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Answers
I calculate
a value of mi to be around 100 yet RocLab limits mi to be between
1 and 50. Why?
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| 1. |
Your values of mi of around 100 are almost certainly
associated with too small a range of confining stresses in your
triaxial testing. This is a problem that I come across very
frequently. The original definition of mi is based on triaxial
tests up to one half of the uniaxial compressive strength of
the intact material. The following quote is from Hoek and Brown,
1997, "Practical estimates of rock mass strength" published
in the Int. J. Rock. Mech. Min Sci.
"Note that
the range of minor principal stress (sig3) values over which
these tests are carried out is critical in determining reliable
values for the two constants. In deriving the original values
of sigci and mi, Hoek and Brown used a range of 0 < sig3< 0.5*sigci
and, in order to be consistent, it is essential that the same
range be used in any laboratory triaxial tests on intact rock
specimens."
For example, if you analyze the following data set for Carrara
marble using RocLab you obtain sigci = 82.28 and mi
= 8.68.
1.72
3.45
5.17
6.9
6.9
10.34
10.34
13.79
13.79
17.24
17.24
20.69
27.59
34.48
34.48
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78.61
89.33
99.81
123.79
125.23
125.65
138.37
137.38
146.6
150.77
160.76
173.79
187.9
205.99
213.58
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Note that the maximum value of sig3 is too low in this case
- it should be about 40 MPa but this is a real data set and
it is all that I have.
On the other hand, if I analyze only the first 4 data points,
up to sig3 = 6.9 MPa, I obtain sigci = 48.92 and mi = 32.13.
If this data set was for hard rock I could easily get mi values
of over 100 by limiting the range of sig3 values.
All the values quoted in the various Hoek-Brown papers are derived
from triaxial test data with the correct range of sig3 values
- this was one of the criteria that we set in determining whether
or not the data were acceptable. The typical range of mi values
if from about 5, for soft ductile rocks, to 35 for very hard
brittle rocks. Hence we set the range of 1 < mi < 50 in RockLab
to cover this range.
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For the
three sigma3, sigma1 laboratory strength data (0,16),(5,105.9),(20,161),
why does RocLab 1.007, RocLab 1.006 and RocData 2.37 using Linear
Regression and Simplex fitting give different values of mi and
sigci when fitting the Hoek-Brown envelope?
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| 2. |
RocLab uses the Levenberg-Marquardt
method, a robust algorithm that has become the standard for
non-linear regression. The method is very reliable in practice,
and has the ability to converge promptly from a wider range
of initial guesses than other typical methods.
RocLab Versions 1.000 to 1.005 used a linearized
form of the Hoek-Brown equation that had the effect of reducing
the Levenberg-Marquardt method to linear regression. This
was modified in Version 1.006 in order to take full advantage
of the power of the Levenberg-Marquardt method.
We looked closely at the varying results provided by different
regression algorithms, which you sent us. We judged the quality
of the results using the ‘Sum Square of Errors”
(SSE) – the sum of the square of the vertical distances
of the given data points from the fitted curve. (In RocLab
Version 1.006 and up this quantity is referred to as “Residuals”;
in some statistical literature it is known as the sum-of-squares.)
The analysis indicated the following:
i. The linear regression results (sigci=50.7 MPa and mi=17.8)
given by RocData Version 2, the published spreadsheet
of Hoek and Brown, and RocLab Versions 1.000 to 1.005,
gave an SSE value of 1490.98
ii. The simplex method in RocData that gave sigci=20.47
MPa and mi=55.46 had an SSE of 663.539
iii. RocLab Version 1.007 results (sigci=20.998 MPa
and mi=49.9) had an SSE of 702.858
These SSEs tell us that RocData and RocLab
1.007 found curves that better fit the data than the linear
regression approaches. It may seem that RocLab 1.007 gave
slightly worse results than the simplex method in RocData,
but this is not so.
RocLab 1.007 allows mi only to range between 1 and
50. When we remove the upper limit, the program gives sigci=20.487,
mi=55.44 and SSE=663.539, nearly identical to the results
of the simplex method.
RocLab 1.007 caps the maximum value of mi at 50 because,
according to Dr. Evert Hoek, mi values higher than this threshold
indicate too small a range of confining stresses in triaxial
testing. The following quote is from the paper “Practical
Estimates of Rock Mass Strength” by Hoek and Brown (International
Journal of Rock Mechanics and Mining Sciences, Vol. 34, No.
8, pp. 1165-1186, 1997):
Note that the range of minor principal stress ( ) values over
which these tests are carried out is critical in determining
reliable values for the two constants. In deriving the original
values of and , Hoek and Brown used a range of and, in order
to be consistent, it is essential that the same range be used
in any laboratory triaxial tests on intact rock specimens.
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