STABILITY MODELS FOR 

 GRANITIC SLOPES 



Infinite Slope Analysis — 

 Natural Slopes 



Slopes fail when the shear stress on any potential failure 

 surface exceeds the shear strength. It is customary to express 

 this balance of forces or stresses in terms of a ratio or factor of 

 safety against sliding. This ratio or factor of safety is commonly 

 defined as follows: 



F = shear resistance along critical surface 



shear forces promoting sliding on the surface 



(1) 



Most natural granitic slopes consist of a shallow, cohesion- 

 less soil mantle overlying an inclined bedrock contact. An ideal- 

 ized soil profile and slope geometry were shown in figure 2. The 

 stability of such slopes can be determined by the so-called 

 "infinite slope analysis." This model assumes that the thickness 

 of the sliding mass is constant and relatively thin compared to its 

 length. This also implies that the sliding surface is parallel to the 

 slope and approximately planar over most of its area. The 

 infinite slope model has been used previously and with good 

 results to analyze the stability of slopes in the Idaho batholith 

 (Gonsior and Gardner 1971; Prellwitz 1975). 



According to the infinite slope model, the factor of safety 

 against sliding can be expressed by the following mathematical 

 equation; 



fCs + C 

 Ltan6' 



Cr "I tanci)' 



(qo + 7H) + (^3-7)HwJ ^ 



[(qo + 7H) + (7sAT-7)Hw] 



F» factor of safety against sliding 



(2) 



H=thickness (depth) of soil mantle (in vertical direction) 

 Hw = height of piezometric surface above bedrock contact 

 Cs = effective cohesion of soil 

 Cr = shear strength increase from root reinforcement 

 expressed as a pseudo cohesion 

 7 = moist density of soil (above piezometric surface) 

 7SAT = saturated density of soil 

 73 = buoyant density of soil (73 = 7sat - 7w) 

 7v» = density of water 



qo = vertical surcharge (from weight of vegetation) 

 P = slope angle or gradient 

 4)' = effective angle of internal friction of the soil 



Equation 2 is a completely general expression for factor of 

 safety, which takes into account the existence or presence of 

 cohesion (Cs) in the soil, a slope surcharge (qo), and a ground 

 water table or piezometric surface in the slope (Hw)- In the event 

 these are absent, the terms in which they appear are removed 

 from the equation. If the soil is completely dry, that is, in the 

 absence of any piezomethc surface (Hw = 0), dry density 

 (^dry) rnay be substituted for moist density (7) in equation 2. 



The relative importance of the various parameters in equation 

 2, such as, cohesion as opposed to friction, and the effect on 

 safety factor of a change in one variable, such as surcharge, is 

 not intuitively obvious. The relative importance of each variable 



in the equation and direction of change that may be produced by 

 altering input variables is best determined by conducting a 

 sensitivity analysis using a realistic range of values for each 

 input variable. This analysis is conducted for the study water- 

 sheds in a subsequent section of the report. 



Slope vegetation and its removal affect several of the input 

 variables in equation 2. Most noticeably or obviously affected 

 will be the slope surcharge (qo). piezometric height (Hw), and 

 root cohesion term (Cr). Reduction in evapo transpiration as a 

 result of ciearcutting may also affect soil density (7) in addition 

 to the piezometric surface elevation. Based on the result of 

 soil-root reinforcement studies conducted to date, the angle of 

 internal friction of the soil (cb) is affected hardly at all by the 

 presence or absence of roots (Gray 1 978; Waldron 1 977). The 

 extent and consequences of forest vegetation and its removal 

 on the stability of granitic slopes in the Idaho batholith are 

 examined further in the next section of the report. Similar 

 studies employing the infinite slope model have been con- 

 ducted by Wu (1976), Brown and Sheu (1975), Gray (1978). 

 and Ward (1976). 



A simple theoretical model of a fiber-reinforced soil was de- 

 veloped by Wu (1 976). This model was used by Wu to estimate 

 the contribution of rooting strength to slope stability in analyses 

 of both forested and cutover slopes in Southeast Alaska. A 

 virtually identical model was developed independently by Wal- 

 dron (1 977) and evaluated in conjunction with direct shear tests 

 run in the laboratory on root-permeated homogeneous and 

 stratified soil. 



A schematic illustration of the root reinforcement model pro- 

 posed by Wu (1 976) is shown in figure 20. The model essential- 

 ly envisions an elastic root or fiber embedded in a soil matrix and 

 initially oriented normal to the shearing surface. Deformation in 

 the soil is resisted by tangential forces which develop along the 

 fiber and which in turn mobilize the tensile resistance of the 

 fiber. These tangential forces (t on fig. 20) are produced by 

 friction or by bonding between the fiber and surrounding soil 

 matrix. The soil friction angle (6) is assumed to be unaffected by 

 the reinforcement. This model also assumes that tensile 

 strength of the fibers or roots is fully mobilized during failure. 

 This requires either fixity of the roots at their ends or roots that 

 are long enough and or frictional enough for the frictional or 



LEGEND 



z = Thickness of shear zone 

 X = Horizontal deflection of root 

 9 = Angle of shear distortion 

 Tr= Root tensile strength 

 f - Skin friction along root 



DEFORMED 

 ROOT^ 



■INTACT 

 ROOT 



SHEAR 

 ZONE 



Figure 20. — Root reinforcement model. Flexible, elastic root is oriented 

 in perpendicular direction to shear surface (after Wu 1976). 



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