Page 62 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

◦
function of xx , yy , xy and
. Determine values of t x , t y for
= 0 , 60 , 90 ,
◦
◦
◦
respectively. Determine the resultant stress on the plane for which
= 60 .
2 The unit free body shown in the figure (left) is subject to the stress components
shown acting parallel to the given reference axes, on the visible faces of the cube.
(a) Complete the free-body diagram by inserting the required stress components,
and specify the six stress components relative to the x, y, z axes.
(b) The l, m, n reference axes have direction cosines relative to the x, y, z axes defined
by

(l x ,l y ,l z ) = (0.281, 0.597, 0.751)
(m x , m y , m z ) = (0.844, 0.219, −0.490)
(n x , n y , n z ) = (−0.457, 0.771, −0.442)

Write down the expressions relating mm , nl to the x, y, z components of stress
and the direction cosines, and calculate their respective values.
(c) From the stress components established in (a) above, calculate the stress invari-
ants, I 1 , I 2 , I 3 , write down the characteristic equation for the stress matrix, and
determine the principal stresses and their respective direction angles relative to
the x, y, z axes.
Demonstrate that the principal stress directions define a mutually orthogonal
set of axes.

3 A medium is subject to biaxial loading in plane strain. Relative to a set of x, y,
co-ordinate axes, a load imposed at the co-ordinate origin induces stress components
defined by

1 8y 2 8y 4
xx = − +
r 2 r 4 r 6
1 4y 2 8y 4
yy = + −
r 2 r 4 r 6
2xy 8xy 3
xy = −
r 4 r 6
2
2
where r = x + y 2
Verify that the stress distribution described by these expressions satisfies the dif-
ferential equations of equilibrium. Note that
1
∂
x
=− etc.
∂x r r 3
4 A medium is subject to plane strain loading by a perturbation at the origin of the
x, y co-ordinate axes. The displacements induced by the loading are given by

1 xy
u x = + C 1
2G r 2
1 y 2
u y = − (3 − 4v) nr + C 2
2G r 2
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