9 T 



sn 



Assuming -- ^ ^ = 0, as in the infinite slope theory, then a = y cos 6, ana 

 equation 41 becomes 



-ri- = Y sin 6 - ^ ^ {(Y n cos e tan <), + c) 2 - t^^2}\ . . . (42) 



This provides a rational, though admittedly oversimplified and slightly inaccurate, 

 means for allowing the active and passive states of stress to occur simultaneously at 

 the top and bottom ends, respectively, of a failure mass in a long slope. It is 

 simplified in that the shear stresses on planes parallel and normal to the slope are 

 constant for a given depth, as in the infinite slope theory; and in that the normal 

 stresses on planes normal to the slope increase at a constant rate from the active to 

 the passive values. It is inaccurate in that it does not enable development of the 

 failure surfaces connecting the critical depth with ground surface at the ends of the 

 failure mass. However, this inaccuracy should not be too serious if L is large 

 relative to the lengths required to develop the connecting failure surfaces. 



The effect of this modification of the infinite slope theory is to increase the 

 critical depth. Unfortunately, attempts to solve equation 42 for have been unsuc- 

 cessful. However, a finite difference approximation has been programed to enable 

 determination of the versus depth relationship as well as the critical depth for 

 any particular slope and soil. 



Figure 12 illustrates the effects of varying L on the t^^^ vs. o.^ relationship 

 and on the critical depth for the slope and soil conditions assumed in figure 5. No 

 general conclusions can be drawn from figure 12 because it represents a particular 

 case. However, the reader can easily conduct a similar analysis for any other slope 

 and soil by programing a finite difference solution to the above differential equation. 

 It will merely be noted that, for the slope and soil conditions assumed in figures 5 

 and 12, transition lengths less than about 100 times the critical depth of 11.85 ft. 

 result in rapidly increasing modified critical depths. 



It will be recalled from figure 5 that the critical depth is reduced from 11.85 tc 

 8.7 ft., or about 25 percent, when = 0, or seepage is occurring throughout the soil 

 profile. However, according to figure 12, if the transition length was about 170 ft. 

 (or approximately 14 times the conventional critical depth of 11.85 ft.), then the 

 modified critical depth would be about 25 percent greater than the conventional criti- 

 cal depth. Thus, a depth of 11.85 ft. on a slope of this length (170 ft.) might be 

 safe under a condition of seepage throughout the soil profile even though the conven- 

 tional theory would predict an unsafe condition. 



Discussion thus far has been limited to uniform, uninterrupted slopes. Usually, 

 the major interest in natural slopes is with regard to their response to excavations, 

 or cuts, and to such other activities as timber harvesting. Recent advances in the 

 application of the finite element method of analysis seem to hold the greatest promise 

 for assessing the response of natural slopes to such alterations. Nevertheless, it 

 still remains necessary to be able to describe the state of stress within the slope 

 before modification as well as the in situ material properties. Having done so, it 

 may be possible to use the simple transition- length approach to assess the stability 

 of many cut slopes. 



Accuracy of any method of analysis remains limited by the ability to measure or 

 predict stress history and soil characteristics. Further, more knowledge about seepage 

 and pore pressures is needed, especially in the realm of unsaturated flow. Also, 

 the mechanisms of creep and progressive failure require considerably more study and 

 elucidation. It is apparent that any accurate, rational method of analysis will 

 ultimately have to combine rheological and soil moisture characteristics with effective 

 strength characteristics in order to fully account for the behavior of natural slopes. 



26 



