 Open Access
 Authors : Mamba Mpele, Enyegue Germain, Bana Abouar, Fry Jean Jacques
 Paper ID : IJERTV10IS110114
 Volume & Issue : Volume 10, Issue 11 (November 2021)
 Published (First Online): 04122021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Simplified Dynamic Method Applied to the Study of Slope Stability
Mamba Mpele 1*, Enyegue Germain 2, Bana Abouar 3, Fry Jean Jacques4
1 Professor, Department of Civil Engineering National Advanced School of Engineering, University of YaoundÃ© I, Cameroon. Po Box: 8390 YaoundÃ© CamerouIJERTV10IS010201n,
2 PhD Researcher, Department of Civil Engineering National Advanced School of Engineering, University of YaoundÃ© I, Cameroon.
3Researcher, Department of Civil Advanced School of Engineering, University of YaoundÃ© I, Cameroon.
4Embankment expert (AFPS, CFMS and CFBR)Chambery France
Abstract:The stability studies of slopes by the classical methods based on the concept of safety coefficient present some insufficiencies. They do not allow the calculation of permanent displacements induced by earth quake. To overcome this limitation, finite elements or finite volumes methods have been developed and introduced in several civil engineering calculation software. But, because the implementation of above methods is complicated, they cannot be used as a design tools but rather as a verification tool. In this paper we shows that simplified dynamic methods for the study of slopes stability under dynamic loads have been developed in the literature and their implementation is very easy, and allow the calculation of irreversible permanent displacements induced by an earthquake, which can be used as a design criterion for slopes or embankments stability.
Key words: Embankments, stability, simplified dynamic method, earthquake.

INTRODUCTION
A stability study of slopes or embankments, whether natural or artificial, consists of two steps: the first one concern to geotechnical studies which permits to calculate the mechanical and physical characteristics of the soils constituting the slopes or embankments and its base. The second one is devoted to the analysis of the stability of the embankment or slope, in order to evaluate their probability of failure. This stability analysis can be performed by two methods: one probabilistic and the other determinist.
The probabilistic approach is based on the spatial uncertainties of the mechanical and physicals properties of soils, which result in an uncertainty on the representativeness of the calculation results. The implementation of this approach is very difficult and it is understandable why it is not often used by civil engineers.
On the other hand, the deterministic approach (energy methods, static and pseudostatic methods) is, with a few exceptions, systematically used. The success of this approach is explained by the fact that its implementation is based on simple theories and the obtained results are easy to interpret. However, these methods have some limitations, especially when the slopes are subjected to dynamic loads.
In this article, we will briefly review the static and pseudostatic methods of slope stability analysis, highlight some of their shortcomings and show how some of them have been overcome in the case of seismic loading.

DETERMINISTIC APPROACH The deterministic approach to slope stability analysis is based on three methods:

Energetic Methods;

Static and PseudoStatic Methods

Dynamic Methods.

Energetic Methods
The objective of the energetic methods is to find the most general analytical form that can be given to the form that can be given to the normal stress function n. To do this, it is assumed that there is a potential function of the failure curve and that, among all the stress distributions which verify the equations of statics, the one that makes the potential minimum is chosen. This energetic approach developed by [15] is not used by engineers. Instead, they prefer static and pseudostatic methods based on: failure calculations and the concept of the safety coefficient.

Static and pseudostatic methods

Concept of safety coefficient
The safety coefficient associated with a failure surface can be defined as the ratio between:

The maximum mobilizable tangential stress max and the mobilized tangential stress on a failure surface;

The resisting forces and the driving forces on a failure surface

The resisting and driving moments of the above forces on a failure surface.
In general, the expression of max is given by the failure criterion of Mohr Coulomb (equation 1). On the other hand, some authors, such as [15], have proposed for noncoherent materials a failure criterion given by the relation (2). The expression of the effectively mobilized stress is obtained by solving the equilibrium equations of the mass located above the failure line or surface.
max
c, , tan ,
(1)
n
n
b
b
,
n
n
max A0
(2)
pref
A0 et b 1 are positives coefficients ;
pref
the reference pressure; c and are cohesion and friction angle of soil.
The equation of the equilibrium problem of a massif is based on the following assumptions:




The soil mass is assumed to be continuous;

The studied problem is plane and there are no edge effects;

The external forces are the weight of the soil, the surcharges and possibly the seismic forces. No account is taken of the driving forces that can be created by water flows;

If there is water flow, it is not disturbed by the slopes failure and the Terzaghi's relation , u is verified;

The mass of the moving soil(above the line or surface failure) is assumed to be rigid and the equilibrium of this mass resting on the rest of the soil is studied;

The seismic forces are taken into account (pseudostatic approach) by introducing the vertical kV and horizontal kH acceleration coefficients at the gravity center of the moving mass;

the safety coefficient is assumed to be constant over the entire rupture area;
Based on these assumptions, the solution of the mechanical equilibrium equations can be done using: global methods or slice methods.

Slice Methods
The principle of the slice method (Figure 1) is that the moving soil mass is broken down into several small elements or slices. The individual equilibrium of each slice is studied (3). The force tensor on the unstable mass is equal to the sum of the forces applied to each slice.
Figure 1: Forces acting on the 'i' slice
Ni sin i Ti cosi kHWi Hi1 Hi1 0 (a)
N cos T sin k 1W V V 0 (b)
(3)
i i i i V i i1 i1
Solving the system of equations (3) gives us the following expressions for Ti and Ni
Ni kV cosi Wi kH sin i Wi cosi Wi Vi1 Vi1 cosi Hi1 Hi1 sin i
Ti kV sin i Wi kH cosi Wi sin i Wi Vi1 Vi1 sin i Hi1 Hi1 cosi
By posing:
(4)
(5)
k Acc cos
H gA
and
k Acc sin , the relations (4) and (5) become:
V g
i A
i A
N cc sin
W cos W V
V cos H

H sin
(6)
T cc gcos
i W i sin Wi
i V
i1V
i1sin i H
i1 H
i1cos i
(7)
i g i i i i i1
i1
i i1
i1 i
The normal effective stress applied to the lice "i" of width is :
, bi N

uibi
(8)
i cos i cos
i i
Using Mohr Coulomb's criterion
b
c, , tan , we obtain the following expression for
max i i i
max i i i
bi
T c, i , i tan ,
(9)
maxi
i cos i cos i
T
T
n i i
The definition (D2) of the safety coefficient gives us the relation:
maxi
F i1
(10)
d n
Writing
Tmax
asfunTci tion of
Ni and
ui using the expression (7) for Ti we obtain this relation (11) which gives the safety
F
F
coefficient
i i1
:
:
d
n , b
A u b ,
i cc i i
i cc i i
ci
sin i Wi cosi Wi Vi1 Vi1 cosi Hi1 Hi1 sin i tan i
(11)
Fd
i1
cosi
g cosi
n
n
Acc cos W sin W V V sin H H cos
i1 g
i i i i i1
i1
i i1
i1 i
Assuming that at the failure the relation (12) is verified; we obtain another expression (14) for Ti .
cos
c, u tan ,
i i
i i
T
i maxi
i i i i
(12)
bi Fd Fd
c, Ni cos u tan ,
c,
u tan ,
i b
i i i
i
i
i i i i i
(13)
Fd Fd
c, bi

uibi
tan , N tan ,
i cos
cos
i i i
i
i
T i i
Fd
(14)
By introducing the relationship (14) into equation (3b) we obtain the following expression for
Ni :
c, u tan , b tan
k 1W ) V
V
i i i i i
V i i1 i1 F
N d
(15)
i sin tan ,
cos i i
i
i
Fd
When we use relation (15) to rewrite the MohrCoulomb criterion, we obtain the following expression for
max .
i
i
c,b u b tan , k 1W ) V V tan , 1
(16)
maxi i i i i i
V i i 1
i 1
i tan tan ,
i
i
b 1
i i
F
F
d
By taking the definition of the safety coefficient (D2) given by equation (17), and introducing the expressions (7), (16) of Ti
i
i
and max we obtain another expression of the safety coefficient Fd which is given by relation (18).
bi
n maxi cos
Fd
i1
,
,
,
,
1
1
i
Ti
(17)
n
n
c b u b tan k
c b u b tan k
,
i i i i
,
i i i i
i
i
V
V
1W V
1W V
i i 1 i 1
i i 1 i 1
V tan
V tan
i
i
1
1
tan tan
tan tan
,
,
i
i
i
i
cos
cos
F
F
F
F
d
d
i
i
(18)
d
i 1
cc
g
cos W sin W V V
sin H

H
cos
A
A
i
i
i
i
i
i
i i 1 i 1
i i 1 i 1
i
i
i 1 i 1
i 1 i 1
i
i
d
i 1
cc
g
cos W sin W V V
sin H

H
cos
In fact;
Fd
= 0 static safety coefficient,
= { 0 pseudo static safety coefficient.

Safety coefficient in the sense of FELLENIUS
If, 1 , +1 = 1 1 = +1 = 0 and using the relation (11) we obtain the pseudostatic safety coefficient of Fellenius (19).
n , b A u b ,
ci i cc sin i Wi cosi Wi i i tan i
F i1
cosi g
cosi
(19)
ps.F n A
cc cosi Wi sin i Wi
i1 g
And if seismic acceleration is null, we obtains a static safety coefficient of Fellenius(20).
n , b
u b ,
ci i cosi Wi i i tan i
F i1
cosi
cosi
(20)
s.F n
sin i Wi
i1

Safety coefficient in the sense of BISHOP
If, 1 , +1 = 1 1 = +1 = 0 and using the relation (18) we obtain the pseudostatic safety coefficient of
c,b u b tan , k 1W tan , 1
i i i i i
V i
i tan tan ,
1 i i cos
i
i
Bishop (21). n
F
(21)
Fps.B
i1
Acc g
cosi Wi sin i Wi
d .B
And if seismic accelerator is null, we obtain a safety static coefficient of Bishop (22).
c,b u b tan , W tan ,
i i i i i i i
tan tan ,
1 i i cos
i
i
(22)
n F
Fs.B
i 1
s.B
sin i Wi
In the literature, there are other assumptions on the variations of : Vi , Hi and zi which have given rise to other methods of calculating the safety coefficient: [24681221]

Global methods
Figure 2: Global method – Massive equilibrium
The moving mass above the failure surface (C) is considered to be a rigid block and its equilibrium is studied. The equilibrium of a small element of the rigid block of center M and length ds is:
– for the equilibrium of forces:
On OX axis:
dT dN .i dW .i 0
C C
(23)
On OZ axis:
dT dN .k dW .k 0
C C
(24)
For the equilibrium of momentums:
OM dT dN dW 0
C
Using the MohrCoulomb's law (26) and writing the vectors equations (28) where the two unknowns are Fd and
(25)
, as a function of and we obtain a system of 3
c,
p
utan,
Fd
Acc W cos
g
W
(26)
1
1
dW
(27)
C W Acc W sin W2
g
c ' F u tan ' tan( ) dx W
1
c Fd
c ' F u tan '
p>
c Fd
tan( ) dx W2
(28)
c ' F u tan '
p x y tan( ) x tan( ) y dx xdW1 ydW2
c
Fd c
To calculate the integrals of the system of equations (28), the rigid block is subdivided into several slices in order to use classical numerical integration methods. The fundamental assumption of the perturbation methods is to admit that the normal stress distribution is a function of the vertical stress applied on the horizontal facet, the parameters: , and the angle that the tangent facet to the failure line does with the horizontal axis, where the normal stress is applied. This hypothesis, yield system of 3 equations with 3 unknowns that gave rise to the perturbation methods 1, 2 and 3 presented by [514].

Shortcomings of static and pseudostatic methods
The shortcomings of the static and pseudostatic methods may be found in assumptions 2, 4 and 6. Assumption 2 assumes that the slope stability problem is planar and that there are no edge effects would be partly true if the length perpendicular to the cross section of the slope is sufficiently large compared to its width. But the dimensions of the slope being finite, it can happen that for some constructions this ratio is close to 1 (bridge piers) in which case the third dimension must be taken into account. This shortcoming led to the development of 3D stability methods by [11122] , Hypothesis 4, states that shear does not cause change in pore pressures at the failure surface. This statement, which is contradicted by the work of [19] has led to the development of slope stability methods that take into account pore pressure variations during shear. Hypothesis 6, which characterizes pseudo static methods, attempts to take into account the effect of earthquakes by introducing seismic acceleration coefficients. This method is not applicable for the analysis of soil masses with a risk of liquefaction and its major problems are:

The choice of seismic coefficients which seems arbitrary, although some authors such as [718] have proposed their maximum values as a function of: Magnitude, focal distance and soil type.

From a theoretical point of view, this method is not satisfactory because the dynamic nature of the seismic action is not taken into account: the eigen periods of the slope are not known and the induced permanent displacements are not calculated although they are used as a second (the first being stressbased) stability study criterion in most engineering problems.
In the end, all the methods mentioned above, have been established on the principle of limits state of resistance calculation. The fact of having a safety coefficient lower than 1 must therefore be interpreted as the resistant forces being exceeded by driving forces (criterion of not exceeding the constraints violated). The second criterion for not exceeding a threshold displacement value, is also important in the dynamic design of geostructures (dams and embankments) with elastoplastic behavior (due to the existence of induced permanent displacements). For embankments, Simplified Dynamic Methods developed by [1013], , were often used in the literature for the verification and dynamic predesign of embankments or dams: Dynamic embankment design tools based on these methods have been developed by : [3920]. In the following pages basics of this method will be presented here.

Simplified Dynamic Methods

Principle
Figure 3: Infinitely long embankment under seismic loading
The principle of the simplified dynamic methods is based on the analysis of the elastic response of a homogeneous triangular or trapezoidal embankment of height "H", based on a rigid foundation subjected to the action of the seismic acceleration (t).
At the end of this analysis the natural frequencies wn , the maximum distortion max of the embankment as well as the maximum
accelerations , kmax applied at the crest of the embankment and on the unstable part of the embankment (P) above the
failure surface (C) are calculated.
The critical acceleration kc associated with the failure surface (C) is then compared to the maximum average acceleration kmax applied (Figure 3) to the unstable part (P):
– If
1 the unstable part (P) of the embankment (Figure 3) does not move (or is stable);
– If
1 he unstable part (P) of the dam (Figure 3) moves along the failure surface. This displacement must be estimated
in order to compare it to the values of the admissible displacements tolerated by the project owner.
The knowledge of these parameters requires that a study of the elastic response of an embankment under dynamic loading be carried out.

Elastic response of an embankment under dynamic loading

Figure 4: Modelling of [19]
The study of the elastic response of an embankment (Figure 4) under dynamic loading by [91013] is based on the following four assumptions:

the material of the embankment body is homogeneous and its mechanical behavior is governed by a linear elastic law;

the influence of stored water is negligible, no significant loss of strength in relation to the generation of pore pressures;

the embankment is infinitely long and modelled by an infinite number of thin horizontal slices interconnected by an elastic spring system;

the mechanical problem considered is the planar type.
According to this model, the equations of motion governing the free vibrations of the [9] embankment (Figure 4) are:
2u G 2u 1 u
t2 y2 y y 0
2v
0 2G 2v
1 v
0
(29)
t2
y2
y y
Where , G and
0 E are: density, shear modulus and lame coefficient of the embankment material.
1 2 1
u(t, y) , v(t, y)
vertical and horizontal displacement of point M (Figure 4).Under the effect of the earthquake, equation (29)
after adding the second member gives the system of equation (30) :
2u G 2u
1 u
2u
2
t2 y
y y
g
t 2
2v
0 2G 2v
1 v
2v
(30)
g
t 2 y2
y y t 2
2
2
Where
2
and
2
2
2
represent the horizontal and vertical component of the acceleration transmitted to the foundation soil
by the acceleration earthquake
g
g
U t . By using =
we obtain the following equations system (31).
2u G 2u
1 u
2u
2
t2 H 2
g
t2
2v
0 2G 2v
1 v
2v
(31)
g
t2 H 2 2
t2
The solutions of the system of equation (31) according to [19] are in the form:
2 j t 2u
u t, 0 n g sin 0 t d
n n
n n
0
0
n1 0 j
1 n
t 2 n
2 j
t 2v
(32)
v t,
0
1
n
j
g
n
n
t 2
sin 1 t d
Where:
n1
n n 1 n 0
0 G
n n
H 2
(33)
1
0 2G 0
0
n n
H 2
n2 G
0 and 1 denotes the nth horizontal and vertical natural frequency of the embankment
0 and 1 are the Bessel functions of order zero and one. the nth root of the Bessel function 0 The first three values of are:
1 2.4048
2
2
5.5201
3
3
8.653
(34)
If we introduce the damping of order , using the fact that it is much less than 1, the relations (32) become:
t 2u
u t, n g exp 0 t sin 0 t d
n
n
0
0
n1 0 t2
n n n
t 2v
(35)
v t, n
1
g
t2
exp 1 t sin 1 t d
n1 n 0
n n n
n n n
2 j0 n
u , t si 0
Where
n
n
j1 n
And then we put U
, t
v , t si 1

Relative acceleration at a point on the embankment
The relative acceleration at time t and at a point M on the coast =
of the embankment is:
2U t,
U t,
t2
n n
2
2
n1
2
(36)
. V t
n n n
n1
Where
V t
t
t
g n n dn
g n n dn
U exp t sin t d
0
(37)
n
n
The velocity associated with the mode n is, according to Mamba [19], a convolution integral which depends on: the relative
acceleration
U t and, the damping and the natural frequency . For a given U t and "n", V t
has a
g g n
maximum called spectral velocity. The associated spectral acceleration S is then equal to: = .
an
Finally the maximum acceleration at any point =
of the embankment is:
2
2
U
S
max
n1
n n vn
(38)
U
U
2
2
max
n an
n an
n1
S
The maximum acceleration at the embankment crest = = 0 has the expressions:
2
2
U
0 S
(39)
max
n1
n n vn
U
U
2
2
max
n an
n an
n1
0 S
(40)
2
U .U
(41)
max
n
n1
max

Relative distortion at a point on the embankment
The distortion caused by the relative displacement at a point M of the embankment located at the coast =
is given by:
t, 1
H
U t,
(42)
Reasoning in the same way as in the previous paragraph we obtain the following relationship of distortion:
t, H V t n
n n
n n
(43)
s
s
v 2 n1
Lets pose (, ) = { distortion if = 0
settlement if = 1
Whatever the time t, the maximum distortion at a point M located at the coast =
in the embankment is:
H
2
s
s
S
n
(44)
max
v 2
an
n1
Where represents the wave velocity in the embankment: =0 if the velocity is horizontal and 1 if it is vertical.
G if 0
v
(45)
s
0 2G
if
1

Maximum acceleration applied to the unstable part (P) of the embankment
In the following, it is assumed that the seismic stress is a pure shear wave (Figure 5). This means that = 0
Figure 5: Principle of calculation of the acceleration applied to the part (P)
The acceleration applied at any point of the unstable part (P) of the embankment located above the failure line and assumed to be rigid is: 0() (P)
If W is the weight of the unstable part (P) which has been divided into "n" slices. If dm() is the elementary mass of the slice "i" located at the coast "y". The average acceleration applied on the part (P) of the embankment is:
g n
.. 0
W U
W U
kmoy t, .g dm yi . t,i
(46)
i1
ymax
1 y ymax
.. 0 y
kmoy t, H
W
dm y.U
t, H
(47)
y0
And its integral form is given by the expression (48).
ymax
ymax ..
1
1
y
y
0
kmoy t, H
W U
t, H dm y
(48)
0
The maximum average acceleration coefficient applied to the gravity center (stiffness assumption) of the unstable part (P) of the embankment is:
k Max k
t, ymax
(49)
max
moy H
In all that follows we assume:
Kc kc .g ,
Kmax kmax.g
et U max umax .g
(50)
Based on the study of the elastic response of embankments under seismic loading, simplified dynamic methods have been developed by several authors:[101316]. In this article we will only present the Makdisi and Seeds method.
4. MAKDISI AND SEED'S METHOD
Figure 6: Solicitation according to [10]
In the of Makdisi and Seeds method, it is assumed that the seismic loading is a pure shear wave. The solutions sought are those
obtained previously provided that = 0. Let be:
U t, ,U t,
U t, ,U t,
,
,
0 0 0
g max
, S 0
(51)
U
U
2
2
0
max
0 U max
n1
0 S 0
an
an
(52)
n an
n an
H
2
0
S 0
n
(53)
max max
v 2
an
s
s
n1
S0 w0.S0 , S 0 Max V 0 t
(54)
an n vn vn
n
n
n
g n n dn
g n n dn
V 0 t
U 0 exp 0 t sin 0 0
t d
t
t
(55)
1 2.4048
G
2
2
5.5201
3
3
8.653
et w0 n
n
n
H
H
(56)

Makdisi Seed Approximations
An observation by [29], of the evolution of the functions () as a function of allowed:

to neglect the contribution of modes greater than 3, for the calculation of the maximum peak acceleration ;
max
max

To neglect the contribution of modes greater than 1, for the calculation of the maximum distortion 0 ();
cycl
cycl

Consider that the equivalent cyclic distortion () is equal to 0.65 times the value of the mean distortion and that
[ 1(0)] 0,3; 
Consider that the dynamic shear modulus and the damping of the embankment material are functions of the cyclic distortion. a result of these approximatons, we deduce the following relationships:
0 3 2
0,5
Umax 0 Umax i 0 Sai
i1
(57)
H
0
1.S
(58)
s
s
max max
V 2 a1
eq
0, 65
0, 65 H .S
(59)
cycl
max
2 a1
moy
V 1
eq cycl
0,195
H
s
s
V 2
.Sa1
s moy
(60)
Umax
1, 6.Sa1
2 1, 06.S
2 0,86.S
0,5
2
2
(61)
a2
a2
a3
a3
1 0 1, 6
2 0 1, 06
1 0 0,86
Using the [23] hypothesis based on the mechanical rigid plastic behavior of the soil and on the basis of the results of cyclic laboratory tests [10], calculated for different accelerations, the permanent displacements induced by earthquakes of magnitude
6,5; 7,5; 8.25 and the whole results are presented as curves representing the ratio versus
Figure (7) and then normalized
time
..0
versus Magnitude and ratio
(Figure 8) which are used as charts.If d denotes the value of the permanent
displacement induced along the failure surface; the variables , , , 0, represent respectively: the maximum acceleration at the embankment crest, the critical acceleration, the acceleration of gravity, the first mode period of the embankment,
the height of the embankment and the maximum average acceleration applied on its unstable part. The ratios:
The ratios:
,
are unit less numbers and d is in meters when g is m., in second and
unit less coefficient for the
0
maximum acceleration of the unstable part of the embankment.
Figures (7) and (8) show the charts of [19] and with a minimum correlation coefficient of 99% [19] proposed the relationships 62 to 67 which can be used instead of the charts.
0
kmax
Umax
y H
0,2
MÃ©thodes des ElÃ©ments Finis
Domaine des valeurs
0.4
y
H
0,6
Valeurs moyennes
0,8
1,0
0 0,2 0,4 0,6 0,8 1,0
Kmax /Umax
Figure 7: Ratio of maximum accelerations to depth of sliding mass according to [10]
10
M 8, 25
1
M 7, 50
en s
en s
0,1
d
kmax.g.T0
d
kmax.g.T0
0,01
0,001
M 6, 50
0,0001
0 0,2 0,4 0,6 0,8 1,0
ky
kmax
Figure 8: Average values of normalized displacement as a function of magnitude and the ratio of critical to maximum acceleration of the unstable block according to [10]
k
1 si
y
y 0
H
max
f 2
(62)
U
1 H
y
y y
max max
1, 07 exp 0, 37261
0, 492
si 0< 1
k
1 si
y
y 0
H
H
H H
max
f 2
(63)
U
2 H
y
y y
max moy
1, 08exp 0, 221
0, 985
si 0< 1
k
1 si
y
y 0
H
H
H H
max
f 2
(64)
U
4 H
y
y y
max min
1,117 exp 0,142
1, 914
si 0< 1
For
0,1 kc
0, 7
H
H H
kmax
Magnitude earthquake 8.25 (average values) :
d k k 2
k
exp 3, 674 1,596Ln c 11, 615 c
6,961 c
(65)
kmax .g.T0
kmax kmax kmax
Magnitude earthquake 7.50 (average values) :
d k k 2
k
exp 1,86313 0, 69815Ln c 7,53273 c
0,95338 c
(66)
kmax .g.T0
kmax kmax kmax
Magnitude earthquake 6.50 (average values)
d k k 2
k
exp 5,334 1, 610Ln c 10,9785 c
7,33983 c
(67)
kmax .g.T0
kmax kmax kmax


Implementation of the Method
The implementation of the Simplified Dynamic Method for the dynamic calculation of embankments according to [17], requires the knowledge of:

the physical and geometrical characteristics of the Embankment;

the dynamic shear strength of the Embankment materials;

the fundamental period (or frequency f0 ) of the embankment (symbol);

the value of the critical acceleration (symbol) and the mean cyclic distortion associated with the earthquake;

the characteristics of the earthquake;

Fill characteristics
The characteristics of the backfill for the use of the Seed and Makdisi method are: The density of the backfill (measured in situ), the maximum shear modulus of the embankment material (obtained from cyclic triaxial tests), the embankment shear modulus G (obtained from geophysical tests) and the embankment height H. These characteristics are illustrated in Figure 9.
Rocky substratum

triangular fill
Rocky substratum

trapezoidal fill
Figure 9: Characteristics of an embankment on bedrock


Dynamic shear strength of the embankment
The dynamic shear strength of the embankment soil (obtained from triaxial tests) is determined not only by the evolution of the stress deviator as a function of the number of cycles, but also by the variations of the shear modulus and damping as a function of cyclic deformation or distortion. When using the Seed Makdisi method, only the variations of the normalized shear modulus
and the average damping factor (%) as a function of the cyclic distortion
(%) are used.
As an indication, Figures 10 and 11 given by [17], show the variations of shear modulus and damping of gravel and gravelly soils as a function of cyclic distortion.
1,0
G/Gmax
G/Gmax
0,8
0,6
0,4
0,2
0
104
5 103 5 102 5
101 5 1
cyclique %
Figure 10: Variation of shear modulus with distortion for gravel and gravelly soils according to [17]
24
Damping coefficient (%)
Damping coefficient (%)
20
16
12
8
4
0
04 5
103 5
102
5 101 5 1
cyclic %
Figure 11: Variation of the damping coefficient with cyclic distortion for sand and gravel soils according to [17]

Basic period first mode T0
For homogeneous fills on bedrock (Figure 9), the fundamental first mode period is according to the formulas (68 or 69) equal to:
T 2 H
1 G
(68)
1
T 2 H
1 .V
(69)
1 s
Where is the shear wave velocity in the embankment. For inhomogeneous fills founded on bedrock, the relationships (70) and
(71) can be used.
T 2 H
1 .V
(70)
1 seq
Where :
s
s
V N
eq M
(71)
with :
N G t n G n n 2B G t
n
(72)
1 1 1 2 1 2 H 3 2 2
M t n n n 2B t
n
(73)
1 1 1 2 1 2 H 3 2 2
Figure 12: Eigenmodes of zoned dams according to April 1989
For homogeneous embankments not resting directly on bedrock the following relationship can be used to calculate the fundamental period 0.
T 2 H1
0 a .V
(74)
1 s1
Figure 13: Calculation of the fundamental period of a homogeneous dam resting on an alluvial layer according to [17].

Critical acceleration Kc
The critical acceleration is obtained by pseudostatic analysis methods. It corresponds to the value of seismic acceleration for which the safety coefficient is equal to 1. And the coast (symbol) being the depth at which the failure surface intersects the face of the slope.
(P) ymax H
x
Kc kc g
y W
Figure 14: Case of any fracture surface
The critical acceleration coefficient for a flat fracture surface (Figure 15) with a static safety factor is:
kc Fs 1tan
(75)
Where is obtained using expression (76).
c, L W cos u L
tan,
Fs
AB ABmoy AB
W sin
A
Kc kc g
(76)
y
y
, 0 H
B
W
Figure 15: Case of a flat fracture surface
4.2.5 Earthquake characteristics
The earthquake characteristics are the accelerogram (Figure 19) at the base of the embankment. The acceleration response spectrum = (, ) associated with the earthquake which is a function of the damping of the embankment material and the period T. This acceleration spectrum is often used in a standard Cartesian representation (Figures 16 and 18), tri logarithmic (Figure 17) or logarithmic (Figure 20). The spectral acceleration is used to calculate: the maximum cyclic distortion.
(equation 58), the maximum peak acceleration
(relation 57) and the periods corresponding to the "i" mode. We
note that, for a given damping, the acceleration spectrum depends on the nature of the soil (Figure 18).
1,4
AccÃ©lÃ©ration en g
AccÃ©lÃ©ration en g
1,2
1,0
0,8
0,6
0,4
0,2
0 0,4 0,8 1,2 1,6 2,0 2,4 2,8
Eigen period (s)
0,00
0,02
0,05
0,10
0,20
Figure 16: Acceleration spectrum (a_max=0,25g) in Cartesian representation according to [17]
DÃ©placement(cm)
DÃ©placement(cm)
AccÃ©lÃ©ration(cm/s/s)
AccÃ©lÃ©ration(cm/s/s)
20
20
0, 02
0 , 05
0,10
Vmax
120c
ms1
0, 02
0 , 05
0,10
Vmax
120c
ms1
500
500
500
Velocity (cm/s)
Velocity (cm/s)
10
10
5
5
200
100
100
50
50
2
2
100
20
20
10
10
5
5
d
d
max
max
90cm
90cm
1
1
0,5
0,5
1g
1g
50
2
2
1
1
a
a
max
max
0,2
0,2
0,1
0,1
0,05
0,05
20
0,5
0,5
0,02
0,02
0,01
0,01
10
0,2
0,2
0,1
0,1
0,05
0,05
5
2
1
0,02
0,05 0,1
0,02
0,02
0,01
0,01
0,2
0,5 1 2
5 10
Period(s)
Figure 17: Acceleration spectrum ( = 1 ) in cartesian trilogarithmic representation according [13]
Steep
soils
Rock
Cohe
gr
sionles
eat de
s soils
pth
Steep
soils
Rock
Cohe
gr
sionles
eat de
s soils
pth
1,8
Acceleration in g
Acceleration in g
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0
0
0, 05
0,5 1,0 1,5 2,0 2,5 3,0
Eigen period (s)
Figure 18: Average spectrum of a magnitude 6.5 earthquake at 8km from the focus for various site conditions according to [17]
1,00
0,75
0,50
AccÃ©lÃ©ration in g
AccÃ©lÃ©ration in g
0,25
0,0
0,25
0,50
0,75
1,00
0,0 7,5 15,0 22,5 30,0
Time (s)
Figure 19: Dynamic calculation of rockfill dams – typical accelerogram according to [9]
ms2
ms2
PSEUDO ACCELERATION IN
PSEUDO ACCELERATION IN
101
100
SPECTRE US A1
0,020
0,050
0,070
0,100
0,150
0,200
0,500
101
101
100
Frequency in HZ
101
102
Figure 20: Dynamic calculation of rockfill dams oscillator spectrum (US A1 standards) according to [9].

Calculation of the dynamic characteristics of embankments
The calculation of the dynamic characteristics of the embankment such as peak acceleration and natural periods depends on the average distortion [9] experienced by the structure. This dependence results from the fact that the dynamic shear modulus and damping vary with the distortion (Figures 10 and 11). The calculation procedure, which is summarized in the flowchart in Figure 21, recommends the following steps:
c
c
max
max
max cycl
max cycl

Step 1, initial data : G , K , G , H , , G G f 1
, g
S f
,T T 0
and
0 0 .
cycl
cycl
a
a
1
1
n
n
eq
eq
Step 2, calculate
to derive 1 , 1 then calculate 1 = (1, 0), 1

Step 3, we check the condition: 1  = 1

If is not verified, we pose 1 = , we calculate the new values of the G , and repeat the procedure in step 2.
If is verified, we take = , = , = , = 1 and go to step 4.


Step 4, the permanent displacements induced by the earthquake are calculated.
Start
H, , ,0, (t), , =g(), = 1(, )
i=0, = 0, (=1,2,3) ,
, , +1 = (+1)
, = (+1, +1),
1
, () = 0,195
+1
()2
1
No Yes
Final Characteristics , ,
,
, , G = 1(+1),
Towards the calculation of induced permanent displacement
Figure 21: Flow chart for calculation of embankment peak acceleration, natural periods and equivalent cyclic distortion [19].


Calculation of induced permanent displacements
Knowing the value of the acceleration at the crest of the embankment, using the chart in Figure 7, the value of the maximum
acceleration coefficient of the unstable part of the embankment is determined. Embankment. The ratio is calculated.
If this ratio is strictly less 1, the induced permanent displacements are given by the charts in Figure 8. If is ratio is greater
than th
than 1, the embankment is stable and the calculation of the induced permanent displacements is not applicable.

APPLICATION
Consider a rockfill dam of 180m height resting on bedrock. The geometric characteristics of this dam are shown in Figure (22). Laboratory tests show that the maximum shear modulus of the dam material is 600MPa, its cohesion is zero and its angle of friction at failure is 400. The tests carried out in situ reveal that the shear modulus of the material in place is 216 MPa and its
density is The variations of the curves 2T/m3 and damping and damping (%) as a function of cyclic distortion obtained
from the cyclic triaxial tests are equal to the average variations of the curves given by [17] and are illustrated in Figures (10) and
(11). The magnitude of the design earthquake is 7,5 on the Richter scale and its maximum acceleration at the base of the structure is equal to 0,25g. The elastic response of the dam is obtained by multiplying the accelerogram spectrum in Figure (16) by an amplification coefficient of 1,6.We evaluate the permanent displacements induced by the design earthquake for the plane failure surfaces given in Figure (22) , all of which pass through point I such that IB=11,5m. We draw your attention to the fact that these flat surfaces are not critical failure surfaces for which the safety coefficient and minimum.
Figure 22: Rockfill dam and slip following flat surfaces

Maximum acceleration at the embankment crest
The values of the shear modulus G = 216 MPA and the density = 2T/m3 allow us to calculate the natural periods of the different modes of vibration of the dam which are given in Table 1.
Table 1: Eigen periods of fill mode 1, 2 and 3
Periods
Values
0 = 1
1,53s
2
0,62s
3
0,39s
The application of the flowchart in Figure (21), requires the calculation of the ratio
= 0,36 which allows us to use the curve
in Figure (10) to find the value of cyclic
= 0,065 % . Moving to Figure (11), we find a damping coefficient = 10%. The
values of the spectral acceleration associated with the different modes are determined using the curve f(T,) (at the Figure (16)) for which = 10% . = 1,6 f (T, =10%) (77)
where is the eigenperiod associated with mode 'i'.Using formular (60) and (61), the peak acceleration of the barrage is found
to be
= 0,62 and the equivalent distortion obtained = 0,060 % is very little different from its initial value
(0.065%). Using the charts of [20] (average values) or the relations proposed by [19]we obtain the values of the induced permanent displacements "d" at different rupture lines which are given in Table 2 and Figure 23.
Permanent displacements induced
in cm
Permanent displacements induced
in cm
Table 2: Permanent displacement 'd' induced by the magnitude 7.5 earthquake
Blocks
(
)
/
d
IBB1
45
23,91Â°
1,57
0,24g
0,25
0,83
0,62g
0,5185g
0,462
1,53s
37,05cm
IBB2
90
25,17Â°
1,49
0,213g
0,5
0,62
0,62g
0,366g
0,5823
1,53s
10,98cm
IBB3
135
25,62Â°
1,46
0,204g
0,75
0,46
0,62g
0,2562g
0,7949
1,53s
0,89cm
IBB4
180
25,85Â°
1,44
0,199g
1,00
0,32
0,62g
0,2013g
0,9874
1,53s
0,05cm
Figure 23: Permanent displacement induced by the Magnitude 7.5 earthquake
From these results, we can see that for flat fracture surfaces, the earthquakeinduced permanent displacements decrease as the
ratio
/ increases. Furthermore, we see that the largest displacement obtained is 37cm when = 0.25


Conclusions


We have just shown that the simpliied dynamic methods in general and in particular that of [20] allow engineers to:

take into account the effects of earthquakes on embankments or slopes

calculate the permanent displacements induced along a failure surface in order to verify the second criterion related to the concept of allowable displacements.
However, the implementation of these methods requires that: the foundation soil and the materials constituting the embankment or slope studied must not only verify the nonliquefaction conditions, but also that the shear strength losses recorded at the level of the failure surface are zero or less than 10% of its initial value.
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