If should be noted that when c?=0 the factor of safety is independent of the depth z. If c? is greater to zero, the factor of safety is a function of z, and ? may exceed ?? provided z is less than a critical value.
For a total stress analysis the shear strength parameters cu and ?u are used and the value of u is zero.
A long natural slope in fissured overconsolidated clay is inclined at 12° to the horizontal. The water table is at the surface and seepage is roughly parallel to the slope. A slip has developed on a plane parallel to the surface at a depth of 5m.The saturated unit weight of the clay is 20 kN/m3. The peak strength parameters are
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c?=10kN/m and ??=26°; the residual strength parameters are c?r=0 and ??r=18°. Determine the factor of safety along the slip plane(a) in terms of the peak strength parameters, (b) in terms of the residual strength parameters.
With the water table at the surface (m=1), at any point on the slip plane: ???satzcos2?
?20?5?cos212?95.5kN/m2 ???satzsin?cos?
?20?5?sin12?cos12?20.3kN/m2 u??wzcos2?
?9.8?5?cos212?46.8kN/m2 Using the peak strength parameters: ?f?c??(??u)tan??
?10?(48.7?tan26)?33.8kN/m2 Then the factor of safety is given by:
F??f33.8??1.66 ?20.3Using the residual strength parameters, the factor of safety can be obtained from equation 1.16:
F? ???tan??r
?sattan?10.2tan18??0.78 20tan121.6 General Methods of Analysis
Morgenstern and Price[1.8] developed a general analysis in which all boundary and equilibrium conditions are satisfied and in which the failure surface
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may be any shape, circle ,non-circle and compound. The soil mass above the failure plane is divided into sections by a number of vertical planes and the problem is rendered statically determinate by assuming a relationship between the forces E and X on the vertical boundaries between each section. This assumption is of the form:
X??f(x)E (1.17) where f(x) is an arbitrary function describing the pattern in which the ratio X/E varies across the soil mass and ? is obtained as part of the solution along with the factor of safety F. The values of the forces E and X and the point of application of E can be determined at each vertical boundary. For any assuming function f(x) it is necessary to examine the solution in detail to ensure that it is physically reasonable (i.e. no shear failure or tension must be implied within the soil mass above the failure surface). The choice of the function f(x) does not appear to influence the computed value of F by more than about 5﹪ and f(x)=1 is a common assumption.
The analysis involves a complex process of iteration for the value of ?and F, described by Morgenstern and Price [1.9], and the use of a computer is essential.
Bell [1.15] proposed a method of analysis in which all the conditions of equilibrium are satisfied and the assumed failure surface may be of any shape. The soil mass is divided into a number of vertical slices and statical determinacy is obtained by means of an assumed distribution of normal stress along the failure surface. Thus the soil mass is considered as a free body as is the case in the ?-circle method.
Sarma [1.16] developed a method, based on the method of slices, in which the critical earthquale accelaration required to produce a condition of limiting equilibrium is determined. An assumed distribution of vertical inter-slice forces is used in the analysis. Again, all the conditions of equilibrium are satisfied and the assumed failure surface may be of any shape. The static factor of safety is the factor by which the shear strength of the soil must be reduced such that the critical acceleration if zero.
The use of a computer is also essential for the Bell and Sarma methods and all solutions must be checked to ensure that they are physically acceptable.
1.7 End-of-Construction and Long-Term Stability
When a slope is formed either by excavation or by the construction of an embankment the changes in total stress result in changes in pore water pressure in the vicinity of the slope and, in particular, along a potential failure surface. Prior to
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construction the initial pore water pressure(u0) at any point is governed either by a static water table level or by a flow net for conditions of steady seepage. The change in pore water pressure at any point is is given theoretically by equation 4.17 or 4.18. The final pore water pressure, after dissipation of the excess pore water pressure, is governed by the static water table level or the steady seepage flow net for the final conditions after construction.
If the permeability of the soil is low, a considerable time will elapse before any significant dissipation of excess pore water pressure will have taken place. At the end of construction the soil will be virtually in the undrained condition and a total stress analysis will be relevant. In principle an effective stress analysis is also possible for the end of construction condition using the pore water pressure (u) for this condition, where :
u?uo??u
However, because of its greater simplicity, a total stress analysis is generally used. It should be realised that the same factor of safety will not generally be obtained from a total stress and an effective stress analysis of the end-of-construction condition. In a total stress and an effective stress analysis of the end-of-construction condition. In a total stress analysis it is implied that the pore water pressures are those for a failure condition: in an effective stress analysis the pore water pressures used are those predicted for a non-failure condition. In the long-term, the fully-drained condition will be reached and only an effective stress analysis will be appropriate.
If, on the other hand, the permeability of the soil is high, dissipation of excess pore water pressure will be largely complete by the end of construction. An effective stress analysis is relevant for all conditions with values of pore water pressure being obtained from the static water table level or the appropriate flow net.
Pore water pressure may thus be an independent variable, determined from the static water table level or from the flow net for conditions of steady seepage, or may be dependent on the total stress changes tending to cause failure.
It is important to identify the most dangerous condition in any practical problem in order that the appropriate shear strength parameters are used in design. Excavated and Natural Slopes in Saturated Clays
Equation 4.17, with B=1 for a fully-saturated clay, can be rearranged as follows:
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?u?11(??1???3)?(A?)(??1???3) (1.18) 22For a typical point P on a potential failure surface(Fig.9.10) the first term in
equation 1.18 is negative and the second term will also be negative if the value of A is less than 0.5. Overall, the pore water pressure change ?u is negative. The effect of the rotation of principal stress directions is neglected. As dissipation proceeds the pore pressure increases to the final value as shown in Fig.1.10. The factor of safety will therefore have a lower value in the long-term, when dissipation is complete, than at the end of construction.
Figure 1.10 Pore press pressure dissipation and factor of safety (After
Bishop and [1.2])
Residual shear strength is relevant to the long-term stability of slopes in over consolidated fissured clays. A number of cases are on record in which failures in this type of clay have occurred long after dissipation of excess pore water pressure hade been completed. Analysis of these failures showed that the average shear strength at failure was bellow the peak value. In clays of this type it is suspected that large strains can occur locally due to the presence of fissures, resulting in the peak strength being reached, followed by a gradual decrease towards the residual value. The development of large local strains can lead eventually to a progressive slope failure. Fissures may not be the only cause of progressive failures: there is considerable nonuniformity of shear stress along a potential failure surface and local overstressing may initiate progressive failure. It should be realised, however, that the residual strength is reached only after a considerable slip movement has taken place and the strength relevant to first-time′ slips lies between the peak and residual values. Analysis of failures in natural slopes in overconsolidated fissured clays has indicated that the residual shear strength is ultimately attained, probably as a result of successive slipping. 1.8 Stability of Earth Dams
In the design of earth dams the factor of safety of both slopes must be determined as possible for the most critical conditions. For economic reasons an unduly conservative design must be avoided. In the case of the upstream slope the most critical stages are at the end of construction and during rapid drawdown of the reservoir level. The critical stages for the downstream slope are at the end of construction and during steady seepage when the reservior is full. The pore water pressure distribution at any stage has a dominant influence on the factor of safety
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and in large earth dams it is common practice to install a piezometer system so that the actual pore water pressures can be measured at any stage and compared with the predicted values used in design(provided an effective stress analysis has been used) . Remedial action can then be taken if the factor of safety , based on the measured values, is considered too bellow. (a) End of Construction
The construction Period of an earth dam is likely to be long enough to allow partial dissipation of excess pore water pressure before the end of construction, especially in a dam with internal drainage. A total stress analysis, therefore, would result in too conservative a design. An effective stress analysis is preferable, using predicted values of ru .
The pore pressure (u) at any point can be written as: u?u0??u
where u0 is the initial value and ?u is the change in pore water pressure undrained conditions. In terms of the change in total major principal stress:
u?u0?B??1 Then:
ru?u0??1?B ?h?hIf it is assumed that the increase in total major principal stress is approximately
equal to the fill pressure along a potential failure surface, then: ru?u0?B (1.19) ?hThe soil is partially saturated when compacted, therefore the initial pore water pressure (u0) is negative. The actual value of u0 depends on the placement water content, the higher the water content, the closer the value of u0 to zero. The value of B also depends on the placement water content, the higher the water content, the higher the value of B. Thus for an upper bound:
ru?B (1.20)
The value of B must correspond to the stress conditions in the dam. Equations 1.19 and 1.20 assume no dissipation during construction. A factor of safety as low as 1.3 may be acceptable at the end of construction provided there is reasonable confidence in the design data.
If high values of ru are anticipated, dissipation of excess pore water pressure
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