foundation moduli and Fig. 16 for different reinforcement stiffness values. The conclusions made previously regarding the relative performance of both typical and modified unit cell models compared to the fullwidth GRCS embankment model also apply using the other parameter values and cases identified in these figures and in Tables 1 and 3. Thus, it may be concluded that using the typical unit cell to model GRCS embankments will result in non-conservative estimates of reinforcement tensile loads.

10 m

sh,l

2.8 m (b)

2.8 m (a)

6.7. Stress reduction ratio

Fig. 12. Calculation of net horizontal displacements of DM column centres from the full-width model. Notes: sh.l and sh.r are the horizontal displacements at the left and right sides of each zone in the full-width model shown in Fig. 10; dh is the net (or relative) horizontal displacement between the left and right sides of each zone in the full-width model.

The soil arching effect and reinforcement membrane effect both reduce the vertical stress acting on the soft foundation surface under the embankment fill self-weight and additional surcharge pressure. In this paper the magnitude of the stress reduction due to these mechanisms is quantified using the stress reduction ratio (SRR; Low et al., 1994) which is defined as the ratio of the equivalent uniform vertical stress acting on the soft foundation surface between the DM columns to the embankment fill self-weight at the foundation surface (surcharge pressure was not considered in the examples reported in this paper). A closed-form solution for estimating the SRR for two-dimensional GRCS embankments has been proposed by Low et al. (1994) and is briefly summarized below. The vertical stress (ss) acting on the top surface of the geosynthetic reinforcement midway between the DM column walls is calculated as (Fig. 17):

Height above DM column base (m)

Fig. 14a shows the reinforcement tensile loads from the fullwidth model zones 1, 2 and 3 at embankment height H ¼ 5 m with base case parameter values. For the full-width model zone 2, the reinforcement tensile load was 43.6 kN/m at left side of the zone and about 45.7 kN/m at the right side of the zone. For fullwidth model zones 1, 2 and 3, the difference in reinforcement tensile load was about 7% at the left side of the zone and 5% at the right side of the zone, and the maximum difference in reinforcement tensile load was about 10%. Therefore, to approximate the continuous tensile load in the full-width model, the applied horizontal load Fx on the right side of the modified unit cell (Fig. 2b) was 90% of the tensile load of the last cable element on the right side of the modified unit cell. The selected magnitude of the horizontal load applied at the right side of the modified unit cell was based on the results of numerical experiments and worked well for all cases using the modified unit cell model reported below. Fig. 14b shows the reinforcement tensile loads from the fullwidth model zone 2, modified unit cell model and typical unit cell model at embankment height H ¼ 5 m with base case parameter values. The numerical results indicate that the reinforcement

10



ss ¼

gef ðs  bÞ Kp  1   2 Kp  2

9

8

7

7

6

6

6

5

5

5

2 1

4

4

x = 2.8 to 5.6 m Zone 2

3

0 1

2

3

4

5

6

gef s 2

 1þ

1 Kp  2



x = 5.6 to 8.4 m Zone 3

3

2

2

1

1

0 0

gef H 

(c)

9

8

x = 0 to 2.8 m Zone 1 Full-width model H=5m Large-strain mode

Kp 1 

10 (b)

7

3

sb s

where gef and H are the unit weight and thickness of the embankment fill, respectively; s is the spacing of DM column walls; b is the thickness of the DM column walls; Kp is the passive earth pressure coefficient and is calculated using the friction angle (fef) of the embankment fill as:

8

4

 þ

(3)

10 (a)

9



0 0

1

2

3

4

5

6

0

1

2

3

4

5

6

Net horizontal displacement, δh (cm) Fig. 13. Net horizontal displacement (dh) of DM column centres from full-width model zones 1, 2 and 3 with embankment fill thicknesses H ¼ 5 m and using large-strain mode (base case).

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

13

100

Reinforcement tensile load (kN/m)

Full-width model zone 1 Full-width model zone 2 Full-width model zone 3 80 H=5m Large-strain mode Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa J = 2000 kN/m Ec = 150 MPa

60

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone, x (m) (a) 100

Reinforcement tensile load (kN/m)

Full-width model zone 2 Modified unit cell model Typical unit cell model 80

H=5m Large-strain mode Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa J = 2000 kN/m Ec = 150 MPa

60

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone or unit cell, x (m) (b) Fig. 14. Reinforcement tensile loads with large-strain mode and embankment fill thickness H ¼ 5 m for base case from (a) the full-width model (zones 1, 2 and 3) and (b) unit cell models and full-width model zone 2.

T ¼ Jε
 1 þ sin fef
 Kp ¼ 1  sin fef

(4)

Low et al. (1994) assumed that the deformed geosynthetic reinforcement is a circular arc with radius (R), subtended angle (2q), and maximum vertical displacement (t) midway between the DM column walls (Fig. 17). The assumed uniform geosynthetic axial tensile load (T) is calculated by:

(5)

where J is the geosynthetic reinforcement stiffness; and ε is the geosynthetic tensile strain. Based on the geometry shown in Fig. 17, the geosynthetic tensile strain can be calculated as:

ε¼

q  sin ðqÞ sin ðqÞ

(6)

where the calculation of one half of the subtended angle is:

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

14

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

100

Reinforcement tensile load (kN/m)

Full-width model zone 2 Modified unit cell model Typical unit cell model 80

H=5m Large-strain mode Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 0.5 MPa J = 2000 kN/m Ec = 150 MPa

60

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone or unit cell, x (m) (a) 100

Reinforcement tensile load (kN/m)

Full-width model zone 2 Modified unit cell model Typical unit cell model 80

H=5m Large-strain mode Efs,1 = Efs,2 = Efs,3 = Efs,4 = Efs,5 = 2.5 MPa J = 2000 kN/m Ec = 150 MPa

60

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone or unit cell, x (m) (b) Fig. 15. Reinforcement tensile loads with large-strain mode and embankment fill thickness H ¼ 5 m from unit cell models and the full-width model zone 2 with (a) foundation soft soil moduli Efs,1 ¼ Efs,2 ¼ Efs,3 ¼ Efs,4 ¼ Efs,5 ¼ 0.5 MPa and (b) Efs,1 ¼ Efs,2 ¼ Efs,3 ¼ Efs,4 ¼ Efs,5 ¼ 2.5 MPa.

sin ðqÞ ¼

4t sb



1þ4

2

(7)

modulus of the soft foundation, respectively. For a layered soft foundation, the equivalent elastic modulus can be calculated by:

t sb

Pn

The magnitude of the geosynthetic axial tensile load must also satisfy vertical force equilibrium, hence:

  tMfs T ¼ R ss  D

(8)

where D and Mfs are the total thickness and equivalent elastic

i¼1 di di ð1þvfs;i Þð12vfs;i Þ i¼1 Efs;i ð1vfs;i Þ

Mfs ¼ P n

(9)

where di, Efs,i and vfs,i (i ¼ 1,2, …n) are the thickness, Young's modulus, and Poisson's ratio of each soft foundation layer, respectively. The radius of the circular arc is calculated as:

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

15

100

Reinforcement tensile load (kN/m)

Full-width model zone 2 Modified unit cell model Typical unit cell model 80

60

H=5m Large-strain mode Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa J = 500 kN/m Ec = 150 MPa

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone or unit cell, x (m) (a) 100

Reinforcement tensile load (kN/m)

Full-width model zone 2 Modified unit cell model Typical unit cell model 80

60

H=5m Large-strain mode Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa J = 5000 kN/m Ec = 150 MPa

40

20

0 0.0

0.5

1.0

1.5

2.0

2.5

2.8

Distance from left side of zone or unit cell, x (m) (b) Fig. 16. Reinforcement tensile loads with large-strain mode and embankment fill thickness H ¼ 5 m from unit cell models and the full-width model zone 2 with (a) geosynthetic axial stiffness J ¼ 500 kN/m and (b) J ¼ 5000 kN/m.

calculated using:

R¼

sb 2 sin ðqÞ

(10)

Iteration is required to minimize the difference between the geosynthetic reinforcement tensile loads calculated using Equations (5) and (8) by adjusting the maximum displacement (t) in Equation (7). Once the maximum displacement is found, the calculation of the other parameters in Equations (6) and (7) is straight forward. The SRR for the GRCS embankments from Low et al. (1994) is

SRR ¼

ass 2J½q  sinðqÞ  gef H ðs  bÞgef H

(11)

where a is an empirical coefficient to calculate the average vertical stress on the geosynthetic reinforcement top surface. The following constraints were considered by Low et al. (1994): 3  Kp  7.5, 0.5  H/s  10, 0.05  b/s  0.3, and 0.8  a  1.0. Fig. 18 shows the calculated SRR values using large-strain mode and modified unit cell model, typical unit cell model, and full-width

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

16

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

width model. Also shown in Fig. 18 are the SRR values using the closed-form solution from Low et al. (1994) for the base case with an equivalent elastic modulus Mfs ¼ 1.31 MPa and thickness D ¼ 10 m of the layered soft foundation. The upper and lower bound values for the empirical coefficient (a ¼ 0.8 and 1.0) in Equation (11) were considered in this investigation based on Low et al. (1994). As shown in Fig. 18, the SRR values from the closedform solution using a ¼ 0.8 and 1.0 bracket the numerical predictions from the full-width model, modified unit cell model, and typical unit cell model. However the predicted geosynthetic reinforcement tensile load when H ¼ 5 m using the closed-form solution (for both a ¼ 0.8 and 1.0) was 67.8 kN/m which is higher than those from the full-width model and modified unit cell model, and is much higher than that from the typical unit cell model (Fig. 14).

T

2θ t

D

H

Embankment fill R

T

Soft clay

7. Conclusions

b s Fig. 17. Schematic showing 2D GRCS embankment subject to soil arching and geosynthetic membrane effects (modified from Low et al., 1994).

model zones 1, 2 and 3. The increase in embankment fill height reduced SRR when other conditions were equal. The calculated SRR values using the modified unit cell were slightly higher than those using the typical unit cell model. However the predictions of SRR using both typical and modified unit cell models were judged to be in good agreement with values at zones 1, 2 and 3 using the full-

Geosynthetic-reinforced and column-supported (GRCS) embankments have been proven to be an effective technique to increase the bearing capacity of soft foundations, reduce differential settlements that may affect adjacent structures when widening existing road and railway embankments, and limit the lateral spreading of the embankment fill. Bhasi and Rajagopal (2015a) and Khabbazian et al. (2015) investigated the performance of GRCS embankments using numerical full-width and typical unit cell models and found that reinforcement tensile loads using the typical unit cell were much lower than those using the full-width model. This discrepancy was ascribed to the inability of the typical unit cell model to capture the lateral spreading of embankment fill and foundation soil due to horizontally fixed side boundaries. To overcome this limitation, Yu et al. (2016b) proposed a modified unit cell approach to simulate the lateral spreading of embankment fill and foundation soil using lateral springs. This paper compares numerical results using both typical and modified unit cell models with results from a matching full-width GRCS embankment model and a closed-form solution (Low et al., 1994). The influence of choice of

Stress reduction ratio, SSR (-)

1.0 Modified unit cell model Typical unit cell model Full-width model zone 1 Full-width model zone 2 Full-width model zone 3 Closed-form solution (α = 1.0) Closed-form solution (α = 0.8)

0.8

0.6

0.4

Closed-form solution from Low et al. (1994) Large-strain mode Efs,1 = Efs,2/2 = Efs,3/3 = Efs,4/4 = Efs,5/5 = 0.5 MPa J = 2000 kN/m Ec = 150 MPa

0.2

0.0 0

1

2

3

4

5

Embankment fill height (m) Fig. 18. Stress reduction ratio (SSR) using large-strain mode from modified unit cell model, typical unit cell model, full-width model zones 1, 2 and 3, and closed-form solution (base case).

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

small- and large-strain mode options using FLAC is also investigated. The main conclusions from this study are summarized below:  Numerical results using both small- and large-strain modes for the same GRCS embankment structure show that the foundation surface settlement and reinforcement tensile load are sensitive to the choice of the large- or small-strain mode option in FLAC. The basic working mechanisms for GRCS embankments are the soil arching effect and geosynthetic membrane effect. The geosynthetic membrane effect can only be numerically modelled when the geometric nonlinearity is considered in the simulation. Thus, numerical modelling of GRCS embankments using FLAC should be carried out using the large-strain mode option (or mesh updating option in FEM software packages).  Increasing the soft foundation modulus reduces the reinforcement tensile loads and reinforcement vertical displacements when other conditions are equal. The use of greater reinforcement tensile stiffness results in greater reinforcement tensile loads but less reinforcement vertical displacements. Elastic modulus values for the deep mixing (DM) column material in the range of 50e250 MPa have negligible influence on the reinforcement tensile loads. However, higher DM column modulus values result in lower reinforcement vertical displacements when other conditions are the same.  Numerical results for zones 1, 2 and 3 below the full-with model were similar with respect to the distribution and magnitude of the reinforcement tensile load which may be expected due to the same fill thickness at these locations. Reinforcement loads generated below the embankment side slopes are less and therefore do not control the selection of the geosynthetic stiffness (and strength) during design.  For the modified unit cell model, the spring stiffness for the embankment fill is dependent on the embankment fill itself, soft foundation soil modulus and reinforcement tensile stiffness. An empirical equation is presented to calculate the spring stiffness for the embankment fill based on the numerical results reported in this paper.  The reinforcement tensile loads using the modified unit cell model generally agree well with those at locations below the full height of the embankment fill while those using the typical unit cell model are poorer.  Assuming that computed reinforcement tensile loads using the full-width GRCS model are representative of the field case, then reinforcement tensile loads computed using the typical unit cell approach are non-conservative for design.  The stress reduction ratio values from the closed-form solution (Low et al., 1994) with the empirical coefficient values (a ¼ 0.8 and 1.0) bound those from the full-width model, modified unit cell model, and typical unit cell model. However the full-width model and modified unit cell model perform better than the typical unit cell model in predicting the reinforcement tensile load when compared to the closed-form solution.

8. Discussion and practical implications This paper investigates the conditions required to achieve a good match between performance predictions using unit cell representations and full-width 2D numerical embankments, and the conditions when unit cells may be practical to reduce computation times. The good agreement between reinforcement tensile loads using the modified unit cell approach proposed by Yu et al. (2016b) and an example matching full-width GRCS model is demonstrated in the current study. This example shows that once the modified

17

unit cell has been calibrated against the results of a limited number of 2D full-width GRCS numerical models (five in this paper) then the unit cell can be used to investigate the same embankment with different geosynthetic reinforcement stiffness values and different foundation soil elastic modulus distributions. For this scenario, significant reductions in computation time are possible (e.g., up to 95% in this study). However, from a practical point of view the benefit of the more accurate modified unit cell proposed by the writers to reduce computation times is limited to an embankment with a fixed geometry. Hence, it appears based on the current study and related prior work that the unit cell approach has the disadvantage of being inaccurate when simple models are used (i.e., typical unit cells with fixed side boundaries) but time-intensive for calibration when an accurate model is used (i.e., modified unit cell in this study). It can be argued that in the near term, full-width 2D numerical models of GRCS embankments will be the method of choice for parametric studies used in design despite long computation times. Acknowledgement The work reported in this paper was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC; Grant number: RGPAS-446148-2013). Notation

Basic SI units are given in parentheses b thickness of the DM column walls (m) D total thickness of the soft foundation (m) di thickness of each soft foundation layer (i ¼ 1, 2, …, n) (m) Efs elastic modulus of foundation soil (Pa) vfs Poisson's ratio of foundation soil (dimensionless) K0 coefficient of lateral earth pressure at rest (dimensionless) Kp coefficient of lateral passive earth pressure (dimensionless) k1 empirical coefficient for spring stiffness of embankment fill (Pa/m) kef spring stiffness for embankment fill of modified unit cell (Pa/m) kfs spring stiffness for foundation soil of modified unit cell (Pa/m) H height (thickness) of embankment fill (m) J axial stiffness of geosynthetic (N/m) q surcharge pressure (Pa) R radius of a circular arc of deformed geosynthetic reinforcement (m) s spacing of DM column walls (m) sh.l horizontal displacement at the left side of each zone in the full-width model (m) sh.r horizontal displacement at the right side of each zone in the full-width model (m) t maximum vertical displacement of geosynthetic reinforcement midway between the DM column walls (m) T uniform geosynthetic axial tensile load (N/m) x, y Cartesian co-ordinates with origin at base of unit cell model, or at bottom and centre of full-width model (m) a empirical coefficient to calculate the average vertical stress on the soft foundation surface (dimensionless) b1 empirical coefficient for spring stiffness of embankment fill (dimensionless)

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002

18

Y. Yu, R.J. Bathurst / Geotextiles and Geomembranes xxx (2017) 1e18

b2

empirical coefficient for spring stiffness of embankment fill (dimensionless) ε geosynthetic tensile strain (dimensionless) ss vertical stress acting on the top surface of the geosynthetic reinforcement midway between the DM column walls (Pa) q half of the subtended angle of a circular arc of deformed geosynthetic reinforcement (degree) dh net (or relative) horizontal displacement between the left and right sides of each zone in the full-width model (m) gef unit weight of embankment fill (N/m3) Other notations appear in the tables. References Ali, K., Shahu, J.T., Sharma, K.G., 2014. Model tests on single and groups of stone columns with different geosynthetic reinforcement arrangement. Geosynth. Int. 21 (2), 103e118. Barden, L., 1962. Distribution of contact pressure under foundations. Geotechnique 12 (3), 181e198. BS8006, 2010. Code of Practice for Strengthened/Reinforced Soil and Other Fills. British Standards Institution (BSI), Milton Keynes, UK. Bruce, M.E.C., Berg, R.R., Collin, J.G., Filz, G.M., Terashi, M., Yang, D.S., 2013. Federal Highway Administration Design Manual: Deep Mixing for Embankment and Foundation Support. U.S. Department of Transportation, USA, 248 pp. Borges, J.L., Gonçalves, M.S., 2016. Jet-grout column-reinforced soft soils incorporating multilayer geosynthetic-reinforced platforms. Soils Found. 56 (1), 57e72. Borges, J.L., Marques, D.O., 2011. Geosynthetic-reinforced and jet grout columnsupported embankments on soft soils: numerical analysis and parametric study. Comput. Geotech. 38, 883e896. Bhasi, A., Rajagopal, K., 2015a. Geosynthetic-reinforced piled embankments: comparison of numerical and analytical methods. ASCE Int. J. Geomech. 15 (5), 04014074. Bhasi, A., Rajagopal, K., 2015b. Numerical study of basal reinforced embankments supported on floating/end bearing piles considering pileesoil interaction. Geotext. Geomembr. 43 (6), 524e536. Briançon, L., Simon, B., 2012. Performance of pile-supported embankment over soft soil: full-scale experiment. ASCE J. Geotech. Geoenviron. Eng. 138 (4), 551e561. Blanc, M., Thorel, L., Girout, R., Almeida, M., 2014. Geosynthetic reinforcement of a granular load transfer platform above rigid inclusions: comparison between centrifuge testing and analytical modelling. Geosynth. Int. 21 (1), 37e52. Budhu, M., 2010. Soil Mechanics and Foundations, third ed. John Wiley & Sons, Inc. EBGEO, 2011. Recommendations for Design and Analysis of Earth Structures Using Geosynthetic Reinforcements e EBGEO. German Geotechnical Society. ISBN: 978-3-433-02983-1, and Digital Version in English, ISBN: 978-3-433-60093-1. Forsman, J., Honkala, A., Smura, M., 1999. Hertsby Case: a Column Stabilized and Geotextile Reinforced Road Embankment on Soft Subsoil. Dry Mix Method for Deep Soil Stabilization. Balkema, Rotterdam, The Netherlands, pp. 263e268. Gu, M., Zhao, M., Zhang, L., Han, J., 2016. Effects of geogrid encasement on lateral and vertical deformations of stone columns in model tests. Geosynth. Int. 23 (2), 129e139. Han, J., Gabr, M.A., 2002. Numerical analysis of geosynthetic reinforced and pilesupported earth platforms over soft soil. ASCE J. Geotech. Geoenviron. Eng. 128 (1), 44e53. Han, J., Oztoprak, S., Parsons, R.L., Huang, J., 2007. Numerical analysis of foundation columns to support widening of embankments. Comput. Geotech. 34 (6), 435e448. Hewlett, W.J., Randolph, M.F., 1988. Analysis of piled embankments. Ground Eng. 21 (3), 12e18. Hosseinpour, I., Almeida, M.S.S., Riccio, M., 2015. Full-scale load test and finiteelement analysis of soft ground improved by geotextile-encased granular columns. Geosynth. Int. 22 (6), 428e438. Huang, J., Han, J., 2009. 3D coupled mechanical and hydraulic modeling of a geosynthetic-reinforced deep mixed column-supported embankment. Geotext. Geomembr. 27, 272e280. Huang, J., Han, J., 2010. Two-dimensional parametric study of geosyntheticreinforced column-supported embankments by coupled hydraulic and mechanical modeling. Comput. Geotech. 37, 638e648. Huang, J., Han, J., Oztoprak, S., 2009. Coupled mechanical and hydraulic modeling of geosynthetic-reinforced column-supported embankments. ASCE J. Geotech. Geoenviron. Eng. 135, 1011e1021. Itasca, 2011. FLAC: Fast Lagrangian Analysis of Continua. Version 7.0 [computer program]. Itasca Consulting Group, Inc., Minneapolis, MN. €bel, C., Alexiew, D., Heitz, C., 2004. German recommendations for Kempfert, H.G., Go the reinforced embankments on pile-similar elements. In: Proceedings of the

3rd European Conference on Geosynthetics, Munich, Germany, pp. 279e284. Khabbazian, M., Kaliakin, V.N., Meehan, C.L., 2015. Column supported embankments with geosynthetic encased columns: validity of the unit cell concept. Geotech. Geol. Eng. 33, 425e442. Lai, Y.P., Bergado, D.T., Lorenzo, G.A., Duangchan, T., 2006. Full-scale reinforced embankment on deep jet mixing improved ground. Ground Improv. 10 (4), 153e164. Liu, H.L., Ng, C.W.W., Fei, K., 2007. Performance of geogrid-reinforced and pilesupported highway embankment over soft clay: case study. ASCE J. Geotech. Geoenviron. Eng. 133, 1483e1493. Liu, K.W., Rowe, R.K., 2015. Numerical study of the effects of geosynthetic reinforcement viscosity on behaviour of embankments supported by deep-mixing method columns. Geotext. Geomembr. 43 (6), 567e578. Liu, K.W., Rowe, R.K., 2016. Performance of reinforced, DMM column-supported embankment considering reinforcement viscosity and subsoil's decreasing hydraulic conductivity. Comput. Geotech. 71, 147e158. Love, J., Milligan, G., 2003. Design methods for basally reinforced pile-supported embankments over soft ground. Ground Eng. 39e43. Low, B.K., Tang, S.K., Choa, V., 1994. Arching in piled embankments. ASCE J. Geotech. Eng. 120 (11), 1917e1938. Nunez, M.A., Briançon, L., Dias, D., 2013. Analyses of a pile-supported embankment over soft clay: full-scale experiment, analytical and numerical approaches. Eng. Geol. 153, 53e67. Prendergast, L.J., Gavin, K., 2016. A comparison of initial stiffness formulations for small-strain soil-pile dynamic Winkler modelling. Soil Dyn. Earthq. Eng. 81, 27e41. Rowe, R.K., Liu, K.W., 2015. 3D finite element modeling of a full-scale geosyntheticreinforced, pile-supported embankment. Can. Geotech. J. 52 (12), 2041e2054. Selvadurai, A.P.S., 1979. Elastic Analysis of Soil-foundation Interaction. Elsevier, Amsterdam. Smith, M., Filz, G., 2007. Axisymmetric numerical modeling of a unit cell in geosynthetic-reinforced, column-supported embankments. Geosynth. Int. 14 (1), 13e22. Van Eekelen, S.J.M., Bezuijen, A., Van Tol, A.F., 2011. Analysis and modification of the British Standard BS8006 for the design of piled embankments. Geotext. Geomembr. 29 (3), 345e359. Van Eekelen, S.J.M., Bezuijen, A., Van Tol, A.F., 2013. An analytical model for arching in piled embankments. Geotext. Geomembr. 39, 78e102. Van Eekelen, S.J.M., Bezuijen, A., Van Tol, A.F., 2015. Validation of analytical models for the design of basal reinforced piled embankments. Geotext. Geomembr. 43 (1), 56e81. Xing, H., Zhang, Z., Liu, H., Wei, H., 2014. Large-scale tests of pile-supported earth platform with and without geogrid. Geotext. Geomembr. 42 (6), 586e598. Xu, C., Song, S., Han, J., 2016. Scaled model tests on influence factors of full geosynthetic-reinforced pile-supported embankments. Geosynth. Int. 23 (2), 140e153. Yapage, N.N.S., Liyanapathirana, D.S., 2014. A parametric study of geosyntheticreinforced column-supported embankments. Geosynth. Int. 21 (3), 213e232. Yoo, C., 2010. Performance of geosynthetic-encased stone columns in embankment construction: numerical investigation. ASCE J. Geotech. Geoenviron. Eng. 136 (8), 1148e1160. Yu, Y., Bathurst, R.J., Miyata, Y., 2015a. Numerical analysis of a mechanically stabilized earth wall reinforced with steel strips. Soils Found. 55 (3), 536e547. Yu, Y., Damians, I.P., Bathurst, R.J., 2015b. Influence of choice of FLAC and PLAXIS interface models on reinforced soil-structure interactions. Comput. Geotech. 65, 164e174. Yu, Y., Bathurst, R., Allen, T., 2016a. Numerical modeling of the SR-18 geogrid reinforced modular block retaining walls. ASCE J. Geotech. Geoenviron. Eng. 142 (5), 04016003. Yu, Y., Bathurst, R.J., 2016. Influence of selection of soil and interface properties on numerical results of two soil-geosynthetic interaction problems. ASCE Int. J. Geomech. http://dx.doi.org/10.1061/(ASCE)GM.1943-5622.0000847 (in press). Yu, Y., Bathurst, R.J., Damians, I.P., 2016b. Modified unit cell approach for modelling geosynthetic-reinforced column-supported embankments. Geotext. Geomembr. 44 (3), 332e343. Yu, Y., Bathurst, R.J., Allen, T.M., Nelson, R., 2016c. Physical and numerical modelling of a geogrid reinforced incremental concrete panel retaining wall. Can. Geotech. J. 53 (12), 1883e1901. Zhuang, Y., Ellis, E.A., Yu, H.S., 2010. Plane strain FE analysis of arching in a piled embankement. Ground Improv. 163 (4), 207e215. Zhuang, Y., Wang, K.Y., Liu, H.L., 2014. A simplified model to analyze the reinforced piled embankments. Geotext. Geomembr. 42 (2), 154e165. Zhang, J., Zheng, J.J., Chen, B.G., Yin, J.H., 2013. Coupled mechanical and hydraulic modeling of a geosynthetic-reinforced and pile-supported embankment. Comput. Geotech. 52, 28e37. Zhang, C.L., Jiang, G.L., Liu, X.F., Buzzi, O., 2016. Arching in geogrid-reinforced pilesupported embankments over silty clay of medium compressibility: field data and analytical solution. Comput. Geotech. 77, 11e25.

Please cite this article in press as: Yu, Y., Bathurst, R.J., Modelling of geosynthetic-reinforced column-supported embankments using 2D fullwidth model and modified unit cell approach, Geotextiles and Geomembranes (2017), http://dx.doi.org/10.1016/j.geotexmem.2017.01.002