1 Introduction

Strip foundations are one of the most cost-effective methods for transmitting loads from light superstructures to the underlying ground. Therefore, bearing capacity is one of the most crucial parameters in designing a strip foundation. It involves a combination of engineering mechanics and the properties of soils. The bearing capacity of a shallow foundation subjected to vertical central loading is typically determined using the bearing capacity equation posed by Terzaghi (1943), expressed as:

$${q}_{u}=\frac{1}{2}{B\gamma N}_{\gamma }+{cN}_{c}+{qN}_{q}$$
(1)

where qu is the ultimate bearing capacity, B is the width of the footing, c is the cohesion, q = γDƒ, γ is the unit weight of soil and Dƒ is the embedment depth of the footing, and Nγ, Nc, and Nq are bearing capacity factors.

A wide range of literature has been published on the undrained bearing capacity of strip footings resting on horizontal surfaces: (1) experimental investigations using full-scale tests, laboratory models, and centrifuge models (e.g., Nova and Montrasio 1991; Briaud and Gibbens 1994; Okamura et al. 1997; Badakhshan, Noorzad, and Zameni, 2018); and (2) theoretical and numerical approaches (e.g., Prandtl 1920; Vesic 1973; Anaswara and Shivashankar 2020; Anaswara et al. 2020; Ghazavi and Dehkordi 2021; Alzabeebee 2022; Chen et al. 2022; Fattah et al. 2022; Nalkiashari et al. 2022). However, shallow foundations are located on or near slopes in several situations, especially for bridge abutments, roads, and pylons in mountainous areas (Shields et al. 1990). The bearing capacity may be significantly reduced according to the slope in these cases. Nevertheless, the ultimate bearing capacity is affected by either local foundation failures or global slope failures, which adversely affect the environment in general (Azzouz and Baligh 1983). Many researchers have investigated the issue of strip footings resting near slopes via different approaches, including the limit equilibrium method. Meyerhof (1957) studied the influence of ground surface inclination on bearing capacity factors and proposed design charts that revealed the significant effect of slope on the bearing capacity of a strip footing. Hansen’s (1970) expressions were modified semi-empirically by Vesic (1975) by incorporating corrective factors for the foundation shape, load, and slope of the ground. Based on the slip line method, Kusakabe et al. (1981) and Graham et al. (1988) investigated the bearing capacity of clay slopes under continuous loads. Yang et al. (2019) in their study presented an analytical approach to evaluating the bearing capacity of a shallow foundation near a slope. The results are presented as dimensionless bearing capacity factors (Ncs, Nγs, and Nqs), allowing for consideration of the contributions of cohesion, soil weight, and embedment in calculating bearing capacity on slopes. They use traditional bearing capacity factors that comply with current geotechnical practice (e.g., AASHTO 2016). A simple upper-bound limit analysis (UBLA) approach is developed to objectively determine bearing capacity and the associated kinematically admissible mechanisms, either bearing or slope stability failure, for shallow foundations embedded near slopes with c–φ parameters. Chen and Xiao (2020) developed an UBLA method based on the failure mechanisms and bearing capacities of rigid strip footings on cohesionless slopes. They included various slope angles, footing distances from slope crests, slope heights, surcharge footings, and footing depths as part of their comparison. Loading tests have been conducted on model slopes with varying slope angles using 100-mm-wide strip footings (Huang 2019); in experiments, it has been shown that bearing capacity increases when a load is eccentric toward the heel of the footing. In contrast, conventional formulas show the opposite trend. This led to the proposal of a procedure for correcting the bearing capacity of a footing setback from the slope crest to address this discrepancy.

In geotechnical problems, numerical analysis, such as finite element difference and the finite element method, has proven extremely useful in analyzing the complex behavior of stress and strain due to external loading. Shiau and Watson (2008) investigate the finite difference in three-dimensional bearing capacity on homogeneous clay. Georgiadis (2010) employed the finite element method (FEM) and proposed a design procedure for calculating the undrained bearing capacity factor Nc. After that, Shiau et al. (2011) proposed design charts based on averaged lower bound and upper bound results based on the finite element limit analysis method for rough and smooth footings placed on purely cohesive slopes and developed a program to forecast bearing capacity. Discontinuity layout optimization was used to investigate the bearing capacity of the foundation established on a (cʹ–ϕʹ) slope by Leshchinsky (2015) and design charts containing critical failure mechanism information by Zhou et al. (2018). Based on a comparison between discontinuity layout optimization (DLO-LA) and the classical ultimate bearing capacity solution, Leshchinsky and Xie (2017) presented design charts in the form of reduction coefficients.

With the rapid development of construction, the existence of slopes may cause the attenuation of the bearing capacity of foundations. Often, engineers design and build the foundation at some depth below the ground’s surface. Therefore, it is necessary to place the foundation as far away from the sloped ground as possible to solve the intricate stability problem. However, the literature on the bearing capacity of embedded footings is very inadequate. Therefore, Qian et al. (2019) conducted two full-scale tests on dry low-sloping ground to compare the performance of two straight-sided piers and two belled piers under combined uplift and lateral loads. In addition, using centrifuge tests, the interaction of reverse faults and shallow foundations embedded at a depth of Dƒ/B has been evaluated by Ashtiani et al. (2015).

Furthermore, Ko et al. (2018) investigated the differences between the cyclic and dynamic behavior of embedded rocking foundations. Baah and Shukla (2019) investigated the performance of an embedded strip footing on an unreinforced and a single-layer geotextile-reinforced sandy slope by conducting laboratory model tests and finite element analysis. Jiang et al. (2018) established a three-dimensional calculation model to study the behavior of piles on the sloping ground under undrained lateral loading conditions. Han et al. (2020) constructed a numerical model based on field test data to discuss the uplift resistance of foundations embedded in the horizontal ground far from slopes. In non-stationary random soil, Wu et al. (2020) investigate the bearing capacity of embedded shallow foundations in non-stationary random soil, with the mean value increasing linearly with depth and a constant coefficient of variation COV. Their study examines the effects of non-stationary features of soil properties on embedded shallow foundations from three perspectives: failure mechanics, statistical characteristics of the bearing capacity factor, and probability of failure. In addition, Li et al. (2020) used finite element analysis to estimate the ultimate bearing capacity of strip footings placed on slopes. Among the parameters affecting the failure mechanism of the footing-on-slope system, embedded depth and edge distance are more significant than any other parameters. Lai et al. (2022) used a multivariate adaptive regression splines and a pseudo-static technique to evaluate the seismic bearing capacity of footing embedded in cohesive soil slopes.

This study evaluates the influence of various parameters on the undrained bearing capacity of strip footings on or near slopes using the finite-difference code Fast Lagrangian Analysis of Continua (FLAC 2005). The parametric studies presented in this section investigate separately the influence of the five parameters Cu/(γB), Dƒ/B, β, λ, and λ*. In addition, a comparison is made between the results of the analyses and the available methods.

2 Problem Definition

Figure 1 shows the geometry of the analyzed problem. This study examines a rigid strip footing of width B = 1 m situated on a homogeneous clay soil with angles of β = 15°, 30°, and 45° and a slope height H/B = 2 at several distances from the foundation edge to the slope crest λ and λ* indicates the distance between the base of the footing and the slope face. Footings are embedded at variable depths (Dƒ/B). Ratios of Dƒ/B of 0, 0.5, 1, and 2 were used. With a shear strength Cu, an undrained Young modulus of Eu = 22.5 MPa, a Poisson ratio of v = 0.49, and an overall soil weight of γ = 18 kN/m3, the Mohr-Coulomb elastic perfectly plastic constitutive model was used to model the soil as a Tresca material. Footings are assumed to be rigid.

Fig. 1
figure 1

Problem geometry

Full size image

3 Numerical Modeling Procedure

A two-dimensional plane-strain finite-difference analysis was performed with Fast Lagrangian Analysis of Continua (FLAC 2005). In engineering mechanics, FLAC (Fast Lagrangian Analysis of Continua) simulates the behavior of structures built out of soil, rock, or other materials that undergo plastic flow when they reach their yield limits by using explicit finite-difference programs. In addition, many researchers have used FLAC to study strip and circular footing-bearing capacities (e.g., Frydman and Burd 1997).

A typical finite element mesh used in the analysis of a B = 1 m wide footing at a distance of λ = 1 m from the crest of a β = 45° inclination and a H/B = 2 m high soil slope is shown in Fig. 2. To minimize possible boundary effects, the size domain contains 11,999 elements, which corresponds to 42 m × 15 m of wide footing, which minimizes the effects of size (a slight angle of slope of 15° without modifying overall mesh density). The slope was constructed by excavating the appropriate soil layers for each analysis. The mesh dimensions were found to be adequate for the majority of the cases analyzed. To achieve this, different mesh sizes were examined for combinations of geometric properties and materials. Meshes have been refined near the crest of the slope, under the base, and adjacent to the foundation boundaries. In contrast, a larger mesh size indicated that extending the boundaries farther away from the footing did not affect the footing’s limit load. The boundary condition for this problem is that the displacement of the left and right vertical sides is constrained horizontally and is fully fixed at the mesh base.

Fig. 2
figure 2

Finite-difference mesh and boundary condition for the case: β = 45°, Dƒ/B = 1, and λ = 1

Full size image

The soil was modeled as a Tresca material using the Mohr-Coulomb elastic perfectly plastic constitutive model with two shear strengths, Cu = 45 and 90 KPa, and the undrained Young modulus, Eu = 22.5 MPa. In order to model undrained conditions without volume changes and ensure numerical stability, Poisson’s ratio was kept constant at v = 0.49 (equivalent to a shear modulus of G = 7.55 MPa and a bulk modulus of K = 375 MPa). A soil’s unit weight is γ = 18 kN/m3, which affects its overall stability. It was assumed that the footing was made of linear elastic material. It has a concrete Young modulus of Ec = 29 GPa and a Poisson ratio of v = 0.21. The footing is connected to the soil via interface elements. It is possible to relate the interface elements’ properties to the properties of the adjacent soil elements, Cu. Having a normal stiffness Kn of 109 Pa/m and a shear stiffness Ks of 109 Pa/m, these parameters do not substantially impact the failure load.

Numerical modeling consists of the following steps:

  • • To balance the initial stress, a model of an embedded foundation on a slope was developed, and the original soil in the deep hole was preserved. Then, a gravity value of 10 was applied to the entire soil model.

  • • The zone representing the footing was subjected to downward velocity (displacement-controlled method). According to conventional calculations, a uniform soil layer causes a progressive movement of the rigid footing caused by vertical velocity applied at the nodes of the footing. This movement is proportional to the distribution of the increase in pressure in the soil. Finally, the pressure under the footing stabilizes at a value that indicates the ultimate vertical load.

Several tests were done to determine the optimal vertical velocity. The value of the velocity applied to the footing area was 1*10–7 m/step. This value is sufficiently small to minimize any inertial effects.

4 Results and Discussions

4.1 Comparison of Available Solutions

4.1.1 Horizontal Ground Surface

In order to verify the reliability predicted by the numerical model, the vertical bearing capacity problem was first validated numerically. Based on Eq. (1), the bearing capacity factor Nc of a strip footing on cohesive soil is as follows:

$${N}_{c}{d}_{c}=\left({q}_{u}-q\right)/{C}_{u}$$
(2)

where Nc is a bearing capacity factor; Cu is a representative undrained shear strength; q = γD is the surcharge at the footing base level; γ is the soil unit weight; D is the distance from the ground surface to the base of the foundation element; and dc is a depth factor.

Excellent compatibility could be seen between the present simulation results using the Nc = 5.19 model and the published data from Prandtl’s solution Nc (= π + 2). The maximum error is 1%, within a reasonable range. It also proves the correctness of the modeling method.

Figure 3 shows the variation of the Nc as a function of the ratio Dƒ/B compared to those calculated by Edward et al. (2005) and Salgado (2004) for the case of a footing placed on the horizontal ground surface. The gradient of the bearing capacity increased significantly with increasing depth Dƒ/B, as seen in the graph. For Dƒ/B = 0, the value of Nc obtained from the present study is close to the upper- and lower-bound solutions predicted by Salgado (2004) and the results of Edward (2005). However, it can be seen that the results of Salgado (2004) via the lower-bound solution are slightly more significant than the results of the present study by up to 1.8% for Dƒ/B = 1, while Edward’s (2005) solution and the upper-bound solution by Salgado (2004) overestimate the value of Nc by up to 3.30%. Furthermore, for Dƒ/B = 2, the results of Salgado (2004) are shown to be significantly better than the value of Nc determined through the current investigation. However, the upper bound solutions of Salgado’s (2004) and Edward’s (2005) solutions overestimate the value of Nc by up to 2.48%.

Fig. 3
figure 3

Comparison of bearing capacity factors of Edward (2005) and Salgado (2004) with the present study

Full size image

4.2 Parametric Analyses

An investigation of the geometrical characteristics and soil properties is presented in this section. The parametric studies presented in this section investigate separately the influence of the five parameters Cu/(γB), Dƒ/B, β, λ, and λ*, specifically Cu/(γB) ratios of 1 and 2.5, three slope angle β = 15°, 30°, and 45°, and normalized slope/footing distance λ = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Finally, λ* corresponds to [λ + (Dƒ / B) * cotβ].

The ultimate bearing capacity would then be stated as

$${N}_{c}=\frac{{q}_{u}}{{c}_{u}}=f\left(\frac{{c}_{u}}{\gamma B},\beta ,\lambda ,{\uplambda }^{*},\frac{{D}_{f}}{B}\right)$$

Additionally, these parameters influence the slope’s failure mode, which can occur in any of the four modes described by Zhou et al. (2018), as shown in Fig. 4.

Fig. 4
figure 4

Different typical failure modes for footing/slope problem (Zhou et al. 2018)

Full size image

(a-1) The first failure mode is the “face failure mode,” which extends along the slope face.

(a-2) Toe failure mode: a failure surface developed from the back corner of the footings to the toe of the slope.

(a-3) Base failure mode: a failure slip extends beneath the toe of the slope, which tends to mobilize a larger volume of shear resistance than face failure and toe failure modes. The passive resistance recovers as the influence of the slope decreases.

(a-4) Prandtl-type failure mode: a general failure mechanism occurs for a footing placed sufficiently far from the slope crest.

“Bearing capacity” failure modes can be referred to as this type of failure. However, the second is as follows:

(b-1) Overall slope failure where the critical shear surface extends beyond the crest and therefore involves part of the slope.

4.2.1 Influence of the Ratio C u/(γB)

A dimensionless strength ratio for slopes with unit weight is defined as Cu/(γB). The unit weight γ plays an essential role in the footing-on-slope problem, unlike the bearing capacity for level ground. Figure 5a shows the effect of the strength ratio Cu/(γB) on the undrained bearing capacity for footing rest on different slope angles β = 15°, 30°, and 45°, respectively, and for two normalized distances λ = 0 and 1, where two strength ratios Cu/(γB) = 1 and 2.5 are considered. It can be seen that the curves show a decrease in Nc linearly with decreasing strength ratio Cu/(γB), and the depth of the footings until the decrease becomes non-linear due to the presence of the slopes due to the reduction of passive soil resistance. According to the curves, the linear portion represents failures occurring within the slope face. However, the non-linear part reflects the interplay between the base failure mode and the toe failure mode. As expected, under the same strength ratio, the factor Nc increases with a decrease in the slope angle β. Factor Nc increases very rapidly for β = 15°, while factor Nc increases very slowly for β = 45°.

Fig. 5
figure 5figure 5

a Variation of Nc for footing at crest of the slope λ = 0. b Variation of dc for footing at crest of the slope λ = 0. c Contours of maximum shear strain for deep foundation Dƒ/B = 1, λ = 4, and β = 45°

Full size image

For a more intuitive representation of the ratio Cu/(γB), the dimensionless parameter depth factor dc may be defined as dividing the footings at depth on the sloping ground by those obtained for the surface footing. Figure 5b shows that the dimensionless parameter depth factor dc continues to increase as the foundation depth Dƒ/B increases, except for the slope angle β = 45°. For Cu/(γB) = 1, the dimensionless parameter depth factors dc gradually increase for β = 15° as the embedded footing increases, ranging from 0.882 to 0.902, a 20.8% increase. The values for β = 15° are significantly higher than those for other slope angles β. Therefore, the dc decreases for β = 30° before Dƒ/B = 0.5, corresponding to 0.777 and 0.763. Therefore, dc remains stable throughout the depth Dƒ/B varying from 0.5 to 2. Consequently, the dc decreased at a faster rate, a decrease of 9.30% in the range of 0.66–0.60 for β = 45°. However, when Cu/(γB) = 2.5, the dimensionless parameter depth factors dc increase slowly for β = 15° with the increase of embedded footing, varying from 0.88 to 0.90, which increases 20.8%.

The values for β = 15° are significantly higher than those for other slope angles β. Therefore, the dc remains stable for β = 30° before Dƒ/B = 1, which is roughly equivalent to 0.77. Subsequently, the dc increased to 0.80, an increase of 4%. However, dc decreases in the range of Dƒ/B = 0–0.5 in the field of 0.68–0.66 for β = 45°, and then slowly increases to 0.68, a 3% increase over the previous value. The dimensionless strength ratio Cu/(γB) effect on the slope angle is much more significant at the lower deep foundation as the slope angle β gets steeper. Some of these failure mechanisms are shown in Fig. 5c for β = 45°, λ = 4, and strength ratios Cu/(γB) = 1 and 2.5, respectively. It is seen that the contours of the maximum shear strain field of deep footing gradually move away from the toe slope to the face slope with the increase of Cu/(γB).

4.2.2 Effect of the Slope Angle β

The effect of slope angle on the undrained bearing capacity of foundations near a slope can be illustrated in Fig. 6 for Cu/(γB) = 1. Since slopes are generally stabilized with an extended embed, the ratio Dƒ/B varies from 0 to 2. Consequently, the normalized footing distance λ measured from the slope crest varies from 0 to 1. The effect of normalized footing distance λ on the foundation response has also been investigated with varying slope angles, especially considering their values from β = 15° to 45°. Meyerhof (1957) revealed that the slope fails because of gravity alone (higher slope angle). This unfavorable effect on bearing capacity can be attributed to reduced resistance in the passive wedge.

Fig. 6
figure 6figure 6

a Variation of Nc with β for Cu/(γB) = 1. b Variation of dc for footing at crest of the slope λ = 0. c Contours of maximum shear strain for deep foundation Dƒ/B = 2 and λ = 0

Full size image

It should be noted that β = 0° refers to level ground. From Fig. 6a in the case where λ = 0, it can be observed that an increase in the value of β results in a rapid reduction of the undrained bearing capacity factor, and the slope angle β affects the bearing capacity factor almost linearly. This is because the horizontal ground surface has a more significant undrained bearing capacity factor than near slopes at all foundation depths.

In cases λ = 1, the bearing capacity decreases as the slope angle increases. However, as the magnitude of β increased, the Nc decrease rate gradually decreased. Considering the behavior of the geometry of the footing, it is clear that the bearing capacity of λ = 0 is smaller than that of λ = 1. Furthermore, Nc decreases faster with larger magnitudes of Dƒ/B. Interestingly, the same Nc factor exists for ratios Dƒ/B of 0 and 0.5 at = 45°; this indicates that the higher slope angle plays a more active role than the ratio Dƒ/B.

In order to more intuitively reflect the effect of slope angle, the dimensionless parameter depth factors dc continue to decrease as slope angle β increases (Fig. 6b).

For λ = 0, the dimensionless parameter depth factors dc are identical for embedded footing for β = 15° and 30°, respectively. Therefore, the dc decreases at a faster rate, a decrease of 9.30% in the range of 0.66–0.60 for β = 45°. However, when λ = 1, the dimensionless parameter depth factors dc gradually decrease for Dƒ/B = 0 as the slope angle decreases from 1 to 0.75, increasing by 14%. Therefore, there is a significant difference for Dƒ/B = 0.5 to 2, which increases to 25%, 28%, and 32%, respectively.

Figure 6c illustrates failure mechanisms for Dƒ/B = 2, λ = 0, and strength ratio Cu/(γB) = 1, respectively. As anticipated, it can be seen that the slip surface deepens, and the failure mechanism changes from the Prandtl-type (for footing placed on a horizontal ground surface, β = 0°). Furthermore, the slip surface expands as the slope angle increases, and the failure mode changes from Prandtl-type failure to face-type failure for small slope angles (less than 30°). It will, however, remain in the toe failure mode (β = 45°).

4.2.3 Effect of the Footing Distance to the Crest λ

Figure 7 shows the variation of Nc computed for different footing distances of λ, for slope angles of β = 15°, 30°, and 45° and Cu/(γB) = 1 and 2.5 respectively. It is evident that the λ significantly affects the failure behavior of a footing-on-slope system. As expected, in all cases, the Nc effect is more dominant when the footing is located near the crest of the slope. In parallel, the gradient of each curve indicates that the larger the λ is, the more significant the Nc effect becomes.

Fig. 7
figure 7

Variation of Nc with λ for Cu/(γB) = 1

Full size image

In addition, the values of effect-bearing capacity are non-linear in most cases. As the slip surface deepens, the failure mechanism transmits from toe failure to surface failure, eventually obeying Prandtl-type failure. In Prandtl-type failure modes, the Nc factor of the footing remains constant when it is far from the slope crest. The Prandtl-type distance λPrandtl that results in the bearing capacity of strip footings not influenced by the slope geometry increases with an increase in the embedded depth.

For Dƒ/B = 0, 0.5, 1, and 2, ratio Cu/(γB) = 1, the Nc factor reaches a constant value at λ = 1, 2, 4, and 6 respectively for β = 15°; λ = 2, 4, 5, and 7 correspondingly for β = 30°; and λ = 3, 4, 7, and 9 accordingly for β = 45°. The Nc factor, on the other hand, reaches a constant value at λ = 1, 2, 3, and 4 for β = 15°; λ = 2, 3, and 6 for β = 30°; and λ = 2, 4, 6, and 8 for β = 45°. A Dƒ/B variation from 0.0 to 2.0 increases Prandtl’s failure type by 1.36 times.

Edge distance λ* is a critical variable in a footing-on-slope system’s failure behavior. With an increase in the distance ratio λ*, the footing-on-slope system finally obeys Prandtl-type failure, and the ultimate bearing capacity remains unchanged. With an increased embedded depth, strip footings not influenced by slope geometry will have a higher bearing capacity when they are farthest from the slope. When the embedded depth is not considered (Dƒ/B = 0), the Prandtl-type distance, in this case, is 2λ*, as mentioned in Fig. 8. Meanwhile, the case’s longest Prandtl-type distance is 13λ*, with the greatest depth Dƒ/B = 2.

Fig. 8
figure 8

Variation of Nc with λ* for Cu/(γB) = 1

Full size image

Figure 9 shows the variation in the (face failure to toe failure to Prandtl-type failure) mechanism with an increase in the footing distance ratio, which is consistent with the variation in the increased gradient.

Fig. 9
figure 9

Contours of maximum shear strain for deep foundation Dƒ/B = 1

Full size image

5 Conclusion

A finite-difference FLAC (2005) code was used to study the effects of undrained footings on slopes under vertical loads. Analyses were conducted with various geometries and soil properties, and results were compared with other solutions. The slope angle β, the distance of the footing from the slope λ, and the embedded depth Dƒ/B all play a role in determining the bearing capacity factor Nc. The following conclusions can be drawn based on a detailed analysis of various charts and distinct failure mechanisms:

  • The unit weight γ plays an extremely significant role in the footing-on-slope problem, unlike the bearing capacity for level ground.

  • As a result, the slope fails due to gravity alone and creates a significant unfavorable effect on the bearing capacity because the passive wedge resists less force.

  • Undrained bearing capacity Nc increases with increasing strength ratio Cu/(γB). The curves show a decrease in Nc linearly with decreasing strength ratio Cu/(γB), and the depth of the footings until the decrease becomes non-linear due to the presence of the slopes due to the reduction of passive soil resistance. According to the curves, the linear portion represents failures occurring within the slope face. However, the non-linear part reflects the interplay between the base failure mode and the toe failure mode.

  • The reduction rate in Nc decreased gradually as the magnitude of slope angle β increased. For λ = 0, the dimensionless parameter depth factors dc are identical for embedded footing for β = 15° and 30°, respectively. Therefore, the dc decreases at a faster rate, a decrease of 9.30% in the range of 0.66–0.60 for β = 45°.

  • Footing distance λ is a significant parameter of the failure behavior of a footing-on-slope system; the bearing capacity factor Nc decreases as footing distance increases λ. For Dƒ/B = 0, 0.5, 1, and 2, ratio Cu/(γB) = 1, the Nc factor reaches a constant value at λ = 1, 2, 4, and 6 respectively for β = 15°; λ = 2, 4, 5, and 7 correspondingly for β = 30°; and λ = 3, 4, 7, and 9 accordingly for β = 45°. The Nc factor, on the other hand, reaches a constant value at λ = 1, 2, 3, and 4 for β = 15°; λ = 2, 3, and 6 for β = 30°; and λ = 2, 4, 6, and 8 for β = 45°. A Df/B variation from 0.0 to 2.0 increases Prandtl’s failure type by 1.36 times.

  • Prandtl-type failure is the expected failure mode for footings on horizontal ground surfaces. As the slope angle β increases, the slip surface expands, and the failure mechanism changes from the Prandtl-type failure mode to the face failure mode or the toe failure mode.