




































In ternationa l
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African Journal of Environmental Economics and Management ISSN 2375-0707 Vol. 7 (6), pp. 001-005, June, 
2019. Available online at www.internationalscholarsjournals.org © International Scholars Journals 

 

Author(s) retain the copyright of this article. 
 
 

Full Length Research Paper 

 

Nomograms for calculating the safety factor of 

homogeneous earth dams in long-term 

stability 
 

Rida Lakehal1*, Lakhdar Djemili1 and Larbi Houichi2 
 

1
Department of Hydraulics, Badji Mokhtar University, Annaba, 23000 Annaba, Algeria. 

2
Department of Hydraulics, Hadj Lakhdar University, Batna, 5000 Batna, Algeria. 

 
Accepted 21 March, 2019 

 
The slope stability analysis is routinely performed by engineers to evaluate the stability of embankment dams, road 

embankments, river training works, excavations and retaining walls. To ensure the geotechnical safety of the dam, 

the slope of embankment must be correctly designed and constructed. In this work, by applying the modified 

method of Bishop, an attempt was made to construct sets of nomgrams for the calculation of the safety factor of 

homogeneous earth dams under long term stability, which allow the user to get the optimal safety factor of the dam, 

immediately, according the material classification and the parameters of design, height and slope. 

 

Key words: Nomograms, safety factor, homogeneous earth dams. 

 
INTRODUCTION 

 
In the state of Annaba, Eastern Algeria, the construction of 
homogeneous earth dams increased, especially in areas 
dominated by agriculture. By considering the questions of 
security, it is absolutely necessary to study their stability in 
the various cases of loading especially in long-term case. 
The most popular method for stability analysis of these 
structures is the limit equilibrium method (LEM), this method 
is widely used by engineers and researchers and it is a 
traditional and well established method. Although the (LEM) 
does not consider the stress strain relation of soil, it can 
provide an estimate of the safety factor of a slope without 
the knowledge of the initial conditions, with the result that the 
(LEM) is favored by many engineers. The LEM is well known 
to be a statically indeterminate problem, and assumptions on 
the distributions of internal forces are required for the 
solution of the safety factor (Cheng, 2006). A relatively large 
number of methods have been developed. Among them, the 
slices method. In this method, the material above the slip 
surface being divided into a number of usually vertical slices.  

Improvements in the methods have aimed on reducing 
the error due to oversimplifying the shape of the slip  
 
 

 
*Corresponding author. E-mail: lakehal@hotmail.fr. Tel: +213-
773-807526. 

 
 
 

 
surface and the resulting incorrect determination of the 
normal stress. The latter is potentially important for frictional 
materials where the shear strength will depend on the 
normal stress. The principal methods are described in the 
papers of Fellenius (1936), Bishop (1955), Morgenstern 
(1965), Janbu (1973), and Sarma (1979). In this work, by 
applying the modified method of Bishop, an attempt was 
made to construct sets of nomograms for calculating the 
safety factor (SF) which characterizes the stability of these 
homogeneous structures. As such, many calculations would 
be carried out starting from the structure of simplified cross 

of the dam. Lastly, nomograms remain a contribution 
approach based on variations of the main properties in 
the long-term stability (Lakehal, 2011). 
 
METHODOLOGY 

 
Data 

 
The data used ar e r elat ed to the geometrical properties of the 
works, mechanical properties of materials, and software employed 
(Lakehal, 2008). 
 
Geometrical properties of the works 

 
In Figure 1, the simplified cross adapted for calculations, which 
represent the inclination (1/X, X in meter) for the upstream and 

mailto:lakehal@hotmail.fr


  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Figure 1. Cross simplified adapted for calculations. 

 

 
downstream slope, are 1/3 and 1/2.5, respectively. A filter of two 
layers was seen for the upstream slope with a thickness of 0.2 m 
fine sand and 0.3 m coarse sand, while a drain with length fixed at 
1/3 of the width of dam with a thickness of (1 m) constituted the 
three layers (0.3 m fine sand, 0.3 m coarse sand, 0.4 m gravel) 
(Lakehal, 2008; Alonso,1996; Colomer, 2009). The geometrical 
characteristics of the embankment taken into account are: 
 
The width of the crest (b) corresponds roughly to the formula of 
Knnapen,  
 

b=1.65  H (1) 

 
H: height of dam in meter, the unevenness {D = Hp + R} 
(Alonso,1996; Lakehal, 2011). 

 

Mechanical properties of materials 
 
The materials used in construction of these homogenous structures 
are intact or compacted fine materials, these materials have values 
of cohesion (c’) and friction angle (φ') seldom out of the following 
natural limits: (5 to 30 kPa for c’ and 15 to 40 for φ'), that is to say a 
fork of 25 kPa and 25°. We can obtain characteristics which are 
very poor (c' = 10 kPa and φ' = 20), excellent (c’ = 25 kPa and φ'  
= 35) or average (c' = 20 kPa and φ' = 25) (Alonso, 1996; Degoutte, 
2002; Duncan, 2005), the value of the density of these materials is 
chosen as the average value (USBR, 2001). In this case the density 

is taken as 20 kn/m
3
 (Alonso, 1996; Colomer, 2009). 

 
Software used 
 
The nomograms are results of several calculations by the Geostab 
software (version 2004) based on limit equilibrium methods. The 
method of calculation is that of Bishop modified (Géostab, 2004; 
Hammouri, 2008). 

 

Methods 
 
In this work a series of nomograms for calculating the safety factor 
of the slope of an homogenous earth dam have been produced 
(Colomer, 2009).  

The first set can be used for dams with height equal 10 m, 20 m and 

30 m, with (1/2.5) value of inclination for downstream slope, the values 

of materials properties varied between: 10° to 35° for effective 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
friction angle (φ’), and 10 kps ,20kps and 30 kps for effective 

cohesion (c’) (Alonso, 1996; Degoutte, 2002; Lakehal, 2008).  
The second sets for height equal 10 to 30 m, with different 

inclinations for downstream slope were analyzed, their values 
being: 1/2 to1/4 (Lakehal, 2008; Colomer, 2009), with the type of 
materials very poor (c' = 10 kPa and φ' = 20) and average (c' = 20 
kPa and φ' = 25) (Degoutte, 2002; Lakehal, 2008) .  

The nomograms developed in this study allow the user to know 
immediately the safety factor (SF) of dam with a known height, 
inclination and mechanical properties of materials.The value of 
safety factor SF =1.5 is set as a minimum value to ensure the 
stability of earth dam in long-term stability (Alonso, 1996; Degoutte, 
2002). The stability of downstream slope should be analyzed (US 
Army, 2003).The first nomograms are represented in Figure 2 for 
downstream slope.  

Figure 2 show results for embankments with a height of less than 
30 m and inclination of (1/2.5), the nomograms have a decimal 
scale to interpolate values of SF in the case when the interpolation 
is required.The second nomograms are represented in Figure 3 for 
downstream slope.  

Figure 3 show the results for embankments with a height of less 
than 30 m with different values of inclination (1/2.5 to 1/4), in this 
case the user can determine the optimal safety factor from the 
height of the dam, type of materials and the inclination of slope. It 
can also determine the optimal inclination adopted for an optimal 
safety factor in the opposite sense. 
 

 

RESULTS AND DISCUSSION 

 

Practical implementation 

 

In Figure 2, four different nomograms are shown, in 
which tow practical implementations have been applied: 
 

(i) The implementation of the nomograms on an earth 
dam with a height of 10 m, the inclination of downstream 
slope is (1/2.5) and the width of the crest is 5 m, the 
effective cohesion: c’ = 22 kps, effective friction angle: φ'  
= 15°, the result of SF is located between: 1.9 to 2.1 and 
by interpolation, the result of SF is 2.056. For the same 
dam, we can obtain the optimal safety factor SF =1,5 with 
the values of materials proprieties: c’ = 17 kps, φ' = 12°. 
 
(ii) The implementation of the nomograms an earth dam 



   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Figure 2. Set of nomograms for calculating the SF of earth dam between 10 to 30 m height. 



  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Figure. 3. Set of nomograms for calculating the SF of earth dam between 10 to 30 m height. 

 
 

 

with a height of 20 m, the inclination of downstream slope 
is (1/2.5) and the width of the crest is 7 m, the effective 
cohesion: c’ = 22 kps, effective friction angle: φ' = 15°, the 
result of SF is located between: 1.3 to 1,5, is lower than 
1,5, to get a value of SF equal 1.5 or  
higher one needs to increase the inclination of 
downstream slope at (1/2.75). In Figure 3, the 
implementation of the nomograms an earth dam with a 
height of 20 m, the inclination of downstream slope is 
(1/2.75) and the width of the crest is 7 m; with the type of 
materials is very poor (c' = 10 kPa, φ' = 20°), the result of 
SF is located between: 1.5 to 1.6 and, by interpolation, 
the result of SF is 1.575, the process is similar as in other 
nomograms. 
 

 

Conclusion 

 

The nomograms shown in this paper are a suitable tool 
for quickly calculating the SF of homogenous earth dams 
(up to thirty meters high). They do not need complicated 
calculations or computer programs. It is only necessary 

 
 
 

 

to know its mechanical and geometrical properties 
(height and inclination of slope).  

Every nomogram allows the inclination of the slope 

in  
a dam to be optimized. Hence, for a given height of dam, 
and according to the type of material and optimal safety 
factor, the nomogram will give the optimum inclination of 
a slope that is stable. Although, the nomograms remain a 
contribution approach based on variations of the main 
properties in the long-term stability. 
 

 
REFERENCES 
 
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the variation of key parameters. French Mag. Geotech., 63: 18-37.  
Bishop AW (1955). The use of the slip circle in stability analysis of slopes.  

Geotechnique, 1: 7-17.  
Colomer M (2009). Safety factor normograms for homogeneous earth 

dams less than ten meters high. J. Eng. Geol., 0013-7952, Elsevier, 
pp. 65-73.  

Cheng YM (2006). Three dimensional slope failure analysis by the 
strength reduction and limit equilibrium methods. J. Comput. 
Geotech. Elsevier, 36:65-73.  

Degoutte G (2002). Small dams. Cemagref editions.179 p, ISBN. 2 85362- 



 
 

 
551-6.  
Duncan J (2005). Soil strength and slope stability, library of congress 

cataloguing-in publication data. 450 p, ISSBN 0-471-69163-1.  
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Géostab Software (2004). Manual de l’utilusateur.135 p.  
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Morgenstern NR, Price VE (1965). The analysis of the stability of 

general slip surfaces. Geotechnique, 15(1): pp. 79-93.  
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Geotech. Eng. DIV. ASCE., 105.N°GT12. pp. 1511-1524.  
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http://www.usbr.gov.  
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Engineering and design - Slope stability. US Army Corps of 
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