Rock Mechanics
Maximiliano Garnier-Villarreal
2026-05-22
Source:vignettes/rock_mechanics_vig.Rmd
rock_mechanics_vig.RmdThis document highlights some functions of the GMisc
package related to rock mechanics problems.
Uniaxial Compressive Strength (UCS)
The UCS is one of the parameters needed to classify the rock mass using Hoek-Brown. There are different ways to obtain this parameter but one simple and cost effective solution is the Point Load Test (PLT). In this test several irregular samples are measured (width and diameter) and their strength is tested using the PLT machine. Afterwards, these data can be analyzed to obtain the I_{50} and the respective UCS.
The function UCS of the package performs the
conventional analysis through a classical (Gaussian) regression of the
failure load with respect to the equivalent diameter (calculated from
the width and diameter of the irregular samples), and predicting the
load and I_{50} for the classical 50 mm
cylinder. The function goes beyond and performs a robust regression to
account for outliers, as well as estimating an I_{50} for each sample and estimating the
mean and median of this value (corrected mean and median).
The arguments for UCS are W for the widths
in millimeters, D for the diameter in millimeters, and
P for the failure load. Also a vertical dashed line is
shown at the 50 mm diameter, as this is the reference diameter.
W = c(70.0, 95.0, 80.0, 90.0, 98.0, 110.0, 100.0, 100.0, 110.0)
D = c(40, 55, 60, 50, 25, 60, 40, 70, 50)
P = c(4.3, 3.4, 4.5, 3.1, 5.0, 6.4, 2.6, 11, 4.1)
UCS(W = W, D = D, P = P)
#> $Graph
#>
#> $Summary
#> Method I50 [MPa] UCS [MPa]
#> 1 Gaussian 0.69 15.83
#> 2 Robust 0.82 18.94
#> 3 Corrected (mean) 1.02 23.41
#> 4 Corrected (median) 0.90 20.71This result could be compared against different methods and a more informative can be made regarding the election of the UCS for the rock mass.
Hoek-Brown
This criterion is used to estimate the mechanical behavior of rock masses (taking in consideration both intact rock and discontinuities) along with strength and deformation parameters. The results are useful to be used in computer models.
The general for of the criterion is:
\sigma_1 = \sigma_3 + \sigma_{ci} \left( m_b \frac{\sigma_3}{\sigma_{ci}} +s \right)^a
where \sigma_{ci} is the compressive strength of the intact rock, m_b, a, and s are material constants.
The discontinuities, geological structures, and general state of the rock mass is assessed and considered through the Geological Strength Index (GSI). Two cases are available, one for a general scenario and one for a flysch scenario.
General GSI
Flysch GSI
To use this criterion there are two functions Hoek_Brown
and Hoek_Brown_plot. The first one calculates the criterion
and the second one plots the results from the first one. The arguments
are: sig.ci the uniaxial compressive strength of the intact
rock in MPa, GSI, mi the intact rock constant,
MR the modulus ratio, D the quality of the
explosive campaign design, unit.weight in kN/m^3, height, in meters, to
which calculate the stress level, and use the expected use
for the rock mass (“general”, “tunnels”, or “slopes”).
Some of these inputs can be obtained from the following tables, where in most cases are rock type dependent.
sig.ci = 16
GSI = 75
mi = 13
MR = 300
D = 0
height = 40
unit.weight = 18.6
HB = Hoek_Brown(sig.ci, GSI, mi, MR, D, height, unit.weight)
HB %>%
purrr::pluck('Results') %>%
as.data.frame()
#> sig.ci GSI mi D mb s a MR Ei sig.3.max sig.t sig.c sig.cm
#> 1 16 75 13 0 5.32 0.0622 0.501 300 4800 4 -0.187 3.98 5.87
#> Erm phi c phi.h c.h
#> 1 3918.48 40.1 1.45 52.5 0.72The results display the input parameters as well as the estimated
strength and deformation parameters (in MPa). The phi.h and
c.h are the strength parameters for the stress level.
Hoek_Brown_plot(HB)
#> $princ.stresses
#>
#> $normal.shear
The plotting function shows two graphs, one in terms of the major and minor principal stresses, and one in terms of the normal and shear stresses. The second one shows the linear fit for the desired stress level, along with the parameters of cohesion (c) and friction angle (\phi) for the whole rock mass and for the desired stress level.
Q
The function Q_93 is very simple in the way that it
plots the Q parameter for the given
equivalent diameter on top of the Q-diagram to determine the quality of
the rock mass and the needed support.
The arguments are De for the equivalent diameter, and
Q. The amount of points have to be same in both vectors, or
one can be a unique value and it will be assigned to all the values of
the other.
In this example different values of Q are plotted for a
single value of De, giving different classifications for
some of the points.
Barton-Choubey
This criterion estimates the strength parameters for discontinuities (fractures, bedding planes, etc.) and has the following form:
\tau = \sigma'_n tan \left[ JRC \ log_{10} \left(\frac{JCS}{\sigma'_n} \right) + \phi_r \right] where \tau and \sigma are the shear and normal stress on the discontinuity plane, JRC is the joint roughness coefficient, JCS is the joint compressive strength, and \phi_r is the residual friction angle of the rock mass.
The JRC can be estimated using:
while the JCS (in MPa) can be calculated using:
JCS = 10^{0.00088 \ \gamma_{rock} \ r \ + 1.01}
where r is the Schmidt hammer’s rebound on the wall of the joint and \gamma_{rock} is the unit weight of the rock in kN/m^3
To use this criterion there are two functions
Barton_Choubey and Barton_Choubey_plot. The
first one calculates the criterion and the second one plots the results
from the first one. The arguments are: JRC,
JCS in MPa, phi.r in degrees,
unit.weight in kN/m^3, and
the depth, in meters, to which calculate the stress
level.
JRC = 10
JCS = 30
phi.r = 26
unit.weight = 18.6
depth = 40
BC = Barton_Choubey(JRC, JCS, phi.r, unit.weight, depth)
BC$parameters
#> # A tibble: 1 × 2
#> c phi
#> <dbl> <dbl>
#> 1 0.102 37.4
Barton_Choubey_plot(BC, units = 'MPa')
The results display the strength parameters, and the plot shows the general criterion (red) and a fit for the desired stress level (blue), and also the parameters of cohesion (c) and friction angle (\phi).
Shinyapp
A shinyapp using these functions can be accessed at https://maximiliano-01.shinyapps.io/rock_mechanics/.