Evaluation of Load Capacity of Stratified Soils (2 Layers) by Means of Numerical Analytical Comparison

The methods for determining load capacity in stratified soils are numerous and differ in their methods and results. These differences in analysis lead to uncertainty in engineering practices or over-dimensioning of the foundation solution. This study seeks to determine three analytical methods of load capacity in stratified soils (2 layers) for shallow foundations: 1) Imaginary foundation, 2) Average parameter method (APM), and 3) Terzaghi's method to compare their results with those obtained from numerical modeling by means of the finite element method using a widely applied software (Abaqus academic version). Within the methodology developed in the finite element modeling, variables were parameterized (modulus of elasticity, depth of deflection, and displacement-load) and two behavioral laws were evaluated (Elastic and Drucker-Prager). The results that were obtained from the analysis show that when performing numerical modeling using the law of elastic behavior in soils of two layers, exaggerated results are generated with respect to analytical methods. Another important result is that when hard soils are on top of soft soils the results of numerical and analytical methods tend to be similar to each other. 1 Universidad del Cauca (Popayán-Cauca, Colombia). jairoalejan@unicauca.edu.co. ORCID: 0000-0003-13657629 2 Universidad del Cauca (Popayán-Cauca, Colombia). bjmartinez@unicauca.edu.co. ORCID: 0000-0002-22218444 3 Ph. D. Universidad del Cauca (Popayán-Cauca, Colombia). lucruz@unicauca.edu.co. ORCID: 0000-00032438-5526 Evaluation of Load Capacity of Stratified Soils (2 Layers) by Means of Numerical Analytical Comparison Most importantly, the variables that have the greatest influence on the load capacity in soils of one and two layers are the angle of friction, yield stress, and in the case of numerical analysis the constraint displacement (load). In addition, it was observed that for numerical modeling better results are obtained when considering an elastoplastic model, such as Drucker Prager.

Most importantly, the variables that have the greatest influence on the load capacity in soils of one and two layers are the angle of friction, yield stress, and in the case of numerical analysis the constraint displacement (load). In addition, it was observed that for numerical modeling better results are obtained when considering an elastoplastic model, such as Drucker Prager.
Dentro de las conclusiones más importantes tenemos que las variables que más

I. INTRODUCTION
Foundations are essential for any infrastructure project [1] which must comply with safety regulations while maintaining a balance between service life and budget restrictions as well as environmental factors and imposed design rules. One of the most important elements in a proper foundation is the stratification of the ground beneath the foundation. Poor characterizations of the ground can lead to increased stresses which may result in failure due to general shear failure [2][3]. Most of the calculations described in previous theories consider soil to be homogeneous [4] while disregarding its heterogeneity and providing results which may be unexpected in the field. As a result, this study evaluates the load bearing capacity [5] in stratified soils (two strata) [6,[7][8] using a comparison between three current analytical methods and results from numerical modeling. The analytical methods utilized in the study included Terzaghi [9], Average imaginary foundation [10], and the average parameter method (APM) [11], which differ considerably in their results. As a result, the results obtained will be compared to numerical modeling using the finite element method [12,[13][14] which is obtained when the Prandtl fault surface is formed by increments in controlled displacement.
Numerical modeling is a widely used and studied method applied to engineering foundations as seen in: [12,[18][19]. It is also accepted as a design methodology in some countries including Colombia with its building construction standard: NSR-10.
In the finite element modeling of this study, different variables (modulus of elasticity, displacement depth, and displacement-load) were parameterized and two behavioral laws were evaluated (Elastic, Drucker-Prager). These behavioral laws were used to identify the parameters that have the most influence on the calculation of a shallow foundation [15]. Additionally, their insight can be extended to other fields since all infrastructure work requires a foundation providing significant benefits and savings which will generate positive impacts on both the economy and on the safety of the structures.

II. METHODOLOGY
The methodology used in this research study is mainly characterized by using models generated in the Abaqus [16] program and analytical calculations of load capacity. Three load capacity methodologies for stratified soils were chosen: APM, Terzaghi's method, and imaginary footing. These methodologies were chosen since they are the best known and most commonly applied methods in engineering.
The study utilized the General Static Load Carrying Capacity (GSLCC) for each stratum. The soil of each stratum was also modeled using finite element modeling in Abaqus while modifying yield stress [17]. The load capacity in stratified soils (two strata) was found using analytical methods, APM, imaginary footing, and Terzaghi's method and then calculated using the elastoplastic behavior law and applying the Drucker Prager method [18].
The soil thicknesses utilized in the study was e=0.15m and e=0.5m while taking into consideration that a thickness below 1m results in a greater probability of the formation of the Prandtl fault [19] which will be used as a point of reference between the analytical and numerical methods.
It is important to note that the ideal values of yield stress are taken from the numerical models carried out using one stratum modeled against two strata. The initial condition resulted in the cohesion in soils of one stratum and two strata being null for both cases Df=0m and Df=0.5m. In the case of Df=0.5m and e=0.15m the foundation will be offset in a similar manner. The data from the thickness of the soil layer using analytical methods was not taken into account since it would result in an overload in the numerical modeling.

A. Methodology for Soil of a Stratum
The analytical methods used to evaluate the soil of a stratum model are shown in   In Table 1 the data used to modify the model of one stratum and two strata with their respective nomenclatures are shown. To create the Drucker Prager elastoplastic model in the finite element program, the following input variables are needed as shown in Table 2.

B. Methodology for Stratified Soil (2 Strata)
Simulations with fine soils on top of thick soils and coarse soils on top of fine soils used the following combinations ( Table 3).

III. RESULTS AND ANALYSIS
The results from the analytical and numerical methods are presented below.

A. Soil of a Stratum
The results of the calculations performed using the numerical and analytical method are presented for a stratum which was modified by yield stress.  Table 4. The graphical results obtained from entering the model into the finite element software are shown in Fig. 3. The contour diagram of the plastic deformations (PE)

1) Calculation of Load
is shown in Fig. 4 and includes a contour diagram of vertical forces. Both Figure 3 and 4 correspond to Soil A, with Df=0 m and e=0.5 m.    Table 5.

3) Calculation of Load Capacity, Df=0.5 m, e=0.15 m. The load capacity results
from the numerical and analytical modeling are presented with a depth of displacement of Df=0.5 m, and a thickness of e=0.15 as shown in Table 6.   Table 7.

B. Stratified Soil (2 Strata)
The results of the analytical methods (APM, Terzaghi, and imaginary foundation) and numerical methods (Abaqus) for two strata with varying depths of displacement and thicknesses are shown below. The numerical models and analytical calculations were calculated for a Df=0 m with a thickness of e=0.15 m are shown in Table 8.  The graphic results of the stratified soil (2 strata) of soil combination 1 Df=0 m and e=0.5 m, are shown as a contour diagram of plastic deformations in Fig. 9 and of vertical forces in Fig. 10.  For the results of the numerical modeling and analytical calculations with a Df=0 m and e=0.5 m, the results are shown in Table 9. The results of using a Df=0.5 m and e=0.5 m are shown in Table 11.  Table 8 to Table 11 and are composed of soils 1, 2, and 3. The load capacity determined using the finite element method and Imaginary foundation are very similar while the results from the APM method are notably higher. This is due to the fact that this method in theory [20], says that better results are obtained when hard soil is on top of soft soil. This clarification is made again in the Terzaghi method which omits a combination of the first three soils since it has not been defined for these soil structures.
An analysis of hard soils on top of soft soils are represented by the combinations of the last three rows of Tables 9 to 12.
When comparing the results obtained from the Terzaghi and APM methods shown in Tabla 10, the APM method has a greater value than the reference value obtained in Abaqus. This is the contrary to the value obtained using the Imaginary foundation method which gives a lower value than that of Abaqus. A unique case is presented in soil combination 5 which demonstrates a difference in the results between the analytical method (Terzaghi's method) and the numerical method (Abaqus) which is less than 1.
The results of soils 4 and 5 and are about the Imaginary foundation and Terzaghi method are lower than those of Abaqus. This contrasts the results using the APM method in which the Abaqus results are higher as shown in Table 8. Furthermore, for soil 6 all of the results from the analytical methods are greater than the numerical methods.
The values obtained from the modeling in Abaqus are lower than the results using analytical methods as shown in Table 11; however, in Table 11 the values are closest to Abaqus. The values increased with the imaginary foundation method being closest, followed by the APM, and the Terzaghi method having the greatest difference. In the case of Table 9, the values from the analytical methods closest to the numerical ones are those obtained using the APM method. This method resulted in a difference between 38 and 49 kPa, followed by the Terzaghi method with a difference between 227 and 1019 kPa, and finally the imaginary foundation method with results which are further from those found using Abaqus.

IV. CONCLUSIONS
After the numerical and analytical calculations had been performed, it was concluded that the variables that primarily influence the loading capacity in soils of one stratum and two strata are the friction angle, yield stress, and failure displacement (loadboundary condition). The most resistant soils do not fully plasticize (high yield stress values) because they do not achieve the levels of deformation necessary to cause failure. This suggests that it would be necessary to reach very high levels of deformation that cannot be represented with the conventional finite element method.
When meshing is performed, finer materials provide better rendering levels, but result in more computational time to create the model in the program. Gravity must be considered in the finite element model in order for the model to have a natural geostatic behavior. The APM method and the numerical modeling method provide the best approximate for what would be physically and semi-empirically expected in a stratified soil structure (2 strata) with a hard soil on top of a soft soil for the evaluation of bearing capacity using elastoplastic behavior. On the other hand, the imaginary footing analytical method and the numerical modeling method are the best for determining what is to be physically expected semi-empirically when there is a soft soil on top of a hard soil. The results of the APM method for soft soils on top of hard soils do not represent results that agree with other methods and the respective values that would be physically expected in the field. As a result, it is not recommended to take these results into account as a design method in the latter case. Finally, this study highlights that the Prandtl fault is reproduced or formed in detail in numerical models when the modeled soil thickness is less than 1m.