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T Analytical Merhodr • proposed by Drucker (1952) is considered to be more representative of the behavior of rocks. This is expressed by the following equation: _ mil)' = k (2) in which material properties a and k may be determined empirically. O[her ideal plasticity criteria can be used. The yielding of rock will primarily occur in the immediate vicinity of the opening where stress concentrations are likely to exist. Brittle fracture-no tension It is assumed that rock is incapable of withstanding significant tensile stresses. This is due primarily to the presence of cracks and fissures which exist in rock in its natural state. Zienkiewicz, et al. (1968) have developed a method of stress analysis, which uses finite element tech- niques, to include this characteristic of rock. The method has been used with success to analyze stress distributions for a number of rock mechanics case studies. Slippage along bedding planes On the basis of observations in shafts on rock move- ments due to mining, Mohr (1951, 1958) has shown that the shearing stresses may overcome the frictional resis- tance between two beds or within the same bed. The rela- • 103 five horizontal displacemen[ of the rock beds resulting from this phenomenon has been found to vary from a few centimeters up to 50 centimeters. This type of relative displacements has been observed over a larger lateral ex- tent and in the same rock bed. Figure 2a shows rock movements above a mined area as observed (ram shafts. Figure 2b illustrates large relative horizontal displace- mentsoccurring within strata as observed from two neigh- boring shafts. This type of rock movement is considered likely to occur in laminated sedimentary rocks, e.g., shale, which have a small resistance to movements along bed- ding planes. Slippage along other geological discontinuities When the opening is situated in the area where joints or faults are preL•alent, the surface subsidence will be sub- stantially afl'ected by the exis[ence of these geological dis- continuities. Examples of this type of subsidence have been reported by Deere (1961) and Lee and Strauss (1969). The analysis for this type of failure can only be performed when the actual distribution of [hese geological discontinuities are ascertained. The shear strength along the joint surface has been studied in detail by Paton (1966) and Deere et al. (1966). They have shown that the shear strength of a rock surface is a function of cohesion c, angle of internal friction ~, and the rouehness of the joint surface. The roughness as shown in Figure 3 may arise from undulations in the joint surface itself or from PMTS ICAL O[sctlR loN er srsreN 4acE PROrnc INrE41AL no.eRTl es G[OLOCiC FACTas LOCATION OF OP[NING, SIZC AMO SHAPE Of OPENING I N P U T I LOAD VAR I•ELES I I ENVIRDMMENT VAR IAELES I I CONSTRUCTION vAR1AELE5 t $ i S i E N 17 r[ST ING A.E40ACN a) FULL SCALE TEST b) NDDrL resr C) IUCROA1NLTiICAI APVRDACN D) 14TME14TICAL DESCE VTION OF STSTCN D) ANALYTICAL NE }MOOS OBJECTIVES PERFORMANCE CRITERIA ~ DE[IS ION ON ACCFPTAEI LITT R E S P O N S E E.O., LIMITING SURFACE DEFLECTION. 1) STRESS MO DAMAGE TO LIMITING SURFACE COMARE VIiN )) 5T4AIN STRUC TURFS, MINIMUM DAMAGE MORlZONTAL SiRAl N, IERfORNANCE U(TER IA" )} OEfL[CTION ', TO ENVIRONMENT LIMITING AREAL -..,,-.~-..-„ i PE4FOPNANCE C41iER1A ARE EASED ON }XE DEFL[CTIONS AND 5}4A INS NNICN STRUCTURES, LOCATED AL THE SURFACE, CAN VITMSTANO SAF[LT. Figure T. A general approach to she subsidence problem.