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This work studies the influence of mechanical and geometrical characteristics of the concrete and the soil on the stresses in a mat foundation. In this study, the soil-structure interaction is modeled by two parameters, the modulus of subgrade vertical reaction (
*k*) and the modulus of subgrade horizontal reaction (2
*T*). These two parameters are dependent on the geometrical and mechanical characteristics of the system. Results of this study show a sensitivity of solicitations to variations of geometrical and mechanical characteristics of the model. Although solicitations in the plate are sensitive to mechanical properties of concrete, these solicitations are strongly influenced by the mechanical and geometrical characteristics of the soil mass. However, it should be noted that the influence of E
_{b} is denoted in the center of the plate whereas the E
_{s} feels almost in the same manner over the entire extent of the plate. This study also shows that for the same load cases, the values of the torsion moment and shear stress are not significant those of bending moments and normal stresses, respectively.

Developments in the construction of civil engineering and especially disorders observed in the supporting structures push the practitioners to better take into account soil-structure interaction in the process of calculating the foundation structures. In addition to this, the structural and geotechnical calculations of foundations are generally conducted separately, that’s why it would be interesting to develop a computational approach that would take into account the geotechnical and structural considerations related to the problematic of the calculation of foundation. It is in this context that an accurate and complete characterization is needed for reliable calculation, hence the interest of this research. A foundation is responsible for transmitting loads from the superstructure to the soil. It provides an interface between the upper part of the structure and the soil. A mat foundation can be considered as a reinforced concrete slab over the whole of the structure whose study is mainly governed by the plate theory [

A mat foundation is considered as a reinforced concrete slab resting on the soil mass. Usually, the thickness is small compared to other dimensions. It is in this context that the Kirchhoff model [

Mat foundations are structures in planar state constraints. They admit vertical displacements along the z axis and the behavioral model may be governed by the model of Kirchhoff for the plates [

Considering the deflection (w) as known, the behavioral model of a plate supported on its periphery can be given by the following equation:

where D is the flexural rigidity of the plate and is given by:

with:

E_{b}: elastic modulus of the material constituting the plate;

e: the thickness of the plate;

ν_{b}: Poisson’s ratio of the plate.

To model the subgrade, the Filonenko-Borodich biparametric model was selected. The model of Filonenko- Borodich [

with:

Both elastic constants of the model are the coefficient of reaction (k) and the tension (T).

The study of influencing factors in this model has been the subject of several scientific publications [

Several authors have had to work on the modulus of reaction k, but all authors coming after Biot gave it a higher value than predicted by Biot [

where:

E_{s} is the modulus of subgrade;

ν_{s} is the Poisson's ratio of the subgrade;

B is the width of the foundation;

E_{b} is the Young modulus of the concrete foundation;

I is the moment of inertia of the cross section of the concrete.

The reaction shear modulus (T) has been proposed by Vlasov and given by the following relationship [

With:

E_{s}: young’s modulus of the soil;

ν_{s}: soil Poisson ratio;

H: thickness of the soil layer (depth of the rigid substrate);

Ф(z): a function which describes the variation of the displacement w(x, y) along the z axis, such that:

To a relatively deep layer of soil where the normal stress may vary with depth, it is possible to use, for the function Φ(z), two types of variation (linear or nonlinear). Selvadurai [

This work shows the displacements along the two variations of Ф(z). The linear variation gives after integration, the following shear parameter:

The hyperbolic variation leads to the following value of T:

To study the reinforced concrete structure, the Kirchhoff model was selected. For analysis of soil foundation, the Filonenko-Borodich model, which assimilates the soil to a spring assembly (elastic modulus k) infinitely close to each other and connected by an elastic membrane (2T voltage), was retained. The superposition of the two previous models leads to the behavior of the plate on the soil mass as shown by the

The theory of plate and taking into account the soil-structure interaction (biparametric model) lead to raft foundations behavioral law that may be governed by Equation (12) below:

With the parameters D, k, T explained above.

In this research, it is assumed a uniform distribution of forces applied to the foundation system. Therefore q(x, y) is a constant value “Q” because it can be assumed that at any point of the foundation, there is a uniform stress distribution. For analytical resolution of the system, the double Fourier series are used. So it is assumed that q(x, y) can be written in the following form:

After analytical resolution, the expression of the deflection is given as follow:

At this value of deflection, it is added the displacements of the points of the interface under the effect of the dead weight of the slab before overload. This displacement is a function of the weight of slab and the value of the elastic modulus of the subgrade and estimated at 25,000 × e/k. From the analytical solution, it is able to demonstrate the influence of the parameters of the model of behavior on the displacement of the plate. A rectangular plate of 20 m × 20 m is considered, with thickness ranging from 20 cm to 80 cm. The plate rests on a soil with elastic modulus ranging between 4 MPa and 8 MPa and Poisson's ratio of between 0.2 and 0.45. The plate is subjected to uniform loading of 200 kN/m^{2}. The influence of all these parameters is illustrated in the following (Figures 4-16).

After determining the expression of w(x, y) at any point (x, y), the solicitations are given for the following expressions.

Bending moments M_{x} M_{y} and torsion moment M_{xy} are given by:

Giving:

And shear forces are given by:

The stress state at the point m(x, y) is given by the following system of equations:

The study shows that even if the displacement is maximal at the center of the plate (_{s} as shown in _{b}, νb) as shown by [

The stress distribution on the upper and lower faces of the plate (_{b} is most felt in the center of the foundation (_{s}) is almost felt in the same way over the entire plate (_{y} for various values of Poisson’s ratio.

The study also shows that even for a distributed load, the values of torsion moment and shear stress were not significant. The results of this study combined with those of Sall [

Results from this research show that even if the displacement is maximal at the center plate, the maximum values of the stresses are not observed at the same point. The study shows a sensitivity of solicitations compared with a variability of geometrical and mechanical characteristics of the model. Although solicitations in the plate are sensitive to mechanical properties of concrete, they are strongly influenced by the mechanical and geometrical characteristics of the soil mass. However, it should be noted that the influence of E_{b} is denoted in the center of the plate whereas the influence of E_{s} is almost the same over the entire extent of the plate. For low values of thickness, the interior efforts are almost constant on the full extent of the foundation. For high values of the depth of the rigid substrate, solicitations are almost constant over the whole of the plate. This study also shows that for the same load cases, the values of torsion moment and shear stress are not significant, compared respectively, with those of bending moments and normal stresses. This study also shows that calculations geotechnical and structural of the foundations should not be carried out separately. The results of this research should allow a good comprehension and a good taking into account of the soil-structure interaction in the calculation of foundations. In the future, it would be interesting to study the influence of the parameters of the model with more realistic laws of behavior.