I suspect that what is required here is probably the mean of the two
principal stresses induced in the face (the third principal stress, normal
to the face, is zero). On this basis, all (!) we need is a solution for the
stresses induced in the planar end of a cylindrical cavity in a
semi-infinite half-space...
I don't know of an analytical solution to this, and so for an accurate
result it must be a case of finding a simple, 3d boundary- or finite-element
program, learning how to use it, and making sure you have appropriate values
for Young's modulus, Poisson's ratio, and the in situ stress field.
Failing that, we could take a stab at using the equations for stresses
around a spherical cavity in an infinite material (see Timoshenko & Goodier,
Theory of Elasticity, or Poulos & Davis, Elastic Solutions for Soil and Rock
Mechanics) in order to generate an estimate. Poisson's ratio and the in situ
stresses are still needed, though.
What does the spherical cavity estimate give us? The maximum circumferential
stress for a uniaxial stress field of magnitude p is
s = p*1.5*(9-5n)/(7-5n) where n=Poisson's ratio.
At 200m depth, the vertical stress will be about 5MPa. With n=0.3, the
induced vertical stress will be about 10MPa. If we guess (and guess is the
operative word) that the horizontal stress is twice the vertical stress,
then the induced horizontal stress will be about 20MPa. The mean of these is
15MPa. Of course, if the horizontal stress is much different from this, and
if the concept of 'mean stress' in this context is nonsense, and if we're
not happy with a spherical cavity model for a cylindrical tunnel, then who
knows...
So, having said all of this, I'm not sure I'd like to guarantee any method
to within probably 50% of the actual in situ value. If the results are
critically dependent on it, then I don't think I'd like to be responsible
for either (a) designing a machine or (b) estimating the job!
Aren't classification schemes that depend critically on undeterminable
factors wonderful?
John H.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Dr John P. Harrison
T.H. Huxley School of Environment, Earth Science and Engineering,
Imperial College, London SW7 2BP, UK
Tel: +44 (0)20 7594 7348 Fax: +44 (0)20 7594 7444
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