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''.1•' rt...sr\\\~pull.Inulusr <br />._ . \ y~ <br />l; Il. N <br />Tht shear font dF, is therefore composed of a teen c dl, independent of <br />the normal stress anti a teen dF, tan ¢, which is proportional to the normal <br />strc». To simplify calculations, wr combine the part dF„ tan ~ with the <br />normal force as shown in Fig. 8.21(6). This' force is designated :u dP' and <br />makes an angle 4 with the normal. The resultant of c dl and dF over the <br />enure failure surface is represented by F, and F, rtspectively. <br />Tht resultant F~ of the cohesion distributed uniformly along the arc ad <br />is equal to the unit cohesion tinter the chord distance nor, or <br />F~ = c(nr/) <br />This can hr undtntood from the forces shown in Fig. 8.21(x/). The cohesion <br />along ~u! is the summation of infinitesimal vectors whose directions follow <br />the arc ud. Thus their resultant vector F~ has a magnitude and direction equal <br />to that of the chord. The line of action of F~ must be such that its moment <br />about any point (for convenience, point 0) is the same as that of the cohesion <br />distributed along the failure surface. <br />Thr moment of the uniformly distributed cohtsion_along section be of <br />the arc is c dl R. Far the rtnire length of the arc, it is c ndR. The moment of <br />the resultant F~ ii simply F,1~. Hence, equating the two we have <br />' F,l~ =e(nd) R <br />or / - c art) R _ e(nd) R - (nd) R <br />(8.35) <br />F c(ad) (ad) <br />ti El:. g.9 ~Lll l'N .\y ALI'SIS ~'1J <br />Thus we see that 1, is larger than R since the arc distance art is always largtr <br />than the chord distance ad. <br />The resultant F of F„ and F, along arc nrl must pass through point p, <br />the intersection of the vectors F and 1N. Furthermore, as noted in Fig. <br />- 6.21(6), dF makes an angle ¢ with the normal to the failure surface. Thus the <br />resultant F must make an angle approximately equal [o ~ with the normal. <br />We may take another approximation and say F should be tangent to a circle <br />with radius R sin ~ [Fig. 8.21(n)]. In [Iris way, the direction and line of action <br />of the force F is determined. At failure, the three forces IV, F~, and Fare <br />in equilibrium, as illustrated by the force polygon in Fig. 8.2I(c). <br />Failure occurs if wr imagine that the weight IV is progressively increased <br />by increasing the density y until the equilibrium conditions shown in Fig. <br />8.21(c) are fulfilled. A more familiar example would be [Itat of applying a <br />load at a point, say c, and increasing this load until failure occurs. Since the <br />slip surface is arbitrarily chosen, many slip circles must br tried. The one <br />which satisfies the failure conditions at the smallest value of density y is the <br />one that will fail first. This is called the <•riricul circle. <br />Stnbilirv number. We note that the equilibrium conditions at failure <br />require that F, F„and )V have the relationship shown in Fig. 8.21{c). Actually <br />five parameters art involved when we consider the three (Drees F, F„ and Iv. <br />They are the shear strength of the material as represented by r and ¢, the <br />unit weight of the material y, and the dinttnsions of the slope r? and fl (Fig. <br />8'3). If four of the five parameters are given, the filth onr can be calculated. <br />0 <br />r--___~- <br />r <br />t d <br />r <br />t 11 <br />t <br />t <br />r <br />Snit Properties Unit weight: y <br />Strength: e = c -•• a tan e <br />Ft~. 8.'_3. Slope parameters included in <br />stability number. <br />Since the five parameters account for all the physical properties of a slope <br />ott a homogeneous ntattrial, their rel:ttionsltip can be calculated once and <br />for all. Taylor (1937) published tltz results of these calculations in the form <br />of charts. To simplify the presentation, three o(tlte parameters, c, y, and H, <br />are combined into a new parameter N„ called the srnhiliry mnnber, defined as <br />N, = 7H (8.39) <br />c' <br />hip. ~._. Photograph of a Blida in the valley of the Gros Ventre River, <br />15'yoming, shoving the topographic change after the soil mass clipped along a <br />curvad failure surface. <br />