The silty sand, the subject of this study, comes from the Tohoué quarry in the Tohouè District, Sèmè-Kpodji Commune. The Commune of Sèmè-Kpodji is located between the parallels 6˚22' and 6˚28' of North latitude and the meridians 2˚28' and 2˚43' of East longitude. The location of the quarry is completed by Figure 1 below.

Figure 2 shows the silty sand sampling area at Tohouè.

Table 1 below shows the geographical coordinates of the various survey wells, silty sand sampling points of Tohouè.

Figure 3 shows a silty sand quarry in Tohoué (a) and a pile (b) of said material.

The equipment used for geotechnical tests complies with the requirements of current standards.

For the sieving granulometric analysis test, the experimental device including the accessories necessary for its implementation is governed by standard NF P 94-056 [9] . Figure 4 shows all of said equipment.

As for the test of the measurement of the weight water content, the experimental device complies with the standard NF P94-050 [10] . The material required for its production is illustrated in Figure 5 below.

For the test to determine the adsorption capacity of silty sand for methylene blue, the experimental device required for its implementation is prescribed by standard NF P 94-068 [11] . Figure 6 below illustrates all of the said equipment.

For the test to determine the organic matter content on the silty sand of Tohouè, the experimental device set up complies with standard XP P 94-047 [12] as presented in Figure 7 below.

Figure 8 below shows the different materials for determining the Sand Equivalent in the 0/2-mm fraction of sands. This test is carried out in accordance with AASHTO T176 [13] , EN 933-8 [14] .

Figure 9 below shows a set of experimental devices for carrying out the Modified Proctor test in accordance with standard NF P94-093 [15] .

Figure 10 below shows a set of experimental devices for carrying out the CBR test in accordance with standard NF P94-078 [16] .

Figure 11 shows a set of experimental devices for carrying out the rectilinear shear test on the box in accordance with standard NF EN ISO 17892-10 [17] .

Figure 12 below shows the entire experimental device for carrying out the oedometric test according to standard XP P 94-091 [18] .

Samples are taken in accordance with ISO 22475-1 [19] .

The various geotechnical tests are carried out in accordance with the standards cited in §1.1.2.

Determination of the friction angle and internal cohesion by the Casagrande box shear the test, go through the calibration of the raw material from the initial condition through the values obtained from the Modified Proctor test, then

Step 1: Determination of the optimal water content and dry density on the material from the quarry (initial state).

Step 2: Determination of the optimal water content and dry density on the class 0/5 test sample.

Step 3: Carrying out the Casagrande box shears the test.

Step 4: Determination of the optimum water content and dry density on the test sample after the test.

Previous studies have shown that the elastic behavior of soils is never linear in reality ( [1] [20] ). Therefore, it is important to focus studies on the nonlinear behavior of soils used in road construction. To do this, several mathematical models, both hyperelastic and hypoelastic, can be used to describe these nonlinear behaviors of soils. However, it has been proven that hypoelastic models are the most recommended when it comes to small deformation studies. Two types of hypoelastic models exist, namely hyperbolic models and variable modulus models, as reported by Babaliyè in 2020. In the context of this study, hyperbolic models mathematically based on a representation of the stress-strain relationship using a hyperbolic or parabolic curve ( [21] ) are best suited to describe the nonlinear elastic behavior of soils ( [1] [20] ).

According to Hardin and Drnevich [22] , the hypoelastic behavior of a material is given by Equation (1).

$\tau =\frac{\gamma }{\frac{1}{G_{\max }}+\frac{\gamma }{{\tau }_{\max }}}$ (1)

where ${\tau }_{\max }$ represents the maximum shear stress, $G_{\max }$ the maximum shear modulus, $\tau$ the shear stress and $\gamma$ the shear strain.

To determine the parameters ${\tau }_{\max }$ and $G_{\max }$, Equation (1) was reformulated by setting: $a=1/G_{\max }$ and $b=1/{\tau }_{\max }$. which gives the following Equation (2):

$\tau =\phi \left(\gamma ,a,b\right)=\frac{\gamma }{a+b\gamma },a,b\in ℝ$ (2)

Using a nonlinear least fit method, the parameters a and b are evaluated. This method consists of fitting the experimental data $y_i$ to the function $\phi$ by minimizing the distance ${\phi }_i$ between $y_i$. and $\phi \left(\gamma ,a,b\right)$:

${\phi }_i=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left[y_i-\phi \left(\gamma ,a,b\right)\right]}^2$ (3)

The implementation of nonlinear regression follows the following steps:

1^st Step: Linearization $\phi \left(\gamma ,a,b\right)$ of around $\left(a_0,b_0\right)$

$\{\begin{array}{l}\frac{\partial \phi }{\partial a}\left(\gamma ,a,b\right)=-\frac{\gamma }{{\left(a+b\gamma \right)}^2}\\ \frac{\partial \phi }{\partial b}\left(\gamma ,a,b\right)=-\frac{{\gamma }^2}{{\left(a+b\gamma \right)}^2}\end{array}$ (4)

$\phi \left(\gamma ,a,b\right)=\frac{\gamma }{a_0+\gamma b_0}-\frac{\gamma }{{\left(a_0+b_0\gamma \right)}^2}\left(a-a_0\right)-\frac{{\gamma }^2}{{\left(a_0+b_0\gamma \right)}^2}\left(b-b_0\right)$ (5)

Let us set: $A=\frac{\gamma }{{\left(a_0+b_0\gamma \right)}^2}$; $B=\frac{{\gamma }^2}{{\left(a_0+b_0\gamma \right)}^2}$ and $C=\frac{\gamma }{a_0+b_0\gamma }$.

Equation (5) becomes:

$\phi \left(\gamma ,a,b\right)=C-A\left(a-a_0\right)-B\left(b-b_0\right)$ (6)

2^nd Step: Determination of a and b.

The minimization of $\phi$ consists of canceling its first derivative with respect to the unknowns a and b. Let:

$\{\begin{array}{l}\frac{\partial \phi }{\partial a}=0\\ \frac{\partial \phi }{\partial b}=0\end{array}$ (7)

With $\phi \left(\gamma ,a,b\right)=C-A\left(a-a_0\right)-B\left(b-b_0\right)$.

Development of the terms of the system of Equation (7).

Case of the first equation:

$\frac{\partial {\phi }_i}{\partial a}=0\iff \frac{\partial }{\partial a}\left[{\displaystyle \sum {\left(y_i-\phi \left(\gamma ,a,b\right)\right)}^2}\right]=0$ (8)

$\iff {\displaystyle \sum \frac{\partial }{\partial a}\left(y_i-\phi \left(\gamma ,a,b\right)\right)\cdot \left(y_i-\phi \left(\gamma ,a,b\right)\right)}=0$

$\iff -{\displaystyle \sum \frac{\partial }{\partial a}\phi \left(\gamma ,a,b\right)\cdot \left(y_i-\phi \left(\gamma ,a,b\right)\right)}=0$

$\iff -{\displaystyle \sum \left(-A\right)\cdot \left(y_i-\phi \left(\gamma ,a,b\right)\right)}=0$

$\iff {\displaystyle \sum A\left(y_i-C+A\left(a-a_0\right)+B\left(b-b_0\right)\right)}=0$

$\iff {\displaystyle \sum A{y}_i}-{\displaystyle \sum A\cdot C}+{\displaystyle \sum A^2\left(a-a_0\right)}+{\displaystyle \sum A\cdot B\left(b-b_0\right)}=0$

$\iff \left(a-a_0\right){\displaystyle \sum A^2}+\left(b-b_0\right){\displaystyle \sum A\cdot B}={\displaystyle \sum A\cdot C}-{\displaystyle \sum A{y}_i}$

$\iff \left(a-a_0\right){\displaystyle \sum A^2}+\left(b-b_0\right){\displaystyle \sum A\cdot B}={\displaystyle \sum A\left(C-y_i\right)}$ (9)

Case of the second equation:

$\frac{\partial {\phi }_i}{\partial b}=0\iff \frac{\partial }{\partial b}\left[{\displaystyle \sum {\left(y_i-\phi \left(\gamma ,a,b\right)\right)}^2}\right]=0$(10)

$\iff {\displaystyle \sum \frac{\partial }{\partial b}\left(y_i-\phi \left(\gamma ,a,b\right)\right)\left(y_i-\phi \left(\gamma ,a,b\right)\right)}=0$

$\iff {\displaystyle \sum \left(-\frac{\partial }{\partial b}\phi \left(\gamma ,a,b\right)\right)\cdot \left(y_i-C+A\left(a-a_0\right)+B\left(b-b_0\right)\right)}=0$

$\iff {\displaystyle \sum -\left(-B\right)\left(y_i-C+A\left(a-a_0\right)+B\left(b-b_0\right)\right)}=0$

$\iff {\displaystyle \sum B\left(y_i-C+A\left(a-a_0\right)+B\left(b-b_0\right)\right)}=0$

$\iff {\displaystyle \sum \left(B{y}_i-B\cdot C+A\cdot B\left(a-a_0\right)+B^2\left(b-b_0\right)\right)}=0$

$\iff {\displaystyle \sum B\left(y_i-C\right)}+\left(a-a_0\right){\displaystyle \sum A\cdot B}+\left(b-b_0\right){\displaystyle \sum B^2}=0$

$\iff \left(a-a_0\right){\displaystyle \sum A\cdot B}+\left(b-b_0\right){\displaystyle \sum B^2}={\displaystyle \sum B\left(C-y_i\right)}$ (11)

So we have the following system:

$\{\begin{array}{l}\left(a-a_0\right){\displaystyle \sum A^2}+\left(b-b_0\right){\displaystyle \sum A\cdot B}={\displaystyle \sum A\left(C-y_i\right)}\\ \left(a-a_0\right){\displaystyle \sum A\cdot B}+\left(b-b_0\right){\displaystyle \sum B^2}={\displaystyle \sum B\left(C-y_i\right)}\end{array}$ (12)

Put into matrix form, the system of Equation (12) becomes:

$\left(\begin{array}{cc}{\displaystyle \sum A^2}& {\displaystyle \sum A\cdot B}\\ {\displaystyle \sum A\cdot B}& {\displaystyle \sum B^2}\end{array}\right)\left(\begin{array}{c}a-a_0\\ b-b_0\end{array}\right)=\left(\begin{array}{c}{\displaystyle \sum A\left(C-y_i\right)}\\ {\displaystyle \sum B\left(C-y_i\right)}\end{array}\right)$ (13)

Thus, the determinant ( $\det M$) of this system of equations is:

$\det M=\left|\begin{array}{cc}{\displaystyle \sum A^2}& {\displaystyle \sum A\cdot B}\\ {\displaystyle \sum A\cdot B}& {\displaystyle \sum B^2}\end{array}\right|$ (14)

$\iff \det M={\displaystyle \sum A^2}{\displaystyle \sum B^2}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\cdot B}$ (15)

Similarly, the determinants associated with a and b are det(a) and det(b) respectively.

Either:

$\det \left(a\right)=\left|\begin{array}{cc}{\displaystyle \sum A\left(C-y_i\right)}& {\displaystyle \sum A\cdot B}\\ {\displaystyle \sum B\left(C-y_i\right)}& {\displaystyle \sum B^2}\end{array}\right|$ (16)

which is worth: $\det \left(a\right)={\displaystyle \sum A\left(C-y_i\right)}{\displaystyle \sum B^2}-{\displaystyle \sum B\left(C-y_i\right)}{\displaystyle \sum A\cdot B}$

So, $a-a_0=\frac{\det \left(a\right)}{\det M}$. (17)

That is: $a=a_0+\frac{{\displaystyle \sum A\left(C-y_i\right)}{\displaystyle \sum B^2}-{\displaystyle \sum B\left(C-y_i\right)}{\displaystyle \sum A\cdot B}}{{\displaystyle \sum A^2}{\displaystyle \sum B^2}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\cdot B}}$ (18)

Also,

$\det \left(b\right)=\left|\begin{array}{cc}{\displaystyle \sum A^2}& {\displaystyle \sum A\left(C-y_i\right)}\\ {\displaystyle \sum A\cdot B}& {\displaystyle \sum B\left(C-y_i\right)}\end{array}\right|$ (19)

which gives: $\det \left(b\right)={\displaystyle \sum A^2}{\displaystyle \sum B\left(C-y_i\right)}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\left(C-y_i\right)}$

So, $b-b_0=\frac{\det \left(b\right)}{\det M}$ (20)

Consequently, $b=b_0+\frac{{\displaystyle \sum A^2}{\displaystyle \sum B\left(C-y_i\right)}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\left(C-y_i\right)}}{{\displaystyle \sum A^2}{\displaystyle \sum B^2}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\cdot B}}$ (21)

The following system is made up:

$\{\begin{array}{l}a=a_0+\frac{{\displaystyle \sum A\left(C-y_i\right)}{\displaystyle \sum B^2}-{\displaystyle \sum B\left(C-y_i\right)}{\displaystyle \sum A\cdot B}}{{\displaystyle \sum A^2}{\displaystyle \sum B^2}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\cdot B}}\\ b=b_0+\frac{{\displaystyle \sum A^2}{\displaystyle \sum B\left(C-y_i\right)}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\left(C-y_i\right)}}{{\displaystyle \sum A^2}{\displaystyle \sum B^2}-{\displaystyle \sum A\cdot B}{\displaystyle \sum A\cdot B}}\end{array}$ (22)

From a Python program, the optimal value of each parameter a and b of Equation (22) is determined by respecting the stopping criterion defined by Equation (23) (Montgomery and Runger, [23] ; Houanou, [24] ; Babaliye, [1] ).

$\frac{a-a_0}{a_0}<{10}^{-6}$ (23)

According to Gérard Degoutte and Paul Royet (2007) and Leipholz [25] , the calculation of Young’s modulus (E) and Poisson’s ratio ( $\upsilon$) can be done from oedometric and shear tests. Thus, the following Equations (24) and (25) are used:

$E=2G\left(1+\upsilon \right)$ (24)

$E=E_{oed}\frac{\left(1+\upsilon \right)\left(1-2\upsilon \right)}{1-\upsilon }$ (25)

where we denote by:

G, the shear modulus,

E, the Young’s modulus,

$\upsilon$, Poisson’s ratio,

$E_{oed}$, the oedometric module.

Equations (24) and (25) allowed us to obtain the following Equation (26):

$2G\left(1+\upsilon \right)=E_{oed}\frac{\left(1+\upsilon \right)\left(1-2\upsilon \right)}{1-\upsilon }$ (26)

Thus, the transformation of Equation (26) becomes:

$\upsilon =\frac{E_{oed}-2G}{2\left(E_{oed}-G\right)}$ (27)

Furthermore, the determination of the oedometric modulus is obtained by Equation (28):

$E_{oed}=\frac{1+e_0}{C_c}\frac{{{\sigma }^{\prime }}_{final}-{{\sigma }^{\prime }}_{initial}}{\log \left(\frac{{{\sigma }^{\prime }}_{final}}{{{\sigma }^{\prime }}_{initial}}\right)}$ (28)

with

$e_0$, Index of voids in the soil in place,

$C_c$, Compression index of the soil in place,

${{\sigma }^{\prime }}_{initial}$, Initial normal stress,

${{\sigma }^{\prime }}_{final}$, Final normal stress.

T results from experimental tests carried out on a series of samples of Tohouè silty sand are as follows:

The samples from the loan were subjected to a granulometric analysis by sieving, Figure 13 below shows the different granulometric curves.

From the curves in Figure 13 below, different parameters are evaluated and the results are recorded in Table 2 below.

From the analysis of Table 2 , it appears that the curvature coefficient C_c is between 1 and 3. Thus, according to the NF P 94-056 [9] standard, the grain size is well spread, therefore, the Tohouè silty sand is well graduated. In addition, the uniformity coefficient is between 2 and 5. It can be deduced that the grain size of said silty sand is spread according to the NF P 94-056 [9] standard. In conclusion, the Tohouè silty sand has a spread grain size.

The percentage at the 0.080-mm sieve pass is 7.67%. This value obtained is lower than those determined by P’Kla et al. [26] , i.e. 12% to 34%, and Tankpinou et al. [7] , i.e. 18.52%. Thus, the silty sand of Tohouè contains fewer fine elements.

The determination of the organic matter content carried out on the silty sand samples gave the results listed in the following Table 3 .

According to Table 3 , the Tohouè sand has a mean organic matter content of 0.13% less than 1.5%, which means that the material is weakly organic [12] .

The results of the methylene blue value are recorded in Table 4 below:

From the analysis of this table, it appears that the methylene blue value of silty sand is 0.41%, between 0.2 and 1.5. Thus, the silty sand of Tohouè is of the sandy-silty type according to standard NF P 94-068 [11] .

The results of the Sand Equivalent test on the Tohouè silty sand are recorded in Table 5 below:

The results from Table 5 show that the Sand Equivalent value of Tohouè silty sand is 23.07%, lower than 60%. Thus, Tohouè silty sand is classified as sandy-clayey with a risk of potential shrinkage or swelling according to AASHTO T176 [13] , EN 933-8+A1 [14] .

These results are presented in the form of a curve with the optimal water content on the abscissa and the maximum dry density on the ordinate (see Figure 14 ).

The results obtained for this test are contained in Table 6 below:

From this table, it appears that the value of the maximum dry density of the

Tohouè silty sand determined at the Modified Proctor Optimum is 1.95 t/m³, a value lower than that determined by Tankpinou et al. [7] , i.e. 2.02 t/m³.

The punching measurements carried out on nine (09) specimens divided into three (03) series according to the molds, are recorded in Table 6 as well as the values of the linear swelling of the specimens. The analysis of the CBR data shows that the CBR index increases when the density increases. The improvement in the CBR is due to the reduction of voids within the sample after compaction.

The CBR index determined for the Tohouè silty sand has a value of 44% to 95% of the OPM ( Table 6 ). This value obtained is between the values of the CBR indices determined by P’Kla et al. (2016) on similar soil types, i.e. 23% and 49%. On the other hand, it is higher than that obtained by Tankpinou et al. [7] , i.e. 3.00%, also on silty sands.

Houngue’s [27] work demonstrated that silty soils with a CBR around 40-50% are often used for subgrades, particularly in tropical climates where humidity varies significantly. He also suggested that stabilizing these soils not only improves the CBR but also durability under climatic conditions.

Similarly, Agossou (2017), studying various soils of Benin including the silty sand of Tohouè, concluded that with a CBR of 44%, these soils can be used as subgrade for moderate traffic roads, but it would be better to stabilize them for higher traffic roads.

As for the linear swelling coefficient of Tohouè silty sand, it is of the order of 0.15%. This linear swelling rate is considered relatively low CEBTP [28] . It indicates that silty sand is not particularly subject to significant dimensional variations in response to variations in water content. This allows us to conclude that Tohouè silty sand has low swelling. This low linear swelling is necessary to ensure the stability and durability of a roadway structure.

Table 7 gives the average values of dry density and optimum content on the run-of-mine material, the calibrated material before and after the shear test. These values are determined to specify the test conditions.

The values obtained made it possible to plot the tangential stresses as a function of the displacement ( Figure 15(a) and Figure 15(b) ), on the one hand, and on

the other hand, the shear stress as a function of the normal stress ( Figure 16(a) and Figure 16(b) ).

From the analysis of Figure 15 , it appears that the curve of the test 1, carried out with a normal stress of 50 kPa, is below that of test 2 relating to a normal stress of 100 kPa. This trend is observed with tests 3 and 4 carried out respectively with a normal stress of 200 kPa and 400 kPa. However, this trend is reversed shortly after the start of the test just before the millimeter of displacement when considering the case of tests 1 and 2, on the one hand, and on the other hand, that of tests 3 and 4. This phenomenon may be due to a reorganization of the material. Also, the same phenomenon is observed on the evolution of the shear stress with the increase of the applied load.

Analyzing Figure 16 , we see that the tangential stress evolves in the same direction as the normal stress. We note that the slope reflecting this increase is of the order of 0.500. The equation of the typical Coulomb line of a shear test is of the form: $\tau =c+\sigma \tan \phi$ where $\tau$ shear stress; c the cphesion; $\sigma$ normal stress and $\phi$ internal friction angle.

By identification, the equations of the line from tests 1 and 2 ( Figure 16 ) made it possible to give the values of c and $\phi$ recorded in Table 8 .

According to Table 8 , we note that the internal friction angle of the Tohouè silty sand is 27.66˚ against 0.54 kPa for internal cohesion.

The results of the oedometric test made it possible to plot the following oedometric curves:

Table 9 lists the parameters taken from the oedometric test using the oedometric curve ( Figure 17 ).

From the analysis of this table, we see that all the determined parameters are constant with the exception of the wet density.

The evaluation of the report $\frac{C_C}{1+e_0}=0.022$ showed that the silty sand of Tohouè is not very compressible ( [29] ). This behavior of the material may be due to its low fine particle content which is around 7.47%.

Furthermore, the swelling coefficient is 0.12%, less than 1%. Consequently, the silty sand of Tohouè is not swelling.

The successive iterations, carried out from the equation system (11) led to the determination of the optimal value of the parameters of the Hardin Drnevich numerical model. These different values are presented in Table 10 below. They concern the normal stress, the shear modulus and the shear stress.

Figures 18(a)-(d) and Figures 19(a)-(d) below represent the Hardin and Drnevich hyperbolic behavior curves of the soil from the Tohouè silty sand.

We observe that the stress-strain curves of the model are very close to those of the observations. Thus, we can say that the model fits the observations well.

Table 11 shows the different values of the calculated coefficient of determination.

According to Table 11 , the coefficients of determination of each compaction energy are close to 100% (varies from 91.99% to 99.40%) except for sample 1 at 50 kPa, this observation may be due to a reworking of the material during the test or a relaxation. This clearly shows that the Hardin and Drnevich model are adequate.

Table 12 presents the results of the oedometric test evaluated on the silty sand of Tohouè.

These results show that the soil studied has low compressibility as the C_c is less than 0.2 according to the NF EN ISO 17892-5 [30] standard and good resistance to deformation under applied loads, which is favorable for its use in construction applications where significant loads can be expected.

Table 13 below gives the Poisson’s ratios and Young’s moduli derived respectively from Equations (24)-(27) and (24) or (25).

The analysis of this Table 13 shows that the shear modulus varies from 64.941 kPa to 103.848 kPa and the Poisson’s ratio varies from 0.392 to 0.484 while the Young’s modulus varies from 51.129 MPa to 289.110 MPa. We note a decrease in the Poisson’s ratio with the increase in normal stress like the normal stress at 50 kPa. In addition, the shear modulus and the Young’s modulus increase with the normal stress at different applications.

Analysis of the data in Table 14 shows that the material contains little water because its average water content, 3.5%, is less than 4%. In fact, according to the NF P 94-093: [15] standard, this material is good for compaction.

According to this Table 15 , the silty sand of Tohouè meets all the criteria for use as a foundation layer for flexible pavements.

According to this Table 16 , Tohouè silty sand does not meet all the criteria for use as a base layer for flexible pavements. To use it as a base layer, it is necessary to make an improvement, either by cement stabilization, litho-stabilization, or other means.