The design of geotechnical structures using numerical methods

The use of numerical simulations in geotechnical engineering today is often limited to the investigation of the serviceability limit state (SLS). Therefore, the main focus up to now was more on the deformation prognosis for geotechnical constructions than the geotechnical design itself. Nevertheless, numerical methods like the Finite-Element-Method (FEM) for example can be a valuable tool to investigate the bearing capacity of geotechnical structures as well. E.g. existing structures which cannot be calculated using classical analytical approaches can be investigated with numerical simulations to estimate the remaining bearing capacity. In the present paper, the use of numerical methods for the design of geotechnical structures will be discussed. First, the approved ϕ - c reduction method is recalculated and verified with the help of a script based on the Python programming language. Afterwards, the Python script is extended so that the diameter of an Embedded Beam Row element can be successively reduced automatically similar to the ϕ - c reduction. Finally, a short outlook is given in which direction the research project should proceed.


Introduction
Numerical methods are increasingly used for complex geotechnical tasks. Therefore, methods for using numerical models for different tasks have to be developed. Especially for the prognosis of deformations within the serviceability limit states (SLS), the application of the Finite Element Method (FEM) has been proved in geotechnical engineering. To use this method in the future to verify the ultimate limit states (ULS) of geotechnical structures, it is the aim of the studies presented in this paper to examine the application of numerical methods for design purposes in more detail. For slopes it is already possible to verify the stability by means of the so-called strength reduction method. This becomes problematic as soon as structural elements such as dowels are located in the slope.
To investigate this issue, this paper presents a comparative calculation. In this study the approved -reduction method implemented in the software Plaxis 2D is recalculated and verified with the help of a script based on the Python programming language. Afterwards, the Python script is extended so that the diameter of an Embedded Beam Row element, that is located in a slope, can be successively reduced similar to the -reduction. This is done for different friction angles of the soil to identify the ultimate limit state. The resulting effects are analysed and presented.
Finally, a short outlook is given in which direction the research project should proceed. The main focus here is on the consideration of structural elements in the area of stability analysis to find a uniform safety factor with the help of the Phi-c reduction.

Strength reduction method
For the studies in this paper, a safety factor SF against failure in a soil mass by using the strength reduction method is used. In principle, an artificial failure mechanism is created in the Finite Element Method (FEM) for this purpose, resulting in a limit state in which equilibrium is no longer present. For this purpose, the characteristic shear parameters (friction angle ′ and cohesion ′) are incrementally decreased (tan ′ and ′) under the assumption of a Mohr-Coulomb failure criterion (e.g. [1] and [2]). This leads to the following equation for the safety factor SF: Accordingly, the characteristic shear parameters are reduced in the same ratio. Therefore, figure 1 shows the limit state conditions for tan ′ and ′ of a soil mass in the shear diagram. After reduction of the shear parameters, the stress state is at the new limit condition. This is done until stress redistribution is no longer possible and an artificial failure is caused.

Comparative calculation
The geotechnical software Plaxis 2D, 2019 is used for the following comparative calculation. Within the software the -reduction is already implemented as a safety analysis. Additionally, it is possible to use the remote scripting interface, which is based on the Python programming language. This allows the user to run the input program via an external Python handler, to execute script files for example. In order to study the -reduction in more detail, it is replicated using a Python script file and calibrated using comparative calculations.
In the present paper's comparative calculations, the safety factor is determined using three different methods: 1. Theoretical safety factor assuming the Mohr-Coulomb failure criterion. 2. Safety factor determined by the safety reduction method implemented in Plaxis. 3. Safety factor calculated using the Python script file.
For the following analysis, the slope angle is varied and the safety factor is then calculated. The soil parameters remain unchanged.

Model information
In the following, the boundary conditions of the investigated model are presented. Figure 2 shows the geometry of the model to be studied. For the calculations, the slope angle s is varied from 29° to 15°. The calculation section was chosen sufficiently large according to the advices from [2], so that external influences can be excluded. This means that no significant stress changes or deformations occur at the boundaries. The soil considered in this study is a cohesionless soil (Material set 1). Thus, slope failure is only dependent on the friction angle. Table 1 lists all relevant soil parameters used.
In the course of the evaluation, it became evident that the relevant sliding circles considering a non-cohesive material tend to locally form in the area near the surface. For this reason, an additional material set 2 (see table 1) is examined, which considers a low degree of false cohesion ( ′ = 0.1 kN m² ⁄ ). To illustrate this phenomenon, the decisive sliding circles for the material sets 1 and 2 for a slope angle of = 25° are shown in figures 3 and 4.    In addition, the discretization of the FE-mesh influences the failure mechanism established within the slope. [4] In order to exclude this, a very fine mesh is used for the comparative calculations (see figure 5).
It should also be noted that all calculations are based on drained conditions.

Results of the comparative calculation
The results of the comparative calculation are shown in figure 6. The calculated safety factors are plotted against the investigated slope angles. Comparing the different methods, the deviations between the calculation methods in relation to the safety factor are within an acceptable range. The results using the Python script deviate from the results using the other calculation methods by a maximum of no more than 5 %.
In addition, the results show that the Plaxis calculation without the approach of a false cohesion (Graph 2.a) provides almost identical results to the theoretical basis (Graph 1). The deviations are below 2 %. A disadvantage here, as already noted, is that the sliding circles are only present in the area near the surface (see figures 3 and 4). Furthermore, it can be seen that the results using the Python script (Graph 3) deviate less from the results of the Plaxis calculation using a false cohesion (Graph 2.b) than without the cohesion approach. This conclusion is also confirmed by a comparison of the critical sliding circles received from different calculations. Figure 7 shows the most critical sliding circle received out of the calculation using the Python script regarding the case of a slope angle of s = 25°. Compared to the results in figure 4 good agreement is visible. In the following table 2, the calculated safety factors regarding the different calculation methods are summarized for three different slope angles.

Influence of structural elements
In this section, the behaviour of an Embedded Beam Row representing structural elements in the slope is examined. The objective of this calculation is to find the optimal cross section of the dowel so that the slope is calculative stable. For this purpose, the Python script from chapter 3 is extended so that the diameter of the used Embedded Beam Row can be successively reduced.

Model information
The slope of figure 2 is enhanced with an additional Embedded Beam Row element at half slope height (see figure 8). The soil parameters used are listed in table 3. In this case, the slope angle remains constant at = 30°.   For these calculations, the Plaxis 2D 2020 version is used. In this new version, the Embedded Beam Row elements are improved. [5] The aim of this calculation is to numerically simulate a failure state in which the failure mechanism is clearly dependent on the structural element. Therefore, a friction angle for the slope ( = 30°) is initially determined using the Python script from section 3 such that failure occurs. In order to prevent near-surface sliding circles, a cohesion of ′ = 1.0 kN/m² is assumed in this case. The calculation shows that the slope fails at a friction angle of ′ = 26.7°. Furthermore, the structural element is modelled fixed in the natural soil below the slope. For this purpose, another material set with increased cohesion representing the natural soil (material set 4) is introduced (see table 3). This ensures that the surrounding soil does not fail due to lateral displacement of the entire pile.
For the Embedded Beam Row, piles with a length of = 10 m and a circular cross section out of steel are considered. The axial distance between the piles was determined based on the investigations of [6], which are summarized in figure 9. Accordingly, the ratio between the axial distance and the diameter of the pile should be at least spacing ⁄ ≥ 2. This value is kept during the successive reduction of the diameter for the axial distance calculated from it. Figure 9. Ratio of axial distance spacing to diameter to describe the range of application of structural elements in 2D and 3D. [6] The following table 4 lists the material parameters for the investigated Embedded Beam Row. It should be noted that the material is modelled as steel using an elasto-plastic formulation so that plastic failure can occur in the element. Based on this, an elastoplastic material behaviour is considered and the plastic moment as well as the plastic normal force are calculated as a function depending on the changing diameter.  For the calculation, as already mentioned, the diameter of the pile is successively reduced until failure occurs. This procedure is carried out for different angles of friction. As already stated in the comparative calculations in section 3, drained conditions are taken into account. Furthermore, a very fine element distribution comparable to the study in section 3 is used (see figure 10).

Results of the calculation
The calculation results have shown that for certain friction angles (26.7°> ′ > 24.1°) slope failure depends on the diameter of the structural element. Accordingly, the optimum situation can be determined by successively reducing the pile diameter. For illustration, figure 11 shows the determined pile diameter in the case of failure as a function of the investigated friction angles of the soil. Furthermore, figures 12 to 14 show the calculation results for the case that the internal friction angle is reduced to ′ = 25.0°. The calculated minimum pile diameter of the structural element in this case is = 0.0504 m. Figure 12 illustrates the deformation of the structural element and indicates that a joint is formed in the upper part of the element. This causes the pile to buckle and fail due to the lateral load of the sliding slope.
In figure 13 the incremental shear strains within the slope in the case of failure are depicted to indicate the failure mechanism. The location of the joint within the Embedded Beam Row corresponds to the location where the sliding circle cuts the structural element.
The resulting moment in the pile is shown in figure 14.
The maximum moment is formed at the joint and exceeds the plastic moment of the Embedded Beam Row ( max = 8,978 kNm > pl,Rd = 5,014 kNm).    Further reduction of the friction angle ( ′ < 24,1°) has no significant influence on the structural elements, as the soil above the element fails before failure of the structural element occurs. Illustrating this, figure 15 shows the incremental strains in the slope for a friction angle of ′ = 23.9°. It can be observed that the critical sliding circle lies above the structural element. From this, it can be concluded that in the case of higher friction angles of the soil the Embedded Beam Row element is decisive for failure. The part of the slope above the structural element is decisive for slope stability if the friction angle lies below a critical value.

Outlook
This study demonstrates that an automated strength reduction calculation to find a failure mechanism using an adaptable Python script is possible. The method implemented using the Python script is verified with acceptable results. This automated script was enhanced to find the critical diameter of an Embedded Beam Row element in a slope. Therefore, the resulting task is to reconcile these two methods and to determine a common safety factor.
Moreover, the studies so far have been carried out using relatively simple pile geometry. Due to this, the future aim is to be able to apply the methods presented in this paper on more complex cases, for example a double T-beam. In addition, it is the aim to be able to include further components in future calculations, such as an anchor layer.
These thoughts can only be realised if the studies are transferred to more complex geotechnical constructions such as a single back-anchored construction pit for example. This extends the challenge for future studies. Beside the existing task to find a uniform safety factor, it is the aim to calibrate and compare the calculation results with models in-situ.
This results in a further research aspect, that has to be dealt with in future. At present, numerous analytical design verifications are required for a single back-anchored excavation pit, all of which are subject to different partial safety factors. In Germany the analyses of the individual components are carried out according to procedure 2 (GEO-2). In order to use FEM for the design of geotechnical structures, it is intended to determine a safety factor that is in line with the current concept of partial factors of safety. [7], [8], [9] 6. Summary In the context of this paper, three different calculation methods for the determination of a safety factor regarding slope stability are examined and compared with each other within the scope of a comparative calculation. The safety factor is determined for different slope angles using identical soil parameters. This allows it to obtain direct information about the safety factor for different angles of repose. The results show that the successive reduction of the friction angle can be implemented in a Python script to obtain comparable results to the -reduction implemented in Plaxis. 10 Subsequently, the knowledge gained is used to investigate the influence of structural elements within a slope. For this purpose, the Python script has been extended, such that it allows to successively reduce the diameter of an Embedded Beam Row element analogous to the -reduction. The critical pile diameters determined in this way have been investigated for different friction angles of the soil. It is shown that there is a certain area where the structural element is decisive for the total failure of the slope. This failure process is characterized by exceedance of the plastic moment of resistance, such that a joint is formed in the element and it buckles.
Using the results of the calculations presented in this paper, the aim is to determine a safety factor that can take into account simple structural elements in the area of slopes in the context of a numerical strength reduction.