The resolution of stresses in rock using stereographic projection

Abstract

This paper shows how to find the shear and normal stress on an arbitrarily oriented plane of weakness in a known three dimensional stress field by manipulations on a stereographic projection. First are found graphically the direction angles of the normal to the plane relevant to co-ordinate axes in principal directions. Knowing the magnitude of the principal stresses and the direction cosines of the plane, the magnitude of the resultant stress across the plane is calculated. The direction of the resultant stress is found by calculating its direction angles and plotting these angles as small circles centered about the principal directions, the point of intersection of the small circles being the stereographic projection of the resultant stress. The angle between the resultant stress and the plane of weakness is read directly from the stereographic projection, facilitating calculation of the magnitudes of the shear and normal stresses of the plane. Finally, the direction of the maximum shear stress in the plane is found graphically, making use of the fact that the normal stress, shear stress, and resultant stress must be coplanar. The principles of stereographic projection are reviewed and an example of stress resolution is worked out.