A Method for Modeling Variation of In Situ Stress Related to Lithology 

J. K. Whyatt 

Spokane Research Laboratory, National Institute for Occupational Safety and Health, 

Spokane, Washington, USA 

ABSTRACT: Assuring ground control safety in many mining and tunneling projects depends, at least in part, on 

an understanding of in situ stress conditions that will be encountered. Yet it is rarely practical to conduct more 

than a very limited number of stress measurements. Stresses along the route of a proposed excavation are 

typically interpolated from available measurements, often assuming a linear variation of stress with depth (or 

elevation). However, projects where multiple stress measurements are conducted often report more complex 

variations of in situ stress, usually in apparent relationship to geologic structures. These structures often include 

lithologies of contrasting elastic properties. A method is proposed for estimating stresses in these cases by first 

back-calculating regional loads from available stress measurements and then modeling the distribution of stress 

throughout the rock mass. The method has been successfully applied to a set of in situ stress measurements from 

the Coeur d’Alene Mining District of northern Idaho, USA. Results provided new insights into district stress 

conditions and the distribution of rockburst hazards along mine drifts and between various mines. This success 

should transfer readily to suitable deep tunneling projects. 

1. INTRODUCTION 

The safety of miners is too often compromised by failures of ground through any of a number of mechanisms, most of which are influenced by the state of in 

situ stress. Thus, a better understanding of how in situ 

stress varies and how these variations control the 

location and severity of hazards was sought as part of 

a research program conducted by the Office for Mine 

Safety and Health of the National Institute for Occupational Safety and Health (NIOSH). The proposed 

method for modeling in situ stress variation described 

in this paper is one product of this program. 

More specifically, the proposed method was 

developed to explore the hypothesis that in situ stress 

variations have had a significant effect on the spatial 

distribution of rockburst hazards encountered during 

driving of development openings in the Coeur d’Alene 

Mining District of northern Idaho. This hypothesis was 

suggested by diverse results from stress measurements 

conducted in the district (Whyatt et al., 1995). It was 

supported by a 3-year study of ramp development 

where rockburst hazards were found to be concentrated in a number of “pockets” that constituted a small 

portion of the ramp system (Whyatt & White, 1998) 

and a case study of a rockburst fatality (Whyatt et al., 

2000). 

The proposed in situ stress modeling method is 

predicated on two assumptions. The first assumption 

is that stress variations within a region of interest arise 

primarily out of contrasts in rock mass properties 

(particularly elastic properties). Stiff portions of the 

rock mass are characterized as isolated inclusions 

within a softer rock mass, a characterization that could 

be extended to stratigraphic geometries. The second 

assumption is that the load path, however complex, 

does not vary by location within the region of interest. 

Since the method does not require definition of a load 

path, it is particularly well-suited for regions with 

complex tectonic histories. Other methods might be 

preferred where the load path is simple (e.g. Martin, 

1990; Konietzky & Marschall, 1996; Homand & 

Souley, 1997) and where discontinuities play an 

important role. 

The paper begins by providing a roadmap to the 

method and applying it to the simple case of an elliptical elastic inclusion in an initially unstressed body. 

The following sections extend the method for application to a rock mass, beginning with refined 

definitions of the relationship between various types of 

load sources and in situ stresses. These definitions 

assure unambiguous modeling of stress distributions 

induced by regional loads. The paper then presents a 

method for generating residual stress fields by relaxation of initial internal stresses, which is used to linearize generation of residual stress field estimates. These 
concepts are then incorporated into a fitting procedure 

that seeks to define a set of uniform regional loads. 

These loads are used to estimate stress conditions 

throughout the rock mass. Experience gained in 

applying this method to stress conditions in the Coeur 

d’Alene district is reviewed. Finally, factors affecting 

the potential usefulness of the method for mining and 

tunneling projects are discussed. 

2. METHODOVERVIEW AND APPLICATIONTO 

AN ELLIPTICAL INCLUSION 

The method proposed for projecting stress along the 

course of an excavation in a naturally variable stress 

field can be illustrated in the simple problem of a 

tunnel passing through an elliptical inclusion, perhaps 

an intrusive stock (Figure 1A). In this example, it is 

assumed that horizontal stress is primarily a function 

of regional loading and there is no residual stress. To 

further simplify the problem, it is assumed that regional loads are aligned with ellipse axes. 

Clearly, the in situ stress field in the vicinity of the 

inclusion will be nonuniform and will depend on the 

the geometry and relative stiffness of the inclusion. 

The relationship between regional loads applied along 

ellipse axes and induced stresses, illustrated in Figure 

2, has been solved exactly (Donnell, 1941). Superposing solutions for each direction of loading provides 

the biaxial solution. This relationship can be used to 

first back-calculate regional loads from the measured 

stresses (Figure 1B), and then carry out the forward 

calculation to determine stresses along the tunnel route 

(Figure 1C). 

Figure 1.—Outline of modeling method. Stressmeasurements (A) 

are an-alyzed in light of lithologic inclusions to find regional loads 

(B), Loads are thus applied to the model to determine stresses in 

the region of interest, in this case, the course of a planned tunnel 

or mine drift (C). 

The robustness, or insensitivity to error, of this 

example benefits from some characteristics of the 

elliptical inclusion solution. For instance, it is sometimes difficult to know the exact elongation of a buried 

inclusion. However, the degree of stress concentration 

in the inclusion increases asymptotically as the ellipse 

is elongated (Figure 2). Thus, errors in estimating the 

degree of elongation become insignificant with elongation. Also, the elliptical problem solution (and its 

geometric limits) has the unique property that stress is 

constant throughout the inclusion (Sendeckyj, 1970). 

Thus, the degree of stress concentration at a measurement site within the inclusion is sensitive to location 

error. This is particularly convenient in cases where 

measurements are located within the hardest available 

rock. This is the case with many overcore measurements, which are best suited for locations with good 

core recovery and linear rock deformation. Of course, 

real geologic structures depart from an ideal ellipse. 

However, these convenient properties should persist to 

the extent that real geologic structures approximate an 

elliptical geometry (and its geometric limits). 

3. SOURCES OF LOAD 

Real geologic settings are more complicated than the 

previous example in both geometry and load path. 

Definition of a precise and direct link between particular load sources and stresses induced within and 

around an inclusion is desirable for unambiguous 

modeling. To this end, the following definitions were 

developed. These definitions are extended and revised 

versions of definitions proposed by Hyett et al. (1986). 

Residual stress: State of stress within an isolated 

rock mass at a uniform temperature of 25' C (77' F) 

that is free from all external tractions, body forces, and 

other load sources (i.e. gravitational, tectonic, thermal, 

and physico-chemical loads). 

Gravitational stress: Reversible change in the state 

of stress caused by gravitational body forces throughout the rock mass while rock mass boundaries are 

maintained as lines of symmetry. A uniaxial strain 

model of gravitational loading is assumed [oh=(1/1-uv) 

ov]. 

Tectonic stress: Reversible change in the stress 

state caused by application of tractions to rock mass 
boundaries. Does not include tractions induced byrestraint of boundary movement in reaction to application of other active load sources. Does include tractions induced by restraint of boundary movement in 

reaction to irreversible consequences of loading. 

Physico-chemical stress: Reversible change in the 

stress state caused by chemical and/or physical 

changes in the rock (e.g. recrystallization, absorption 

of water, and fluctuation of groundwater levels) while 

rock mass boundaries are maintained as lines of symmetry. 

Thermal stress: Reversible change in the state of 

stress caused by variation of rock temperature from a 

uniform temperature of 25' C (77' F) while rock mass 

boundaries are maintained as lines of symmetry. 

Figure 2.—Stress-concentrating effect of elliptical inclusion (plane stress). 

In developing these definitions, a uniaxial strain 

model of gravitational loading was specified. However, it is not uncommon for regions relatively untouched by tectonic activity to develop horizontal 

stresses in excess of this model. This might occur for 

one of two reasons. First, it may be that an alternative 

model of reversible generation of horizontal stress, 

such as a global model of gravitational loading (e.g. 

Sheorey, 1994), is simply superior to the uniaxial 

strain model. In this case, it might be useful to apply 

that specific model. 

Second, any tendency toward viscous deformation 

in such a rock mass will tend to increase the value of 

horizontal stress (i.e. the apparent Poisson’s ratio in 

the uniaxial strain model). Such deformations are 

driven by a reduction in the total potential energy of 

the rock mass. This reduction results from the combination of an increase in stored strain energy due to 

increased horizontal stress with a greater loss in gravitational potential energy (i.e. the rock mass slumps or 

settles during this viscous deformation). In a rock mass 

with little or no long-term shear strength, the stress 

state will trend toward a lithostatic distribution 

(horizontal stress equal to the vertical gravitational 

stress in a homogeneous rock mass) over time. 

However, by the definitions introduced here, 

gravitational stresses are only those that will disappear 

when gravitational loading is removed. Thus, the 

apparent increase in Poisson’s ratio is just that– 

apparent–and the change in stress must involve other 

load sources that become apparent after gravitational 

loading is removed. In this case, the remaining stress 

is removed in two parts. First, remaining tractions on 

rock mass boundaries are removed as tectonic loads. 

This removal is consistent with the definition of tectonic loading as tractions applied to rock mass boundaries. Second, the remaining internal stress distribution is removed as a residual stress state, leaving a 

null stress state. 

These definitions also consider scale, since models 

necessarily address load sources and stress variations 

in the context of their interaction with an engineering 

project. As a practical matter, then, load sources must 

be defined relative to this scale and are seen as acting 

either as tractions on the boundaries of this rock mass, 

or internally as body forces. Since residual stress is 

defined as “what’s left” within this rock mass after 

loads are removed, residual stress systems larger than 

the rock mass in question will contribute to the 

tectonic load component. Similarly, the influence of 

any kind of loading applied outside the rock mass of 

interest will be applied as a tectonic load. 

Thus, some rules for behavior of rock mass 

boundaries are also required. These are— 

• Boundaries are lines of symmetry for application of 

load sources within the rock mass. For example, 

heating of the rock mass will be mirrored in neighboring portions of the crust so that rock mass boundaries will not be displaced. 

• Changes in boundary tractions caused by application of loads within neighboring sections of the crust 
will be passed through rock mass boundaries as tectonic loads. For example, heating of a neighboring 

portion of the crust by intrusion of a batholith will 

exert tractions on the rock mass that are functionally 

equivalent to tectonic boundary tractions. 

• Changes in boundary tractions caused by irreversible processes (i.e. that are not removed with removal 

of load sources) are, in essence, part of a residual 

stress system that is larger than the rock mass of 

interest. They are treated as tectonic loads. 

These rules broaden the definition of tectonic 

loading and the tectonic stress field that results. The 

broadening arises from the fact that the exact source of 

boundary loads is not material to understanding the 

state of stress in the subject rock mass. 

4. REFERENCE STATE GENERATION OF A 

RESIDUAL STRESS FIELD 

The definitions developed for various load sources 

provide direct and linear methods for applying all but 

residual stress. All the nonlinearity associated with 

evolution of the in situ stress field is assigned to the 

residual stress state. Given the complex tectonic 

history of many regions, it is often difficult, if not 

impossible, to model generation of residual stresses 

accurately. The obvious alternative—directly mapping 

the stress field—is usually impractical. However, if a 

residual stress field were proposed, it would be 

possible to determine how closely it (in combination 

with stresses induced by other load sources) matched 

measured stresses. Then, if a number of alternatives 

were proposed, an error measure could be used to 

choose which provided the best representation of the 

in situ stress field. This is the approach that has been 

taken. 

This approach requires that a large number of 

alternative residual stress fields be generated that are 

relevant to the problem being considered. Furthermore, residual stress fields are much more likely to be 

useful if they are linked to the geometry of relevant 

geologic structures in a consistent and physically 

meaningful way. In other words, similar inclusions 

within a geologic setting should be associated with 

similar patterns of residual stress. 

The analysis would be greatly simplified if a linear 

procedure could be developed for generating these 

residual stress distributions, i.e. a method whereby a 

limited number of residual stress states could be 

superposed to fit a desired distribution of residual 

stress. In such a case, residual stress states could be 

included in a fitting routine on an equal footing with 

gravitational and tectonic loads. Such a method can be 

developed by simply assigning a uniform initial stress 

field to the region of interest and then allowing the 

model to adjust elastically in the absence of load 

sources. This procedure is illustrated with a simple 

example in Figure 3. 

Figure 3.—Tectonic (B) and residual (C) components of a uniform stress field in an elastic model of an elliptical soft inclusion (A). 

The essential point of this example is that a unique 

residual stress state (Figure 3B) is generated by 

removing the boundary load, Px, from a body with a 

uniform and equivalent initial internal stress field. The 

residual stress state is revealed as the model adjusts to 

the absence of loads and reaches equilibrium. This 

residual stress state can be quite complex, depending 

on inclusion geometry. Since the residual stress state 

can also be described by the original stress field that 

created it (Figure 3A), it may be more convenient to 

describe the residual stress state in terms of its 

generating initial internal stress field or “reference 

state.” The reference state for the residual stress 

distribution of Figure 3B is the constant initial internal 

stress field of Figure 3A. 

The reference state concept has a number of 

interesting advantages. First, a complex, complete and 

self-consistent residual stress field is specified by the 

very few parameters required to define a uniform 

initial stress state. Second, the linear, reversible relationship between a reference state and a related residual stress field is much easier to handle than the 

irreversible processes that create a residual stress field. 

Third, it provides a framework for studying how the 
residual stress field relates to the lithology and 

geometry of geologic structures. 

However, it is highly unlikely that accurate 

representations can be developed from reference states 

for all residual stress distributions that occur in nature. 

The hope is simply that residual stress fields can be 

generated that are sufficiently close to reality to be 

useful. When such a procedure is used in solving an 

inverse problem, statistical measures of the quality of 

fit should provide insight into whether reasonable 

residual stress fields are being generated. Poor approximations of significant residual stress fields will result 

in poor matches between model and measurement. 

In exploring the potential for reference state 

generation of residual stress fields, it is useful to 

consider some characteristics of the relationship 

between reference state and residual stress field. These 

include— 

Existence. All reference states will generate a 

particular residual stress state when load sources are 

removed. However, not all conceivable residual stress 

fields can be represented by a reference state. For 

example, a simple prestressed concrete beam can be 

imagined that contains several bars with various levels 

of prestress. This system will never reach a completely 

uniform stress state without inelastic deformation (i.e. 

changing the residual stress state). Residual stress 

states that cannot be attained exactly through relaxation of a reference state might be attainable through 

inelastic deformation induced by an applied load 

history or definition of a nonuniform initial stress 

state. 

Uniqueness. A reference state is not necessarily a 

unique generator of a residual stress field in a heterogeneous rock mass. In the trivial case of a homogeneous rock mass, every uniform initial stress field is a 

reference state. However, the number of alternative 

reference states would seem to diminish greatly with 

the addition of geologic complexity. 

Linearity. Since the reference state is defined in the 

context of elastic adjustment to the removal of all load 

sources, the relationship between reference state and 

residual stress field is linear. Small changes in 

reference state should, therefore, cause only small 

changes in the residual stress field. 

Finally, the type of reference state that fits a given 

rock mass might provide some insight into how 

residual stresses in a rock mass were generated. For 

instance, if— 

Low-modulus rocks are more highly stressed than 

high-modulus rocks, then residual stress more than 

nullifies the concentration of stress in high-modulus 

rocks that result from application of applied loads. The 

reference state is a higher state of stress than the 

applied loads. Thus, the model will expand against the 

applied loads, allowing high-modulus rocks to shed 

stress more quickly (i.e. with less deformation) than 

low-modulus rocks. One possible geologic interpretation of this reference state is formation of rock under 

high pressures followed by elastic relaxation during 

erosion and uplift in the absence of other sources of 

load. 

Rocks carry the same level of in situ stress regardless of elastic modulus, then residual stress exactly 

counteracts structural stress induced by loading. Thus, 

the current state of stress is the reference state. One 

possible geologic interpretation is that long-term viscous processes have eliminated or greatly reduced 

stress contrasts between rock types. Such viscous 

deformation would lead toward a lithostatic stress 

field. 

High-modulus rocks tend to be more highly 

stressed than low modulus rock, then residual stress 

incompletelycounteracts or reinforces structural stress 

patterns. Thus, the reference state will be less than the 

applied loads. If residual stresses actually reinforce 

structural stress patterns, the reference state loads act 

in the opposite sense from applied loads. A tensile 

reference state for a rock mass, while not intuitive, is 

appropriate and necessary for achieving a residual 

stress state that amplifies structural stresses developed 

by compressive loading. One possible geologic interpretation is that long-term viscous processes concentrated in low-modulus rocks shift loads from soft to 

hard rock. 

5. PROCEDURE FOR ESTIMATING LOAD 

SOURCES AND STRESS VARIATION 

This procedure consists of (1) discovering the set of 

load sources currently applied to a region of interest, 

and then (2) projecting the state of stress caused by 

these loads at similar sites throughout the region. That 

is, the inverse problem (estimating applied loads from 

measured stresses) is first solved, and then the forward 

problem (estimating stresses from applied loads) is 

solved. Both steps use models of individual inclusions 

within the region of interest to determine relationships 

between local stresses and components of various 

regional loads (including the regional reference state). 

If various inclusions are effectively isolated from 

each other within the region of interest, the modeling 

task can be broken into a number of simpler models of 

each inclusion. That is, if significant stress perturbations induced by the inclusions do not overlap, particularly at points of interest, there should be no 

difference between a regional model encompassing all 

inclusions and a limited number of smaller models 
constructed for the inclusions of interest. 

These models are then loaded by unit increments of 

each load source component. For a three-dimensional 

problem with gravitational, tectonic, and residual loading, models would be run with 10 different load components. One component is the acceleration of gravity 

(or density of rock), which covers gravitational stress. 

Three traction components are pertinent, all in the 

horizontal plane, which may be considered to vary 

linearly with depth. The final six are components of 

the reference stress field. These components may also 

be considered to vary linearly with depth. 

The linear relationship between unit increments of 

each load source component (including the reference 

state) and local stresses is the key to solving the inverse problem. In the forward problem, the principle 

of superposition can then be used to calculate the 

induced stress field within the model as a linear combination of stresses induced by these unit load components. In the backward problem, the load sources are 

fit to measured stresses. Fitting can be accomplished 

with any of a number of routines that reduce estimate 

error (e.g. a squared error measure1

)

1

Defined as the sum of squares of normal stress component error plus twice the sum of squares of shear component error. Double 

weighting preserves invariance with respect to coordinate system. 

 and the solver 

supplied in the Excel spreadsheet program. 

6. APPLICATION TO THE COEUR D’ALENE 

DISTRICT OF NORTHERN IDAHO 

The proposed modeling method has applied with good 

results in an analysis of in situ stress variation in the 

Coeur d’Alene Mining District of northern Idaho 

(Whyatt, 2000). This analysis sought to reconcile 

widely varying measurements of in situ stress (Table 

1) into a stress model that would be valid throughout 

the district and, hopefully, help explain observed 

spatial variations in the intensity of rockburst hazards. 

These measurements do not suggest a linear 

relationship with depth or elevation (e.g. Figure 4). 

Table 1.—Summary of in situ stress measurements, Coeur d’Alene Mining District, USA, megapascals 

Crescent Star Lucky Friday Sunshine Mine breakouts 

3300 level 7300 level 4250 level 5300 level 12 shaft 10 shaft 

Stress 0h1 ............ 54.0 66.7 89.0 113.4 

Bearing ............ .. N 20° W N 21° W N 38° W S 80° W N 80° W N 65° W 

Stress 0h2 ............ 43.3 49.8 49.0 73.1 

*Measured/estimated 0v 1.0 0.7 1.2 1.8 

*Measured vertical stress divided by an estimate of vertical stress based on depth of overburden. 

Figure 4.—Observed magnitude of vertical 

stress from four in situ stress measurements at 

various depths. Variability that can be attributed 

to topography is indicated by the shaded region. 

The regional geology of the Coeur d’Alene district 

is well suited to the assumptions of this method. Four 

mechanically significant rock types have been 

identified (Whyatt et al., 1996). Three of these occur 

in stratigraphic units while the fourth is characterized 

by silicification found in alteration halos emanating 

from quartz veins. The softest of these rock types, 

siltite-argillite, makes up over 80% of the accessible 

rock mass. Harder rock types (sericitic, vitreous, and 

silicified vitreous quartzite) are two to three times 

stiffer than siltite-argillite and are associated with 

economic portions of veins. In addition, these rock 

types typically provide better core recovery and are 

more isotropic than siltite-argillite rock. As such, they 

have been preferred host rocks for in situ stress measurements. 
Over the tectonic history of the district, which can 

be summarized as two periods of intense folding 

followed by three periods of intense faulting, harder 

rock strata have been deformed and faulted to form 

isolated inclusions of various shapes and sizes. The 

contemporary stress field is assumed to be controlled 

primarily by gravitational and tectonic loads, along 

clay-rich rock should behave over geologic time. 

Moreover, the method provided the best fit when 

intense residual stress fields were allowed to develop 

only at well-silicified sites. with a residual stress field. 

Application of this method to these stress 

measurements provided a much improved stress 

model. For example, the squared error measurement 

for load sources in the horizontal plane of the measurements was more than an order of magnitude less 

than linear models while successfully anticipating 

stress orientations at the two breakout sites (which 

were not used in the fitting procedure). The set of 

loads developed in this analysis (Table 2) generated a 

weakly biaxial in situ stress state, close to lithostatic, 

in the siltite-argillite rock mass far from inclusions 

(Table 3). This result is well in line with how this 

Table 2.—Inferred regional loads at a depth of 1500 m 

Tectonic strain: 

8h1 = 1654 microstrain 

Bearing = N 41° W 

8 = 1239 microstrain h2

Reference state in silicified rock: 

0 = 152 MPa (tension) h1

Bearing = N 67° W 

0 = 49.6 MPa (tension) h2

Table 3.—Estimated stress field in siltite-argillite rock far 

from quartzitic inclusions at a depth of 1500 m 

0 = 33.2 MPa h1

Bearing = N 41° W 

0 = 40.7 MPa h2

0 = 40.7 MPa v

The tensile reference state for residual stress 

reflects the fact that silicified rocks are more highly 

stressed (according to both measurements and rockburst experience) than their elastic modulus would 

imply. This result suggests that the observed increase 

in rockburst hazard at these sites is due to heightened 

stress levels as well as to the impressive strength and 

brittleness of this rock type. Moreover, it suggests that 

the silicification process and/or resulting alteration of 

rock properties is associated with development of the 

residual stress field. As such, these silicified zones 

may provide interesting sites for further exploration of 

residual stresses. 

Finally, this analysis shows that criteria for locating 

future stress measurement sites in the district must 

consider the potential for improving stress model 

accuracy as well as the potential for an accurate measurement. 

7. CONCLUSIONS AND DISCUSSION 

A method has been proposed for back-calculating 

regional load sources from a variety of stress measurements within a naturally varying in situ stress field. 

The method assumes that stress variation arises primarily from contrasts in rock properties, particularly 

elastic modulus, and that unusually soft or hard portions of the rock mass exist in isolated inclusions. In 

formulating this method, a number of definitions and 

boundary condition rules are proposed that clarify 

links between load sources and resulting reversible 

stresses while lumping all irreversible effects into the 

residual stress field. These definitions, and an approximate method of generating a residual stress field by 

relaxation of an initial uniform stress state (the reference state), allow back-calculation of loads with 

modest computational resources. 

The method has been applied to an analysis of 

stress variation in the Coeur d’Alene Mining District 

of northern Idaho. It successfully found a set of 

regional loading conditions consistent with a diverse 

set of stress measurements. These loads provided 

reasonable estimates of stress characteristics at other 

points. While this model is based on a sparse data set 

and is far from perfect, it does provide significant 

advantages over linear models of stress variation with 

depth and elevation. Most importantly, it provides new 

insight into the spatial distribution of rockburst 

hazards. 

On a more general level, these results suggest that 

the scatter evident in stresses measured in many 

regions is likely a real variation that is associated with 

geologic structures. Thus, consistency should be 

sought in load sources rather than in measured 

stresses. That is, stress estimates based on regional 

loads will often reflect stress variations that will be 

overlooked by direct extrapolation from available 

measurements. Proper investigation of load sources 

and modeling of associated stress variation should 

prove beneficial to most underground engineering 

projects sensitive to in situ stress conditions, but will 

be particularly well suited to tunnels and mine drifts 

extending through diverse geologic conditions. 

These results also suggest that the number and 
spatial distribution of stress measurements may often 

be more important than the absolute accuracy of the 

measurements. 

8. ACKNOWLEDGMENTS 

The author is indebted to Dr. Charles Fairhurst for his 

guidance in developing, formalizing and documenting 

the proposed method. Assistance from my colleagues 

at the Spokane Research Laboratory (NIOSH), particularly Brian White and Ted Williams, in applying this 

method to the Coeur d’Alene District, is also gratefully 

acknowledged. 

9. REFERENCES 

Donnell, L.H. 1941. Stress concentrations due to elliptical 

discontinuities in plates under edge forces. Contributions to 

Applied Mechanics and Related Subjects: Von Karman Anniversary Volume. Pasadena: California Institute of Technology, pp. 293-309. 

Homand, F. & Souley, M. 1997. Interpretation of stress measurements in a Provence mine using a block modelling. In 

Rock Stress. Proceedings of the International Symposium on 

Rock Stress, K. Sugawara and Y. Obara Kumamoto, eds. 

(Japan, Oct. 7-10, 1977). Rotterdam: Balkema, pp. 205-210. 

Hyett, A.J., Dyke, C.G. & Hudson, J.A. 1986. A critical examination of basic concepts associated with the existence and 

measurement of in situ stress. In Rock Stress and Rock Stress 

Measurements: Proceedings of the International Symposium 

on Rock Stress and Rock Stress Measurements, O. Stephansson, ed. (Stockholm, Sept. 1-3, 1986). Lulea, Sweden: Centek 

Publ., pp. 387-396. 

Konietzky, H. & Marschall, P. 1996. Excavations disturbed zone 

around tunnels in fractured rock—Example from the Grimsel 

Test Site. Geomechanics '96, Z. Rakowski, ed. Rotterdam: 

Balkema, pp. 235-240. 

Martin, C.D. 1990. Characterizing in situ stress domains at the 

AECL Underground Research Laboratory. Can. Geotech. J. 

27:631-646. 

Sendeckyj, G.P. 1970. Elastic inclusion problems in plane 

elastostatics. Intern. J. of Solid Structures, 6:1535-1543. 

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

transversely isotropic rock. Int. J. Rock Mech. Min. Sci. & 

Geomech. Abstr. 311:23-34. 

Whyatt, J.K. 2000. Influence of geologic structures on stress 

variation and the potential for rock-bursting in mines with 

particular reference to the Lucky Friday Mine, Idaho. Ph.D. 

dissertation. Minneapolis: University of Minnesota, 203 pp. 

Whyatt, J.K. & White, B.G. 1998. Rock bursting and seismicity 

during ramp development, Lucky Friday Mine, Mullan, 

Idaho. In Proceedings, 17th International Conference on 

Ground Control in Mining, Syd S. Peng, ed. (Morgantown, 

WV, Aug. 4-6, 1998.) Morgantown: Univ. of West Virginia, 

pp. 317-325. 

Whyatt, J.K., White, B.G. & Johnson, J.C. 1996. Strength and 

deformation properties of Belt strata, Coeur d’Alene Mining 

District, ID. U.S. Bur. Mines Report of Investigations 9619, 

65 pp. 

Whyatt, J.K., Williams, T.J. & Blake, W. 1995. In situ stress at the 

Lucky Friday Mine (In four parts): 4. Characterization of 

mine in situ stress field. U.S. Bur. Mines Report of Investigations 9582, 26 pp. 

Whyatt, J.K., Williams, T.J. & White, B.G. 2000. Ground 

conditions and the May 13, 1994, rock burst, Coeur d’Alene 

Mining District, northern Idaho. In Pacific Rocks 2000. Rock 

Around the Rim: Proceedings of the Fourth North American 

Rock Mechanics Symposium (NARMS 2000), J. Girard, M. 

Leibman, C. Breeds, and T. Doe, eds. (Seattle, WA, July 31­

Aug. 3, 2000). Rotterdam: Balkema, pp. 313-318.