8.8 Pseudo-static approach
In pseudo-static methods, the cyclic earthquake motion is replaced with a constant horizontal acceleration equal to kc(g), where kc is the seismic coefficient, and g is the acceleration of gravity. A force is applied to the soil mass equal to the product of the acceleration and the weight of the soil mass.
The limit equilibrium method is modified to include horizontal and vertical static seismic forces that are used to simulate the potential inertial forces due to ground accelerations in an earthquake. In pseudostatic slope stability analyses, a factor of safety against failure is computed using a static limit equilibrium stability procedure, in which a pseudostatic, horizontal inertial force, which represents the destabilizing effects of the earthquake, is applied to the potential sliding mass.
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 out of slope direction. This pseudostatic lateral force Fh is calculated as follows:
Fh = horizontal pseudostatic force acting through centroid of sliding mass out of slope direction.
m=total mass of slide material
W=total weight of slide mass
a= acceleration, maximum horizontal acceleration at ground surface due to earthquake
amax = peak ground acceleration
A considerable limitation of the pseudostatic approach is that the horizontal force representing the effects of seismicity is constant and acts in only one direction. With dynamically applied loads, the force may be applied for only a few tenths of a second before the direction of motion is reversed. The result of these transient forces will be a series of displacement pulses rather than complete failure of the slope.
Seismicity subjects sliding mass to vertical as well as horizontal pseudostatic forces. However, the effect of vertical pseudostatic force on sliding mass is ignored as it has very little effect on its slope stability. Although, the weight of the sliding mass W can be readily estimated, selection of seismic coefficient requires considerable experience and judgement. Certain guidelines regarding selection of seismic coefficient are as follows:
· Higher the value of peak ground acceleration, higher is the value of
· is also determined as function of earthquake magnitude.
· When items 1 and 2 are considered, should never be greater than .
· Sometimes local agencies suggest minimum value of seismic coefficient.
· For small slide mass, =.
· For intermediate slide mass, =0.65.
· For large slide mass, =0.1 for sites never generating 6.5 magnitude earthquake and, =0.15 for sites near faults generating 8.5 magnitude earthquake.
· =0.1 for severe earthquake, = 0.2 for violent and destructive earthquake and=0.5 for catastrophic earthquake.
This is simplest type of slope stability analysis. A wedge failure has planar slip surface, inclined at an angle α to horizontal. Analysis could be performed for the case of planar slip surface intersecting the face of slope or passing through toe of slope. Factor of safety for pseudostatic analysis is obtained as follows:
FOS= factor of safety for pseudostatic analysis
u= average pore water pressure along slip surface
c’ and ’ are the effective cohesion and friction angle material
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 this purpose 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 case of sloping water table, flow net can be used to estimate pore water pressure below water table.
Method of SlicesIn 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 a rotational 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. Factor of safety can be calculating the resisting forces to driving forces for each slice and then summed to obtain factor of safety. However, this method involves 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, α. 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.