In this part of this article, the objective was to present the methodology of this research work. The general objective of this work has been to analytically and numerically study the hydro-mechanical behavior of SSI in Cantilever retaining walls in case of overturning failure. The analytical methodology and the numerical methodology were presented. The Mathematical equations which form the criterion for design and numerical model which forms the basis for simulating the hydro-mechanical SSI interaction in Cast3m. The physical, mechanical and geometrical parameters of soil and concrete which will help us to simulate this multi-physical system were presented. This protocol of research helped us to study Concrete Cantilever retaining walls of 3 m height, retaining Sandy Clayey soils in the presence of rising water table (upward seepage). In this part of the work, the parameters were exploited, the heel length, rising water table, resulting Active Earth pressure and Settlement were presented.

The Analytical and Numerical methods were chosen because it helped us manipulate various parameters to obtain varied structure-soil behavior which is difficult to obtain from an experimental work. The Analytical gives us discrete solutions and the Numerical gives us numerical solutions that approach the reality. The Cast3m software and MS. Excel were employed to attain this research objective.

The analytical method applied here was used to compute values of active Earth pressures, Horizontal thrusts, overturning moments, and resistant moment against overturning with the goal of obtaining Factors of safety for heel length variation and water height rise. Microsoft Excel spreadsheet was used to compute these values.

In order to achieve the objective of modelling the hydro-mechanical behavior of the retaining wall analytically. The following materials were used:

1) Concrete parameters (Density, geometrical parameters of wall);

2) Clayey-Sand soil parameters (Saturated density, humid density, submerged density, Angle of internal friction of soil);

3) Water (density);

4) Excel spread sheet (for analytical computation).

A table of values for different variations of the heel by considering the geometrical parameters of the wall was developed. The physical properties of the materials were defined, the densities, angle of internal frictions were defined. The parameters used for computing the active earth pressure coefficient were defined. The Coulomb’s method was chosen to compute active earth pressure coefficient because the wall is not frictionless. The overturning moments, resistant moments, Factors of safety for varied heel lengths and water altitude behind the wall were calculated. This method is presented below:

The height of the retaining wall was considered fixed at 3 m. This height is preferable because it is the height that suits this kind of retaining wall contrary to other walls (mass retaining walls, MSE) which can go to heights of over 3 m. Tables 1-3 respectively show the geometrical parameters and computational value, the physical properties of soil and retaining wall and the coulomb’s parameters for commutating active earth pressure coefficient.

Continued

The above parameters enabled us to calculate the Coulomb Active Earth pressure coefficient K_a using equation.

The different weights, lever arm and moments that tend to stabilize the wall, were calculated using the MS Excel spread sheet.

For Water level = 0 m.

$M_{RES}i=Wi\times Li$ (2.1)

For water level (H_w ≠ 0 m).

Total Resisting moment ( $M_{RES}$) is given by:

$M_{RES}=\underset{i=1}{\overset{3}{{\displaystyle \sum }}}Wi\times Li$ (2.2)

The overturning moment was computed following the table as shown below. In Case of Water presence and absence of water. The Vertical pressures were calculated, and the horizontal pressures were calculated using the Active Earth pressure coefficients. The Horizontal thrust was then calculated together with their lever arm from the Point C at the base of the wall about which the Overturning moments were calculated.

The Active Earth Pressure force is given by:

$P_a=\frac{1}{2}{K}_a\times {\rho }_{us}\times g\times H^2$ (2.3)

When there is no rising ground water: $Hw=0$:

Active Earth Pressure Force (N) $P_{a1}=\frac{1}{2}{K}_a\times {\rho }_{us}\times g\times H{T}^2$ (2.4)

Lever Arm (m) $L4=\frac{HT}{3}+H3$ (2.5)

When there is rising ground water: $Hw\ne 0$. Table 4 show the active earth pressure computation.

$P_{a11}=\frac{1}{2}{K}_a\times {\rho }_{us}\times g\times {\left(HT-Hw\right)}^2$ (2.20)

$L5=\frac{HT-Hw}{3}+Hw+H3$ (2.6)

$P_{a2}=\left({\rho }_{sat}-{\rho }_{us}\right)\times g\times K_a\times H{w}^2\times \frac{1}{2}$ (2.7)

$L6=\left(0.5\times Hw\right)+H3$ (2.8)

$L7=\left(Hw/3\right)+H3$ (2.9)

$P_{a3}=\frac{1}{2}{\rho }_w\times g\times H{w}^2$ (2.10)

$M_{ov}=\left(P_{a11}\times L5\right)+\left(P_{a11}\times L6\right)+\left(P_{a2}\times L7\right)+\left(P_{a3}\times L7\right)$ (2.11)

For a retaining wall to be considered safe against overturning, the Factor of safety is a ratio between the Moment tending to resist overturning and the moment tending to cause overturning. This Value is said to be greater than or equal to 2 to 3.

$2\le FO{S}_A=\frac{M_{RES}}{M_{ov}}\le 3$ (2.12)

From this methodology of the analytical work, the values of safety factor against overturning for different heel lengths and water levels were calculated. Overturning moments, Resisting moments with varied heel lengths and water level.

The numerical method was also used to simulate the same system. As peculiar as this method is, it helped us to numerically obtain values of stresses in the soil, settlements which enabled us to calculate Overturning moment from resulting stresses and Resisting moments from the resulting weights of that favor the stability of the wall. However, this method takes into consideration several constraints which the analytical model does not. This method used three models, concrete, soil, and interface, to simulate the hydro-mechanical interaction. Below is a presentation of the materials and method of the numerical method.

The materials used for this method are given below:

CAST3M does not have any particular system of measuring units. It is to the user to provide the data in a coherent system checking the fundamental law of dynamics: $F=M\cdot a$. Where F: force, M: mass and a: acceleration. Once the measuring units used in the data are defined, all the results will be expressed in these same units. There is an exception to this rule concerning the measurement of the angles which must always be expressed in degrees. On the other hand, the temperatures and the thermal dilation coefficient must be expressed in coherent units. Presented below is a methodology of how the software functions. The first part is the definition of the problem. The second part consists of resolving the problem.

The model chosen for soil in the numerical approach is the CamClay Model while the analytical model takes into consideration, the soil densities, in saturated and unsaturated conditions, as well as the internal angle of friction. These parameters were used to represent the hydro-mechanical conditions of the soil in the case of rising groundwater. The soil retained by the wall and the foundation soil are of the same type but vary in their cohesions, pre-consolidation pressures, void ratios, Poisson’s coefficients. The type of soil used for this work is Clayey Sand. Soil is an elasto-plastique Material which typically follows the CamClay criteria. Parameters are defined in Table 5 and Table 6 above.

A 2D model to simplify this work was considered. The Cast3m software by default considers a default thickness of 1 m. Calculation was 2-dimensional with lines segment of 2 nodes and a triangular finite surface. The density of the finite element was 0.1 which implies that for a unit length, the segment is broken into ten other line segments.

The geometry of the model was programmed using the Cast3m software ( Figure 1 ).

The materials mode of behavior under loadings, as already defined above in materials was defined. These material properties where varied in case of water influence in soil. The case of submerged soil.

The Cast3m software calculates the stiffness matrix of the Cantilever Retaining wall and the Soil in order to obtain a solution. Since the work is done in 2 dimensions, with three degrees of freedom: translation in ox axis, oy axis and rotation. Figure 2 show the boundary condition of the model.

The Cast3m software makes provision or use to load the model in two kinds of load application: by imposing a force or by imposing a displacement. In this work, loading by force was suitable for the numerical model because it permitted us to know the exact load to apply on the wall and soil in conformity with the loads computed in the numerical work. This method of load application permitted us to produce the corresponding displacements and stresses. After the loads are applied, the software then computes the solution in two ways: non-linear and linear calculation [8] . The interest relates only to the maximum values of displacements, constraints and forces. A linear solution method in which the model’s own weight is applied in a single iteration was chosen. The software to calculate the self-weight of each constituent element of model and distribute it evenly to all finite elements of that element was applied. Figure 3 and Figure 4 respectively show the self-weight of cantilever wall (Cast3m) and self-weight of soil (Cast3m).

As specified in the objectives, the graphs of active Earth pressure, safety factor with heel length was plotted. The digital method involves the judicious choice of points from which data will be extracted. The horizontal stress from three main points was extracted as: P20, P21A, and P22 at the base of the wall, at the level of the water table and at the summit, respectively. In order to see the effect of increasing the heel length on the wall displacement, the horizontal displacement at the summit of the wall on point P16 was extracted. Figure 5 shows how data were extracted.

In order to calculate the overturning moment base on the numerical result. The horizontal stresses from the different points given above were extracted, then the principle of vertical stress increase as depth increases was applied. Two cases are possible in case of vertical stresses. The principle is explained in Figure 6 below.

The pressures: Pres1 and Pres2 were used to calculate the horizontal thrust in both submerged and unsubmerged cases, just as the figure above. From the software, the overturning moments, resisting moments, factor of safety were calculated.

The peculiarity with the Cast3m software was that it would enable us to get results of different stresses in the soil and the retaining wall for different water conditions. Most importantly, the stress evolution from the base of the heel to the top of the backfill soil was observed. The analysis with pictures was performed, the displacements and deformation of the wall for different water conditions. This will help us analyse the von Mises stresses which is calculated for a particular failure criterion.

Cast3m software permitted us to have stress fields and displacement fields of different hydro-mechanical conditions. For these analyses, water depths of 0, 1, 2 and 3 m which simultaneously corresponds to 0, H/3, 2H/3, and H were chosen.

Yielding walls are walls that have the tendency to move forward under lateral loads. Below is presented horizontal displacement fields for varying water depths ( Figures 7-10 ).

From the results, extract horizontal displacement values at top of the wall was collected. Figure 11 shows the influence of water depth on the horizontal displacement of the wall.

Between a submerged depth of 0 to 2H/3 the displacement is constant, but experiences a great increase from 2H/3.

The software also enables us to analyze horizontal stresses that act in the soil and in the retaining wall. Figures 12-15 below show the horizontal stress fields for different submerged conditions.

The water level height was varied from 0 to 3 m. The overturning moment was computed analytically and numerically. The results are presented below in Figure 16 .

Here, graphs of FOS against water level (H_w) were plotted to see the effect of increasing water level on safety against overturning. The following results were obtained in Figure 17 below.

$H_w\le \frac{H}{3}$ (3.1)

No influence of water on overturning

$\frac{H}{3}<H_w\le H$ (3.2)

water begins to influence the stability aginst overturning.

Figure 18 below shows the factor of safety for different heel lengths and water depth.

At different water levels behind the soil, the length of the heel was varied, computed the factor of safety for both the numerical and analytical work ( Figure 19 ).

Figure 20 below show different overturning moments, resisting moments, factors of safety taken, for different heel length for both the numerical and analytical solutions for water depth of 1 m.

Figure 21 below show the values for resistant moments and overturning moments for different heel lengths for a water depth of 2 m.

Figure 22 below shows different Overturning moments, resisting moments, factors of safety taken, for different heel length for both the numerical and analytical solutions for water depth of 1 m.

In order to better appreciate the influence of heel length on the global variation of water depth, Figure 23 combines the solutions obtained analytically and numerically.

Based on the results obtained, empirical norms for heel lengths which can be adopted when designing or pre-dimensioning retaining walls to be able to counter overturning in unsaturated, partially saturated and fully saturated conditions with little or no drainage system provided. The lower boundary values are chosen for a factor of safety of 2, against overturning while the upper boundary values are chosen for a factor of safety of 3, against overturning ( Figures 25-27 ).

Figures 28-30 show the numerical pre-dimensioning.

From the equations obtained above, the relating factor of safety against overturning and k values for both Numerical and analytical solutions, where Lc = kH. A pre-dimensioning norm against overturning for upper and lower bound values of safety factors, which are between 2 and 3 ( Table 7 ).