An analytical and numerical study on influence of the load transfer shallow mixing layer on stress-induced and settlement of ground improved by CDM columns

In the literature, there are a few analytical methods for evaluating stress induced on the CDM column heads and most of them involve the use of geosynthetics embedded layers, except the method of ALiCC. However, the ALiCC method still has some limitations in actual designs. This paper presents numerical analysis (by using Plaxis software) and analytical analysis (by using the ALiCC method) on the influence of thickness and stiffness of the SM layer (without the use of geosynthetics embedded layers) on stress induced on column heads and settlement of the improved ground. A typical parametric case and actual project case were taken into analyses. Analysis results from the examined cases indicate that, as expected, when the thickness or stiffness of the SM layer increases, the settlement of the ground decreases and the stress induced on the column head increases. An important finding from this study is that the maximum stress induced in the CDM column is typically not on the head of the columns but at the middle depths where soil layers are softer than the SM layer and the bearing layer at the column toes.


Introduction
In recent years, Cement Deep Mixing (CDM) method has been commonly used as an effective method to improve soft ground throughout the world [1,2].This method is used to reduce settlement, high water content, vibration, as well as to increase the stiffness of soil and the stability of an embankment.The method is also used to mitigate liquefaction and provide lateral support of the ground.Under the effect of embankment loads, CDM columns tend to collapse from external failure or internal failure [3].However, they do not fail simultaneously but instead one by one at different times [3].As it works individually, the strength and stiffness of all individual columns cannot be mobilized simultaneously.
To improve the workability of the improved ground, a load transfer layer, which can be a shallow mixing (SM) layer or a combined layer of aggregates and geogrid, is often designed and constructed on top of the CDM columns to effectively transfer the load to the improved ground.By using a load transfer layer on top of the columns, they can work simultaneously, thereby increasing the stability of the column system and reducing the differential settlement of the columns and the surrounding soil [4].However, with the application of the load transfer layer, which is often stiffer than the soil beneath, the CDM columns carry more stress than the surrounding soil.
Bearing capacity of the individual columns is an important issue in the design, especially when a load transfer layer is included.If the stress induced in the column exceeds the compressive strength of 1289 (2023) 012104 IOP Publishing doi:10.1088/1757-899X/1289/1/012104 2 the column material, the column will fail locally, even if the ground as a whole remains stable.In such a case, the stress induced in the column should be strictly controlled.In the literature, several methods have been proposed to estimate the stress induced on the column head (ALiCC method [5], British Standard BS 8006 [6], German method (EBGEO) [7], Guido method [8], Low method [9], etc.).Regarding the stress induced on the columns under SM layer, ALiCC is the only method that takes into account the value.However, ALiCC method still has some limitations, such as the method does not take into account the effect of the stiffness of the SM layer as well as the ratio of the stiffness of column to the stiffness of soil (E c /E s ) to calculate the stress induced in column and soil.Moreover, this method only estimates stress induced on the top of the column and surrounding soil.It is therefore very necessary to study on the effect of the SM layer on settlement of the improved ground and stress induced in the columns by using both analytical and numerical analyses, in which several aspects can be considered from the numerical analysis.
This paper presents analytical and numerical analyses on the influence of the SM layer (load transfer layer) on the stress induced on head of as well as along the CDM columns and settlement of the improved ground under one-dimensional loading conditions.The influence of variation of thickness and stiffness of the SM layer as well as of the improvement area ratio on the stress induced and settlement was numerically investigated by using PLAXIS 2D (V21.01)carried out for parametric study cases and an actual project case in Vietnam.Findings from this study would help geotechnical engineers have a better understanding on stress induced in the CDM columns under an SM layer.

Analytical and numerical theories
2.1.Analytical theories 2.1.1.Load transfer Mechanisms.In the literature, there are two extreme load conditions: equal stress and equal strain.The equal stress condition refers to the case when the improved soil layer is under ideally flexible load (Figure 1(a)), and the equal strain condition refers to the case under ideally rigid load (Figure 1(b)).The stress concentration ratio (n) is defined as the ratio of the stress on the column (q c ) to that on the soil (q s ).In the equal stress condition, n = q c /q s = 1.0 but the soil settles more than the columns (S s > S c ), whereas in the equal strain condition, n = q c /q s > 1.0, but the columns and the soil settle equally (S s = S c ) [10].When the improved layer is under 1D equal strain condition, the strain can be calculated as follows [10]: where  z is vertical strain at a depth of z, M c and M s are constrained moduli of the column and the soil, respectively.
3 2.1.2.Theory of induced stress according to the ALiCC method.The ALiCC method is a method of using CDM columns combined with a shallow mixed reinforcement layer on top of the columns [5].This method has been proposed by the Public Works Research Institute of Japan.The structure of the load transfer layer from the ALiCC method is shown in Figure 2. The stress applied to the CDM column head (q c ) and the stress applied on the soil (q s ) between the columns are given as follows [5]: where  is the unit weight of the embankment soil; d c is the diameter of the CDM column; V c is the embankment volume acting on a single column; and V s is the embankment volume acting on the soil zone between the columns, and they are estimated as follows: where s is the distance between two columns; θ is the arch angle value; H e is thickness of embankment; H sm is the thickness of the SM layer.

Theory of settlement evaluation.
When CDM columns are used to improve soft ground under applied load (L, B, >> H CDM ), the settlement of the improved soil layers as well as the soil layers below improved zone can be calculated using 1D theory.The calculation theory of 1D is applied if one of the following conditions is satisfied [11]: (1) the width of the loading area is greater than 4 times the thickness of the compressive soil layer (B > 4H c ); (2) the depth to the top of the compressive soil layer is greater than 2 times the width of the loaded area (D > 2B); and (3) the compressive layer lies between two stiffer soil layers.Under 1D conditions, the total settlement of the ground (S t ) is calculated by [2].
S S S  (7) where S comp is the settlement of the composite (improved) layers; S untr is the settlement of the untreated soil layer (under the improved layers).The settlement of the improved layers is calculated by the following formula [12,13].
where q is the surcharge load applied to the improved ground; h i , M s,i , and a s,i are the thickness, constrained modulus, and improvement area ratio of sub-layer i, respectively; E c is the elastic modulus of the soil cement column.

Numerical theories
Analytical solutions typically have many limitations as the solutions cannot consider practical aspects of the improved ground.Thus, in practice, numerical methods, especially the finite element method (FEM), are often applied to evaluate bearing capacity and settlement of the improved ground.In the case of FEM analysis, the choice of the calculation model and input parameters are very important and greatly affect the result of calculating the stress distribution on the column.
In case of CDM columns or other types of columns applied to reinforce soft ground, two common models are often considered: (1) plane strain model; and (2) axisymmetric model.For the 1D conditions in this study, the axisymmetric model is applied for analysis.Each CDM column and the influence zone around the column in this model are idealized as cylindrical masses (Figure 3).The size of the CDM column in the model is taken as the actual diameter of the CDM column.The radius of the unit cylinder is given by R = 0.564s, where s is the distance between the columns [14,15].[14,15]).

A comparative study on analytical and numerical analyses
2.3.1.Analysis purpose.This section presents settlement and stress induced values from simple analytical and numerical analyses of equivalent conditions.The purpose of the comparative study is to verify whether input parameters of numerical model (e.g., soil domain, boundary conditions, mesh refinement) are reasonable.Once the model is verified, the strength characteristics of soil and columns can be changed to examine the effect of considered aspects.

Input parameters.
For equivalent conditions in both analytical and numerical analyses, the following assumptions are applied in this comparative study: (1) all soils and columns are elastic materials; (2) the improved layer (clayey soil and CDM columns) is considered as composite elastic material, in which the equivalent constrained (M) is calculated by the denominator of Equation ( 8); and (3) simply, all layers are homogeneous and have constant parameters with depth.Figure 4(a) shows an illustration of the soil profile used in the analyses.
It is assumed that CDM columns and the SM layer have the same physical and mechanical properties.The columns and the clay layer have the following parameters: column diameter d c = 0.8 m, column spacing s = 2 d c = 1.6 m, unit weight  = 20 kN/m 3 , unconfined compressive strength, q u = 1000 kPa, elastic modulus E c = 300q u [2] = 300,000 kPa, and Poisson ratio  c = 0.35.The clayey soil layer has: thickness = 10.0 m, E s = 7500 kPa,  s = 0.30.In this case, all soil layers and CDM columns were modelled by using the Linear elastic model (to be able to compare with analytical theories).
For numerical analyses, the finite element method (FEM) is applied by using PLAXIS 2D (V21.01)software.The CDM columns were installed in square pattern (Figure 3).Following this, a 2D unit cylinder cell was modelled with the radius of the cell R = 0.564s = 0.902 m.The vertical boundaries of the model were horizontal fixities, and the bottom boundary was full of fixities.Figure 4(b) shows the unit cell modelled in the program.Table 1 shows a summary of input parameters for the analyses.In the analyses, the thickness of the SM layer was varied from 0.4 m to 1.0 m, but the total thickness of compacted layer and SM layer was kept constant of 1.5 m. Figure 5(a) shows a comparison of settlement on the surface of the compacted layer obtained from analytical method (Equation ( 7)) and numerical method.The figure indicates that settlement values from the analytical and numerical methods are very similar, with a small gap of about 1.0 mm.This insignificant gap is unavoidable since a FEM model cannot produce exact results as obtained from analytical solution of ideal assumptions.Figure 5(b) shows a comparison of the stress induced in the soil layers with depth under 1D conditions.Note that, for 1D conditions, the stress increment is constant with depth, and this feature was also obtained from the numerical method.The results indicate that the selected soil domain size, mesh refinement, and boundary conditions in the numerical analyses are reasonable.The parametric studies were carried out to evaluate the effect of the SM layer and other aspects on settlement of the improved ground and stress induced in the columns.Basically, the same soil profile and column specifications used in the comparative study were again used in this section.However, there were two distinctive points in the numerical model of this section: (1) soil and CDM columns were modelled as separated materials (i.e., not the equivalent material), as graphically shown in Figure 6; and (2) for soil layer and CDM columns, more exact constitutive models (e.g., Soft soil model and hardening soil model) can be used.However, for simplicity and repeatability, both soil and CDM columns were modelled by using the Mohr-Coulomb (MC) model.For evaluating the influence of thickness and stiffness of the SM layer, CDM columns of diameter d c = 0.8 m were installed in the square pattern, with the distance between two columns (in the same row) s = 2d c .This pattern results in an improvement area ratio of a s = A c /A s = 19.6%.Table 2 shows input parameters for the soil layers and columns (as well as the SM layer) in the numerical analyses.The influence of the thickness of the SM layer on the settlement of the improved ground was investigated by varying the thickness (t sm ) from 0.4 m to 1.0 m and keeping the same stiffness E sm = 300,000 kPa. Figure 7(a) shows the variation of settlement at different points.As shown, the total settlement on top of the compacted layer and on top of the clay layer decreases with the increases in the thickness, whereas the settlement on top of the columns is almost constant.The influence of the stiffness of the SM layer on the settlement of the improved ground was investigated by varying the stiffness (E sm ) from 100,000 kPa to 400,000 kPa and keeping the constant thickness of t sm = 0.4 m. Figure 7(b) shows the variation of the settlement at different points.As expected, the settlement on top of the compacted layer and on top of the clay layer also decreases with the increase in the stiffness, and similarly, the settlement on top of the columns is almost constant.(3) of AliCC method.It is interesting to note from Figure 8(a) that when the thickness of the SM layer increases from 0.4 m to 1.0 m, the stress induced on top of the columns increases from around 305 kPa to 367 kPa in numerical analysis.However, the stress induced on top of the clay layer remains relatively constant of around 50 kPa.Figure 8(a) also shows that when the thickness of the SM layer increases from 0.4 m to 1.0 m, the stress induced on top of the columns increases from around 219 kPa to 223 kPa in analytical analysis (ALiCC method).Similarly, Figure 9(a) shows that when the stiffness of the SM layer increases from 100,000 kPa to 400,000 kPa, the stress induced on top of the columns increases from around 268 kPa to 316 kPa.However, the stress induced on top of the clay layer remains relatively constant of around 55 kPa.The variation of total stress induced along the columns and in the surrounding soil is shown in Figure 8(b) and Figure 9(b).The total stress in the columns shows two distinctive features.Firstly, the thickness or stiffness of the SM layer has insignificant effect on the stress induced along the column.Secondly, the stress profiles show a trend that the induced stress is smaller at the two ends of the columns, where the columns sockets in stiffer materials than the surrounding soil, and is larger in the middle portion of the layer when the column is surrounded by softer soil layer.

Influence of improvement area ratio.
The influence of the improvement area ratio (a s ) on the settlement and stress induced is also investigated herein.In this case, the thickness of the compacted 100,000 kPa 200,000 kPa 300,000 kPa 400,000 kPa 100,000 kPa 200,000 kPa 300,000 kPa 400,000 kPa q = 100 kPa q u = 1000 kPa t sm = 0.4 m Along column;E sm In clay layer;E sm layer and the SM layer was kept constant of 1.1 m and 0.4 m, respectively.Three values of the column diameter d c = 0.8 m, 1.0 m, and 1.2 m were applied, resulting in values of a s = 19.6%,30.7%, and 44.17%, respectively.The variation of settlement on top of columns and layers with respect to the change of a s is given in Figure 10.This figure indicates that the settlement decreases significantly with the increase in the a s ratio.These results reflect the fact that when a s increases, the general stiffness of the improved ground increases, and therefore the settlement of the system decreases.Figure 11(a) shows the variation of the stress induced on top of the columns and clay layer.The figure indicates that the stress on both materials decreases when a s increases; however, the variation of stress on top of the clay layer is insignificant compared with that on the column head.The significant variation of the stress on the column head is attributed to the fact that, under the same load intensity and stiffness of the system of compacted layer and SM layer, the total load portion carried by the column system increases insignificantly because the clay layer is much softer than the columns and therefore its carried load portion varies insignificantly.Under such conditions, smaller columns are subjected to larger stress induced.
Figure 11(b) shows the variation of total stress along the columns and in the clay layer with respect to different a s values.Note that, the stress induced along the columns and in the clay layer has similar trend observed in Figure 8(b), and 9 (b), however, the magnitude of the stress in the column decreases pronouncedly with the increase in a s .As the clay is much softer, the stress induced in the clay layer varies very insignificantly.[16].Figure 12 shows the plan view of the storage yard with a total area of B  L = 229 m  420 m and a cross-sectional view of soil profile and design of CDM columns and the SM layer.The yard was designed to store coal up to 17.0 m high, resulting in a maximum applied pressure of q = 196 kPa.The design discussed herein was not the final approval yet but it was an economical approach to be refined for the construction.
In this design approach, CDM columns were designed to have diameter of d c = 1.6 m and length of L c = 6.0 m (from the bottom of the SM layer) installed in square pattern of s = 3.2 m (i.e., a s = 19.6%).The CDM column bases were designed to locate in the clayey sand layer with an average SPT N = 17 (medium dense).The compressive strength of the CDM column material was designed to have q u = 800 kPa.
As shown in Figure 12, the structural layer and soil profile include: Floor structure layer, t (thickness) = 0.65 m; SM layer, t sm = 1.35 m; sandy clay layer, t = 1.29 m, average SPT N = 11; sandy clay, t = 2.4 m, average SPT N = 3; clayey sand layer, t = 6.6 m, average SPT N = 17; sandy clay layer, t = 5.0 m, average N = 17.Table 3 shows input parameters for 1D numerical analyses, which represent working conditions at the center of the yard.Note that the ground stability at the edge of the yard is not discussed herein.13 (a) and 13 (b) illustrate the soil domain applied in the numerical analysis and the total stress domain in colour under the applied load.It is interesting to note that the maximum stress zone was located at the depths of around 3.3 m to 5.6 m, the depths of the soft sandy clay layer.Figure 13(c) shows the total stress profiles with depth extracted at the center of the column and center of the soil space (or at the right edge of the domain) of Figure 13(b).Interestingly, the variation trend of stress in the column is similar to that obtained from parametric study shown in Figures 8(b), 9(b), and 11(b).Note that, the total stress in the depth from 2.5 m to 6.5 m was over the compressive strength of the column (q u = 800 kPa), indicating that the column would be locally failed.This result indicates that, although the general capacity of the improved ground may be sufficient, the columns could be locally failed due to stress concentration.Analysis results from the parametric studies and this case study show that the maximum stress induced in the column may not be on top of the column, but the location very much depends on the stiffness of soil layers and the stiffness of the SM layer.

Conclusion
This paper presents analytical and numerical studies on the influence of the SM layer (load transfer layer) on the stress induced on head of as well as along the CDM columns and settlement of the improved ground under one-dimensional loading conditions.The numerical analyses were carried out by using PLAXIS 2D (V21.01) for parametric study cases and an actual project case in Vietnam.Some key findings from the study can be drawn as follows.
For the typical examined parametric case study (d c = 0.8 m, a s = 19.6 %, E c /E s = 40, CDM column length = 10 m = thickness of soft soil layer, applied pressure = 100 kPa), when the thickness of the SM layer increases from 0.4 to 1.0 m, the settlement on top of the compacted layer decreases insignificantly from 29.9 to 29.0 mm and the stress induced on the column head increases from 305 kPa to 367 kPa.Similarly, when the stiffness of the SM layer increases from 100 MPa to 400 MPa, the settlement decreases insignificantly from 30.5 to 29.9 mm and induced stress increases from 268 kPa to 316 kPa.These results indicate that more stress is transferred to the columns when the thickness or stiffness of the SM layer increases.However, the stress induced along the column is almost constant with the variation of the thickness and stiffness.
When the improvement area ratio (a s ) increases from 19.6% to 44.2%, the settlement on top of the compacted layer decreases significantly from 30.0 to 22.0 mm and the stress induced on head of the CDM column columns decreases significantly from 306 kPa to 165 kPa.This behavior is attributed to the fact that, under the constant applied load and unchanged SM layer, the larger columns result in smaller induced stress on head of and along the columns.
An important finding from this study is that the maximum stress induced in the CDM column is typically not on head of the columns but at the middle depths where soil layers are softer than the SM and the bearing layer at the column toes.This finding was consistently indicated both from typical soil profile of the parametric studies and from the actual soil profile of the project case in Vietnam.

Figure 2 .
Figure 2. Structure of the load transfer layer from the ALiCC method (modified after [5]).

Figure 4 .
Ground profiles in comparative study: (a) Analytical model, (b) Numerical model.

Figure 5 .
(a) Comparison of total settlement profile, (b) Stress increment profile obtained from analytical and numerical analyses.

Figure 6 .
Figure 6.Improved ground of parametric study case.

Figure 7 .
(a) Influence of thickness of the SM layer on settlement of the ground, (b) Influence of stiffness of the SM layer on settlement of the ground.The influence of thickness and stiffness of the SM layer on the stress induced on top of the columns and clay layer is shown in Figures 8(a) and 9(a).The influence of thickness and stiffness of the SM layer on the stress induced along the columns and in the clay layer is shown in Figures 8(b) and 9(b).

Figure 8 (
a) also shows the stress induced on top of the columns and clay layer obtained from Equation (2) and Thickness of SM layer (m)

Figure 8 .Figure 9 .
Influence of thickness of the SM layer on: (a) stress induced on the top of the columns and clay layer, (b) stress induced along the columns and in the clay layer.Influence of stiffness of the SM layer on: (a) stress induced on the top of the columns and clay layer, (b) stress induced along the columns and in the clay layer.

Figure 10 .
Figure 10.Influence of improvement area ratio on settlement of the ground.

Figure 11 . 1 .
Figure 11.Influence of improvement area ratio on: (a) stress induced on top of the columns and clay Layer, (b) stress induced along the columns and in the clay layer.

Figure 12 .
Figure 12.The cross-sectional view of improved ground and the plan view of the storage yard of Quang Trach 1 Thermal Power Plant (TPP).

Figure 13 .
(a) Numerical soil domain, (b) colour spectrum of total stress in the column and surrounding soil, (c) distribution of total stress with depth in the center of CDM column and in the center soil portion.

Table 1 .
Input parameters for the comparative study.

Table 2 .
Input parameters for the parametric study.

Table 3 .
Input parameters for Quang Trach 1 project.