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# Seismic Response and Stability Analysis of Landfills

(6) Slope Stability Analysis

The two conventional approaches to evaluate the stability of landfill structural systems consist of the pseudo-static method of analysis and the Newmark Sliding-Block analysis.

Pseudo-static analysis –

The pseudo-static method gives the limit state equilibrium which produces factor of safety values of slopes during seismic loading. A minimum FS of 1 is desired and essentially means that the driving force is equal to the resisting force since,

FS = 1 = resisting force / driving force

thus, resisting force = driving force if FS = 1

FS = ( ( W * cos(α) – k * W * sin(α) * tan(φ) ) / ( W * sin(α) + k * W * cos(α) )

where k = ah / g

Where,

ah = pseudo-static accelerations in horizontal directions

k = coefficients for horizontal and vertical directions

W = weight of failure mass

g = Earth’s gravitational constant

φ = Friction angle soil property

α = See Figure 1.8

With this said, literature indicates choosing the k coefficients is based on very subjective criteria and descriptions. This can significantly affect the accuracy of this method.

Also, the seismic accelerations exist in the horizontal and vertical directions, however, the horizontal component is the only one considered in design since the vertical forces average out to 0 anyway, thus, becoming insignificant (Kramer,1996).

Figure 1.8: Force diagram depicting a sliding mass on an inclined plan. Force diagram of a landslide mass sitting on an inclined planar slip surface (Terzaghi, 1950)

As per the diagram,

W = weight / unit length of landslide mass

k = seismic coefficient

s = shear resistance along slip surface

α = angle of slip surface inclination

Note in this scenario, the whole landslide mass is considered as a whole, from the toe of the slope to the surface above. This limits our selection and generalizes the failure planes.

Newmark analysis –

The sliding-block analysis was introduced by Nathan Newmark in 1965. The Newmark analysis gives the cumulative, permanent displacement of the slopes during seismic loading. The method utilizes the basic physics scenario of the rigid block on an inclined plane, as shown in Figure 1.9 (Kramer, 1996). Just as there is a certain amount of friction that a wooden block on an inclined surface will need to overcome to be in motion, likewise, the yield acceleration is the acceleration required for the landslide mass to initiate motion, or sliding in this case. The yield acceleration is ay = kyg. When the ay is exceeded, permanent deformation occurs. This method is used to compare between the displacement of the slopes and allowable displacement of the landfill components in the event of a major earthquake. This method is more refined than the pseudo-static method and the end result of this analysis is more applicable to design since it produces a direct physical displacement.

Figure 1.9: Forces acting on a block on an inclined plane in dynamic conditions. (Kramer, 1996)

Figure 1.10: Illustrated the integration of the accelertion above the yield acceleration to render the velocity which is integrated again to obtain the displacement (Krishna, 2009).