Size Effect on the Unconfined
Compressive Strength and Modulus of
Elasticity of Limestone Rock
Alaa H.J. Al-Rkaby1, and Zuheir M.S. Alafandi2
1 Department of Civil Engg., Curtin Uni., Perth, Australia
2 Building & Const. Dept., Duhok Tech. Inst.,
Duhok Polytechnic Uni., Kurdistan Region, Iraq
ABSTRACT
This paper discusses the dependence of common rock properties on size and shape of the
specimen. The phenomenon, known as scale effect, is performed in limestone. Uniaxial
compression test was carried out using specimens in three sizes, with different shapes (different
lateral /height (l/h) ratio).Compressive strength (qu) and Modulus of elasticity (E) were evaluated.
The results showed that qu and E depend on specimen size. Also, qu is increased with l/h ratio and
decreased as the size increases, while E is increased when the size increases. The results also
showed that E (concerning shape effect) is minimum when l/h =1, and E increased when l/h
become more or less than one. Such testing results should be considered when suggesting
methods for determining uniaxial compressive strength and deformability of stone and rock
materials are revised.
KEYWORDS:
Limestone, Compressive strength, Modulus of elasticity, Shape effect,
Size effect.
INTRODUCTION
The unconfined compressive strength is the most direct means of determining rock strength. The
choice of specimen size and shape is determined by design requirements, rock conditions and cost.
Evidently, small specimen of intact rocks are the least expensive to obtain and test (Franklin and
Dussealt 1983).
Various codes and standards specify the requirements of test samples. Commonly, they
recommend that sample diameter or lateral dimension should not be less than 50 mm (Hawkins
1998).
The strength of uniaxial compression decreases with the increase of height:
diameter (L/D)
ratio till it tends to a stable value, which implies that the lower bound of L/D should be located at the
point on which the negative slope of the curve of strength versus L/D increase most rapidly. For
sedimentary rocks and concrete ,this point is located at L/D≈1,while for some other rocks ,this point
is located at L/D≈2-3(Zhen and Yong-en 2005). Hence, ISRM (International society of rock
mechanics) (1979) suggested that rock strength under uniaxial compression, the ratio of L/D should
be (2.5-3). For the same case the ASTM (American society for Testing and Materials) (1989)
suggested a value of L/D≥1. About the size effect, Hoek and Brown (1980) collected a number of
strength measurement taken from the literature for different rock types of cylindrical shape and found
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that the sample strength increases progressively as the sample diameter decreases. In contrast to that,
Hawkins (1998) mentioned that the maximum strength of sedimentary sample was obtained on cores
of approximately (40-60) mm in diameter, while before and after this limit the strength was decrease.
Thuroet al. (2001) studied scale effects of cylindrical shape samples on rock ,they found that the
shape had the greatest impact on rock strength properties, while Kourkouliset al. (2005) compared
the effect of size on cubical and cylindrical building stones properties they mentioned that the
dependence of mechanical properties on sample size is not the same for the cylindrical and the cubic
specimens.
It's obvious that, the rock scale affects the results of the unconfined compressive strength, but
little data published for non-cylindrical shape or prismatic shape.
Accordingly, this paper is
concerned with scale effect on:
1-Uncofined compressive Strength (qu)
2-Modulus of Elasticity (E).
TEST PROGRAM
The test was performed using limestone rock, prismatic shape samples classified into three groups
(See Table 1, figure 1 and figure 2). The height and lateral dimensions were changed from 50mm to
300mm and the lateral dimension to height ratio (l/h) was changed from 0.5 to 2. To study the scale
effect, the samples are split up into two groups, first: to study the shape effect, in which the l/h ratio
changed from 0.5 to 2, second: to study the size effect in which l/h remains constant and only the
influence of absolute volume is taken.
Table 1: Dimensions and sizes of samples used in the study
Variable
l/h=0.5
l/h=0.67
l/h=1
l/h=1.5
l/h=2
Small (S)
50*50*100
50*50*75
50*50*50
75*50*50
100*50*50
Medium(M)
100*100*200
100*100*150
100*100*100
150*100*100
200*100*100
Large(L)
150*150*300
150*150*225
150*150*150
225*150*150
300*150*150
Note. Dimensions in mm and the prism dimensions as l*w*h as shown in fig.(1)
h
l
w
Figure 1: Dimensional variables for the tested samples
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5145
Figure 2: Samples used in the study
MATERIAL
Table 2: Physical properties of the rock
Specific gravity
Gs
2.66
Moist density
gm/cm3
1.96
Dry density
gm/cm3
1.85
Porosity n
%
30
Water content%
6
The rock samples used in this work is limestone, Yellowish to white color, fine crystalline and
has the physical properties shown in Table 2
TEST PROCEDURE
Special care was applied during the preparation of the specimen to ensure that the bases of the
specimen were parallel to each other and perpendicular to the load direction. The unconfined
compression test was performed using a testing machine Wagtech Model C089-02 with capacity 2000
kN and sensitivity of 0.1 kN with a stiff frame digital monitoring device. qu and E were determined
according to the ISRM (1979) and ASTM (1989). Figure (3) shows the method used to calculating qu
and E after drawing the complete stress-strain curve of the tested rock. The test was performed by
placing the rock sample in a loading device and the axial deformation of the specimen under the
uniaxial compression was recorded. The modulus of elasticity is determined from the stress-strain
curve by taking the average modulus of linear portion from this curve (See figure3). The
deformation was measured under loading using dial gauge sensitive to 0.01mm (See figure 4a and
4b).
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5146
Figure 3: Modulus of linear
portion of
(a) Large size,l/h=1 (b) Large
i l/hloading.
05
Figure 4: Samples under
RESULTS AND DISCUSSION
Shape Effect
The results of shape effect on the compressive strength are given in figure (5).The mean value of
three samples for the rock properties is plotted against the (l/h).The figure shows the effect of shape
with three sizes (S,M and L) on qu. The results of modulus of elasticity is shown in figure(6). It is
clearly exhibited that E is strongly dependent on the variation of l/h ratio. It is obvious that when l/h
varies from (0.5-1), the decrement percentage were 51.49%, 73.83% and 58.8% for S,M and L sizes
respectively. Also the figure indicate that when l/h varies from (1-2),the increment percentage were
from 50.2%, %,54.91 % and 51.05% for S,M and L sizes respectively .
The ASTM C170 states that correction of the shape for unconfined compressive strength
expressed by the following equation:
σ
σc
= 0.778 + 0.222 (l/h)
(1)
To compare the results with the above equation, the average of three values of the sizes (S, M, L)
taken at different l/h, and then all values are divided by the value of qu at l/h=1(cubical shape),also the
equation of the ASTM is drawn(linear equation) (Figure 7). As shown in the figure, the variation of
the correction factor is (0.93-1.18) for the present study, while it is arranged from (0.88-1.22) for
ASTM correction. It is clear that the results of this study are close to ASTM and the shape of the
variation is close to linear shape (shape of correction equation for ASTM). The same procedure is
followed for modulus of elasticity, the two correction curves are presented in the same figure (See
figure 8).
The correction curve for E differs from qu correction .It is obvious that if the correction for the
two properties (qu and E) is taken into consideration and to minimize the effect of shape on qu and E,
the l/h should be ≤1.
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18
17
16
15
14
13
12
11
10
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3500
3000
E (MPa)
qu (MPa)
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w =50 mm (S size)
2500
2000
1500
w=50 mm(S size)
w =100 mm (M size)
w=100 mm(M size)
1000
w =150 mm (L size)
w=150 mm(L size)
500
0
0.5
1
1.5
2
0
2.5
0.5
1
1.5
2
2.5
l/h ratio
l/h ratio
Figure 5: Effect of Shape on qu
Figure 6: Effect of Shape on E
1.3
1.2
1.1
1
Present Study
0.9
ASTM
0.8
0
0.5
1
1.5
l/h ratio
2
2.5
Correction Factor
qu Correctio Factor
1.7
1.5
1.3
1.1
E
0.9
qu
0.7
0
0.5
1
1.5
l/h Ratio
2
2.5
Figure 7: Relation between qu correction Figure 8: Correction curves for qu and E
and l/h
Size Effect
The effect of size can be taken by the change in dimensions without any change in the shape of
the specimen. For l/h=0.5,and l=w, they changed from (50,100,150)mm while (h) changed
from(100,200,300),for l/h=2 and ,h=w changed from (50,100,150) while (l) changed from
(100,200,300) and so on. The size effect on qu can be seen in figure(9) which shows a clear decrease
in qu as the size of the specimen increased ,while the modulus of elasticity increased as shown in
figure (10). The results also indicate that the size affects on E slightly more than qu. Size effect is
not like the shape effect specially in the behavior of E with shape (figure(6) and figure(10)).
To draw correction factor for size effect (See figure11) ,the mean values for both qu and E for
each l/h values are taken then these values are divided by the value of w=100mm. The figure shows
that the range of correction factors for w=100 are (0.9-1.12),(0.84-1.15) for qu and E respectively.
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l/h=2
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16
15
14
13
12
11
10
l/h=0.5
l/h=1.5
3500
l/h=1
l/h=0.67
l/h=1
3000
l/h=0.67
l/h=1.5
l/h=0.5
E (MPa)
qu (MPa)
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2500
l/h=2
2000
1500
0
100
200
1000
0
50
width (w)(mm)
100
150
200
width(w)(mm)
Figure 9: Effect of size on qu
Figure 10: Effect of size on E
correction factor
1.2
1.1
1
0.9
0.8
qu
0.7
E
0.6
0
50
100
150
width (w)(mm)
200
Figure 11: Effect of size on qu and E correction factors
CONCLUSIONS
1. The effect of shape and size are different in both qu and E.
2. The effect of shape on E is more than that in qu and it is minimum in l/h=1 ,while the qu is
increased as l/h increased.
3. It is recommend to use l/h<=1 to minimize the effect of shape if both E and qu are taken into
consideration.
4. The effect of size on both E and qu works in opposite way; as E increases qu decreases with size.
5. As qu is corrected for shape ,the same thing could be done for E .
6. Size may be taken into consideration and correct for qu and E.
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REFERENCES
ASTM, C170, Standard Test Method for Compressive Strength of Natural Building Stone,
Philadelphia ,USA.,04.08,13-15. 1989.
Franklin, J. A. and DussealtM. B, Rock Engineering, McGraw-Hill ,USA. 1989.
Hawkins, A.B., Aspect of rock strength, Bull. Eng. Geol. Env.57:17-30. 1998.
Hoek, E. and Brown, E.T, Underground excavation in rock ,Inst. Min. Metall. London :
Chapman and Hall. 1980.
ISRM, Suggested Methods for Determining Uniaxial Compressive Strength and
Deformability of Rock Materials, ISRM Committee on Standardization of Laboratory
Tests , Int. J. Rock Mech. Min. Sci., Vol. 16:137-140. 1979.
Kourkoulis, S. K.Caroni, C., Papageogiou E. A , Contribution to the study of the size effect
for natural building stones,5th GRACM International Congress on Computational
Mechanics, Limassol, 29 June-1 July. 2005.
Thuro, K., Plinningr, R.J.,Zah, S., Scale effect in rock strength properties, Eurock 2001:Rock
Mechanics a challenge for society, Espoo ,June , pp.169-174. 2001.
Zhen, FU and Yong-en,CAI ,Numerical analysis on the influence of rock specimen size on
crack stress field, ACTASEISMOLOGICAL SINICA, Vol. 18, No.3:322-330,2005.
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