Stresses within elastic circular cylinders loaded uniaxially and triaxially

Abstract

The elastic deformation of circular cylinders of rock is one of the basic problems in rock mechanics; it is fundamental to the understanding of standard laboratory tests on rock specimens and to design of pillars in mines. A closed-form solution has been derived for stresses within cylindrical elastic specimens subjected to end-boundary conditions encountered in laboratory testing. The boundary conditions are: Perfect confinement, in which radial expansion of the ends of the specimen is prevented; direct contact, in which the specimen is placed directly against the platens of the loading machine; uniform loading, in which steel inserts of a certain design are placed between the specimen and the platens and in which strains on the cylindrical surfaces of the specimen are uniform; teflon inserts, in which wafers of teflon, 0·005 in. thick, are placed between the specimen and the platens; and neoprene inserts, in which wafers of neoprene rubber, 0·063 in. thick, are placed between the specimen and the platens. The experimental and theoretical analyses of stresses and strains in rock and steel cylinders indicate that ‘uniaxially-loaded’ specimens in most laboratory tests are actually triaxially stressed. The stresses are different in different directions and they vary from point-to-point within specimens. For ‘frictional’ boundary conditions, where the specimens tend to expand more than the platens of the loading machine, including perfect confinement and direct contact, the absolute values of maximum and minimum stress concentrations and the stress gradients increase with increasing friction. Highly nonuniform stress distributions are developed under end conditions of perfect confinement. The magnitudes of the radial and circumferential stresses decrease as the expansion of the platens or the inserts approaches that of the specimens. For teflon and neoprene end-boundary conditions, involving platens or inserts that expand more than the specimen, the rate of change of stress increases as the expansional effect of the end-inserts increases. The more expansive the end-inserts, the more nonuniform are the stress distributions within specimens.