A Solid-Inclusion 

Borehole Probe to 

Determine Three-fDimensional 

Stress Changes at ;a 

Point in a Rock Mass

GEOLOGICAL SURVEY BULLETIN 1258-C

A Solid-Inclusion 

Borehole Probe to 

Determine Three-Dimensional 

Stress Changes at a 

Point in a Rock Mass

By THOMAS C. NICHOLS, JR., JOHN F. ABEL, JR., and FITZHUGH T. LEE

CONTRIBUTIONS TO ENGINEERING GEOLOGY

GEOLOGICAL SURVEY BULLETIN 1258-C

Development and laboratory testing of a 

solid-inclusion borehole instrument and the 

mathematical bases for reduction of the 

three-dimensional data produced by the 

instrument

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 15 cents (paper cover)
CONTENTS

Page

Abstract______ _____________________________________________________ Cl

Introduction. ______________________________________________________ 1

Scope of investigation.______________________________________________ 4

Instrumentation-__ _ ________________________________________________ 4

Probe development.____________________________________________ 4

Monitoring system.____________________________________________ 8

Grout__ _ ______________________________________________________ 9

Stiffness. ______________________________________________ 10

Tensile strength.___________________________________________ 11

Elastic range._____________________________________________ 11

Creep properties__________________________________________ 11

Electrical insulation characteristics___________________________ 11

Testing______________________________________________________ 12

Hydrostatic program__________________________________________ 12

Elastic model program._________________________________________ 14

First test____________________________________________ 16

Second test-_______________________________________________ 16

Mathematical analysis of probe measurements__..______-________--___- 19

Computation of lateral stresses in second elastic model test._____________ 24

Field investigations.________________________________________________ 26

References cited..__________________________________________________ 27

ILLUSTRATIONS

Page 

FIGURE 1. Diagram showing position of strain-gage rosettes on steel ball__ C7

2. Photograph of encapsulated three-dimensional borehole probe. _ 8

3. Diagram of laboratory strain-monitoring system____________ 9

4. Photograph of glass vial molds with cured epoxy casts._______ 12

5. Photograph of block of cast acrylic showing three-dimensional 

borehole probe grouted into central core hole and strain 

gages on external surface.______________________________ 15

6. Diagram showing stresses determined from elastic model test

1. under a 200,000-pound load________________________ 17

7. Diagram showing stresses determined from elastic model test

2. under a 300,000-pound load-___---__----_------------ 18

8. Diagram showing surface stress on model test 2_____________ 20

in
IV

NOTE

In this report a theoretical cylinder was used for the elastic approximation of a spherical inclusion in an elastic epoxy host. The actual 

theoretical spherical approximation for a rigid sphere in a semi-infinite 

mass of low-modulus material produces a stress concentration factor of 

about 1.9 instead of the 1.5 factor for a rigid cylinder (Goodier, 1933, 

p. 40; Edwards, 1951, p. 26-27). The apparent cause of this higher stress 

concentration factor is the lack of as much adjacent rigid material to 

distribute the available stress.

It must be understood that no complete theoretical three-dimensional 

elastic analysis exists for the exact case of the probe in rock, because 

(1) the probe is surrounded by only one diameter of low-modulus material 

across the borehole and 6M diameters along the borehole and (2) no rock 

mass rigorously meets the requirements of isotropy and homogeneity 

required for the application of theoretical elasticity.

An approximate theoretical elastic solution is possible by combining 

the results of two plane strain analyses, one across the borehole and one 

along the borehole. This can be done, using the finite element technique.
CONTRIBUTIONS TO ENGINEERING GEOLOGY

A SOLID-INCLUSION BOREHOLE PROBE TO DETERMINE 

THREE-DIMENSIONAL STRESS CHANGES AT A POINT 

IN A ROCK MASS

By THOMAS C. NICHOLS, Jr., JOHN F. ABEL, Jr., and FITZHTJGH T. LEE

ABSTRACT

In order to relate changing stress conditions within rock masses to geologic 

factors, a welded high-modulus solid-inclusion borehole probe to determine threedimensional stress changes at a point in a rock mass was developed and tested.

The probe consists of a 1-inch-diameter chrome alloy steel ball, mounted with 

three strain-gage rosettes, and is encapsulated in a filled waterproof epoxy. 

The reliability of the probe is established by conducting a series of tests in which 

the probe was subjected to changing hydrostatic and triaxial stress fields. In all 

these tests the probe responded as predicted by elastic theory.

Because reduction of field measurements is difficult and time consuming by 

use of a desk calculator, computation of the three-dimensional stress tensor has 

been programed for computer analysis.

The probe, when adequately field tested, will be capable of resolving data that 

will aid in the understanding of anisotropic rock behavior associated with 

tunneling and other underground or near-surface engineering operations.

INTRODUCTION

The U.S. Geological Survey started research in July 1965 on the 

behavior of rock masses under changing stress conditions caused by 

the excavation of underground openings.

The natural stress field that exists at a point within a rock mass, 

prior to any excavation, is dependent on the depth below the surface 

and the physical characteristics of the medium. The physical characteristics of a rock mass are affected by all the geologic processes to 

which it has been subjected. Geologic processes may produce folding, 

fracturing, faulting, jointing, chemical alteration, recrystallization, 

foliation, lineation, and other rock anisotropies. Changes in the preexisting stress field at any point in a rock mass, which result from 

excavation, are controlled by the physical characteristics of the rock 

mass, the proximity of the point of measurement to the excavation, and 

the geometry of the excavation.

01
C2 CONTRIBUTIONS TO ENGINEERING GEOLOGY

Theoretical methods have been developed for estimating the stress 

distribution near different types of excavations within an idealized 

elastic body (Obert and others, 1960, p. 8-29). However, field investigations and measurements of rock strains have demonstrated that 

divergencies from elastic theory occur in even the most elastic rock 

masses (Obert, 1962, p. 8; Merrill and others, 1964, p. 32-37; Dare, 

1962, p. 16; Utter, 1962, p. 17-19; Hast, 1958, p. 74-79; Panek and 

Stock, 1964, p. 50-52; Wisecarver and others, 1964, p. 12-20 Morgan 

and others, 1965, p. 45-47; and Eyder and Officer, 1965, p. 182-188). 

This suggests that most rock masses are less competent, less homogeneous, and behave less elastically than the idealized models used in 

theoretical investigations, many of which are based on laboratory physical property measurements of small samples (Leeman, 1964, p. 82- 

114). It is reasonable to assume that the greater the deviation of a 

rock mass from a perfectly elastic material, the greater will be the 

deviation of its behavior from that predicted by elastic theory. The 

stress conditions that exist adjacent to an excavation within a fractured rock mass cannot be estimated with confidence by any existing 

technique. Consequently, most past measurements have been made 

under the most favorable geologic conditions in relatively homogeneous media such as massive granite, gneiss, rock salt, trona, magnetite, 

anhydrite, and thick-bedded sandstone.

The problem of measuring the actual stress changes that are induced in the rock surrounding an underground opening has intrigued 

and challenged many investigators. Efforts to make measurements of 

these stress changes in natural materials and on full-size underground 

structures have been largely unsuccessful until the last decade because 

of the lack of suitable instrumentation.

The availability of new instrumentation (Hast, 1958) has permitted 

the approximate determination of the rock stresses acting in the 

plane perpendicular to the axis of an instrumented borehole. Analysis 

of these stresses required the assumptions of (1) isotropy, (2) perfect elasticity, and (3) either plane stress or plane strain. Subsequent 

developments have made it possible to measure the degree of anisotropy 'and the variations of the elastic moduli of the rock at the 

point where the instrumentation has been applied (Fitzpatrick, 1962). 

Although the necessity for assuming plane stress or plane strain has 

not been eliminated-in any previous methods of instrumentation and 

interpretation, the theoretical works of Sechler (1952) and Panek 

(1966) have shown that these assumptions are not required; thus 

these assumptions will undoubtedly be eliminated in the future.

The mathematical analysis of stress changes induced around openings in theoretically elastic bodies has been accomplished (Goodier,
BOREHOLE PROBE, STRESS CHANGES, EN A ROCK MASS C3

1958) assuming either plane stress or plane strain. The planar analysis 

assumes that the opening can be represented by a slice (plane) of 

effectively infinite extent in two directions and of finite (negligible) 

extent in the direction of the long axis of the opening. The plane 

stress approximation assumes that the slice is subjected to no restraint 

in the direction of the opening axis when stresses are applied in the 

plane. The plane strain approximation assumes that all strain in the 

direction of the opening is prevented as stresses are applied in the 

plane. These analyses have provided the basic background for predicting stress levels which can be anticipated in rock surrounding 

underground structures.

Two possible methods of instrumentation to eliminate the necessity of assuming plane stress or plane strain in interpreting measurement data have been proposed (Panek, 1966; Leeman, 1964) : (1) 

Measure the extension of the borehole as well as the diametrical change 

resulting from three-dimensional stress changes, and (2) measure 

the diametrical borehole change iat the point under investigation in 

two or more boreholes with different bearings and inclinations. The 

first method was used in the present investigation because of the difficulty of accomplishing the second. Prior to construction of the 

probe described in this paper, instrumentation for the first method was 

not available. For this instrumentation two types of probes may be constructed one of low-elastic-modulus material and the other of highelastic-modulus material. Both types of probes may be used successfully to determine plane strain; however, because of the probability of 

the instrument being disturbed by shock waves, the low-modulus 

type is not practical for investigating stress changes around an active 

underground opening being excavated by explosives. Consequently, 

a high-modulus instrument grouted in a drill hole was developed for 

use in the present investigation. High-modulus instrumentation also 

eliminates the need for accurately determining the anisotropic variation of rock strength and modulus provided that the instrument 

modulus is approximately five times greater than the rock modulus 

(Leeman, 1964, p. 61) ; such a high-modulus instrument effectively 

acts as a rigid inclusion in the medium.

Stresses in an ordinary jointed or faulted rock mass are universally 

compressive within the earth. Tensile stresses cannot be transmitted 

across breaks in a rock mass, but relative tensile strains can be produced and measured. These strains result either from decreases in compression as the opening is approached or from Poisson strain in a 

plane normal to the direction of an increased compressive stress.

The main objective of the investigation was to use the high-modulus 

three-dimensional borehole probe to monitor the stress changes around 

an active underground opening during excavation. It was believed
C4 CONTRIBUTIONS TO ENGINEERING GEOLOGY

that ,by use of these measurements, together with geophysical and conventional rock mechanics measurements, relationships could be established that would provide a basis for making accurate predictions of the behavior of a rock mass during tunnel construction. With 

such predictions, more efficient and economical tunnel support (temporary and permanent) could be installed as a tunnel was being 

driven.

Although the following discussion is concerned primarily with a 

solid-inclusion probe for use in rock, a similar probe for installation 

in soil and intensely altered rock masses (for example a fault gouge) 

is currently being developed in the laboratory. It is anticipated that 

such a probe will be valuable for determining the three-dimensional 

stress field changes associated with landslides, building foundation 

materials, earthfill dams, and many engineering-construction projects.

SCOPE OF INVESTIGATION

The long-range objectives of this investigation are (1) to determine 

the three-dimensional changes in the stress field that occur in a rock 

mass when a part of that rock mass is excavated and (2) to determine 

how geologic conditions alter or influence these stress changes. The 

initial laboratory phases of probe development, calibration, and model 

studies have been completed and are reported here. Preliminary measurements of the three-dimensional stress changes are presently being 

made in a rock mass close to an active underground excavation. These 

measurements permit the determination of stress changes around the 

excavation. The magnitude, direction, and location of stress concentrations resulting from the excavation are being obtained. This information by comparison makes it possible to determine hbw the theoretical elastic stress pattern adjacent to the excavation has been altered by 

the geologic conditions at the test location.

The probe described here can be employed over a wide range of 

geologic conditions, grossly unsuited to existing low-modulus stress 

determination instrumentation, because it is designed so that measurements can be made in the presence of joints, foliation or bedding, and 

lithology anisotropies that weaken a rock mass. The probe is designed to average out the variations in rock mass characteristics, competency, and anisotropy.

INSTRUMENTATION

PROBE DEVELOPMENT

The first step in developing the high-modulus three-dimensional 

borehole probe was to review the results of related studies made by 

previous investigators.
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C5

A search of the literature revealed that a probe having the desired 

capabilities had not been constructed previously. According to Leeman 

(1964, p. 54-55), a probe capable of monitoring strain for determination of stress changes within a material would have the following 

characteristics. It would consist of an effectively rigid, ideally elastic 

material. The probe would have strain-sensing devices oriented so as to 

obtain complete information on the magnitudes and directions of the 

principal stresses acting on it. The probe would be welded as an inclusion into the host material so that any changes of stress in the host 

material would be accompanied by proportional stress changes acting 

in the same direction in the probe. The strain changes induced in the 

probe should be large enough to be measured accurately. If the modulus 

of the probe material is approximately five times that of the host material, stress changes in the probe can be expected to be as much as 2 times 

those in the host material.

The geometry of the probe and the boundary conditions that affect 

the stress distributions were important factors that had to be considered in the development of the borehole probe.

Two geometric forms, the cube and sphere, were thought to be the 

best possible choices for the strain-sensing elements. The cube was 

considered because orthogonal planes necessary for determining a 

three-dimensional stress field (Leeman, 1964, p. 49) are defined by the 

sides of the cube and because strain instrumentation of these sides is 

very easily achieved. However, anomalous stress distributions located 

at the edges and corners of the cube, as well as possible distortion of 

transmitted stresses across the edge boundaries, discouraged its use. 

The sphere does not have orthogonal planes on which to mount strainsensing devices, but it has the advantage that stress distribution is 

relatively uniform and free of directional distortion. The stress level 

in the spherical probe would be greater than that in the host medium 

because of the difference in stiffness (modulus) between the two materials. As previously stated, small changes in the modulus of the host 

medium are of little consequence provided the probe modulus remains 

at least five times greater than the modulus of the rock mass.

For a high-modulus inclusion, welded to the host material, the strain 

in a given direction at the center of the sphere is equal to the strain 

measured in the same direction at a point on the surface of the sphere 

(Edwards, 1951, p. 26). Therefore, strain-measuring devices (straingage rosettes) can be placed on a sphere in such a manner that the 

strain components in orthogonal planes can be determined.

An error is introduced because the strain gages occupy a finite area 

on the surface of the sphere, instead of a point as postulated by the 

theory. However, the magnitude of this error is less than 1 percent for

306-521 O 68
C6 CONTRIBUTIONS TO ENGINEERING GEOLOGY

the 1-inch ball and %-inch strain gages. This small error is eliminated 

in the hydrostatic calibration step.

In order to resolve the magnitude and direction of the stress changes 

in the three-dimensional stress field at a point within a material, it is 

necessary to know six components of the stress tensor (Leeman, 1964, 

p. 48). Of these components, three are the normal stresses acting perpendicular to mutually orthogonal planes and the other three are the 

shear stresses acting on the same orthogonal planes. Thus strain-sensing devices must be placed on the probe so that these six components of 

stress can be calculated from strain measurements. The strain of the 

sphere can then be used to calculate the stress in the adjacent rock.

As a result of the previous consideration it was decided to design 

and construct a spherical probe with three strain-gage rosettes 

mounted orthogonally on the surface of the sphere. The three straingage rosette sensors represent planes tangent to the probe and are normal to a set of three orthogonal axes. These normals have been given 

the designation of a?, y, and 0, and are called the reference axes. The z 

axis is coincident with the borehole axis and is normal to the plane tangent to rosette 3, the a? axis is parallel to the normal of the plane tangent to rosette 2, and the y axis is parallel to the normal of the plane 

tangent to rosette 1 (fig. 1). Because of this particular orientation 

of the rosettes, two strain gages on each rosette are parallel to two of 

the reference axes.

One rosette could have been eliminated if strain measurements had 

been made by two nonorthogonal rosettes placed on the probe. The 

theoretical basis for this simplified technique is presented in detail by 

Panek (1966). Difficulties encountered in the field application of this 

technique (Wentworth, 1966) and the lack of redundancy of strain 

measurements used to check the normally directed strains induced the 

present authors to employ the three-rosette system.

The detailed mathematical analysis begins on page C19.

For the core of the probe, a 1-inch-diameter chrome alloy steel ball 

bearing (sphere) was used. These balls have a roundness tolerance 

of less than 0.0005 inch and an elastic modulus of 30.5 X106 psi (pounds 

per square inch). The balls were first etched in a dilute solution of 

nitric acid to prepare the surface for epoxy bonding agents and also 

to check the surface for structural homogeneity. Then, the balls were 

cleaned with acetone to remove grease and foreign substances which 

could hinder bonding action of epoxies. Strain-gage rosettes consisting 

of three strain gages, 45° apart, were applied to the balls with an epoxy 

adhesive. Three rosettes were applied to each ball with the centers 

of the rosettes 90° apart as shown in figure 1. Electrical connections 

to the gages were made with 10-inch lengths of enamel-insulated copper wires. A oast acrylic rod approximately 8 inches long, used as a
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C7

Steel ball

y

Acrylic rod

FIGURE 1. Position of strain-gage rosettes on steel ball.

handle, was cemented to the ball with epoxy. Then the entire ball, with 

mounted gages and extending wires and handle, was sealed in a cylindrical capsule of epoxy grout approximately iy2 inches in diameter 

and 1% inches high (fig. 2). The probe was then ready for testing.
C8 CONTRIBUTIONS TO ENGINEERING GEOLOGY

"Enamel-iniulatecJ 

copper wires

'Epoxy 

encapsulation 1.5 INCHES

FIGUBE 2. Encapsulated three-dimensional borehole probe.

MONITORING SYSTEM

For testing in the laboratory, it was necessary to develop a monitoring system that would simultaneously and continuously measure the 

responses of as many as 27 strain gages. Several racks of strain-gage 

amplifiers, power supplies, and two-axis (a?-y) recorders were used 

to provide a multichannel monitoring system that gave automatic 

recording of strain responses within the specimens being tested. The 

essential elements of this system are shown in figure 3.

For field tests, an ordinary battery-operated strain indicator, which 

is portable and easy to use and avoids the common problem of fluctuat-
BOREHOLE PROBE, STREiSS CHANGES, IN A ROCK MASS C9

Power 

supply

Active gage

J--Dummy gage

Wheatstone-bridge circuit

Amplifier

Signal on x oryaxis-

(Gain = 200)

FIGURE 3. Diagram of laboratory strain-monitoring system.

ing 110-volt alternating-current power in tunnels, was selected. The 

disadvantages of using this type of instrument are:

1. Only single readings can be taken, thus a switching device must be 

used to take all the readings at one station.

2. Because continuous readings cannot be taken, some stress anomalies 

may be missed if they do not exist at the time of scheduled 

readings.

Even with these readout disadvantages, enough information is being 

obtained to evaluate the effectiveness of the solid inclusion probe as 

a stress monitoring device.

GROUT

One of the practical problems which had to be solved as part of 

the development of the three-dimensional borehole probe was the formulation of a satisfactory grout. It was considered desirable, in order 

to minimize boundary effect errors, to use the same type of grout to 

encapsulate the probe and to couple the capsule to the host medium. 

Consequently, the following requirements were established: 

1. The grout must act either as a rigid inclusion or as a low-modulus 

inclusion when placed in the drill hole.
CIO CONTRIBUTIONS TO ENGINEERING GEOLOGY

2. The grout must form a high tensile strength bond between itself, 

the rock mass, and the probe.

3. The grout must behave elastically over the range of anticipated 

stress changes.

4. The grout must be virtually free of creep over the anticipated range 

of loading.

5. The grout must be waterproof in order to achieve electrical insulation of the probe.

6. The grout must exhibit a minimum amount of volume shrinkage 

during the curing period.

STIFFNESS

In order for the grout plug to act as a rigid inclusion between the 

probe and the rock mass it must be characterized by a stiffness (rigidity) of about five times the rock mass stiffness. The stiffness determined for rock specimens taken from various field test sites reached a 

level of strain of 6 psi per juin. per in. (pounds per square inch per 

microinch per inch). Field measurements from one of the test sites 

indicate that the stiffness of the rock mass does not exceed 2 psi per 

juin. per in. and generally is much lower because of geologic weakness 

of the rock mass not present in prepared laboratory specimens. Therefore, a stiffness on the order of 10 psi per /uin. per in. was considered 

adequate for the grout.

Epoxy grouts of various formulations were tested in an extensive 

laboratory testing program. The stiffnesses of the unfilled (pure) 

epoxy materials tested were in the range of 0.3 to 0.4 psi per juin. per 

in. The addition of steel, carborundum, and fused alumina fillers to 

the epoxy generally resulted in increased stiffness. The maximum 

stiffness obtained was 1.5 psi per /uin. per in. It was concluded that 

a grout plug sufficiently rigid to satisfy the requirements of a highmodulus bond could not be obtained by use of a pure epoxy or an 

epoxy-base grout. A filled epoxy grout, however, would satisfy the 

requirements of a low-modulus bond between the probe and the rock 

mass.

A requirement for a low-modulus grout is that the approximate 

stiffness of the grout and of the rock mass be known. This requirement can be met by either filled or unfilled epoxy materials.

Other observations made during the laboratory grout-testing program showed that:

1. Excessive air bubbles that were entrapped by the steel-filled epoxy 

decreased the stiffness.

2. Angular particles (crushed carborundum) resulted in higher stiffness and viscosity than rounded particles (fused alumina).
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS Cll

3. Grout viscosity decreased as the particle-size range of the fillers 

was successively restricted.

4. Introduction of only 3 percent minus 325-mesh particles drastically

reduced the grout stiffness.

The actual selection of grout formulation should be made only 

after the conditions of emplacement have been completely evaluated. In any case, the selection of an optimum grout formulation 

would require the maximum stiffness attainable.

TENSILE STRENGTH

The tensile bond formed by the various epoxy grouts exceeded the 

tensile strength of the glass in the glass vials used to form the test 

specimens (fig. 4). This was true even when the surface of the vial 

was purposely dirtied in an effort to determine the influence of debris 

on the rock surface of a borehole. The epoxy appears to have the 

ability to absorb rock particles and to penetrate thin films of grease 

and oil.

The laboratory study included testing of grout under wet conditions as well as dry conditions. The field conditions will undoubtedly 

be damp, if not wet; therefore an acceptable grout must function properly under water.

ELASTIC RANGE

The elastic response of the various grouts was tested in the range 

of compression 1,500 to 7,000 psi. A rigid grout must be capable of 

transmitting the anticipated field stress changes of as much as 4,000 

psi and strains of 2,000 juin. per in. (microinches per inch). A lowmodulus grout, on the other hand, must be capable of transmitting 

strains of 2,000 /mi. per in. The laboratory grout program has demonstrated that the requirements for a low-modulus grout, which we 

anticipate employing in the field investigations, are not difficult to 

achieve.

CREEP PROPERTIES

The grout presently being used has been tested for a period of 6 

months under a constant static load and without a load. Temperatures 

have been allowed to vary as much as 7e F. Over this period there has 

been virtually no creep caused by loading. The only changes in strain 

which were recorded were in response to temperature changes and were 

attributed to thermal expansion and contraction.

ELECTRICAL INSULATION CHARACTERISTICS

The various epoxies with no fillers all had good insulating characteristics. However, the epoxy grouts with metal fillers lost most of 

their insulating characteristic when highly stressed. This indicates
C12 CONTRIBUTIONS TO ENGINEERING GEOLOGY

Via! A Vial B Vial C VlalO

FIGURE 4. Glass vial molds with cured epoxy casts. Molds are % inch in 

diameter and 2 inches long. Vials A and D: Epoxy containing no filler. Vials 

B and C: Epoxy containing carborundum filler. No attempt was made to 

remove the glass from vials A and C, and curing stresses caused extensive 

fracture of vial A. An attempt was made to remove the glass from vials B 

and D, and the effect of high bond strength is apparent.

that the filling aggregate develops increased grout stiffness by making 

physical contact between the filling particles. This property tends to 

rule out the use of an epoxy with metal filler in contact with the 

probe. The use of strain gages requires a minimum of 50-megohm 

resistance to ground. The resistance to ground of partly-metal-filled 

grouts in contact with the probe was only about 100 ohms when stressed 

close to failure. The actual resistance depends on both the formulation 

of the epoxy and the stress level. The epoxy formulation with carborundum filler and fused silica filler developed resistance well above 

the 200-megohm level and maintained it under stress.

TESTING

HYDROSTATIC PROGRAM

In order to adequately calibrate the probe, encapsulated as shown 

in figure 2, and to determine if it was behaving in an essentially 

elastic manner, it was first tested in a hydrostatic stress field under 

varying stress increments and durations.
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS CIS

Each probe was placed in a triaxial chamber capable of withstanding 

pressures of 8,000 psi. The probe was then subjected to the following 

series of hydrostatic pressure tests.

Initially the pressure was increased in increments of 500 psi up to 

2,500 psi. The 2,500-psi stress level was chosen as a maximum test 

pressure because it seemed doubtful that changes of stress more than 

2,500-psi compression would occur within the rock at any of the 

potential field test sites. The pressure was then dropped back to 0 psi 

in one increment.

The resulting strains of pairs of the sensing elements of the straingage rosettes were automatically plotted on cc-y recorders. Thus, the 

line plot on each x-y recorder represented the strain of two strain 

gages plotted against each other. If the probe was behaving in an 

elastic manner and if each strain gage was perfectly applied to the 

sensing sphere, a straight 45° line would be plotted. The x-y plots of 

the gages on all the probes tested so far have been straight-line plots 

having slopes very close to 45°. Slight deviations from 45° probably 

are caused by the strain gages not conforming precisely to the curvature of the ball, thus a small error is introduced in the strain response. 

Since the stress-strain ratio of the probe is known, it is a simple matter 

to correct for this error by calibrating each gage in a hydrostatic stress 

field.

Another test was performed on one probe to test the ability of the 

probe to accurately measure stress changes over a long period of time. 

The probe was placed in the triaxial chamber and the stress was 

raised to 2,000 psi by 500 psi increments. The probe was then allowed 

to remain under this stress for 2 weeks. Once each day during this 

period the stress was reduced to 0 psi and then returned to the upper 

limit to see if the gage calibrations remained constant and if the x-y 

plots were repeatable. Over the entire 2-week period the probe behaved 

in a uniform and repeatable manner.

A third hydrostatic test was performed on one probe to find the 

approximate stress at which the epoxy encapsulation would fail. The 

probe was placed in the triaxial cell and subjected to pressure in 

increments of 500 psi until failure occurred at 4,500 psi. Failure was 

presumably caused by high stress concentrations on the surface of 

the probe. After failure, the probe continued to record stress changes 

but the x-y plots were no longer linear in any range of stress application. Thus the probe is of no apparent value if the encapsulation 

material fails.

After the favorable performance of the probe in a hydrostatic stress 

field, the next step was to test it in a uniaxial or triaxial stress field 

to further determine its elastic behavior. This was done by grouting

306-521 O 68 3
C14 CONTRIBUTIONS TO ENGINEERING GEOLOGY

it in a cast acrylic block and loading the block as described in the next 

section.

ELASTIC MODEL PROGRAM

The three-dimensional solid-inclusion borehole probe is in the process of being tested in laboratory models. The initial laboratory model 

study has been completed and has proved the feasibility of the probe. 

Measured strain changes in a virtually elastic model have been 

converted to stress changes.

The laboratory elastic model testing program is partly complete. 

Two tests have been conducted in which a probe (shown in fig. 2) was 

employed to determine the stress changes occurring within the cast 

acrylic model (fig. 5). These measurements were reduced by the mathematical method described on page C19. The results were compared 

with the stress applied, and the stress changes were predicted by 

elastic theory (p. C24). The comparison indicated good agreement 

between the stress calculated from measurements and the stress 

predicted theoretically.

The elastic model was constructed from a single block of cast 

acrylic 19 inches high, 13 inches wide, and 12 inches deep. A 2-inchdiameter borehole was drilled 6 inches into the block and the probe was 

grouted into the hole with a nonfilled epoxy for the first test and with 

a filled epoxy for the second test (fig. 5). The minimum distance 

between the surface of the probe and the free-surface boundaries of the 

block was 8y2 probe diameters. These conditions resulted in only 

partial confinement of the probe by the surrounding acrylic.

Two difficulties were encountered in the first elastic model test:

1. Bubbles as much as 14 millimeters in diameter developed between 

the drill-hole walls and the epoxy plug as the result of skrinkage 

of the epoxy during curing. This difficulty was eliminated in 

the second elastic model by the use of a filled epoxy.

2. Air was entrapped within the second of two layers of epoxy used to 

cement the probe in the acrylic block. This condition was attributed to the heat generated by a poorly controlled epoxy formulation. This problem was solved by the elimination of the double 

pour and by using a slower setting epoxy in the second model test. 

The elastic model was placed in a 400,000-pound press and loaded 

axially to 200,000 pounds in the first test and 300,000 pounds in the 

second test to determine what stress changes occurred in the probe as 

a result of the loading. One surface of the model was instrumented 

with strain-gage rosettes for the second test (fig. 5) in order to obtain 

control information for interpretation of test results.

The elastic model tests have shown close overall agreement with 

elastic theory. The differences measured have been well within the
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C15

FIGURE 5. Block of cast acrylic showing three-dimensional borehole probe 

grouted into central core hole. A-E, strain gages on external surface.
C16 CONTRIBUTIONS TO ENGINEERING GEOLOGY

variations anticipated as the result of the acrylic block not behaving 

as a perfectly elastic body. Tensile strains as the result of the Poisson 

effect, a term used to describe the lateral deformation resulting from 

longitudinal stress (Coates, 1965, p. B14; Seldenrath and Gramberg, 

1958, p. 80-91; and Jenkins, 1958, p. 102), were present in the proper 

direction and magnitude as theoretically predicted with respect to the 

physical constants for the acrylic and the applied loading. The principal stress changes were calculated for the two completed tests. The 

measured surface compressive stress pattern and stress trajectories 

were then plotted (fig. 8).

FIRST TEST

The results of the first elastic model test indicated that the probe 

was situated in a triaxial stress field (fig. 6). This stress field is highly 

directional, and the maximum principal compressive stress change is 

almost parallel to the applied compressive load. This compressive stress 

resulted in a lateral strain in all directions in planes normal to the 

loading direction. Having measured Young's modulus and Poisson's 

ratio for the acrylic and the epoxy (which were almost identical) 

and knowing that no external restraint was applied to the model, we 

were able to predict the tensile stresses which should have been obtained from probe measurements. The lateral stress obtained from 

probe measurements should have been about 0.39 times the maximum 

principal compressive stress change for the finite but unrestrained 

model employed. Because of the bubbles which formed along parts of 

the drill hole, the lateral stresses obtained from the probe measurements were not equal along and across the drill hole. However, the 

average value of these stresses indicated a lateral tensile stress of 

0.35 times the maximum principal compressive stress change.

The fact that the magnitude of the maximum principal compressive 

stress change obtained from the probe measurements was twice the 

applied compressive stress is attributed to the confinement of the 

probe in the model. It should be remembered that uniaxial compression does not imply uniform compression. This aspect of the model 

and probe response was investigated as part of the second elastic model 

test.

SECOND TEST

In the second elastic model test the probe was grouted in place with 

a metal-filled epoxy having a static Young's modulus of 1.55 X106 psi 

compared to the acrylic static modulus of 0.327 X106 psi. The probe 

was virtually confined on three sides by acrylic and one one side by 

the metal-filled epoxy. The principal stress changes which were determined are shown in figure 7. This stress field was also highly directional, and the maximum principal compressive stress change alined
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C17

Vertical

North

Acrylic block

x -axis bears S. 45° E.

and plunges 10° 

j/-axis bears S. 45° W.

and plunges 80° 

z-axis bears S. 45° W.

and is horizontal

West

STRESSES ACTING ON PROBE

Principal 

stress

°l (maximum) 

°2 (intermediate) 

°3 (minimum)

Magnitude (psi) 

( + ) compres,sion 

( ) tension

 + 2640 

- 113 

-1728

Bearing

S. 11° W. 

S. 64° E. 

S. 25° W.

Plunge

86° 

1° 

4°

FIGURE 6. Stresses determined from elastic model test 1, under a 200,000-pound 

load. Vertical planes are stippled.

itself with the applied load. The lateral stresses obtained from probe 

measurements were influenced by the physical properties of the acrylic 

block and the filled epoxy grout. Because the grout had a greater stiffness than the. acrylic, one of the remaining principal stresses was alined 

parallel to the drill hole and the other was alined normal to the drill 

hole. Based on the directional variations of stiffness, theoretical stress 

calculations were made to check the validity of the lateral stresses 

obtained from probe measurements (p. 024). The agreement of the 

theoretically computed stresses and the probe stresses was within 7 

percent.

Muskhelishvilis' mathematical plane stress or plane strain analysis 

of the influence of a circular welded elastic inclusion within an elastic 

body provides a theoretical means of demonstrating the absence of any
CIS CONTRIBUTIONS TO ENGINEERING GEOLOGY

Vertical

North

East

x -axis bears S.45" E.

and is horizontal 

y -axis is vertical 

z -axis bears S. 45° W.

and is horizontal

STRESSES ACTING ON PROBE

Principal 

stress

°i (maximum) 

°2 (intermediate) 

(*3 (minimum)

Magnitude (psi) 

( + ) compression 

( ) tension

+ 3876 

-1885 

-3610

Bearing

S. 19° E. 

S. 40° E. 

S. 43° W.

Plunge

71° 

16° 

10°

FIGUBB 7. Stresses determined from elastic model test 2, under a 300,000-pound 

load. Vertical planes are stippled.

stress or strain concentration influences resulting from the borehole 

in which the probe is placed (Leeman, 1964, p. 54-56). This analogy 

holds for any such inclusion provided the inclusion is elastic and 

welded, regardless of the relative stiffness of the inclusion and the host 

material. (See note facing page Cl.)

The field borehole probe has been designed so that in a 3-inchdiameter borehole the stress-carrying member extends on each end at 

least 7 inches (7 ball diameters) from the center of the probe. When 

the probe is grouted into a borehole, the stresses contained within the 

host material are transmitted uniformly through it except at the very 

ends where stress decay is expected. If the 60° approximation of the 

Boussinesq stress-influenced zone is used to define the zone surrounding the probe, then a uniform nondecayed stress field would be expected 

to exist around and beyond the 'sensing ball. Thus this part of the
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C19

probe should act as a solid welded inclusion, nearly perfectly confined, and there should be no effect on the transmitted stresses induced 

by the geometry of the borehole.

The discrepancies between stresses obtained from measurements in 

the acrylic block and those calculated from theoretical considerations 

can be explained by the fact that neither of the theoretical boundary 

conditions (complete restraint or complete nonrestraint) was effective. 

The acrylic model block was considerably too narrow to act as a semiinfinite restraining mass to the probe, but was not too narrow (infinitesimal) to offer a certain amount of restraint to the probe. The 

effectiveness of this restraint in confining the central parts of the 

model can be seen in figures 7 and 8.

The fact that the stress obtained from probe measurements was 

double the applied stress in that direction led to the emplacement of 

the strain-gage rosettes on one surface of the model as described previously. The principal stress trajectories were constructed from the rosette results, and the maximum compressive stress, present on that surface of the model was contoured (fig. 8). From these results it can be 

seen that the size and geometry of the elastic model resulted in increased restraint. This was indicated by high values of compression at 

the center of the model. Consequently, the load-carrying capacity increased toward the center of the model. The geometry of the model employed in these tests apparently resulted in a stress concentration at 

the probe location of twice the applied stress. This; stress concentration 

results from the triaxial confinement of the probe by the surrounding 

acrylic and grout.

MATHEMATICAL ANALYSIS OF PROBE 

MEASUREMENTS

The strain gages that are attached to the probe measure the induced 

strains in three directions at each of three positions on the surface of 

the spherical probe. The three principal stress changes in the rock mass 

are transmitted to the probe, and each strain gage measures the resultant strain induced in the direction of the gage orientation. Therefore, 

each strain rosette permits the calculation of the strain ellipse that was 

produced by all three principal stress changes in the plane of measurement. In addition, the normal stresses on the surface of the sphere 

in the direction of the strain gages and the shear stresses normal to the 

direction of the strain gages may be calculated if Young's modulus and 

Poisson's ratio for the probe are known.

The stresses and strains in the probe are not the same as those in the 

host material because the relative stiffness of the two materials is very 

different. Because the probe is virtually rigid, the stress in it will ap-
C20 CONTRIBUTIONS TO ENGINEERING GEOLOGY

| Boundary of 

acrylic block

EXPLAN ATI ON

Principal stress trajectory

Line of equal maximum 

principal stress

Dashed where inferred

Compressive Tensile

Locations of measured principal

stresses

Relative magnitude shown

FIGURE 8. Surface stress on model test 2.

proach the limit of about 1.5 times the stress in the host material (Leeman, 1964, p. 55). (See note facing page Cl.)

The calculation of the plane stress or plane strain ellipse from 45° 

rosette data is a standard stress analysis technique (Hetenyi, 1958, p. 

412), based on the fact that strain measurements in any three nonparallel directions precisely determine one strain ellipse. Once the stress
.BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C21

ellipse has been calculated, normal and shear stresses can be computed 

for any direction in the plane of the ellipse. In this study the directions 

of interest are those parallel to the arbitrary a?, y, and z axes of the 

probe.

After the shear and normal stresses have been calculated (Hetenyi, 

1958, p. 412) for each of the strain-gage directions on each of the three 

orthogonal stress ellipses that are parallel to the reference axes, the 

components of the stress tensor for either the probe or the host material 

can be calculated. The rosettes produce two stress determinations for 

each of the three orthogonal normal stresses and one determination of 

the shearing stress acting in the direction of each of the arbitrary axes, 

a?, I/, and 2. The duplicate normal stresses are checked for compatibility 

and are averaged as part of the calculation. The six shearing stresses in 

the stress tensor are actually only three different shearing stresses, 

because:

and

(Note: TXV is the shear stress on the yz plane parallel to the y axis and 

normal to x axis.)

The stress tensor is determined by the following stresses:

&xx TXV Txz 

Tyx ffyy Tyz

(Note: ffxx is the normal stress acting on the yz plane parallel to the 

direction of the x axis.) The stress tensor is then solved as a determinant for that set of three orthogonal stresses denoted by S (Sechler, 

1952) which, when rotated to a unique position, reduce the shearing 

stresses to zero.

^ Vxx) Txy TXZ

= 0
C22 CONTRIBUTIONS TO ENGINEERING GEOLOGY

The expanded determinant reduces to :

I O 22 I "TxyTyzTzx ^xx^yz VyyTzx

-oWHO, (1)

which in turn can be solved for its real roots Si, S2 , and 83, the magnitudes of the principal stresses.

The next step in the mathematical analysis is to determine the 

direction cosines (lt, mi} and n^ of the angles between each of the 

three principal stress change axes (i) and the reference probe axes 

(x, y, z). This can be accomplished by substituting the principal stress 

change values and the normal and shear stress changes determined 

from the rosettes in the following simultaneous equations (Sechler, 

1952, p. 23) :

T^n^O, (2)

 TvJL t+ (Si <rvv)mt Tyeni= 0, (3) 

<ru)nt=Q, (4)

and

lt*+mt*+nt=l, (5)

where

«Sfi=respective maximum (1), intermediate (2), or minimum (3)

principal stress change; 

lt= direction cosine of the angle between the arbitrary x axis and

the i principal stress change axis; 

m^= direction cosine of the angle between the arbitrary y axis and

the i principal stress change axis; and 

ft ^direction cosine of the angle between the arbitrary z axis and

the i principal stress change axis.

After calculating the nine direction cosines, for the angles between 

the reference (x, y, z) axes and the three principal stress change axes, 

one must calculate the bearing and plunge of the principal stress 

change axes. This is accomplished by first calculating the angles 

between the reference probe axes and the directions north, east, and 

vertical. This calculation is based on the properties of right spherical 

triangles. The cosine of the face angle of a right spherical triangle 

opposite the right diedral angle equals the product of the cosines of
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS <C23

the other two face angles (Kells and others, 1940, p. 271-273). From 

this the following formulas are derived:

IK=C,OS Nj cos Vj, (6)

m*=cos (90°-Fy), (7) 

and

njc=cos Ej cos Vj, (8) 

where

y=respective arbitrary x, y, or z axis and

&=respective north (A7), vertical (V) or east (E) spatial axis.

lk=direction cosine of the angle between arbitrary x axis and the

respective k spatial axis. 

Misdirection cosine of the angle between the arbitrary y axis

and the respective k spatial axis. 

%=direction cosine of the angle between the arbitrary z axis

and the respective k spatial axis. 

NJ=horizontal angle between north andj axes. 

Vj=vertical angle between horizontal and j axes (downward

plunge is positive). 

Ej=horizontal angle between east and j axes.

The direction cosines for the angles between the principal stress 

change axes and the north, east, and vertical axes can then be calculated. This follows from the fact that the cosine of the angle between 

any two intersecting lines in any three-dimensional system is equal to 

the sum of the products of their respective direction cosines, as follows:

cos eki=lkli+mkmi-i-nkni) (9) 

where

6ki= angle between respective spatial axis north (N), vertical 

(V), or east (E) and respective principal stress change 

direction maximum (1), intermediate (2), or minimum 

(3).

These angles, between directions of the principal stress changes and 

north, east, and vertical, are then reduced to horizontal and vertical 

angular components, again by using the properties of spherical triangles with one right angle. The horizontal angle between any principal stress change direction and north is the bearing of that principal 

stress change, and the complement of the vertical angle is the plunge 

of that principal stress change.
C24 CONTRIBUTIONS TO ENGINEERING GEOLOGY

cos $Nt=cos Ni cos Vi. (10) 

cos 0vt=cos (90° Vi). (11)

cos 0£i=cos Ei cos Vi, (12) 

where

0Ni= angle between north axis and the respective i principal stress

change axis (1, 2, or 3), 

evl= angle between vertical axis and the respective i principal

stress change axis (1, 2, or 3), 

BBi= angle between east axis and the respective i principal stress

change axis (1, 2, or 3), 

Ni= horizontal angle between north axis and the respective i

principal stress change axis (1, 2, or 3), 

Vi= vertical angle between horizontal axis and the respective i

principal stress change axis (1, 2, or 3) (downward plunge

is positive), and 

Ei= horizontal angle between east axis and the respective i

principal stress change axis (1, 2, or 3).

The last step in the mathematical analysis consists of the determination of the magnitude, bearing, and plunge of the maximum 

shearing stress changes which accompany the previously calculated 

principal stress changes.

(13) 

where

$i=maximum principal stress change, 

$3= minimum principal stress change, and 

Tmax= maximum shearing stress change.

The preceding calculations have been programed for computer 

analysis in the interest of economy, speed, and accuracy.

COMPUTATION OF LATERAL STRESSES IN SECOND 

ELASTIC MODEL TEST

Static testing of the elastic constants of the acrylic block and the 

partly metal-filled epoxy grout, in addition to an accurate knowledge of the geometric distribution of these materials in the laboratory model, permitted calculation of the effective moduli in any direction. The calculated effective moduli are determined by making a 60° 

approximation of the Boussinesq stress-influenced zone from the sides
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C25

of the spherical probe for each of the stress directions. The maximum 

principal stress (o-i) zone has only a small (% inch) annulus of metalfilled epoxy with the remaining 8% inches on each side in the acrylic. 

The intermediate principal stress (<j2 ) zone has about the same annul us 

of metal-filled epoxy but only 5% inches of acrylic on each side. The 

minimum principal stress (o-3 ) zone, which contains the borehole, has 

a rather complex geometric distribution of metal-filled epoxy and 

acrylic. This distribution also differs from side to side. Accurate measurement of the geometric material distribution within the 60° stress 

zone permitted an estimate to be made for the effective elastic modulus in the direction of the minimum principal stress.

Following are the material moduli, Poisson's ratios, and calculated 

effective elastic moduli:

1. Modulus of the acrylic block=0.327 X106 psi.

2. Modulus of the filled grout=1.55X106 psi.

3. Poisson's ratio (all materials) = 0.39

4. Effective modulus in the direction of the maximum principal stress, 

E!=0.330 X106 psi.

5. Effective modulus in the direction of the intermediate principal 

stress, E'2=0.347X106 psi.

6. Effective modulus in the direction of the minimum principal stress, 

E3=0.706 X106 psi.

By using these constants, the stress relationships can be calculated as 

follows:

Assume the principal stresses o-x , <r2 , or

Assume the principal strains ei^e2

Then by using the relationships from Sechler (1952, p. 55),

and

=J_

Eg

it can be shown that

| JB
C26 CONTRIBUTIONS TO ENGINEERING GEOLOGY

By substituting the effective moduli and Poisson's ratio, the above 

relationship can be expressed as follows:

o-2= -0.226o-!+1.35o-3, (14) 

or

0-2+0.2260-1 n= L35~' 

or

1.35o"3 0*2

0.226 (16)

By substituting measured values of o-i and 0-3 into equation 14, 

0-2= 2,015 psi of tension compared with the actual measured value 

of 1,885 psi of tension. Similarly, 0-3 as computed becomes 3,420 psi 

of tension as compared with a measured value of 3,610 psi of tension.

FIELD INVESTIGATIONS

The laboratory investigations and probe development were, in part, 

an outgrowth of field studies performed by the U.S. Geological Survey over a period of several years. During the course of these studies, 

both at the surface and underground, the need for more adequate 

stress determinations was recognized. For example, in the Straight 

Greek Tunnel pilot bore (Robinson and Lee, 1964, 1967) the rock 

load on the steel sets was measured by the contractor with prop-load 

cells and the deflection of the walls and roof was measured with borehole extensometers. More than 40 instrument stations were established through the 8,350-foot-long bore. This provided a large amount 

of data on rock behavior during excavation. Nevertheless, very little 

new knowledge was gained of fundamenetal rock mass behavior. The 

stress conditions within the rock mass prior to tunneling 'and the behavior of the stress field in response to tunneling operations are still 

virtually unknown. The writers believe that a more accurate understanding of these factors will increase the efficiency and economy of 

underground excavation. For example, in addition to the loads carried 

by steel sets, it would be helpful in underground mining or tunneling to know the changing state of stress adjacent to the opening and 

for a considerable distance within the rock surrounding the opening.

Changes in the state of stress adjacent to an active underground 

opening could be monitored by placing and grouting borehole probes 

into holes drilled into the walls and ahead of an advancing tunnel 

face. Further, the state of stress in the rock mass prior to excavation 

could conceivably be estimated by overcoring the borehole probe.
BOREHOLE PROBE, STRESS CHANGES, IN A ROCK MASS C27

Other investigators (Varnes, 1962, p. 48) have mentioned the need 

for a method which would detect the buildup of stresses preceding the 

development of faults or preceding movement along preexisting faults. 

Accurate knowledge of the stress buildup in an area would permit the 

prediction of the possible orientations of fault structures resulting 

from the indicated stress fields, whether they be related to shear or 

tensile failure of the rock mass. It should be possible to monitor such 

a buildup in stress at some distance from a fault because the stresses 

producing a fault are undoubtedly not restricted in their extent, as 

faults are not restricted in their extent. It appears feasible, because 

the probe can be placed at depth in a rock mass, to foresee the use of 

the probe, in relation to active faults, as a means of predicting impending hazardous movements.

The scarcity of any studies regarding the influence of geologic factors on the distribution of stresses in rock masses makes it imperative 

that efforts be made to bring about a union between stress analysis and 

geology in the future of rock mechanics.

REFERENCES CITED

Coates, D. F., 1965, Rock mechanics principles: Canada Dept. Mines and Tech. 

Surveys, Mines Br. Mon. 874, 338 p.

Dare, W. L., 1962, Measuring changes in pillar strain during pillar recovery 

Experiments in a New Mexico uranium mine: U.S. Bur. Mines Kept. Inv. 

6056,17 p.

Edwards, R. H., 1951, Stress concentrations around spheroidal inclusions and 

cavities: Jour. Applied Mechanics, Am. Soc. Mech. Engineers Trans., v. 73, 

p. 19-30.

Fitzpatrick, John, 1962, Biaxial device for determining the modulus of elasticity 

of stress-relief cores : U.S. Bur. Mines Kept Inv. 6128,13 p.

Goodier, J. N., 1933, Concentration of stress around spherical and cylindrical 

inclusions and flaws: Jour. Applied Mechanics, Aon. Soc. Mech. Engineers 

Trans., v. 55, p. 39-44.

 1958, Elasticity and plasticity: New York, John Wiley & Sons, Inc., 152 p.

Hast, Nils, 1958, The measurement of rock pressure in mines : Stockholm, Sveriges 

Geol. Undersokning Arsb., v. 52, no. 3, 183 p.

Hetenyi, M., 1958, Handbook of experimental stress analysis: New York, John 

Wiley & Sons, Inc., 1077 p.

Jenkins, J. D., 1958, Discussion of Seldenrath, T. R., and Gramberg, J.. Stressstrain relations and breakage of rocks, in Walton, W. H., ed., Mechanical 

properties of non-metallic brittle materials: New York, Interscience Publishers, Inc., p. 102-103.

Kells, L. M., Kern, W. F., and Bland, J. R., 1940, Plane and spherical trigonometry : New York, McGraw-Hill Book Co., 529 p.

Leeman, E. R., 1964, The measurement of stress in rock: South African Inst. 

Mining and Metallurgy Jour., v. 65, no. 2, p. 45-114.

Merrill, R. H., Williamson, J. V., Ropchan, D. M., and Kruse, G. H., 1964, Stress 

determinations by flatjack and borehole-deformation methods: U.S. Bur. 

Mines Rept. Inv. 6400, 39 p.
C28 CONTRIBUTIONS TO ENGINEERING GEOLOGY

Morgan, T. A., Fisher W. G. and Sturgis W. J. 1965, Distribution of stress in the 

Westviaoo Trona mine, Westvaco, Wyoming: U.S. Bur. Mines Kept. Inv. 6675, 

58 p.

Obert, Leonard, 1962, Effects of stress relief and other changes in stress on the 

physical properties of rock: U.S. Bur. Mines Kept. Inv. 6053, 8 p.

Obert, Leonard, Duvall, W. I., and Merrill, R. H., 1960, Design of underground 

openings in competent rock: U.S. Bur. Mines Bull. 587,36 p.

Panek, L. A., 1966, Calculation of the average ground stress components from 

measurements of the diametral deformation of a drill hole: U.S. Bur. Mines 

Rept. Inv. 6732, 41 p.

Panek, L. A., and Stock, J. A., 1964, Development of a rock stress monitoring 

station based on the flat slot method of measuring existing rock stresses: 

U.S. Bur. Mines Kept. Inv. 6537,61 p.

Robinson, O. S., and Lee, F. T., 1964, Geologic research on the Straight Creek 

Tunnel site, Colorado: Highway Research Board, National Research Council 

[U.S.], Highway Research Rec. 57, p. 18-34.

 1967, Results of geologic research, Straight Creek Tunnel pilot bore, Colorado: Highway Research Board, National Research Council [U.S.] Highway 

Research Rec. 185, p. 9-19.

Ryder, J. A., and Officer, N. G., 1965, An elastic analysis of strata movement 

observed in the vicinity of inclined excavations, in Symposium on rock 

mechanics and strata control in mines: Johannesburg, South African Inst. 

Mining and Metallurgy Proc., p. 168-193.

Sechler, E. E., 1952, Elasticity in engineering: New York, John Wiley & Sons, 

Inc., 419 p.

Seldenrath, T. R-, and Gramberg, J., 1958, Stress-strain relations and breakage of rocks, in Walton, W. H., ed., Mechanical properties of non-metallic 

brittle materials: New York, Interscience Publishers, Inc., p. 79-102.

Utter, Stephen, 1962, Determination of stresses around an underground opening, 

Climax molybdenum mine, Colorado: U.S. Bur. Mines Rept. Inv. 6137, 

26 p.

Varnes, D. J., 1962, Analysis of plastic deformation according to Von Mises' 

theory, with application to the South Silverton area, San Juan County, 

Colorado: U.S. Geol. Survey Prof. Paper 378-B, 49 p.

Wentwotftih, D. H., 1966, Some factors influencing strain relief overcoming: 

Golden, Colorado School of Mines, M.S. thesis T-1097,136 p.

Wisecarver, D. W., Merrill, R. H., Rausch, D. O., and Hubbard, S. J., 1964, Investigation of in situ stresses, Ruth mining district, Nevada, with emphasis 

on slope design problems in open-pit mines: U.S. Bur. Mines Rept. Inv. 

6541, 21 p.

U. S. GOVERNMENT PRINTING OFFICE : 1968 O - 306-521