Rock Mechanics – a Challenge for Society, Särkkä & Eloranta (eds.),
 2001 Swets & Zeitlinger Lisse, ISBN 90 2651 821 8
Scale effects in rock strength properties.
Part 1: Unconfined compressive test and Brazilian test
K.Thuro
Engineering Geology ETH, Swiss Federal Institute of Technology, Zurich, Switzerland
R.J. Plinninger, S. Zäh, S. Schütz
Department of General, Applied and Engineering Geology, Technical University of Munich, Germany
ABSTRACT: Performing a testing program to evaluate the impact of shape and size on common rock properties, some surprising results were encountered. Firstly, the shape effect had the greatest impact on rock
strength properties. In the typical range of length/diameter ratios varying from 1 to 3, the influence on destruction work, modulus of elasticity and tensile strength is quite significant. In comparison, the effect on unconfined compressive strength is much lower. Depending on the measuring technique used for longitudinal displacement along the core sample, the modulus of elasticity increases and the strain destruction work decreases
with core length. Size, very often considered to have a controlling influence on strength, in contrast to shape
only had a marginal effect within the tested diameter range from 50 to 110 mm. Such testing results should be
considered when the suggested methods for determining the uniaxial compressive strength, deformability of
rock materials and the tensile strength are revised.
1 SCOPE
In rock mechanics and engineering geology, the unconfined compressive test and the Brazilian test are
considered to be the most wide spread methods to
obtain rock strength properties. Well known is the
scale effect concerning the unconfined compressive
strength but little data has been published since the
early 1980s (with the exception of Hoek & Brown
1980, Hawkins 1998) and even less when considering other important rock properties.
Since modern testing devices today allow sophisticated servo-contolled stress-strain paths and PCbased monitoring to obtain all data, a state-of-the-art
approach would be favourable. Now, the results of
the classic testing methods can be updated as well as
the determination of properties not yet included.
Strong efforts have led to a research program studying the principles of rock strength properties in different rock types such as metamorphic, igneous and
sedimentary rock (Schütz 1995, Zäh 1999). In this
contribution emphasis is placed on scale effects of
− unconfined compressive strength
− modulus of elasticity (Young´s modulus)
− destruction work (strain energy; defined as the integral of the stress-strain curve)
− and tensile strength
The tests were performed using core diameters
between 50 and 110 mm, since most of the core
samples taken during typical site investigations are
of this size. In this context, the so-called “scale effect” is split up into two categories: shape and size.
1 The shape effect describes the impact of variation
of the length/diameter ratio of a cylindrical
specimen (“core”) on rock strength properties.
2 The size effect is defined by the influence of the
absolute size (i.e. diameter) of the core sample
where the length/diameter ratio is left constant.
2 CORE TESTING
2.1 Unconfined compressive test
Figure 1. PC-based testing arrangement for determining complete stress-strain curves during unconfined compressive tests
with a digital monitoring equipment.
The unconfined compressive tests were performed
using a servo-controlled testing machine with a stiff
frame and a digital monitoring device (Figure 1).
Figure 2 shows the core specimen during loading,
calculating strain along the entire length of the core
169
following the ISRM (1978a) suggested methods and
those of Fairhurst & Hudson 1999. In addition to the
standard values of unconfined compressive strength
(UCS) and modulus of elasticity (Young´s modulus;
E, Figure 3), the complete stress-strain curve was
measured. The value defined as the integral of the
curve was calculated, an termed the “specific destruction work” W [kJ/m³], in short: destruction
work (after Thuro & Spaun 1996a, b, Thuro 1997,
Figure 3). This parameter is sometimes also described as “strain energy” by other authors.
which is still assoziated with failure. The strength of
a more or less fractured rock sample, which only reacts upon friction with an increase of strength, is excluded in the determination of destruction work.
2.2 Brazilian test
The Brazilien test was performed to determine the
indirect tensile strength according to ISRM (1978b)
suggested methods. The same testing equipment was
used as that for the unconfined compressive tests using 5 x 20 mm hard fibre stripes for load distribution
of the diametrically tested specimen (Figure 4).
σ1
UCS
L
Stress σ
σ2 = σ3 = 0
∆L
L
Stress-strain curve
D
ε = ∆L / L
Strain ε
D
Unconfined Compressive
Strength UCS
Unconfined Compressive Test
Figure 2. Unconfined compressive test – core specimen with
failure under unconfined compression. Terms and abbreviations.
Stress σ
Pre-failure-section
E = δσ/δε
δσ
δε
Post-failure-section
UCS
Unconfined
compression
Destruction work
W= σdε
Strain ε
εmax
Figure 3. Unconfined compressive test – complete stress-strain
curve and determining modulus of elasticity (Young´s modulus
E) and specific destruction work W (strain energy).
As can be observed, the integral of the envelope
curve is an energy (or work) related to the volume,
required for destruction of the rock sample. As a
product of both stress and strain, destruction work
represents the work of deformation including the
post-peak section. Whereas the modulus of elasticity
submits the gradient (derivation) of the linear section, the destruction work is estimated out of the area
(integral) under the stress-strain-envelope.
The value of the maximum strain εmax taken for
determining destruction work is the one strain value
Figure 4. Brazilian testing arrangement for determining indirect
tensile strength. Core specimen and abbreviations.
2.3 Testing setup
Three rock types were tested which could be obtained in high quality out of quarries, meaning that
emphasis has been placed on homogeneity and isotropy of the samples:
− A coarse-grained two-mica granite of the Bavarian forest near Passau, Germany.
− A fine-grained, pyroxen- and amphibole-rich kersantite (a type of mafic dyke) of the south Bohemian massif near Vienna, Austria.
− A fine- to medium-grained clastic limestone of
the northern alps near Salzburg, Austria.
The following testing program was performed:
1 For the shape effect during unconfined compression, common length-diameter ratios between 1
and 3 in steps of ¼ were drilled out of ∅ 50 mm
cores, and a mean value was calculated from 3-5
samples for each step.
2 For the shape effect during the Brazilian test,
common length-diameter ratios between 0.5 and 2
in steps of ¼ were cut out of ∅ 70 mm cores, calculating a mean value from 3-4 samples for each
step.
3 For the size effect under unconfined compression,
the diameter of the cores was varied between 45
and 80 mm with an exception of 110 mm for the
granite. This resulted in length-diameter ratios
170
W=2646-1008·ln(L/D) yσ(n-1)=93.4 kJ/m³ n=36 R2=93%
3.500
1.8
Kersantite
1.6
3.000
1.4
2.500
1.2
2.000
1.0
1.500
0.8
0.6
1.000
W(L/D=x) / W(L/D=2.0)
Destructioin work [kJ/m³]
constant with 2.0. Mean values were calculated
from 3-5 samples for each diameter. In our experience these core diameters represent typical
core sizes in geotechnical testing practice.
4 For the size effect during the Brazilian test, the
diameter of the cores was varied between 45 and
80 mm leaving the length-diameter ratio constant
with 1.0 and calculating a mean value out of 4
samples for each diameter.
0.4
500
3 RESULTS
1.0
1.5
1.20
Kersantite
1.15
500
1.10
475
1.05
450
1.00
0.95
425
0.90
400
0.85
375
0.80
350
2.0
2.5
E=46.29+17.03·ln(L/D) yσ(n-1)=1.40 GPa n=36 R2=95%
70
1.2
65
1.1
60
1.0
55
0.9
50
0.8
45
E(L/D=x) / E(L/D=2.0)
Modulus of elasticity [GPa]
Kersantite
0.7
40
1.0
1.5
2.0
2.5
3.0
Length/diameter ratio
Figure 7. Modulus of elasticity of the kersantite samples in correlation with core shape (length/diameter ratio).
TS=12.98-3.302·ln(L/D) yσ(n-1)=0.69 MPa n=26 R2=84%
17
1.3
16
1.2
15
1.1
14
13
1.0
12
0.9
11
0.8
10
9
TS(L/D=x) / TS(L/D=1.0)
550
UCS(L/D=x) / UCS(L/D=2.0)
Unconfined compressive strength [MPa]
UCS=474.1-27.40·ln(L/D) yσ(n-1)=9.10 MPa n=36 R2=51%
1.5
3.0
Figure 6. Destruction work of the kersantite samples in correlation with core shape (length/diameter ratio).
Tensile strength [MPa]
Various results from the performed testing program
can be seen in the Figures 5 to 8. The mean values of
the rock properties are plotted against the
length/diameter ratio together with their maximum
and minimum. Since the interesting point is the
variation of the received values, the right hand scale
is normalized for a L/D=2.0 equals 1 (compression
tests) and L/D=1.0 equals 1 (Brazilian tests). For all
the diagrams a logarithmic regression function has
been chosen in the common form f(x) = a + b ln (x)
where x = L/D. The relationship found for all three
rock types resemble one another very closely, suggesting a geometrical factor for the correlation.
Surprisingly, the influence on the destruction
work (Figure 6) , the modulus of elasticity (Figure 7)
and the tensile strength (Figure 8) is quite significant
with a standardizing factor between app. 0.7 and 1.3
meaning about 30% variation. In comparison, the effect on unconfined compressive strength is much
lower with a standardizing factor between approx.
0.95 and 1.05 meaning only 5% variation.
Remarkably the modulus of elasticity increases
and the destruction work decreases with core length,
which will be discussed in section 4. Also the indirect tensile strength is strongly dependent on the
variation of the length to diameter ratio.
1.0
2.5
Length/diameter ratio
3.1 Shape effect
525
2.0
0.7
Limestone
8
0.5
1.0
1.5
2.0
Length/diameter ratio
Figure 8. Indirect tensile strength of the limestone samples in
correlation with core shape (length/diameter ratio).
3.0
Length/diameter ratio
Figure 5. Unconfined compressive strength of the kersantite
samples in correlation with core shape (length/diameter ratio).
3.2 Size effect
Results of the performed testing program can be
seen in the Figures 9 to 13. The mean values of the
171
1.2
Limestone
230
220
1.1
210
200
1.0
190
180
0.9
170
160
0.8
150
40
50
60
70
550
1.3
1.2
500
1.1
450
1.0
400
0.9
350
0.8
300
0.7
250
0.6
40
50
60
70
W(D=x) / W(D=50mm)
Destruction work [kJ/m³]
1.4
Limestone
80
Diameter [mm]
Figure 11. Destruction work of the limestone samples in correlation with core diameter.
E=37.61+3.625·ln(D) yσ(n-1)=1.46 MPa n=30 R2=18%
65
Limestone
1.2
60
1.1
55
1.0
50
0.9
45
E(D=x) / E(D=50mm)
240
UCS(D=x) / UCS(D=50mm)
Unconfined compressive strenght [MPa]
UCS=153.0+10.71·ln(D) yσ(n-1)=7.33 MPa n=30 R2=7.2%
W=619.9-51.05·ln(D) yσ(n-1)=31.3 kJ/m³ n=30 R2=8.8%
600
Modulus of elasicity [GPa]
rock properties are plotted against the diameter together with their maximum and minimum. Since the
interesting point is the variation of the received values, the right hand scale is normalized against a
D=50mm (equals 1). As for the shape, for all the
diagrams a logarithmic regression function has been
chosen in the standard form f(x) = a + b ln (x) where
x = D. The relationships found for all three rock
types closely resemble one another, suggesting a
geometrical correlation.
The interesting result is that size, normally considered to have a significant influence on all strength
properties, in contrast to shape only has a marginal
effect in the tested diameter range of the unconfined
compressive strength (Figure 9, Figure 10, note the
opposite dip of the curves), destruction work (Figure
11), modulus of elasticity (Figure 12) and the indirect tensile strength (Figure 13). The normalized
variation (see right hand scale of figures) of less than
approx. 5% nearly always stays within the min/max
range of the data set. As a conclusion it can be
stated, that there was no size effect observed in the
tested diameter range.
0.8
40
40
50
60
70
80
Diameter [mm]
Figure 12. Modulus of elasticity of the limestone samples in
correlation with core diameter.
80
Diameter [mm]
TS=12.10-0.284·ln(D) yσ(n-1)=0.66 MPa n=24 R2=0.7%
Figure 9. Unconfined compressive strength of the limestone
samples in correlation with core diameter.
150
UCS [MPa]
1.1
140
130
1.0
120
0.9
110
UCS(D=x) / UCS(D=50mm)
1.2
0.8
100
40
60
80
100
1.3
14
1.2
1.1
12
1.0
10
0.9
0.8
8
0.7
TS(D=x) / TS(D=50mm)
160
1.4
Limestone
Tensile strength [MPa]
UCS=148.8-4.544·ln(D) yσ(n-1)=3.28 MPa n=17 R2=13%
16
0.6
6
40
50
60
70
80
Diameter [mm]
Figure 13. Indirect tensile strength of the limestone samples in
correlation with core diameter.
120
Diameter [mm]
Figure 10. Unconfined compressive strength of the granite
samples in correlation with core diameter.
172
Ca
0.88 + (0.24 ⋅ D / L)
(1)
where C = calculated compressive strength of an
equivalent 2:1 length/diameter specimen; Ca =
measured compressive strength of the specimen
tested; D = test core diameter; L = test core height.
The variation of C for a length/diameter ratio between 1 and 3 would be 0.89 to 1.04.
Using the diagram of Figure 14, the variation for
the data presented in this contribution is 0.96 to 1.03
which is less, but quite close to ASTM. In addition,
correction curves for the modulus of elasticity and
the destruction work are proposed. The functions are
simplified from the ones presented in the chapter before for convenience but without loosing accuracy.
Notice that the E correction is only valid when strain
(or displacement) is measured between the load plattens!
To obtain a correction curve for the indirect tensile strength, Figure 8 may be taken as an approximation. But more testing must be performed to validate the function parameters found.
1,3
UCS* = UCS·(0.925+0.036·L/D)
W* = W·(0.397+0.304·L/D)
E*
= E·(1.24-0.33·ln(L/D))
high
strain
Region of high
stresses
Specimen 1:1
high
strain
low
strain
high
strain
Region of high
stresses
low
strain
high
strain
Figure 15. Conceptual explanation for the relationship between
shape effect and strain measurement.
1,1
1,0
0,9
0,8
Correction curve UCS
approx. 2/3 of core length
Correction factor
1,2
Specimen 3:1
(elasto-)
plastic
C=
elastic
According to ASTM 1986, the shape correction for
the unconfined compressive strength is:
(elasto-)
plastic
4.1 Shape effect
total core length (Figure 16). The influence of shape
on the modulus of elasticity should then be minimized. Note that in the case of determining destruction work, the strain measurement must be performed between the load plattens (i.e. along the
entire core) to gain a value for the entire deformability. Therefore, such testing must be carried out
on cores with a L/D ≥ 2.
(elasto-)
(elasto-)
plastic elastic plastic
4 DISCUSSION & CONCLUSIONS
Correction curve W
Correction curve E
0,7
1,0
1,2
1,4
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
Length/diameter ratio
Figure 14. Shape correction curves for unconfined compressive
strength UCS, destruction work W and modulus of elasticity E.
The final question to be answered is the reason for
the significant impact of shape on strain measurement. This can be visualized by Figure 15, where it
is suggested that high stresses below the load plattens due to friction result in local plastification and
higher strain in those regions. If this does occur, it is
recommended that the strain (or displacement)
measurement for determining the modulus of elasticity should be picked only along the central 2/3 of the
Strain / displacement
measurement along
approx. 2/3 of core length
recommended for
determination of the
modulus of elasticity
Figure 16. Recommendation for the strain measurement along
approx. 2/3 of the specimen length during unconfined compression testing.
173
4.2 Size effect
For comparison, the obtained data range has been
plotted in the published diagrams of Hoek & Brown
1980 (Figure 17) and Hawkins 1998 (Figure 18). Although the tested size range of this contribution is
not as wide, in our experience the selected core diameters represent typical core sizes for geotechnical
testing practice. Thus, within our tested data range,
no size effect could be proved – quite in contrast to
the quoted authors!
Figure 17. Relationship between UCS and specimen size plotted as dimensionless values (after Hoek & Brown 1980).
REFERENCES
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Fairhurst, C.E. & Hudson, J.A. 1999. Draft ISRM suggested
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in uniaxial compression. Int. J. Rock Mech. Min. Sci. 36:
279-289.
Hawkins, A.B. 1998. Aspects of rock strength. Bull. Eng. Geol.
Env. 57: 17–30.
Hoek, E. & Brown, E.T. 1980. Underground excavations in
rock. Inst. Min. Metall. London: Chapman & Hall.
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Tests. Int. J. Rock Mech. & Min. Sci. & Geomech. Abstr.
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Figure 18. Relationship between UCS and specimen (core) size
plotted as dimensionless values (after Hawkins 1998). Note that
due to standardization at a core diameter of 54 mm, the data
range of this contribution is shifted by a factor of 50/54=0.93!
174