The Triangular Stress Diagram 

A Graphical Representation of 

Crustal Stress Measurements

GEOLOGICAL SURVEY PROFESSIONAL PAPER i2»i

The Triangular Stress DiagramA Graphical Representation of 

Crustal Stress Measurements

By HENRI S. SWOLFS

GEOLOGICAL SURVEY PROFESSIONAL PAPE^R 1291

A method is devised that uses a ternary 

diagram to interpret point measurements of 

three-dimensional stress and to assess 

structural stability of crustal rocks

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1984
UNITED STATES DEPARTMENT OF THE INTERIOR 

WILLIAM P. CLARK, Secretary

GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Library of Congress Cataloging in Publication Data

Swolfs, Henri S.

The Triangular Stress Diagram.

(Geological Survey Professional Paper ; 1291)

Bibliography: 19 p.

1. Rock deformation Graphic methods. 2. Earth Crust Graphic methods. I. Geological Survey

(U.S.) II. Title. III. Series.

QE604.S96 1983 551.8 82-21047

For sale by the Branch of Distribution, U.S. Geological Survey 

604 South Pickett Street, Alexandria, VA 22304
CONTENTS

Page

Abstract ............................................ 1

Introduction .......................................... 1

The triangular stress diagram ................................ 3

Failure surfaces ........................................ 9

Stress paths .......................................... 11

Total stress paths .................................... 12

Effective stress paths .................................. 14

Discussion and conclusions .................................. 16

References cited ........................................ 17

ILLUSTRATIONS

Page

FIGURE 1. Summary plot of positive stress differences between vertical stress component and minimum horizontal stress component

as a function of depth ................................................... 2

2. Summary plot of minimum horizontal to vertical stress ratios as a function of depth ................. 3

3. Diagrammatic representation of a state of stress 0 .................................... 6

4. Construction and properties of triangular diagram ..................................... 7

5. Triangular diagram of experimentally determined failure envelopes or surfaces ..................... 10

6. Composite plot consisting of two panels of triangular diagram that combines test results for 0=60° ........ 11

7. Plot of distribution of total stress states and effective stress states measured in shales and marlstones ...... 12

8. Plot of distribution of total stress states measured in quartzitic beds of Witwatersrand Group, South Africa, and vertical

section along total stress path ............................................... 13

9. Plot of stress path associated with increasing hydrostatic pressure ........................... 14

10. Plot of probable relation between stress states measured at shallow depths and focal-mechanism solutions ..... 16

TABLES

Page

TABLE 1. Measured stresses in the United States ............................................ 4

2. In situ stress parameters, Rangely, Colo ........................................... 15

Hi

THE TRIANGULAR STRESS DIAGRAM A GRAPHICAL 

REPRESENTATION OF CRUSTAL STRESS MEASUREMENTS

By HENRI S. SWOLFS

ABSTRACT

A new graphical method is introduced that facilitates the representation and interpretation of the generally anisotropic in situ states-ofstress measured in rocks. The purpose of the method is to clearly 

and easily display these stress states, each measured in terms of 

three values, in two-dimensional space; this is done by projecting 

three-dimensional stress space onto a triangular-coordinate (ternary) 

diagram and plotting the measured stress states as points. Failure 

surfaces determined either by experiment or by invoking hypothetical 

failure criteria are plotted on the diagram to enclose the stability 

field. Thus, an anisotropic stress condition measured at a given depth 

in the Earth's crust is represented by a single point on the triangular 

diagram and its relative position in the enclosed stress space may 

yield useful information on the degree of mechanical stability of the 

rock mass at the point of measurement. The method also allows the 

representation of any number of measured stress states in a given 

locality, each plotted in its own position. This method is most useful 

as a reconnaissance tool, provided complete measurements of in situ 

stress as a function of depth are available, to correlate them with 

information obtained by different, more indirect methods. A variety 

of stress paths either associated with the measured stress-depth distribution, or modified by fluid injection or mined excavations, among 

others, can be traced or extrapolated to establish whether the imposed disturbance leads to failure of crustal rocks.

INTRODUCTION

The development and subsequent use of field techniques to obtain stress measurements in rocks over a 

significant range of crustal depths has contributed important information on the in situ state of stress. Many 

of these results have been reported by Renalli and 

Chandler (1975), McGarr and Gay (1978), Brown and 

Hoek (1978), among others. Because an adverse change 

in crustal stresses may result in the eventual instability 

of the rock mass, knowledge of their present distribution in reasonable detail is important. From this detail 

can be derived the kind of information that is particularly relevant to the design and long-term maintenance 

of underground structures and civil engineering works, 

and to the understanding of earthquake mechanics.

A state of stress is completely defined by three principal stress components and their orientations, 

Si>S2>S3 (total stresses) or o-1 >o-2>o'3 (effective

stresses). Compressive stresses are reckoned positive 

and effective stresses are equal to the total stresses 

less the interstitial pore pressure (P) provided the rocks 

are sufficiently permeable; that is, cr=S-P. Failure 

criteria generally are expressed in terms of effective 

stresses. With reference to crustal stresses, a common 

assumption (Anderson, 1951) is that the vertical total 

stress, Sv, is also a principal stress, Si, S2 , or S3. 

Likewise, Shmax and Shmin are the maximum and minimum horizontal total stresses and principal stresses.

Figures 1 and 2 summarize the author's compilation 

(Swolfs and others, 1977) of the then available stress 

measurements obtained from hydraulic-fracturing tests 

in the United States (see also McGarr and Gay, 1978). 

Only the data for Sv (overburden pressure) and Shmin 

(assumed equal to the measured instantaneous shut-in 

pressure, ISIP) are plotted against depth to provide 

geologic perspective. No attempt at averaging the two 

horizontal stress components, Shmax (if known) and 

Shmin, has been made because otherwise useful information becomes somewhat obscure. A complete listing of 

the data used in figures 1 and 2 may be found in table 

1.

When the field data are ranked with the type and 

geologic age of the rock in which the measurements 

were made, several observations can be made. For example, at or near the Earth's surface, the horizontal 

stresses tend to equal or exceed the vertical stress, 

whereas at greater depths the vertical stress becomes 

the more dominant component (figs. 1 and 2). This general phenomenon, recognized in many localities around 

the world, is perhaps associated with the effects of erosional unloading (denudation) of the rock exposed at the 

Earth's surface (Voight, 1966) and is especially pronounced in the crystalline, brittle rocks. The data also 

provide cogent evidence for the proposition (Griggs, 

1936) that rocks possess long-term strengths; that is, 

in the absence of external influences, rocks are capable 

of sustaining finite stress differences over geologically 

significant times without continuously deforming. The
THE TRIANGULAR STRESS DIAGRAM

Shmin ~ Sv (MPa)

5

0.5

1.0

1.5

2.0

2

0)

o>

1 2.5 

^_

t

LU

Q 3.0

3.5

4.0

4.5

5.0

0 40 30 20 10 0 10 20 30 4

I II III .

.0£?isw

NORMAL AND iC^Ss 

STRIKE-SLIP Ss v yFAULTING /

vSs r/ -

vSs /

TQz vSs vSs /

-

vDol Shv v sh ^ 

*Gr t

CJ

/

"

-

:

I 

i i v i i i i i i i

^*Ss Ss * Gn

Q3DSS 

-2

v3 

Sa

~ 4 THRUST FAULTING

-5

-6

-7

 8 03

-9 o 

-10

-11

-12

-13

-14

-15

-16

I

I I I

EXPLANATION

Above 600 meters (2000 feet) 

D Paleozoic rocks

 Tertiary shale, sandstone (Sh, Ss)

o Tertiary oil shale (Ms) 

4 Tertiary tuff (Tf)

* Granite, granodiorite, gneiss 

Below 600 meters

D Paleozoic rocks

v Sedimentary rocks 

Sa-salt, Ls-limestone, Sh-shale 

Ss-sandstone, Dol-dolomite

T Crystalline, Precambrian rocks 

Qz-quartzite, Gr-granite

FIGURE 1. Summary plot of the positive stress differences in megapascals (MPa) between the vertical stress component (=pgh) 

and the minimum horizontal stress component (=ISIP) as a function of depth. Lithologies are identified by the abbreviations: 

Gr, granite; Gn, gneiss; Qz, quartzite; Dol, dolomite; Ss, sandstone; Ls, limestone; Sh, shale; So, salt; Tf, tuff; and Ms, marlstone. 

In the left panel, the magnitude of the vertical stress component is either intermediate or greatest in value; in the right panel 

the vertical stress magnitude is least in value. The line marked "calcite" indicates the stress differences inferred for limestones 

in the Gulf coastal region (Friedman and Heard, 1974).

magnitudes of the stress differences (fig. 1) or stress 

ratios (fig. 2) depend in large measure on rock type 

and increase with prt3sure (depth) under normal geothermal conditions. At one end of the spectrum (fig. 

2), the strong "brittle" rocks, including dolomite, granite, and quartzite, maintain stresses that approach the 

short-term failure strength (see also Handin and others, 

1963, figs. 26 and 27). At the other end of the spectrum, 

such "ductile" rocks as salt, limestone, and some shales 

persist in supporting low stresses that probably are related to their respective long-term yield or creep 

strengths. In support of this latter observation, Friedman and Heard (1974) found, by analyzing the growth 

of mechanical twins in calcite, that Lower Cretaceous 

limestones in the Gulf Coast area have maintained a 

nearly constant horizontal-to-vertical stress ratio of 0.91

over a depth range of 6.5 km (calcite line in figs. 1 

and 2); this ratio is considered to be a maximum.

Hypothetically, therefore, the suggestion may be entertained that in a typical vertical rock sequence composed of, say, granite, shale, dolomite, limestone, and 

sandstone, the stresses in each rock unit are limited 

over the long term by the respective strengths, resulting in an inhomogeneous horizontal stress distribution 

throughout modified to varying degrees by the processes of burial, denudation, and tectonics. At the same 

time, the vertical stresses and, for reasons of equilibrium and continuity, the shear stresses, must be smooth 

varying functions of depth. The evidence, clearly, is 

mixed for the very reason that the already limited data 

base also is incomplete (figs. 1 and 2). The resolution 

of the ambiguities would require the analyses of com-
THE TRIANGULAR STRESS DIAGRAM

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.8 3.0

EXPLANATION

Above 600 meters (2000 feet)

Paleozoic rocks

Tertiary shale, sandstone (Sh, Ss)

Tertiary oil shale (Ms)

Tertiary tuff (Tf)

Granite, granodiorite, gneiss 

Below 600 meters

Paleozoic rocks

Sedimentary rocks

Sa-salt, Ls-limestone, Sh-shale 

Ss-sandstone, Dol-dolomite

Crystalline, Precambrian rocks 

Qz-quartzite, Gr-granite

5.0 -

FIGURE 2._Summary plot of the minimum horizontal to vertical stress ratios as a function of depth. Lithologies are identified by 

the abbreviations: Gr, granite; Gn, gneiss; Qz, quartzite; Dol, dolomite; Ss, sandstone; Ls, limestone; Sh, shale; So, salt; Tf, tuff; 

and Ms, marlstone. The shaded curved band in the far left panel represents the short-term rupture strengths of intact crystalline 

rocks tested to failure under conventional triaxial stress conditions and strain rates. The frictional strengths of preexisting faults 

and fractures are bounded by two values for the coefficient of sliding friction ((x=0.6 and 1.0).

plete states of stress (three-dimensional) and their distribution in the rock column.

The purpose of this paper is to introduce an alternative to the three-dimensional space coordinate plot 

(Meldahl, 1944) in which the states of stress are uniquely represented as points. The present method involves 

the transformation of rectangular-coordinate stress 

space into a triangular-coordinate system and plotting 

the measured states of stress in the Earth's crust in 

dimensionless units in a triangular (ternary) diagram 

familiar to geologists.

THE TRIANGULAR STRESS DIAGRAM

A particular compressive state of stress, 0, defined 

by three principal stresses and their orientations, can 

be represented in a rectangular system of coordinates 

as a point in principal stress space (fig. 3). Let the 

coordinates of the Cartesian set of axes be defined by 

the principal stress components S%,Sy, andS 2 . The 

position of stress state 0 relative to the space diagonal 

H can be represented by the resultant vector r (fig.
THE TRIANGULAR STRESS DIAGRAM 

TABLE 1. Measured stresses in the United States

[MPa, megapascals; leaders ( ) indicate no data]

State

CaliforniaColorado 

Idaho 

Illinois 

Michigan 

Lithology Age

Sandstone Tertiary 

Granite Jurassic 

Marl stone- Tertiary 

shale.

Sandstone Pennsyl vanianPermian.

Sandstone Cretaceous 

Granite Precambrian 

Quartzite Precambrian 

Limestone Pennsyl vanianLimestone Devonian 

Dolomite Ordovician 

Sandstone Cambrian 

Shale Precambrian 

Depth

(meters)

240

287

157

210

234

240

265

292

308

37

158

190

267

338

385

406

442

451

64

92

110

61

425

443

449

269

319

384

423

430

313

378

402

453

476

72

108

237

248

320

385

216

1914

2303

2322

3660

2300

94

97

97

100

1230

2806

3660

5105

sv

(MPa)

5.1

6.5

4

n .2

5.7

6.3

6.5

7.2

7.9

8.3

.9

3.6

4.3

6.0

7.6

8.7

9.2

10.2

10.2

1.4

2.1

2.5

1.4

9.6

10.0

10.1

6.1

7.2

8.7

9.6

9.7

7.1

8.6

9.1

10.2

10.8

1.6

2.4

5.4

5.6

7.2

8.7

4.9

42.7

52.1

52.5

82.8

CO fi Oc.U

2.1

2.2

2.2

2.3

32.0

73.0

95.0

1-J9 C 1 *>£. . ~J

S hmi n S hmi n /Sv

(MPa)

11.4 2

8.2 1

4.4 1

4.9

4.4

5.1

4.9

5.4

5.2

1.3 1

3.3

3.9

3.5

5.3

6.0

7.0

7.4

7.5

2.2 1

2.5 1

2.3

1.9 1

7.5

7.5

7.8

4.1

4.9

5.2

5.0

4.7

5.7

6.8

7.1

7.1

7.0

2.7 1

3.1 1

4.3

6.3 1

5.7

7.4

2.6

31.4

34.4

40.4

35.9

tye r\ cO.U

2.3 1

2.3 1

2.3 1

2.7 1

29.5

40.0

64.5

96.0

.22

.26

.05

.86

.70

.78

.68

.69

.63

.44

.92

.91

.58

.70

.69

.76

.74

.74

.57

.19

.92

.36

.78

.75

.77

.67

.68

.60

.52

.48

.80

.79

.78

.70

.65

.69

.29

.80

.13

.79

.85

.53

.74

.68

.77

.43

A O

.42

.10

.05

.05

.17

.92

.55

CO. .00

.72

hmax

(MPa)

22.5

5.2

7.0

7.4

7.5

8.0

10.9

10.0

1.3

4.8

4.8

4.0

6.8

8.3

9.1

8.0

9.5

4.0

 

3.6

11.8

11.3

11.3

6.6

5.6

8.6

10.6

9.5

9.5

8.3

5.0

5.6

5.8

 

 

13.0

3.1

55.2

A O Q 4t.O

 

 

 

50.5

56.0

QQ C OO.3

147.0

Fracture References*

direction

.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

N.

15°

_.._

 

 

 

 

25°

65°

 

 

 

 

90°

 

 

__

 

E-W

__

 

E-W

E-W

 

 

 

 

 

 

 

 

87°

 

 

 

 

75°

45°

75°

70°

20°

15°

60°

7 C<> 

/O

66°

58°

72°

62°

E.

E.

W.

E.

E.

W.

W.

W.

E.

W.

W.

W.

W.

E.

E.

E.

E.

27.

Do.

10,11,14.

Do.

Do.

Do.

Do.

Do.

Do.

4,25.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

7,10,19,20.

23.

Do.

20.

13.

5,6,10.

Do.

Do.

Do.

10,11,15.

Do.

Do.

Do.

^References:

1.

2.

3.

4.

5.

6.

7.

8.

Aamodt (1977).

Blair and others (1977).

Brechtel and others (1976).

Bredehoeft and others (1976).

Haimson and Stahl (1969).

Haimson and Fairhurst (1970).

Haimson (1973).

Haimson (1974).

9.

10.

11.

12.

13.

14.

15.

Haimson

Haimson

Haimson

Haimson

and others

(1977a).

(1976).

and others

(1974).

(1976).

B. C. Haimson (written commun.,

Haimson

Haimson

(1977b).

(1977c).

1976)
THE TRIANGULAR STRESS DIAGRAM 

TABLE 1. Measured stresses in the United States Continued

State

Nevada 

New Mexico-- 

New York ----

Ohio 

s.c. 

S. Dak. 

Texas 

Utah 

y «a - ___ -

Wisconsin 

Wyoming 

Lithology

Tuff 

Sandstone 

Granite 

Sandstone 

Sandstone 

Shale 

Gneiss 

Granite 

Granite 

Shale 

Shale 

Salt 

Granite 

Shale 

^anHctnno--

Age

T *'

Cretaceous 

Precambrian 

Devonian 

Silurian 

Tertiary 

Precambrian 

Precambrian 

Precambrian 

Cretaceous 

Devonian 

PennsylvanianPrecambrian 

Tertiary 

Depth Sv 

(meters) (MPa)

380 

268 

263 

400 

153 

165 

183

214 

235 

249 

326 

309 

437

437 

454 

420 

420 

420 

408 

416

934 

2150

765 

1990 

2930

491

512

510 

800

812

815

194

210

120

150

180

215

245

250

1890

 )AK o*tu

2750

840

914

135

150

169

174

188 

2769

ddflA

7.0 

4.9 

4.8 

7.3 

2.8 

3.0 

3.3

3.9 

4.3 

4.1 

5.9 

5.6 

7.9

7.9 

8.2 

7.6 

7.6 

7.6 

7.3 

7.5

21.1 

48.6

18.0 

50.4 

75.3

11.1

11.6

11.6

1 ft ft lo »o 

18.4

18.5

4.4

4.7

 ) O

 J £

4.1

4.9

5.8

6.6

6.9

55.2

8

C

.0

65.0

22.1

20.7

3.5

3.9

4.4

4.5

4.9

CO C

UC.O 

1 1K 1

shmin 

(MPa)

3.1 

4.1 

3.3 

3.4 

2.3 

2.8 

3.6

3.2 

3.3 

4.1 

3.4 

3.1 

3.4

3.0 

3.1 

6.0 

6.4 

6.0 

7.7 

7.0

14.5 

32.3

14.7 

33.3 

36.7

1 9 G Li. .0

13.9

14.9 

15.4

i4!g

15.6

3.0

3.2

7.0 

10.5

11.0

15.5

15.8

20.0

18.0

7.6

51.5

16.3 

21.2

6.8

7.5

6.2

6.3

7.5 

52.0

«1 -5

shmin/sv

.44 

.84 

.69 

.47 

.82 

.93 

1.09

.82 

.77 

1.00 

.58 

.55 

.43

.38 

.38 

.79 

.84 

.79 

1.05 

.93

.69 

.66

.82 

.66 

.49

1.15

1.20

1.28

oc

,OO

.81

.84

.68

.68

2.19

2.56

2.24

2.67

2.39

2.90

.33

 oj ftQ

.79

.74 

1.02

1.94

1.92

1.41

1.40

1.53

QO

.OO

71

^hmax 

(MPa)

10.7 

9.2 

9.1 

7.3 

8.0 

6.8

6.9

7.4

12.5 

13.1 

12.3

___

4.8

8 Q.3

15.0

14.0

27.3

17.5

35.9

9ft 7

£O «O

56.9

OQ C

oo.b

15.9

18.5

16.0

17.5

11.7

Fracture 

direction

N.

N.

N.

N. 

N.

N. 

N.

N.

N.

N. 

N.

N.

N.

N.

N.

N.

N.

35°

14°

45°

53° 

50°

55° 

35°

ft7° O /

65°

74° 

62°

68°

67°

51°

. _ .

60°

50°

67°

ff\ObO 

50°

_

 

25°

E.

E.

E.

E. 

E.

E. 

E. 

E.

E.

E.

E. 

E.

E.

E.

E.

E.

E.

W.

E.

E.

References

9,24. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

18. 

Do.

1,2. 

Do. 

Do.

5

Q

,O*

Do.

Do.

5,6,8,10. 

Do.

Do.

26.

Do.

10,11,14. 

Do.

Do.

Do.

Do.

Do.

28.

21.

3.

3,16. 

29.

10,11,12,17.

Do.

Do.

Do.

Do. 

1 a10. 

 )">

16. Haimson (1977d).

17. Brown and Hoek (1978).

18. Power and others (1975).

19. Raleigh and others (1972).

20. Raleigh (1972).

21. Roegiers and Fairhurst (1973).

22. Simonson and Jones (1976).

23. Smith and others (1976).

24. Tyler and Vollendorf (1975).

25. Wolff and others (1974).

26. Zoback and Healy (1977).

27. Zoback and others (1977).

28. E. R. Hoskins (written commun., 1977).

29. A. H. Jones (oral commun., 1977).
THE TRIANGULAR STRESS DIAGRAM

3) normal to the space diagonal whose magnitude is 

given by (Meldahl, 1944)

r=.

or, in terms of the second stress-deviation invariant, 

/2> as

 =V2/9

(2)

Both 0 and r lie in a plane defined by 

S% = constant and which is perpendicular to H. 

The intersection of this plane with the compressive 

quadrant of principal stress space results in thej,riangle 

ABC whose sides make an angle of arc cos (V2)~1= 45° 

with all three rectangular axes. By projecting the stresses S%, s*j, and S cz onto the sides of the triangle AB, 

CA, and BC, respectively, a new set of stress coordinates are obtained such that

S,= V2 Scx, Sy = V2 Sjj and Sg = V2 (3)

This new projection implies that the origin of rectangular space is projected onto the apices of the triangle 

rather than along the space diagonal as it is in the 

axionometric projection of Meldahl (1944). An important 

result of this new projection is that stress states that 

vary only by added hydrostatic pressure occupy different positions in the plane of the triangle; other consequences are described later in this section. In terms 

of the new triangular stress coordinates, the vector r 

is given by

H

FIGURE 3. Representation of a state of stress 0 in rectangular- coordinate space ( S^,Sc ,and S° ) and, alternatively, in triangular 

coordinates (Sx, Sy, and Sz) on a plane ABC through 0 perpendicular to the space diagonal H. The position of 0 relative to H, either 

in the positive (compressive) quadrant of rectangular stress space 

or in the plane ABC, is given by the vector r, which is a measure 

of the stress deviation. Direction of shading indicates the direction 

of projection of the rectangular axes onto the sides of the triangular 

plane.

An added convenience is obtained by normalizing the 

stress components with respect to their sum 

(Ii=Sx + Sy + Sz). The state of stress can now be plotted 

in dimensionless units (fractions or percent) on triangular-coordinate graph paper (for example, Dietzgen No.

r- -±- 1 \/fe V (£>, ie v)t 

V6

- + ( e e^2,/o -S,)2 -V/2 . (4)

340-TC ; see also fig. 4A). Each of the normalized com-

'Any use of trade names is for descriptive purposes only and does not constitute 

endorsement by the U.S. Geological Survey.

FIGURE 4 (facing page). Construction and properties of the triangular stress diagram. A, Procedure for plotting the point O in the triangular 

diagram representing the state of stress SV>SX>SZ in dimensionless units (fractions or in percent). B, Distribution of six points in 

different positions in the triangular diagram obtained by permutating the three stress components magnitudes along equivalent directions 

in an isotropic material. Loci of points where two stress components are equal in magnitude are the lines Aa, Bb, and Cc. The 

point of intersection of these three lines defines the state of hydrostatic stress (H). C, Structural definition of the six panels of the 

triangular diagram in terms of the preferred mode and orientation of faulting. The seismological focal-mechanism symbols in each 

panel indicate the preferred motion on preexisting faults; the P-axis (greatest compression direction) bisects the unshaded quadrant. 

D, Definition of states of stress that differ by a hydrostatic pressure. Point O defines the state of total stress (P=O) and point O' 

defines the state of effective stress (P=P0). The separation between points O and O' is proportional to the added hydrostatic pressure 

(P0). E, Definition of the stress deviation associated with the state of total stress at 0 in terms of a dimensionless parameter r; 

that is, r=(J2) //i, where /! and J2 are the first stress invariant and the second stress-deviation invariant, respectively. In an isotropic 

material f is the radius of a circle that contains all six equivalent stress states shown in figure 45. Because an added hydrostatic 

pressure does not affect the stress deviation, points O and 0' in figure 4D are equivalent in terms of J2, but differ in position by 

the ratio fe/f=/1/(/1 -3P). F, Definition of the failure surfaces for frictional sliding on suitably oriented faults and fractures. The position 

of these surfaces in the triangular diagram vary only with the value of the coefficient of sliding friction (|x=0.6 and 1.0).
THE TRIANGULAR STRESS DIAGRAM

0.1 0.2 0.3 0.4 c 0.6 0.7 0.8 0.9

0.2

0.8

-e 0.2 0.4 c 0.6 0.8

0.6

0.8

0.2

0.4

0.6

0.8

0.6

0.4

0.8

0.2

0.4

0.6

0.8

0 10 20 30 PERCENT

0.2 0.8

0.8

5, 0.2 0.4 c 0.6 0.8
8 THE TRIANGULAR STRESS DIAGRAM

pressive stress components may vary from zero to one 

(100 percent), but all three combine in the triangular 

plane of projection at a single point, whose position varies with the relative magnitudes of the stress components.

Let Sx, Sy, and Sz be the total normal stress components measured relative to a system of rectangular axes 

and transformed into the triangular diagram ABC (fig. 

4). If, for the present, the magnitudes of the stresses 

are such that Sy>Sx>Sz, then this triad of values can 

be represented in a triangular-coordinate graph (fig. 

4A) as a single point, 0, by (1) normalizing each stress 

component with respect to the first stress invariant, 

Ii=Sx + Sy + Sz, to obtain the ratios

and S,= (5)

(2) finding the values of each ratio, either as a fraction 

or in percent, along the appropriate sides of the diagram, AB, CA, and BC, respectively, and, finally, (3) 

projecting each value into the interior of the diagram 

as shown in figure 4A to plot the point 0 that now 

defines the stress condition Sy>Sx>Sz. The loci of 

points where two stress components are equal in magnitude are the lines Aa (SX=SZ), Bb (Sy=Sz), and Cc 

(Sx=Sy). The intersection (H) of these three lines defines the condition of hydrostatic stress, Sx=Sy=Sz, it 

is the normal projection of the space diagonal in rectangular coordinates and, therefore, retains the same definition. Stress states for which one stress component 

is identically zero are plotted along the sides of the 

diagram: for example, along AB for SZ =Q, along CA 

for SX =Q, and along BC for ^=0 (fig. 45). Negative 

(tensile) stresses are plotted outside the diagram.

Five additional points, P, Q, R, S, and T (fig. 45), 

can be plotted at different positions in the diagram, 

provided the rock material is isotropic, by permutating 

the same relative magnitudes among the stress components. Thus, letting Sy>Sx, the magnitude assigned to 

Sz increases from smallest value at 0, intermediate at 

P to largest at Q. Similarly, but now letting Sx>Sy, 

points R, S, and T are identified by decreasing magnitudes assigned to Sz.

The triangular diagram (fig. 4C) is now divided into 

six panels in which the states of stress are distinguished 

by magnitude and orientation. Provided that Sz is vertical and Sx and Sy are horizontal, the panels can be identified with three types of faulting (Anderson, 1951). For 

example, the panels that include the points 0 and T 

are associated with thrust faulting, those that include 

P and S with strike-slip (wrench, transcurrent) faulting, 

and those that include Q and R with normal faulting. 

Likewise, motion on preexisting faults and fractures 

characterized by the orientation and inclination of the

slip vector can be classified accordingly in the appropriate panels. The pairs of laterally adjacent, similarly 

named panels differ only by the orientation of the consequent or preexisting structures (faults and fractures). 

The effective stresses defined by <r=S P, where 

P=P0 is the ambient interstitial pore pressure, can be 

plotted in the diagram (fig. 4D) using the relationship

= a= s~ p 

7,-3P

(6)

and following the plotting procedures as outlined earlier. The new point on the triangular diagram, for example 0' representing the effective stress condition 

vy>vx>vz, is plotted outward from point 0 on a line 

that passes through points 0 and H. The separation 

between points 0 and 0' is proportional to the magnitude of the ambient pore pressure P0. Thus, as the 

pore pressure is increased relative to P0, the new effective stress states are plotted progressively farther outward from point 0' along the same line that passes 

through 0', 0, and H. This important property of the 

triangular diagram is analogous to the translation of 

two-dimensional Mohr circles of constant radius (shear 

stress) along the normal stress axis in response to 

changing pore pressures.

Any point in the triangular diagram also defines a 

set of deviatoric stresses. It can be shown (Meldahl, 

1944) that, in rectangular principal stress space, the 

deviatoric stresses are defined by a vector r normal 

to the space diagonal whose magnitude is proportional 

to the second stress-deviation invariant, J2 (equation 

2). By transformation into the triangular diagram (fig. 

4#) the normalized vector f originating at H remains 

proportional to J2 (equation 4); that is,

r/i=(/2) 1/2 (7)

Consequently, point 0 and, by permutation of the three 

normalized principal total stresses (fig. 45), points P 

through T lie on a circle of radius vector f; the circle 

being the projection of a cylindrical surface coaxial with 

H. The following caveat should be entered at this point: 

because fluid pressures do not affect the magnitude of 

the stress deviation, points such as 0' (fig. 4D) that 

designate the effective stresses must have the same 

value of «/2 as point 0. However, because of the way 

in which the triangular diagram is constructed, the 

radius vector (fe) terminating at point 0' is greater 

than f by the factor /!/(/! -3P). This is the only exception to the general rule regarding the fractional length 

of the radius vector f and its relationship to the magnitude of </2 (equation 7).
FAILURE SURFACES

Mechanical or structural stability can be assessed by 

the proximity of points on the triangular diagram to 

failure surfaces or envelopes. The simplest of these, the 

Coulomb criterion, can be used to approximate the 

limits of mechanical stability and is given, in terms of 

two effective principal stresses (a!>a3), by the equation

(8)

where C0 is the cohesive strength of intact rock (free 

of throughgoing fractures or faults), g=((jx2{ +l)1/2 +jxi)2 , 

and jxi is the coefficient of internal friction (Jaeger and 

Cook, 1968). For fractured or faulted rock with negligible cohesion (C0X)), equation 8 reduces to:

(9)

where jx is the coefficient of sliding friction.

Figure 4F shows a family of failure envelopes corresponding to values of the coefficient of sliding friction 

that range from 0.6 to 1.0 (valid for a wide variety 

of rocks (Byerlee, 1978)). Points E (o^a^ and C 

(03=02) can be found by using equation 9, assigning 

an arbitrary value to either o^ or cr3 , and substituting 

the appropriate value for jx. By rotating the principal 

effective stress components into different orientations, 

the envelopes close to form a polygonal set of curves 

whose positions in the triangular diagram vary only 

with the value assigned to jx; these envelopes are the 

projections of polygonal cylinders coaxial with H. The 

straight-line segments connecting points E and C and 

so on around (finding their origin at the apices (A, B, 

and C) of the diagram) reflect the implicit assumption 

of the Coulomb criterion that the magnitude of the intermediate principal stress, cr2 , has no effect on the frictional strength of rock. Fracture-strength (rupture) envelopes can be plotted in a similar fashion using equation 8, except that their positions in the diagram will 

vary with C0 , M-;, and stress.

This completes the introduction to the triangular 

stress diagram. In summation, a particular state of 

compressive stress can be represented as a single point 

in the diagram. Its position relative to the center of 

the diagram (H) defines the operating stress deviation 

(that is, r/i=(«72) 1/2), whereas its proximity to an appropriate failure surface determines the degree of mechanical stability in the dry state. The incorporation of pore 

pressure P shifts the point closer to the failure surfaces 

along a prescribed path. Clearly, a succession of complete stress determinations with depth in a certain locality can be represented by an equal number of points 

distributed within the diagram; the distribution of 

stress states thus represented perhaps being characteristic of the tectonic setting of the locality. Finally, 

the use of failure surfaces is not in the least restricted 

to the simplest ones alluded to earlier; more complex 

but informative ones may be constructed point by point 

provided a sufficient number of experiments involving 

different stress states have been made. These latter 

items will be elaborated in more detail in the following 

sections.

FAILURE SURFACES

The combinations of stresses at which rocks, on the 

one hand, fail either by fracture or frictional sliding 

or, on the other hand, by intracrystalline or cataclastic 

flow are best explored with careful experimentation in 

the laboratory (Jaeger, 1967, for review). The experimental results, often involving only two stresses in conventional triaxial compression (oi>a2=a3) or extension 

(a1 =a2>a3) tests, are conveniently represented as failure envelopes in the two-dimensional Mohr circle diagram. The fact, however, that the envelopes for compression and extension tests on the same material 

rarely coincide (predicted otherwise by the Coulomb 

criterion) indicates the important and significant effect 

of the intermediate principal effective stress, cr2 , on 

rock behavior, particularly with respect to fracture 

propagation (Paterson, 1978, p. 39). Unfortunately, little is known about the fracture strengths of intact rock 

under truly triaxial stress states (cr1 >a2>a3), and almost no knowledge exists about the frictional strengths 

of discontinuous rock under the same general stress 

conditions despite its obvious importance to practical 

rock mechanics.

The shear strengths of intact rock under true triaxial 

stress conditions have been investigated using specialized techniques and equipment (for example, Mogi, 

1971). In triangular coordinates figure 5A shows, as an 

example, the measured failure strengths of Mizuho 

trachyte (Mogi, 1971, fig. 7b) as a function of cr2 at 

two constant values of cr3 . The corresponding Coulomb 

failure envelopes are shown in figure 5A as thin straight 

lines for comparison purposes.

The two failure envelopes measured for the trachyte 

(fig. 5A) are only slightly convex outward, but are 

otherwise in fair agreement with the "straight-line" 

Coulomb envelopes. Taken together as contours, the 

measured failure envelopes define a portion of a failure 

surface that, with increasing stress levels, converges 

slowly toward the hydrostatic axis (H). The generalized 

failure surface or its approximate equivalent (equation 

8) for intact, isotropic rock material wraps around but 

remains symmetric about the hydrostatic axis (fig. 5A 

shows only one of the six panels).

The failure strengths of anisotropic model materials
10 THE TRIANGULAR STRESS DIAGRAM

0.2

FIGURE 5. Experimentally determined failure envelopes or surfaces. A. Panel of the triangular diagram showing the experimentally determined fracture strengths of Mizuho trachyte as a function of a2 at constant a3 =45 MPa (solid circle) and a3 =100 MPa (solid square) 

(replotted from Mogi, 1971). Coulomb envelopes (thin straight lines, equation 8) are shown for comparison. B. Panel of the triangular 

diagram showing the variation in strength of a planar anisotropic rock model as a function of the angle p (see inset) between the 

least-stress component (a3) and the joint set. The intermediate stress component (a2) is parallel to the joint set. The constant applied 

stresses are: data line I, a2=(T3=0.4 MPa; data line II, a2=1.0 MPa, a3=0.6 MPa; data line III, a2=1.2 MPa, a3=0.8 MPa. Lowest 

strength is achieved by frictional sliding at (3=60° (data replotted from Reik and Zacas, 1978). Thin straight lines are the failure 

envelopes obtained by equation 9 (see also fig. 4F) for ^,=0.6, 0.8, and 1.0. C. Panel of the triangular diagram showing the variation 

in strength of a planar anisotropic rock model as a function of the angle (see inset) between the intermediate stress component (a2) 

and the joint set, and increasing magnitudes of a2. The least-stress component (a3) is parallel to the joint set and constant at 0.4 

MPa. Dashed lines connect data points for the same angle p for values of a2 equal to 0.4 MPa, 0.8 MPa and 1.2 MPa (data replotted 

from Reik and Zacas, 1978). Thin straight lines are the Coulomb envelopes (equation 9) that imply no effect of the intermediate stress 

component (a2) on strength; this condition is true only where (3=0°.

under general states of stress have been reported by 

Reik and Zacas (1978). Two examples will be considered 

here: in the first example the intermediate stress component (cr2) is parallel to the throughgoing set of discontinuities (fig. 5B), whereas in the second example the 

least-stress component (a3) is parallel to the planar discontinuities in the testing material (fig. 5C). In the first 

sequence of tests (o-2 joint parallel), the usual reduction 

in strength is observed as the material anisotropy assumes angles (3=60°) for which sliding along preexisting joints is enhanced (fig. 5.6). With reference to the 

sliding-friction envelopes obtained using equation 9 (see 

also fig. 4F), an approximate value of |x=0.8 appears 

valid for the model material composed of lightweight 

concrete blocks (Reik and Zacas, 1978, table 1). The 

influence of the intermediate stress component is, at 

most, rather slight. The coefficient of sliding friction 

under general states of stress (data lines II and III, 

fig. 5.6) is only slightly greater compared with that obtained in the conventional tests (data line I, fig. 5B), 

and perhaps is attributable to differences in surface 

roughness and misfit of the model blocks rather than 

to differences in the applied states of stress. Thus, sliding-friction coefficients determined by conventional

compression testing of precut rock samples 

(0-3=<T2 =confining pressure) may be readily extrapolated to more general stress states provided no change 

in the stress field orientation with respect to the 

preexisting planes of weakness has occurred. Notice 

should be made that the present interpretation may 

change as more experimental results of the same general nature become available.

The interchange of the least and intermediate stress 

directions relative to the preexisting planes of weakness 

(fig. 5C) leads to entirely different results. This situation may result, for example, from the superposition 

of mining-induced stresses on the preexisting state of 

stress or, simply, by a reorientation of the stress field. 

The preliminary experiments by Reik and Zacas (1978, 

fig. 12) on a model material made up of hard plaster 

blocks show that, as expected under conventional triaxial stress conditions (line H-C, fig. 5C), the strengths 

decrease to a minimum value as the joint set assumes 

an angle of 30° with respect to CTI. However, increasing 

(j2 by a factor of 3 above the constant o-3=0.4 MPa, 

increases the strength of the rock model by about 200 

percent, even though the joint set is most favorably 

oriented to ov In fact, no sliding occurs at 3=60° at
STRESS PATHS 11

elevated magnitudes of o-2; rather, a new shear fracture 

parallel to o-2 is formed in compliance with the Mohr 

hypothesis. Clearly, in the 03 joint parallel example, 

the effect of the intermediate stress component is to 

diminish the material anisotropy to the point of "isotropy" (P=0°, fig. 5C) accompanied by a substantial increase in the shear strength; a condition not covered 

by the Coulomb criterion.

Although it may not be prudent to compare or even 

combine the information contained in figure 5Z? and 5C 

because of the two completely different model materials 

used in the tests (Reik and Zacas, 1978, table 1), just 

such an attempt will be made to illustrate a possible 

geologic situation (fig. 6). Suppose the effective principal stresses at failure are relabeled as follows: the 

greatest stress component is vertical (crj and the two 

remaining stresses are horizontal and are oriented 

either parallel to the joint set (vht) or at an obtuse angle 

to the same (tr/^,); a stress state associated with normal 

faulting. The two failure conditions of a rock mass 

whose anisotropy (planes of weakness) is fixed in space 

at a small angle (30°) to CTV , may then be combined 

synthetically into a short section of a single failure envelope (dashed line, fig. 6) whose overall shape is not 

known, but, for an anisotropic material, may no longer 

be symmetric about the hydrostatic axis (H). Depending upon the magnitude of CT^ , either frictional sliding 

(JJL=constant) along preexisting normal faults is the preferred mode of failure (left panel) or hardening takes 

place (right panel) that eventually leads to the formation of a new normal fault that cuts across the preexisting planes of weakness.

This brief exposition has been limited by lack of data 

to the brittle types of failure, friction and fracture, 

which are predominant in the shallow regions of the 

Earth's crust. At greater depths, as pressures and temperatures increase and depending upon the rock types 

involved, a transition to ductile types of failure modes 

will take place. This general change in behavior will 

manifest itself in the shapes of the failure envelopes 

or surfaces; they will become more rounded (to some 

extent shown in fig. 5C) until, in the purely plastic 

state, they assume a nearly circular shape.

STRESS PATHS

The portrayal of complete stress determinations as 

a function of depth in crustal rocks will be limited to 

a few examples. The data, replotted in the triangular 

stress diagram (or part of it), have been taken from 

the literature without modifications of any sort. As is 

customary, the measured stress components are labeled 

by their presumed orientations and, in part, by their 

relative magnitudes; also, they are assumed to be prin0.4

FIGURE 6. Composite plot consisting of two adjacent panels of 

the triangular diagram that combines the test results for (3=60° 

shown in figures 5B and 5C. Coordinates of the plot are relabeled as <jv normalized vertical effective stress, a/^ normalized horizontal effective stress parallel to joint set, and 

&ho normalized horizontal effective stress oriented at 60° to 

the joint set. With increasing values of a/,0 from a/,0<a^ to 

Vho>Vht, the failure mechanism of the joint set changes from 

frictional sliding (left panel) to the formation of a new fault that 

cuts across the preexisting joint set (right panel). Dashed line 

indicates the approximate position and shape (contour) of a failure envelope that, in the right panel, must bend as it approaches the Coulomb (isotropic) failure envelope O=0°, see fig. 

5C).

cipal stresses. Accordingly, the vertical and horizontal 

total stresses are defined as Sv ^Shmax ^Shmin- Because 

Shmin is always smaller than Shmax m magnitude, only 

one-half of the triangular diagram will be used to display the data; in other words, the right-hand side of 

the complete diagram (fig. 4) is deleted, leaving the 

intervals along line Cc to measure Sv (or CTV). In the 

resulting half diagram, the three panels may be labeled 

with reference to the coordinate stress components by 

the letters T, S, and N to indicate the three main types 

of faulting or slip along suitably oriented, preexisting 

structures (fractures and faults). In the case of slip, 

the appropriate seismological focal-mechanism symbols 

(see fig. 4C), if known, may also be used to identify 

the panels; in fact, by their abundance and distribution 

in certain localities, the choice of failure surfaces may 

be constrained.
12 THE TRIANGULAR STRESS DIAGRAM

TOTAL STRESS PATHS

Figures 7 and 8A present the field measurements of 

in situ total stress in sedimentary rocks (Bredehoeft 

and others, 1976) and in metasedimentary rocks (Gay, 

1977), respectively. The range of depth in both sets 

of data differ considerably: 37-475 m in the shales and 

marlstones of the Piceance Creek basin in Colorado, 

and 110-2500 m in the predominantly quartzitic beds 

of the Witwatersrand Group in South Africa. The 

former were measured with the hydraulic-fracturing 

technique in vertical boreholes from the surface (table 

1); the latter by overcoring in deep mines (Gay, 1977, 

table 1).

Both sets of data show considerable scatter in the 

measurements, but when summarized in each of the diagrams, some general features become apparent that are 

a direct consequence of the rate at which each total 

stress component increases with depth. Depth is taken 

to be colinear with the hydrostatic axis (FT); the data 

are thus projected on the diagram such that the shallow 

measurements appear near the bottom of the diagram 

and migrate upward with increasing depth. For example, in the sequence of measurements in sedimentary 

rocks (solid symbols, fig. 7), the shallow ones appear 

in the T panel and progressively deeper ones traverse 

through the S panel into the N panel. The overall trend, 

whose projection in the diagram is the dashed line in 

figure 7, incorporates the majority of total stress states. 

This trend is called the "total stress" path.

The South African data (Gay, 1977), summarized in 

figure &A, show a somewhat different distribution. As 

before, the measured stress states traverse the three 

panels with increasing depth following a generalized 

total stress path (dashed line, fig. &A). The reflection 

of the total stress path at R merely indicates that an 

overall azimuthal rotation of the horizontal stress components is taking place with increasing depth (see also 

Gay, 1977). More important is the observation that the 

deviatoric stress values (proportional to f see fig. 4E 

and equation 7) diminish at first with depth to a minimum at R (about 500 m) and then increase once again 

(fig. 8B). The implication is that the mechanical stability 

of this rock system is greatest at those depths where 

the deviatoric stresses are smallest and virtually vanish 

near the hydrostatic axis (H). In this respect, note that 

the frequency of mining-induced earthquakes (tremors) 

did not reach a near-steady level of 300 tremors per 

year since 1915 until the mining operations exceeded 

a depth of about 500 m (Gane and others, 1946).

Several studies have indicated that the otherwise tectonically stable region in southern Africa has been locally disturbed to a significant degree by the gold-mining 

process (Cook and others, 1966; McGarr and others,

FIGURE 7. Plot of the distribution of the total stress states 

(solid symbols) and effective stress states (arrowheads) measured in shales and marlstones, Piceance Creek basin, Colorado (modified from Bredehoeft and others, 1976). Shallow 

measurements are projected in the T panel (bottom of the 

half diagram) and progressively deeper ones traverse through 

the S and N panels along a total stress path (dashed line). 

Surfaces connoting frictional sliding along suitable oriented 

faults are included in the S and N panels to approximate the 

upper limits to the effective stress distribution.

1979). The induced stress magnitudes at the head of 

longwall mines are estimated to be of the order of 100 

MPa (vertical), which, when superimposed on the in 

situ total stresses at depth, are sufficient and of such 

orientation to generate new extensional fractures 

(cracks coalescing into rupture zones oriented at small 

angles to the greatest, vertical stress component 

(McGarr and others, 1979, fig. 4)) rather than slip on 

preexisting fractures and faults. Because the mines are 

apparently free of ground water, the combined stresses 

(total plus induced) are effective stresses, and failure
STRESS PATHS 13

0.2

0.6

- 0.2

500

1000

1500

2000

2500

\

\

\

\

0.2 0.4 20 10 0 

r (percent)

10 20

FIGURE 8. A, Plot of the distribution of total stress states (solid symbols) measured in the quartzitic beds of the Witwatersrand Group, 

South Africa (modified from Gay, 1977). Reflection of the total stress path (dashed line) at R indicates the azimuthal rotation of the 

horizontal stress components with increasing depth. B, Vertical section along the total stress path of figure 8A showing the depth 

distribution of total stress states as a function of the dimensionless parameter f (equation 7). The magnitude of the deviation stress, 

Vjz, at first decreases with depth from about 5 MPa at shallow depth to near zero (500 m) and then increases to about 20 MPa 

at 2.5 km depth.

of intact rock predominates with perhaps one horizontal 

stress component approaching tension. An appropriate 

failure surface would lie almost at the left border of 

the diagram (not shown in fig. 8A). The mode of failure 

in these deep mines probably is similar to that observed 

in the model experiments (fig. 5C) by Reik and Zacas 

(1978).

Although the presently available data base is too 

sparse to justify any firm conclusions, the apparent differences in the total stress paths trends shown in figures 7 and 8A may reflect differences in the tectonic 

environments; that is, sedimentary basins versus crystalline basement complexes. For instance, the four 

measured total-stress states in the Michigan basin 

(table 1) despite the scatter aline themselves in a manner similar to that shown in figure 7. The partial sets 

of stress measurements in the crystalline (granites and 

gneiss) rock masses in California, South Carolina, and 

Wisconsin (table 1), however, tend to group themselves 

in different positions in the diagram, but, on the average, follow trends (total stress paths) more like that 

shown by the South African data (fig. 8A). The data 

from these three different localities, all obtained in 

about the same depth interval (100-300 m) below the
14 THE TRIANGULAR STRESS DIAGRAM

surface, also may be partitioned according to the 

geologic age of the host rocks; that is, the data from 

the Precambrian rocks (South Carolina and Wisconsin) 

are contained within the T panel and those from the 

Jurassic rocks (California) fall in the S panel. Considerably more work and data are needed before these matters and their importance or lack thereof can be considered seriously.

EFFECTIVE STRESS PATHS

Knowledge of the interstitial fluid-pressure distribution in the Piceance Creek basin sediments (Bredehoeft 

and others, 1976, table I) enables the calculation (equation 6 ) and plotting of the effective stresses with depth 

(arrowheads, fig. 7). Each point representing a measured state of total stress is shifted in the direction of 

the "effective stress" path (arrow), a distance that is 

proportional to the ambient pore-pressure magnitude 

(see fig. 4D). All effective stress paths used extrapolate 

back through the hydrostatic axis (H). The distribution 

of effective stress state so obtained is closer to the 

limits of the stability field.

In conjunction with the sliding-friction envelopes (fig. 

4F), the mechanical stability of the faulted sedimentary 

sequence at depths greater than 400 m appears to be 

tenuous at best (panel N, fig. 7). This general condition 

is similar to that inferred in the coastal plains of Texas 

and South Carolina (Zoback and others, 1978). A significant external disturbance could enhance slow, but over 

the longterm, continuous movement along preexisting 

and suitably oriented normal faults in the region. 

Likewise, at shallower depths, long-term creep may be 

accelerated by mined excavations (Agapito and Page, 

1976) that cause the threshold creep strength to be exceeded.

Because the normal faults in the Piceance Creek 

basin are known to cut the surface, the use of the sliding-friction envelopes is restricted to the N and S 

panels (fig. 7). At these depths, the stress field is favorably oriented to the existing faults (Bredehoeft and 

others, 1976, fig. 16). At shallow depths, however, 

some other failure surface" not associated with frictional 

sliding must operate whose general shape and position, 

although not known, may in part be located close to 

the shallow stress distribution in the T panel.

The probable connection between fluids injected 

under pressure through boreholes and the generation 

of earthquakes in the subsurface, as well as the accompanying mechanisms, have been amply demonstrated 

by the controlled field experiments near Rangely, Colo. 

(Raleigh and others, 1972, 1976). The essential parameters of the experiment are reproduced in figure 9. Data 

point I in the S panel represents the state of total stress

FIGURE 9. Plot of the stress path associated with increasing 

hydrostatic pressure (dotted line). Data points I, II, and III 

identify the total, effective and critical stress states, respectively, determined by Raleigh and others (1972) in the Weber 

Sandstone, Rangely, Colo. Failure surface for |x=0.81 is 

shown in the S panel as a limit to the increasing hydrostatic 

pressure at which point a preexisting fault system fails and 

earthquakes with predominantly strike-slip motions are generated.

measured by Haimson (1973) at 1.86 km depth in the 

Weber Sandstone (table 2). Incorporating the measured 

value of the ambient pore pressure (P0=16.2 MPa) in 

equation 6 gives data point II, the state of effective 

stress, which lies on an effective stress path that originates at point I and extrapolates back through H, the 

hydrostatic axis.

The existence of a subsurface high-angle fault system 

and the orientation of the operative stress field was 

inferred from the predominantly strike-slip focal 

mechanisms and the subsurface geology, and suggested 

that the failure mechanism was primarily frictional slid-
STRESS PATHS 15

TABLE 2. In situ stress parameters, Rangely, Colo.

[MPa, megapascals; leaders ( ), not applicable]

Stress 

orientation

None 

Vertical 

Horizontal 

maximum -

Horizontal

S

(MPa)

42.7

55.2

! 1 S 

(MPa) (percent)

129.3

33.0

42.7

__ 24 3

p o 

(MPa)

16.2

_

0 /Jo

(percent) (MPa)

11.9

32.9

48.3

IP Q

r

(percent)

9.2

...

ing (Raleigh, 1972). The coefficient of sliding friction 

of the sandstone was measured as |x=0.81 (Byerlee,

1975). By use of this value in equation 9, the appropriate failure envelope can be drawn in the diagram as 

shown in figure 9.

Increasing the fluid pressure in the sandstone reservoir by forceful injection from the surface led to the 

discovery of a critical pressure (Pc =25.7 MPa) above 

which a series of swarmlike earthquakes were triggered 

(Raleigh and others, 1972). By substituting this value 

of Pc in equation 6, data point III can be plotted on 

the extension of the effective stress path that connects 

point I and II, but slightly beyond the failure envelope. 

The discrepancy between point III and the intersection 

of the effective stress path with the straight friction 

envelope (|x=0.81) is small. As mentioned earlier, anywhere along this effective stress path the shear stresses 

remain unchanged and equal to the value at point I; 

that is, f/!=(/2)1/2 (table 2).

The general problem of earthquake triggering, either 

by fluid injection, filling of surface reservoirs, or even 

earth tides, has been treated rather exhaustively in the 

literature (Gupta and Rastogi, 1976; Klein, 1976). Provided that fluid pressures are transmitted unencumbered from the surface into initially highly stressed 

rocks at depth, a rather tight juxtaposition of several 

geologic factors is required to enhance the likelihood 

of induced earthquakes: (1) the presence of near-vertical 

faults or fractures, (2) an in situ stress state favorably 

oriented with respect to these structures, and (3) the 

intermediate stress component oriented in the plane of 

the fractures. The predominance of strike-slip motions, 

and to a lesser extent dip-slip (Gupta and Rastogi,

1976), associated with hydraulically induced earthquakes provides compelling evidence that Sv (the intermediate stress component) exerts little or no influence 

on the frictional behavior of near-vertical fractures or 

faults. Evidently, all that is required to resolve this 

problem is a measure of the least and greatest effective 

stress components (vhmin and crhmax) at the hypocentral 

depths and an estimate of the frictional resistance 

(Byerlee, 1978) of the fractured mass. In these special 

cases (fig. 9), the triangular stress diagram merely reproduces the same mechanical information that can be 

obtained by different constructions (that is, the Mohr 

circle diagram, among others). However, the new diagram allows the reconciliation, although speculative depending on the quality and quantity of available data, 

of conflicting information obtained by different methods 

at the same location.

A case in point is the Lake Jacossee area in South 

Carolina. Shallow hydraulic-fracturing measurements 

between 120 and 250 m depth in metamorphic rock 

(gneiss) obtained by Haimson (1977b) resulted in the 

delineation of the states of stress and the orientation 

of Shmin:N. 30° W. Composite focal-plane solutions (Talwani, 1977) of swarmlike earthquakes triggered by the 

filling of Lake Jacossee, on the other hand, indicated 

predominant strike-slip motions on near-vertical fractures or faults at a depth of about 1 km and dip-slip 

motions at greater depths to 3 km. The average orientation ofShmin (^-axis) at these depths is N. 35° E.

Much of the available information is reproduced in 

figure 10. The sequence of measured stress states plotted in the T panel suggests a likely, but admittedly 

speculative, extrapolation of the stress distribution with 

depth by means of a total stress path (dashed line, fig. 

10) that, after reflecting at R, continues into the S 

panel. With increasing depths, the horizontal stress 

components assume new azimuthal directions. The consequent increase of the pore pressure at depth due to 

the filling of the surface reservoir may then shift the 

stress states closer to the failure surfaces along the effective stress path (dotted line, fig. 10) where the likelihood of slip along preexisting, near-vertical and northsouth-trending faults (see focal-mechanism symbol) is 

good. The preponderance of dip-slip events at greater 

depths (3 km) is indicated by the dashed arrow (fig. 

10). At still greater depths, the effect of the intermediate stress component (Shmax) may become important and diminish the likelihood for further slip because 

of the increased shear strength (hardening) of the fractured rock mass.

This interpretation is in general agreement with the 

observed facts (Talwani and Stevenson, 1980). The apparent disagreement between the shallow and deep 

stress directions obtained by entirely different methods 

may be resolved by assuming that the rate of increase 

of the near-surface greatest stress component is smaller 

than that of the other two components so that at hypocentral depths it becomes the least-stress component. 

The two remaining stress components increase at about 

the same rate. The resulting state of stress at depth 

is also sufficiently high and of the proper orientation 

in a relatively small interval of depth to culminate in 

the generation of earthquakes following a rather small 

increase in the fluid pressure. Further occurrence of
16 THE TRIANGULAR STRESS DIAGRAM

0.2,

FIGURE 10. Plot of the probable relation between stress states 

measured at shallow depths (solid symbols) by Haimson 

(1977b) and focal-mechanism solutions for strike-slip events at 

1 km depth (Talwani, 1977; Talwani and Stevensen, 1980) near 

Lake Jacossee, S.C., by means of a total stress path (dashed 

line) and an effective stress path (dotted line). Dashed arrow 

indicates the approximate failure path leading to normal-faulting events at 3 km depth.

earthquakes at much greater depths (>3 km) may be 

prevented by the continued rotation of the stress field 

with respect to the "fixed" orientation of the prevailing 

structures.

One of the main differences between the two examples discussed in this section is the magnitude and rate 

of application of fluid pressures necessary to trigger 

the earthquakes. At Lake Jacossee, the earthquake sequence commenced about 7 months after the reservoir 

had reached its maximum head of 102 m (Talwani, 

1977). The surface water-pressure of a hypothetical reservoir required to overcome the frictional resistance of

the subsurface fault at Rangely could have been generated only by constructing a dam more than 900 m high. 

This improbable comparison, nevertheless, serves to 

show that the moderately stressed sandstone responded 

as it did (elastically) primarily because of the rapidity 

of the imposed disturbance; that is, high rates of injection through a borehole. Had the injection rate been 

much lower, as it would be in the hypothetical example 

of a manmade lake, then the threshold creep strength 

of the sandstone would have been exceeded only incrementally and, on the average, would have attenuated 

the protracted disturbance. The crystalline rocks under 

Lake Jacossee apparently are stressed near their shortterm strength and readily fail in response to small, but 

slow increases in fluid pressure.

DISCUSSION AND CONCLUSIONS

A new graphical method has been introduced that 

provides a clear and concise method of representing the 

measured stress states and their distribution in crustal 

rocks. Point measurements of compreSsive states of 

total stress in rocks, defined by the magnitudes and 

orientations of three mutually orthogonal stress components, are plotted in dimensionless units as points in 

a triangular plane of projection that is normal to the 

axis of hydrostatic stress (hydrostat or space diagonal 

in Cartesian coordinates). Each measured state of total 

stress occupies a unique position in the diagram and 

carries information on the relative depth, spatial ordering of the stress components in terms of relative magnitudes and orientations by their appearance in six different panels, and the magnitude of the stress deviation 

(stress difference, shear stress).

The dimensionless form of the triangular diagram 

provides for the convenient one-to-one comparison of 

the measured stress distributions in different structural-geologic environments (figs. 7 and 8A). Given sufficient data as a function of depth, the stress distribution 

may assume, with due allowance for scatter, an overall 

trend (total stress path) whose direction in the diagram 

may, in turn, be characteristic of the structural-geologicor tectonic environment. Furthermore, the very reason 

for the existence of these trends must reside in the 

fact that the conditions of equilibrium and continuity 

of stress in the rock column are satisfied. An additional 

point of interest is the observation, although uncertain 

at this time due to insufficient data, that the stress 

deviation as measured by the radius-vector f (fig. 4E) 

at first decreases with increasing depth to a minimum 

value and then increases again (figs. 7, 8A, and 10). 

The point (depth) of closest approach to the hydrostatic 

axis (center of the diagram) varies with each locality,
REFERENCES CITED 17

but implies the attainment of maximum mechanical stability, particularly with regard to the strong, brittle 

rock types (fig. 8A).

Another advantage of the triangular diagram is that 

stress states that differ only by an added hydrostatic 

pore pressure are plotted as separate points (effective 

stresses) along an effective stress path that always rectilinearly extrapolates to the hydrostatic axis (figs. 4F 

and 9). All these stress states, however, share the exact 

same stress deviation associated with the state of total 

stress (pore pressure, P=0) from which the path originates, even though with increasing hydrostatic pore 

pressure the points are plotted farther outward from 

the hydrostatic axis.

The stress distributions are limited in the extreme 

by some type of surface beyond which the stresses cannot increase without the rock material rupturing or 

yielding. Such surfaces can be constructed only point 

by point by conducting experiments or they can be determined approximately by invoking a suitable failure 

hypothesis or criterion. The latter approach appears 

iustified in characterizing the rupture strengths of isotropic rocks (fig. 5A) and the frictional strengths of 

properly oriented planar discontinuities (fig. 55). The 

experimental approach evidently is required to determine fully the strength properties of anisotropic rock 

material under general states of stress; a problem of 

great practical importance to rock mechanics because 

no failure hypothesis of valid generality yet exists.

The main properties of the triangular diagram have 

been illustrated, to the extent possible, by using published measurements of total stress in crustal rocks 

(table 1). Most of these measurements have been obtained by the hydraulic-fracturing method that, in the 

least, provides reliable data on the magnitude of the 

stress component normal to the induced fracture; there 

is a correspondingly greater uncertainty in estimating 

the magnitude of the stress component parallel to the 

fracture. The vertical stress (only rarely measured) is 

calculated directly from the weight of the overburden. 

A further restriction stems from the working assumption that everywhere the principal stress components 

are vertical and horizontal. Thus direct comparisons 

with the results from other techniques (for example, 

overcoring) or, in terms of orientations, from earthquake focal-plane solutions are often only approximate. 

Nevertheless, the triangular diagram provides a practical means of combining these bits of information and, 

by so doing, perhaps improves and furthers the understanding of the behavior of crustal rocks.

Also, the examples used here have been restricted 

to the localities in which the measurements were obtained. Clearly, no longer can all data be combined 

from, say, all hard-rock localities (see figs. 1 and 2)

because each data set plots in different positions or 

panels in the diagram and along different trends (compare figs. 8A and 10). The situation is somewhat better 

in sedimentary environments that appear or are known 

to be tectonically quiescent.

The division of the triangular diagram into six panels 

(fig. 4C), or three panels in the half-diagram (figs. 7, 

8A, 9, and 10), each connoting a precise arrangement 

of stress magnitudes and orientations, allows for the 

structural or tectonic analysis of the interaction and 

probable consequences of the prevailing stress field on 

the existing structures, if known. Strictly speaking, the 

analysis is a qualitative proposition, with the quantitative aspects improving as the data base increases.

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