No abstract

In open-cut mining the sides of quarries often deform when in weak contact with their bases. When the angle of inclination of the contact is less than the angle of friction, the caving of the rocks occurs over a complex slip surface beginning at the top of the surface of detachment of height Hgo. Below the zone of detachment the slip surface has a linear section with an angle of inclination of ~ = (45 ~ + ~/2), and then acquires a curvilinear form close to a circular cylinder, intersecting the contact at an angle 0 [i], Here [sin (p' 1/t --C.ctg (p--C' ctg (P'~-I 0=45o+r , arcsin sin~o ' ..... --~ , where ~ and ~' are the angles of internal friction in the rock and at the contact, in degrees, C and C' are the cohesions in the rock and at the contact in tons per square meter, In Fig. 1 the slip surface is shown by the line C'CMDA. The angle of inclination of the curved surface ~ changes from m at point_ M to (e + ~) at point D, where ~ is the angle of inclination of the contact.On section DA the slip surface coincides with the contact surface, At limiting equilibrium the whole prism of possible caving can be divided into three parts; the prism of settlement C'CMBB'C', the prism of active pres__sure B_~'BMDEKB', and the support prism KEDAK.The settlement prism has plane slip areas CM and BM with angles of inclination m. The boundary of the active pressure prism and the support prism is a curved surface ED coinciding with one of the lines of the second family of slip surfaces, This surface, starting from a depth Hgo from the slope, forms with surface MD an angle (90 ~ --~) and is approximated as a circular cylinder. Its angle of inclination at the top is m, and at the bottom (2m-~ + 8).The settlement prism of weight P:, the active pressure prism of weight P2, and the support prism of weight Pa are balanced by the reactions from the unmoved rock RI', R3' and from the contact, Rs', by the__interbloc__k reactions R2' and R~', and b~ the forces of cohesion over the plane slip__aareas C.MC and C-MB, over the curved sections C~ and C-DE, and over the contact surface C'.AD.By successively considering the conditions of limiting equilibrium of the prisms with projection of all the forces onto the axes of a rectangular coordinate system XAY and compilation of a general equation, we can find the minimum height of the stable slope H and the corresponding width of the prism of possible caving r by means of the equations It = H " It9o;(2) r=r"H (3) 90~ where H" = (be--ad) +V(be--a.d)2--(d~--e.f).(a2--e.m). (a ~ -~" I) ]+ a = k. ~ + 2'sin (8 --e)ieos (8 --8 + ~') cos ~ (8 --~ --~) ~-(~ --8).cos 2 ~ [ sin e.sin (~ --8) sin (~ --8 +_. ~ --q~) ] C" .cos q7-sin ~-eos s -1-"tSt)~.sin (~--~).sin (e --~) [.sin 8 --cos (8 --~ --q~)-eos (8 --6 -+-qr J --(;.sin (e --e?).eo8 (g --5 + ~') ;