Stability analysis and design charts for a sandy soil slope supporting an embedded strip footing
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Abstract
In several field situations, especially in the hilly terrains, the construction of footings on slopes becomes essential. The factor of safety of a slope supporting a loaded footing on the crest is dependent on the position of the footing from the crest edge. Most studies in the past have focused on analysing the bearing capacity and settlement behaviour of a footing resting on a slope crest, but foundations in most infrastructure projects are usually built at some depth below the ground surface. Therefore, the availability of some form of design charts for determining the factor of safety of a slope supporting an embedded footing will be highly useful for the practising engineers. In the current work, the finite element analysis of a sandy soil slope supporting an embedded footing was carried out using the Plaxis 2D, a finite elementbased commercial software, in order to examine the effect of slope geometry, soil properties, and footing locations on the stability of slope in terms of factor of safety. The results of the analysis show that the factor of safety of the slope increases with an increase in the footing edge distance, footing depth and soil relative density, but it decreases with an increase in the slope angle and applied pressure on the footing. Some design charts have been developed along with an illustrative example to explain how these charts can be used by the practising engineers.
Keywords
Factor of safety Slope stability Strength reduction technique Failure zoneIntroduction
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 loadbearing 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 nondimensional bearing capacity factors which are influenced by the soil friction angle, slope inclination and footing edge distance. Narita and Yamaguchi [5] carried out a logspiral 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 loadbearing 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 loadcarrying 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 loadbearing 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 footingslope 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
Numerical simulation
A series of twodimensional 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 elastoplastic model and firstorder 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 \( EA \).
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 geosyntheticreinforced 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 loadsettlement 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 geotextilereinforced slope from a centrifuge test data reported by Zornberg et al. [30, 31].
Validation with failure surface and factor of safety data
Validation with loadsettlement data
Results and discussion
Parameters used in the numerical analysis
Parameter  Values  

Slope  
Angle (°)  40, 50, 60  
Height (m)  6  
Relative density (%)  50  70  90 
Total unit weight (kN/m^{3})  14.88  15.30  15.78 
Cohesion (kPa)  3.75  6.5  7.25 
Internal friction angle (°)  36  37  38 
Footing  
Width (m)  1.0  
Normal stiffness (kN/m)  5,000,000  
Flexural rigidity (kN/m^{2}/m)  8500 
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
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
Effect of slope angle
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
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
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
Applied pressure pattern
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 6m 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 \( D/B = 0,\,0.5,\,1.0 \).
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
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 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 (D_{r}) increases, F significantly improves until D_{r}= 70% beyond which further increase in D_{r} 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.
Notes
Authors’ contributions
EBF carried out numerical modelling of the research problem, and presented the results in graphical forms. SKS worked with EBF to analyze the results and presented the discussion along with an illustrative example. Both authors read and approved the final manuscript.
Competing interests
The authors declare that there is no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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