17.1.8 Hoek-Brown Rock Plasticity Model

The Hoek-Brown criterion defines the stress condition under which a rock mass will deform inelastically and, if not supported adequately, collapse. The criterion applies for isotropic behavior.

17.1.8.1 Parameters

The parameters defining the Hoek-Brown criterion are estimated from a combination of laboratory tests on intact rock cores and an `adjustment' procedure using an empirical chart that accounts for the quality of rock mass. This empirical chart defines the so-called geological strength index (GSI), based on visual inspections of surface condition and structure type of the rock mass. Figure 17.14 shows the GSI empirical chart, adapted from Hoek & Brown [39].

Figure 17.14: Estimate of GSI based on geological descriptions

The surface conditions in this chart are:

VERY GOOD:

very rough, fresh unweathered surfaces.

GOOD:

rough, slightly unweathered, iron stained surfaces.

FAIR:

smooth, moderately weathered or altered surfaces.

POOR:

slickensided, highly weathered surfaces with compact coatings or fillings of angular fragments.

VERY POOR:

slickensided, highly weathered surfaces with soft clay coatings or fillings.

The structure types of the rock mass are:

BLOCKY:

a very well interlocked undisturbed rock mass consisting of cubical blocks formed by three orthogonal discontinuity sets.

VERY BLOCKY:

interlocked, partially disturbed rock mass with multifaceted angular blocks formed by four or more discontinuity sets.

BLOCKY/DISTURBED:

folded and/or faulted with angular blocks formed by many intersecting discontinuity sets.

DISINTEGRATED:

poorly interlocked, heavily broken rock mass with a mixture of angular and rounded rock pieces.

From these two parameters you must pick the appropriate box in the chart of Figure 17.14. Then estimate the average value of the GSI from the contours. Do not attempt to be too precise, quoting a range of GSI from 36 to 42 is more realistic than stating that GSI=38.

17.1.8.2 Formulation

The Hoek-Brown failure criterion for intact rock samples can be formulated by means of a yield function f_i as follows:

$$\begin{split}f_{\mathrm{i}} &= S_{1} - S_{3} - \sqrt{ m_{\mat... ...{3} = \frac{ \sigma _{3} }{ \sigma _{\mathrm{ci}} } \end{split}$$ (17.156)

where $\sigma_{{1}}^{}$ is the most compressive principal stress, $\sigma_{{3}}^{}$ is the least compressive principal stress, $\sigma_{{\mathrm{ci}}}^{}$ is the unconfined compressive strength of th rock sample, and m_i is the so-called Hoek-Brown constant deduced from test results for a particular (intact) rock type. Both parameters $\sigma_{{\mathrm{ci}}}^{}$ and m_i can be determined by means of fitting the $\sigma_{{1}}^{}$ versus $\sigma_{{3}}^{}$ curve defined by (17.156) to the scattered ( $\sigma_{{1}}^{}$,$\sigma_{{3}}^{}$ ) data obtained from test results of a particular intact rock sample.

Figure 17.15: Hoek-Brown yield condition in S₁ -S₃ plane

Due to joints and defects in a rock mass the strength of the mass is reduced below the strength of intact specimen of the same rock type (see Hoek et al. [40]). The Hoek-Brown failure criterion for rock mass in situ, taking into account for rock mass quality and internal imperfections, can be presented by means of a modified yield function f as follows [Fig.17.15]:

$$\left(\vphantom{ m_{\mathrm{b}} S_{3} + s }\right. \left.\vphantom{ m_{\mathrm{b}} S_{3} + s }\right)^{{a}}_{}$$ (17.157)

$$\left(\vphantom{ \frac{ \mathrm{GSI} - 100 }{ 28 } }\right. {\frac{{ \mathrm{GSI} - 100 }}{{ 28 }}} \left.\vphantom{ \frac{ \mathrm{GSI} - 100 }{ 28 } }\right)$$ (17.158)

where

$$\begin{cases} \exp\left( \dfrac{ \mathrm{GSI} - 100 }{ 9 } \... ... [\medskipamount] 0 & \text{if $\mathrm{GSI} < 25$} \end{cases}$$ (17.159)

and

$$\begin{cases} 0.5 \qquad & \text{if $\mathrm{GSI} \ge 25$} \... ...athrm{GSI} }{ 200 } & \text{if $\mathrm{GSI} < 25$} \end{cases}$$ (17.160)

The plastic strain rate can be derived from the non-associative flow rule, in which the plastic potential is given as (Carranza-Torres & Fairhurst [14])

$${\frac{{ 1 + \sin \psi }}{{ 1 - \sin \psi }}}$$ (17.161)

where $\psi$ is the dilatancy angle. The principal plastic strain rate vector is obtained as

$$\dot{{\boldsymbol{\varepsilon}}}^{{\mathrm{p}}}_{} \left\{\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilo... ...\\ [\medskipamount] \dot{\varepsilon }_{3} \end{array} \negthickspace }\right. \begin{array}{c} \dot{\varepsilon }_{1} \\ [\medskipamount] \dot{\varepsilon }_{2} \\ [\medskipamount] \dot{\varepsilon }_{3} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{\varepsilon... ...\ [\medskipamount] \dot{\varepsilon }_{3} \end{array} \negthickspace }\right\} \dot{{\lambda}} \left\{\vphantom{ \negthickspace \begin{array}{c} { 1 }/{ \sigma ... ...ount] - { K_{p} }/{ \sigma _{\mathrm{ci}} } \end{array} \negthickspace }\right. \begin{array}{c} { 1 }/{ \sigma _{\mathrm{ci}} } \\ [\medskipamount] 0 \\ [\medskipamount] - { K_{p} }/{ \sigma _{\mathrm{ci}} } \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} { 1 }/{ \sigma _... ...unt] - { K_{p} }/{ \sigma _{\mathrm{ci}} } \end{array} \negthickspace }\right\}$$ (17.162)