bonding strength between roots and soil matrix to exceed the 

 tensile strength of the roots. This assumption appears justified 

 from field examination of broken, exposed roots in shear planes 

 of landslides (Wu 1976; Megahan and others 1978). 



According to this model the tensile force that develops in the 

 fiber can be resolved into a lateral component that directly 

 resists shear and into a normal component that increases the 

 normal or confining stress on the shear plane. (Vlathematically 

 this translates into an increase in shear resistance or "root 

 cohesion" Cr as follows: 



CR = tR [cos e tan 4) + sine] (3) 



where Cr = shear strength increase from root 



or fiber reinforcement 

 6 = angle of shear distortion 

 <t) = angle of internal friction 

 tR = average tensile strength of roots per unit 



area of soil 



The average tensile strength of fibers or roots per unit area 

 of soil (Ir) can be determined by multiplying the tensile strength 

 of the roots (Tr) by the fraction of the soil cross section filled or 

 occupied by roots (Ar/A). Thus: 



tR= S Tj ni ai (4c) 

 A 



where T| = tensile strength of roots in size class i. 



Equations 3 and 4 and the model from which they are derived 

 provide a basis for estimating the contribution of roots to soil 

 shear strength. The only uncertain or indeterminate variable in 

 the equations is the angle of shear distortion (9). This angle will 

 vary with the amount of horizontal shear displacement and the 

 thickness of the shear zone. 



From results of laboratory direct shear tests conducted by 

 Waldron (1977) on various root-permeated soils, the angle of 

 shear distortion varied between 40 to 50 degrees. From the 

 results of field observations of failures in root-permeated soil 

 masses on slopes (Wu 1976), the angle appeared to vary 

 between 45 and 70 degrees at most. By running a parametric 

 variation or sensitivity analysis on equation 3, Wu (1976) 

 showed that the bracketed term is relatively insensitive to all 

 expected values of either friction angle (4)) or shear distortion 

 angle (9). The bracketed term only varied from 0.92 to 1 .31 for 

 20 4) « 40 and 40 9 « 70. Thus assuming the midpoint of 

 the range to be the most probable value for the bracketed term, 

 the shear strength increase from root or fiber reinforcement may 

 be estimated to an average or first approximation simply by 



t, = TR-AR/A (4a) 

 where Tr = tensile strength of roots, psi 



AR/A = root area ratio or fraction of soil cross 

 sectional area occupied by roots 



This is an extremely useful and important relationship because 

 it states that the root contribution to soil strength can be deter- 

 mined solely from measuring the tensile strength of the roots, 

 Tr, and the fraction of the soil cross section occupied by roots, 

 Ar/A. The root cross sectional area, Ar, can be found by count- 

 ing the number of roots in different size classes, nj, in a given soil 

 cross sectional area, A, and then by summing the product of the 

 root numbers in each size class times their corresponding aver- 

 age cross sectional area, ai, for that size class. Thus: 



tR =Tr 1 r\, a, (4b) 

 A 



where ni = number of roots in size class i 



aj = average cross sectional area of roots in 



size class i, square inches 

 A = area of soil in sample count, square inches 

 Tr = average tensile strength of roots, psi. 



This linear relationship expressed in equation 4 between root 

 tensile strength per unit area of soil and root area ratio has been 

 validated by Waldron (1977) for herbaceous plant roots (barley 

 and alfalfa) and Wu (1976) for woody plants (spruce and hem- 

 lock). 



In the case of natural root systems, the tensile strength tends 

 to vary with the size or diameter of the root (Wu 1 976; Burroughs 

 and Thomas 1977; and Gray 1978). Accordingly, the root ten- 

 sile strength term must be included inside the summation and 

 the average root tensile strength per unit area of soil com- 

 puted by the following relationship: 



CR-1.12tR (5a) 

 or Cr«1.12Tr-Ar (5b) 

 A 



or Cr-=1.12 I T| ni a, ^ (5c) 

 A 



This simple theoretical model or relationship permits an esti- 

 mate of rooting contribution to shear strength based solely on a 

 determination of the root area ratio or concentration of roots in a 

 soil cross section and on measurements of tensile strength or 

 resistance of the roots themselves. The validity of the model is 

 supported by results of direct shear tests run on root-permeated 

 soils in both field and laboratory (Endo and Tsuruta 1969; 

 Waldron 1977). 



The root reinforcement model can be used to obtain esti- 

 mates of root shear-strength increase or "root cohesion" (Cr) in 

 granitic soils of the Idaho batholith. Root density and tensile 

 strength data from the studies by Burroughs and Thomas 

 (1977) and fVlegahan and others (1978) were employed for this 

 purpose. Calculated "root cohesions" for Rocky Mountain 

 Douglas-fir at various elapsed times after cutting are summa- 

 rized in table 4. Root cohesions were calculated using equation 

 (5a) and root tensile strength per unit area of soil data for 

 Rocky Mountain Douglas-fir reported by Burroughs and 

 Thomas (1977). They included data for all roots in the 

 size interval to 0.4 inch (0 to 1 cm). As noted previously 

 inclusion of roots up to 0.4 inch (1 cm) in diameter corresponds 

 to a root area ratio of 0.045 percent. This ratio is a reasonable 

 lower bound estimate for root concentration across a potential 

 failure surface in the shallow, coarse-textured granitic soils of 

 the Idaho batholith. With increasing time after felling, both the 

 number of roots and the tensile strength of the remaining roots 

 decrease; this accounts for the decrease in root cohesion (table 

 4). 



The tabulated data indicate that shear strength increases 

 from root reinforcement up to 1 .5 Ib/in^ (10.3 kPa) are possible in 

 granitic soils for initial root area ratios as low as 0.045. Further- 



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