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Critical State Soil Mechanics Phần 10

Tham khảo tài liệu 'critical state soil mechanics phần 10', khoa học tự nhiên, công nghệ môi trường phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 197 formulae will have to be developed to meet special difficulties at some boundaries but for all such developments direct reference should be made to Sokolovski s texts. 9.11 Sokolovski s Shapes for Limiting Slope of a Cohesive Soil We now consider Sokolovski s use of integrals of the equations of limiting equilibrium in regions throughout which either Ẹ or n has constant values. We consider only regions where Ẹ is constant since those with n constant are the exact converse and the families of characteristics become interchanged. In the regions where Ẹ is constant everywhere and n varies then 5 _ Y Ộ Ẹ o constant 9.46 2k and the one constant ặ0 applies to every -characteristic. The family of -characteristics for which dy dx tan - n 4 must become a set of straight lines since n -Y.- Í-2 f0 - 2 9.47 so that dn -2dộ and ệwill be constant along each a-characteristic. The equation of any -characteristic becomes y l tanU n 9.48 x - x 0 4 J where x ộ y ộ are the coordinates of some fixed point on the characteristic. For the special case when all a-characteristics pass through one point we can choose it as the origin of coordinates so that x ệ y ộ 0 and the family of a-characteristics is simply a fan of radial straight lines. The curved -characteristics having I n dy dxtanl Ộ 4 are then segments of concentric circles orthogonal to these radii. Solutions to specific problems can be constructed from patchwork patterns of such regions as we see for the general case of the limiting stability of a slope of a cohesive soil shown in Fig. 9.34 and as we saw for the case of high-speed fluid flow in Fig. 1.8 . The essential feature of these solutions is that the values of one parameter Ẹ in Fig. 9.34 which are imposed at one boundary OA0 remain unchanged and are propagated through the pattern and have known values at a boundary of interest 0A3 . WAW7 ------- Soil not in a limiting state Fig. 9.33 Limiting Shape of Slope in Cohesive Soil 2k 7 nk 7 A limiting shape for a slope in .