SLOPE STABILITY ANALYSES FOR EARTHQUAKES
9.2 INERTIA SLOPE STABILITY-PSEUDOSTATIC METHOD
This method is easy to understand and is applicable for both total and effective stress slope stability analyses. The method ignores cyclic nature of earthquake. It assumes that additional static force is applied on the slope due to earthquake. In actual analysis, a lateral force acting through centroid of sliding mass is applied which acts in out of slope direction.
This pseudostatic lateral force Fh is calculated as follows:
where, Fh = horizontal pseudostatic force acting through centroid of sliding mass in out of slope direction. For two dimensional analysis, slope is usually assumed to have unit length.
m = total mass of slide material.
W = total weight of slide mass.
a = acceleration, maximum horizontal acceleration at ground surface due to earthquake. ( = amax)
amax = peak ground acceleration.
amax/g = seismic coefficient.
Earthquake subjects sliding mass in general to vertical as well as horizontal pseudostatic forces. Since vertical pseudostatic force on sliding mass has very little effect on its stability, it is ignored.
Based on the results of field exploration and laboratory testing, unit weight of soil or rock can be determined. Consequently, weight of sliding mass, W can be readily calculated.
On the other hand, selection of seismic coefficient takes considerable experience and judgement.
Certain guidelines regarding selection of seismic coefficient is as follows:
1. Higher the value of peak ground acceleration, higher the value of kh. 2. kh is also determined as function of earthquake magnitude.
3. When items 1 and 2 are considered, kh should never be greater than amax/g.
4. Sometimes local agencies suggest minimum value of seismic coefficient.
5. For small slide mass, kh = amax/g
6. For intermediate slide mass, kh = 0.65amax/g
7. For large slide mass, kh = 0.1 for sites near faults generating 6.5 magnitude earthquake and , kh = 0.15 for sites near faults generating 8.5 magnitude earthquake.
8. kh = 0.1 for severe earthquake, = 0.2 for violent and destructive earthquake and
= 0.5 for catastrophic earthquake.
9.2.1 Wedge Method
This is simplest type of slope stability analysis (refer Fig. 9.1). Failure wedge has planar slip surface, inclined at an angle A to horizontal. Analysis could be performed for the case of planar slip surface intersecting the face of slope or passing through toe of slope.
Fig. 9.1 Wedge method (Courtesy: Day, 2002)
As per pseudostatic wedge analysis of Fig. 9.1, four forces are acting:
W = weight of failure wedge = total unit weight Gt times cross-sectional area of failure wedge for assumed unit length of slope.
Fh= khW = horizontal pseudostatic force acting through centroid of sliding mass in out of slope direction.
N = normal force acting on slip surface.
T = shear force acting along slip surface.
For total stress analysis:
T = cL + NtanF = suL
Slope Stability Analyses for Earthquakes 93
caFa = shear strength parameters for effective stress analysis Na = effective normal force acting on slip surface
driving forces Wsin F cos W sin F cos
...(9.2a)
where, FS = factor of safety for pseudostatic analysis u = average pore water pressure along slip surface
For total stress analysis, total stress parameters of soil should be known and is often performed for cohesive soils. For effective stress analysis, effective stress parameters of soil should be known and is often performed for cohesionless soils. For effective stress analysis, pore water pressure along slip surface should also be known. For soil layers above water table, pore water pressure is assumed zero. If the soil is below water table and water table is horizontal, pore water pressure below water table is hydrostatic. In the case of sloping water table flow net can be used to estimate pore water pressure below water table.
9.2.2 Method of Slices
In this method, failure mass is subdivided into vertical slices and factor of safety is determined based on force equilibrium equations. A circular arc slip surface and rotational type of failure mode is often used in this method.
The resisting and the driving forces are calculated for each slice and then summed to obtain factor of safety of the slope. The equation to calculate factor of safety is identical to Eq. (9.2), with driving and resisting forces calculated for each slice and then summed to obtain factor of safety. However, there are more unknowns than equilibrium equations in the method of slices. Consequently, an assumption is to be made concerning interslice forces. In ordinary method of slices, resultant of interslice forces is parallel to average inclination of slice, A. Bishop simplified, Janbu simplified, Janbu generalized, Spencer method and
Morgenstern-Price method are other methods of slices. Because of the tedious nature of calculations, computer programs are routinely used to perform the pseudostatic slope stability analysis using the method of slices. It has not been discussed in detail in this book.
9.2.3 Other Slope Stability Considerations
Important factors which are needed in the cross section to be used for pseudostatic slope stability analysis is as follows:
Different soil layers: If the slope contains different soil or rock type, with different engineering properties, it must be incorporated in the analysis. For all soil layers, either effective shear strength or shear strength in terms of total stress parameters must be known.
Horizontal pseudostatic force is specified for every layer.
Slip surfaces: Either planar or composite type slip surface may be needed for analysis.
Tension cracks: Tension cracks at the top of slope can reduce factor of safety of a slope by as much as 20 percent. This should be included in the analysis. Destabilizing effects of water in tension cracks should also be included in the analysis.
Surcharge loads: Surcharge loads (at top or even on slope face) as well as tie-back anchors should be included in the analysis.
Nonlinear shear strength envelope: If shear strength envelope of soil is non linear, it should be included in the analysis.
Plane strain condition: Long uniform slopes are plane strain condition. Friction angle in this case is about 10% higher than the friction angle obtained in triaxial experiment. This should be included in the analysis.
These considerations are incorporated in Eq. (9.2) to complete the analysis as per actual conditions.