•

PREDICTION OF COMPRESSIVE STRENGTH FROM OTHER ROCK PROPERTIES

By D. V. D'Andrea, R. L. Fischer, and D. E. Fogelson

• • • • • • • • • • • report of investigations 6702

UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU OF MINES

\965 ..

This publication has been cataloged as follows:

•

D'Andrea, Dennis V Prediction of compressive strength from other rock proper­

ties, by D. V. D'Andrea, R. L. Fischer, and D. E. Fogelson. [Washingtonl U.S. Dept. of the Interior, Bureau of Mines [1965]

23 p. illus., tables. (U. S. Bureau of Mines. Report of investiga­tions 6702)

Includes bibliography.

1. Rocks. 2. Rock pressure. I. T itle. (Series)

TN23.U7 no. 6702 622.06173

U.S. Dept. of the Int. Library

T

CONTENTS

Abstract ...... . .. .. ... . .. . •... . .... . ................ . . . ........ . . . . . ... .. Page

1 1 2 2 4 6 6 6 6 8 9

Introduction .... . ............ .. .. . ..... ... . . . . . .... . . . . ..... . ........... . Acknowledgments . . .... . .... . ... .. .. . .... . . . .. . . . . .. ... . . . . . .......... . .. . . Rock properties . .... . ... .. ... ... .. . . .. ...... ... .... . . . . . .. .. . • . . . ... . .. . . Experimental data . .... .... .... . . . . ... . ... . ........... . . . ... ... . . . . . . . . . . . Analysis of data ... . .. . . .. ... .. .... . . . ... ..... ................... . . . .. . . .

Plots of compressive strength versus the other properties . . ..... . ..• Regression analysis .. ... • . .. ... . ...... .... .. .. . .. .... ... . . . . . ....• . .

Description of program . .. . .. . . . .... . . ... . .... . . . ..... .... . ... . . Results ·of linear analysis . . ..................... . ... .. . ..... . . Results of curvi l inear analysis . . .... . ...... . . . .... . ... .. .. . .. .

Other relationships ... .. . .. . . . . .... . ...... . .....•... . ....... ... . . . . . Summary and conclusions . .. ... . . ...... . .. . . . ... .. .. . . . . . . . .. . ... . . . .... . . .

14 21 22 23

References . . . ... ... .... .... .. . . .... . . . .... . . . . . . .. . . . . .... . . . . .......... . Appendix . ........................ , . . . . .. . . . .. . ... . ... ... . . . ... ... . ...... .

Fig . 1. 2. 3. 4. 5.

6 . 7. 8. 9.

10. 11. 12. 13 .

1. 2.

3 . 4 . 5 . 6. 7.

A-1.

ILLUSTRATIONS

Point load tensile testing apparatus . ..•... . .......... . .... . ..... . . Plots of compressive strength versus the other properties .. . . .. . .. . Measured versus predicted compressive strength from equation (1) . . . Meas~red versus predicted compressive strength from equation (8) . . . Measured versus predicted compressive strength from equation of

step 25 . . • .. •.•. .• .. . . • .. • . . .. •••••.. . • • •. • . •. . • .• . •• ...•. . . •.. .. Measured versus predicted compressive strength (Judd and Huber) .. . . Plots of specific gravity versus other properties . . ... . .. . ........ . Plots of shear velocity versus other properties . . . .. . ..... . ....... . Plots of Young ' s modulus . .. . • .. . . ...... . ...... . . ........ . . . . . .... . . Plots of modulus of rigidity versus other properties .. . . . ......... . Plots of bar velocity versus other properties ..... . ............... . Plots of longitudinal velocity versus other properties . . . . ..... ... . Plots of point load tensile strength versus other properties ..... . •

TABLES

3 7

12 13

14 15 16 16 17 17 18 19 20

Physical property data . . .. . . . . .... .. . .. . . ... .. . .... .. .. . . . . . ..... . . 5 Number of tests and coefficients of variation of the measured

rock properties . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Quantities used in linear analysis ..................... . ........... 8 Simple correlati on coefficients. . ................. . ... .... ..... ... . 8 First and second order terms used in curvilinear analysis ......•.. . 10 Prediction equation of step 25 . . . .• . • . • . . • . ••••••••• . ••• . ••• • •.•••. 11 Multiple correlation coefficients of curvilinear regression

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Averages, standard deviations, standard errors , and coefficients

of variation of compressive strength and point load tensile strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . 23

PREDICTION OF COMPRESSIVE STRENGTH FROM OTHER ROCK PROPERTIES

by .

D. V. D'Andrea, 1 R. L. Fischer, 2 and D. E. Foge lson 3

ABSTRACT

Nine rock properties were determine d for rocks coming from 49 locations and having a wide range of compressive strengths . Plots of the nine rock properties versus each other were prepared . A stepwise multiple linear regression analysis was performed to obtain several equations fo r predicting compress.i ve strength . Both linear and curvilinear relationships \17ere assumed between compressive strength and the other rock properties . The prediction equations had multiple correlation coefficients ranging from 0 . 947 for an equation with 1 variable (point load tensile strength) to 0.986 for an equa­tion with 25 variabl e s . Of the properties tested only point load tensile strength could be used a lone to estimate compressive strength with fair accuracy.

INTRODUCTION

The use of multiple regression analysis to develop pr ediction equations for rock properties as functions of othe r rock properties is a relatively new a pproach to the study of the physical properties of rock . The purpose of these studies is to determine if it is possible to predict particular proper­ties , such as compressive strength, from measurements of other more eas ily measured properties. Obviously, for any practical use in engineering problems, the deviations of the predicted from the actua l values must be as small as possible .

Any analysis of this type has a better chance for success if it uses data obtained under uniform test conditions. Such data had been taken previously

1 Geophysicist, Minneapolis Mining Research Center, Bureau of Mines, Minneapolis, Minn .

2 Physicist, Minneapolis Mining Research Center, Bureau of Mines , Minneapolis, Minn.

3 Research geophysicist, Minneapolis Mining Research Center, Bureau of Mines, Minneapolis, Minn.

Work on manuscript completed February 1965 .

I •

2

by the Bureau of Mines (1-£, 10-1!) .4 These data we r e used by Judd and Huber (l) to develop a prediction equation expr essing compressive strength as a function of impact toughness, scleroscope hardness, and the modulus of rigid­ity. While the deviation of the predicted from the measured values of com­pressive strength was too large for most practical applications , their results suggested that a better prediction equation for compressive strength might resul t if other physical properties were used in the analys is or if a curvi­linear r elati onsh i p was assumed .

Dur ing the pas t 2 years, in conjunction with the deve l opment and study of new me t hods of measuring rock properties, the Bureau of Mines performed a simil ar analysis in an attempt to improve the prediction equa tion for compres­sive s trength . Bo t h l inear and curvilinear relationshi ps tvere assumed in an a nalysis that included several different properties from those u sed by Judd and Huber.

The properties analyzed in both studies a r e listed below.

Bureau of Mines

Compress i ve strength Young's modulus (dynamic) Modulus of rigidity Bar velocity Specific gravity Point load tensile strength Longitudinal velocity Shear ve l ocity Poisson ' s ra t io

Judd and Huber

Compressive strength Young ' s modulus (dynamic) Modulus of rigidity Bar velocity Specific gravity Porosity Modulus of rupture Impact toughness Scleroscope hardness Specific damping capac ity

ACKNOWLEDGMENTS

The author s a r e indebted to John Shaw, research director, Bureau of Mines Denver Mining Resear ch Center, for his coopera t ion and the use of the GE 225 computer, and to Miss Pauline Vi r ciglio, mathematician , Bureau of Mines Minneapolis Mini ng Research Center, for invaluable assistance with the compu­tations and sta t is t ical analysis in this report.

ROCK PROPERTI ES

All rock property tests tvere performed on rock cores having standard dril l cor e diameters of 1-1/8, 1-5/8, or 2-1/8 inches .

Compressive strengths were determined in the conventional manner by plac­ing a rock core between the loading heads of a compression testing machine and loading the specimen to failure . The cores tested had lapped ends and length­to - diameter ratios of approximate l y 2 to 1.

4 Under lined number s in parentheses refer to items in the bibl iography at the end of this report.

and

FIGURE 1.- Point Load Tensile Testing Apparatus.

Point load tensile strengths were obtained by applying a compressive point load to the surface of a cylindrical core perpendicular to the axis of the core . Figure 1 shows the geometry of this test. The point l oad tensile strength is com­puted from the following expression .

T = 0 . 96 ~3

3

where F is the breaking f orce and d is the core diameter . This test requires no special core preparation and therefore can be performed rapidly .

Specific gravity was computed by dividing the weight of an air-dried spec­imen by the specimen volume determined from its exterior dimensions. Longi tud­inal velocity was obtained by using an ultrasonic pulse technique . A resonance method was used to determine bar and shear velocities. A more complete description of these test methods is given by Johnson and Fischer (~).

The modulus of rigidity,~' Young ' s modulus, E, and Poisson 's ratio, \) , ~vere computed from the measured velocities and specific gravity by the following formulas:

~ = k p V/a [3 (C/V5 )2

- 4] (C/V5 ) 2 1

\) 1/4 [J[l- (V9 /C) 2 ] [ 9- (V9 /C) 2 ] - [ 1 - (VB I c) 2 J J ' ~mere p specific gravity,

c longitudinal velocity,

vs shear velocity,

4

V8 =bar velocity,

and k =a constant depending on the units used .

Two methods based on different ve l ocity measurements were used to compute Young's modulus . Because they gave different results for many rocks and because it was not known which method would be most useful for predicting com­pressive s trength, the two were treated as separate variables in the analysis.

EXPERIMENTAL DATA

Rock property measurements were made on rock cores from 49 different locations . The loca.tions were sel ected so that rocks with a wide range of properties could be tested . .The 49 locations included 19 rock types, and com­pressive strengths ranged f rom 1,500 to 45,000 psi . Table 1 gives the physical property data and the rock types for the 49 l ocations . The number of digits shown in table 1 for each property does not indicate the e rrors in the meas­urements. Each property was determined by averaging several measurements, and these averages were no~ rounded f or the computer anal ysis .

The number of measurements performed to determine a property was not the same for al l l ocations because the amoun t of core available from each location varied. The average number of measurements and the average coefficient of variation for each measured property are listed in tabl e 2 . The coefficient of variation , C, in percent, for a measured property of a rock from a particu­lar location is given by

c = 100 ~ X

lmere S is the standard deviation and X is the average value of the measured property . The measurements of t he breaking strength of the rock , point l oad tensil e strength, and compress i ve strength have coefficient of variation of about 20 percent. The coefficients of variation ranged from 4 . 5 to 47 . 8 per­cent for compressive strength and from 6.6 to 54 .9 percent for point load tensile strength . Table A-1 (appendix) gives the average, s tandard deviation, number of measurements , standard error, and coefficient of variation of t he compressive strength and poin t l oad t ensi l e strength for each of the 49 loca­tions . The average c~efficient of variation for the nondes t ructive tests was l ess than 6 percent.

TABLE 1. - Physical property data

Rock tvoe Location

l'innt load

tensile s trength, 103 psi

Speci fic gravity

Longitudinal ve l ocity, 1cf! fps

Shear velocity, 103 fps

Bar velocity, 103 fos

Granite gneiss .... 1 2.054 2 . 65 18.367 11.217 17 . 483 Do....... .... 2 1.985 2.80 18 . 800 11.267 17 .633

Slate..... . . ...... 3 1.518 2 . 71 18.100 10.700 16 . 817 Limestone....... . . 4 , 1.:,_31:_4 2. 74 20.928 12.006 18 .950

Grani te . ... . ... . . . 5 ,---1'.308 2. 70--[-1-5.888 --9-:-233 -r~ Limes t one • • • •••• • • 6 1.408 2.71 20.808 11.650 18.612 -racon~~-- 1 2 .474- 2. 95 - 2o.14o n.s6o 18 .4oo

Do...... . .... 8 2 .483 3.07 20.071 11.428 18.428 Granite . ......... 9 1.153 2.68 16.571 10. 143 15.367

Do. .......... 10 1.882 2 .65 17.200 10.328 15 .786 Syenite pegmatite 11 1.582 2.72 19.183 11. 150 17 . 283 Granite .••.. ••• • • 12 1.688 2. 70 16 . 871 10.314 15. 467 Limestone • • . ··~--13 1.340 2. 84 19. 550 11. 567 17.520 Anor t hosite ...... 14 1.. 680-- 2. 64' - 17. 533 - ro.840 [6.375

~lodulus of ri§id ity, 10 psi

Young 's modulus

(El), 106

PSi

Young' s modulus

<.!'~) ' 10' psi

Poisson's ratio

Compressive strength, 103 nsi

4.50 10.93 10 .82 0. 195 30.233 4.80 11.75 11. 70 .217 32.551 4 . 19 10 . 35 10 . 3 2 . 230 24. 98 9 5.33 13 . 28 13 . 36;_ , ___ ..... 25;..;8,__. 22 .158 3 .1-1-- 6 . 17 --1~i4 .327 27. 02'0-1-4 .96 .. 12 . 67 . 12. 61 .270 21 . 265 5.32 13 . 48 13.34 . 249' 36.401 5. 41 14.07 13. 63 .244 33.892 3.72 8.54 8.93 .232 19.153 3.82 8 . 92 9.30 .244 30 . 361 4.56 10 . 97 11.35 . 263 18.814 3 . 88 8 . 72 9. 33 . 245 31 . 415 5 .13 11.77 12.63 . 268 26.022 4.19 - - 9.56 '9-:-97 .223 34 .871~+--·---

Limestone .. . ..... 15 .911 · 2.65 16 .242 9 .716 13.933 --sanGStone ~~ - .o·6o- - 1.88---·-s :-534 - 2. 923 3~9Cf3 --

3.38 6.94 8.26 .303 1-16 . 4~~ -----.u- - . 39 --.57 . 3'91- j . 4o7

Limestone ··· · · .:..:...:. 17 .963 2.70 16.300 9 .433 14.650. Do ........... --r8 - .9'3,- 2.57 - 1.4 .15o-r--s.-600' -l2~7oo Do . .......... 19 . 736 2.61 16 . 067 9 . 450 13 .840 Do ...... ... .. 20 .827 2.61 16. 140 9 . 333 15 .250

Basalt • • • • • • •• • • • 21 1.552 2. 88 - 17. 150 - 10.075 !).2'50-R:hyolite porphyry 22 . 987 2.49 13. 583 8.483 12.417 Serpentine •.•.• •• 23 .740 2.75 17.633 9. 117 14.417 Limestone •• . . • ••• 24 . 708 2. 54 14.183 8 . 533 12.883

olomite .••••••• :- --25 .50~ 2 . 51 -- 16.392 -s. 63) 13:'Io7

3.24 7.82 8.09 .266 14.074 2.60 5.65 li'. 24 . 262 16.633 3 .15 6.75 7.78 . 300 15.559 3 . 07 8 . 19 7.66 . 206 19 . 519.--1---3.95 - 9.CJ4 - '9-:-n- - ---:-2..,.,- - 21.63'o 2. 42 5 . 18 5 . 70 .248 13 . 934 3 .08 7.72 8.12 .333 16 . 351 2. 50 5 .69 6.08 .255 17.0~3.;.0_..__ 2.S'2 --- s.87 o .59 . 342- n.796

Limestone. ....... 26 . 670 2.~~ 13. 517 8.291 12. 508 sandstone . • • • • • • • i l -:-l:r8r- 1 . 8~ _ -- 5. 1'33 -f--3~76'o s:·rr.s-Dolomite . • • • • • • • • --28 . 366 2. 54 13 . 200 7. 250 10 . 933 Sandstone .• • • • • • • 29 .045 1.86

1_ 4.088 2.535 3 .352

-sv3te ::. .. . . . . . . . 3'o . 92r 2. 64 16 . 950 9. 380 16 .375

2.35 __ 1__5 .3~ ___ 5_. ?_3 .235 13.330:_+-.36 .68 . 81 . 261 4.334

1. 80 4 . 10 4 . 6 2 . 325 7. 943 . 16 • .28 .38 .331 2. 451

3.14 9.56 8 . 03 .167 12.390 Granite gneiss • • • 31 .854 2.62 12.140 7.075 8 . 854 Taconite • • •.•.. • • 32 2.317 3.62 16.456 8 . 925 13.525 Limestone ••••• • •• 33 .322 2.34 11 . 032 6.708 9 . 067 Chalk . ••••..••.•. ~ --. fJ-2--1.68 - 8 . 205 1-4:-o-52 --5-:/73

1.77 2.77 4.40 . 381 19.710 3.89 8.94 10 . 05 .330 44.201 1.42 2.60 3.43 -~ 5 . .;.11;;1;..--.J--

. 37- .76 . . 99 - . .)~ £ - 2.169 Do.......... . 35 .075 1.60 6.472 3.333 4 .567 .24 .45 . 63 .391 2.765

Sandlli!.le .• • •• • • • 36 .165 2. 18 I ~ 6.907 3.616 4 .688 Grani t e ... .. .. ... 37 1.800 2.72 16 .700 10 . 000 14 . 600

.38 • . 65 1.00 .402 5.925 3 .67 7 . 83 8.96 . 289 28.400

Do .. . . .. . .... 38 1. 340 2. 67 19. 500 10 . 600 16.500 4.05 9.81 10 . 20 .312 19.400 Basalt •• • • • • ••••• 39 2. 290 2 . 96 21. 700 11 . 900 19.100 5 . 65 14.58 14.51 . 284 42.349 Limestone 40 •. 510 2 . 30 13.100 7.480 11 . 500 M8 r ble ••••• •• •••• 1--·4r.f.;..-.1-lf--.2:-2~ 3 . 04 21. 992 - 12.617 19~350

1.74 - 4.11 4 . 38 .286 7.680 6.54 15.39 16 .41 .284 36.40tl -i--

Schist •••••• . .••• 42 1.273 2. 74 18 .478 10 . 663 16 .488 4 . 21 10 . 06 10 . 53 . 272 20.625 Per i dotite ....... 43 .999 3 . 30 21.650 12. 533 18.900 Quartzite . . ... . . . 44 1.518 2.59 14.407 9 . 343 13.257 Schist .• • • ••••• • • 45 1. 330 2. 85 17.980 10.300 17 . 020 Sandstone • • •• . •. • 4.6 .040 1.87 6 . 870 3.907 5 .845 Quartzite 47 2 . 530 2 . 63 16 . 049 11.0 21 14 . 775 Gabbro.. . ...... . . 48 1 . 305 3 . 10 23. 133 12 . 000 18.725 Greenstone . . ..... 49 1 . 612 2 . 79 20 . 178 11.000 17 .857

7 . 00 15 . 91 17.46 . 291 17 . 750 3 . 05 6.ll, 6.94 . 241 31. 167 4.08 11. 15 10 . 24 . 203 24 .010

.39 .86 . 98 . 309 1. 540 4.31 7.75 9 . 08 .241 45 . 172 6.03 14.66 15 . 86 .338 29.047 4.56 12.01 11 . 76 . 279 16 . 546

6

TABLE 2. - Number of tests and coefficients of variation of the measured rock properties

Property

Compre ssive strength ......... . Point load tensile strength .. . Specific gravity ............. . Longi tudinal velocity . . ...... . Bar velocity .... . ....... . .... . Shear velocity . . .. . .. . . ... ... .

Average number of tests per

location 11 17

5 7 6 6

ANALYSIS OF DATA

Average coefficient of variation,

percent 22 . 2 19 . 6 1.9 5 . 4 5 . 8 5 .4

Plots of Compressive Strength Versus the Other Properties

The rock property data were put on standard keypunch cards for computer analysis . A computer and high-speed printer were used to plot compressive strength versus each of the other properties. These plots are shown in figure 2. ~.fuile most of these plots have substantial sea tter, some trends are apparent. For example figure 2! indicates that a linear relationship exists between compressive strength and point l oad tensile s trength . The plots of compressive strength versus s peci fic gravi ty and compressive strength versus shear ve l ocity, shown in figur es 2~ and 2~, respectively , indicate cur vilinear trends .

After examining these plots it was decided to perform a stepwise multiple linear regression analysis and attempt first to predic t compressive strength as a linear function of the other rock properties and then to see if a better prediction equation coul d be obtained by assuming a curvilinear relationship .

Regression Analysis

Description of Program

The multiple linear regression program used was written6 from a method outlined in 'Mathematical Methods for Digital Computers"~). The input data for this program are a set of observations of several independent variables and a dependent variable . The progr am output l ists the means and standard devia­tions of all variables and the s imple correlation coefficients between all pairs of variables. It then lists a series of pr ediction equations , each one containing one more independent variable than the last . With this stepwise procedure a new variab l e is added to the variables of the preceding step until some final prediction equation is obtained . The new variable sel ected at any step is the one that resul ts in the prediction equation with t he smallest error of estimate . I f two variables are very highly correlated, the final prediction equation will contain only one of them . The addition of a new

5 computer pr ogram was written by D. R. Falconer and P. A. Hunter of the Southwes t Research I nstitut e .

. ·- 40 I .. 0. .. .

"' 2 /

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~ 30 ~

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I ' / UJ a: 1- .. .. V> 20 .. J • UJ > ~ .. 1 ;;, <ll / ~ UJ · l"" • a: -Cl. 10 ::E

.0 0 v.· <..>

I 0.5 1.0 1.5 2.0 2.5 0

POINT LOAD TENSILE STRENGTH, 103 p$l

-;;; 40 Q.

"' 2 :£ 1- 30 ~ z w a: 1-<ll

w 20 > <ll <ll w a: Cl. 10 ::E 0 <..>

·;;;40 0.

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"' "' w g: 10 :::E 0 <..>

.. j . . . . .. . . . • . . . . . . . '·. . , . ..

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0 4 8 12 SHEAR VELOCITY, 103 f p s

. . ,.

. . . .. . . . .

0 0 5 10 15 YOUNG'S MOOULUS(£1), 106p s i

• 40 .. ..

0.

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0 I 2 3 SPECIFIC GRAVITY

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20 . w .. ~ I :• <J) <I)

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0 5 10 15 BAR VELOCITY, 103 f p s

·;;; 40 f-a.

"'o :£ ::; 30 z w a: 1-(/)

w20 > ;;, <ll UJ

g: 10 :::E 0 <..> . .

·'•

,

. . . . .

0 5 10 15 YOUNG'S MODULUS(£2), I06p s i

7

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20 . UJ .. , ~ .. <J) . <ll , . w •• a: 10 ... ::E 0 0 <..> . .. · ..

0 5 10 15 20 LONGITUDINAL VELOCITY, 103 f ps

·~ 40

"' 0

:£ 1-~ 30 z w a: 1-<I)

w 2 0 > <I) <I)

w a: 0. 10 :::E 0 <..>

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w 20 ~ (/) <J) w g: 10 ::E

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: . . . . . . . ..

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0 2 4 6 MODULUS OF RIGIDITY, I06 psl

0

0

. .

. . .. . . . . . .·· ..

0.1 0.2 0.3 POISSON$ RATIO

' 0 .4

FIGURE 2. ·Plots of Compressive Strength Versus the Other Propert ies ..

8

variable that is very highly correlated 'vith a variable in the equat ion of the preceding step will not result in a better prediction equation because the new variable in this case is only a coded value of a variable already in the analysis.

Results of Linear Analysis

The firs t step in the regression analysis was to see how well compressive strength could be predicted from a linear function of the other rock proper­ties. The quantities included in this analysis are shown in table 3 . The independent variables will be referred to as first order terms.

TABLE 3 . - Quantities used in linear analysis

Variable Symbol Property Units Yp c.s. Compressive str ength . . .. . . ... ... 10~ psi. xtl) T Point load tensile strength . .. .. 103 psi. X(2) p Specific gravity ............ .... -X(3) c Longitudinal velocity ...... . ... . 163 fps. X(4) vs Shear ve locity .... ... .. .......•. 103 fps . X(5) Va Bar velocity . . .. .. . .. ...••. .. ... 103 fps. X(6) 1-L Modulus of rigidity ..... . ..•.... 106 psi. X(7) El Young' s modulus (method 1) .. .... 106 psi. X(8) Ea Young's modulus (method 2) .. .. .. 106 psi. X_{9) \) Poisson ' s ratio . .. ..... . ... .••. . -

Simple correlation coefficients between all pairs of variables listed in table 3 are given in table 4. Compressive strength is most highly correlated with point l oad t ensile strength and least highly correlated with Poisson's ratio. Very high correlations exist between the modulus of rigidity and Young's modulus computed by method 2 and between the two methods of computing Young's modulus .

TABLE 4. - Simple correlation coefficients

..--l Cd Q.) c: 4-1 ........ ........ :>

·r-l 0 ..--l N Ul •r-l C) '"d I>. I>. :>-. :>-. - Ul,.t:

•r-l :>-. ;:I .j.J .j.J .j.J Ul .j.J Ul(f)'"d Ullll'"d c: Ul .j.J 4-l.j.J .j.J ·r-l ·r-l •r-l ;::l •r-l - ;:I 0 - ;:I 0 0 Q.) bO ·r-l •r-l ·r-l (.) ,_. (.) (.) r--l'"d bO..--l ..t: bO..--l ..t: II) 0 ,_. r:: (.) :> bOO Cd 0 0 ;:I ·r-l c: ;:I .j.J C:;::l.j.J Ul ·r-l P.,Q.) Q.) Cd C:r--l Q.)..--l ,_...--l '"d bO ;::l'"d Q.) ;::l'"d Q.) ·r-l .j.J l:i ,_. p..,_. .3 Q.) ..t: Q.) Cd Q.) 0 •r-l o o a ~~s 0 Cd 0 .j.J

C/) bO :> C/) :> j:(l :> ~ ,_. :;... a ......... P-< ,_. (.) Ul

Tensile strength.*.' .... 0 . 760 0.761 0 . 808 0 . 785 0.813 0 . 784 0. 790 -0.491 0 . 947 Specific gravity .. • .... - . 866 .856 .842 . 864 .841 . 869 - . 361 .761 Longitudinal velocity . . - - . 980 . 985 . 960 .959 .964 -.462 .710 Shear velocity .. . ...... - - - .988 .961 .940 . 951 -.568 .761 Bar velocity . .. ... .. . . . - - - - .955 .961 . 953 -.595 .720 Modulus of rigidity . ... - - - - - .984 . 997 - .465 .767 Young 1 s modulus

(method 1) . ••••. •. .... - - - - - - .990 -.487 . 718 Young's modulus

(method 2) ...... .. .... - - - - - - - - .431 .742 Poisson's ratio . . . . . ... - - - - - - - - -. 451

9

The prediction equations obtained in the first four steps of this analy-sis follow with their standard error s of estimate, SY:

<_ \ .:: ~/c• c = IE> f. 4- IS 13 T ~~c..~ SteE 1 ) ........

y "P = 2.374 + 15.295 X(l); c~ 2,4oo + 15 . 3 T "PS l (1)

sy = 3 . 744; ( T= PSI) I<C./c.M 'Z. c e.) t7o+ 17 . 3 T

SteE 2 ( T= K~/cM7) = pott-lT LOA!) STQ

Y.., = - 3 . 840 + 14.092 X( l) + 2.924 X(2); (2)

sy = 3 .709;

Step 3

y p

:::: -9 .989 + 15.200 X(1) + 6.371 X(2) - 0.525 X(7); (3)

sy 3 .561;

Ste p 4

y p -7 .442 + 14 . 445 X(l) + 4 . 669 X(2) + 3.877 X(6)- 1.818 X(7); (4)

sy = 3 . 431.

The first variable selected was point load tensile strength, X(l), fo llowed by s pecific gravity, X(2), Young ' s modulus (E 1 ), X(7), and the modulus of rigidity, X(6). F tests were performed to determine whether one of these equations was a statistically signif icant improvement over another. At the 5-percent significance level , equat ion (2) is not a significant improve ­ment over equation (1) . However, equation (3) is significantly bette r than equation (1) , and equation (4) is a s ignificant improvement over equation (3).

Results of Curvilinear Analys is

Figures 2] and 2Q, the plots of compressive strength versus speci f ic gravity and shear velocity, indicate curvilinear trends. A more accurate pre­dicti on equation was expected, i f the squares and cross products of the nine first orde r terms were also entered as variables. All of the 54 first and second order terms could not be included , because the computer used had only an 8,000 word memory, limiting the r egression program to 35 independent vari­ables. All of the terms could not have been entered even on a larger computer because there were only 49 measurements of compressive streng th . However, the t e rms eliminated were highly correlated with others that wer e i ncluded . Table 5 lists the 35 quant i ties that \olere entered as independent var iables.

10

TABLE 5. - First and second order terms used in curvilinear analysis

Variable Quantity Variable Quantity Variable Quantity

X(l) T X(l3) T ~ X(25) C E1 X(2) p X(l4) T E1 X(26) v2 s X(3) c X(l5) p2 X(27) Vs Va X(4) Vs X(l6) p c X(28) Vs ~ X(5) Ve X(l7) p vs X(29) Vs El X(6) ~ X(l8) p Va X(30) \)

X(7) El X(l9) p 1-.1.

X(31) Va !.L X(8) T2 X(20) p El X(32) VB El X(9) T p X(21) c2 X(33) ~2 X(lO) T C X(22) C V5 X(34) ~ El X(ll) T 'V5 X(23) C V8 X(35 Ela X(l2) T V8 X(24) c ~

In this analysis, point load tensile strength was again the first vari­able selected, and the prediction equation of step 1 was again equation (1) from the linear analysis . From step 1 on, other variables were selected resulting in prediction equations with lower errors of estimate than those obtained in the linear analysis. The prediction equations for steps 2 through 5 follo~o1 :

Step 2

y p = 1.353 + 23 .335 X(l)- 0 .449 X(l2); (5)

sy = 3.566;

SteE 3

yp = 2 .010 + 27.718 X(l) - 1.321 X(l2) + 2 .138 X(13); (6)

s y = 3 .351;

SteE 4

y p 7.849 + 28.994 X(l) - 1 . 671 X(l2) + 2.956 X(l3) (7)

- 17.223 X(30);

Sy = 3 .305;

SteE 5

yp = 1.216 + 26.403 X(l) + 1 .210 X(lO) - 2. 771 X(l2) (8)

+ 2 . 542 X(l3) - 31.463 X(30);

sy 3.218 .

In these equa tions

X(l) = T,

11

X(lO) = TC,

X(l2) = TV8 ,

X(l3) = T~,

and X(30) = \).

The analysis continued until 31 steps had been completed . The four vari­ables that were not selected by the computer were highly correlated with others already in the prediction equation of step 31 . The equation with the smallest error of estimate was obtained at step 25. The output of step 25 is listed in table 6·. Point load tensile strength appears by itself or as a multiplier of another property eight times in this equation.

TABLE 6. - Prediction equation of step 25

Step 25 sy = 2.819 Intercept = 30.231

Variable Coefficient Variable Coefficient

X(l) - 240 . 644 X(l7) -26.974 X(3) - 8.484 X(20) -34 .082 X(4) 79 . 227 X(21) 3.538 X(5) - 37.885 X(22) -5.576 X(7) 221 . 605 X(23) -5 . 852 X(8) 5 . 953 X(25) -.604 X(9) 62 . 962 X(26) -4.897 X(lO) 8 . 659 X(27) 9.117 X(ll) 34.419 X(28) 5.290 X(l2) -20. 610 X(30) -68 .065 X(l3) -51. 429 X(32) - 5 . 383 X(l4) 9.101 X(35) 3 . 157 X_(_l6) 10.198

Again F tests were used to compare the prediction equations . There is no improvement after equation (6) at a significance level of 5 percent. At the 10-percent l evel, equation (8) is an improvement over equation~) . The equa­tion of step 25 is an improvement over equation (8) at a significance level of 20 percent.

Figure 3 is a plot of measured versus predicted compressive strength for equation (1). The prediction of compressive strength from point load tensile strength alone is sufficien tly accurate for some applications involving compressive strength .

A plot of measured versus predicted compressive strength from equa tion (8) is shown in figure 4. This plot shows that a good estimate of compres ­sive strength can be obtained from a simple function of other rock properties.

Figure 5 is a pl ot of measured versus predicted values for the equation of step 25. For this plot the deviations of the predicted from the measured

12

50

·- 4 0 "'

Equa t i o n (I)

c. Sy =3.744 ,t)

0 • .. . / I • t-<!)

"/ z I.&J 30 • • a::

I •

t- • CJ) • • w

I -/. > CJ) I • CJ) V. · • w 20 - ---··-a:: , ;,· a. • :?! • 0 ··' • u 0 ' w a:: • ::::> 10 CJ)

<t w • • ::?!

-~ •• • • 0 10 20 30 40 50

PRE DICTED CO MPRE SS IVE ST RENGTH, 103

ps I

FIGURE 3. · Measured Versus Predicted Compressive Strength F rom Equation (1) .

values are probably as small as could be expect ed with this set of data , where the average coefficient of variation of the measured compressive strength was 22 percent.

These results can be compared with those obtained by Judd and Huber shown in figure 6. The prediction equa t ion for the plot in f igure 6 is

where

YP = -4. 477 + 0.672 ~ + 0.288 X5 + 2.878 X7

,

Y = compressive strength, p

(9)

13

50

40 Equo t ion(8) {I)

Q. Sy= 3 .218 .., 0

-r." t-(!)

z 30 UJ 0::: t-(f)

UJ > • (f) (f)

UJ 20 a:

a. • :!: • 0 • u Q

UJ a: ::> (f) 10 <t w • • ~

. /· ., •

0 10 20 30 4 0 50 PREDICT ED COMPRE SSI VE ST RENGTH, 10

3 p si

FIGURE 4. • Measured Versus Predicted Compressive Strength From Equat ion (8) .

x4 =impact toughness,

X5 = scl eroscope hardness,

and X7 = the modulus of rigidity.

Equation (9) was obtained using physical property data from 126 locations and assuming a linear relationship between compressive strength and the other rock properties . This equation , developed by Judd and Huber, has a standard

•

14

50

·- 40 en a.

If)

0

I~

1-(!)

z 30 w Q::

1-(f)

w > (f) (f)

w 20 a:: Cl. :E 0 (..)

Cl w Q::

::::> 10 (f)

<[ w :E

0

Equ a tion of step 25

Sy=2.8 19

• • •

-

10 20 30 40 PR EDIC TED CO MPR ESSI VE STRE NG TH, 103

p si

5 0

F IGURE 5. ·Measured Versus Predicted Compressive Strength From Equation of Step 25.

error of estimate of 5.615, compared with 3.744 for equation (1) and 2.819 for the equation of step 25.

Table 7 lists the multiple correlation coefficients for the curvilinear equations of t his report. The multiple correlation coefficients range from 0.947 for equation (1) to 0.986 for the equation of step 25.

Other Relationships

A computer and a high-speed printer were used to plot all of the rock properties versus each other . The plots involving compressive strength are

15

shown in figure 2 and were discussed earlier. The plots not involving com­pressive strength are shown in figures 7 t o 13.

Ill

Q.

.., 0

J: .... ~ z w a: .... (/)

w > (/)

(/)

w a: 0.. ~ 0 u 0 w a: :::> (/)

<t w ~

90

80

70

60

50

4 0

30

20

10 .

TABLE 7. - Multiple correlation coefficients of curvilinear regress ion equations

Regression equation

(1) . . . . .. . . . .. . . .. . . . (5) .. . . ... .. .... . . .. . (6 ) ............ . ..•.. (7) . . . • . . .. •..• . . .... (8) ... '• .. .. . .. . .. ... . Step 25 .. . .. . . . .. ... .

. . . . . . /. ... . . . . . . . ·: .· ·~ . . . /."- .. .. , .. . ·' . . / .. . .. ;~-~- . .. ' . . :\ .

~~ ~ .. . . .... ,:. • .. . . . . •

Number of variables

•

1 2 3 4 5

25

Multiple correlati on coefficient

0 . 947 . 953 . 960 . 962 .964 .986

The plots of figures 7 to 13 were examined to determine whether any of them indicate relationships that could be used to obtain a good estimate of t he Y-value from the X- value. As t11as the case with the plots involving compress ive strength, most of the plots in figures 7 to 13 have substantial scatter. However, some trends are a pparent. Some of t he plots show definite trends because the Y-value was der ived from the X-value .

0 10 20 3 0 4 0 50 60 7 0 PR E D I CTED CO M PRE SS I V E STREN GT H, 103 p sI

F IGUR E 6. · Measured Versus Pred icted Compress i ve Strength (Judd and Huber) .

An example of this is modulus of rigidi ty versus shear veloci ty­shown in figure 8A. Of the plots wi t h definite trends t~ere the Y­value was not derived from the X-value , only a few indicate relat ion­ships that could be used to obtain a good esti­mate of one property from the measurement of another .

16

·- 6 .. a.

2 ,.: ..... 0 4 § a: ... 0 til

3 2 ..... :;:) c 0 :::E

-.. ,-

....

~· . . -I -1

" 0

1

0 4 8 12 SHEAR VELOCITY,I03 fp s

15

~10 til :;:)

-' :;:) a 0 :::E

~ 5 z :;:)

0 >-

..

. • I .. . .. . .. ' .. ) . . ....

.. 0 0 4 8 12

SHEAR VELOCITY, 103 f p l

0 .40 .----• • -, • ...-~ ----r---..,

.30 0 ..... c a:

~ .20 c til !!? 0 a.

.1 0 1-

•

. .. .. .,. • • ..

I • •. •. . . ·' . . '· . .

0 0 4 8 12

SHEAR VELOCITY, 103 fps

FIGURE 8.- P lots of Shear Velocity Versus Other Properties.

20 Ill a.

Ul 0 16 . .....

N I.LJ ~

(/) 12 ::::> _j

::::> 0 8 0 ~ (/)

(!) 4 z ::::> 0 >-

15

Ill a.

'f:, ..... l4J 10 (/)

::::> _J

::::> 0 0 ~ (/) 5 .(!)

-: z ::::> 0 >-

• I •

• • ... ,~,. ..

.:-.. .I

• • • ...

•

0 0 4 8 12 16 20

YOUNG'S MODULUS (£1), 106 psi

17

0.40 ~ I I

•

• • • • • • • • .30 • .. • t- -

• • • •• 0 • • • • • • • t- • •• • <( .- •• 0: • • • • • (/) • z .20 .... • • -• 0 (/) • (/)

0 ~

.10 .... -

0 I _l _l

0 5 10 15 YOUNG'S MODULUS (£1), I06psi

FIGURE 9. • Plots of Young's Modulus.

I I I 0.40 .: I I I f- ·- • • • • • • • • • • • • •

• .30 • •• • •• 1- -• • • • • • - 0 • • • •• -r- •• - ...... • I •

• • <( • • • ' ·' 0: • • • (/) • • •• • z .20 .... • • -• •

•• 0 (/) •

t • (/)

.... - 0 ~ - .10 1- -

• •

0 0 11" I I I I .1 _l

0 3 4 6 0 2 4 6 MODULUS OF RIGIDITY. J06 p s i MODULUS OF RIGIDITY, I06psi

FIGUR E 10. · Plots of Modulus of Rigidity Versus Other Properties.

18

20

Ill Q. - 16 ., 0

> .... 12 0 0 _J

w 8 > 0:: <{ w ::r 4 (/)

0

15

"' Q. 4D 0 . -~10 (/)

:::> _J :::> c 0 ~

!/) C)

5 z :::> 0 >-

0

·- 6 "' Q.

4D 0

> .... c 4 C)

•• :-/ , ..

~<' ~ !I • • 25

0:: ~ 0 (/)

:::> _J 2 :::> c 0 ~

4 8 12 16 20 0 BAR VELOCITY, I03 fp s

0 .40 r-

• •• •

• .30 • -.. 0

t 1-

• <t

• •• 0::

I !/) .20 -• z • 0

(J)

I (J) -0 a.. .. .1 0 f-

\ 0 5 10 15 0

BAR VELOCITY, I03 f ps

••• ••

. .. •

• • -. " . '· ••

•• , ....

® 5 10 15

BAR VELOCITY,I03 fps

• ;t • I I

•

• • • • .,. . • • •

• •

~ • •• •

20

-

•

. ' • ••••• • ·' • •• ... • . ,

• • • -• •

-

0 I I I

5 10 15 BAR VELOCilY, I03 fps

FIGURE 11. • Plots of Bar Velocity Versus Other Properties.

.

"' Q. ~ ,., 0

> !:: u 0 ...J w > a: ct w :I: (/)

20

16

12 • ·~·

8 r·· 4 J...-Y'~ .. 25 0 0 L---~--~----~--~--~

4 8 12 16 20 2 4 LONGITUDINAL VELOCITY,I03 f ps

·; 6 -· 0.

~ > ~

!:?4 (!)

0:: ~ 0 (/)

3 2 ::::> 0 0 ::e

•

• •• • • • ••• • • -4 ••

I

. L. .I •

• •

0 5 10 15 20 LONGITUDINAL VELOCITY, I0 3 f ps

0.4 0 .--- -. •• - • ...-.- r-, - -.,-_--,.-,--,

. 30 0 ~ ct 0:: (/)

·z .20 0 (/) (/)

0 0...

. 10

• 1-

1-

1-

•

•

•

• . . ... •

• • • ••

, . -·. . ,. . ... : .• I .... . ~ -•

-

I I I I

0 5 10 15 20 LONGITUDINAL VELOCITY,I03 fps

"' Q. -.., 0

> ~

0 0 _, w > a: ct en

"' 0.. fQ 0

20

16

12

8

4 11•.25 0 0 L---~--~----~---L---J

15

4 8 12 16 20 24 LONGITUDINAL VELOCITY, 103 f ps

• ••

• •• • ••

'• •• • ••

:1 ,.. -

•

0 5 10 15 20

• •

LONGITUDINAL VELOCITY,I03 fps

FIGUR E 12. • P lots of Longitudinal Velocity Versus Other Properties .

19

20

:!'>

,.. 1-

> <(

~2 -~ . ... . 0 w CL Ul

:of! .. - ·. ..

I

..

0 0 0.5 1.0 1.5 2.0 2.5

POl NT LOAD TENSILE STRENGTH,IO:!psi

.. ... -Q ~

15

= 10 u 0 .J ... > ~ 1-(1) 5 ~ -

J

. . : ..

•• . : • • .. . . . . .

.. .. .

0 0.5 1.0 1.5 2.0 2.5 POINT LOAD TENSI LE STRENGTH,IO:!p sl

~ 20 .. 2 ~ 1- 15 0 0 .J w > .J 10 < z 0

~ · ;::) 1- 5 , ;:; z io 0 ..J

• • ... , . . . . . . •

.. •• • • • • • . ~ .. .. . .

0 0 .5 1.0 1.5 2.0 2.5 POINT LOAD TENSILE STRENGTH,I03p$l

-;;6 ... ID

2 ~ 1-

04 ;:; cr ... 0 Ul ;::)

5 2 C)

0 :IE

~ .. ~ . , ·. ..

· ... ....

0

.

0 0.5 1.0 1.5 2.0 2.5 POINT LOAD TENSILE STRENGTH,I03 pal

0.40 1-" I

0 ;:: <( a:

.30

~ .20 0 VI VI

0 Q.

.10

" I •

I

..... . . . .. ..

••

0 I

0 0.5 1.0 1.5 2.0 2.5 POINT LOAD TENSILE STRENGTH, IOSpsl

12

.. ~ .. 2 ~ 8 1-u g ... > a: c 4 ... X Ul

• • :

•

• .. .. ·' . . . .· • •

.. • • •

0 0.5 1.0 1.5 2.0 2.5 POINT LOAD TENSILE STRENGTH,t03psl

.. .. • 2

15

~to VI ;::) .J :::l C) 0 2

~ 5 z ;::)

0 ,..

.. . . .. ·.

0 ~·· I I

0 0.5 1.0 1.5 2.0 2.5 POINT LOAD TENSILE STRENGTH,IO:!psl

FIGURE 13. - Plots of Point Load Tensi le Strength Versus Other Properties.

21

The plots of shear velocity versus longitudinal ve l oc ity, bar velocity ver sus longitudinal velocity , and shear velocity versus bar velocity, shown in figures 12~, 12]!, and llA, respectively, indicate linear relationships. These plots could be used to estimat e one velocity from another. However, this estimat e woul d not be much better than that which would be obtained by assum­ing Poisson's ratio equal to 0 . 25 and using the follm11ing equations (2_):

vs = c fljj,

Va = c ~5/6,

and vs = Va ~2/5)

~11here c longitudinal ve l ocity,

Va = bar ve locity,

and Vs shear velocity .

Equations (10), (11) , and (12) were used to cons truct the l ines on figures 12A, 12~, and l lA . These lines show that an assumed value of 0 .25 for Poisson's ratio fits the data very ~11ell.

(10)

(11)

(12 )

Some of the plots could be used to obtain estimates of t he dynamic con­stant s . ~11o examples are figure 12g, modulus of rigidity versus longitudinal velocity , and figur e 12Q, Young ' s modu l us versus longitudinal ve l oci t y .

Figur e 9~ is a plot of Young ' s modulus (E1 ) det er mined by method 1 versus Young ' s modul us (E2 ) determined by method 2. Although method 2 usua l ly yie l ds a slightly higher value than method 1 , the di fference is not large enough to be important in most app l ications .

SUMMARY AND CONCLUSIONS

Measurements of compressive strength, point load t ens ile strength, spe­cific gravity, longitudinal velocity, shear velocity, and bar velocity were made on rocks having a wide range of compr essive strengths . From these meas­ured values, Young ' s modulus, the modulus of rigidity, and Poisson ' s ratio were calculated . Regression analyses were used to obtain equations for pre­dicting compr essive strength from the other properties. The prediction equa­tions obtained by assuming a linear relationship between compressive strength and the other pr operties were improved by assuming a curvilinear relationship and by using the eight properties with some of their squares and cross pr od­ucts as independent variables . Point load tensile strength a l one could be used t o estimate compressive strength with fair accuracy . Equations that improve this estimate and that contain less than six variables were also found . Finally a prediction equati on containing 25 variables was ob t a ined. The pre ­dicted values from this equation ~11ere as close to the measured values as could be expected considering the error in the measured values . All of the derived equations have smaller errors of estimate than t hat of the equation obtained in an ear l ier study by Judd and Huber . These results are encouraging consid­ering that the measurements were made on a wide variety of rocks including many that >~Tere neither homogeneous nor isotropic .

22

BIBLIOGRAPHY

1 . Blair , B. E. Physical Properties of Mine Rock, Part III. BuMines Rept. of I nv . 5130, 1955, p . 5.

2 . _____ Physical Properties of Mine Rock, Par t IV . BuMines Rept . of I nv. 5244 , 1956, p. 4.

3. Crow, E. L. , F. ~. Davis, and M. W. Maxfield . Statistics Manual . Dover Publications Inc . , New York, 1960 , pp . 147-194 .

4 . Efroymson, M. A. Multiple Regression Analysis. Ch. in Mathematical Methods for Digital Computers, ed . by Anthony Ralston and Herbert Wi l f. John Wil ey & Sons , Inc., New York , 1960, pp . 191- 203.

5 . Rueter, T. F., and R. H. Bo l t . Sonics . John Wiley & Sons , Inc ., New York, 1960, pp . 26- 27 .

6 . Johnson, J. B., and R. L. Fischer . Effects of Mechanical Properties of Material s on Cratering : A Laboratory Study. BuMines Rept. of I nv . 6188, 1963, pp. 2- 8 .

7. Judd, W. R. , and C. Huber . Correl ation of Rock Properties by Statistical Methods. Internat. Symp . Min. Res ., Rolla, Mo . , Feb . 22-25 , 1961 . Pergamon Press, New York, 1962, p. 62.

8. Obert, L., s . L. Windes, and W. I . Duval l . Standardized Tests for Determining the Physical Properties of Mine Rock . BuMines Rept . of Inv. 3891, 1946, p . 67 .

9. Ostle, B. Statistics in Research . I owa St ate College Press, Ames , Iowa , 1954, pp. 202-231 .

10. Windes, S . L. Physical Properties of Mine Rock . Part I . BuMines Rept. of I nv . 4459, 1949, p . 8.

11. Physical Properties of Mine Rock. Part II . BuMines Rept . of Inv. 4727, 1950, p. 8.

APPENDIX

TABLE A-1. - Averages, standard deviations, standard errors, and coefficients of variation of compressive strength and point l oad tensile strength

Compressive strength Point load tensile stren th

23

Standard Standard Coefficient Standard Standard Coefficient Location Average, deviation, N error,~ of Average, deviation N error,~ of

103 psi 103 psi 103 psi variati on, 103 psi 103 psi 103 psi variation, percent percent

1 30.233 2 . 26 9 0. 75 7.5 2.054 0.177 16 0 .044 8.6 2 32.551 6 . 72 10 2.10 20.6 1 .985 .501 13 .140 25 . 2 3 24 . 989 7.44 11 2.20 29.8 1.518 .553 17 .130 36 . 4 4 22 . 158 5.54 12 1.60 25.0 1.314 .252 14 .067 19.2 5 27.020 5 .15 10 1.60 19.1 1.308 .232 13 .064 17.7 6 21.265 5 .13 11 1.50 24.1 1 .408 .234 14 .063 16 .6 7 36.401 '6 .04 15 1.60 16.6 2.474 .392 29 .073 15.8 8 33 .892 9 . 77 16 2.40 28.8 2.483 .628 18 .150 25.3 9 19.153 3 .31 11 1.00 17 .3 1.153 .214 15 .055 18.6

10 30.361 3.84 12 1.10 12 . 6 1.882 . 124 12 .036 6 .6 11 18 .814 5 .32 12 1.50 28.3 1.582 .454 20 .100 28 . 7 12 31.415 7.32 10 2.30 23.3 1.688 .339 15 . 088 20.1 13 26.022 5.16 15 1.30 19.8 1.340 .195 17 .047 14.6 14 34 .871 7.37 7 2.80 21.1 1.680 .402 9 .130 23.9 15 16.402 3.03 13 .84 18 .5 .9ll .312 13 .087 34.2 16 3.407 .32 7 .12 9.4 .060 .014 12 .004 23.3 17 14.074 5.59 6 2.30 39.7 .963 .253 12 .073 26.3 18 16.633 5. 77 9 1.90 34.7 . 937 .158 13 .044 16.9 19 15 .559 2.33 9 .78 15.0 .736 . 126 11 . 038 17.1 20 19.519 5.92 5 2.60 30.3 .827 .107 11 .032 12 . 9 21 21.630 10.34 5 4 . 60 47.8 1.552 .186 13 . 052 12.0 22 13.934 2.66 17 .65 19 . 1 .987 .166 18 .039 16.8 23 16.351 6.58 9 2 . 20 40.2 . 740 .245 14 .065 33.1 24 17.030 4.05 11 1.20 23.8 .708 .084 12 . 024 11 . 9 25 11.796 5.10 8 1.80 43.2 .506 .151 11 .046 29.8 26 13.330 5.37 10 1. 70 40.3 .670 .134 14 . 036 20.0 27 4.334 .26 8 .09 6.0 . 086 .020 13 . 005 23.2 28 7.943 2.90 10 .92 36 .5 .366 .201 11 .061 54 . 9 29 2 . 451 .28 13 . 08 11.4 .045 .005 13 .001 11.1 30 12 .390 5.35 4 2.70 43.2 . 927 .206 7 .078 22.2 31 19.710 3.09 21 .67 15.7 .854 .122 23 .025 14 .3 32 44.201 8.14 6 3.30 18.4 2 .317 . 542 16 .140 23 .4 33 5.111 .90 13 .25 17.6 .322 .034 15 . 009 10 .6 34 2.169 .30 12 .09 13.8 . 132 .019 16 . 005 14.4 35 2 . 765 .13 13 .04 4 . 7 .075 .014 16 .004 18.7 36 5. 925 1.20 14 .32 20.2 .165 .015 18 . 004 9.1 37 28.400 1.27 4 .64 4.5 1.800 .192 100 .019 10.7 38 19 . 400 3.04 7 1.10 15.7 1.340 .202 28 . 038 15.1 39 42.349 4.32 10 1.40 10.2 2.290 .304 22 . 065 13.3 40 7 .680 .725 13 .20 9.4 .510 .049 33 .009 9 .6 41 36.400 11.25 9 3 . 70 30 .9 2.206 .315 15 .081 14.3 42 20.625 4.27 17 1.00 20.7 1.273 .185 17 .045 14 .5 43 17.750 . 4.39 14 1.20 24.7 . 999 . 331 16 .083 33.1 44 31.167 5 .53 17 1.30 17.7 1.518 .313 17 .076 20.6 45 24 .010 4.22 14 1.10 17.6 1.330 .255 11 .077 19.2 46 1.540 .29 5 .13 18 . 8 .040 .008 11 .002 20.0 47 45 . 172 8.55 26 1. 70 18.9 2.530 .242 27 .047 9.6 48 29 .047 5.25 6 2.10 18 . 1 1.305 .235 14 .063 18.0 49 16 . 546 5.89 15 1.50 35.6 1. 612 .450 15 .120 27.9

~The standard error of an average J.s g1ven by S/./N, \vhere S l.S the standard dev1at1on of the average and N is the number of observations.

INT.- BU.O F MINES,PCH .,P A . 8873