Three-Dimensional Mohrs Circles

The stress state in Fig. 1.23 is associated with the principal stresses located at the position where the shear stresses vanish. The stress tensor for such a condition may be written as  

As in the two-dimensional case, the direct stresses are on the horizontal axis and the shear stresses are on the vertical axis. For the construction of the Mohr s circle, three circles are required. The stresses on any plane at any rotation, when plotted in the three-dimensional Mohr s circle diagram, are represented by a point located either on one of the three circles or within the area between the largest and the two smaller circles. The maximum shear stress is given by the radius of the largest circle. When constructing the Mohr s circle, the angle of rotation is double that of the real stress system. Shear stresses are positive if they cause clockwise rotation, 

The center of each Mohr s circle which lies on the a axis is given by  

Note that the center, C2, may be evaluated from the values of a and ay, which is equivalent to the center obtained from the principal stresses, ffj and ffn. These principal stresses are obtained at a certain orientation of the stress system of an element in space, in which all the shear-stress components equal zero. In this special orientation, the normals of the faces correspond to the principal directions and the normal stresses associated with these faces are the principal stresses. 

In Fig. 1.24, the solution of a sample problem solved is also indicated by the data below. The following measurements were performed on a body under stress, 

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