Page 60 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

Figure 2.13 Construction of a Mohr
circle diagram, appropriate to the
geomechanics convention of stresses.

axes, by known values of xx , yy , xy . A set of reference axes for the circle diagram
construction is defined by directions n and , with the sense of the positive axis
directed downwards. If O is the origin of the n − co-ordinate system, a set of
quantities related to the stress components is calculated from

1
OC = ( xx + yy )
2
1
CD = ( xx − yy )
2
DF =− xy
Points corresponding to C, D, F are plotted in the , plane as shown in Figure 2.13,
using some convenient scale. In the circle diagram construction, if xy is positive,

the point F plots above the n axis. Construction of the line FDF returns values
of = xy and n = xx which are the shear and normal stress components acting
on the surface cb of the element. Suppose the surface ed in Figure 2.13 is inclined
at an angle
to the negative direction of the y axis, or, alternatively, its outward
normal is inclined at an angle
to the x axis. In the circle diagram, the ray FG
is constructed at an angle
to FDF , and the normal GH constructed. The scaled

distances OH and HG then represent the normal and shear stress components on the
plane ed.
A number of useful results can be obtained or verified using the circle dia-
gram. For example, OS 1 and OS 2 represent the magnitudes of the major and minor
principal stresses 1 , 2 . From the geometry of the circle diagram, they are given
by

1,2 = OC ± CF
1
2
1

= ( xx + yy ) ± + ( xx − yy ) 2 1/2
2 xy 4
confirming the solution given in equation 2.24a. The ray FS 1 defines the orientation
of the major principal plane, so FS 2 , normal to FS 1 , represents the orientation of the
major principal axis. If this axis is inclined at an angle 1 , to the x axis, the geometry
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