A slope’s stability is a function of the material properties and geometry of the slope. Pore water pressure has been included in the material properties section of this chapter. The effect of material properties can be investigated using the equations for the stability of an infinite slope, while the role of geometry can be looked at through a simple method of slices analysis. The mechanics of slope stability can be divided into two forces: the driving forces and the resisting forces. These two forces oppose each other and the state of stability existing in a slope can be thought of as their ratio:
When the material properties and geometry of a slope are examined, this simplified ratio becomes an equation called the factor of safety (Fs) against landsliding and is defined as (Al-Khafaji and Andersland, 1992):
Equation 1
A Fs > 1 would theoretically be a stable slope because the shear strength (resisting forces) would be greater than the actual shear stress. A Fs < 1 would theoretically be an unstable slope because the actual shear stress (driving forces) would be greater than the shear strength. A critically stable slope would have a Fs = 1. Because of the inability to know all the conditions present within a slope, Senneset (1996) recommends that slopes with a factor of safety less than 1.3-1.5 be considered potentially unstable. Since most of the failures examined in the study area failed very quickly and the general material is composed of semiconsolidated clays and silts, undrained conditions were assumed in all stability analysis, unless otherwise mentioned. When the forces acting on a slope are examined in detail, the parameters and geometry that contribute to a stable, unstable, or critically stable slope become apparent.
The resistance of a material to sliding along a slip surface is its shear strength (Easterbrook, 1993). Shear strength of materials varies considerably depending on their physical properties. A simple model for a slopes stability, the limit equilibrium equation, is derived using the Mohr-Coulomb Law, is illustrated in equation 2 (Figure 14)(Lambe and Whitman, 1969). Using the infinite slope equation, derived from equation 2 (equation 3), which is a good approximation for landslides that are about ten times longer than they are thick and have generally uniform soil conditions, the factor of safety is defined as (Figure 14)(Lambe and Whitman, 1969):
Equation 2
Figure 14: Infinite slope diagram illustrating parameters and equations for the factor of safety and the critical slope thickness for a slope with seepage parallel to the slope.
Equation 3*
*Note - for saturated slopes with seepage parallel to the slope. See Figure 14 for deviation of equation 3, from equation 2, including definition of variables.
Examining equations 2 and 3 and Figure 14, one can see that the parameters which influence material properties are cohesion, angle of internal friction, slope angle, pore-water pressure, and thickness. From these equations, one can see that a decrease in cohesion (c) and/or the angle of internal friction (f) and/or an increase in slope angle (q) will decrease the factor of safety, potentially causing failure (Easterbrook, 1993). These parameters are for conditions on the failure plane. Many slopes are colluvial slopes, which are natural slopes formed by movement of soil from its original place by landsliding and creep (Al-Khafaji and Andersland, 1992). For natural slopes, the residual strength should be used (Skempton, 1985). The residual strength occurs when the strength drops off because of the formation of a shear plane by continued shear deformation (Spangler and Handy, 1982). Slip surfaces usually occur at the bedrock-soil interface, where colluvial slip surfaces generally form (Richards, 1980).
A useful form of the infinite slope equation is to solve for the critical thickness (Figure 14, Equation 4). This is the thickness required to give a factor of safety of one
and is calculated as:
Equation 4
From equations 1 through 3, it is clear that anything that either decreases the shear strength or increases the shear stress will move the slope closer to failure. There are two conditions that can cause a reduction of shear strength (Skempton, 1985):
1) a temporary reduction in shear strength, usually caused by an increase in water content, which will eventually dissipate, or
2) a permanent reduction in shear strength, decreasing the strength to the residual value, usually due to shearing deformation.
It is a common observation that much of the shallow landsliding in the Portland area occurs during the winter rainy seasons. This increased precipitation results in higher groundwater tables, saturation of soils, and consequent pore pressure increases and temporary shear strength reduction (Spangler and Handy, 1982). Through equations 2 it is clear that an increase in water content or saturation of a slope increases the pore water pressure, in turn decreasing the frictional resistance through the angle of internal friction (Spangler and Handy, 1982). Once even several centimeters of movement have taken place, no more than the residual strength can be developed along a surface, resulting in permanent strength reduction (Skempton, 1985). This observation is confirmed by the approximately 700 landslides that occurred in the Portland metropolitan area during the greater than normal rainy seasons of 1996 and 1997 (Burns et al., 1998).
The relationship between time of year and landslide movement is not as clear for very large or deep-seated landslides (defined in Chapter 6), but it is expected that deep-seated landslides would potentially be most active during the winter (Richards, 1982). Furthermore, it may take prolonged periods of rainfall to activate or reactivate deep-seated landslides (Baum et al., 1993).
In developed areas, local slope saturation may be a result of concentrated runoff from man-made structures. This higher than normal concentration of runoff from buildings and paved surfaces can raise the water table on slopes that might not normally have such a high water table, even during the rainy seasons. Trees can act as pumps to reduce the water table and consequent pore water pressure.
Another human related cause of shear strength reduction is the removal of vegetation from a slope. Trees act as water pumps, reducing the pore water in the soil. Woody vegetation also increases the resisting force through tree root strength on shallow colluvial slopes (Riestenberg and Sovonick-Dunford, 1983). The tree roots play the same role as cohesion in equation 2-4. Their role in increasing shear strength can be seen through equation 5:
Equation 5
Where: A=area of slip surface penetrated by roots
F=normal
force required to break each root
The simplest stability model illustrating the role of geometry is the Bishop method (Figure 15)(Bishop, 1955). This type of analysis displays the role of different portions of a landslide. Other methods of slices, like Janbu, are similar, however they allow for a more complex geometry (Janbu, 1973).
In the modified Bishop analysis, a circular slip surface is assumed and the area above this slip surface is divided into slices of various widths that can be examined individually (Figure 15B) or together (Figure 15A) to calculate the factor of safety (Bishop, 1955).
Figure 15: Method of slices (modified Bishop Method) parameters and equation for calculating the factor of safety: (A) general diagram displaying circular slip surface and slices and (B) detail of forces acting on a single slice. Side forces not included in Bishop Model, added in Janbu (Bishop, 1955).
The factor of safety in this type of analysis is the resisting moments divided by the driving moments, where moments are about the circular center (Figure 15). With this analysis in mind, the addition of weight (or surcharge), generally to the higher numbered slices (right of center) will increase the driving moments about the center. In the study area, this addition of weight can be caused by human or natural agencies such as the following:
1) loads imposed by fill and buildings or construction at the top of slopes and
2) weight of water from heavy rains and concentrated man-made structure runoff.
The removal of weight from the lower numbered slices (left of center) will decrease the resisting moments. The following are the three main ways that these resisting moments are being reduced in the study area:
1) bank erosion by creeks and streams, which removes support from the base of slopes and from toes of preexisting landslides,
2) human excavation and cuts of slopes for constructing roads and building lots, and
3) landslides on the lower portion of the slope, decreasing the stability of the upper part of the slope.
The stability of a slope is governed by these two factors: material properties and the slope geometry. It should now be easy to see that a change in either of these properties by human and natural agencies can affect the factor of safety of the slope or an existing landslide on that slope.
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