Stability Analysis of Soil Slopes Subjected to Foundation

Loads during Earthquakes

Sukanta Das [1] and B. K. Maheshwari [2]

Department of Earthquake Engineering, Indian Institute of Technology Roorkee, India

sukanta1993das@gmail.com

Abstract. The stability of slopes subjected to foundations loads near the top of
the slope has been investigated in the present study. The c- soil slope with dif-
ferent width and embedment depth of foundation has been considered here. In
the present study, the building load is considered as a surcharge load on the
slopes. Strength reduction method has been adopted to investigate the variation
of Factor of safety (FOS) of slope against the different parameters of the im-
posed load and slope angle. The earthquake force has been considered as fixed
body force within the soil mass. From this study, the FS decreases with the in-
crease of building or foundation loads and at a certain distance from the edge of
the slope, the FOS remained constant. The improvement of FOS has been ob-
served with the increase of embedment depth of footing for all surcharge loads.
The rapid reduction of FOS has been noted when foundation load is increased
on slopes in both static and seismic case.

Keywords: Slope Stability, Foundation loads, Strength reduction method, Set-

back distance, Earthquakes.

1 Introduction

The design of shallow foundations in hill slopes is a challenging task for geotechnical
engineers since ancient days. Recently due to lack of land or increase in population,
the construction of building in hilly areas are increasing. Several researchers have
reported that the bearing capacity decreases with the increase of slope angle for both
static and seismic conditions. The slopes get more unstable which are marginally
stable before earthquake and the existing or newly constructed building loads may
create a vulnerable situation. Last few earthquakes for example 2005 Kashmir earth-
quake and 2011 Sikkim earthquake have shown the vulnerability of commercial
buildings in slopes and hilly areas. Therefore, stability of the existing slopes should be
checked before designing any structure on slopes.
It is expected that when a footing placed at the crest of the slope the stability of the
slope should be reduced. Although it depends on edge distance of footing, embedment
depth and imposed loads on footing. Meyerhof [1] proposed an analytical formula for
estimation of bearing capacity of footing on the face and near the top of the slope.
Several researchers performed experimental and theoretical investigation to determine
bearing capacity of footing on slopes [2-8]. In most of the study, it has been observed
2

that the bearing capacity decrease with the increase in the slope angle. However, with
the increase of edge distance (3-5times of footing width) of footing from crest of the
slope the bearing capacity is not dependent on slope angle. Keskin and Laman [8]
performed small scale experimental test and numerical analysis to determine the bear-
ing capacity of footing on slopes. They also checked stability of that slope after apply-
ing ultimate load on the footing. Numerical model study has been also carried by
many researchers for the evolution of bearing capacity factors of footing on slopes [9-
13]. Archaryya and Dey [13] studied bearing capacity of footing near the top of the
soil using Finite Element Method and compared with past experimental results. Raj et
al. [14-15] investigated the stability of the slope under the building loads. They re-
ported that stability of slopes increases with the increase of footing width and its dis-
tance from the slope. Although, they considered footing on the face of the slope.
Baah‑Frempong and Shukla [16] studied the stability of cohesionless soil slope sup-
porting an embedded strip footing. This study is limited on static and sandy type of
soil.

In the present study, the stability of the c- soil slope subjected to strip footing loads
has been studied for both static and seismic conditions. Different influence factor on
stability of slopes has been studied such as loading factor [q/H, where, q = Load
intensity (kN/m2), = Unit weight (kN/m3) and H= Height of slope (m)], Embedment
depth ratio [Df/B, where, Df = Depth of footing from ground level, B = Footing width]
and setback distance [De/B, where, De = Distance of footing edge from crest of slope,
B = Footing width]. Duncan [17] reported, the shear strength of soil must be divided
by factor of safety (FOS) to bring the slopes on the verge of failure point. Which is
commonly known as strength reduction method (SRM). A finite element limit analy-
sis (FELA) uses optimization techniques to directly compute the upper or lower
bound plastic collapse load (or limit load) for a mechanical system rather than time
stepping to a collapse load, as might be undertaken with conventional non-linear finite
element techniques [18].

2 Numerical Modeling

In the present study, two homogeneous soil slopes (slope angle,  200 and 300) under
the foundations loads have been studied. The soil properties have been taken from
Panigrahi et al. [19] where the site is located at hilly terrain within the Mizoram state
as shown in Table. 1. The strip footing width (B=1, 3 and 5m) at an edge distance
(De=0, 3, 6, 9, 12, 15, 18 and 21m) from the crest of the slope has been considered.
The footing embedment depth (Df =0.5, 1 and 2m) effect on stability of slopes also
studied. A schematic diagram of slope subjected to footing loads is presented in Fig.
1.

The 2D finite element limit analysis (FELA) based on strength reduction method
(SRM) has been adopted to check the stability of slopes under the footing loads. In
 3

SRM methods, the soil strength reduced by a factor and at verge of failure factors are
represented by strength reduction factor or factor of safety of the slope [20-22]. An
elastoplastic constitutive model based on Mohr-Coulomb failure criterion and follow-
ing associated flow rule has been used for soil modeling in FELA. The strip footing
modeled as a steel plate with Young’s modulus (E) compare to soil. A uniformly dis-
tributed load (UDL)has been applied to represent the super structure loads. The
boundary conditions are considered in such a way that the bottom is fixed against
both (horizontal and vertical) directions. While vertical edge is fixed laterally (hori-
zontal) and free to move vertically. The model dimensions have been chosen by sev-
eral trial, where the effect of boundary condition is insignificant. The sensitivity anal-
ysis has been performed for optimum (number of element at which FOS are not
changing much) meshing and approximately at 8000 elements are sufficient for the
present study. The 15noded triangular plane strain elements has been considered.
Loukidis and Salgado [23-24] reported that the 15-node elements have the fastest
convergence with adaptive free meshing where it yields particular location of failure
slip surface in Optum G2. In this meshing additivity system needs lesser number of
elements to produced accurate results. A fixed body force within the soil mass has
been considered in the present study.

Fig. 1. Schematic diagram of slope subjected to footing loads.

Table 1: Basic Soil Properties (after Panigrahi et al. [20])

Descriptions Values
3
Unit weight,  (kN/m ) 20
Poisson’s ratio,  0.3
Cohesions, c (kPa) 15
Angle of internal friction, Degree 30
Young’s modulus, E (MPa) 250

3 Results and Discussions

In the present study, using FELA method the stability of slopes in terms of factor of
safety (FOS) with different condition of footing has been studied. Keskin and Laman
[8] performed experimental test for bearing capacity of footing on slopes and checked
FOS by limit equilibrium method. For the FELA and keeping same parameters, a
comparison has been shown in Table 2. It can be observed that there is a good
agreement between the test results and the present study. The maximum value of q
considered in the analysis is 800kPa.

(a)
2.5
B=5m
β =20
0
B=3m
2.1 B=1m

1.7 αh=0
β =30
0
FOS

1.3

0.9

0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
q/g
H
4

Fig. 2. Variation of FOS vs. Loading Factor for different slope angle for De=3m, Df= 0.5m: (a)
Static case (b) Seismic case.
Table 2: Comparison of bearing capacity, qu (kPa) and FOS
Qu (kPa) FOS at 60kPa load
Df/B (K&L) Present Study K&L Present Study
0 31.50 33.61 1 1.03
1 56.70 57.33 1.86 1.84
Note: Results of Footing Located at Different Locations from the Slope Crest (β=30o, Dr=65%,
B=70 mm), K&L means Keskin and Laman [8].
A strip footing located at the top of the slope (200 and 300) for different footing width
and load factor has been analyzed. It is clear from the Fig. 2 (a, b) that with the in-
crease of footing load, factor of safety (FOS) of the slope reduced for both static and
seismic cases. At 300 slope the load carrying capacity quite less compare to 200 slope.
It can be observed that for small surcharge i.e. upto load factor 0.12 there is not much
effect of footing width on FOS. This phenomenon is known as global (overall stabil-

ity of slope) and local failure (bearing capacity of footing) problem as explained by
Paul and Kumar [25]. In local failure problem, when footing width increases the
bearing FOS increase. Therefore, the FOS of the slope also increases with the footing
width. A total displacement of slopes for loading factor 1 for 200 Slopes has been
presented in Fig. 3 (i-iii). In Fig. 3 (iii) for footing width 5m the deformation is more.
As footing width increases the contact pressure is also increase and deformation in-
creases as loading intensity is fixed (q in kN/m2).
 5

(i)
(ii)

(iii)
Fig. 3. Slope failure pattern for different footing with for q=300kPa, h=0 and Df=0m:
(i) B=1m (ii) B=3m and (iii) B=5m.
With the increase of setback distance the effect of slope angle on bearing capacity of
footing and FOS of slopes gets decreases. At a certain setback distance of the bearing
capacity of footing is constant or not influenced by slope angle and this distance is
called as critical setback distance. For bearing capacity problem, the critical setback
distance 2-5 times of footing width for static case and 3-9 times of footing width for
seismic case [13, 26]. The same trend has been found in this study but different criti-

(a)
1.5

β =20
0
1.3
αh=0
β =30
0
1.1
FOS

0.9

0.7 (b)
1.5
0.5 β =20
0

1.3 0 1 2 3 4 5 6 7
Setback Distance (De/B)
αh=0.1
1.1 β =30
0
FOS

0.9

0.7

0.5
0 1 2 3 4 5 6 7 8
Setback Distance (De/B)
6

cal setback distance.

Fig. 4. Variation of FOS vs. Setback Distance (De/B) for different slope angle for q=300kPa,
Df=0m and B=3m: (a) Static case (b) Seismic case.
From the Fig. 4 (a, b), the critical setback distance is found 2 times of footing width
for static case and 2.5 times of footing width for seismic case for 200 slopes. For 300
slopes, 5 and 5.5 times of the footing width respectively. An overall critical setback
distance can be identified from this study and past research based on bearing capacity
problems. For seismic cases the critical distance may vary for different seismic coef-
ficient. Therefore, at 200 slope after setback distance 2-2.5 times of footing width
goes into a local failure problem. Similarly, 300 slope after 5-5.5 times of the footings
width the failure get transfer from global failure to local failure.
 7

Fig. 5. Variation of FOS vs. Embedment Ratio (Df/B) for different slope angle for q=300kPa,
De=0m and B=3m: (a) Static case (b) Seismic case.

Table 3: Critical setback distance (m)

Slope angle 200 Slope angle 300
Static 2 5
Seismic 2.5 6

The effect embedment depth (Df) on stability of slopes has been presented in Fig. 5 (a,
b). The FOS or stability of the slope increase with the increase of embedment ratio for
all cases as shown in Fig. 5. The shear zones increase with the increase of embedment
ratio and the failure shape get modified as presented in Fig. 6.

(i) (ii)
8

(iv)

(iii)

Fig. 6. Failure pattern with different embedment ratio for q=300kPa, h=0 and B=3m, i)
Df=0m, ii) Df=0.5m, iii) Df=1m and iv) Df=2m

4 Conclusions

A series of numerical analyses has been performed to check the stability of the slope
subjected to foundation loading for static and seismic condition. The primary focus of
the present study is to identify global and local failure problem when footing placed
near the top of slope. And influence of other effects that are footing loads, Embed-
ment ratio, Footing width. From the above study the following conclusions can be
drawn:
1. For a given slope angle, after a certain footing load the FOS gets reduced
drastically. This threshold load indicates the change of global to local
failure problem. However, with the increase of slope angle the threshold
load decreases.
2. Within the local failure problem, the FOS increases with the increase in
footing width but within global failure problem the effect of footing width
on FOS is insignificant.
3. At a particular embedment and slope angle, the FOS of the slope is re-
mained constant after certain setback distance. Moreover, at 200 slope,
The FOS of the slope is remained constant after setback distance 2 and 2.5
times of footing width in static and seismic cases. Similarly, for 300 slope
after 5 and 6 times of the footings width the FOS of the slope is remain
constant in static and seismic cases.
4. The FOS increases with the increase in embedment depth of footing but
initially the rate of increase of FOS is higher for a constant slope angle,
footing width and Load. In both static and seismic cases, with the increas-
es of slope angle the FOS decreases and the trend remained same.
5. The influence zone increased with the increases of Footing width as well
as embedment depth in conditions. Therefore, irrespective of slope angle
 9

and seismic conditions the FOS is directly proportional to the footing

width and embedment depth of footing.

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