Mohr's Circle For Two-dimensional Stress States
A two-dimensional Mohr's circle can be constructed if we know the normal stresses, and the shear stress . The following sign conventions are usually used:
- Tensile stresses (positive) are to the right.
- Compressive stresses (negative) are to the left.
- Clockwise shear stresses are plotted upward.
- Counterclockwise shear stresses are plotted downward.
The reason for the above sign convention is that, in engineering mechanics, the normal stresses are positive if they are outward to the plane of action (tension), and shear stresses are positive if they rotate clockwise about the point in consideration. In geomechanics, i.e. soil mechanics and rock mechanics, however, normal stresses are considered positive when they are inward to the plane of action (compression), and shear stresses are positive if they rotate counterclockwise about the point in consideration.
To construct the Mohr circle of stress for a state of plane stress, or plane strain, first we plot two points in the space corresponding to the known stress components on both perpendicular planes, i.e. and (Figure 1 and 2). We then connect points and by a straight line and find the midpoint which corresponds to the intersection of this line with the axis. Finally, we draw a circle with diameter and centre at .
The radius of the circle is, and the coordinates of its centre are .
The principal stresses are then the abscissa of the points of intersection of the circle with the axis (note that the shear stresses are zero for the principal stresses).
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