10

CHAPTER 3. STABILITY OF SLOPESA slope can be defined as the inclined surface of the ground.

Slopes of different nature are represented in the fig. 3.1

a. natural slopes, when the inclined surface is the result of geological processes

b. slopes formed by excavation

c. slopes of embankments and earth dams

Fig. 3.1

A main problem in the design and construction is to ensure the stability of slopes. The principal factors which can cause instability of slopes are:

reduction of the shear strengths of cohesive soils as a result of an increase of the water content,

seepage forces,

surcharge on the upper part of the slope

removal of the soil at the base of the slope

seismic or other dynamic actions.

There is a great variety of types of slope failure. The most important ones are illustrated in fig. 3.2

a. rotational slip, with failure surface as a circular arc, characteristic for homogenous soil conditions;

b. rotational slip, with failure surface as a non-circular arc, characteristic for non-homogenous soil conditions;

c. translational slip, where the form of the failure surface is influenced by the presence of an adjacent stratum of significantly different strength at a relatively shallow depth below the surface; the failure surface could to be plane and roughly parallel to the slope or could be assimilated with as a polygonal one;

d. compound slip, where the adjacent stratum of different strength is at greater depth, the failure surface consisting of curved and plane sections.

Defining the forces which provide stability and of those who trigger the instability and of the factors which modify the ratio between these forces represents a compulsory requirement for understanding of natural phenomena, quite often transformed in real disasters, called in Engineering Geology landslides and for adopting appropriate measures to prevent and to control them.

Fig. 3.2

3.1 Stability of slopes in non-cohesive homogenous soil mass

3.1.1 Non-cohesive soil in dry or saturated state

By expressing the limiting equilibrium conditions for a soil particle at the surface of a slope in a non-cohesive soil in dry or saturated state, the following relation is obtained:

tan

(3.1)

where is the angle of stable slope and is the angle of internal friction of the soil.

The stability condition to be used in the design is:

(3.2)

where FS is the factor of safety.

The very important conclusion to be drawn from the relation (10.1) is that for non-cohesive soils in dry or saturated state the angle of stable slope is independent of the height of the slope.

3.1.2 The influence of a seepage on the stability of slope in a non-cohesive soil

A slope in a non-cohesive soil is considered, with a flow line tangential to the slope (fig. 10.3). A unit volume of soil at the surface of the slope is subjected to the following forces:

own weight , of two components:

- tangential T = sin

- normal N = cos

seepage force: j =

friction force: F = N tan

The stability condition is:

T + j

tan

(3.3)

For usual values of and n, is about 10 kN/m3. If is taken also 10 kN/m3; the relation (3.3) becomes tan.

Fig. 3.3

By comparing (3.1) with (3.3) one can realize that the seepage force reduces to half the magnitude of the stable slope.

3.2 Stability of slopes in cohesive homogenous soil mass

3.2.1 Case of the plane failure surface

A cohesive homogenous soil mass is considered, limited by a slope of height H and angle to the horizontal. It is required to check the stability of the slope.

In order to outline the forces involved, one can assume as a working hypothesis the loss of stability along a plane failure surface with an angle to the horizontal (fig. 3.4).

Fig. 3.4

The weight G of the wedge ABC which tends to slip has two components:

N = G cos

T = G sin

The failure condition is fulfilled in each point of the surface BC.

By summing up the stresses along the area BC (plane strain), the total resistance S opposed by the soil to the tendency of the prison ABC to slip is:

S = G cos tan

The stability condition is:

T

G sin

(3.4)

From the sinus rule in the triangle ABC:

= L

G =

(3.5)

By replacing (10.5) in (10.4):

(3.6)

For a given height H, the slope of angle to the horizontal is stable of the condition (10.6) is met for any values of between .

The most unfavorable situation corresponds to the value of for which the first member of the relation (10.6) becomes maximum.

If the following notation is introduced:

sin

The condition of maximum requires:

- cos(

sin

(3.7)

Considering (3.7), the relation (3.6) becomes:

(3.8)

The stability condition (3.8) shows that, unlike the non-cohesive soils, in the case of cohesive soils the angle of stable slope depends on the height of the slope.

At limit, by making equal the two terms of the relation (3.8), one can obtain the maximum height for which a slope with an angle to the horizontal can be stable, named critical height Hcr:

Hcr =

(3.9)

Fig. 3.5

If the angle decreases, Hcr increases (fig. 3.5).

From the relation (10.9) it results that in the stability condition of a slope of cohesive soil there are involved 5 parameters:

shear strength parameters , c

unit weight

geometrical characteristics of the slope H,

The expression (10.9) can be rewritten as:

(3.10)

Ns is a non-dimensional factor named stability number.

For given and for a certain shape of the failure surface, Ns has a defined value.

For instance, for the particular case of vertical slope = 90o:

Ns = 4 tan (45o +

(3.11)

Hcr =

(3.12)

If

Ns = 4

(3.13)

Hcr =

(3.14)

Relations (3.12) and (3.14) have been obtained by applying Rankines theory in the case of limiting equilibrium in the soil mass limited by horizontal ground surface. It was established that in the case of cohesive soil, the depth at which the active earth pressure becomes zero is:

z0 =

and the critical height Hcr, the height on which a vertical cut in a cohesive soil can remain unsupported, is:

Hcr = 2z0 =

3.2.2 Case of circular failure surface. Method of friction circle

The potential failure surface is assumed to be a circular arc (fig. 3.6).

Fig. 3.6

Forces which should be in equilibrium are:

the weight G of the soil mass which slips ABC;

the resultant C of the cohesion mobilized along the failure surface: , acting parallel to the chord at a distance from the centre 0;

the total reaction Q on the failure surface tangential at the -circle of radius

r = R sin.

There are three ways in which the factor of safety can be expressed: with respect to the cohesion

For a given , the -circle is drawn and from the intersection of lines of action of forces G and C a tangential to the circle defines the direction of Q. The triangle of forces is drawn by decomposing G to the directions of C and Q.

By this way is obtained the total cohesion force required for the limiting equilibrium Creq which, divided by the area , gives the required cohesion creq.

The factor of safety is:

(3.15)

with respect to the internal friction angle

For a given c, the forces polygon is built, with known values of G and C, resulting the value and the direction of the force Q. From the point of intersection of the forces G and C a parallel is drawn to the direction of Q.

The normal drawn from the centre of the circle to the direction of Q defines the radius of the -circle. By knowing Rsin, tanreq can be established.

The factor of safety:

(3.16)

with respect to both cohesion and internal friction

For a given , the limiting equilibrium requires creq; for a given c, the limiting equilibrium requires tanreq. There is an infinite number of pairs of values (tanreq, creq) which in the system of coordinates (c 0 tan) defines a curve (fig. 10.7)

Fig. 3.7

A point M corresponds to the parameters and c of the shear strength. By linking this point to the origin, the line OM intersects the curve in point P.

The factor of safety in respect to both internal friction and cohesion is:

FS =

(3.17)

The same curve allows to obtain the two other factors of safety:

Analysis for the case = 0

In the case of a fully saturated clay under undrained conditions, an analysis in terms of total stress is appropriate. Only moment equilibrium is considered in the analysis. The potential failure surface is a circular arc (centre 0, radius R, length La (fig. 3.8). Potential instability is due to the total weight of the soil mass (G) above the failure surface. For the equilibrium, the shear strength required to be mobilized along the failure surface is expressed as:

(3.18)

where Fc is the factor of safety with respect to shear strength.

Moment about 0 is taken:

G

(3.19)

To determine the minimum factor of safety Fc, it is necessary to analyse the slope for a number of trial failure surfaces.

Taylors stability coefficients

Using the method of -circle for soil having both internal friction and cohesion and the analysis for the case =0, Taylor established stability coefficients NS for the analysis of homogenous slopes in terms of total stress (fig.10.9), in the hypothesis of a circular failure surface.

Fig. 3.8

Fig. 3.9

As one can see, for the particular case =90o and for =0, NS=3.85 (as compared to NS=4 for the hypothesis of plane failure surface).

For the use of Taylors graphs, there are two problems encountered in practice.

Design of a stable slope

, c, , NS are given. The height H for a given factor of safety FS is sought for.

The height required for the limiting equilibrium is Hreq. Then, in order to comply with the factor of safety FS, Hreq = FS

NS =

(3.20)

From the value of NS on the ordinates axis, a horizontal is drawn until intersects the curve corresponding to the given. The vertical drawn from the point of intersection meets the needed abscissa.

Checking the stability of a slope

,c, are given. FS is required

From the abscissa, a vertical is drawn until intersects the curve of the given . A horizontal is drawn and the ordinate NS is found.

NS =

FS =

(3.21)

3.2.3 The influence of the groundwater on the stability of slopes in cohesive soils

The presence of the groundwater can significantly modify the conditions of stability of slopes in cohesive soils. Depending on the water level inside and in front of the slope, three distinct situations are occurring.

a. Slope completely immersed, the soil in submersed state

There are two ways in which the presence of water can be considered (fig. 3.10).

1. When computing the weight G of the sliding soil mass the unit weight of the soil is taken .

The weight G is composed with the resultant of the water pressure on the side AB and with the resultant of the water pressure on the failure surface BC. The resultant is obtained, which then is decomposed, as shown at p. 3.2.2, on the directions of C and R.

2. The weight of the sliding mass is obtained directly by considering the unit weight in submersed state . Then, the procedure outlined in the variant 1 is used.

Fig. 3.10b. Rapid drawdown of the water level in front of the slope

This situation can take place in the case of dikes or earth dams facing an accidental discharge of the water reservoir (fig.3.11). Since the drawdown is rapid and sudden as a result of the low permeability of the soil, it is not followed immediately by the draw down of the water inside the soil mass. The total weight G of the soil in saturated state is the same as in the previous case, but there is no longer water pressure on the slope and the resultant G of the water pressure on the failure surface BC is smaller due to the lowering of the water level in front of the slope.

Fig. 3.11G and G are composed and their resultant B is decomposed on the direction of C and R.

The stability conditions in this case are much more severe than in the previous case.

c. Stability of slope under conditions of steady seepage

After a sufficient time elapsed to allow the consolidation (dissipation of excess pose water pressure will have taken place) a steady seepage is generated through the soil mass directed toward the base of the slope, accompanied by the lowering of the ground water level.

If stability conditions are checked with the Swedish method, the factor of safety can be expressed in terms of effective stress with the following relation:

(3.22)

The pore water pressure ui at the centre of the base of each slice is taken to be , where zw is the vertical distance of the centre point below the water table. This procedure slightly overestimates the pore water pressure, which strictly should be , where zc is the vertical distance below the point of intersection of the water table and the equipotential through the centre of the slice base (fig. 10.12). The error involved is on the safe side.

Fig. 3.12A series of trial failure surfaces must be chosen in order to obtain the minimum factor of safety.

3.3 Stability of slopes in non-homogenous soil mass

3.3.1 The method of slices

The potential failure surface, in section, is assumed to be a circular arc with centre 0 and radius R. The soil mass (ABCD) above a trial failure surface (BD) is divided by vertical planes into a series of slices of width b (fig. 10.13). The base of each slice is assumed to be a straight line. The inclination of the base to the horizontal is and the height, measured on the centre-line, h.

The factor of safety is defined as the ratio of the available shear strength () to the shear strength () which is required to be mobilized to maintain a condition of limiting equilibrium:

FS =

(3.23)

The factor of safety is taken to be the same for each slide, implying that there must be mutual support between slices, meaning that forces must act between the slices.

Fig. 3.13The forces (per unit dimension normal to the section) acting on a slice are:

the total weight of the slice, G = if the case);

the total normal force on the base, N, equal to . In general, this force has two components: the effective normal force , equal to , and the boundary water force U, equal to ul, where u is the pore water pressure at the centre of the base and l is the length of the base;

the shear force on the base T =

the total normal forces on the slides, E1 and E2;

the shear forces on the sides, X1 and X2.

Any external forces, such as a surcharge pressure on the ground surface, must be also included in the analysis.

The problem is statically indeterminate and, in order to obtain a solution, assumptions must be made regarding the interslice forces E and X and the resulting solution for factor of safety is not exact.

The Fellenius solution

Known also as the Swedish method, this solution implies that for each slice the resultant of the interslice forces is zero.

The following example refers to a stratified soil mass (fig. 3.14).

When the sliding mass ABCD is divided in slices, it is required that the base of each slice belongs to only one layer.

The forces acting on the slice i are considered.

The total weight of the slice i, Gi, is decomposed in a normal component Ni and a tangential one Ti.

Ni = Gi cos

Ti = Gi sin

The force Ti tends to produce the sliding of the slice i along the failure surface.

The force opposing to the sliding is the shear force due to the shear strength mobilized on the base of the slice i.

Fig. 3.14

Si =

ci, represent the shear strength parameters corresponding to the soil layer in which is located the base of the slice i.

The equilibrium of the slice i can be expressed considering the moment of the forces Ti and Si about 0.

The factor of safety FS can be defined as the ratio between the stability moment MS of forces opposing to the sliding and the overturning moment M0 of forces producing the sliding.

(3.24)

A particular case occurs when the vertical passing through 0 divides the sliding mass (ABCD) in two parts.

As one can see, for the slices located at the left of the vertical line, the tangential component of the weight W is directed against the sliding, therefore it must be included among the stability forces. The expression of the factor of safety becomes:

(3.25)

Where sincorresponds to the slices located at the left of the vertical and sinto the slices located at the right of the vertical.

In relations (3.24) and (3.25) the factor of safety is expressed in terms of the total stresses.

The Swedish method underestimates the factor of safety; the error, compared with more accurate methods of analysis, is usually within the range 5-20%.

An approximative method to obtain the minimum factor of safety is the following one:

It is assumed that the centres of circular failure surfaces susceptible to induce low factors of safety are located on a line O1M (fig. 3.15). The point O1 is found at the intersection of lines making angles with the slope, respectively the horizontal passing through point E. Values of in function of the slope are given in the table 3.1.

Fig. 3.15Table 3.1tan

1.75.11:11:1.51:21:31:5

60o45o33o45o26o34o18o25o11o19o

29o28o26o25o25o25o

40o37o35o35o35o37o

The point M has the coordinates (4.5H) and (H), in respect with the point B.

On the line O1M are chosen several points as centres of circular failure surfaces. Using the slices method, the factor of safety is computed for each surface and is drawn at a convenient scale normal to the line O1M, taken as a reference. A curve of variation of FS is obtained. The tangential to the curve parallel to the reference line defines .

If is larger than a preestablished allowable value , the stability condition of the slope is fulfilled.

If < , appropriate measures have to be considered in order to improve stability conditions, such as diminishing the slope (angle ), removing soil from the upper part of the slope or creating an embankment at the base of the slope (fig.3.16).

Fig. 3.163.3.2 Translational slip in the case of a plane failure surface parallel to the surface of the slope

It is assumed that the potential failure surface is parallel to the surface of the slope and is at a depth that is small compared with the length of the slope. The slope can be considered as being of infinite length, therefore end effects can be ignored. The slope is inclined at angle to the horizontal and the depth of the failure plane, which usually is the thickness of the soil layer covering a rock layer, is z (fig. 3.17). The water table is taken to be parallel to the slope at a height of m Z (0 < m < 1) above the failure plane. Steady seepage is assumed to be taking place in a direction parallel to the slope. The forces on the side of any vertical slice are equal and opposite and the stress conditions are the same at every point in the failure plane.

Fig. 3.17In terms of effective stress, the shear strength of the soil along the failure plane is:

(3.26)

and the factor of safety is:

FS =

(3.27)

For a slice of width 1 and height z, the weight G is:

G =

The vertical pressure on the base is:

p =

The expressions for and u are:

u =

By replacing and u in the relation (3.26) and then in the relation (3.27), the factor of safety FS is found.

Special cases

If = 0 and m = 0, when the soil between the surface and the failure plane is not fully saturated, then:

FS =

(3.28)

The relation (3.28) is identical to the one previously established.

If = 0 and m = 1, when the water table coincides with the surface of the slope, then:

F =

(3.29)

As already shown in the p. 10.1, when = 0, in the case of non-cohesive soils, the factor of safety is independent of the depth Z.

For a total stress analysis the shear strength parameters cu and are used and u is taken zero.

3.3.3 Translational slip on a predetermined failure surface taken as a polygonal surface

As in the previous case, the failure surface is predetermined and is represented by the surface of a rock layer covered by a mass of soil. Quite often this is a diluvium made primarily of cohesive soils deposited over a rock.

To check the stability, the contact surface between the deluvial mass and the rock is assumed to be a polygonal surface (fig. 10.18.a). Vertical planes are drawn in the points of change in slope of the contact surface, dividing the sliding soil mass in a number of blocks. Horizontal interblock forces are assumed.

The analysis starts with the block at the top of the sliding mass, on which the following forces are acting and must be in equilibrium (fig. 10.18.b):

Fig. 3.18 the weight G1, of known magnitude and direction;

the reaction Q1 on the base of the slice, with an angle to the normal, where is the angle of internal friction of the soil;

the cohesion force C1 on the base, known in magnitude and direction; C1 = cl1, where c is the cohesion of the soil;

the normal force E2-1 on the side, representing the interblock force, known only as direction.

Forces G1 and C1 are composed and their resultant is decomposed on the directions of Q1 and E2-1.

Moving to the second block, besides the forces G2, Q2, C2 and E3-2, the normal force transmitted by the slice 1, E1-2 = E2-1 should be considered.

The known forces E1-2, G2 and C2 are decomposed and their resultant is decomposed on the known directions of Q2 and E3-2.

In the same manner, the thrust E transmitted by each block to the next block is found.

At the block at the bottom, there are Gn, Cn and En-1-n known in magnitude and direction and Qn known in direction. The polygon of the forces is constructed.

There are three possibilities (fig. 10.19):

Fig. 3.19a. the polygon closes; this situation corresponds to the limiting equilibrium;

b. in order to close the polygon, is needed a force E directed in the direction of the slide; the slope is stable, has a reserve of stability proved by the force E; to define the factor of safety FS, the analysis is repeated by gradually increasing the weight of the slices until the situation a of limiting equilibrium is met; the factor of safety is then equal to the factor to which the weight of the slices was increased;

c. in order to close the polygon, is needed a force E opposed to the direction of the slide; the slope is not stable and the force E, multiplied by a factor of safety FS could be used for the dimensioning of a wall placed at the bottom of the slope (fig. 3.20).

Fig. 3.203.4 Choice of the shear strength parameters to be used in the stability of slopes analysis

The use of the appropriate shear strength parameters, in correlation with the construction conditions, is essential for the analysis of the stability of slopes.

In what follows, two examples will be considered.3.4.1 Stability of an embankment on a saturated clay

Even if the slopes of the embankment are stable, the loss of stability can occur along a failure surface originating in the foundation soil (fig. 3.21).

Fig. 3.21A point M is taken in the possible failure surface. Diagrams of variation in time of the load exerted on the foundation soil by placing successive layers of soil within the embankment, of the shear stress on a plane passing through the point M, of the shear strength f of the clay and the factor of safety FS.

The initial water pressure is:

uo =

(3.30)

where ho is the depth of the considered point below the ground water table.

Due to the very low permeability of the clay and to the relatively short construction period of the embankment, it can be assumed that there is no drainage of the pore water and that no significant dissipation is likely during construction. The clay will be loaded in the undrained conditions, responding with the shear strength in undrained conditions.

After the end of construction, total stresses in the point under consideration remain unchanged, while the excess pore water pressure dissipates gradually until becoming zero after a time t2. The reduction in the pore water pressure is joined by a decrease of the soil porosity and an increase of the effective stresses and of the shear strength of the foundation soil.

Checks of the stability should be made for two characteristic moments of the construction:

a. end-of-construction (time t1)

A total stress analysis is performed using the parameter cu obtained from U U shear tests (unconsolidated-undrained tests) on saturated samples.

b. long-term stability (time t2)

An effective stress analysis is performed, using the shear strength parameters , obtained in drained conditions.

If one examines the diagram of variation in time of the factor of safety FS = , (fig. 3.22) realizes that in this example the most dangerous condition corresponds to the end-of-construction time t1. In the long term, following t1, the stability conditions are improving.

Fig. 3.223.4.2 Stability of an excavated slope

An excavated slope is considered (fig.3.23) in a cohesive soil mass. The excavation is joined by a lowering of the water table.

A point P is taken on a potential failure surface.

As in the previous example, it is assumed that the excavation takes place over a short period of time; therefore undrained conditions are valid for the clay. As the excavation proceeds the shear stress in the point P increases.

The reduction of the geological pressure on the point P leads to a decrease of the pore water pressure (the pore water pressure change is negative).

It was shown that for an increment of isotropic stress associated with a major principal stress increment, the excess pore water pressure is:

(3.31)

Fig. 3.23For a saturated clay in undrained conditions B = 1

(3.32)

In the case of an excavated slope, the minimum principal stress decreases more than the maximum principal stress , therefore is negative, ( is positive and is negative, with a value depending on A.

In time, the excess pore water pressure dissipates producing a heave of the clay and a reduction of the shear strength.

Checks of the stability for the two characteristic moments of the construction:

a. end-of-construction (time t1)

As in the previous example, a total stress analysis is performed, using the parameter cu.

b. long-term stability (time t2)

An effective stress analysis is performed (by knowing the pore water pressure corresponding to the final position of the ground water) using the parameters and .

One can realize that, unlike the previous example, the most dangerous conditions appears after a long time t2 because the shear strength decreases until the excess negative pore water pressure become zero (fig. 10.24).

Fig. 10.24In conclusion, when choosing the shear strength parameters the following recommendations have to be followed:

for the end-of-construction, condition when the construction means both the loading or the unloading of the saturated clay susceptible to loose its stability, and the construction period is short in comparison to the time required for the dissipation of the excess pore water pressure, a total stress analysis is performed using the parameter cu;

similarly, but on partly saturated clay, analysis can be done either in total stresses with parameter obtained from U U tests or in effective stresses (which implies establishing the parameter pressure) with parameters , ;

for the long term stability analysis, the effective stress are used, after establishing the pore water pressure corresponding to the final level of the ground water with the parameter and .

PAGE 89

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