Digital solutions in the determination of bearing capacity for straight and stepped mat foundations on sloping ground

On sloped sites where a mat foundation is to be used, because of the limited bearing capacity of the ground, it is necessary to cut into the sloped surface to provide a level base for the mat foundation. To reduce the amount of excavation and backfill a stepped mat may be used down the slope. Traditionally, determining the bearing capacity values for spread foundations on sloping ground has been a complex and time-consuming process. Several factors must be considered, including various variables (soil type, soil profile, slope angle and geometrical configuration of the infrastructure). The digital solution used in this study involves the use of advanced software that takes into account various input parameters and performs complex calculations to determine the bearing capacity of the foundation on horizontal and sloped ground. For this research, the analyses were done for three different slope angles (β = 5°, 10°, 15°) and a comparison is made between the bearing capacity of the ground for straight and stepped mat foundations using both analytical methods and finite element method.


Introduction
Building on sloping ground can present a number of challenges, including issues with foundation stability or ground general instability, but with proper design, planning and execution, it is possible to create safe and stable buildings that are visually appealing and blend in with the surrounding landscape.One way to address these challenges is through the use of set-back foundation configurations.Set-back foundations are designed that each level of the building is set back from the one below it, following the slope of the ground [1].These types of foundations allow the building to follow the natural slope and the ground will provide adequate support and stability.However, the bearing capacity of these foundations must be carefully considered.
The computation of the soil bearing capacity is one of the most important problems in geotechnical engineering and has a decisive role in the design of safe and economic spread foundations.Terzaghi was the first to evaluate the bearing capacity of a strip footing on a horizontal and homogeneous ground [2].The computation relationship includes the influences of the soil properties (internal friction angle, cohesion and unit weight) and the features of the foundation (foundation depth and the width of the footing).This well-known relationship was updated by different authors [3][4] [5] [6] taking into account new contribution factors: base inclination (bc, bq, bγ), embedment depth (dc, dq, dγ), ground surface inclination (gc, gq, gγ), load inclination (ic, iq, iγ), foundation base shape (sc, sq, sγ) and load eccentricity.This paper focuses on the ground surface inclination factors named also slope factors.The first estimation of the bearing capacity factors for a strip foundation on sloping ground was proposed by Meyerhof in 1957 [4].After that, Brinch Hansen [5] and Vesić [6] proposed new relationships for the slope factors.These slope factors are independent from the internal friction angle.Later, some researchers have published research about the bearing capacity of foundations on sloping ground, but was mostly limited to purely cohesive massifs [7], [8], [9] or to purely cohesionless massifs [10].Some investigations showed that, in the case of cohesionless soils, the bearing capacity is always governed by the foundation soil failure, while in cohesive soils the bearing capacity is dictated by the slope stability 1304 (2024) 012033 IOP Publishing doi:10.1088/1757-899X/1304/1/012033 2 [11].Van Baars proposed new slope factors based on the numerical calculations, taking into account the influence of the internal friction angle [12].
Considering the increased number of buildings positioned on natural slopes but also the multitude of situations caused by architecture design and asymmetric shapes, the evaluation of the soil bearing capacity becomes more complex.In this paper the bearing capacity for buildings' foundations located on sloping grounds is investigated using the finite element method and compared with the existing analytical equations.

Context and description of the foundation design conditions
One footing width (B = 30 m) in two configurations was considered.The geometries of the problem analysed is shown in Figure 1.Cases (a1 to a4) represent a straight mat foundation that transmits a uniform vertical pressure to the foundation ground and cases (b1 to b4) represent a stepped mat foundation.A step with the height of 3.0 m was considered, which divides the foundation into a 10 m and 20 m openings.Four slope angles were considered (β = 0°, 5°, 10°, 15°).For each geometry, six soil types with different geotechnical characteristics are considered, traced from NP 122:2010 [13].The detailed geotechnical parameters for the six types of soils (S1 to S6) are presented in Table 1.The ultimate bearing capacity of the ground is determined using both analytical methods and finite element method.

Analytical methods
A total of 72 analytical calculation has been performed for comparison purpose.This calculation has been done using Brinch Hansen, Vesic and Van Baars methods [5; 6; 12].The general equation used for evaluating the ultimate bearing capacity qu is: where B is the width of the foundation; B' is the reduced width of the foundations; is the soil unit weight; c is the cohesion of the soil; ' q is the effective overburden lateral to the footing level; , ,  2.

Finite element method (FEM)
The FEM involves the application of the Mohr-Coulomb model, which considers a linearly elasticperfect plastic behaviour for the soil.This model requires five input parameters, which include two parameters related to elasticity (i.e.soil modulus Es and Poisson's ratio ν) and three parameters related to strength (internal friction angle ϕ, cohesion c and the dilatancy angle ψ).The values for the soil modulus and Poisson's ratio were kept constant (Es = 20 MPa and ν = 0.4990) for all types of analyzed soils.
This study includes finite element analyses that are exclusively plane strain analyses and were carried out using the Plaxis2D software.The dimensions of the model have been selected in such a way that the model boundaries do not intersect with the significant isobars, as shown in Figure 2. The finite element model has implemented the "standard fixity" condition, whereby the lateral vertical boundaries are prevented from moving in the horizontal direction, and the bottom boundary of the model is restrained from both vertical and horizontal movements.
The discretization of the model was accomplished using triangular elements with 15 nodes.The meshing was performed through a fully automated finite element generation technique that was integrated into the Plaxis2D software.The mesh coarseness was set to "medium".In order to obtain the most accurate results, a mesh with medium element distribution was generated, with a coarseness factor of 0.10 for the area next to the foundations (60.00 m to the right, 30.00 m to the left and 45.00 m on the bottom).Figure 2 displays the standard meshing that was generated for the numerical model.

Figure 2. A model generated in Plaxis2D
In the Initial Phase zero initial stresses are generated by using the gravity loading procedure with ΣMweight equal to zero.The prescribed load is activated in a separate phase.The calculation type is Plastic Analysis and a tolerated error of 0.01 is defined.The applied pressure was selected so as to obtain the collapse of the ground.The bearing capacity qu was obtained by multiplying the considered pressure with the reached phase proportion ΣMstage.

Results and discussions
Four different methods to evaluate the bearing capacity of spread foundations have been used, and their effectiveness has been studied.The results obtained using different methods may differ significantly, leading to different values on the bearing capacity.

Analytical methods
The bearing capacity values determined using analytical methods are presented in Table 2.For horizontal ground (β = 0°) the smallest values for qu are obtained using Brinch Hansen method for all type of analysed soil.The largest values for qu are obtained in the case of Vesić method.The percentage difference of qu between Vesić and Brinch Hansen methods is between 35.30% and 45.05%.For sloped ground the smallest values for qu are obtained using Van Baars method.Here is the largest rate of decrease of the bearing capacity with the increase of the slope angle.

Finite element method
The detailed results obtained in the finite element analyses are presented in Appendix A. For soil type S2, when the slope increases from 0° to 15° the bearing capacity decreases from 702 to 265 kPa (Figure A2).For soil type S3, when the slope increases from 0° to 10° the bearing capacity decreases from 475 to 270 kPa (Figure A3).For soil type S4, when the slope increases from 0° to 15° the bearing capacity decreases from 883 to 485 kPa (Figure A4).For soil type S5, when the slope increases from 0° to 15° the bearing capacity decreases from 450 to 306 kPa (Figure A5).For soil type S6, when the slope increases from 0° to 10° the bearing capacity decreases from 480 to 256 kPa (Figure A6).This shows that the decrease of the bearing capacity of the soil varies considerably for different soil types with the increase of the slope degree.

The impact of the analysis method on the bearing capacity
The values of the bearing capacity on horizontal ground for the six types of soils determined using one numerical method (FEM) and three analytical methods (Brinch Hansen, Vesić and Van Baars) were analysed (Figure 3).It was observed that the bearing capacity values varied significantly between different types of soils and methods.The main reason for these differences is that the quantitative evaluation for N  are different in all the analytical methods used.The FEM and Vesic method produced similar values for the bearing capacity in all 6 soil types with higher values, in general, obtained using Vesic method.Additionally, Brinch Hansen and Van Baars methods also produce similar values for the bearing capacity in all 6 soil types, but their values were generally lower than those produced by FEM and Vesic method.

The influence of the slope inclination angle on the bearing capacity
Variations in the slope inclination angle (β) significantly affect both the stability conditions and the bearing capacity values of a footing founded on sloping terrain.As the inclination of the slope increases, the bearing capacity of the footing decreases.This study considers three distinct slope angles: β = 5°, 10°, and 15°, and Appendix A illustrates that qu decreases as the slope angle increases.This phenomenon occurs because steeped slopes result in a smaller passive zone, which in turn provides less resistance from the soil adjacent to the slope face (as shown in Figure 4).In addition, when a footing is situated on a slope, the passive zone formation is one-dimensional and confined by the slope face, resulting in a significant reduction of the confinement pressure and a corresponding decrease in bearing capacity.The rate of decrease in bearing capacity for foundations on a 5° slope, when compared to foundations on horizontal ground, is presented in Figure 5.The results showed that the rates of decrease in bearing capacity using Van Baars method are approximatively double than those from the other three methods.A possible reason for the Van Baars method giving values for the bearing capacity lower than other methods is that the factors of the bearing capacity were calibrated using only two values for the internal friction angle (0° and 30°) while the beta angle has three values (10°, 20° and 30°).The rates of decrease of the bearing capacity for foundations on a 10° slope, when compared to foundations on horizontal ground, are presented in Figure 6.The results showed that the rates of decrease of the bearing capacity using Van Baars method are higher than those from the other three methods.
The rate of decrease of the bearing capacity for foundations on a 15° slope, when compared to foundations on horizontal ground, is presented in Figure 7.In the case of Van Baars method, the rate of decrease was computed only for soil type S1.In the case of soil types S2, S2, S3, S4, S5 and S6, it was not possible to calculate the bearing capacity because the Van Baars method can only be applied when the internal friction angle is higher than the slope angle.It was also observed that, in the cases where FEM was used for soil type S1 S6, the bearing capacity could not be calculated due to the slope collapsing before the application of pressure from foundation.It was observed from the results that, in general, the rates of decrease in bearing capacity are higher when Van Baars method is used.This indicates that the utilization of Van Baars method leads to a higher reduction in the overall bearing capacity of the foundation on sloping ground compared to horizontal ground.

Bearing capacity for stepped footings
It was observed that, for horizontal ground (β = 0°), the bearing capacity of the stepped mat is approximately equal to that of the straight mat, as shown in Table 4.This suggests that the dependence of steps of the foundation mat does not significantly affect the overall bearing capacity when the ground is horizontal.This can be explained by the fact that, on horizontal ground, the load is evenly distributed across the foundation, regardless of whether it is a stepped or straight mat.Therefore, it can be concluded that, for horizontal ground conditions, the use of stepped mat does not provide any significant advantage in terms of bearing capacity over a straight mat.
In the case of slopes of 5° and 10°, it was observed that the bearing capacity of the stepped mat is lower than that of the straight mat, and the rate of decrease is between 16% and 27%.This indicates that, on slopes, the use of a stepped mat can lead to a significant reduction in bearing capacity.The reason for this decrease can be explained by the fact that, on slopes, the load distribution is uneven, and the stepped mat may not be able to effectively redistribute the load.Therefore, it can be concluded that the effectiveness of the stepped mat in increasing bearing capacity is dependent on the slope angle and soil type.It was observed that there are no analytical methods available for evaluating the bearing capacity of stepped foundations.To address this issue, one possible approach is to use the base inclination factors (b), where the selected angle is the one between the line joining the extremities of the footing and the horizontal direction.The use of the base inclination factors can provide a useful approximation of the bearing capacity of a stepped foundation, without the need for complex analytical methods.The reason for this is that the base inclination factors take into account the effect of the steps on the load distribution, and provide a means of estimating the reduction in bearing capacity due to the presence of the steps.

Conclusions
In this study, numerous finite element simulations of straight and stepped mat foundations on sloping ground have been made in order to compare the bearing capacities with the common analytical equations for the bearing capacity proposed by Brinch Hansen, Vesić and Van Baars.The embedding of digital solutions in the determination of bearing capacity for foundations on sloping ground has greatly improved the accuracy and efficiency of the process.By utilizing advanced software, engineers are able to quickly and accurately determine the bearing capacity of a foundation, resulting in safer and more reliable structures.This is demonstrated by the fact that the values for the bearing capacity obtained from the model footing tests from the literature are all on the upper bound of various proposed theoretical solutions [14].
Overall, these results indicate that it is important to consider multiple methods when determining the bearing capacity of soil for foundation design, as different methods may give different results for different soil types.
Further research will be conducted to deal with different types of foundation configurations and different types of soils.The results presented in this paper provides a global understanding of the bearing capacity of foundations on sloping ground and offer useful data and references for the bearing capacity computation.

Figure 1 .
Figure 1.The geometries of the analysed footing's cases: a -straight mat, b -stepped mat  are the bearing capacity factors which are functions of the friction angle  ; , , cq b b b  are the base inclination factors; , , cq d d d  are the depth factors; , , cq g g g  are the ground inclination factors; , , cq i i i  are the load inclination factors; , , cq s s s  are the shape factors for the footing geometry.In this study the values for base inclination factors (b), depth factors (d), load inclination factors (i) and shape factor (s) are set to one.The values for the bearing capacity factors and slope factors are determined the equations from Table

Figure 3 .
Figure 3.The values for the bearing capacity of foundations on horizontal ground for different soil types (in kPa) using FEM and analytical methods

Figure 4 .
Figure 4. Comparison between passive zones for foundations placed on horizontal ground (left) and foundations placed on slopes of 5° (middle) and 10° (right)

Figure 5 .
Figure 5.The rate of decrease of bearing capacity for the foundation on sloping ground with a slope angle of 5° compared with the foundation on horizontal ground

Figure 6 .
Figure 6.The rate of decrease of bearing capacity for the foundation on sloping ground with a slope angle of 10° compared with the foundation on horizontal ground

Figure 7 .
Figure 7.The rate of decrease of bearing capacity for the foundation on sloping ground with a slope angle of 15° compared with the foundation on horizontal ground

10 Figure 11 Figure
Figure A2.FEM incremental displacement plots at failure for soil type S2

Table 3 .
Soil bearing capacity values (in kPa) determined using analytical methods

Table 4 .
Soil bearing capacity values (in kPa) determined using FEM for straight mat and stepped mat