COMPARISON OF INTACT ROCK FAILURE CRITERIA USING VARIOUS STATISTICAL METHODS Sarat Kumar Das* (Corresponding author) Assistant Professor, Civil Engineering, National Institute of Technology, Rourkela, Orissa, India, Email: [email protected]; [email protected]Phone: 91-9437390601. Prabir Kumar Basudhar Professor, Indian Institute of Technology Kanpur, Kanpur, India, Email: [email protected]ABSTRACT This paper describes comparison of four different rock failure criteria based on triaxial test data of ten different rock strength data using various statistical methods. Least square, least median square and re-weighted least square techniques are used to find out the best fit parameters utilizing the experimental data that describes the failure state for each criterion. The least median square method could identify the scattered data and these scattered data points are observed at higher confining stress. It was observed that the fitting of failure criteria to different rock strength data depends upon the statistical methods used. The prediction of unconfined compressive strength and failure strength for different rock, estimated using various statistical methods are discussed in terms of different statistical performances of the prediction. Key words: Rock failure criteria; parameter estimation; least square method; least median square method; re-weighted least square method. Symbols: 1 Major principal stresses 3 Minor principal stresses C 0 Uniaxial compressive strength of intact rock for Hoek-Brown failure criteria m, s The material parameters for Hoek-Brown failure criteria
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COMPARISON OF INTACT ROCK FAILURE CRITERIA USING
VARIOUS STATISTICAL METHODS Sarat Kumar Das* (Corresponding author)
Assistant Professor, Civil Engineering, National Institute of Technology, Rourkela,
Published in Acta Geotechnica, Vol 4, No 3, 2009, 223-231 DOI: http://dx.doi.org/10.1007/s11440-009-0088-1 Archived with Dspace@nitr, http://dspace.nitrkl.ac.in/dspace
2
^
Standard deviation for Least Median Square method
ri Residuals from Least Median Square fit
C0, b and Material parameters for Yudhbir et al. failure criterion
C0, t and Material parameters for Sheorey failure criterion
1predicted Predicted major principal stresses
1Experiment Experimental major principal stresses
Mean value
Standard deviation
1. INTRODUCTION
For constructing engineering structures through rocks, it is essential to know the
stress-strain-time behavior and the strength characteristics of the concerned rock
mass. However, for stability problems like deep excavations and excavation by
tunneling, the primary goal is to avoid failure state, defining the ultimate stress
condition influenced by the induced stress and strength of intact rock which can be
predicted by using some failure criteria. The strength criteria used in rock mechanics
are either empirical or based on mechanics (Hoek and Brown 1980). There are nearly
twenty rock failure criteria (Yu 2002) to predict the strength of rocks, though each
one is suitable for a particular type of rock depending upon its physical and
mineralogical characteristics.
All such predictive models involve some material/ empirical parameters which
need to be determined from the available experimental data. The predictive capability
of the strength theories are then compared with the laboratory test data based on the
derived parameters. Sometimes estimation of such parameters from actual
measurement is difficult and can be derived by trial and error procedure (Carter et al.
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1991). But as the failure envelopes are drawn using the above parameters, it should be
predicted accurately. For nonlinear strength models, more systematic approach is
used by minimizing an error (objective) function constructed with the purpose to find
the error in estimation from the observed values.
The least square (LS) (Shah and Hoek 1992; Li et al. 2000) and weighted least
square (Desai 2001) techniques are widely used to frame the objective function and
different mathematical optimization methods available in literature are used to solve
the objective function. Colmenares and Zoback (2002) used the grid search method to
find out the parameters using minimization of the mean standard deviation misfit to
the test data. They discussed the applicability of different rock failure criteria for
different rock strength data and observed that applicability of strength theories
depends upon the type of laboratory rock strength data. Abdullah and Dhawan (2004)
have also drawn similar conclusions and emphasized the need for re-examination of
semi-empirical correlations.
But in general, scatter-ness of experimental data is almost unavoidable due to
material nonhomogeneities, equipment/procedural error, and random testing effects.
The least square error method has problems with such scattered data (known as
outliers in statistics). A single outlier, if severe, can significantly affect the results. In
case of geomaterials, the parameters are very often found from limited data due to the
cost involved and the data are scattered. Rousseeuw (1998) suggested the use of the
least median square (LMS) method, which is more robust and stable against
perturbations due to outliers. The method also helps in identifying the outliers, which
may be either corrected or deleted and the revised LS method is known as re-weighted
least square (RLS) method (Rousseeuw 1998).
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With the above in view, studies have been made to find out the model parameters of
different failure criteria applicable to different rock strength data based on LS, LMS
and RLS methods. The accuracy of different failure criteria are compared in terms of
errors associated with the prediction of unconfined compressive strength value. The
failure envelopes as obtained from above parameters and comparison of predicted
data for different failure theories using different statistical methods is discussed. The
strength, material parameters comparison and suitability of failure criteria to a
particular rock is as per the laboratory test data and should not be extrapolated for
field rock masses. The strength of rock mass may be reduced due to structural
discontinuity, in which case above failure criteria may not hold good (Pariseau 2007).
2.0 ANALYSIS
2.1 Adopted failure criteria
The failure criteria due to Hoek and Brown (1980), Yudhbir et al. (1983),
Ramamurthy et al. (1985) and Shereoy (1997) has been considered for the present
study and are presented in Equ 1, 2, 3 and 4 respectively as follows.
0)()( 200331 CsCm (1)
where 1, 3 are major and minor principal stresses respectively, C0 the uniaxial
compressive strength of intact rock, m and s are the material parameters. The value of
s is unity for intact rock.
0
3
0
1
Cba
C (2)
For intact rock a = 1.0, the parameters need to be determined are C0, b and .
b
tt
CB
3
0
3
31 (3a)
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Ramamurthy (1994) also observed that for most of the rock the value of b = 2/3 and 3/1
03.1
t
CB
.
So equation (3a) can be rewritten as 3/1
3031 13.1
t
C
(3b)
The parameters to be determined are C0 and uniaxial tensile strength (t).
t
C 301 1 (4)
The parameters need to be determined are C0, t and .
All the above are triaxial nonlinear failure criteria ignoring the effect of intermediate
stress and the advantage of nonlinear failure criteria over linear failure criteria is well
recognized (Mostyn and Douglas, 2000; Pariseau 2007)
2.2 Adopted statistical methods and the error function
The error function for the above failure criteria, while using LS method, with n
number of experimental data can be written as (Shah and Hoek 1992; Li et al. 2000):
nj
jpredictedalExperimentfERR
1
2
11 )()( (5)
When LMS method is used the objective is to minimize the median of square of error
instead of the conventional sum of square of errors as in LS method. Thus the
objective function can be written as (Rousseeuw 1998)