Practical_Rock_Engineering - Part III (2007) by E.Hoek

In situ and induced stresses

Introduction

Rock at depth is subjected to stresses resulting from the weight of the overlying strata

and from locked in stresses of tectonic origin. When an opening is excavated in this

rock, the stress field is locally disrupted and a new set of stresses are induced in the

rock surrounding the opening. Knowledge of the magnitudes and directions of these

in situ and induced stresses is an essential component of underground excavation

design since, in many cases, the strength of the rock is exceeded and the resulting

instability can have serious consequences on the behaviour of the excavations.

This chapter deals with the question of in situ stresses and also with the stress

changes that are induced when tunnels or caverns are excavated in stressed rock.

Problems, associated with failure of the rock around underground openings and with

the design of support for these openings, will be dealt with in later chapters.

The presentation, which follows, is intended to cover only those topics which are

essential for the reader to know about when dealing with the analysis of stress

induced instability and the design of support to stabilise the rock under these

conditions.

In situ stresses

Consider an element of rock at a depth of 1,000 m below the surface. The weight of

the vertical column of rock resting on this element is the product of the depth and the

unit weight of the overlying rock mass (typically about 2.7 tonnes/m3 or 0.027

MN/m3). Hence the vertical stress on the element is 2,700 tonnes/m2 or 27 MPa. This

stress is estimated from the simple relationship:

σv=γz (1)

where σv is the vertical stress

γ is the unit weight of the overlying rock and

z is the depth below surface.

Measurements of vertical stress at various mining and civil engineering sites around

the world confirm that this relationship is valid although, as illustrated in Figure 1,

there is a significant amount of scatter in the measurements.

In situ and induced stresses

Figure 1: Vertical stress measurements from mining and civil engineering projects

around the world. (After Brown and Hoek 1978).

The horizontal stresses acting on an element of rock at a depth z below the surface are

much more difficult to estimate than the vertical stresses. Normally, the ratio of the

average horizontal stress to the vertical stress is denoted by the letter k such that:

σh=kσv=kγz (2)

Terzaghi and Richart (1952) suggested that, for a gravitationally loaded rock mass in

which no lateral strain was permitted during formation of the overlying strata, the

value of

k is independent of depth and is given by

k=ν(1−ν), where

ν is the

Poisson's ratio of the rock mass. This relationship was widely used in the early days

of rock mechanics but, as discussed below, it proved to be inaccurate and is seldom

used today.

Measurements of horizontal stresses at civil and mining sites around the world show

that the ratio

k tends to be high at shallow depth and that it decreases at depth (Brown

and Hoek, 1978, Herget, 1988). In order to understand the reason for these horizontal

stress variations it is necessary to consider the problem on a much larger scale than

that of a single site.

2

In situ and induced stresses

Sheorey (1994) developed an elasto-static thermal stress model of the earth. This

model considers curvature of the crust and variation of elastic constants, density and

thermal expansion coefficients through the crust and mantle. A detailed discussion on

Sheorey’s model is beyond the scope of this chapter, but he did provide a simplified

equation which can be used for estimating the horizontal to vertical stress ratio

k. This

equation is:

1·§

k=0.25+7Eh¨0.001+¸ (3)

©z¹

where z (m) is the depth below surface and Eh (GPa) is the average deformation

modulus of the upper part of the earth’s crust measured in a horizontal direction. This

direction of measurement is important particularly in layered sedimentary rocks, in

which the deformation modulus may be significantly different in different directions.

A plot of this equation is given in Figure 2 for a range of deformation moduli. The

curves relating k with depth below surface z are similar to those published by Brown

and Hoek (1978), Herget (1988) and others for measured in situ stresses. Hence

equation 3 is considered to provide a reasonable basis for estimating the value of k.

Figure 2: Ratio of horizontal to vertical stress for different deformation moduli based

upon Sheorey’s equation. (After Sheorey 1994).

3

In situ and induced stresses

As pointed out by Sheorey, his work does not explain the occurrence of measured

vertical stresses that are higher than the calculated overburden pressure, the presence

of very high horizontal stresses at some locations or why the two horizontal stresses

are seldom equal. These differences are probably due to local topographic and

geological features that cannot be taken into account in a large scale model such as

that proposed by Sheorey.

Where sensitivity studies have shown that the in situ stresses are likely to have a

significant influence on the behaviour of underground openings, it is recommended

that the in situ stresses should be measured. Suggestions for setting up a stress

measuring programme are discussed later in this chapter.

The World stress map

The World Stress Map project, completed in July 1992, involved over 30 scientists

from 18 countries and was carried out under the auspices of the International

Lithosphere Project (Zoback, 1992). The aim of the project was to compile a global

database of contemporary tectonic stress data.

The World Stress Map (WSM) is now maintained and it has been extended by the

Geophysical Institute of Karlsruhe University as a research project of the Heidelberg

Academy of Sciences and Humanities. The 2005 version of the map contains

approximately 16,000 data sets and various versions of the map for the World,

Europe, America, Africa, Asia and Australia can be downloaded from the Internet.

The WSM is an open-access database that can be accessed at (Reinecker et al, 2005)

The 2005 World Stress Map is reproduced in Figure 3 while a stress map for the

Mediterranean is reproduced in Figure 4.

The stress maps display the orientations of the maximum horizontal compressive

stress. The length of the stress symbols represents the data quality, with A being the

best quality. Quality A data are assumed to record the orientation of the maximum

horizontal compressive stress to within 10°-15°, quality B data to within 15°-20°, and

quality C data to within 25°. Quality D data are considered to give questionable

tectonic stress orientations.

The 1992 version of the World Stress Map was derived mainly from geological

observations on earthquake focal mechanisms, volcanic alignments and fault slip

interpretations. Less than 5% of the data was based upon hydraulic fracturing or

overcoring measurements of the type commonly used in mining and civil engineering

projects. In contrast, the 2005 version of the map includes a significantly greater

number of observations from borehole break-outs, hydraulic fracturing, overcoring

and borehole slotting. It is therefore worth considering the relative accuracy of these

measurements as compared with the geological observations upon which the original

map was based.

4

In situ and induced stresses

Figure 3: World stress map giving orientations of the maximum horizontal

compressive stress. From .

5

In situ and induced stresses

Figure 4: Stress map of the Mediterranean giving orientations of the maximum

horizontal compressive stress. From .

6

In situ and induced stresses

n discussing hydraulic fracturing and overcoring stress measurements, Zoback

(1992) has the following comments:

‘Detailed hydraulic fracturing testing in a number of boreholes beginning very

close to surface (10-20 m depth) has revealed marked changes in stress

orientations and relative magnitudes with depth in the upper few hundred

metres, possibly related to effects of nearby topography or a high degree of

near surface fracturing.

Included in the category of ‘overcoring’ stress measurements are a variety of

stress or strain relief measurement techniques. These techniques involve a

three-dimensional measurement of the strain relief in a body of rock when

isolated from the surrounding rock volume; the three-dimensional stress

tensor can subsequently be calculated with a knowledge of the complete

compliance tensor of the rock. There are two primary drawbacks with this

technique which restricts its usefulness as a tectonic stress indicator:

measurements must be made near a free surface, and strain relief is

determined over very small areas (a few square millimetres to square

centimetres). Furthermore, near surface measurements (by far the most

common) have been shown to be subject to effects of local topography, rock

anisotropy, and natural fracturing (Engelder and Sbar, 1984). In addition,

many of these measurements have been made for specific engineering

applications (e.g. dam site evaluation, mining work), places where

topography, fracturing or nearby excavations could strongly perturb the

regional stress field.’

Obviously, from a global or even a regional scale, the type of engineering stress

measurements carried out in a mine or on a civil engineering site are not regarded as

very reliable. Conversely, the World Stress Map versions presented in Figures 3 and 4

can only be used to give first order estimates of the stress directions which are likely

to be encountered on a specific site. Since both stress directions and stress magnitudes

are critically important in the design of underground excavations, it follows that a

stress measuring programme may be required in any major underground mining or

civil engineering project.

Developing a stress measuring programme

Consider the example of a tunnel to be driven a depth of 1,000 m below surface in a

hard rock environment. The depth of the tunnel is such that it is probable that in situ

and induced stresses will be an important consideration in the design of the

excavation. Typical steps that could be followed in the analysis of this problem are:

The World Stress Map for the area under consideration will give a good first

indication of the possible complexity of the regional stress field and possible

directions for the maximum horizontal compressive stress.

7

In situ and induced stresses

1.

During preliminary design, the information presented in equations 1 and 3 can

be used to obtain a first rough estimate of the vertical and average horizontal

stress in the vicinity of the tunnel. For a depth of 1,000 m, these equations

give the vertical stress

σv = 27 MPa, the ratio k = 1.3 (for Eh = 75 GPa) and

hence the average horizontal stress

σh= 35.1 MPa. A preliminary analysis of

the stresses induced around the proposed tunnel shows that these induced

stresses are likely to exceed the strength of the rock and that the question of

stress measurement must be considered in more detail. Note that for many

openings in strong rock at shallow depth, stress problems may not be

significant and the analysis need not proceed any further.

For this particular case, stress problems are considered to be important. A typical next

step would be to search the literature in an effort to determine whether the results of

in situ stress measurement programmes are available for mines or civil engineering

projects within a radius of say 50 km of the site. With luck, a few stress measurement

results will be available for the region in which the tunnel is located and these results

can be used to refine the analysis discussed above.

Assuming that the results of the analysis of induced stresses in the rock surrounding

the proposed tunnel indicate that significant zones of rock failure are likely to

develop, and that support costs are likely to be high, it is probably justifiable to set up

a stress measurement project on the site. These measurements can be carried out in

deep boreholes from the surface, using hydraulic fracturing techniques, or from

underground access using overcoring methods. The choice of the method and the

number of measurements to be carried out depends upon the urgency of the problem,

the availability of underground access and the costs involved in the project. Note that

very few project organisations have access to the equipment required to carry out a

stress measurement project and, rather than purchase this equipment, it may be worth

bringing in an organisation which has the equipment and which specialises in such

measurements.

2.

Where regional tectonic features such as major faults are likely to be

encountered the in situ stresses in the vicinity of the feature may be rotated

with respect to the regional stress field. The stresses may be significantly

different in magnitude from the values estimated from the general trends

described above. These differences can be very important in the design of the

openings and in the selection of support and, where it is suspected that this is

likely to be the case, in situ stress measurements become an essential

component of the overall design process.

Analysis of induced stresses

When an underground opening is excavated into a stressed rock mass, the stresses in

the vicinity of the new opening are re-distributed. Consider the example of the

stresses induced in the rock surrounding a horizontal circular tunnel as illustrated in

Figure 5, showing a vertical slice normal to the tunnel axis.

8

In situ and induced stresses

Before the tunnel is excavated, the in situ stressesσv,

σh1and

σh2 are uniformly

distributed in the slice of rock under consideration. After removal of the rock from

within the tunnel, the stresses in the immediate vicinity of the tunnel are changed and

new stresses are induced. Three principal stresses

σ1,σ2 and

σ3acting on a typical

element of rock are shown in Figure 5.

The convention used in rock engineering is that

compressive stresses are always

positive and the three principal stresses are numbered such that

σ1 is the largest

compressive stress and

σ3 is the smallest compressive stress or the largest tensile

stress of the three.

Figure 5: Illustration of principal stresses induced in an element of rock close to a

horizontal tunnel subjected to a vertical in situ stressσv, a horizontal in situ stress

σh1 in a plane normal to the tunnel axis and a horizontal in situ stress

σh2 parallel to

the tunnel axis.

9

In situ and induced stresses

Figure 6: Principal stress directions in the rock surrounding a horizontal tunnel subjected to a

horizontal in situ stress

σh1equal to 3σv, where

σv is the vertical in situ stress.

Figure 7: Contours of maximum and minimum principal stress magnitudes in the rock

surrounding a horizontal tunnel, subjected to a vertical in situ stress of

σv and a horizontal in

situ stress of 3σv .

10

In situ and induced stresses

The three principal stresses are mutually perpendicular but they may be inclined to

the direction of the applied in situ stress. This is evident in Figure 6 which shows the

directions of the stresses in the rock surrounding a horizontal tunnel subjected to a

horizontal in situ stress

σh1 equal to three times the vertical in situ stressσv. The

longer bars in this figure represent the directions of the maximum principal stressσ1,

while the shorter bars give the directions of the minimum principal stress

σ3

at each

element considered. In this particular case,

σ2 is coaxial with the in situ stressσh2,

but the other principal stresses

σ1

and

σ3are inclined to

σh1and

σv in the immediate

vicinity of the tunnel.

Contours of the magnitudes of the maximum principal stress

σ1

and the minimum

principal stress

σ3are given in Figure 7. This figure shows that the redistribution of

stresses is concentrated in the rock close to the tunnel and that, at a distance of say

three times the radius from the centre of the hole, the disturbance to the in situ stress

field is negligible.

An analytical solution for the stress distribution in a stressed elastic plate containing a

circular hole was published by Kirsch (1898) and this formed the basis for many early

studies of rock behaviour around tunnels and shafts. Following along the path

pioneered by Kirsch, researchers such as Love (1927), Muskhelishvili (1953) and

Savin (1961) published solutions for excavations of various shapes in elastic plates. A

useful summary of these solutions and their application in rock mechanics was

published by Brown in an introduction to a volume entitled

Analytical and

Computational Methods in Engineering Rock

Mechanics (1987).

Closed form solutions still possess great value for conceptual understanding of

behaviour and for the testing and calibration of numerical models. For design

purposes, however, these models are restricted to very simple geometries and material

models. They are of limited practical value. Fortunately, with the development of

computers, many powerful programs that provide numerical solutions to these

problems are now readily available. A brief review of some of these numerical

solutions is given below.

Numerical methods of stress analysis

Most underground excavations are irregular in shape and are frequently grouped close

to other excavations. These groups of excavations can form a set of complex three-dimensional shapes. In addition, because of the presence of geological features such

as faults and dykes, the rock properties are seldom uniform within the rock volume of

interest. Consequently, closed form solutions are of limited value in calculating the

stresses, displacements and failure of the rock mass surrounding underground

excavations. A number of computer-based numerical methods have been developed

over the past few decades and these methods provide the means for obtaining

approximate solutions to these problems.

11

In situ and induced stresses

Numerical methods for the analysis of stress driven problems in rock mechanics can

be divided into two classes:

•

Boundary discretization methods, in which only the boundary of the

excavation is divided into elements and the interior of the rock mass is

represented mathematically as an infinite continuum. These methods are

normally restricted to elastic analyses.

•

Domain discretization methods, in which the interior of the rock mass is

divided into geometrically simple elements each with assumed properties. The

collective behaviour and interaction of these simplified elements model the

more complex overall behaviour of the rock mass. In other words domain

methods allow consideration of more complex material models than boundary

methods.

Finite element and

finite difference methods are domain techniques

which treat the rock mass as a continuum. The

distinct element method is also

a domain method which models each individual block of rock as a unique

element.

These two classes of analysis can be combined in the form of

hybrid models in order

to maximise the advantages and minimise the disadvantages of each method.

It is possible to make some general observations about the two types of approaches

discussed above. In domain methods, a significant amount of effort is required to

create the mesh that is used to divide the rock mass into elements. In the case of

complex models, such as those containing multiple openings, meshing can become

extremely difficult. In contrast, boundary methods require only that the excavation

boundary be discretized and the surrounding rock mass is treated as an infinite

continuum. Since fewer elements are required in the boundary method, the demand

on computer memory and on the skill and experience of the user is reduced. The

availability of highly optimised mesh-generators in many domain models has

narrowed this difference to the point where most users of domain programs would be

unaware of the mesh generation problems discussed above and hence the choice of

models can be based on other considerations.

In the case of domain methods, the outer boundaries of the model must be placed

sufficiently far away from the excavations in order that errors, arising from the

interaction between these outer boundaries and the excavations, are reduced to an

acceptable minimum. On the other hand, since boundary methods treat the rock mass

as an infinite continuum, the far field conditions need only be specified as stresses

acting on the entire rock mass and no outer boundaries are required. The main

strength of boundary methods lies in the simplicity achieved by representing the rock

mass as a continuum of infinite extent. It is this representation, however, that makes it

difficult to incorporate variable material properties and discontinuities such as joints

and faults. While techniques have been developed to allow some boundary element

modelling of variable rock properties, these types of problems are more conveniently

modelled by domain methods.

12

In situ and induced stresses

Before selecting the appropriate modelling technique for particular types of problems,

it is necessary to understand the basic components of each technique.

Boundary Element Method

The boundary element method derives its name from the fact that only the boundaries

of the problem geometry are divided into elements. n other words, only the

excavation surfaces, the free surface for shallow problems, joint surfaces where joints

are considered explicitly and material interfaces for multi-material problems are

divided into elements. n fact, several types of boundary element models are

collectively referred to as ‘the boundary element method’ (Crouch and Starfield, 1983).

These models may be grouped as follows:

Indirect (Fictitious Stress) method, so named because the first step in the solution is

to find a set of fictitious stresses that satisfy prescribed boundary conditions. These

stresses are then used in the calculation of actual stresses and displacements in the

rock mass.

Direct method, so named because the displacements are solved directly for the

specified boundary conditions.

Displacement Discontinuity method, so named because the solution is based on the

superposition of the fundamental solution of an elongated slit in an elastic continuum

and shearing and normal displacements in the direction of the slit.

The differences between the first two methods are not apparent to the program user.

The direct method has certain advantages in terms of program development, as will

be discussed later in the section on Hybrid approaches.

The fact that a boundary element model extends ‘to infinity’ can also be a

disadvantage. For example, a heterogeneous rock mass consists of regions of finite,

not infinite, extent. Special techniques must be used to handle these situations. Joints

are modelled explicitly in the boundary element method using the displacement

discontinuity approach, but this can result in a considerable increase in computational

effort. Numerical convergence is often found to be a problem for models

incorporating many joints. For these reasons, problems, requiring explicit

consideration of several joints and/or sophisticated modelling of joint constitutive

behaviour, are often better handled by one of the domain methods such as finite

elements.

A widely-used application of displacement discontinuity boundary elements is in the

modelling of tabular ore bodies. Here, the entire ore seam is represented as a

‘discontinuity’ which is initially filled with ore. Mining is simulated by reduction of

the ore stiffness to zero in those areas where mining has occurred, and the resulting

stress redistribution to the surrounding pillars may be examined (Salamon, 1974, von

Kimmelmann et al., 1984).

13

In situ and induced stresses

Finite element and finite difference methods

In practice, the finite element method is usually indistinguishable from the finite

difference method; thus, they will be treated here as one and the same. For the

boundary element method, it was seen that conditions on a domain boundary could be

related to the state at

all

points throughout the remaining rock, even to infinity. In

comparison, the finite element method relates the conditions at a few points within

the rock (nodal points) to the state within a finite closed region formed by these

points (the element). In the finite element method the physical problem is modelled

numerically by dividing the entire problem region into elements.

The finite element method is well suited to solving problems involving heterogeneous

or non-linear material properties, since each element explicitly models the response of

its contained material. However, finite elements are not well suited to modelling

infinite boundaries, such as occur in underground excavation problems. One

technique for handling infinite boundaries is to discretize beyond the zone of

influence of the excavation and to apply appropriate boundary conditions to the outer

edges. Another approach has been to develop elements for which one edge extends to

so-called 'infinity' finite elements. In practice, efficient pre- and post-processors allow the user to perform parametric analyses and assess the influence of

approximated far-field boundary conditions. The time required for this process is

negligible compared to the total analysis time.

Joints can be represented explicitly using specific 'joint elements'. Different

techniques have been proposed for handling such elements, but no single technique

has found universal favour. Joint interfaces may be modelled, using quite general

constitutive relations, though possibly at increased computational expense depending

on the solution technique.

Once the model has been divided into elements, material properties have been

assigned and loads have been prescribed, some technique must be used to redistribute

any unbalanced loads and thus determine the solution to the new equilibrium state.

Available solution techniques can be broadly divided into two classes - implicit and

explicit. Implicit techniques assemble systems of linear equations that are then solved

using standard matrix reduction techniques. Any material non-linearity is accounted

for by modifying stiffness coefficients (secant approach) and/or by adjusting

prescribed variables (initial stress or initial strain approach). These changes are made

in an iterative manner such that all constitutive and equilibrium equations are satisfied

for the given load state.

The response of a non-linear system generally depends upon the sequence of loading.

Thus it is necessary that the load path modelled be representative of the actual load

path experienced by the body. This is achieved by breaking the total applied load into

load increments, each increment being sufficiently small, so that solution

convergence for the increment is achieved after only a few iterations. However, as the

system being modelled becomes increasingly non-linear and the load increment

14

In situ and induced stresses

represents an ever smaller portion of the total load, the incremental solution technique

becomes similar to modelling the quasi-dynamic behaviour of the body, as it responds

to gradual application of the total load.

In order to overcome this, a ‘dynamic relaxation’ solution technique was proposed

(Otter et al., 1966) and first applied to geomechanics modelling by Cundall (1971). In

this technique no matrices are formed. Rather, the solution proceeds explicitly -

unbalanced forces, acting at a material integration point, result in acceleration of the

mass associated with the point; applying Newton's law of motion expressed as a

difference equation yields incremental displacements, applying the appropriate

constitutive relation produces the new set of forces, and so on marching in time, for

each material integration point in the model. This solution technique has the

advantage that both geometric and material non-linearities are accommodated, with

relatively little additional computational effort as compared to a corresponding linear

analysis, and computational expense increases only linearly with the number of

elements used. A further practical advantage lies in the fact that numerical divergence

usually results in the model predicting obviously anomalous physical behaviour.

Thus, even relatively inexperienced users may recognise numerical divergence.

Most commercially available finite element packages use implicit (i.e. matrix)

solution techniques. For linear problems and problems of moderate non-linearity,

implicit techniques tend to perform faster than explicit solution techniques. However,

as the degree of non-linearity of the system increases, imposed loads must be applied

in smaller increments which implies a greater number of matrix re-formations and

reductions, and hence increased computational expense. Therefore, highly non-linear

problems are best handled by packages using an explicit solution technique.

Distinct Element Method

In ground conditions conventionally described as blocky (i.e. where the spacing of the

joints is of the same order of magnitude as the excavation dimensions), intersecting

joints form wedges of rock that may be regarded as rigid bodies. That is, these

individual pieces of rock may be free to rotate and translate, and the deformation that

takes place at block contacts may be significantly greater than the deformation of the

intact rock. Hence, individual wedges may be considered rigid. For such conditions it

is usually necessary to model many joints explicitly. However, the behaviour of such

systems is so highly non-linear, that even a jointed finite element code, employing an

explicit solution technique, may perform relatively inefficiently.

An alternative modelling approach is to develop data structures that represent the

blocky nature of the system being analysed. Each block is considered a unique free

body that may interact at contact locations with surrounding blocks. Contacts may be

represented by the overlaps of adjacent blocks, thereby avoiding the necessity of

unique joint elements. This has the added advantage that arbitrarily large relative

displacements at the contact may occur, a situation not generally tractable in finite

element codes.

15

In situ and induced stresses

Due to the high degree of non-linearity of the systems being modelled, explicit

solution techniques are favoured for distinct element codes. As is the case for finite

element codes employing explicit solution techniques, this permits very general

constitutive modelling of joint behaviour with little increase in computational effort

and results in computation time being only linearly dependent on the number of

elements used. The use of explicit solution techniques places fewer demands on the

skills and experience than the use of codes employing implicit solution techniques.

Although the distinct element method has been used most extensively in academic

environments to date, it is finding its way into the offices of consultants, planners and

designers. Further experience in the application of this powerful modelling tool to

practical design situations and subsequent documentation of these case histories is

required, so that an understanding may be developed of where, when and how the

distinct element method is best applied.

Hybrid approaches

The objective of a hybrid method is to combine the above methods in order to

eliminate undesirable characteristics while retaining as many advantages as possible.

For example, in modelling an underground excavation, most non-linearity will occur

close to the excavation boundary, while the rock mass at some distance will behave in

an elastic fashion. Thus, the near-field rock mass might be modelled, using a distinct

element or finite element method, which is then linked at its outer limits to a

boundary element model, so that the far-field boundary conditions are modelled

exactly. In such an approach, the direct boundary element technique is favoured as it

results in increased programming and solution efficiency.

Lorig and Brady (1984) used a hybrid model consisting of a discrete element model

for the near field and a boundary element model for the far field in a rock mass

surrounding a circular tunnel.

Two-dimensional and three-dimensional models

A two-dimensional model, such as that illustrated in Figure 5, can be used for the

analysis of stresses and displacements in the rock surrounding a tunnel, shaft or

borehole, where the length of the opening is much larger than its cross-sectional

dimensions. The stresses and displacements in a plane, normal to the axis of the

opening, are not influenced by the ends of the opening, provided that these ends are

far enough away.

On the other hand, an underground powerhouse or crusher chamber has a much more

equi-dimensional shape and the effect of the end walls cannot be neglected. In this

case, it is much more appropriate to carry out a three-dimensional analysis of the

stresses and displacements in the surrounding rock mass. Unfortunately, this switch

from two to three dimensions is not as simple as it sounds and there are relatively few

16

In situ and induced stresses

good three-dimensional numerical models, which are suitable for routine stress

analysis work in a typical engineering design office.

EXAMNE3D () is a three-dimensional boundary element

program that provides a starting point for an analysis of a problem in which the three-dimensional geometry of the openings is important. Such three-dimensional analyses

provide clear indications of stress concentrations and of the influence of three-dimensional geometry. In many cases, it is possible to simplify the problem to two-dimensions by considering the stresses on critical sections identified in the three-dimensional model.

More sophisticated three-dimensional finite element models such as

FLAC3D

() are available, but the definition of the input parameters and

interpretation of the results of these models would stretch the capabilities of all but

the most experienced modellers. It is probably best to leave this type of modelling in

the hands of these specialists.

It is recommended that, where the problem being considered is obviously three-dimensional, a preliminary elastic analysis be carried out by means of one of the

three-dimensional boundary element programs. The results can then be used to decide

whether further three-dimensional analyses are required or whether appropriate two-dimensional sections can be modelled using a program such as

PHASE2

(), a

powerful but user-friendly finite element program that

generally meets the needs of most underground excavation design projects.

Examples of two-dimensional stress analysis

IA boundary element program called

EXAMINE2D is available as a free download

from . While this program is limited to elastic analyses it can

provide a very useful introduction for those who are not familiar with the numerical

stress analysis methods described above. The following examples demonstrate the use

of this program to explore some common problems in tunnelling.

Tunnel shape

Most contractors like a simple horseshoe shape for tunnels since this gives a wide flat

floor for the equipment used during construction. For relatively shallow tunnels in

good quality rock this is an appropriate tunnel shape and there are many hundreds of

kilometres of horseshoe shaped tunnels all over the world.

In poor quality rock masses or in tunnels at great depth, the simple horseshoe shape is

not a good choice because of the high stress concentrations at the corners where the

sidewalls meet the floor or invert. In some cases failures initiating at these corners

can lead to severe floor heave and even to failure of the entire tunnel perimeter as

shown in Figure 8.

17

In situ and induced stresses

Figure 8: Failure of the lining in a horseshoe shaped tunnel in a highly stressed poor

quality rock mass. This failure initiated at the corners where the invert meets the

sidewalls.

Figure 9: Dimensions of a 10 m span

modified horseshoe tunnel shape

designed to overcome some of the

problems illustrated in Figure 8.

The stress distribution in the rock mass surrounding the tunnel can be improved by

modifying the horseshoe shape as shown in Figure 9. In some cases this can

eliminate or minimise the types of failure shown in Figure 8 while, in other cases, it

may be necessary to use a circular tunnel profile.

18

In situ and induced stresses

In situ stresses:

Major principal stress σ1 = 10 MPa

Minor principal stress σ3 = 7 MPa

Intermediate principal stress σ2 = 9 MPa

Inclination of major principal stress to

the horizontal axis = 15º

Rock mass properties:

Friction angle φ = 35º

Cohesion c = 1 MPa

Tensile strength = zero

Deformation modulus E = 4600 MPa

Figure 10: Comparison of three tunnel

excavation profiles using

EXAMINE2D.

The contours are for the Strength Factor

defined by the ratio of rock mass strength

to the induced stress at each point. The

deformed boundary profile (exaggerated)

is shown inside each excavation.

19

In situ and induced stresses

The application of the program

EXAMINE2D to compare three tunnel shapes is

illustrated in Figure 10. Typical “average” in situ stresses and rock mass properties

were used in this analysis and the three figures compare Strength Factor contours and

deformed excavation profiles (exaggerated) for the three tunnel shapes.

It is clear that the flat floor of the horseshoe tunnel (top figure) allows upward

displacement or heaving of the floor. The sharp corners at the junction between the

floor and the tunnel sidewalls create high stress concentrations and also generate large

bending moments in any lining installed in the tunnel. Failure of the floor generally

initiates at these corners as illustrated in Figure 8.

Floor heave is reduced significantly by the concave curvature of the floor of the

modified horseshoe shape (middle figure). In marginal cases these modifications to

the horseshoe shape may be sufficient to prevent or at least minimise the type of

damage illustrated in Figure 8. However, in severe cases, a circular tunnel profile is

invariably the best choice, as shown by the smooth Strength Factor contours and the

deformed tunnel boundary shape in the bottom figure in Figure 10.

Large underground caverns

A typical underground complex in a hydroelectric project has a powerhouse with a

span of 20 to 25 m and a height of 40 to 50 m. Four to six turbine-generator sets are

housed in this cavern and a cutaway sketch through one of these sets is shown in

Figure 11. Transformers are frequently housed in a chamber or gallery parallel to the

powerhouse. Ideally these two caverns should be as close as possible in order to

minimise the length of the bus-bars connecting the generators and transformers. This

has to be balanced against the size and hence the stability of the pillar between the

caverns. The relative location and distance between the caverns is explored in the

series of EXAMINE2D models shown in Figure 12, using the same in situ stresses

and rock mass properties as listed in Figure 10.

Figure 11: Cutaway sketch of the

layout of an underground powerhouse

cavern and a parallel transformer

gallery.

20

In situ and induced stresses

21

In situ stresses:

Major principal stress σ1 = 10 MPa

Minor principal stress σ3 = 7 MPa

Intermediate stress σ2 = 9 MPa

Inclination of major principal

stress to the horizontal axis = 15º

Rock mass properties:

Friction angle φ = 35º

Cohesion c = 1 MPa

Tensile strength = zero

Deformation modulus E = 4600

MPa

Figure 12: Comparison of three

underground powerhouse and

transformer gallery layouts,

using

EXAMINE2D. The

contours are for the Strength

Factor defined by the ratio of

rock mass strength to the

induced stress at each point. The

deformed boundary profile

(exaggerated) is shown inside

each excavation.

In situ and induced stresses

Figure 13: Displacement vectors and deformed excavation shapes for the

underground powerhouse and transformer gallery.

A closer examination of the deformations induced in the rock mass by the excavation

of the underground powerhouse and transformer gallery, in Figure 13, shows that the

smaller of the two excavations is drawn towards the larger cavern and its profile is

distorted in this process. This distortion can be reduced by relocating the transformer

gallery and by increasing the spacing between the galleries as has been done in Figure

12.

Where the combination of rock mass strength and in situ stresses is likely to cause

overstressing around the caverns and in the pillar, a good rule of thumb is that the

distance between the two caverns should be approximately equal to the height of the

larger cavern.

The interested reader is encouraged to download the program

EXAMINE2D (free from

) and to use it to explore the problem, such as those illustrated in

Figures 10 and 12, for themselves.

22

In situ and induced stresses

References

Brown, E.T. 1987. ntroduction.

Analytical an computational methos in

engineering rock mechanics, (ed. E.T. Brown), 1-31. London: Allen and Unwin.

Brown, E.T. and Hoek, E. 1978. Trends in relationships between measured rock in

situ stresses and depth.

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.

15,

pp.211-215.

Crouch, S.L. and Starfield, A.M. 1983. Boundary element methods in solid

mechanics . London: Allen and Unwin.

Cundall, P.A. 1971. A computer model for simulating progressive large scale

movements in blocky rock systems. Fracture , Proc. symp. ISRM, Nancy

I1, Paper 2-8.

Engelder, T. and Sbar, M.L. 1984. Near-surface in situ stress: introduction.

J.

Geophys. Res.

89, pp.9321-9322. Princeton, NJ: Princeton University Press.

Herget, G. 1988.

Stresses in rock. Rotterdam: Balkema.

Hoek, E., Carranza – Torres, C. and Corkum, B., 2002. Hoek - Brown failure

criterion –

2002 edition. In

Proceedings of NARMS-TAC 2002,

Toronto (eds.

Bawden,R.W., Curran, J., Telesnicki, M) pp. 267-273. Download from

Kirsch, G., 1898. Die theorie der elastizitat und die bedurfnisse der festigkeitslehre.

Veit. Deit. Ing.

42 (28), 797-807.

Lorig, L.J. and Brady, B.H.G. 1984. A hybrid computational scheme for excavation

and support design in jointed rock media. n

Design and performance of

underground excavations, (eds E.T. Brown and J.A. Hudson), 105-112. London:

ddBrit. Geotech. Soc.

ILove, A.E.H. 1927. A treatise on the mathematical theory of elasticity. New York:

Dover.

Muskhelishvili, N.I. 1953.

Some basic problems of the mathematical theory of

elasticity. 4th edn, translated by J.R.M. Radok. Gronigen: Noordhoff.

Otter, J.R.H., Cassell, A.C. and Hobbs, R.E. 1966. Dynamic relaxation.

Proc. Instn

Civ. Engrs

35, 633-665.

Reinecker, J., Heidbach, O., Tingay, M., Sperner, B., & Müller, B. 2005: The release

2005 of the World Stress Map (available online at ).

Salamon, M.D.G. 1974. Rock mechanics of underground excavations. In

Advances

in rock mechanics

, Proc. 3rd ., Denver

1B,

951-1009. Washington,

DC: National Academy of Sciences

Savin, G.N. 1961.

Stress concentrations around holes. London: Pergamon.

Sheory, P.R. 1994. A theory for in situ stresses in isotropic and transversely isotropic

rock.

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.

31(1), 23-34.

Terzaghi, K. and Richart, F.E. 1952. Stresses in rock about cavities.

Geotechnique

3,

57-90.

23

In situ and induced stresses

von Kimmelmann, M.R., Hyde, B. and Madgwick, R.J. 1984. The use of computer

applications at BCL Limited in planning pillar extraction and the design of mine

layouts. In

Design and performance of underground excavations, (eds E.T. Brown

and J.A. Hudson), 53-64. London: Brit. Geotech. Soc.

Zoback, M. L. 1992. First- and second-order patterns of stress in the lithosphere: the

World Stress Map Project.

J. Geophys. Res.

97(B8), 11761-11782.

24

Rock mass properties

Introduction

Reliable estimates of the strength and deformation characteristics of rock masses are

required for almost any form of analysis used for the design of slopes, foundations and

underground excavations. Hoek and Brown (1980a, 1980b) proposed a method for

obtaining estimates of the strength of jointed rock masses, based upon an assessment of

the interlocking of rock blocks and the condition of the surfaces between these blocks.

This method was modified over the years in order to meet the needs of users who were

applying it to problems that were not considered when the original criterion was

developed (Hoek 1983, Hoek and Brown 1988). The application of the method to very

poor quality rock masses required further changes (Hoek, Wood and Shah 1992) and,

eventually, the development of a new classification called the Geological Strength Index

(Hoek, Kaiser and Bawden 1995, Hoek 1994, Hoek and Brown 1997, Hoek, Marinos and

Benissi, 1998, Marinos and Hoek, 2001). A major revision was carried out in 2002 in

order to smooth out the curves, necessary for the application of the criterion in numerical

models, and to update the methods for estimating Mohr Coulomb parameters (Hoek,

Carranza-Torres and Corkum, 2002). A related modification for estimating the

deformation modulus of rock masses was made by Hoek and Diederichs (2006).

This chapter presents the most recent version of the Hoek-Brown criterion in a form that

has been found practical in the field and that appears to provide the most reliable set of

results for use as input for methods of analysis in current use in rock engineering.

Generalised Hoek-Brown criterion

The Generalised Hoek-Brown failure criterion for jointed rock masses is defined by:

'§·σ''3¨σ1=σ3+σcimb+s¸

¨¸σci©¹a (1)

'' and

σ3 are the maximum and minimum effective principal stresses at failure,

where

σ1mb is the value of the Hoek-Brown constant m for the rock mass,

s and a are constants which depend upon the rock mass characteristics, and

σci is the uniaxial compressive strength of the intact rock pieces.

Rock mass properties

Normal and shear stresses are related to principal stresses by the equations published by

Balmer1

(1952).

'σn=''σ1+σ32−'''σ1−σ'3dσ1dσ3−12dσ'1⋅dσ1dσ3+1'3''

(2)

(3)

'τ=σ1−σ('3)dσ'3dσ'1

dσ+1where

'''dσ1dσ3=1+ambmbσ3σci+s()a−1 (4)

In order to use the Hoek-Brown criterion for estimating the strength and deformability of

jointed rock masses, three ‘properties’ of the rock mass have to be estimated. These are:

• uniaxial compressive strength

σci of the intact rock pieces,

• value of the Hoek-Brown constant

mi for these intact rock pieces, and

• value of the Geological Strength Index GSI for the rock mass.

Intact rock properties

For the intact rock pieces that make up the rock mass, equation (1) simplifies to:

'§·σ3''¨σ1=σ3+σcimi+1¸¨σci¸©¹0.5 (5)

The relationship between the principal stresses at failure for a given rock is defined by

two constants, the uniaxial compressive strength

σci and a constant

mi. Wherever

possible the values of these constants should be determined by statistical analysis of the

results of a set of triaxial tests on carefully prepared core samples.

') values over which these tests are

Note that the range of minor principal stress (σ3carried out is critical in determining reliable values for the two constants. In deriving the

'original values of

σci andmi, Hoek and Brown (1980a) used a range of 0 <σ3< 0.5σci

and, in order to be consistent, it is essential that the same range be used in any laboratory

triaxial tests on intact rock specimens. At least five well spaced data points should be

included in the analysis.

1 The original equations derived by Balmer contained errors that have been corrected in equations 2 and 3.

2

Rock mass properties

One type of triaxial cell that can be used for these tests is illustrated in Figure 1. This cell,

described by Franklin and Hoek (1970), does not require draining between tests and is

convenient for the rapid testing on a large number of specimens. More sophisticated cells

are available for research purposes but the results obtained from the cell illustrated in

Figure 1 are adequate for the rock strength estimates required for estimating

σci and

mi.

This cell has the additional advantage that it can be used in the field when testing

materials such as coals or mudstones that are extremely difficult to preserve during

transportation and normal specimen preparation for laboratory testing.

Figure 1: Cut-away view of a triaxial cell for testing rock specimens.

3

Rock mass properties

Laboratory tests should be carried out at moisture contents as close as possible to those

which occur in the field. Many rocks show a significant strength decrease with increasing

moisture content and tests on samples, which have been left to dry in a core shed for

several months, can give a misleading impression of the intact rock strength.

Once the five or more triaxial test results have been obtained, they can be analysed to

determine the uniaxial compressive strength

σci and the Hoek-Brown constant

mi as

described by Hoek and Brown (1980a). In this analysis, equation (5) is re-written in the

form:

y=mσcix+sσci (6)

'''2 and

y=(σ1−σ3) where

x=σ3

For n specimens the uniaxial compressive strength

σci, the constant and

mi the

coefficient of determination

r2are calculated from:

¦yª¦xy−(¦x¦yn)º¦x2σci= (7) −«»22n«¬¦x−((¦x)n)»¼n

1ª¦xy−(¦x¦yn)º (8)

mi=«»

22σci«¬¦x−((¦x)n)»¼

¦xy−(¦x¦yn]2[2 (9)

r=2222[¦x−(¦x)n][¦y−(¦y)n]

A spreadsheet for the analysis of triaxial test data is given in Table 1. Note that high

quality triaxial test data will usually give a coefficient of determination

r2of greater than

0.9. These calculations, together with many more related to the Hoek-Brown criterion can

also be performed by the program RocLab that can be downloaded (free) from

.

When laboratory tests are not possible, Table 2 and Table 3 can be used to obtain

estimates of

σci and

mi.

4

Rock mass properties

Table 1: Spreadsheet for the calculation of

σci and

mi from triaxial test data

Triaxial test dataxsig3sig1038.3572.47.580.515115.620134.347.5sumx441.1yxyxsq0.025.056.3225.0400.0706.3sumxsqysq281sumysq1466.890.04542.7622713.85329.0039967.510120.36151805.413064.49261289.834523.50475776.5sumysumxyCalculation resultsNumber of tests n =Uniaxial strength sigci =Hoek-Brown constant mi =Hoek-Brown constant s =Coefficient of determination r2 =537.415.501.000.997Cell formulaey =(sig1-sig3)^2sigci =SQRT(sumy/n - (sumxy-sumx*sumy/n)/(sumxsq-(sumx^2)/n)*sumx/n)mi =(1/sigci)*((sumxy-sumx*sumy/n)/(sumxsq-(sumx^2)/n))r2 =((sumxy-(sumx*sumy/n))^2)/((sumxsq-(sumx^2)/n)*(sumysq-(sumy^2)/n))

Note: These calculations, together with many other calculations related to the Hoek-Brown criterion, can also be carried out using the program RocLab that can be

downloaded (free) from .

5

Rock mass properties

Table 2: Field estimates of uniaxial compressive strength.

Uniaxial

Comp.

Grade* Term Strength

(MPa)

R6 Extremely > 250

Strong

Point

Load

Index

(MPa)

>10

Field estimate of

strength

Specimen can only be

chipped with a

geological hammer

Examples

R5 Very 100 - 250 4 - 10

strong

R4 Strong 50 - 100

2 - 4

25 - 50

R3 Medium

strong

1 - 2

R2 Weak 5 - 25

**

R1 Very

1 - 5

weak

**

R0 Extremely 0.25 - 1 **

weak

* Grade according to Brown (1981).

** Point load tests on rocks with a uniaxial compressive strength below 25 MPa are likely to yield highly

ambiguous results.

Fresh basalt, chert,

diabase, gneiss, granite,

quartzite

Specimen requires many Amphibolite, sandstone,

blows of a geological basalt, gabbro, gneiss,

hammer to fracture it granodiorite, limestone,

marble, rhyolite, tuff

Specimen requires more

Limestone, marble,

phyllite, sandstone, schist,

than one blow of a

shale

geological hammer to

fracture it

Claystone, coal, concrete,

Cannot be scraped or

schist, shale, siltstone

peeled with a pocket

knife, specimen can be

fractured with a single

blow from a geological

hammer

Chalk, rocksalt, potash

Can be peeled with a

pocket knife with

difficulty, shallow

indentation made by

firm blow with point of

a geological hammer

Highly weathered or

Crumbles under firm

altered rock

blows with point of a

geological hammer, can

be peeled by a pocket

knife

Indented by thumbnail Stiff fault gouge

6

Rock mass properties

Table 3: Values of the constant

mi for intact rock, by rock group. Note that values in

parenthesis are estimates.

7

Rock mass properties

Anisotropic and foliated rocks such as slates, schists and phyllites, the behaviour of

which is dominated by closely spaced planes of weakness, cleavage or schistosity,

present particular difficulties in the determination of the uniaxial compressive strengths.

Salcedo (1983) has published the results of a set of directional uniaxial compressive tests

on a graphitic phyllite from Venezuela. These results are summarised in Figure 2. It will

be noted that the uniaxial compressive strength of this material varies by a factor of about

5, depending upon the direction of loading.

10090Compressive

strength

-

MPa8Angle of schistosity to loading direction

Figure 2: Influence of loading direction on the strength of graphitic phyllite tested by

Salcedo (1983).

In deciding upon the value of

σci for foliated rocks, a decision has to be made on

whether to use the highest or the lowest uniaxial compressive strength obtained from

results such as those given in Figure 2. Mineral composition, grain size, grade of

metamorphism and tectonic history all play a role in determining the characteristics of the

rock mass. The author cannot offer any precise guidance on the choice of

σci but some

insight into the role of schistosity in rock masses can be obtained by considering the case

of the Yacambú-Quibor tunnel in Venezuela.

This tunnel has been excavated in graphitic phyllite, similar to that tested by Salcedo, at

depths of up to 1200 m through the Andes mountains. The appearance of the rock mass at

8

Rock mass properties

the tunnel face is shown in Figure 3 and a back analysis of the behaviour of this material

suggests that an appropriate value for

σci is approximately 50 MPa. In other words, on

the scale of the 5.5 m diameter tunnel, the rock mass properties are “averaged” and there

is no sign of anisotropic behaviour in the deformations measured in the tunnel.

Figure 3: Tectonically deformed and sheared graphitic phyllite in the face of the

Yacambú-Quibor tunnel at a depth of 1200 m below surface.

Influence of sample size

The influence of sample size upon rock strength has been widely discussed in

geotechnical literature and it is generally assumed that there is a significant reduction in

strength with increasing sample size. Based upon an analysis of published data, Hoek and

Brown (1980a) have suggested that the uniaxial compressive strength

σcd of a rock

specimen with a diameter of

d mm is related to the uniaxial compressive strength

σc50 of

a 50 mm diameter sample by the following relationship:

σcd

This relationship, together with the data upon which it was based, is shown in Figure 4.

9

§50·=σc50¨¸©d¹0.18 (10)

Rock mass properties

Uniaxial

compressive

strength

of

specimen

of

diameter

dUniaxial

compressive

strength

of

50

mm

diameter

specimen1.51.41.31.2MarbleLimestoneGraniteBasaltBasalt-andesite lavaGabbroMarbleNoriteGraniteQuartz diorite1.11.00.90.80.7250300Specimen diameter d mm

Figure 4: Influence of specimen size on the strength of intact rock. After Hoek and

Brown (1980a).

It is suggested that the reduction in strength is due to the greater opportunity for failure

through and around grains, the ‘building blocks’ of the intact rock, as more and more of

these grains are included in the test sample. Eventually, when a sufficiently large number

of grains are included in the sample, the strength reaches a constant value.

The Hoek-Brown failure criterion, which assumes isotropic rock and rock mass

behaviour, should only be applied to those rock masses in which there are a sufficient

number of closely spaced discontinuities, with similar surface characteristics, that

isotropic behaviour involving failure on discontinuities can be assumed. When the

structure being analysed is large and the block size small in comparison, the rock mass

can be treated as a Hoek-Brown material.

Where the block size is of the same order as that of the structure being analysed or when

one of the discontinuity sets is significantly weaker than the others, the Hoek-Brown

criterion should not be used. In these cases, the stability of the structure should be

analysed by considering failure mechanisms involving the sliding or rotation of blocks

and wedges defined by intersecting structural features.

It is reasonable to extend this argument further and to suggest that, when dealing with

large scale rock masses, the strength will reach a constant value when the size of

individual rock pieces is sufficiently small in relation to the overall size of the structure

being considered. This suggestion is embodied in Figure 5 which shows the transition

10

Rock mass properties

from an isotropic intact rock specimen, through a highly anisotropic rock mass in which

failure is controlled by one or two discontinuities, to an isotropic heavily jointed rock

mass.

Figure 5: Idealised diagram showing the transition from intact to a heavily jointed rock

mass with increasing sample size.

Geological strength Index

The strength of a jointed rock mass depends on the properties of the intact rock pieces

and also upon the freedom of these pieces to slide and rotate under different stress

conditions. This freedom is controlled by the geometrical shape of the intact rock pieces

as well as the condition of the surfaces separating the pieces. Angular rock pieces with

clean, rough discontinuity surfaces will result in a much stronger rock mass than one

which contains rounded particles surrounded by weathered and altered material.

The Geological Strength Index (GSI), introduced by Hoek (1994) and Hoek, Kaiser and

Bawden (1995) provides a number which, when combined with the intact rock properties,

can be used for estimating the reduction in rock mass strength for different geological

11

Rock mass properties

conditions. This system is presented in Table 5, for blocky rock masses, and Table 6 for

heterogeneous rock masses such as flysch. Table 6 has also been extended to deal with

molassic rocks (Hoek et al 2006) and ophiolites (Marinos et al, 2005).

Before the introduction of the GSI system in 1994, the application of the Hoek-Brown

criterion in the field was based on a correlation with the 1976 version of Bieniawski’s

Rock Mass Rating, with the Groundwater rating set to 10 (dry) and the Adjustment for

Joint Orientation set to 0 (very favourable) (Bieniawski, 1976). If the 1989 version of

Bieniawski’s RMR classification (Bieniawski, 1989) is used, then the Groundwater rating

set to 15 and the Adjustment for Joint Orientation set to zero.

During the early years of the application of the GSI system the value of GSI was

estimated directly from RMR. However, this correlation has proved to be unreliable,

particularly for poor quality rock masses and for rocks with lithological peculiarities that

cannot be accommodated in the RMR classification. Consequently, it is recommended

that GSI should be estimated directly by means of the charts presented in Tables 5 and 6

and not from the RMR classification.

Experience shows that most geologists and engineering geologists are comfortable with

the descriptive and largely qualitative nature of the GSI tables and generally have little

difficulty in arriving at an estimated value. On the other hand, many engineers feel the

need for a more quantitative system in which they can “measure” some physical

dimension. Conversely, these engineers have little difficulty understanding the

importance of the intact rock strength

σci and its incorporation in the assessment of the

rock mass properties. Many geologists tend to confuse intact and rock mass strength and

consistently underestimate the intact strength.

An additional practical question is whether borehole cores can be used to estimate the

GSI value behind the visible faces? Borehole cores are the best source of data at depth

but it has to be recognized that it is necessary to extrapolate the one dimensional

information provided by core to the three-dimensional rock mass. However, this is a

common problem in borehole investigation and most experienced engineering geologists

are comfortable with this extrapolation process. Multiple boreholes and inclined

boreholes are of great help the interpretation of rock mass characteristics at depth.

The most important decision to be made in using the GSI system is whether or not it

should be used. If the discontinuity spacing is large compared with the dimensions of the

tunnel or slope under consideration then, as shown in Figure 5, the GSI tables and the

Hoek-Brown criterion should not be used and the discontinuities should be treated

individually. Where the discontinuity spacing is small compared with the size of the

structure (Figure 5) then the GSI tables can be used with confidence.

12

Rock mass properties

Table 5: Characterisation of blocky rock masses on the basis of interlocking and joint

conditions.

13

Rock mass properties

Table 6: Estimate of Geological Strength Index GSI for heterogeneous rock masses such

as flysch. (After Marinos and Hoek, 2001)

14

Rock mass properties

One of the practical problems that arises when assessing the value of GSI in the field is

related to blast damage. As illustrated in Figure 6, there is a considerable difference in the

appearance of a rock face which has been excavated by controlled blasting and a face

which has been damaged by bulk blasting. Wherever possible, the undamaged face

should be used to estimate the value of GSI since the overall aim is to determine the

properties of the undisturbed rock mass.

Figure 6: Comparison between the results achieved using controlled blasting (on the left)

and normal bulk blasting for a surface excavation in gneiss.

The influence of blast damage on the near surface rock mass properties has been taken

into account in the 2002 version of the Hoek-Brown criterion (Hoek, Carranza-Torres

and Corkum, 2002) as follows:

§GSI−100·mb=miexp¨ (11)

¸

©28−14D¹

15

Rock mass properties

§GSI−100·s=exp¨¸

©9−3D¹and

(12)

a=11−GSI/15−20/3+e−e

26() (13)

D is a factor which depends upon the degree of disturbance due to blast damage and

stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very

disturbed rock masses. Guidelines for the selection of D are presented in Table 7.

Note that the factor D applies only to the blast damaged zone and it should not be applied

to the entire rock mass. For example, in tunnels the blast damage is generally limited to a

1 to 2 m thick zone around the tunnel and this should be incorporated into numerical

models as a different and weaker material than the surrounding rock mass. Applying the

blast damage factor D to the entire rock mass is inappropriate and can result in

misleading and unnecessarily pessimistic results.

'The uniaxial compressive strength of the rock mass is obtained by setting

σ3=0 in

equation 1, giving:

(14)

σc=σ

and, the tensile strength of the rock mass is:

sσσt=−ci

mb (15)

''Equation 15 is obtained by setting

σ1=σ3=σt in equation 1. This represents a

condition of biaxial tension. Hoek (1983) showed that, for brittle materials, the uniaxial

tensile strength is equal to the biaxial tensile strength.

Note that the “switch” at GSI = 25 for the coefficients s and a (Hoek and Brown, 1997)

has been eliminated in equations 11 and 12 which give smooth continuous transitions for

the entire range of GSI values. The numerical values of s and a, given by these equations,

are very close to those given by the previous equations and it is not necessary for readers

to revisit and make corrections to old calculations.

16

Rock mass properties

Table 7: Guidelines for estimating disturbance factor D

Appearance of rock mass

Description of rock mass

Excellent quality controlled blasting or

excavation by Tunnel Boring Machine results

in minimal disturbance to the confined rock

mass surrounding a tunnel.

Suggested value of D

D = 0

Mechanical or hand excavation in poor quality

rock masses (no blasting) results in minimal

disturbance to the surrounding rock mass.

Where squeezing problems result in significant

floor heave, disturbance can be severe unless a

temporary invert, as shown in the photograph,

is placed.

Very poor quality blasting in a hard rock tunnel

results in severe local damage, extending 2 or 3

m, in the surrounding rock mass.

D = 0.8

D = 0

D = 0.5

No invert

Small scale blasting in civil engineering slopes

results in modest rock mass damage,

particularly if controlled blasting is used as

shown on the left hand side of the photograph.

However, stress relief results in some

disturbance.

Very large open pit mine slopes suffer

significant disturbance due to heavy production

blasting and also due to stress relief from

overburden removal.

In some softer rocks excavation can be carried

out by ripping and dozing and the degree of

damage to the slopes is less.

D = 1.0

Production blasting

D = 0.7

Mechanical excavation

D = 0.7

Good blasting

D = 1.0

Poor blasting

17

Rock mass properties

Mohr-Coulomb parameters

Since many geotechnical software programs are written in terms of the Mohr-Coulomb

failure criterion, it is sometimes necessary to determine equivalent angles of friction and

cohesive strengths for each rock mass and stress range. This is done by fitting an average

linear relationship to the curve generated by solving equation 1 for a range of minor

principal stress values defined by

σt

<

σ3 <σ3max, as illustrated in Figure 7. The fitting

process involves balancing the areas above and below the Mohr-Coulomb plot. This

results in the following equations for the angle of friction

φ' and cohesive strengthc' :

'ºª6amb(s+mbσ3)a−1−1«'n»

φ=sin

(16)

'a−1»«2(1+a)(2+a)+6amb(s+mbσ)3n¼¬'a−1σci(1+2a)s+(1−a)mbσ'3n(s+mbσ3)nc='[(1+a)(2+a)1+6amb(s+mbσ3n)(]

'a−1)((1+a)(2+a))

(17)

'where

σ3n=σ3maxσci

Note that the value of

σ’3max, the upper limit of confining stress over which the

relationship between the Hoek-Brown and the Mohr-Coulomb criteria is considered, has

to be determined for each individual case. Guidelines for selecting these values for slopes

as well as shallow and deep tunnels are presented later.

The Mohr-Coulomb shear strength

τ, for a given normal stress

σ, is found by

substitution of these values of

c' and

φ' in to the equation:

τ=c'+σtanφ' (18)

The equivalent plot, in terms of the major and minor principal stresses, is defined by:

σ1='2c'cosφ'1−sinφ'+1+sinφ'1−sinφ'σ'3 (19)

18

Rock mass properties

19

Rock mass properties

However, there are times when it is useful to consider the overall behaviour of a rock

mass rather than the detailed failure propagation process described above. For example,

when considering the strength of a pillar, it is useful to have an estimate of the overall

strength of the pillar rather than a detailed knowledge of the extent of fracture

propagation in the pillar. This leads to the concept of a global “rock mass strength” and

Hoek and Brown (1997) proposed that this could be estimated from the Mohr-Coulomb

relationship:

σcm='2c'cosφ'1−sinφ'

(20)

'with c' and

φ' determined for the stress range

σt<σ3<σci/4 giving

σcm

'Determination of

σ3max

'(mb+4s−a(mb−8s))(mb4+s)a−1=σci⋅

2(1+a)(2+a)(21)

'The issue of determining the appropriate value of

σ3max for use in equations 16 and 17

depends upon the specific application. Two cases will be investigated:

'Tunnels

− where the value of

σ3max is that which gives equivalent characteristic curves

for the two failure criteria for deep tunnels or equivalent subsidence profiles for shallow

tunnels.

Slopes – here the calculated factor of safety and the shape and location of the failure

surface have to be equivalent.

For the case of deep tunnels, closed form solutions for both the Generalized Hoek-Brown

and the Mohr-Coulomb criteria have been used to generate hundreds of solutions and to

'find the value of

σ3max that gives equivalent characteristic curves.

For shallow tunnels, where the depth below surface is less than 3 tunnel diameters,

comparative numerical studies of the extent of failure and the magnitude of surface

subsidence gave an identical relationship to that obtained for deep tunnels, provided that

caving to surface is avoided.

The results of the studies for deep tunnels are plotted in Figure 8 and the fitted equation

for both deep and shallow tunnels is:

20

Rock mass properties

'σ3max'σcm'§σcm=0.47¨¨γH©·¸¸¹−0.94 (22)

'where

σcm is the rock mass strength, defined by equation 21,

γ is the unit weight of the

rock mass and H is the depth of the tunnel below surface. In cases where the horizontal

stress is higher than the vertical stress, the horizontal stress value should be used in place

of

γH.

21

Rock mass properties

Similar studies for slopes, using Bishop’s circular failure analysis for a wide range of

slope geometries and rock mass properties, gave:

'σ3max'σcm'§σcm=0.72¨¨γH©·¸¸¹−0.91 (23)

where H is the height of the slope.

Deformation modulus

Hoek and Diederichs (2005) re-examined existing empirical methods for estimating rock

mass deformation modulus and concluded that none of these methods provided reliable

estimates over the whole range of rock mass conditions encountered. In particular, large

errors were found for very poor rock masses and, at the other end of the spectrum, for

massive strong rock masses. Fortunately, a new set of reliable measured data from China

and Taiwan was available for analyses and it was found that the equation which gave the

best fit to this data is a sigmoid function having the form:

ay=c+ (24)

−((x−x0)/b)1+e

Using commercial curve fitting software, Equation 24 was fitted to the Chinese and

Taiwanese data and the constants a and b in the fitted equation were then replaced by

expressions incorporating GSI and the disturbance factor D. These were adjusted to give

the equivalent average curve and the upper and lower bounds into which > 90% of the

data points fitted. Note that the constant a = 100 000 in Equation 25 is a scaling factor

and it is not directly related to the physical properties of the rock mass.

The following best-fit equation was derived:

§·1−D/2Erm(MPa)=100000¨(25)

¸

¨((75+25D−GSI)/11)¸©1+e¹

The rock mass deformation modulus data from China and Taiwan includes information

on the geology as well as the uniaxial compressive strength (σci) of the intact rock This

information permits a more detailed analysis in which the ratio of mass to intact modulus

(Erm/Ei) can be included. Using the modulus ratio MR proposed by Deere (1968)

(modified by the authors based in part on this data set and also on additional correlations

from Palmstrom and Singh (2001)) it is possible to estimate the intact modulus from:

22

Rock mass properties

Ei=MR⋅σci

(26)

This relationship is useful when no direct values of the intact modulus (Ei) are available

or where completely undisturbed sampling for measurement of

Ei is difficult. A detailed

analysis of the Chinese and Taiwanese data, using Equation (26) to estimate

Ei resulted

in the following equation:

§·1−D/2Erm=Ei¨0.02 (27)

+¸

¨((60+15D−GSI)/11)¸1+e©¹

This equation incorporates a finite value for the parameter c (Equation 24) to account for

the modulus of broken rock (transported rock, aggregate or soil) described by GSI = 0.

This equation is plotted against the average normalized field data from China and Taiwan

in Figure 9.

Figure 9: Plot of normalized in situ rock mass deformation modulus from China and

Taiwan against Hoek and Diederichs Equation (27). Each data point represents the

average of multiple tests at the same site in the same rock mass.

23

Rock mass properties

Table 8: Guidelines for the selection of modulus ratio (MR) values in Equation (26) -

based on Deere (1968) and Palmstrom and Singh (2001)

24

Rock mass properties

Table 8, based on the modulus ratio (MR) values proposed by Deere (1968) can be used

for calculating the intact rock modulus

Ei. In general, measured values of Ei are seldom

available and, even when they are, their reliability is suspect because of specimen

damage. This specimen damage has a greater impact on modulus than on strength and,

Ihence, the intact rock strength, when available, can usually be considered more reliable.

Post-failure behaviour

When using numerical models to study the progressive failure of rock masses, estimates

of the post-peak or post-failure characteristics of the rock mass are required. In some of

these models, the Hoek-Brown failure criterion is treated as a yield criterion and the

analysis is carried out using plasticity theory. No definite rules for dealing with this

problem can be given but, based upon experience in numerical analysis of a variety of

practical problems, the post-failure characteristics, illustrated in Figure 10, are suggested

as a starting point.

Reliability of rock mass strength estimates

The techniques described in the preceding sections of this chapter can be used to estimate

the strength and deformation characteristics of isotropic jointed rock masses. When

applying this procedure to rock engineering design problems, most users consider only

the ‘average’ or mean properties. In fact, all of these properties exhibit a distribution

about the mean, even under the most ideal conditions, and these distributions can have a

significant impact upon the design calculations.

n the text that follows, a slope stability calculation and a tunnel support design

calculation are carried out in order to evaluate the influence of these distributions. In each

case the strength and deformation characteristics of the rock mass are estimated by means

of the Hoek-Brown procedure, assuming that the three input parameters are defined by

normal distributions.

Input parameters

Figure 11 has been used to estimate the value of the value of GSI from field observations

of blockiness and discontinuity surface conditions. ncluded in this figure is a

crosshatched circle representing the 90% confidence limits of a GSI value of 25 ± 5

(equivalent to a standard deviation of approximately 2.5). This represents the range of

values that an experienced geologist would assign to a rock mass described as

BLOCKY/DISTURBED or

DISINTEGRATED and

POOR. Typically, rocks such as flysch,

schist and some phyllites may fall within this range of rock mass descriptions.

25