- Email: [email protected]
A correction equation on the influence of length-to diameter ratio on the uniaxial compressive strength of rocks
Engineering Geology, 22 (1986) 293--300
293
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
A CORRECTION EQUATION ON THE INFLUENCE OF LENGTH-TODIAMETER RATIO ON THE UNIAXIAL COMPRESSIVE STRENGTH OF ROCKS
N. T U R K and W.R. DEARMAN
Department of Geotechnical Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne (Great Britain) (Received February 28, 1985;accepted after revision October 29, 1985)
ABSTRACT Turk, N. and Dearman, W.R., 1986. A correction equation Qn the influence of length-todiameter ratio on the uniaxial compressive strength of rocks. Eng. Geol., 22: 293--300. A general equation, derived from three published equations, is proposed for the correction of uniaxial compressive strength test results to a length-to-diameter ratio of two. In addition, an equation is given for standardization of test results to a length-to-diameter ratio of two and 50 mm diameter. The methods have been tested by analysis of over thirty sets of test data from the literature. Results for correction to length-to-diameter ratio of two fall within a 10% error band of the experimental value equivalent to a length-to-diameter ratio of two. The twostage correction to a length-to-diameter ratio of two and 50 mm diameter, from a range of specimen lengths and diameters, gives results which are lower than the direct test values for specimens with these dimensions. INTRODUCTION
Uniaxial compressive strength is one of the most important mechanical properties of rocks which is mainly used for the design of structures and characterization of intact rock materials. In rock engineering, the uniaxial compressive strength of rocks is generally defined as the failure strength of an intact rock specimen, having a diameter of 48 or 54 mm and a length-todiameter ratio of at least 2, preferably 2.5--3 (ASTM 2983-79; ISRM, 1979). There are both internal and external factors influencing uniaxial compressive strength. Important internal factors are defects, mineralogy, grain size, porosity, degree of weathering or alteration and anisotropy. The external factors are specimen shape and size, type of platen, rate of loading and degree of saturation. During testing, the influence of these factors should be recognized and results should be interpreted accordingly. Otherwise, the test results may be misleading or virtually useless or both. Different authorities have proposed standard methods for uniaxial compressive testing of rock specimens (ASTM 2938-79; ISRM, 1979). One of these procedures should be followed to eliminate the influence of the external 0013-7952/86/$03.50
© 1986 Elsevier Science Publishers B.V.
294
factors and increase the reliability and repeatability of the test results. These procedures generally specify the size and shape of the test specimens. (1) ISRM (1979) requires that the test specimen should be a right circular cylinder, having a diameter preferably not less than NX core size, approximately 54 mm, and a height-to-diameter ratio of 2.5 to 3. (2) ASTM (D 2938-79) specifies that the test specimens shall be circular cylinders with a diameter of not less than N× wireline core size, approximately 48 mm, and a length-to-diameter ratio of 2 to 2.5. C O R R E C T I O N EQUATIONS
In practice, it is not always possible to obtain test specimens of the required size. Correction equations have been proposed for standardizing the results obtained from non-standard specimens. Two types of correction equation have been proposed in order: (1) to standardize the result for a length-todiameter ratio of 2; (2) to standardize for a 50 mm diameter. Widely accepted correction equations for the former are the following. (a) Hobbs (1964) and Szlavin (1974} recommended the following equation for Coal Measures rocks in the U.K. O___c= 1 em 0.848 + 0.304(D/L)
(1)
where e¢ is the corrected uniaxial compressive strength to a length to diameter ratio of 2, om is the measured uniaxial compressive strength of nonstandard size rock specimens, D is the diameter of the specimen, and L is the length of the specimen. (b) ASTM (D2938-79) recommends the following correction formula for standardizing non-standard size uniaxial compressive strength test results: ~c_ Om
1 0.88 + 0.24(D/L)
(2)
where the parameters are the same as for eq.1. (c) Protodyakonov (1969) has recommended the use of the following formula for standardization of non-standard size, uniaxial compressive strength test results, based on Russian experience: o__¢_= 1 em 0.875 + 0.25(D/L)
(3)
where the parmheters are the same as for eq.1. The plots of the above equations as oc/Om versus D/L are shown in Fig.1. Even though each curve has a different path, they have a similar trend. These correction equations are empirical and based on practical experience. They also represent the experimental results of different countries. The mean of the above equations could be expected to give a new correction equation applicable generally:
295 1.2
•
1.1
-'~.~,,~
1.0
~
O'c] O"m 0.9
- -
a
....
b
~
~-~.~
~'~-.~.
0.8 i 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
D/L
Fig.1. %/o m versus D/L f o r different correction equations: (a) Hobbs ( 1 9 6 4 ) and Szlavin
(1974); (b) ASTM (D2638-79); (C) Protodyakonov (1969). (~c -
am
1
0.86766 + 0.26466(D/L)
(4)
On simplification, eq. 4 becomes: oc _ 1.15 om - 1 + 0.3(D/L)
(5)
where the parameters are the same as in eq.1. The plot of this equation is shown on Fig.2. It is worth noting that ISRM (1979) does not give any correction equation for the uniaxial compressive strength of non-standard size specimens. Hoek and Brown (1980) have proposed the following correction equation for standardizing the uniaxial compressive strength of rock specimens to 50 mm diameter specimen strength, based on curve fitting to test results from the literature:
Oso= (n.~°"s o,
\5-0)
(6)
where os0 is the uniaxial compressive strength of a 50 mm diameter rock specimen, o, is the uniaxial compressive strength of a rock specimen having a different diameter, and D is the diameter of the specimen. If eq.4 and 6 are combined then a general correction equation for the uniaxial compressive strength of rocks is obtained: 05._.00=
om
DO.iS 1.754 + 0.535(D/L)
(7)
where the parameters are the same as in eqs.1 and 6. This proposed new equation would enable the required corrections to be made to the uniaxial compressive strength of non-standard size specimens both for a length-to-diameter ratio of 2 and for 50 mm diameter. Additionally, eq.7 has been plotted in graphic form for practical application in Figs.3 and 4.
296 1.2
1.1
1.0 O"c/ O'm 0.9 0.8 0
I
0.2
014
01.6
0.8'
11.0
112
1.4l
11.6
D/L
Fig.2. %/am versus D/L for the proposed mean correction equation.
1.3
1.2
1.1
1.0
0.9
~d~m 0.8
0.7 0.6 0.5 0
0.5
1.0
D/L
1.5
2.0
2.5
Fig.3. The plot of aso/o m against D/L for different specimen diameters. DISCUSSION
To explore their limitations, the proposed new correction equations 4 and 7 have been applied to some easily available uniaxial compressive test results on different sized cores. Available test results can be grouped under the following three headings. (1) 50 m m diameter and varying length-to-diameter ratios Dreyer and Borchert (1962) have published the results o f uniaxial compressive strength determinations on 50 mm diameter marble cylinders with different length-to-diameter ratios. The test results are plotted in Fig. 5 as
297
D/L 0.25
1.2
0.50 1.1
~
1.0 0.9 0"5o/O'm 0.8
0.75 1.00 1.25 1.50 1.75 2.00
~ ~
0.7 t 0.6
// /
0"50
2;
4; 610 8=0 Diameter(ram)
1;0
Fig.4. The plot of os0/Om against specimen diameter for different D/L ratios.
200
Marble Diameter 50mm =Experimental r e s u l t s oCorrected using Eqn.4 10% error band
150
o
o'C MPa 1000
[] o -o--
I ---'-|-I--
I
--
I-I
i
I
1 2 3 Length/Diameter
I
4
Fig.5. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 50 mm diameter marble specimens (Dreyer and Borchert, 1962).
original test values and as corrected values, using eq.4, to a length-to-c]iameter ratio of 2. The horizontal solid line in Fig.5 represents the original uniaxial compression strength at L/D = 2. A 10% error band, shown in the figure, covers all the corrected values, and except for the lowest L/D the error o f the corrected results is much less than 10%.
298
(2) A diameter other than 50 mm and varying length to diameter ratios For massive Ormonde Sandstone (Fig.6) specimen diameter was 25.4 mm and length-to-diameter ratio varied from 0.25 to just under 4 {Hobbs, 1964). The experimental data have been corrected to a length-to-diameter ratio of 2 (eq.4) and to 50 mm diameter and length-to-diameter ratio of 2 (eq.7}. Except for the lowest length-to-diameter ratio, the corrected results to a L/D of 2 lie within the 10% error band on the diagram. Corrections using eq.7 show a wider scatter, and the resultant unconfined compressive strength falls below those for a length-to-diameter ratio of 2 derived from the uncorrected test results. For trachyte (Mogi, 1966) low length-to-diameter ratios were not tested (Fig.7) and the variation of L / D from 1 to 3 is not great. The corrected results to L/D of 2 using eq.4 are well within a 10% error band, and close to the original experimental value for L/D of 2. Using eq.7, the corrected results 150
Sandstone Diameter Experimental Corrected ,~Corrected
25.4mm values using Eqn.4 using Eqn.7
100
10%
dc MPa
error
band
[]
-b
G-•
-
-
500
•
~'
I
i
2
3
4
Length/Diameter
Fig.6. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 25.4 mm diameter specimens of massive Ormonde Sandstone (Hobbs, 1964). Trachyte
Diameter
12.7mm
140
10%
•
error
band
Lm
-
120 100
..............
• ......
& --.-i--
-A' _ ~L.__AL_ - - - - -&
80
~c
60
MPa
4O 20
%
• Experimental Corrected • Corrected
|.
05
values using Eqn.4 using Eqn.7
|
|
1
1,5
2
!
I
2.5
3
Length/Diameter
Fig.7. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 12.7 mm diameter trachyte specimens (Mogi, 1964).
299 } Experimental v a l u e s , with standard deviation • C o r r e c t e d using Ecln.7 15
d C
lO
MPa
i
fi _ _ _ i
5
i
20
i
i
30 40 50 D i a m e t e r (ram)
J
60
Fig.8. Relation between experimental test results and corrected values of uniaxial compressive strength and diameter of a gypsum--plaster mix, for length-to-diameter ratio of 2 (Einstein et al., 1970).
for unconfined compressive strength are uniform and much lower than the results obtained using eq.4. The chain-link line on Fig.7 passes through the corrected value at L / D = 2. (3) Different diameters with a length to diameter ratio o f two Plaster cylinders, prepared to different diameters but at a standard lengthto-diameter ratio of 2, were tested by Einstein et al. (1970). Test results are plotted in Fig.8 with the standard deviations given by the authors, and the results corrected using eq.7. The standard deviation on each of the three results is about + 2.5%. Corrected values fall within the standard deviation for 50 mm diameter specimens, with the result for the smallest diameter showing greatest divergence from the mean 50 mm diameter results. CONCLUSIONS In testing the proposed correction equations, although the analyses of only four sets of experimental results have been presented (Figs.5, 6, 7, 8), over thirty examples from the literature have been analysed. Results are similar to those presented here. The original test results and their corrected values have been plotted as graphs on the same diagram by plotting the uniaxial compressive strength against length to diameter ratios of the specimen. The following general observations can be made. (1) The correction equation 4 gives results which are well within what may be regarded as acceptable error limits for specimens having a length to diameter ratio between 1 and 4. Within this range, the error is generally less than 10% (Figs.5, 6, 7). Any specimen tested with a length-to-diameter ratio less than 1 would have a very complex stress distribution and this would affect the results. On the other hand, an attempt would not normally be made to test specimens having length-to-diameter ratios greater than 4. Instead, it would be wiser to cut such a core and get two test specimens instead of one.
300
(2) Correction of the test results to 50 mm diameter has given, in general, lower uniaxial compressive strength values than the direct experimental values for specimens having diameters less than 50 mm. The reverse is the case for larger diameter specimens. It must be pointed out that there are mixed findings on the influence which the specimen diameter has on the uniaxial compressive strength of rocks. While data compiled by Hoek and Brown (1980) from the literature clearly indicate the influence of diameter on the uniaxial compressive strength of rocks, it is difficult to reach the same conclusion from some other test results (Hodgson and Cook, 1970). However, as demonstrated here, when the results of Einstein et al. (1970) on plaster specimens were corrected to 50 mm diameter, the corrected uniaxial compressive strength values fell well within the standard deviation of the test values of 50 mm diameter specimens (Fig.8). As most laboratory testing is done on rock specimens having a diameter less than 50 mm, such a correction would give lower results than the original values obtained in the laboratory. Thus, if the corrected values are used in design calculations, this would increase the safety factor. The general observation is that the smaller the diameter and the smaller the length-to-diameter ratio of the specimen, the higher is the scatter of the test results (Figs.5, 6 and 8). However, considering the variability of rock types and testing conditions, the correction equations 4 and 7 give acceptable estimates of the uniaxial compressive strength of rocks for a length-to-diameter length ratio of 2, and 50 mm diameter, within the limitations 1 < diameter < 4. REFERENCES ASTM D-2938-79. Standard method of test for unconfined compressive strength of rock core specimens. In: 1980 Annual Book of ASTM Standards, Part 19, pp.440--443. Dhir, R.K., Sangha, C.M. and Munday, J.G.L., 1972. Influence of specimen size on unconfined rock strength. Colliery Guardian, Jan. 1972, pp.75--78. Dreyer, W. and Borchert, H., 1962. Kritische Betrachtung zur PriifkSrperformel von Gesteinen. Bergbautechnik, 129(5): 265--272. Einstein, H.H., Baecher, G.B. and Hirschfeld, R.C., 1970. The effect of size on strength of a brittle rock. Proc. Congr. Int. Soc. Rock Mech., 2nd, Belgrad, Vol. 2(3--5), pp.7--13. Hobbs, D.W., 1964. Rock compressive strength. Colliery Eng., 41 : 287--292. Hodgson, K. and Cook, N.G.W., 1970. The effects of size and stress gradient on the strength of rocks. Proc. 2nd Congr. Int. Soc. Rock Mech., 2nd, Belgrad, Vol. 2(3--5), pp.31--34. Hoek, E. and Brown, E.T., 1980. Underground Excavations in Rock. The Institution of Mining and Metallurgy, London, pp.527. ISRM, 1979. Suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int. J. Rock Mech. Min. Sci. Geomeeh. Abstr., 16: 135--140. Mogi, K., 1966. Some precise measurements of fracture strength of rocks under uniform compressive stress. Rock Mech. Eng. Geol., IV: 41--55. Protodyakonov, M.M., 1969. Method of determining the strength of rocks under uniaxial compression. In: M.M. Protodyakonov, M.I. Koifman and others, Mechanical Properties of Rocks. Translated from Russian, Israel Program for Scientific Translations, Jerusalem, pp.l--8. Szlavin, J., 1974. Relationships betweea some physical properties of rock determined by laboratory test. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 11: 57--66.
293
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
A CORRECTION EQUATION ON THE INFLUENCE OF LENGTH-TODIAMETER RATIO ON THE UNIAXIAL COMPRESSIVE STRENGTH OF ROCKS
N. T U R K and W.R. DEARMAN
Department of Geotechnical Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne (Great Britain) (Received February 28, 1985;accepted after revision October 29, 1985)
ABSTRACT Turk, N. and Dearman, W.R., 1986. A correction equation Qn the influence of length-todiameter ratio on the uniaxial compressive strength of rocks. Eng. Geol., 22: 293--300. A general equation, derived from three published equations, is proposed for the correction of uniaxial compressive strength test results to a length-to-diameter ratio of two. In addition, an equation is given for standardization of test results to a length-to-diameter ratio of two and 50 mm diameter. The methods have been tested by analysis of over thirty sets of test data from the literature. Results for correction to length-to-diameter ratio of two fall within a 10% error band of the experimental value equivalent to a length-to-diameter ratio of two. The twostage correction to a length-to-diameter ratio of two and 50 mm diameter, from a range of specimen lengths and diameters, gives results which are lower than the direct test values for specimens with these dimensions. INTRODUCTION
Uniaxial compressive strength is one of the most important mechanical properties of rocks which is mainly used for the design of structures and characterization of intact rock materials. In rock engineering, the uniaxial compressive strength of rocks is generally defined as the failure strength of an intact rock specimen, having a diameter of 48 or 54 mm and a length-todiameter ratio of at least 2, preferably 2.5--3 (ASTM 2983-79; ISRM, 1979). There are both internal and external factors influencing uniaxial compressive strength. Important internal factors are defects, mineralogy, grain size, porosity, degree of weathering or alteration and anisotropy. The external factors are specimen shape and size, type of platen, rate of loading and degree of saturation. During testing, the influence of these factors should be recognized and results should be interpreted accordingly. Otherwise, the test results may be misleading or virtually useless or both. Different authorities have proposed standard methods for uniaxial compressive testing of rock specimens (ASTM 2938-79; ISRM, 1979). One of these procedures should be followed to eliminate the influence of the external 0013-7952/86/$03.50
© 1986 Elsevier Science Publishers B.V.
294
factors and increase the reliability and repeatability of the test results. These procedures generally specify the size and shape of the test specimens. (1) ISRM (1979) requires that the test specimen should be a right circular cylinder, having a diameter preferably not less than NX core size, approximately 54 mm, and a height-to-diameter ratio of 2.5 to 3. (2) ASTM (D 2938-79) specifies that the test specimens shall be circular cylinders with a diameter of not less than N× wireline core size, approximately 48 mm, and a length-to-diameter ratio of 2 to 2.5. C O R R E C T I O N EQUATIONS
In practice, it is not always possible to obtain test specimens of the required size. Correction equations have been proposed for standardizing the results obtained from non-standard specimens. Two types of correction equation have been proposed in order: (1) to standardize the result for a length-todiameter ratio of 2; (2) to standardize for a 50 mm diameter. Widely accepted correction equations for the former are the following. (a) Hobbs (1964) and Szlavin (1974} recommended the following equation for Coal Measures rocks in the U.K. O___c= 1 em 0.848 + 0.304(D/L)
(1)
where e¢ is the corrected uniaxial compressive strength to a length to diameter ratio of 2, om is the measured uniaxial compressive strength of nonstandard size rock specimens, D is the diameter of the specimen, and L is the length of the specimen. (b) ASTM (D2938-79) recommends the following correction formula for standardizing non-standard size uniaxial compressive strength test results: ~c_ Om
1 0.88 + 0.24(D/L)
(2)
where the parameters are the same as for eq.1. (c) Protodyakonov (1969) has recommended the use of the following formula for standardization of non-standard size, uniaxial compressive strength test results, based on Russian experience: o__¢_= 1 em 0.875 + 0.25(D/L)
(3)
where the parmheters are the same as for eq.1. The plots of the above equations as oc/Om versus D/L are shown in Fig.1. Even though each curve has a different path, they have a similar trend. These correction equations are empirical and based on practical experience. They also represent the experimental results of different countries. The mean of the above equations could be expected to give a new correction equation applicable generally:
295 1.2
•
1.1
-'~.~,,~
1.0
~
O'c] O"m 0.9
- -
a
....
b
~
~-~.~
~'~-.~.
0.8 i 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
D/L
Fig.1. %/o m versus D/L f o r different correction equations: (a) Hobbs ( 1 9 6 4 ) and Szlavin
(1974); (b) ASTM (D2638-79); (C) Protodyakonov (1969). (~c -
am
1
0.86766 + 0.26466(D/L)
(4)
On simplification, eq. 4 becomes: oc _ 1.15 om - 1 + 0.3(D/L)
(5)
where the parameters are the same as in eq.1. The plot of this equation is shown on Fig.2. It is worth noting that ISRM (1979) does not give any correction equation for the uniaxial compressive strength of non-standard size specimens. Hoek and Brown (1980) have proposed the following correction equation for standardizing the uniaxial compressive strength of rock specimens to 50 mm diameter specimen strength, based on curve fitting to test results from the literature:
Oso= (n.~°"s o,
\5-0)
(6)
where os0 is the uniaxial compressive strength of a 50 mm diameter rock specimen, o, is the uniaxial compressive strength of a rock specimen having a different diameter, and D is the diameter of the specimen. If eq.4 and 6 are combined then a general correction equation for the uniaxial compressive strength of rocks is obtained: 05._.00=
om
DO.iS 1.754 + 0.535(D/L)
(7)
where the parameters are the same as in eqs.1 and 6. This proposed new equation would enable the required corrections to be made to the uniaxial compressive strength of non-standard size specimens both for a length-to-diameter ratio of 2 and for 50 mm diameter. Additionally, eq.7 has been plotted in graphic form for practical application in Figs.3 and 4.
296 1.2
1.1
1.0 O"c/ O'm 0.9 0.8 0
I
0.2
014
01.6
0.8'
11.0
112
1.4l
11.6
D/L
Fig.2. %/am versus D/L for the proposed mean correction equation.
1.3
1.2
1.1
1.0
0.9
~d~m 0.8
0.7 0.6 0.5 0
0.5
1.0
D/L
1.5
2.0
2.5
Fig.3. The plot of aso/o m against D/L for different specimen diameters. DISCUSSION
To explore their limitations, the proposed new correction equations 4 and 7 have been applied to some easily available uniaxial compressive test results on different sized cores. Available test results can be grouped under the following three headings. (1) 50 m m diameter and varying length-to-diameter ratios Dreyer and Borchert (1962) have published the results o f uniaxial compressive strength determinations on 50 mm diameter marble cylinders with different length-to-diameter ratios. The test results are plotted in Fig. 5 as
297
D/L 0.25
1.2
0.50 1.1
~
1.0 0.9 0"5o/O'm 0.8
0.75 1.00 1.25 1.50 1.75 2.00
~ ~
0.7 t 0.6
// /
0"50
2;
4; 610 8=0 Diameter(ram)
1;0
Fig.4. The plot of os0/Om against specimen diameter for different D/L ratios.
200
Marble Diameter 50mm =Experimental r e s u l t s oCorrected using Eqn.4 10% error band
150
o
o'C MPa 1000
[] o -o--
I ---'-|-I--
I
--
I-I
i
I
1 2 3 Length/Diameter
I
4
Fig.5. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 50 mm diameter marble specimens (Dreyer and Borchert, 1962).
original test values and as corrected values, using eq.4, to a length-to-c]iameter ratio of 2. The horizontal solid line in Fig.5 represents the original uniaxial compression strength at L/D = 2. A 10% error band, shown in the figure, covers all the corrected values, and except for the lowest L/D the error o f the corrected results is much less than 10%.
298
(2) A diameter other than 50 mm and varying length to diameter ratios For massive Ormonde Sandstone (Fig.6) specimen diameter was 25.4 mm and length-to-diameter ratio varied from 0.25 to just under 4 {Hobbs, 1964). The experimental data have been corrected to a length-to-diameter ratio of 2 (eq.4) and to 50 mm diameter and length-to-diameter ratio of 2 (eq.7}. Except for the lowest length-to-diameter ratio, the corrected results to a L/D of 2 lie within the 10% error band on the diagram. Corrections using eq.7 show a wider scatter, and the resultant unconfined compressive strength falls below those for a length-to-diameter ratio of 2 derived from the uncorrected test results. For trachyte (Mogi, 1966) low length-to-diameter ratios were not tested (Fig.7) and the variation of L / D from 1 to 3 is not great. The corrected results to L/D of 2 using eq.4 are well within a 10% error band, and close to the original experimental value for L/D of 2. Using eq.7, the corrected results 150
Sandstone Diameter Experimental Corrected ,~Corrected
25.4mm values using Eqn.4 using Eqn.7
100
10%
dc MPa
error
band
[]
-b
G-•
-
-
500
•
~'
I
i
2
3
4
Length/Diameter
Fig.6. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 25.4 mm diameter specimens of massive Ormonde Sandstone (Hobbs, 1964). Trachyte
Diameter
12.7mm
140
10%
•
error
band
Lm
-
120 100
..............
• ......
& --.-i--
-A' _ ~L.__AL_ - - - - -&
80
~c
60
MPa
4O 20
%
• Experimental Corrected • Corrected
|.
05
values using Eqn.4 using Eqn.7
|
|
1
1,5
2
!
I
2.5
3
Length/Diameter
Fig.7. Relation between experimental test results and corrected values of uniaxial compressive strength and length-to-diameter ratio for 12.7 mm diameter trachyte specimens (Mogi, 1964).
299 } Experimental v a l u e s , with standard deviation • C o r r e c t e d using Ecln.7 15
d C
lO
MPa
i
fi _ _ _ i
5
i
20
i
i
30 40 50 D i a m e t e r (ram)
J
60
Fig.8. Relation between experimental test results and corrected values of uniaxial compressive strength and diameter of a gypsum--plaster mix, for length-to-diameter ratio of 2 (Einstein et al., 1970).
for unconfined compressive strength are uniform and much lower than the results obtained using eq.4. The chain-link line on Fig.7 passes through the corrected value at L / D = 2. (3) Different diameters with a length to diameter ratio o f two Plaster cylinders, prepared to different diameters but at a standard lengthto-diameter ratio of 2, were tested by Einstein et al. (1970). Test results are plotted in Fig.8 with the standard deviations given by the authors, and the results corrected using eq.7. The standard deviation on each of the three results is about + 2.5%. Corrected values fall within the standard deviation for 50 mm diameter specimens, with the result for the smallest diameter showing greatest divergence from the mean 50 mm diameter results. CONCLUSIONS In testing the proposed correction equations, although the analyses of only four sets of experimental results have been presented (Figs.5, 6, 7, 8), over thirty examples from the literature have been analysed. Results are similar to those presented here. The original test results and their corrected values have been plotted as graphs on the same diagram by plotting the uniaxial compressive strength against length to diameter ratios of the specimen. The following general observations can be made. (1) The correction equation 4 gives results which are well within what may be regarded as acceptable error limits for specimens having a length to diameter ratio between 1 and 4. Within this range, the error is generally less than 10% (Figs.5, 6, 7). Any specimen tested with a length-to-diameter ratio less than 1 would have a very complex stress distribution and this would affect the results. On the other hand, an attempt would not normally be made to test specimens having length-to-diameter ratios greater than 4. Instead, it would be wiser to cut such a core and get two test specimens instead of one.
300
(2) Correction of the test results to 50 mm diameter has given, in general, lower uniaxial compressive strength values than the direct experimental values for specimens having diameters less than 50 mm. The reverse is the case for larger diameter specimens. It must be pointed out that there are mixed findings on the influence which the specimen diameter has on the uniaxial compressive strength of rocks. While data compiled by Hoek and Brown (1980) from the literature clearly indicate the influence of diameter on the uniaxial compressive strength of rocks, it is difficult to reach the same conclusion from some other test results (Hodgson and Cook, 1970). However, as demonstrated here, when the results of Einstein et al. (1970) on plaster specimens were corrected to 50 mm diameter, the corrected uniaxial compressive strength values fell well within the standard deviation of the test values of 50 mm diameter specimens (Fig.8). As most laboratory testing is done on rock specimens having a diameter less than 50 mm, such a correction would give lower results than the original values obtained in the laboratory. Thus, if the corrected values are used in design calculations, this would increase the safety factor. The general observation is that the smaller the diameter and the smaller the length-to-diameter ratio of the specimen, the higher is the scatter of the test results (Figs.5, 6 and 8). However, considering the variability of rock types and testing conditions, the correction equations 4 and 7 give acceptable estimates of the uniaxial compressive strength of rocks for a length-to-diameter length ratio of 2, and 50 mm diameter, within the limitations 1 < diameter < 4. REFERENCES ASTM D-2938-79. Standard method of test for unconfined compressive strength of rock core specimens. In: 1980 Annual Book of ASTM Standards, Part 19, pp.440--443. Dhir, R.K., Sangha, C.M. and Munday, J.G.L., 1972. Influence of specimen size on unconfined rock strength. Colliery Guardian, Jan. 1972, pp.75--78. Dreyer, W. and Borchert, H., 1962. Kritische Betrachtung zur PriifkSrperformel von Gesteinen. Bergbautechnik, 129(5): 265--272. Einstein, H.H., Baecher, G.B. and Hirschfeld, R.C., 1970. The effect of size on strength of a brittle rock. Proc. Congr. Int. Soc. Rock Mech., 2nd, Belgrad, Vol. 2(3--5), pp.7--13. Hobbs, D.W., 1964. Rock compressive strength. Colliery Eng., 41 : 287--292. Hodgson, K. and Cook, N.G.W., 1970. The effects of size and stress gradient on the strength of rocks. Proc. 2nd Congr. Int. Soc. Rock Mech., 2nd, Belgrad, Vol. 2(3--5), pp.31--34. Hoek, E. and Brown, E.T., 1980. Underground Excavations in Rock. The Institution of Mining and Metallurgy, London, pp.527. ISRM, 1979. Suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int. J. Rock Mech. Min. Sci. Geomeeh. Abstr., 16: 135--140. Mogi, K., 1966. Some precise measurements of fracture strength of rocks under uniform compressive stress. Rock Mech. Eng. Geol., IV: 41--55. Protodyakonov, M.M., 1969. Method of determining the strength of rocks under uniaxial compression. In: M.M. Protodyakonov, M.I. Koifman and others, Mechanical Properties of Rocks. Translated from Russian, Israel Program for Scientific Translations, Jerusalem, pp.l--8. Szlavin, J., 1974. Relationships betweea some physical properties of rock determined by laboratory test. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 11: 57--66.