Page 38 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

Figure 2.2 Free-body diagram for
establishing the stress transformation
equations, principal stresses and their
orientations.

∗
[ ], defined by
⎡ ⎤
ll lm nl
∗
[ ] = ⎣ lm mm mn ⎦
nl mn nn
∗
The analytical requirement is to express the components of [ ] in terms of the
components of [ ] and the direction cosines of the l, m, n axes relative to the x, y, z
axes.
Figure 2.2 shows a tetrahedral free body, Oabc, generated from the elementary
cubic free body used to define the components of the stress matrix. The material
removed by the cut abc has been replaced by the equilibrating force, of magnitude
t per unit area, acting over abc. Suppose the outward normal OP to the surface abc
is defined by a row vector of direction cosines ( x , y , z ). If the area of abc is A,
the projections of abc on the planes whose normals are the x, y, z axes are given,
respectively, by

Area Oac = A x = A x
Area Oab = A y = A y
Area Obc = A z = A z

Suppose the traction vector t has components t x , t y , t z . Application of the
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