art%3A10.1007%2Fs00603-011-0161-6(1).pdf

ORIGINAL PAPER

The Effect of Specimen Size on Strength and Other Propertiesin Laboratory Testing of Rock and Rock-Like CementitiousBrittle Materials

William J. Darlington Pathegama G. Ranjith

S. K. Choi

Received: 24 January 2011 / Accepted: 27 May 2011 / Published online: 17 June 2011

Springer-Verlag 2011

Abstract The effect of specimen size on the measured

unconfined compressive strength and other mechanical

properties has been studied by numerous researchers in the

past, although much of this work has been based on

specimens of non-standard dimensions and shapes, and

over a limited size range. A review of the published liter-

ature was completed concentrating on the presentation of

research pertaining to right cylindrical specimens with

height:diameter ratios of 2:1. Additionally, new data has

been presented considering high strength (70 MPa) cement

mortar specimens of various diameters ranging from 63 to

300 mm which were tested to failure. Currently, several

models exist in the published literature that seek to predict

the strengthsize relationship in rock or cementitious

materials. Modelling the reviewed datasets, statistical

analysis was used to help establish which of these models

best represents the empirical evidence. The findings pre-

sented here suggest that over the range of specimen sizes

explored, the MFSL (Carpinteri et al. in Mater Struct

28:311317, 1995) model most closely predicts the

strengthsize relationship in rock and cementitious mate-

rials, and that a majority of the empirical evidence supports

an asymptotic value in strength at large specimen

diameters. Furthermore, the MFSL relationship is not only

able to model monotonically decreasing strengthsize

relationships but is also equally applicable to monotoni-

cally increasing relationships, which although shown to be

rare do for example exist in rocks with fractal distributions

of hard particles.

Keywords Specimen size Compressive strength Rock Rock-like materials Scale effect Size effect

1 Introduction

For most engineering design, material properties such as

strength, Youngs modulus and Poissons ratio are of

critical importance. Establishing these parameters often

proves problematic when considering materials such as

concrete, and more acutely rock, where the size of the

engineering structure far exceeds the size of any laboratory

test specimen.

A great deal of research has been focused on upscaling

laboratory measured strength parameters to field problems.

This has been done in a number of ways. Empirical studies

have concentrated on finding size-dependent relations

between laboratory measured properties using small size

laboratory specimens (e.g. 30150 mm diameter cylin-

ders). To establish the properties of specimens larger than

this, and thus make sizestrength comparisons with labo-

ratory measured values, in situ testing or back analysis of

large structures is often used.

The results of these are generally well known and

accepted. The most commonly cited; the general work of

Hoek and Brown (1980), Brace (1981) and the original

work of Weibull (1951) form either the basis or what is

directly used in many rock mechanics designs today.

W. J. Darlington (&) P. G. RanjithDepartment of Civil Engineering, Monash University,

Clayton, VIC, Australia

e-mail: [email protected]

P. G. Ranjith


S. K. Choi

CSIRO Earth Science and Resource Engineering,

Bayview Avenue, Clayton, VIC, Australia


123

Rock Mech Rock Eng (2011) 44:513529

DOI 10.1007/s00603-011-0161-6

Figure 1 shows the data compiled, and corresponding

relationship established by Hoek and Brown (1980)

between specimen diameter and the strength of intact rock.

This relation is represented by Eq. 1 where rcd is theuniaxial compressive strength of a sample with diameter, d;

and rc50 is the uniaxial compressive strength of a 50 mmdiameter sample.

rcd rc5050

d

0:181

A similar relation has also been proposed by Cunha

(1990):

rcd rc5050

d

0:222

It is generally accepted that there is a significant

reduction in strength with increasing specimen size.

Equations 1 and 2 are the current benchmarks when

quantifying this phenomenon in rock. Hoek (2000)

suggests this reduction in strength is due to the increased

probability that failure of rock grains will occur as the

specimen size increases. Rock strength will reach an

asymptotic minimum value at a certain specimen size that

will depend on the type and condition of the rock. Hoek

(2000) goes further, hypothesising that the strength of a

rock mass will reach a constant minimum value if the size

of the rock blocks are considerably smaller than the rock

mass under consideration. However, importantly the power

law proposed by Hoek and Brown (1980) includes a

horizontal asymptote of zero as diameter tends to infinity.

It is important to note that the literature discussed and all

results presented in this paper are exclusively based on

cylindrical specimens with a height to diameter ratio of 2:1.

Although data exists on specimens of other ratios and

shapes, the results of these are outside the scope of this

paper. From a general perspective, with most, if not all,

strengths obtained for both rock and concrete design

obtained from specimens with height to diameter ratios of

2:1, and this being a widely accepted standard, a study

using samples with these geometric properties seems most

useful.

1.1 Cementitious Materials

Carpinteri et al. (1999) noted a significant lack of research

regarding the compressive strengthsize relationship of

laboratory size concrete specimens and explored the

applicability of the size effect law (SEL) (Eq. 3) proposed

by Bazant (1984) and the multifractal scaling law (MFSL)

(Eq. 4) proposed by Carpinteri et al. (1995) through com-

parisons with a variety of published data (Fig. 2). In rela-

tion to Eqs. 3 and 4, rN is the normal strength; d is thespecimen diameter; ft is a strength parameter; b and k0 aretwo empirical constants; dmax is the maximum aggregate

size; fc and lch are constants that represent the nominal

compressive strength of an infinitely large specimen and an

internal material length, respectively; these are determined

by non-linear least squares fitting:

rN bftffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 d=k0dmax p 3

rN fcffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 lch

d

r4

Considering concrete, Carpinteri et al. (1999) noted

Eqs. 3 and 4 as the only published relations to describe the

strengthsize relationship. Furthermore, they showed that

SEL and MFSL generate opposite predictions. The SEL

predicts infinitely large specimens to have zero strength

similar to Eqs. 1 and 2, while the MFSL predicts large

specimens to have a finite asymptotic minimum strength.

Considering the MFSL further, it was found to underesti-

mate large specimen strength by 10%, while the strength of

small specimens (where d ffi 10dmax) are overestimated by10%.

Symons (1970) also assesses the scale effect in cement

stabilised materials. Here specimens were made using three

grades of aggregate (well-graded sand, crushed rock and

gravel-sand-clay) and varying cement content percentage.

Four specimens of each aggregate grade at each percentage

of cement were tested. Several height to depth ratios were

assessed in a range of cylindrical specimen diameters and

square prism sizes; the average results obtained for the

cylindrical specimens with a height to diameter ratio of 2:1Fig. 1 Influence of specimen size on the strength of intact rock(Hoek 2000)

514 W. J. Darlington et al.

123

are presented in Figs. 3 and 4. There appears to be no

clear or significant scale relationship for the well-graded

sand-cement mortar specimens, while the cemented cru-

shed rock specimens show a decrease in strength with size

modelled accurately by both the Hoek and Brown and

MFSL relationships. Figure 5 shows the results produced

by Hoskins and Horino (1969) for Plaster of Paris. Con-

sidering only their data for specimens with diameters [50mm the MFSL relationship models the experimental data

well, while the relationship of Hoek and Brown (1980)

significantly underestimates the strength of larger diame-

ter specimens.

(b)

(a)

(c)

Fig. 2 Plots showing the data of Blanks and McNamara (1935)for concrete of different water to cement ratios a w/c = 0.53;b w/c = 0.55; and c w/c = 0.54), and the strengthsize relations ofBazant (1984) (SEL), Carpinteri et al. (1995) (MFSL), and Hoek and

Brown (1980) (after Carpinteri et al. 1999)

Fig. 3 Normalised UCS of well-graded sand-cement mortar corre-lated with specimen diameter (after Symons 1970). The size effect

relations of Hoek and Brown (1980) and Carpinteri et al. (1995) are

also plotted (the MFSL has only been fitted to the 14% well-graded

sand data in order to preserve clarity of the figure)

Fig. 4 Normalised UCS of cemented crushed rock correlated withspecimen diameter (after Symons 1970). The size effect relations of

Hoek and Brown (1980) and Carpinteri et al. (1995) are also plotted

(the MFSL has only been fitted to the 4% crushed rock data)

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0 50 100 150

Fig. 5 UCS of Plaster of Paris in correlation with specimen diameter(error bars indicate standard error of the mean) (after Hoskins andHorino 1969). The size effect relations of Hoek and Brown (1980)

and Carpinteri et al. (1995) are also plotted for comparison. The

MFSL line has been fitted only to data [50 mm

The Effect of Specimen Size on Strength and Other Properties 515

123

1.2 Sedimentary Rocks

Natau et al. (1983) demonstrated a trend that followed the

same form as Hoek and Browns (1980) model (Eq. 1), but

with a significantly different power (Fig. 6). However, it is

important to note that Fig. 6 depicts the results for jointed

yellow limestone with an unspecified joint spacing and

load orientation which may influence the magnitude of the

decrease in strength with specimen size.

Pells (2004) summarised the effect of specimen size on

the strength of Hawkesbury Sandstone (Fig. 7) by testing

specimens ranging from 18 to 144 mm in diameter. Using

Eq. 1, Pells (2004) expected the 150 mm specimens to

have a strength of around 85% of that of the 50 mm

specimens. However, this was not found to be the case, and

no clear scale effect was seen in this rock. The conclusions

of this research do point out that almost all the rock types

used in the derivation of Hoek and Browns (1980) Eq. 1

were igneous or crystalline and therefore would inherently

contain micro-cracks. Following the theory of Weibull

(1951) these micro-cracks are the cause of a strength scale

effect. Although most sedimentary rocks do exhibit some

scale effect, Hawkesbury Sandstone appears to be an

exception rather than the rule. Figure 7 displays a mono-

tonically increasing MFSL relationship for the WPO data

due to the increasing nature of the original dataset and the

fact that the MFSL is fitted to each dataset using non-linear

least squares fitting to find the correct values of the fitting

constants. It is hypothesised that the monotonically

increasing relationship seen here is due to the fractal dis-

tribution of weathering effects in this moderately weath-

ered Sandstone. In modelling rock behaviour where there is

a monotonically increasing strengthsize relationship the

MFSL has the advantage of being able to model both

increasing and decreasing trends.

Thuro et al. (2001) tested two rock types under uniaxial

compression: a coarse-grained two-mica granite; and a

fine- to medium-grained clastic limestone. These results are

shown in Figs. 8 and 12, respectively. No dramatic scale

effect for either UCS or Youngs modulus is shown and the

MFSL relationship plotted in Fig. 8 monotonically

increases due to the increasing nature of the original dataset

being modelled. As the authors point out, this contradicts

what had been previously published by Hoek and Brown

(1980) (Fig. 1), and Hawkins (1998) (Fig. 11).

Hawkins (1998) critically reviews the commonly

accepted relation proposed by Hoek and Brown (1980)

(Eq. 1). In doing so, he presents new data pertaining to the

strengthsize relationship of a range of sedimentary rocks.

Hawkins results are plotted in Fig. 11. They show that

over a sample of specimen diameters, peak strength will be

Fig. 6 UCS of jointed yellow limestone correlated with specimendiameter and line of best fit in the form of Eq. 1 with a power of 1.6

(instead of 0.18). The size effect relation of Carpinteri et al. (1995) is

also plotted for comparison. The UCS of a 50 mm specimen was

obtained for the purpose of normalisation using Eq. 1 (after Natau

et al. 1983)

Fig. 7 UCS of Hawkesbury Sandstone correlated with specimendiameter (error bars indicate 1.0 standard deviation from the mean)(after Pells 2004). The size effect relations of Hoek and Brown

(1980), and Carpinteri et al. (1995) are also plotted for each sandstone

outcrop. WPO West Pymble Outcrop, GQ Gosford Quarry

Fig. 8 UCS of limestone in correlation with specimen diameter(error bars indicate minimum and maximum values in data set) (afterThuro et al. 2001). The size effect relations of Hoek and Brown

(1980) and Carpinteri et al. (1995) are also plotted for comparison


123

seen at a specimen diameter of 4060 mm, and that at

diameters smaller or larger than this the strength of the

specimen will decrease. If only considering strength pre-

dictions for specimens with a diameter[54 mm, the Hoekand Brown (1980) relation seems to provide a reasonable

prediction for Purbeck Limestone and Hollington Sand-

stone. This cannot be said for the other rock types pre-

sented in Figs. 9, 10, 11.

1.3 Igneous Rocks

The recent work presented in Thuro et al. (2001) largely

contradicts the Weibull (1951) expectation of a strength

scale effect in materials containing micro-cracks i.e.

igneous or crystalline rock types. The experimental data

presented by Thuro et al. (2001) shows no such relationship

(Fig. 12). If proved by further studies, the implications of

these results are immense and far reaching; forcing a

reassessment of a large amount of the empirical theory that

is currently relied upon in rock mechanics design.

Aside from sedimentary rocks and plaster of Paris,

Hoskins and Horino (1969) present results for salida

granite which show an obvious size effect (Fig. 13).

Figure 14 shows the results of research carried out by

Jackson and Lau (1990). Their work on Lac du Bonnet

granite shows a decrease in strength with increasing

specimen size, although it is not as significant as what is

predicted by Eq. 1.

Yuki et al. (1995) studied the anisotropic behaviour of

Ohya Stone (welded tuff) over varying diameters from 30

to 150 mm. They showed that no strength decrease was

evident when specimens were loaded either parallel, or

Fig. 9 UCS of Longmont Sandstone in correlation with specimendiameter (error bars indicate standard error of the mean) (afterHoskins and Horino 1969). The size effect relations of Hoek and

Brown (1980) and Carpinteri et al. (1995) are also plotted for

comparison. The MFSL line has been fitted only to data [50 mm

Fig. 10 UCS of Kansas Limestone in correlation with specimendiameter (error bars indicate standard error of the mean) (afterHoskins and Horino 1969). The size effect relations of Hoek and


comparison. The MFSL line has been fitted only to data [50 mm

Fig. 11 UCS of various sedimentary rocks in correlation withspecimen diameter (after Hawkins 1998). The size effect relations

of Hoek and Brown (1980) and Carpinteri et al. (1995) are also

plotted (the MFSL has only been fitted to the Pennant Sandstone data,

and only to data [54 mm)

Fig. 12 UCS of granite in correlation with specimen diameter (errorbars indicate minimum and maximum values in data set) (after Thuroet al. 2001). The size effect relations of Hoek and Brown (1980) and

Carpinteri et al. (1995) are also plotted for comparison


123

normal to the stratification, moreover, a slight increase in

strength was seen (Figs. 15, 16). This finding is extremely

pertinent and remains the only study to this date and to the

authors knowledge to consider an anisotropic strength

size effect in specimens with height to diameter ratios of

2:1. The unusual monotonically increasing nature of the

MFSL seen fitted to these increasing data sets is due to the

non-linear least squares fitting requirement inherent in the

MFSL function. Yuki et al. (1995) concludes that the

increase in UCS with increasing specimen diameter is

related to the percentage of pumice fragments contained in

the specimen. As the specimen size increases, so does the

percentage of pumice.

Nishimatsu et al. (1969) conducted sizestrength studies

on several igneous rocks (Figs. 17, 18, 19, 20, 21). The

saajome andesite shows a strong size effect and excellent

correlation with both the MFSL and Eq. 1. Nishimatsu

et al. (1969) tested specimens between 13 and 70 mm, but

aside from the aforementioned saajome andesite which has

a good spread of specimen diameters within this range, the

other rocks tested in the study fail to include data from the

3070 mm range. This makes it difficult to compare these

datasets to those of other researchers as it is impossible to

say if the low strength values seen at the 70 mm diameter

are preceded by a general decline, as would be modelled by

the MFSL or Eq. 1, with a peak strength at a diameter

Fig. 13 UCS of Salida granite in correlation with specimen diameter(error bars indicate standard error of the mean) (after Hoskins andHorino 1969). The size effect relations of Hoek and Brown (1980)

and Carpinteri et al. (1995) are also plotted for comparison

Fig. 14 UCS of Lac du Bonnet granite in correlation with specimendiameter (error bars indicate 1.0 standard deviation from the mean)(after Jackson and Lau 1990). The size effect relations of Hoek and


comparison

Fig. 15 UCS of Ohya Stone (welded tuff) (loaded horizontally to thedepositional surface) in correlation with specimen diameter (after

Yuki et al. 1995)

Fig. 16 UCS of Ohya Stone (welded tuff) (loaded vertically to thedepositional surface) in correlation with specimen diameter (after

Yuki et al. 1995)

Fig. 17 UCS of Saajome andesite in correlation with specimendiameter (after Nishimatsu et al. 1969)


123

within this untested range (similar to what has been seen in

some sedimentary rocks) or if the low strength is an outlier

(although this seems unlikely).

1.4 Metamorphic Rocks

The only example of strengthsize effect testing regarding

metamorphic rocks reported in the published literature was

produced by Hoskins and Horino (1969) (Fig. 22) and

relates to Carthage marble. Obviously it is not possible to

make any broad conclusions as to the existence and qual-

ities of a sizestrength relationship in metamorphic rocks

due to a distinct lack of published data. However, the

sample supports a decrease in strength of approximately

7% when moving from a 50 to 127 mm specimen diameter.

Here again the power relation of Hoek and Brown (1980)

overpredicts this decrease in strength significantly. Clearly

more research is required into the strengthsize effect in

metamorphic rock types in order to reach sound and sig-

nificant conclusions.

1.5 Youngs Modulus and Poissons Ratio

Aside from the works of Thuro et al. (2001) (Fig. 23),

Jackson and Lau (1990) (Fig. 24), Yuki et al. (1995)

(Figs. 25, 26) and the new results presented in this paper,

few have explored the existence of a size effect in Youngs

modulus or Poissons ratio. When considering the Youngs

modulus results of Thuro et al. (2001) a significant size

effect is not obvious. While Jackson and Lau (1990) show a

Fig. 18 UCS of Ogino tuff in correlation with specimen diameter(after Nishimatsu et al. 1969)

Fig. 19 UCS of Inada granite in correlation with specimen diameter(after Nishimatsu et al. 1969)

Fig. 20 UCS of Shinkomatsu andesite in correlation with specimendiameter (after Nishimatsu et al. 1969)

Fig. 21 UCS of Aoishi sandy tuff in correlation with specimendiameter (after Nishimatsu et al. 1969)

Fig. 22 UCS of Carthage marble in correlation with specimendiameter (error bars indicate standard error of the mean) (after Hoskinsand Horino 1969). The size effect relations of Hoek and Brown (1980)

and Carpinteri et al. (1995) are also plotted for comparison


123

significant decrease (*10%) in Youngs modulus with anincrease in specimen diameter (from 45 to 300 mm).

Conversely, Yuki et al. (1995) shows an increase in

Youngs modulus with specimen size, of approximately

27% for Ohya Stone (welded tuff) loaded horizontally to

the depositional surface and only 4% for Ohya Stone loa-

ded vertically to the depositional surface. However, as

mentioned previously Yuki et al. believes this to be related

to the percentage of pumice fragments contained in the

specimen.

In the only published dataset pertaining to Poissons

ratio, Jackson and Lau (1990) observed a decrease in

Poissons ratio for Lac du Bonnet with increasing specimen

size (Fig. 27). Interestingly, a decrease in strength and

Youngs modulus with increasing size was also seen for the

same rock.

2 Experimental Method

In order to explore the effect of specimen size, high

strength cylindrical mortar specimens with a range of

diameters were manufactured. The diameters of the spec-

imens are listed in Table 1. The 63.5 and 83.5 mm diam-

eters were chosen in line with standard diamond drill rock

core sizes; HQ and PQ, respectively, while 150 mm

specimens are sometimes used in the initial stages of deep

drilling or in weak, weathered or fractured rock, and are

also commonly used during concrete testing. The 300 mm

Fig. 23 Youngs modulus of limestone in correlation with specimendiameter (error bars indicate minimum and maximum values in dataset) (after Thuro et al. 2001)

Fig. 24 Youngs modulus of Lac du Bonnet granite in correlationwith specimen diameter (error bars indicate 1.0 standard deviationfrom the mean) (after Jackson and Lau 1990)

Fig. 25 Youngs modulus of Ohya Stone (welded tuff) (loadedhorizontally to the depositional surface) in correlation with specimen

diameter (error bars indicate 1.0 standard deviation from the mean)(after Yuki et al. 1995)

Fig. 26 Youngs modulus of Ohya Stone (welded tuff) (loadedvertically to the depositional surface) in correlation with specimen

diameter (error bars indicate 1.0 standard deviation from the mean)(after Yuki et al. 1995)

Fig. 27 Poissons ratio of Lac du Bonnet granite in correlation withspecimen diameter (error bars indicate 1.0 standard deviation fromthe mean) (after Jackson and Lau 1990)


123

diameter specimens were chosen as the largest diameter

that could be tested at the current testing facility; with the

required failure load approaching the maximum load

capacity of the loading frame. Each cylindrical specimen

tested had lengths twice their diameter.

The mortar mix was designed to produce samples which

closely resemble sandstone in terms of peak strength,

Youngs modulus and Poissons ratio. As a high cement

water ratio was used, a plasticiser (Glenium 27) was added

to the cement mortar mix to increase workability and

prevent segregation. Additionally, the Glenium 27 aided by

increasing early and ultimate compressive strength and

Youngs modulus, while also decreasing any shrinkage.

These characteristics were desirable as it was found that

small amounts of shrinkage could cause precision speci-

men end preparation to differentially shrink to unac-

ceptable levels of flatness and perpendicularity if not tested

soon after being prepared.

The cement mortar was mixed in an 85-l pan mixer

using the proportions outlined in Table 2. Initially half of

the total amount of sand required was placed in the mixer

and the total amount of cement was added followed by the

balance of the sand. The dry mix was then mixed for 60 s.

Half the volume of water was then added and it was mixed

for a further 60 s during which time the plasticiser was

poured in, followed by the remaining volume of water. The

mix was then rested for 120 s before it was finally mixed

for another 120 s.

The particle size distribution of the sand used in the

mortar is shown in Fig. 28. It is to be noted that the particle

size is not typical of natural sandstonethe effect of sand

particle size distribution will be examined in future studies.

After being left to cure for approximately 24 h the

specimens were removed from their individual moulds,

relocated to a misted curing room and left to cure for

28 days. Due to the large size of some of the specimens it

was not possible to use curing tanks.

Once the specimens had cured for 28 days, they were

prepared in accordance with ASTM D4543-01 (ASTM

2001). End flatness of the 63.5 and 83.5 mm diameter

specimens was achieved by machining the specimen ends

in a specially constructed v-block clamp fixed to a high

quality tool and cutter grinder. A diamond cup wheel with

a 126-grit size suitable for roughing and finishing precision

tools was used to grind the end surfaces. Once the speci-

men was clamped in the v-block it was not removed until

both ends had been ground. This was achieved by rotating

and relocating the entire v-block clamping system on the

tool and cutter grinder feeder table while the specimen was

secured. These techniques and procedures ensured that the

ends were parallel. The ends of the larger specimens were

trimmed parallel using a 900 mm diameter diamond saw

and large v-block clamping system.

The tests were conducted in accordance with ASTM

D7012-04 (ASTM 2004). A strain-controlled Amsler

loading frame with a 5,000-kN capacity was used during all

testing. Two Schaevitz LVDTs were used to measure the

axial displacement to a reported accuracy of 0.375% of the

full range. To calculate the applied strain, the respective

displacements were averaged.

In addition to this measure of axial strain four 68 mm

long Kyowa strain gauges were attached to the specimen;

two axially and two laterally around the specimen mid-

height at 908 intervals such that the axial strain gaugeswere separated by 180. To avoid strain readings resultingfrom artificial strength variations caused by the gauge

adhesive (where strain measurements may have been more

representative of the adhesive rather than the mortar under

examination or defects and voids in the mortar surface) a

two-stage application process was used. After accurate

centres and alignment lines were marked on the specimen,

a layer of epoxy resin was applied to the mortar surface

over these markings. The resin filled any surface irregu-

larities and pore spaces. After the epoxy had cured it was

Table 1 Cylindrical sample dimensions

Number of specimens Diameter (mm) Length (mm)

10 63.5 127

10 83.5 167

3 150 300

3 200 400

3 300 600

Table 2 Mix design

Sand

(kg)

Cement

(kg)

Water

(kg)

Plasticiser

(Glenium 27) (ml)

85 34 13.6 150

0

10

20

30

40

50

60

70

80

90

100

10 100 1000 10000

Particle Size (m)

Per

cent

age

Pas

sing

Fig. 28 Particle size distribution of sand used in cement mortar mix(following methodology outlined in ASTM D422-63 (ASTM 2007)


123

sanded back to the original surface level of the mortar,

leaving in situ any resin that had filled irregularities and

pore voids of the mortar along the gauging area. These

areas were then cleaned and neutralised prior to the

application of the strain gauges using the recommended

low temperature curing adhesive. A dataTaker DT8000

data acquisition system recorded the strain, displacement

and force readings at a rate of 20 Hz.

Published data pertaining to laboratory testing of cyl-

inders of rock and rock-like brittle materials with height

to diameter ratios of 2:1 were collected from the pub-

lished literature. Statistical analysis was carried out on the

new experimental data presented in this paper as well as

the data collected from the literature (as described ear-

lier). The analysis was completed using five different

relational forms comprising a linear relationship between

UCS and diameter, a linear relationship between UCS and

1/diameter, a power relationship in the form of Hoek and

Brown (1980) relationship (i.e. a power relationship

between UCS and 1/diameter), the MFSL relationship

(Carpinteri et al. 1995) and an exponential relationship

between UCS and diameter. This comprehensive series of

statistical analysis had two aims. Firstly, to assess the

applicability of the currently accepted Hoek and Brown

(1980) relationship and the similar Cunha (1990) rela-

tionships for rock strength variation with specimen size.

Secondly, to explore the goodness of fit of other relational

forms.

Using simple transformations, all the commonly used

theoretical and empirical relationships along with other

mathematical forms to describe the strengthsize effect

could be reduced to linear equations enabling the use of

linear regression analysis methods. In addition to calcu-

lating the coefficients of the different relational forms, R2

values, p values, 95% confidence intervals and residual

plots were generated to assess the goodness of fit of the

models listed hereunder.

To test the fit of Hoek and Brown (1980), the following

transformation was used to enable the use of simple linear

regression analysis:

rN rc5050

d

k

Y ln rcd ln rc5050

d

k ! ln 50krc50

k ln 1

d

a bX5

After completing the linear regression, the original

coefficients of the power curve were found using:

rc50 ea

50k; and k b

Similarly the MFSL was transformed as follows:

rN fcffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 lch

d

r

Y r2N f 2c 1 lchd

a bX 6

After completing the linear regression, the coefficients

of the original equation were found using:

fc ffiffiffia

p

lch bf 2c

The fit of an exponential model was also assessed using

the following transformation:

rN Aekd

Y ln rN ln Aekd

ln A kd a bX 7

After completing the linear regression, the coefficients

of the original equation were found using:

A ea

k b

A simple linear model of the following form was also

assessed:

rN A kd 8

In addition to the simple linear model, the following

linear relationship between UCS and 1/diameter was also

tested:

rN A k1

d: 9

3 Results and Discussion

3.1 StrengthSize Effect

The average 7-day strength of the mortar was found to be

41 MPa. To establish the consistency between the different

batches of cement mortar used in making the range of

various sized specimens, three 100 mm diameter reference

samples were taken from each mortar batch. The

mechanical properties of these cylinders were found to be

very consistent with an average UCS of 66.1 MPa

(SD = 3.8 MPa) and Youngs modulus of 23.4 GPa

(SD = 0.4 GPa).

Considering the specimens made to explore the size

effect; the average UCS and Youngs modulus across all of

the specimens were found to be 66.0 MPa (SD = 2.8 MPa)

and 23.7 GPa (SD = 1.5 GPa), respectively. The stress

strain results of a typical cylinder are presented in Fig. 29.

Comparing these standard deviations with the results of the


123

100 mm batch reference samples the figures are very

similar, suggesting that the variance seen in the strength

size specimens can almost entirely be explained by the

variation in mortar batch properties.

Figures 30, 31 and 32 show the findings of the current

study. The relationship between UCS and test specimen size

(Fig. 30) shows a decrease in the strength of a 63.5 mm

specimen when compared to that of a 300 mm specimen of

approximately 10%. The relations proposed by Hoek and

Brown (1980) (Eq. 1), Cunha (1990) (Eq. 2), and the MFSL

of Carpinteri et al. (1995) (Eq. 4) have been plotted

alongside the experimental data. The first two of these

relations fit the data only when small specimen diameters

are considered (63.5100 mm diameter), but grossly un-

derpredict the UCS of 150, 200, and 300 mm diameter

specimens. The MFSL line of best fit shows an exceptional

fit across the range of specimen diameters tested.

For the current dataset, the decrease in strength with size

is not as strong as is predicted by Eqs. 1 and 2 (also plotted

in Fig. 30). A line of best fit has also been plotted in the

form of Eq. 10 resulting in a fit of reasonable quality

(R2 = 0.26). Here a power of 0.08 has been used instead of

0.16, as suggested originally by Hoek and Brown (1980).

rcd rc6363

d

0:0810

During the statistical analysis process, 95% confidence

intervals were estimated for each of the models under

investigation. In order to assess the applicability of the

models represented by Eqs. 1 and 2, the relevant

confidence intervals for cemented materials have been

listed in Table 3. The powers suggested by Hoek and

Brown (1980) and Cunha (1990) of 0.18 and 0.22,

respectively, clearly fall outside the 95% confidence

intervals of half the listed datasets suggesting these

powers are inappropriate when considering cement

stabilised materials. It can also be seen that most of these

results are not statistically significant, with the 95%

confidence intervals encompassing zero.

Some of the datasets, namely those of Hoskins and

Horino (1969) and Hawkins (1998) include data pertaining

to relatively small diameter specimens (i.e. \50 mm).Considering these data sets, peak strength is seen in spec-

imens ranging from 38 to 76 mm in diameter. Specimens

with diameters smaller or larger than these show lower

strengths, contradicting Weibulls (1951) theory. This has

consequences when trying to fit any of the relational

models commonly attributed to sizestrength effects in

rock [i.e. exponential, power, MFSL, linear (UCS vs.

1/diameter), etc.]. Fitting these models to complete data

sets (i.e. sets that include data pertaining to diameters less

than the peak strength diameter (generally around 50 mm)

can cause extremely poor statistical fits and large diameter

specimen strength predictions (which are of most interest

in terms of large scale design). This phenomenon is only

seen in the sedimentary rocks analysed. However, the

strength of some of the granite samples appears to

asymptote when the specimen diameter is \50 mm.Assessing the 95% confidence intervals for Eqs. 1 and 2

for sedimentary rocks (Table 4), it can be seen that the

powers suggested by Hoek and Brown (1980) and Cunha

(1990) of 0.18 and 0.22, respectively, are included in many

0

10

20

30

40

50

60

70

-1500 -1000 -500 0 500 1000 1500 2000 2500 3000 3500

Axi

al S

tres

s (M

Pa)

MicroStrain

Fig. 29 Typical stress strain relation for a 83.5 mm diameter cementmortar cylinder

0.70

0.80

0.90

1.00

1.10

1.20

50

55

60

65

70

75

80

85

90

0 50 100 150 200 250 300 350 400 450

Fig. 30 Experimental datashowing the relationship

between specimen diameter and

average UCS for high strength

cement mortar specimens. The

size effect relations of Hoek and

Brown (1980), Cunha (1990),

and a line of best fit in the form

of Carpinteri et al.s (1995)

MFSL (Eq. 3) are also plotted

for comparison


123

of the confidence intervals, especially those that only

include specimens of diameters larger than 54 mm. It can

also be seen that in datasets that include a maximum

strength at diameters larger than the smallest diameters

tested, the model becomes statistically significant when

these small diameter data points are omitted.

With regards to Table 5 and igneous rock types, it can

be seen that the powers suggested by Hoek and Brown

(1980) and Cunha (1990) of 0.18 and 0.22, respectively are

included in a majority of the confidence intervals, but zero

is also included in all but one of the intervals which sug-

gests that this power relationship is not statistically sig-

nificant at a 5% level.

The original empirical sizestrength relationship pro-

posed by Hoek and Brown (1980) was based on data from

several sources. Each source employed varying test pro-

cedures and experimented using specimens that did not

necessarily conform to a single standard. For example, the

test results of Pratt et al. (1972) relating to quartz diorite

pertained to a mix of triangular and circular prisms each

with a height to depth ratio of 1.5, although consideration

of the varied cross-sections was not made. The results of

Mogi (1961) were also used by Hoek and Brown (1980) to

establish the size strength relation apparent in marble. The

specimens used in the aforementioned research were rect-

angular prisms.

Thuro et al. (2001), amongst others, have discussed

testing specimens of non-standard diameter to height ratios

and/or geometric volumes. Variations in specimen geom-

etry can induce significant differences in stress distribu-

tions within a test specimen and lead to variation in

measured peak strength. As with in situ testing, although

these tests have a sound purpose and can provide valuable

information under specific circumstances, the appropriate-

ness of using the results of these various methods in one

Fig. 31 Relationship between specimen diameter and averageYoungs modulus (error bars indicate 1.0 standard deviation fromthe mean)

Fig. 32 Relationship between specimen diameter and Poissons ratio

Table 3 95% confidence intervals for the power relationship in the form of Hoek and Brown (1980) and Cunha (1990) considering cementstabilised materials

Reference Material tested Lower 95% Upper 95%

Blanks and McNamara (1935) Concrete (w/c = 0.53, Agre. max = 38.1 mm) -0.005 0.123

Blanks and McNamara (1935) Concrete (w/c = 0.55, Agre. max = 19.05 mm) 0.074 0.125

Blanks and McNamara (1935) Concrete (w/c = 0.54, Agre. max = 9.52 mm) 0.058 0.136

Symons (1970) Cement mortar (WGS 2%) -4.353 4.273




Symons (1970) Concrete (CR 2%) -0.425 0.724




New data presented in this paper Cement mortar 0.017 0.100

Hoskins and Horino (1969) Plaster of Paris -0.078 0.060

Hoskins and Horino (1969)a Plaster of Paris 0.039 0.080

Restricted data set: a includes only specimens with diameter [50 mm


123

cohort to generate a general strengthsize relation needs

further investigation.

In order to assess the fit of the different models to the

new and published datasets R2 and residual plots enabled

a simple method applicable to the given relational forms.

Table 6 shows the R2 values for the five different models

applied to the various datasets pertaining to cemented

materials. It should be noted that R2 is not an absolute

measure of a models fit and that regression residual plots

were also analysed to ensure their randomness, but for the

purpose of discussion in this context R2 provides a simple

measure with which to compare the models fit to the

data. Another issue associated with the regression analysis

carried out on the published datasets is that they are

generally diameter grouped mean results, not entire sets

of experimental data. The issue of using mean grouped

data is highlighted by Rey et al. (2001) who show it will

artificially increase R2 values resulting from any regres-

sion analysis. For this reason the above tables are mis-

leading as it is not possible to estimate the artificial

increase in R2 values without access to the original

complete datasets.

Table 4 95% confidenceintervals for the power

relationship in the form of Hoek

and Brown (1980) and Cunha

(1990) considering cement

sedimentary rocks

Restricted data sets: a includes

only specimens with diameter

[54 mm, b[38 mm, c[50 mm


Hawkins (1998) Pilton Sandstone -0.138 0.127

Hawkins (1998) Clifton Down Limestone -0.318 0.241

Hawkins (1998) Purbeck Limestone -0.437 0.134

Hawkins (1998) Pennant Sandstone -0.340 0.122

Hawkins (1998) Bath Stone -0.313 0.119

Hawkins (1998) Burrington Oolite -0.311 0.139

Hawkins (1998) Hollington Sandstone -0.335 -0.047

Hoskins and Horino (1969) Kansas Limestone -0.068 0.046

Hoskins and Horino (1969) Longmont Sandstone -0.047 0.013

Natau et al. (1983) Limestone (yellow) 0.951 1.714

Pells (2004) Hawkesbury Sandstone (West Pymble Outcrop) -0.244 0.039

Thuro et al. (2001) Limestone -0.329 0.220

Hawkins (1998)a Pilton Sandstone 0.139 0.421

Hawkins (1998)b Clifton Down Limestone 0.249 0.493

Hawkins (1998)b Purbeck Limestone 0.177 0.444

Hawkins (1998)a Pennant Sandstone 0.185 0.547

Hawkins (1998)a Bath Stone 0.132 0.727

Hawkins (1998)a Burrington Oolite 0.274 0.461

Hawkins (1998)a Hollington Sandstone 0.053 0.253

Hoskins and Horino (1969)c Kansas Limestone -0.061 0.119

Hoskins and Horino (1969)c Longmont Sandstone -0.021 0.024

Table 5 95% confidenceintervals for the power

relationship in the form of Hoek

and Brown (1980) and Cunha

(1990) considering cement

Igneous rocks

a A single metamorphic rock

has also been included as only

one example exists in the

literature

Restricted data set: b includes

only specimens with diameter

[50 mm


Hoskins and Horino (1969) Salida granite -0.380 0.567

Jackson and Lau (1990) Lac du Bonnet granite -0.003 0.157

Thuro et al. (2001) Granite -0.251 0.314

Yuki et al. (1995) Ohya Stone (welded tuff) loaded horizontally

to the depositional surface

-0.308 0.133

Yuki et al. (1995) Ohya Stone (welded tuff) loaded vertically


-0.134 -0.012

Nishimatsu et al. (1969) Saajome andesite -0.010 0.282

Nishimatsu et al. (1969) Ogino tuff -0.088 0.242

Nishimatsu et al. (1969) Inada granite -0.098 0.354

Nishimatsu et al. (1969) Shinkomatsu andesite -0.163 0.262

Nishimatsu et al. (1969) Aoishi sandy tuff -0.128 0.134

Hoskins and Horino (1969) Carthage marblea -0.027 0.080

Hoskins and Horino (1969)b Carthage marblea -0.042 0.165


123

Analysis of Table 6, which compares the models fit to

cementitious datasets shows that of all the models, the

power relationship consistently had the third highest R2

value, while the simple linear (UCS vs. 1/diameter) and the

theoretically derived MFSL relationship generated the first

and second highest. However, for these three models the R2

values were generally very similar, and in some cases the

exponential and a simple linear form fit the data most

closely. The same can also be said when considering the

published data sets featuring sedimentary rocks (Table 7),

although here there is stronger, more consistent support for

a linear (UCS vs. 1/diameter) or MFSL relationship fitting

the experimental data most closely.

Aside from evaluating the model fit solely on the

grounds of statistical parameters it is also important to

consider the intrinsic properties of a given relational form

when assessing the validity of its application to the size

strength relationship of rocks and other brittle materials.

The power and exponential relationships have horizontal

asymptotes of zero, so as the specimen diameter is

increased the predicted strength of the rock tends to zero.

Similarly unrealistic, a negative linear relationship will

predict strength values less than zero. Obviously the

strength of an infinitely large rock will never be zero (or

negative). Conversely both the theoretically derived MFSL

and the empirical linear (1/diameter) relation incorporate a

horizontal asymptote. As the rock diameter tends to infin-

ity, both relations predict a rock strength which tends to a

constant. In light of the relative success of the MFSL

model to predict the strength of a majority of rocks, its

theoretical basis, and its close relation to what is logically

expected in a physical respect, it appears to be the best

model to use when modelling the decrease in strength of a

rock or cemented specimen with an increase in diameter.

Table 8 shows that the scale effect in igneous rocks is

best modelled by exponential or linear models, however,

given the aforementioned practical considerations and the

nature of these models to predict zero or negative strength

values at large specimen sizes, it seems unlikely they are

the most appropriate predictive tools. Figures 12, 13, 14,

15, 16, 17, 18, 19, 20, 21 and 22 show the MFSL producing

an acceptable fit when plotted alongside the experimental

datasets pertaining to igneous rocks.

Aside from the majority of experimental results that

indicate a strengthsize effect (with specimen strength

decreasing with size), there are some exceptions to the size

law including the samples tested by Symons (1970) (14%

well-graded sand-cement mortar), Pells (2004) (Hawkes-

bury Sandstone), Thuro et al. (2001) (limestone and gran-

ite), Hoskins and Horino (1969) (Longmont Sandstone),

and Yuki et al. (1995) (welded tuff) that show an increase

in strength with specimen size or no significant strength

size relationship. Yuki et al. (1995) justifies this mono-

tonically increasing relationship due to the fractal distri-

bution of hard particles within the rock. This is an

important point and highlights the possible role of micro

structure and rock fabric in any size effect relationship. The

lack of a clearly consistent size effect makes application of

any derived relationship or model difficult and warrants

further study to quantify which materials demonstrate a

Table 6 R2 values and comparative model ranking for various relational forms obtained from regression analysis of publish data of cementstabilised materials

Reference Material tested R2 (rank)

Linear Linear

(1/diam)

Power

(1/diam)

MFSL Expon.

Blanks and McNamara (1935) Concrete (w/c = 0.53, Agre. max = 38.1 mm) 0.518 (3) 0.286 (4) 0.525 (2) 0.273 (5) 0.533 (1)



Symons (1970) Cement mortar (WGS 2%) 0.001 (5) 0.063 (2) 0.014 (3) 0.065 (1) 0.001 (4)




Symons (1970) Concrete (CR 2%) 0.985 (1) 0.847 (5) 0.916 (3) 0.863 (4) 0.979 (2)




New data presented in this paper Cement mortar 0.215 (5) 0.294 (1) 0.258 (3) 0.293 (2) 0.216 (4)

Hoskins and Horino (1969) Plaster of Paris 0.004 (4) 0.148 (1) 0.029 (3) 0.137 (2) 0.002 (5)

Hoskins and Horino (1969)a Plaster of Paris 0.932 (5) 0.944 (3) 0.967 (1) 0.947 (2) 0.934 (4)

Restricted data set: a includes only specimens with diameter [50 mm


123

Table 7 R2 values and comparative model ranking for various relational forms obtained from regression analysis of publish data of sedimentaryrocks


Linear Linear

(1/diam)

Power

(1/diam)

MFSL Expon.

Hawkins (1998) Pilton Sandstone 0.073 (3) 0.105 (1) 0.002 (5) 0.100 (2) 0.072 (4)

Hawkins (1998) Clifton Down Limestone 0.051 (3) 0.117 (1) 0.018 (5) 0.061 (2) 0.020 (4)

Hawkins (1998) Purbeck Limestone 0.009 (5) 0.320 (1) 0.220 (2) 0.188 (3) 0.045 (4)

Hawkins (1998) Pennant Sandstone 0.003 (5) 0.373 (1) 0.182 (3) 0.285 (2) 0.016 (4)

Hawkins (1998) Bath Stone 0.009 (5) 0.246 (1) 0.167 (3) 0.176 (2) 0.026 (4)

Hawkins (1998) Burrington Oolite 0.000 (5) 0.314 (1) 0.128 (3) 0.240 (2) 0.003 (4)

Hawkins (1998) Hollington Sandstone 0.301 (5) 0.794 (1) 0.638 (3) 0.723 (2) 0.327 (4)

Hoskins and Horino (1969) Kansas Limestone 0.001 (5) 0.200 (1) 0.069 (3) 0.195 (2) 0.001 (4)

Hoskins and Horino (1969) Longmont Sandstone 0.573 (5) 0.881 (1) 0.749 (3) 0.881 (2) 0.573 (4)

Natau et al. (1983) Limestone (yellow) 0.307 (5) 0.625 (2) 0.638 (1) 0.525 (4) 0.540 (3)

Pells (2004) Hawkesbury Sandstone (West Pymble Outcrop) 0.923 (1) 0.571 (4) 0.830 (3) 0.561 (5) 0.913 (2)

Thuro et al. (2001) Limestone 0.053 (5) 0.088 (1) 0.071 (3) 0.086 (2) 0.055 (4)

Hawkins (1998)a Pilton Sandstone 0.959 (2) 0.894 (5) 0.930 (3) 0.914 (4) 0.960 (1)

Hawkins (1998)b Clifton Down Limestone 0.795 (5) 0.974 (1) 0.947 (3) 0.952 (2) 0.858 (4)

Hawkins (1998)b Purbeck Limestone 0.746 (5) 0.959 (1) 0.912 (3) 0.934 (2) 0.802 (4)

Hawkins (1998)a Pennant Sandstone 0.888 (5) 0.940 (2) 0.932 (3) 0.943 (1) 0.914 (4)

Hawkins (1998)a Bath Stone 0.758 (5) 0.911 (1) 0.876 (3) 0.892 (2) 0.799 (4)

Hawkins (1998)a Burrington Oolite 0.920 (5) 0.988 (2) 0.981 (3) 0.989 (1) 0.941 (4)

Hawkins (1998)a Hollington Sandstone 0.888 (3) 0.878 (5) 0.888 (2) 0.892 (1) 0.887 (4)

Hoskins and Horino (1969)c Kansas Limestone 0.376 (2) 0.139 (3) 0.257 (5) 0.136 (4) 0.381 (1)

Hoskins and Horino (1969)c Longmont Sandstone 0.250 (5) 0.429 (2) 0.339 (3) 0.429 (1) 0.250 (4)

Restricted data sets: a includes only specimens with diameter [54 mm, b [38 mm, c [50 mm

Table 8 R2 values and comparative model ranking for various relational forms obtained from regression analysis of publish data of igneousrocks


Linear Linear

(1/diam)

Power

(1/diam)

MFSL Expon.

Hoskins and Horino (1969) Salida granite 0.954 (1) 0.772 (5) 0.863 (3) 0.782 (4) 0.948 (2)

Jackson and Lau (1990) Lac du Bonnet granite 0.474 (3) 0.379 (4) 0.478 (1) 0.377 (5) 0.474 (2)

Thuro et al. (2001) Granite 0.103 (2) 0.097 (4) 0.104 (1) 0.095 (5) 0.102 (3)

Yuki et al. (1995) Ohya Stone (welded tuff) loaded horizontally


0.731 (1) 0.433 (5) 0.594 (3) 0.452 (4) 0.708 (2)

Yuki et al. (1995) Ohya Stone (welded tuff) loaded vertically


0.963 (1) 0.830 (4) 0.930 (3) 0.815 (5) 0.963 (2)

Nishimatsu et al. (1969) Saajome andesite 0.801 (2) 0.628 (4) 0.745 (3) 0.618 (5) 0.813 (1)

Nishimatsu et al. (1969) Ogino tuff 0.396 (2) 0.062 (4) 0.224 (3) 0.048 (5) 0.427 (1)

Nishimatsu et al. (1969) Inada granite 0.662 (2) 0.272 (4) 0.520 (3) 0.243 (5) 0.700 (1)

Nishimatsu et al. (1969) Shinkomatsu andesite 0.329 (2) 0.013 (4) 0.154 (3) 0.008 (5) 0.354 (1)

Nishimatsu et al. (1969) Aoishi sandy tuff 0.001 (4) 0.009 (1) 0.002 (3) 0.008 (2) 0.000 (5)

Hoskins and Horino (1969) Carthage marblea 0.615 (2) 0.280 (4) 0.451 (3) 0.279 (5) 0.617 (1)

Hoskins and Horino (1969)b Carthage marblea 0.783 (1) 0.745 (5) 0.767 (3) 0.751 (4) 0.783 (2)

a A single metamorphic rock has also been included as only one example exists in the literature

Restricted data set: b includes only specimens with diameter [50 mm


123

size effect whether it be a positive or negative trend and

which do not.

3.2 Youngs Modulus and Poissons Ratio-Size Effect

When considering Fig. 31 it can be concluded that for the

material tested in this study there is a slight increase in

Youngs modulus (approximately 8%), although consider-

ing the standard deviations of the data this trend is difficult

to support statistically. Poissons ratio shows large vari-

ability with little evidence of a strong size effect, although

in this contribution only very few data were obtained

(Fig. 32). More generally there seems to be a general lack

of data pertaining to this area of research, and in light of the

inconsistencies seen between the published datasets it is

not possible to make a sound conclusion regarding the size

effect in Youngs modulus or more acutely Poissons ratio.

4 Conclusion

Assessing the available Youngs modulus and Poissons

ratio data it cannot be confirmed whether a size effect

exists. Given the limited quantity of data on these prop-

erties relationship with specimen size no conclusion can

be considered statistically significant and further research

is warranted in this area.

Considering the new and previously published data

presented in this paper on the strengthsize effect, it is

clear that large variations in the magnitude of any rela-

tionship exist. Possible reasons for this variation must be

established. It is hypothesised that no consistent result can

be seen in the published experimental data due to one or

more of the following issues: (a) the testing method/

apparatus used; (b) specimen preparation; and/or (c) the

type of material under examination (not-excluding the

possibility of anisotropic or load orientation strengthsize

relationships).

The particulars of any testing method and apparatus will

inevitably generate some variation in results. It is possible

that the variations seen in the published data are due to a

high sensitivity to the testing methodologies. Specifically,

high sensitivities may be associated with boundary condi-

tions including platen friction, the effect of capping

materials (if used) and specimen end preparation (including

flatness, perpendicularity and smoothness). Other experi-

mental peculiarities such as test rig stiffness, load rate, etc.

may also be a cause for this variability between different

researchers results. It is hypothesised that these factors

have a strong influence in causing some of the high vari-

ability, and low strengths seen in specimens \50 mm indiameter, where stress concentrations due to poor end

preparation will lead to a comparatively dramatic effect on

specimen strength (when compared to the effect a stress

concentration may have on a specimen larger than 50 mm

in diameter).

It appears that any scale relationship is highly material

dependent. It seems probable that the nature of any

strengthsize effect is determined to some degree by the

materials general structural classification (igneous, sedi-

mentary or otherwise). This paper presents results showing

high variations of the strengthsize effect even within single

material classifications. Igneous rocks appear to fit all of the

tested sizestrength relations relatively poorly when com-

pared to sedimentary rocks fitted to the same models.

Generally speaking, a majority of the results analysed

are modelled best by a linear (1/diameter) relationship or

the MFSL. Both models incorporate a horizontal asymptote

dictating large scale rock strength. Furthermore, it is

impossible to model the scalestrength effect of all rocks

using a power law model with a single fixed general power.

More research is clearly needed to fully understand this

phenomenon. Such research should aim at generating

consistency across new datasets and aim to generate con-

sistency within a dataset, using a single rock type or

material over a complete scale range (i.e. testing

20300 mm specimens) while maintaining consistent

boundary conditions. The results of specimens at, or

exceeding, 300 mm are of crucial importance given a

specific and distinct lack of these data in publications to

date and their practical relevance to large scale design,

improving statistical relevancy and confirming the exis-

tence of a horizontal asymptote and the threshold diameter

at which asymptotic strength is met.

The approach used in this paper is based mainly on

statistical analysis of existing data. The results have indi-

cated that the scale relationship depends strongly on the

type (or structural classification) of the rock. It may be

worthwhile to study the geometric and physical properties

of the heterogeneities that are generally observed in each

type of rock; how evolution of the failure process may be

influenced by the nature of the heterogeneity and the

applied load; how other factors such as specimen size and

boundary conditions may affect the measured peak

strength. This may help to explain the rock type-dependent

scale relationship based on a mechanistic approach.

Acknowledgments The authors would like to extend their thanksand appreciation to the undergraduate student Sajmir Bella for his

assistance with the laboratory work associated with this project.

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http://www.rocscience.com/education/hoeks_corner

The Effect of Specimen Size on Strength and Other Properties in Laboratory Testing of Rock and Rock-Like Cementitious Brittle MaterialsAbstractIntroductionCementitious MaterialsSedimentary RocksIgneous RocksMetamorphic RocksYoungs Modulus and Poissons Ratio

Experimental MethodResults and DiscussionStrength--Size EffectYoungs Modulus and Poissons Ratio-Size Effect

ConclusionAcknowledgmentsReferences

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