The projected structure is a circular flour storage tank transmitting uniform surface loads that push us to choose a deck as a foundation system. The 20m diameter flush to 1m, transmits a uniform load of 150 kPa corresponding to a permissible settlement of 20 cm.

The geotechnical survey of the site showed a stratigraphy of the soil made up of:

The results of the pressure-measuring test are shown in Figure 1 .

The load capacity calculation using the Terzaghi [10] approach gives a value of 125 kPa, which is significantly lower than the load transmitted by the container. The estimated settlement corresponds to a value of 57 cm which is much higher than the permissible settlement of 20 cm. In order to increase the load-bearing capacity and reduce the settlement of the tank to a value admissible (which allows to guarantee its stability during service), a reinforcement by ballasted columns was decided.

The technique of improvement by ballasted columns is equivalent to incorporating into an initial soil, whose mechanical characteristics (C, $\varphi$, P_LM or peak resistance) and deformability (E, Pressiometric Module) are low, a grained material called ballast compacted using a vibrating needle ( Figure 2 ).

Ballast columns are generally distributed in the form of a regular mesh under structures usually designed on a [11] .

The assumptions regarding the choice of deformation modulus, diameter and length of the columns ( Figure 3 ) depend in part on the chosen implementation material and its performance [12] . However, in the case of non-floating columns, the length depends on the depth of the substrate.

The choice of a method of improvement is always accompanied by an objective expressed in terms of incorporation rate, the number of landfills and bearing capacity is determined by the characteristics of the soil in place. This requires the carrying out of surveys, identification tests and soil characterization.

The geotechnical study program is summarized as follows:

In parallel, the pressure-measuring test [13] and the penetrating test [14] were carried out to allow in situ estimation of bearing capacity and compaction.

The method of sizing used is that of Priebe (1976-1995) [15] which introduces the notion of overall improvement of the mechanical characteristics of the treated medium and corrections inherent to the relative compressibility (soil-column) and the effect of depth. Priebe was based on the principle of the unit cell with the assimilation of elastic deformations of the soil surrounding the column to that of a thick tube, The same drained characteristics as the compressible soil.

Once the dimensioning is validated, it will be necessary to check the internal loads corresponding to:

$q_{re}={\sigma }_r\ast {\tan }^2\left(\frac{\text{π}}{4}+\frac{{\varphi }^{\prime }c}{2}\right)$(1)

$q_{rp}=9\cdot C_u+L_c\cdot \left(2\cdot \frac{C_u}{R_c}-{\gamma }_{col}\right)$(2)

where,

C_u represents the undrained cohesion of the soil.

L_c: length of the column.

R_c: average radius of the column.

${\sigma }_r$: represents the radial embrace.

Within the case of pressiometric test

$\begin{array}{c}{\sigma }_r\approx Pl{e}^{*}\end{array}$(3)

where,

${\sigma }_r\approx 4\cdot Pl{e}^{*}\left(s_i{\varphi }^{\prime }c\ge {38}^{\circ }\right)$(4)

Ple* = min (Ple*[z]) on column height in each layer.

The vertical stress of rupture q_r in the column is equal to:

q_r = min (q_re; q_rp; 1.6 MPa).

In this case, the following relationships can be established:

$C_u\approx \frac{P{l}^{*}}{5.5}\text{when}P{l}^{*}<0$(5)

$C_u\approx \frac{P{l}^{*}}{10}+0.025\text{when}P{l}^{*}\ge 0.3\text{MPa}$(6)

The weighting coefficients of 2 and 1.5 are assigned in the calculation to the ULE and the SLE respectively.

The settlement shall be calculated by the method of homogenization under a uniform load [16] where ballasted columns are stopped on a more compact layer.

After the construction of the ballasted columns, the settlement of each layer i in the center of the work is written

$w_i=h_i\cdot {\sigma }_t{a}_i\cdot E_{col}+\left\{\left(1-a_i\right)\cdot E_{si}\cdot \frac{1-{\nu }_{si}}{1-{\nu }_{si}-2{\nu }_{si}^2}\right\}$(7)

The value of the constraint in the column at layer i (σ ci) can be given by:

${\sigma }_{ci}=E_{col}\cdot \frac{{\sigma }_t}{a_i}\cdot E_{col}+\left\{\left(1-a_i\right)\cdot E_{si}\cdot \frac{1-{\nu }_{si}}{1-{\nu }_{si}-2{\nu }_{si}^2}\right\}$(8)

where:

a_i: percentage of incorporation (ratio of sections), in the layer i considered;

E_col: Young’s module of the column;

E_si: Young’s module of the layer i considered;

ν_si: Poisson coefficient of the layer i considered;

σ_t: mean vertical stress provided by the structure;

h_i: layer thickness i.

In the case where pressure tests are available

$\begin{array}{c}\begin{array}{c}{w}_i=\frac{h_i\cdot {\sigma }_t}{a_i}\cdot E_{col}+\left\{\left(1-a_i\right)\cdot \frac{E_{Mi}}{{\alpha }_i}\right\}\end{array}\end{array}$(9)

The value of the constraint in the column at layer i (σ_ci) can be given by:

${\sigma }_{ci}=E_{col}\cdot \frac{{\sigma }_t}{a_i}\cdot E_{col}+\left\{\left(1-a_i\right)\cdot \frac{E_M{}_i}{{\alpha }_i}\right\}$(10)

With:

E_Mi: pressiometric module of the layer i considered.

α_i: Coefficient linking the Eeod of the layer i considered.

In order to better visualize the internal behavior of the column and its interaction with the soil, a numerical simulation is performed under the software Plaxis 2D.

Operations on PLAXIS can be divided into three main parts which are Pre-processing (using the Input subroutine), Calculations (using the Calculations subroutine) and Post-processing (using the Output subroutines and Curves subroutines). First, the initial soil model (without inclusion) is defined in order to know the real behavior.

The purpose of pre-processing is to create our model on Plaxis 2D. This involves defining our problem (physical representation), followed by defining the construction procedure corresponding to the creation of the physical model with dimensions 30 × 15 m with defined boundary conditions ( Figure 4 ). The model considered is an axisymmetric with a six-nodded triangular element mesh [17] .

When defining the construction procedure, it is up to us to establish the initial situation and the different stages of the construction.

This is followed by the definition of the material properties ( Figure 5 ).

In this case, the procedure is as follows:

The definition of the characteristics of the initial material and the chosen behavior law corresponding to that of Mohr-coulomb which is adapted to the available data [18] .

The characteristics of our terrain model on the geometric, physical and geotechnical plane are made from the following data ( Table 1 ).

After material definition, application of loading follow. In the case of this present study, the foundation is considered to be rigid. To this end, the settlement of our foundation will be simulated as a uniform vertical downward movement [19] .

In modelling consolidation, it is important to consider boundary conditions. This means specifying how the load is applied, soil drainage conditions, and other factors that influence the consolidation process. In principle, all boundaries must have one boundary condition in each direction. That is to say, when no explicit boundary condition is given to a certain boundary, the natural condition applies, which is a prescribed force equal to zero and a free displacement. To avoid the situation where the displacements of the geometry are undetermined, some points of the geometry must have prescribed displacement [17] . The simplest form of a prescribed displacement is a fixity (zero displacement), but non-zero prescribed displacement may also be given. When the geometry model is complete, the finite element model (or mesh) can be generated. Plaxis allows for a fully automatic mesh generation procedure, in which the geometry is divided into elements of the basic element type and compatible structural elements, if applicable.

The following step is the calculation phase which contains all the elements to define and initiate a finite element calculation. It is to choose the project for which the calculations will be defined ( Figure 6 ). Each calculation step is characterized by a series of iterations to reduce the solution balance errors.

Then the Output subroutine contains several very important menus such as the deformation menu that allows the visualization of the different deformations through options like Total Displacements, Total Deformations. The Constraints (stresses) menu It allows us to visualize the total and effective stresses in our model ( Figure 7 ).

Finally, the Curves subroutine is used to display stress-strain, load-displacement or time-displacement curves, stress paths or strain of selected points in geometry. These curves represent the evolution during the different phases of calculation, which gives an overview of the global and local behavior of the soil. The

points whose curves are generated must be selected in the calculation program before starting the calculation process.

The settlement representing a uniform vertical displacement to the base will be simulated by the “Prescribed displacement” option ( Figure 8 ).

The idea of fixing a vertical displacement of 20 cm is to start from the allowable base to obtain the minimum stress that can induce this settlement

( Figure 9 ) and compare it to the load applied by the storerver.

To achieve a 20 cm settlement on our model, a force of 2168.1 kN was applied. In 2D plaxis, for axisymmetric models the force Y is expressed as a unit of force

per radiant ( Figure 5 ). Therefore, in order π to obtain the total force necessary to produce a displacement of 20 cm under our foundation, it is necessary to multiply the force obtained on plaxis by 2, so that:

The stress obtained by simulation being lower than the load provided by the net (150 kPa), it is clear that our reservoir induces unacceptable deformations in this soil.

This translates into the deformed mesh model below ( Figure 10 ).

We observe deformations at the level of the ground underlying our structure ( Figure 11 ).

It is therefore up to us to determine the settlement produced by the application of a stress equal to 150 kPa. As before, the Prescribed displacement option is adopted on Plaxis 2D to simulate the settlement. We will prescribe a uniform displacement of 5 cm in an increasing manner on Plaxis 2D, until a displacement corresponding to the application of a stress equal to 150 kPa is reached.

The vertical deformation model obtained is shown in Figure 12 .

Figure 12 allows to identify vertical displacements corresponding to a stress of 150 kPa such that the first under the structure which are oriented downwards are bumps which have a value of 70 cm. The second ones on the adjacent side of the structure are smaller displacements corresponding to soil uplifts.

The simulation shows that the soil in place is not suitable to support this tank. What will happen with the ballasted column improvement solution chosen? To answer this question, we present the results of the simulation obtained after improvement ( Figure 13 ).

The same logic of the displacement of 20 cm prescribed in the case of the floor in place was adopted. To obtain a settlement of 20 cm on our model reinforced by the ballasted flaps, a force of 13,255 kN was applied ( Figure 13 ). In 2D plaxis, for axisymmetric models the force Y is expressed as unit of force per radiant. Therefore, to obtain the total force necessary to produce a displacement of 20 cm under our foundation, it is necessary to multiply the force obtained on plaxis by 2π, such as:

The corresponding deformed mesh is presented in Figure 14 .

After applying a load equal to 265.1 kPa, a vertical settlement of 20 cm ( Figure 15 ) and a horizontal displacement of 17 cm were obtained ( Figure 16 ), therefore, when a stress of 150 kPa on the reinforced ground, the deformations will remain within the tolerable range, that is to say, less than 20 cm (Figure 16).

Potential limitations and assumptions of this simulation study include the challenge of converting stone column grids into 2D structures for modeling. Different approaches like enhanced soil parameters, embedded beam elements, and equivalent column methods are available for modeling stone columns, but selecting the most suitable approach can be complex due to limited literature guidance. Additionally, the software assumes specific material models like the Mohr-Coulomb model for soil analysis, which may not fully capture the complex behavior of soft soils under varying conditions. Furthermore, the accuracy of the analysis heavily relies on the input soil parameters obtained from laboratory tests, which can introduce uncertainties in the simulation results.

The reservoir has a diameter of 20 meters and block measures 22 meters. The mesh under the chosen grid is a regular square-shaped mesh and characterized by a reference mesh A_ref = 3.61 m² (1.9*1.9 m²) according to the CFMS recommendations [8] .

Either A the surface of the right-of-way of the row and n the number of columns under this surface such that:

A = (22/2)² × π = 379.94 m²;

n = 379.94/3.61 = 106 columns.

The effective breaking stress q_re by lateral expansion is:

$q_{re}=0.375\cdot {\tan }^2\left(\frac{180}{4}+\frac{19}{2}\right)$

$q_{re}=0.737\text{MPa}=737\text{kPa}$

The Constraint at SLE is:

Qa,ELS = 737/2 = 369 KPa

The Constraint at ULE is:

Qa,ELU = 737/1.5 = 491 KPa

For settlement calculation, the homogenization method is chosen ( Table 2 ):

$S=\frac{E_{col}\cdot H}{a\cdot E_{col}+\left\{\left(1-a\right)\cdot E_{soli}\right\}}$

The settlement obtained by the homogenization method is of the order of 15 cm. The homogenization method is also used for the load-bearing capacity ( Table 3 ).

$\begin{array}{c}{\sigma }_c=\frac{E_{col}\cdot {\sigma }_t}{a\cdot E_{col}+\left\{\left(1-a\right)\cdot E_{sol}\right\}}\end{array}$

σ_c = 257 KPa.