Introduction

The stability analysis of slopes has been a challenging task for geotechnical engineers since ancient days. In assessing the stability of any slope, the focus is mainly on calculating the factor of safety to estimate the degree of closeness of the slope from the failure condition. Some infrastructure projects, particularly in hilly terrains, involve the construction of footings/foundations on slopes. When a foundation is built on a slope, the factor of safety is expected to reduce depending on the foundation location relative to the crest edge and its depth. In the past, several studies have been carried out on the bearing capacity and settlement behaviour of footings resting on unreinforced slopes. Some investigators have developed analytical formulations to estimate the bearing capacity of a footing positioned on a slope [1,2,3,4,5,6]. Meyerhof [4] developed an analytical formulation to estimate the bearing capacity of a footing on a slope face and crest for both completely cohesive and cohesionless soils. For a footing on top of the slope, it was reported that the bearing capacity factors decreased with higher slope inclination and increased with the footing distance from the slope edge. The bearing capacity of the footing is independent of the slope angle when the footing is located at a distance, greater than 2–6 times the footing width, from the slope crest edge. Graham et al. [3] used the stress characteristics to analyse the load-bearing pressure of a footing on cohesionless soil slope. They compared their results with the experimental data of Shields et al. [7] and found a good agreement between them. Saran et al. [6] used the limit equilibrium and limit analysis methods to obtain the bearing capacity of a foundation close to a slope. It was demonstrated that the two methods produced almost the same results presented in the form of non-dimensional bearing capacity factors which are influenced by the soil friction angle, slope inclination and footing edge distance. Narita and Yamaguchi [5] carried out a log-spiral analysis of the bearing capacity of footings located on slopes. They compared their results with other analytical and experimental works and found that the method overestimates the bearing capacity in comparison to other analytical solutions. A good agreement was however observed between the ultimate bearing capacity and the form of failure surface from the log spiral analysis and the laboratory model test. Buhan and Garnier [1] analysed the load bearing behaviour of a footing close to a slope by the yield design theory and compared the predictions with a full scale and a centrifuge test results. Castelli and Motta [2] utilised the limit equilibrium method to investigate the load bearing capacity of a foundation on a ground surface with a slope. A parametric study, under static and seismic conditions, was conducted to examine the effect of footing edge distance, slope inclination, footing embedment depth and seismic coefficients on the footing load bearing pressure.

Some investigators have analysed the bearing capacity of footings on slopes by laboratory model tests [8,9,10]. Castelli and Lentini [8] experimentally evaluated the load-bearing behaviour of footings on slopes, and proposed modified bearing capacity factors by considering the effect of sloping ground. Keskin and Laman [9] studied the bearing capacity of a strip footing positioned on a sand slope. They demonstrated that the ultimate bearing capacity improved with an increase in the footing edge distance, sand relative density, and decrease in slope inclination. Patil and Chore [10] demonstrated that the bearing capacity of a strip footing on a fly ash and furnace slag slope is influenced by the footing edge distance and slope angle. They also established a good agreement between the ultimate bearing capacity obtained from the experimental work and analytical solution.

Numerical modelling has also been utilised by some researchers to study the bearing capacity of footings on slopes [9, 11,12,13]. Georgiadis [12] presented a finite element analysis of the load-carrying capacity of foundations situated on slopes and compared the results with available analytical methods. Keskin and Laman [9] validated the results from their laboratory test with a numerical modelling data. Archaryya and Dey [11] numerically studied the collapse process and bearing capacity of a foundation on top of a slope that has no cohesion. They analysed the relationship between the footing bearing capacity and the following parameters: internal friction angle of the soil, slope angle, footing embedment depth, footing width, unit weight of the soil and elastic modulus of the soil. Zhou et al. [13] analysed the load-bearing capacity and collapse process of an axially loaded footing on slopes and established six types of collapse.

Most of the past studies have analysed the case of a footing on top of a slope. However, in most infrastructure projects, footings are usually built at some depth below the ground surface; and in the case of sloping ground below the slope crest. Therefore, the analysis of the behaviour of footings embedded in slopes will give the geotechnical/civil engineers more insight into the design of such foundations. There are however very few experimental studies on embedded strip footings supported by slopes [7, 14]. Shields et al. [7] performed a laboratory test to analyse the load bearing pressure of a footing placed on and below the crest of a sand slope constructed in a large tank having a length, width and height of 15 m, 2 m and 2.2 m, respectively. Bauer et al. [14] conducted a laboratory experiment to extend the work of Shields et al. [7]. They investigated the effect of the footing width and inclined applied load on the bearing capacity of the footing at different locations below the crest and face of a dense sandy slope. The test was conducted in a tank having the same internal dimensions as that used by Shields et al. [7].

Previous investigations on the performance of a footing on or in a slope have mainly focused on the bearing capacity and settlement characteristics of the footing, but the behaviour of a footing on or in a slope is governed either by the bearing capacity of the footing or the overall stability of the slope. A combination of the footing bearing capacity behaviour and the slope stability analysis will enable the engineers to gain a better understanding of the design of a footing-slope system. It is widely known that a number of studies have been conducted on the stability of unreinforced slopes/embankments with and without a footing/surcharge load on the crest. However, the literature reveals that the stability analysis of slopes, in terms of factor of safety, carrying embedded footing loads have not been greatly investigated. Also, the stability charts for such slopes are not available. Therefore, in this paper, an attempt is made to develop a numerical model to analyse the stability of a sandy soil slope supporting an embedded footing subjected to loads. The objective of this work is to establish the relationship between the factor of safety of the slope and the following slope and footing parameters: soil properties, slope inclination, footing locations, and applied pressure to the footing. The pattern of failure surfaces developed with respect to the footing locations and applied pressure have also been analysed. Additionally, some practical design charts have been developed for the routine use by the practising engineers.

Slope geometry and foundation details

The geometry of the slope and foundation details used in the present study are illustrated in Fig. 1. The slope has a height \( H \) and angle of inclination to the horizontal \( \beta \). A vertical load is applied to the footing having a width B and located at a distance e from the crest edge with an embedment depth D below the crest. The load applied to the footing is translated to a pressure q at the base of the footing. The slope comprises of a sandy soil with a limited cohesion. It has a unit weight \( \gamma \), cohesion c and angle of internal friction \( \phi \). The slope is compacted to various density states expressed in terms of relative density Dr. A preliminary analysis indicated that a berm width (2–14 m) and a foundation thickness (2–14 m) has a negligible effect on the numerical modelling. Consequently, a berm width of 10 m and foundation layer thickness of 10 m was adopted for this study to ensure that the failure surfaces developed remain within the slope boundary. It should be noted that other researchers while analysing the stability of unreinforced and reinforced slopes, by numerical modelling, utilized berm width and foundation thickness ranging from 5 to 15 m [15,16,17,18].

Fig. 1
figure 1

Geometry of the slope used in the study

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Numerical simulation

A series of two-dimensional finite element slope stability analysis was conducted on a slope supporting an embedded footing (Fig. 1), using Plaxis 2D (2016) software. Plaxis 2D has been developed for the analysis of various geotechnical engineering problems, including slope stability assessment [19]. The finite element analysis has been utilised by many researchers, including Chok et al. [20], for slope stability evaluation. The factor of safety modelling, in Plaxis 2D, utilizes five sequential modes, namely soil, structures, mesh, flow condition and staged construction. The software has several models to simulate soil behavior. Out of these, the Mohr–Coulomb model (MC model), which is an elasto-plastic model and first-order of approximation of soil behavior, was selected due to the availability of required data [21, 22]. The MC model requires six input parameters, namely Young modulus of elasticity (E), Poisson’s ratio (\( \mu \)), total unit weight (\( \gamma \)), friction angle (\( \phi \)), cohesion (c), and dilatancy angle (\( \psi \)). Griffiths and Lane [23] observed that, using \( \psi \)= 0 allowed a model to predict a reliable factor of safety, and a realistic form and position of the potential failure surface. They further noted that \( E \) and \( \mu \) had a negligible influence on the predicted factor of safety. This observation was substantiated by Hammah et al. [24] when they investigated the effect of \( E \) (2000–200,000 kPa), \( \mu \) (0.2–0.48) and \( \psi \) (0–35) on the factor of safety of a homogeneous soil slope. Cheng et al. [25] have also found that \( \psi \) and \( E \) are not sensitive to the factor of safety calculation and concluded that these parameters are not important in the slope stability analysis. Therefore, \( \psi = 0 \), \( \mu = 0.3 \) and \( E = 30,000\;{\text{kN/m}}^{ 2} \) have been adopted for the present study. According to Griffiths and Lane [23], the most critical parameters required for the stability assessment of a slope by finite element method are the soil unit weight, friction angle and cohesion. The footing was modelled as an elastic beam element using the “create plate” option in the structures mode of the software. The “create plate” option is utilised to specify the footing properties, including the flexural rigidity \( EI \) and normal stiffness \( EA \).

Plaxis 2D has five options for mesh generation ranging from very coarse to very fine, with the medium mesh being the default. Preliminary analysis was conducted to determine and reduce the effect of mesh on the finite element modelling. The boundary conditions set for the model restrained the vertical boundary horizontally and the bottom horizontal boundary in both vertical and horizontal directions. The slope surface is not horizontal, therefore the primary state of stress was determined by using a gravity force due to the soil mass. The finite element stability analysis uses the strength reduction technique, also known as the strength reduction method, to calculate the factor of safety [23]. In this method, the original strength parameters, c and \( \phi \), of the soil are reduced by dividing them by a factor called the strength reduction factor (SRF). As the SRF increases, the slope soil strength gradually reduces until the slope attains the state of equilibrium. The value of SRF at the equilibrium state is considered as the factor of safety of the slope. The factored or reduced shear strength parameters cf and \( \phi_{f} \) are expressed as follows:

$$ c_{f} = \frac{c}{SRF} $$
(1)
$$ \phi_{f} = \tan^{ - 1} \left( {\frac{\tan \phi }{SRF}} \right) $$
(2)

It should be noted that the SRF is represented as the incremental multiplier \( \left( {\sum {Msf} } \right) \) in Plaxis 2D. When the value of \( \left( {\sum {Msf} } \right) \) obtained during the analysis is generally constant for a number of successive steps, then a limit equilibrium state has fully been attained.

Validation of the numerical model

The present study, as already mentioned, is focused on assessing the factor of safety of an unreinforced slope supporting an embedded footing. The literature has revealed that numerous experimental data on the deformation of unreinforced slopes have been reported, but the literature on the stability analysis of such slopes, in terms of factor of safety, is not available. There is, however, a very limited laboratory (centrifuge) test data reported on the deformation and factor of safety of a geosynthetic-reinforced sand slope. It should be noted that most factor of safety analysis of slopes are based on either the limit equilibrium or numerical methods. As a result, some investigators have validated their works with published numerical data [26,27,28]. Therefore, the validity of the numerical simulations utilized in this study has been verified by using the finite element method to predict the load-settlement behaviour of a footing resting on an unreinforced slope as reported by Gill et al. [29] as well as the failure surface and associated factor of safety of a geotextile-reinforced slope from a centrifuge test data reported by Zornberg et al. [30, 31].

Validation with failure surface and factor of safety data

Zornberg et al. [30, 31] conducted a centrifuge test on a geotextile-reinforced sand slope and determined the factor of safety of the slope at various acceleration levels. This reinforced slope has been employed for the numerical model validation because there is no experimental reported data, to the best knowledge of the authors, on the factor of safety of unreinforced slopes. The slope used by Zornberg et al. [30, 31] has a height \( H = 228\,{\text{mm}} \) and inclination angle \( \beta = 63^\circ \). Underlying the slope is a sand foundation layer with a thickness \( H = 25.4\,{\text{mm}} \). Eighteen geotextiles having an equal length \( L = 203\,{\text{mm}} \) were installed at an equal vertical interval of 12.7 mm below the slope crest. The sand used for the slope and foundation construction was compacted to a relative density \( D_{r} = 55\% \) with a corresponding friction angle \( \varphi = 35^\circ \). A wide-width tensile test, carried out, on the geotextile resulted in an ultimate tensile strength of 0.063 kN/m at 17.7% strain. A 2-dimensional finite element model was developed, using Plaxis 2D package, as well as the sand and geotextile properties reported by Zornberg et al. [30, 31] to predict the failure surface and factor of safety of the slope at failure. The size of the slope, as reported by Zornberg et al. [30, 31], was increased by a scale factor of 10 for the purpose of modelling. Figure 2 compares the slip surfaces obtained from the present study and that of Zornberg et al. [30, 31]. It can be noticed that the results are in good agreement. Also, the factor of safety of 0.99 calculated from the current study is comparable to the factor of safety of 0.9 determined by Zornberg et al. [30], for the slope at failure.

Fig. 2
figure 2

Comparison of failure surfaces between the present study and works of Zonberg et al. [30, 31]

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Validation with load-settlement data

Gill et al. [29] investigated the performance of a footing on an unreinforced fly ash slope. The test was performed in a masonry tank having internal dimensions of 2400 mm (length), 310 mm (width) and 1200 mm (depth). The fly ash used for the study comprise 4% clay, 68% silt and 28% sand. It was compacted to a maximum dry unit weight (MDU) of 9.34 kN/m3 at an optimum moisture content (OMC) of 48%. An unconsolidated undrained triaxial test conducted on the fly ash at MDU state resulted in a cohesion, \( c = 20\,\,{\text{kPa}} \) and an angle of internal friction, \( \varphi = 1 4^{ \circ } \). The slope was constructed to a height, \( H = 7 5 0\,\,{\text{mm}} \) from the base of the tank and an angle, \( \beta = 4 5^{ \circ } \) to the base of the tank. A 2-dimensional finite element numerical model was established in this study to represent the load-settlement behaviour of the footing, having a width, \( B = 1 0 0\,\,{\text{mm}} \) and located at an edge distance, \( D_{e} \) to width ratio, \( D_{e} /B = 1 \), on the fly ash slope crest as reported by Gill et al. [29]. The fly ash was modelled using the MC model with the strength parameters reported by Gill et al. [29]. It should be noted that the MC model, as stated in the previous section, is the first-order of approximation of the behaviour of soil. It assumes the soil stiffness to be constant and for that reason does not take into account the stress nor strain dependency of the soil stiffness during the soil deformation. Figure 3 compares the load-settlement data of the footing from the laboratory experiment [29] and the numerical modelling (present study). It can be observed that the numerical results closely agree with the experimental results for low settlements, particularly for \( s/B < 4\% \) within the limitations of the MC model used for the modelling. Kazi et al. [32, 33] validated their experimental works on the load-settlement analysis of a footing on unreinforced and geotextile-reinforced sand bed using Plaxis 2D along with the MC model, and also found that the numerical and experimental results match for lower settlements.

Fig. 3
figure 3

Comparison of load-settlement curves from Gill et al. [29] and the present study

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Results and discussion

After validating the numerical simulation, a parametric study was conducted to analyse the effect of the following factors on the stability, defined in terms of factor of safety (F), of the sandy slope supporting the embedded footing loads, as presented in Fig. 1: slope angle (\( \beta \)), sand relative density (Dr), footing edge distance (e), footing depth (D) and applied footing pressure (q). The properties of the slope and footing used for the numerical modelling are presented in Table 1. The unit weight and strength parameters of the slope were adopted from laboratory test data reported by Kazi et al. [32, 33]. The correlation of the slope soil relative density \( D_{r} \) with the cohesion c and friction angle \( \phi \), based on the laboratory test results in Table 1, are presented in Fig. 4a, b, respectively. The line in Fig. 4a can be described by the following equation:

Table 1 Parameters used in the numerical analysis
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Fig. 4
figure 4

Correlation of the slope soil relative density, \( D_{r} \) with: a cohesion, \( c \), b friction angle, \( \varphi \)

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$$ D_{r} = 10.307c + 9.8773\;\quad(R^{2} = 0.9018) $$
(3)

The line in Fig. 4b may be presented in the equation:

$$ D_{r} = 20\phi - 670\;\quad(R^{2} = 1) $$
(4)

As the values of \( R^{2} \) for the correlations in Eqs. (3) and (4) are close to unity and equal to 1, respectively, the equations can be used to reasonably estimate the relative density of sands with similar properties as those used in the present study.

Although numerous data have been generated from several analyses, in this study, typical results have been presented and discussed in the subsequent sections. It should be pointed out that a slope height H = 6 m and a footing width B = 1 m were kept constant in all the analysis in the present work.

Effect of footing depth and applied pressure

The effect of the footing embedment depth D on F was investigated for the slope with \( \beta = 40^{ \circ } ,\,\,50^{ \circ } ,\,\,60^{ \circ } \), H = 6 m and \( D_{r} = 70\% \). In this analysis, the depth of the footing, with a width B = 1 m, was varied at depth ratios \( D/B = 0.0,\,0.5,\,1.0,\,1.5,\,2.0,\,2.5 \) for each edge distance ratio \( e/B = 0,\,1,\,2,\,3,\,4 \). It should be noted that \( D/B = 0 \) and \( e/B = 0 \) represent the footing on the slope crest and crest edge, respectively. The values of q varying between 0 and 240 kPa were applied to the footing for each combination of depth and edge distance ratio to observe the stability of the slope. A typical finite element model with mesh and boundary conditions is presented in Fig. 5. The variation of F with \( D/B \) for the footing at \( e/B = 0 \) and \( e/B = 2 \) for the 40° and 50° slopes are, respectively, shown in Figs. 6 and 7. It is observed that F increases with increasing \( D/B \) for all the slope angles and footing edge distance. A similar trend on the bearing capacity of a sand slope carrying an embedded footing in a large tank was reported by Shields et al. [7] and Bauer et al. [14]. Although Patra et al. [34] and Shin et al. [35] studied the performance of a footing placed in a sand bed, they as well found that the bearing capacity of the footing increased with increasing \( D/B \).

Fig. 5
figure 5

Atypical finite element model with mesh and boundary conditions

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Fig. 6
figure 6

Variation of factor of safety, \( F \), with footing depth ratio \( D/B \) for 40° slope: a \( e/B = 0 \), b \( e/B = 2 \)

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Fig. 7
figure 7

Variation of factor of safety, \( F \), with footing depth ratio \( D/B \) for 50° slope: a \( e/B = 0 \), b \( e/B = 2 \)

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It is noticed in Figs. 6 and 7 that the rate of improvement in F, generally reduces slightly when the footing is located at \( D/B \ge 1 \) and \( e/B = 0 \) as well as at \( D/B \ge 0.5 \) and \( e/B = 2 \). It is again observed in Figs. 6 and 7 that the slope behaves as if it is not carrying any footing load \( (q = 0\,{\text{kPa}}) \) when the footing is placed at a certain \( e/B > 0 \), \( D/B > 0 \) and \( q > 0 \). For example, it is demonstrated in Figs. 6a and 7a that locating the footing at \( e/B = 0 \) and \( D/B = 2.5 \) and subjecting it to a load \( q = 40\,{\text{kPa}} \) yields almost the same factor of safety as the slope without a footing load. A similar observation is made in Figs. 6b and 7b when the footing is positioned at \( e/B = 2 \) and \( D/B \ge 1 \) then loaded with \( q = 40\,{\text{kPa}} \). It is further noted in Figs. 6 and 7 that \( F \) reduces with increasing \( q \). Considering the contour interval of the factor of safety with respect to the applied load, it can be observed that, the rate of decrease of F reduces for \( q > 200\,{\text{kPa}} \) regardless of the footing location.

Effect of footing edge distance

An analysis was carried out to determine the influence of the footing edge distance on the 40°, 50° and 60° slopes. In this case, H = 6 m, B = 1 m, and \( D_{r} = 70\% \) were kept constant while the footing was loaded up to 240 kPa at edge distance ratio \( e/B = 0 ,\, 1 ,\, 2 ,\, 3 ,\, 4 \) for each \( D/B \) value of 0.0. 0.5, 1.0, 1.5 and 2.0. The variation of F with \( e/B \) for the various \( D/B \) values is presented in Fig. 8. It is observed that for \( D/B = 0 \) (footing on top of the crest), F improves with increasing \( e/B \) up to \( e/B = 3 \), then remains stable for \( e/B > 3 \). Thus, it can be stated that the maximum \( e/B \), for the surface footing, beyond which the slope has a negligible effect on F is 3. Other researchers [9, 29, 35,36,37,38,40] who investigated the load bearing pressure of a footing on unreinforced and geosynthetic-reinforced slopes reported a trend comparable to the observation made in the present study. They found that the ultimate bearing capacity of the footing improved with increasing the footing edge distance from the crest. The maximum \( e/B \) within which the slope has a significant influence on the ultimate bearing capacity of the footing ranged from 2 to 5. It is again noted in Fig. 8 that when the footing is placed at \( D/B \ge 0.5 \), F improves steadily with an increase in \( e/B \). This observation is similar to the trend reported by Shields et al. [7] and Bauer et al. [14].

Fig. 8
figure 8

Variation of factor of safety, \( F \), with edge distance ratio, \( e/B \): a \( q = 120\,{\text{kPa}} \), b \( q = 160\,{\text{kPa}} \)

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Effect of slope angle

In order to investigate the influence of the slope angle \( \beta \) on F, the footing, having a width B = 1 m, was positioned at each combination of \( e/B = 0 ,\, 1 ,\, 2 ,\, 3 ,\, 4 \) and \( D/B = 0.0,\,0.5,\,1.0,\,1.5,\,2.0,\,2.5 \) in the 40°, 50° and 60° slopes having H = 6 m and \( D_{r} = 70\% \). The values of \( q \), applied to the footing, were varied from 0 to 240 kPa to observe the corresponding F for each slope angle. The variation of F with \( \beta \) for each \( D/B \) value is presented in Fig. 9. It is observed that F reduces with increasing \( \beta \) regardless of the footing depth and edge distance. This outcome agrees with the findings of Khan et al. [41] and Rai et al. [42] who respectively analysed the stability of a plastic pin-reinforced slope and a geogrid-reinforced mine waste dump and determined that the factor of safety reduced with increasing the slope inclination. The results, obtained in this study, are also consistent with the observation made by Lee and Manjunath [39], Choudhary et al. [37] and Keskin and Laman [9] while investigating the load-bearing capacity of a footing positioned on an unreinforced and a geosynthetic-reinforced slope crest. They found that the bearing capacity of the footing decreased with increasing the slope angle.

Fig. 9
figure 9

Variation of factor of safety, \( F \), with slope angle, \( \beta \): a \( e/B = 0 \), b \( e/B = 2 \)

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To overcome the reduction effect of the slope angle on \( F \), the footing could be moved further away from the crest edge and/or below the crest into the slope. For example, in Fig. 9a, the factor of safety of the slope for a footing located at \( D/B = 0 \) and \( e/B = 0 \) reduces from 1.22 to 0.97 (becoming unstable) when the slope angle is increased from 40° to 60°. However, relocating the footing to \( D/B = 0.5 \) and \( e/B = 2 \) (Fig. 9b) in the 60° slope improves the factor of safety, by 54%, from 0.97 to 1.49. It is also demonstrated in Fig. 9b that having the footing at \( D/B = 0 \) and \( e/B = 2 \) in the 50° slope, as well as \( D/B = 0.5 \) and \( e/B = 2 \) in the 60° slope yields the same factor of safety.

Effect of relative density

Sandy soils have been classified into various density states ranging from loose to very dense based on the degree of compaction, which has a direct effect on the strength of the sand. To establish the relationship between the slope soil relative density (Dr = 50%, 70%, 90%) and  F, the footing (\( B = 1\;{\text{m}} \)) was placed within the slope, having a height H = 6 m and inclined at angles \( \beta = 40^\circ ,\,50^\circ ,\,60^\circ \), at the edge distance ratio, \( e/B \) of 0, 1, 2 and 3 for each depth ratio \( D/B \) of 0, 0.5, 1.0, 1.5 and 2. A pressure ranging from 120 to 200 kPa was applied to the footing at each footing location, defined by \( e/B \) and \( D/B \), to observe the corresponding factor of safety. Figure 10 shows the relationship between F and \( D_{r} \). It is noticed that \( F \) improves with increasing \( D_{r} \) from 50 to 70%, then the rate of improvement in F reduces for \( D_{r} > 70\% \), regardless of the footing depth and edge distance. Keskin and Laman [9] observed, while studying the load-settlement performance of a footing placed on an unreinforced sand slope crest, that the ultimate bearing capacity of the footing improved with increasing the slope relative density. Kazi et al. [33] studied the effect of relative density on the bearing capacity of a footing on an unreinforced sand layer and determined a trend similar to that established in the present study.

Fig. 10
figure 10

Variation of factor of safety, \( F \), with relative density, \( D_{r} \): a \( e/B = 0 \), b \( e/B = 2 \)

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The improvement in F resulting from increasing the slope soil relative density \( D_{r} \) could be attributed to the fact that increasing \( D_{r} \) improves the shear strength of the slope which correspondingly increases the shearing resistance along the critical failure surface and hence improves the factor of safety. It is also noted that for a particular \( e/B \) and \( D_{r} \), F increases with an increase in \( D/B \). Combining \( e/B \), \( D/B \) and \( D_{r} \) can significantly affect F. From Fig. 10a, F = 1 when the footing is located at \( e/B = 0 \), \( D/B = 0 \) and the slope is compacted to a relative density \( D_{r} = 50\% \). When the footing is placed below the crest at \( e/B = 0 \), \( D/B = 0.5 \), and \( D_{r} \) increases from 50 to 70% (Fig. 10a), F increases by 40% from 1 to 1.4. If the footing is again moved to \( e/B = 2 \), \( D/B = 0.5 \) and keeping \( D_{r} = 70\% \) (Fig. 10b), F further increases by 40% to 1.8.

Failure zone and critical slip surface pattern

The stability (factor of safety) of a slope is very much related to the failure (shear) zone including the critical slip surface developed within the slope. The size of the failure zone and the corresponding length of the critical slip surface influence the shearing resistance along the failure surface. As the size of the shear zone and the length of the failure surface increase, the resistance to the footing failure offered by the shear zone, from the slope face, and the shearing resistance along critical slip surface increase and enhance the overall stability of the slope. This section presents and discusses the typical pattern of shear zones and failure surfaces generated within the slope (\( H = 6\;{\text{m}} \) and \( D_{r} = 70\% \)) as a result of varying the footing location (edge distance and depth) and the applied pressure. The footing width \( B = 1\;{\text{m}} \) was used for the analysis.

Edge distance pattern

The critical failure surface developed from moving the footing away from the crest edge on the slope \( (D/B = 0) \) and within the slope \( (D/B = 1) \) are respectively displayed in Fig. 11a, b. The failure surfaces for the surface footing (Fig. 11a) start from the crest and end on the slope face for \( e/B = 0,\,1,\,2 \). The distance from the crest edge to the starting point of these slip surfaces on the crest increases with \( e/B = 0,\,1,\,2 \). However, these slip surfaces exit at the slope face with a decreasing distance from the crest. Figure 11a depicts that the overall size of the sliding wedges including the length of the failure surfaces for \( e/B = 0,\,1,\,2 \) increases from \( e/B = 0 \) to \( e/B = 2 \), with a corresponding improvement in F.

Fig. 11
figure 11

Slip surface pattern with edge distance ratio, \( e/B \): a \( D/B = 0 \), b \( D/B = 1 \)

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It is further observed that the slip surfaces generated for \( e/B = 3,\,4 \) are different from those obtained for \( e/B = 0,\,1,\,2 \). The sliding surfaces for \( e/B = 3,\,4 \) are symmetrical about the vertical axis, starting from a point within the slope and propagate to the crest. The size of the failure zone is almost the same for \( e/B = 3,\,4 \) and larger than the size of the shear zones obtained for \( e/B = 0,\,1,\,2 \). Therefore the values of F obtained for \( e/B = 3,\,4 \) are higher than those established for \( e/B = 0,\,1,\,2 \). It can be stated that the slip surfaces and shear zones for \( e/B = 0,\,1,\,2 \) are influenced by the slope and the crest whereas the sliding wedges for \( e/B = 3,\,4 \) are entirely influenced by the crest (horizontal ground).

Figure 11b shows the pattern of failure surfaces and shear zones developed, from moving, the footing within the slope \( (D/B = 1) \), away from the crest edge at the edge distance ratio \( e/B = 0,\,1,\,2,\,3,\,4 \). It can be observed that the sliding surfaces, in this case, start from the crest and terminate on the slope face for all \( e/B \) values considered. The distance from the crest edge to the starting point of the slip surface on the slope crest and the distance from the crest to the ending point of the sliding surface along the slope face increases with increasing \( e/B \) from 0 to 4. Therefore, increasing \( e/B \) for a footing located below the crest increases the length of the sliding surface and size of the failure wedge and hence F. A comparison between the size of the failure zone for each \( e/B \) value presented in Fig. 11a, b, indicates that the size of the failure wedge for the embedded footing is larger than the surface footing for all \( e/B \) values considered for the analysis.

Embedment depth pattern

Figure 12 displays the pattern of sliding surfaces and failure zones generated from increasing the footing depth \( (D/B = 0,\,0.5,\,1.0,\,1.5,\,2.0) \) at \( e/B = 1 \). This failure pattern is similar to that observed from varying the footing edge distance \( e/B = 0,\,1,\,2,\,3,\,4 \) within the slope \( (D/B = 1) \) as shown in Fig. 11b. It can be clearly seen that as the footing depth ratio increases, the length of the slip surface and size of the shear zone increases and correspondingly improves F.

Fig. 12
figure 12

Slip surface pattern with embedment depth ratio, \( D/B \)

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Applied pressure pattern

Figure 13 presents the pattern of sliding surfaces and size of failure zones resulting from varying the footing applied pressure from q = 0 kPa to q = 240 kPa. It is observed that the slip surfaces for all the applied pressure considered, in the analysis, commence from almost the same location on the crest but end at different locations on the slope face, ascending the slope with increasing q. The length of the sliding surface and size of the sliding wedge reduce with an increase in q, resulting in a decrease in F.

Fig. 13
figure 13

Slip surface pattern with applied pressure, \( q \)

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Design charts

Slope design charts are tools used by geotechnical engineers for quick and preliminary assessment of slope stability. The estimated factor of safety from these charts can be used as a quality control check for detailed analysis. The charts can also be utilised in estimating the strength parameters of failed slopes for remedial works design [43].

Taylor [44] produced the first set of stability charts for unreinforced homogenous soil slopes without a loaded footing on the crest. Since then, several other charts for unreinforced slopes, not subjected to footing loads, have been developed by investigators [20, 44,45,46,47,48,49,50,51,52]. Review of previous works shows that design charts for estimating the factor of safety of slopes carrying embedded footing loads are limited. Consequently, it was considered to develop design charts for estimating the factor of safety of a 6-m high, 40°, 50° and 60°, slopes supporting a footing having a width B = 1 m, and subjected to a pressure \( q = 100\,{\text{kPa}} \). The locations of the footing considered are \( e/B = 0,\,1,\,2 \) and \( D/B = 0,\,0.5,\,1.0 \).

For each combination of \( e/B \) and \( D/B \), the slope soil strength parameters c and \( \phi \) were varied from 0 to 10 kPa and 30° to 50°, respectively to determine the resultant F for each slope angle. A dimensionless stability number (N) defined in Eq. (5) [48] was calculated for each analysis and the relation between F and N are presented as design charts in Figs. 14, 15, and 16.

Fig. 14
figure 14

Design charts for \( \beta = 40^\circ \): a \( D/B = 0 \), b \( D/B = 0.5 \), c \( D/B = 1 \)

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Fig. 15
figure 15

Design Charts for \( \beta = 50^\circ \): a \( D/B = 0 \), b \( D/B = 0.5 \), c \( D/B = 1 \)

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Fig. 16
figure 16

Design Charts for \( \beta = 60^\circ \): a \( D/B = 0 \), b \( D/B = 0.5 \), c \( D/B = 1 \)

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$$ N = {c \mathord{\left/ {\vphantom {c {\gamma H\tan \phi }}} \right. \kern-0pt} {\gamma H\tan \phi }} $$
(5)

It is observed that F improves with an increase in N values. When required, other design charts can be produced for various combination of soil strength, foundation depth, edge distance, slope angle and applied pressure following the methodology developed in this paper.

Illustrative example

An example is presented in this section to explain how the design charts presented in Figs. 14, 15, and 16 can be utilized by geotechnical engineers. Let us consider the following problem:

A footing having a width of 1 m is to be constructed on a slope crest, at a distance of 1 m from the crest edge. The footing pressure is \( q = 100\;{\text{kPa}} \). Consider the following: H = 6 m, \( \beta = 50^{ \circ } \), \( \gamma = 15.3\,\,{\text{kN/m}}^{ 3} \), \( c = 3\;{\text{kPa}} \) and \( \phi \) = 35°. Determine the factor of safety of the slope.

Solution

From Eq. (5),

$$ N = \frac{c}{\gamma H\tan \phi } = \frac{3}{(15.3)(6)(\tan 35)} = 0.05. $$

Using the slope angle, \( \beta = 50^{ \circ } \), the footing edge distance ratio \( e/B = 1 \), and the footing depth ratio \( D/B = 0 \), the factor of safety of the slope can be estimated from Fig. 15a as \( F = 1.3 \), which indicates that the slope is stable.

Conclusions

The stability analysis of a sandy slope supporting an embedded footing has been evaluated by developing a numerical model with focus on the factor of safety. A series of 2D finite element analysis has been conducted to determine the effect of the following parameters on the factor of safety (F) of the slope: slope angle (\( \beta \)), relative density (Dr), footing edge distance (e), footing depth (D) and applied pressure (q). Based on the results and discussion, the following conclusions can be drawn:

  • The factor of safety (F) of the slope reduces with an increase in the applied footing pressure and the slope angle.

  • Increasing the footing depth ratio (\( D/B \)) improves F irrespective of the slope angle (\( \beta \)) and the edge distance ratio (\( e/B \)).

  • The slope stability also improves with increasing footing edge distance ratio (\( e/B \)), regardless of the slope angle (\( \beta \)) and the depth ratio (\( D/B \)). For a surface footing, \( F \) increases to a critical value at \( e/B = 3 \) then remains constant for \( e/B > 3 \).

  • As the relative density (Dr) increases, F significantly improves until Dr= 70% beyond which further increase in Dr results in a marginal increase in F.

  • Design charts for estimating F for 40°, 50° and 60° slopes, having a height H = 6 m, and supporting an embedded footing with parameters q = 100 kPa, \( e/B = 0,\,1,\,2 \), \( D/B = 0,\,0.5,\,1.0 \), and B = 1 m have been provided. An illustrative example has been given to explain how the developed charts can be utilized by the geotechnical/civil engineers. As required, the design charts for other cases may be created following the methodology developed in this paper.