Generic transformation models for some intact rock properties

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Generic transformation models for some intact rock

properties

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2017-0537.R2

Manuscript Type: Article

Date Submitted by the Author: 23-Feb-2018

Complete List of Authors: Ching, Jianye; National Taiwan University, Li, Kuang-Hao; National Taiwan University Phoon, Kok-Kwang; National University of Singapore, Department of Civil & Environmental Engineering Weng, Meng-Chia; National Chiao Tung University

Is the invited manuscript for

consideration in a Special Issue? :

N/A

Keyword: intact rock properties, ROCK/9/4069, uniaxial compressive strength, Young’s modulus, transformation model

https://mc06.manuscriptcentral.com/cgj-pubs

Canadian Geotechnical Journal

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Submitted to Canadian Geotechnical Journal

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Generic transformation models for some intact rock properties

Jianye Ching1, Kuang-Hao Li2, Kok-Kwang Phoon3, and Meng-Chia Weng4

ABSTRACT

A global intact rock database of nine parameters, including uniaxial compressive strength and

Young’s modulus, is compiled from 184 studies. This database, labeled as “ROCK/9/4069”, consists

of 27.5% igneous rock, 59.4% sedimentary rocks and 13.1% metamorphic rock. The vast majority

(> 95%) of intact rocks in the database are in their natural moisture contents. About 14% of the data

points are for weathered rocks, and about 4% are foliated metamorphic rock. It is found that most

existing transformation models are data-specific or site-specific in the sense that they fit well to

their own calibration databases but do not necessarily fit well to ROCK/9/4069. One can infer that

transformation models for intact rocks are more data/site dependent than those for soils. It is evident

that ROCK/9/4069 has coverage wider than most existing transformation models. The

ROCK/9/4069 database is then adopted to calibrate the bias and variability of existing

transformation models. Transformation models with relatively large application ranges and

relatively small transformation uncertainties are selected as generic transformation models. These

generic models can be valuable for scenarios where site-specific models are not available, e.g.,

1 (Corresponding author) Professor, Dept of Civil Engineering, National Taiwan University, Taiwan. Email: [email protected]. Tel: +886-2-33664328. 2 Graduate Student, Dept of Civil Engineering, National Taiwan University, Taiwan. 3 Professor, Dept of Civil and Environmental Engineering, National University of Singapore, Singapore. 4 Professor, Dept of Civil Engineering, National Chiao Tung University, Taiwan.

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construction projects with insufficient budget or the preliminary design stage of a project.

Key words: intact rock properties; ROCK/9/4069; uniaxial compressive strength; Young’s modulus;

transformation model.

INTRODUCTION

The mechanical properties of intact rock, uniaxial compressive strength (σc) and Young’s modulus

(E), are essential parameters to estimate the strength and deformability of rock mass for engineering

design (Hoek and Brown 1997). However, during preliminary design stages, rock samples may not

be available to determine σc and E, so the two properties are often estimated based on correlation

equations. These correlation equations are referred to as “transformation models” in the

geotechnical literature (Phoon and Kulhawy 1999a). Tables 1 and 2 summarize some transformation

models for intact rocks that are related to σc and E. They are referred to as the σc models and the E

models in this study. The survey for transformation models in these tables is not exhaustive. Useful

compilations of these transformation models are provided by Zhang (2016, 2017). All

transformation models are suitable for the range of conditions found in the calibration databases.

However, a transformation model calibrated by one database may be significantly different from a

model calibrated by another database. Figure 1 illustrates two σc models based on point load

strength (Is50) and their calibration databases. It is clear that each model fits well to its own

calibration database, but the two calibration databases have different trends, hence different

transformation models. Figure 2 further shows several Is50-σc models in the literature. The

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difference among the transformation models is evident and significant for intact rocks. Many Is50-σc

models in the literature are dataset-specific or site-specific. They are not “generic” in the sense that

their usefulness as first-order estimates under more general conditions beyond the calibration range

is questionable.

Dataset-specific or site-specific models have their merits. For instance, for a construction

project with sufficient budget, it is feasible to conduct tests on a number of intact rock samples to

establish the transformation model for the target site. The resulting site-specific model can be

adopted to estimate σc or E for that target site. However, for a construction project with insufficient

budget or for the preliminary design stage of a project, rock samples may not be available, so it may

not be possible to develop site-specific models. One possible strategy is to adopt a transformation

model found in the literature, but its applicability may be questionable because the model was not

tailored for the target site. In this circumstance, a generic transformation model calibrated by a

global rock database may be desirable.

In this study, a global database, named ROCK/9/4069, for nine intact rock parameters,

including porosity (n), unit weight (γ), L-type Schmidt hammer hardness (RL), Shore scleroscope

hardness (Sh), Brazilian tensile strength (σbt), point load strength index (Is50), P-wave velocity (Vp),

uniaxial compressive strength (σc), and Young’s modulus (E), is compiled from 184 studies in the

literature. Prakoso (2002) and more recently Aladejare and Wang (2017) compiled global rock

databases, but their focuses were mainly on quantifying the uncertainties of individual rock

properties rather than on developing generic transformation models that require correlation

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information between two or more properties. It will be shown that the ROCK/9/4069 database is

indeed “global” and that most transformation models in Tables 1 and 2 are site-specific because the

coverage of the database is wider than that of each model. This ROCK/9/4069 database is then

compared with the transformation models in Tables 1 and 2. The models that fit ROCK/9/4069 well

will be selected as generic transformation models.

Transformation models are not exact, as shown by the data scatter in Figure 3, and the

discrepancy between model prediction and actual design property is called transformation

uncertainty (Phoon and Kulhawy 1999a). Transformation uncertainty deserves more explicit and

rigorous characterization, because it can be more influential than other sources of geotechnical

uncertainties (e.g., Honjo 2011; Honjo and Otake 2014). A general characterization of the

transformation uncertainty will require the calibration of its bias (difference between model

prediction and average of the data) and variability (data scatter about its average). In this study,

generic transformation models will be selected based on the calibration results by ROCK/9/4069.

With the selected generic models, not only the point estimate of σc or E can be predicted, but its

probability distribution will also be developed. The practical importance of determining the

probability distribution is that the uncertainty of the point estimate can be explicitly presented in the

form a 95% confidence interval. The 5% quantile can also be obtained as a statistical-based

characteristic value.

Besides the development of generic transformation models, global rock databases themselves

have other values. They can provide statistics of individual rock properties similar to those

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presented by Prakoso (2002) and Aladejare and Wang (2017). These statistics are essential inputs to

reliability-based design. Also, for Bayesian analysis (e.g., Feng and Jimenez 2014, 2015; Ng et al.

2015; Wang and Aladejare 2015, 2016), a global rock database can also be used to construct the

prior probability density functions (PDF) for rock parameters. With the prior PDF, site-specific data

can be used to obtain the posterior PDF using Bayesian analysis. In the companion paper (Ching et

al. 2017a), the multivariate prior PDF for the nine intact rock parameters will be developed to

facilitate the further Bayesian analysis. Probabilistic transformation models allowing multivariate

input variables will be developed in the companion paper. These multivariate models potentially can

have higher accuracy.

DATABASE ROCK/9/4069

This study compiles a global database (ROCK/9/4069) from the literature consisting of a significant

number of data points for nine parameters of intact rocks. In the literature, global univariate and

multivariate databases have been compiled for clays, sands, and rocks. Table 3 shows some such

databases, labelled as (material type)/(number of parameters of interest)/(number of data points).

Univariate databases are presented in the first 4 rows of Table 3. They can be distinguished from the

multivariate databases by the absence of “number of data points” in their labels. The reason is that

statistics are calculated at the “data group” level (can be broadly interpreted as site-specific level)

and each data group consists of different number of data points. The data points in ROCK/9/4069

will be compared with the existing transformation models in Tables 1 and 2. This serves as the basic

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consistency check for the database. The ROCK/9/4069 database is then adopted to calibrate the bias

and variability of the existing transformation models, and recommendation on suitable generic

transformation models will be made.

The ROCK/9/4069 database consists of 4069 intact rock data points from 184 studies. Jointed

rock masses are not covered by this database. The number of data points associated with each study

varies from 1 to 163 with an average 23.6 data points per study. The ROCK/9/4069 database is

dominated by igneous and sedimentary rocks (27.5% and 59.4%, respectively). The remaining

(about 13.1%) data points are metamorphic rock. About 14% of the data points are for weathered

rocks, and about 4% are foliated metamorphic rocks (e.g., argillite, gneiss, orthogneiss, phyllite,

paragneiss, schist, and slate). This paper did not delve into more detailed classifications of igneous

rocks (intrusive, extrusive, pyroclastic) and sedimentary rocks (clastic, chemical). The vast majority

(> 95%) of intact rocks in the database are in their natural moisture contents. There is no data point

for saturated rocks. Prakoso (2002) noted that the mean strength (Brazilian tensile strength, point

load strength index, and uniaxial compressive strength) for saturated rocks is about 80% of that of

dry rocks. He did not find any effect of moisture content on the coefficient of variation of the three

strength parameters. The geographical regions cover 44 countries/regions, including Afghanistan,

Australia, Austria, Brazil, Canada, China, Egypt, France, Germany, Greece, Hong Kong, Hungary,

India, Indonesia, Iran, Israel, Italy, Japan, Macao, Malaysia, Mexico, Morocco, Nepal, Netherlands,

New Zealand, Nigeria, North Sea, Norway, Pakistan, Portugal, Russia, Saudi Arabia, Singapore,

South Africa, South Korea, Spain, Sweden, Taiwan, Thailand, Turkey, United Kingdom, United

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States, Ukraine, and Uruguay. The properties of the data in ROCK/9/4069 cover a wide range of

unit weight (γ) (15 to 35 kN/m3), porosity (n) (0.01 to 55%), uniaxial compressive strength (σc) (0.7

to 380 MPa), Young’s modulus (E) (0.03 to 120 GPa), and P-wave velocity (Vp) (0.4 to 8 km/sec).

The values of σc cover the full range of weak (σc < 20 MPa), medium (20 < σc < 100 MPa), and

strong rocks (σc > 100 MPa), based on classification on unweathered rock material strengths

proposed by Kulhawy et al. (1991). The details for this global database are presented in the

Appendix (Table A1). In this table, the 5th column “Rock type” shows the classes of the rocks as

well as the weathering grade. The weathering grade in the 5th column, if available, is based on either

the International Society of Rock Mechanics method (ISRM 1981) or the Geological Society

Engineering Group Working Party Reports (GSE-GWPR 1995). Many papers do not report the

weathering grades for the rock samples, but it is expected that most rock samples in ROCK/9/4069

should have ISRM grades ranging from Grade I (fresh) to Grade IV (highly weathered), because

samples with grade V or above would have disintegrated into soils.

Nine parameters are of primary interest, including n, γ, RL, Sh, σbt, Is50, σc, E, and Vp. They are

categorized into four groups:

1. Index properties: porosity (n), unit weight (γ), L-type Schmidt hammer hardness (RL), and Shore

scleroscope hardness (Sh). There are two types of Schmidt hammer: L-type and N-type. In

ROCK/9/4069, RL data dominate (84% are RL) because the L-type hammer is recommended for

rocks (ISRM 1981). For the RN data in ROCK/9/4069, they are converted to RL using the

following empirical equation (Aydin and Basu 2005):

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N LR 1.0646R 6.3673= +

(1)

Regardless of rock types, this empirical equation provides adequate fit to the data points in

ROCK/9/4069 with simultaneous information for RN and RL, as seen in Figure 4.

2. Strengths: Brazilian tensile strength (σbt), point load strength index (Is50), and uniaxial

compressive strength (σc). In general, strengths are size dependent. It is customary to correct Is

to a standard diameter of 50 mm (Franklin 1985), because point load test can be conducted over

a wide range of diameter:

0.45

s50 s

dI I

50

= ×

(2)

where Is is the result before correction; d is the sample diameter (in mm). In contrast, there is

usually no need to correct for σbt and σc, because these tests are usually conducted on samples

with about the standard diameter, d = 50 mm. For the σc data in ROCK/9/4069, 50% have

diameters between 48 mm and 54 mm, and 95% have diameters between 21 and 110 mm. Hoek

and Brown (1997) proposed a sample-size correction equation for σc that is very similar to Eq.

(2), but a much smaller exponent of 0.18 is adopted. With the 0.18 exponent, the correction ratio

(d/50)0.18 is mild: it is less than 2% for the cases with diameters between 48 mm and 54 mm and

less than 15% for those with diameters between 21 mm and 110 mm. As a result, sample-size

correction is not carried out for the σc data in ROCK/9/4069. For the σbt data in ROCK/9/4069,

50% have diameters between 50 mm and 54 mm, and 95% have diameters between 28 and 76

mm. Sample-size correction is not carried out for the σbt data because the diameter range is

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relatively narrow. Prakoso (2002) studied the effect of sample diameter on laboratory and

field-scale uniaxial compressive strength (σc) and point load strength (Is) results. He suggested

an exponent of 0.60 in Eq. (2) for Is. For σc, the trend is not clear for diameter less than 50 mm.

For diameter greater than 50 mm, he suggested an exponent of 0.25 in Eq. (2) for σc. It should

be noted that Eq. (2) refers to an average correction. Prakoso (2002) did not observe an effect of

sample diameter on the COV of rock strengths (σc, σbt, and Is).

3. Stiffness: Young’s modulus (E). In ROCK/9/4069, 50.6% of the E data are Et50 (tangent

modulus at 50% of the peak strength), 10.1% are Es50 (secant modulus at 50% of the peak

strength), and 39.3% are Eav (average modulus for the linear portion of stress-strain curve).

Because there are abundant data for Et50 and Eav, the difference between Et50 and Eav can be

verified. Taking the σc-E relationship as an example, Figure 5 shows that σc-Et50 and σc-Eav

relationships have similar trends. This suggests that the difference between Et50 and Eav is not

significant. Although there are not many Es50, Tamrakar et al. (2007) reported Et50 versus Es50

data for more than 40 rocks in Central Nepal, and their values differ by 20% on average. This

difference is significantly less than the transformation uncertainty related to E to be presented

later. Because the difference among the E data with different definitions is not significant

compared with the transformation uncertainty in E, they are combined to form the entire E data.

For the E data in ROCK/9/4069, 50% have diameters between 50 mm and 54 mm, and 95%

have diameters between 25 and 76 mm. Sample-size correction is not carried out for the E data

because the diameter range is relatively narrow. Prakoso (2002) did not study the effect of

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sample diameter on modulus. An ideal database is one that contains σc and E values for a

standard sample diameter of 50mm. This is not possible for a database assembled from the

literature. The more practical approach is to apply a sample size correction such as Eq. (2), but

Eq. (2) can be viewed as a transformation model with its own transformation uncertainty. In

other words, while sample size correction is the right thing to do from a physical perspective, it

is not guaranteed that a transformation model with a corrected property will be less uncertain

than the corresponding model with an uncorrected property. Some judgment is needed here on

whether the correction is sufficiently significant to warrant the cost of incurring additional

transformation uncertainty in the correction step. This study specifies whether a property is

corrected for sample size or other effects explicitly (a recommended approach for all statistical

characterization studies of this nature). For uncorrected properties such as σc, σbt, and E, the

associated transformation models may carry additional uncertainty due to this lack of correction.

There is insufficient data to clarify the approximate percentage change in uncertainty.

4. Dynamic property: P-wave velocity (Vp).

There are in total 4069 data points in the database. Each data “point” stored is one row in the

Excel worksheet that consists of a set of values measured for the “same” intact rock. Intact rock

samples are considered to be the “same” if they are from nearby spatial locations (proximity at the

same site). The ideal is to measure all nine parameters from the same intact specimen, but this is not

possible in practice. The resulting database is also not a genuine multivariate database in the sense

that the nine parameters are not simultaneously measured. Although 14% of the data points are

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reported as “weathered”, there is no strong evidence that these data points exhibit significantly

different transformation relationships (Figure 6). Prakoso (2002) showed that the effect of moisture

content (dry versus saturated) on unweathered and weathered rock strength is similar. Kulhawy and

Prakoso (2003) further showed that the decrease in intact rock strength and stiffness with increasing

degree of weathering is relatively consistent for different test methods. This suggests that the

strength-stiffness transformation models may be relatively independent of weathering degree.

Although 4% of the data points are for foliated metamorphic rocks, there is no strong evidence that

foliated rocks exhibit significantly different transformation relationships (Figure 7), either.

Therefore, all data points (weathering grade known or unreported; foliated or non-foliated) are

combined together to form the ROCK/9/4069 database. This does not imply that weathering and

foliation are not important. Prakoso (2002) commented that the coefficient of variation of σbt, Is σc,

and Et50 tends to increase with the weathering grade. However, this observation refers to

site-specific data and to an individual parameter rather than to the transformation relationship

between two or more parameters. In contrast, Prakoso (2002) showed that the core orientation

(including some foliated metamorphic rock data) may have only small influence on site-specific

coefficient of variation. In general, Figures 6 and 7 indicate that the effects of weathering and

foliation may be difficult to discern against the significant transformation uncertainties found in the

global bivariate relationships. The effects of weathering and foliation may be more discernable at a

site-specific level where transformation uncertainties are typically smaller. However, site-specific

transformation models are not studied in this paper.

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The basic statistics of for the nine parameters in the ROCK/9/4069 database are listed in Table

4. The statistics are the mean value, coefficient of variation (COV), minimum value (min), and

maximum value (max). Note that the mean and COV are not for a specific site but for the entire

ROCK/9/4069 database that covers numerous sites. They should not be used for design, which

requires statistics at the site level. The site-level mean and COV are given in Table 5. Overall, the

percentages for igneous, sedimentary, and metamorphic rocks are 27.5%, 59.4%, and 13.1%,

respectively. About 14% of the data points are weathered rocks, and about 4% are foliated

metamorphic rocks. There is no data point for saturated intact rocks. Prakoso (2002) indicated that

the effect of saturation is similar for all three strength parameters (Is50, σbt, and σc): (saturated

strength) is on average 80% of (dry strength), regardless of the rock class. Prakoso and Kulhawy

(2011) further showed that the actual data spread is (saturated strength) ≈ 50% to 90% of (dry

strength). Zhang (2016) also indicated that (saturated σc) ≈ 50 ~ 90% of (dry σc) for most rocks,

whereas this range can become lower (30 ~ 90%) for carbonate rocks (e.g., gypsum, limestone, and

marble). Zhang (2016) further indicated that (saturated E) ≈ 30 ~ 80% of (dry E) for most rocks.

The numbers of available data points (N) for the nine rock parameters are shown in the second

column of Table 4. The number of data points is further divided into the numbers of igneous,

sedimentary, and metamorphic data points. For instance, there are in total 1288 data points with γ

information. Among them, 238 are igneous rocks, 879 are sedimentary rocks, and 171 are

metamorphic rocks.

It is useful to note that Table 4 is different from similar tables presented by Prakoso (2002) and

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Aladejare and Wang (2017). The COVs presented in these papers refer to variability at a site level,

while Table 4 refers to COV at a global level (all sites). The COV and range (Max – Min) are

expected to be larger at the global level compared to corresponding statistics at a site level. Tables 5

to 7 shows the site-level statistics for the igneous, sedimentary, and metamorphic intact rock data

points in ROCK/9/2069 for comparison with the univariate rock databases (ROCK/8 and

ROCK/13). One “data group” in the tables can be broadly interpreted as data from one site. To

obtain meaningful second-order site-level statistics (e.g., COV), only data groups with more than 3

data points are considered. Pepe et al. (2017) showed that with 3 or less data points in a data group,

it is not possible to reliably estimate population mean. These site-level statistics are broadly

consistently with those presented by Prakoso (2002) (Table 8) and Aladejare and Wang (2017)

(parenthesized values in last column of Tables 5 to 7).

Comparison to existing transformation models

The data points in ROCK/9/4069 are compared with the transformation models in Tables 1 and 2.

Only transformation models with relatively broad application ranges in the input variables, i.e., the

ranges of the calibration databases, are compiled in these tables. The 6th column in Tables 1 and 2

shows the application ranges of the transformation models. Models with relatively narrow

application ranges are not investigated, unless there are no competitive models. The transformation

models are further labeled using the template: (primary input parameter)-(target parameter) (1st

column in Tables 1 and 2). There are ten types of models: n-σc, RL-σc, Sh-σc, σbt-σc, Is50-σc, Vp-σc,

RL-E, Sh-E, σc-E, and Vp-E models. Figure 2 and Figures 8~16 show the comparison results (results

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for Is50-σc models have already been presented in Figure 2). In these figures, transformation models

are numbered according to the numbers in the square brackets [.] shown in the 5th column in Tables

1 and 2. For instance, for the first model in Table 1 proposed by Tuğrul and Zarif (1999), the 5th

column shows “Fig. 8a [1]”, which means this transformation model is the number 1 curve in

Figure 8a. For each model, only the transformation equation within its application range is plotted

in the figures, i.e., no extrapolation outside its own calibration database. For models with a

secondary input parameter, e.g., the RL-σc model proposed by Cargill and Shakoor (1990) has a

secondary input parameter ρ (density), they are not plotted in the figures. The “igneous”,

“sedimentary”, and “metamorphic” subplots show the models developed specifically for each rock

class, together with the ROCK/9/4069 data points pertaining to each rock class. The “mixed”

subplot shows the models developed for multiple rock classes, together with all ROCK/9/4069 data

points. There are models with very similar transformation relationships, e.g., the Sh-σc model

proposed by Altindag and Guney (2010) (σc ≈ 0.1821×Sh1.5833) and the one proposed by Kilic and

Teymen (2008) (σc ≈ 0.159×Sh1.6269) are very similar in their application ranges. For very similar

models, only one model is investigated, and the others are presented in 7th column in Tables 1 and 2

for reference. The investigated model is the oldest model (if their application ranges are similar) or

the model with the largest application range.

The following observations are obtained from Figure 2 and Figures 8-16:

1. For all models, the transformation uncertainty with respect to ROCK/9/4069 is fairly large. This

is in contrast to the smaller transformation uncertainties with respect to their own local

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calibration databases. Figure 1 shows two examples with smaller transformation uncertainties:

the Is50-σc models fit quite well to their own local calibration databases. Although the

transformation uncertainties with respect to their own local calibration databases are smaller

(Figure 1), these two models exhibit fairly large transformation uncertainties with respect to

ROCK/9/4069 (see #3 and #5 models in Figure 2d).

2. Many transformation models do not fit the average trend of the ROCK/9/4069 database. This is

possibly because each transformation model was developed by its own calibration database, and

the calibration database is not as global as ROCK/9/4069 in terms of the ranges of parameters

covered. In short, many transformation models are dataset-specific or site-specific. A separate

evidence for this dataset-specificity or site-specificity is that many transformation models are

fairly distinct from one another. Figure 1 shows two fairly distinct transformation models – this

distinction implying significantly different predicted values is quite evident throughout Figure 2

and Figures 8-16.

3. Most transformation models are well covered by the ROCK/9/4069 “data cloud”, meaning that a

subset of ROCK/9/4069 data points can follow each transformation model well. This indicates

the ROCK/9/4069 database may have sufficient global coverage. There are only few exceptions.

One possible exception is the #2 transformation model in Figure 12.

4. In general, there is no strong evidence indicating that the transformation relationships exhibited

by ROCK/9/4069 data points depend on rock classes (igneous, sedimentary, and metamorphic).

There are very few exceptions, e.g., Vp-σc data trend for sedimentary rocks is slightly lower

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than those for igneous and metamorphic rocks (Figure 12). However, the distinction is not very

significant. The n-σc relationship data trend for metamorphic rocks (Figures 8c) is unclear

because they have relatively small porosity.

QUANTIFICATION OF TRANSFORMATION UNCERTAINTY

Although dataset-specific or site-specific models have their merits, there are scenarios where such

models are not available, e.g., construction projects with insufficient budget or the preliminary

design stage of a project. In this case, a generic transformation model developed based on a global

database may be desirable. A site-specific model carries smaller transformation uncertainties, but it

may produce a significantly biased predicted value when applied to a different site (highly

inaccurate although more precise). A global model carries much larger transformation uncertainties,

but the predicted value is less biased for different sites (relatively more accurate, but imprecise). At

first glance, both site-specific and global models appear to be equally limited, except in different

ways. However, it is not possible to correct for bias unless there is a global database such as

ROCK/9/4069 to compare against (this is carried out in the following section), while a large

transformation uncertainty can be handled rationally by using a more conservative characteristic

value (this is automatically adjusted through a 5% quantile). Ching and Phoon (2012b) discussed

the establishment of generic transformation models using global databases and their relative

strengths and limitations. In this section, the transformation uncertainties for the models in Tables 1

and 2 will be quantified. In particular, the bias and variability for each model will be calibrated by

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ROCK/9/4069. Models with relative small bias and variability with respect to ROCK/9/4069 will

be selected as generic transformation models.

The data scatter about the transformation model, as illustrated in Figure 3, can be quantified

using probabilistic methods. The transformation model is typically evaluated using regression

analyses, and the spread of the data about the regression curve can be modeled in many instances as

the following multiplicative form (Ching and Phoon 2014a):

actual target value actual target value

b predicted target value unbiased predictionε = =

× (3)

where the actual target value = measured value of the design property, and predicted target value =

estimated value of the design property from a transformation model. The product of a constant b

(bias factor) and the predicted target value produces an unbiased prediction on average. The bias of

the prediction is captured by b, whereas ε only captures the variability of the prediction, not the bias.

Ching and Phoon (2014a) also considered the following additive form:

actual target value b predicted target value actual target value unbiased predictionε = − × = −

(4)

However, it is found in this study that the multiplicative form is more suitable than the additive

form, because the former typically produces a larger p-value (to be discussed later). Therefore, only

the multiplicative form is considered in this study.

For ε in Eq. (3) to only capture the variability without the bias, ε must have a unit mean. The

standard deviation of ε, denoted by σ, quantifies the transformation uncertainty. Here, the standard

deviation of ε is the same as its coefficient of variation (COV), denoted by δ, because the mean

value for ε is unity. From a conceptual point of view, the multiplicative form is identical to the

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“model factor” in the geotechnical reliability literature (e.g., Dithinde et al. 2016), which is

typically defined as the ratio of a measured response (e.g., pile capacity) to the calculated response.

Calibration of the bias and variability

The bias and variability of all transformation models in Tables 1 and 2 are calibrated by the

ROCK/9/4069 database. Consider the Is50-σc model proposed by Bieniawski (1975) as an example

(the fifth Is50-σc model in Table 1; σc ≈ 23×Is50). The bias factor (b) can be estimated as the sample

mean of the ratio (actual target value)/(predicted target value). For the Is50-σc model proposed by

Bieniawski (1975), the actual target value is σc, and the predicted target value is 23×Is50. The data

points in the ROCK/9/4069 database with simultaneous information of (Is50, σc) are extracted.

However, not all these data points are accepted because the Bieniawski’s model is only applicable

to sedimentary rocks with application range Is50 = 0.28~14 MPa (see the 6th column in Table 1). To

avoid extrapolation, 601 data points with simultaneous (Is50, σc) information and with Is50 = 0.28~14

MPa are finally adopted, and 601 ratios σc/(23×Is50) are computed. The sample mean of these ratios

is equal to 0.81 (b ≈ 0.81). This means that b×predicted target value = 0.81×23×Is50 is the unbiased

prediction for σc. The variability term ε = σc/(0.81×23×Is50) is computed for all 601 data points. The

sample COV (sample standard deviation divided by sample mean) of these ε values is 0.636 (δ ≈

0.636). If sedimentary rock Is50-σc data points outside the range Is50 = 0.28~14 MPa are also

incorporated in the calibration of (b, δ), the calibrated (b, δ) will involve extrapolation for

Bieniawski (1975)’s model. The resulting b is now 0.97 and resulting δ is now 1.425: the previous

small COV (namely 0.636) cannot be maintained.

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In general, the calibrated δ values of the transformation models for intact rocks are

significantly larger than those for clays and sands. For transformation models that predict the

undrained shear strength of a clay, Ching and Phoon (2014a) showed that the calibrated δ values

range from 0.34 to 0.55. For transformation models that predict the friction angle of a sand, Ching

et al. (2017b) showed that the calibrated δ values range from 0.05 to 0.13. It is remarkable that the

calibrated δ values for the σc models of intact rocks range from 0.46 to more than 4 and those for

the E models range from 0.45 to more than 6. It is evident that the transformation uncertainties for

intact rock models are significantly higher than those for soil models.

Selected generic transformation models

Tables 9 and 10 show the calibrated bias and variability for various σc and E transformation models

under the multiplicative form in Eq. (3). The number of available calibration data points (N) that are

within the application range is shown in the 6th column. According to the calibration results, the

following criteria are adopted to select the most suitable generic transformation models that fit the

ROCK/9/4069 database well: (i) it is preferable that the model has a broad application range. The

larger the number of available calibration data (N), the broader the application range; (ii) it is

preferable that δ is small, because this indicates that the model has less transformation uncertainty

with respect to the ROCK/9/4069 database. It is also desirable that b is close to 1 (less biased), but a

biased model can be correct to an unbiased model if a global database is available as shown in this

study. As a result, models with large N (criterion i) and small δ (criterion ii) can be considered as

“better” generic transformation models. When these two criteria conflict with each other, judgment

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is adopted to select the generic model as explained below. Consider model 1 with (N1, δ1) and

model 2 with (N2, δ2). Suppose N1 > N2 (model 1 has more application range) but δ1 > δ2 (model 2

has less uncertainty). If N1 is significantly greater than N2 but δ1 is only slightly greater than δ2,

model 1 will be selected. It turns out that models developed for a single rock class are rarely

selected, because their N values are typically significantly less than the N values of models

developed for multiple rock classes.

The selected generic models are highlighted in bold in Tables 9 and 10 and are reproduced in

Table 11 for ease of application. The formula shown in Table 11 is in the format of (actual target

value) = b×(predicted target value). The COV value (δ) and the application range for each model is

also shown in Table 11. Caution should be taken if the input parameter value is outside the range.

For σc transformation models (Table 9), there seems to be a general trend that newer transformation

models (in terms of the publication year) are selected. However, for E transformation models (Table

10), this trend does not exist: two older models are actually selected. It is evident that the

transformation uncertainty for the generic E models is more significant than that for the generic σc

models (Table 11). Among the generic σc models, the generic Is50-σc model has the lowest

transformation uncertainty. This suggests that the Is50 information is the most effective in terms of

estimating σc. Among the generic E models, the generic Sh-E and Vp-E models have lower

transformation uncertainty. This suggests that the Sh and Vp information is the most effective in

terms of estimating E.

Probability distribution of the actual target value

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It is possible to characterize the probability distribution of the target value (σc or E) based on

available input parameters (e.g., Is50 or Vp information). For instance, for the Is50-σc models, the

model proposed by Mishra and Basu (2013), σc ≈ 14.63×Is50, is selected as the generic

transformation model. According to Table 11, b = 1.18 and δ = 0.445 are calibrated by the

ROCK/9/4069 database. For the multiplicative form, the actual target value can be expressed as

Actual target value = b predicted target value× ×ε

(5)

where ε is a random variable with mean = 1 and COV = δ = 0.445. This means that

( )c s50Actual = 1.18 14.63 Iσ × × ×ε

(6)

The distribution type for ε can be examined by the K-S (Kolmogorov-Smirnov) test (Conover 1999).

The common null hypothesis for the multiplicative form is that ε follows the lognormal distribution.

If the p-value for the K-S test is larger than 0.05, the hypothesis is deemed acceptable. Otherwise,

there is sufficient evidence to reject the lognormal hypothesis. The p-values for all transformation

models are listed in Tables 9 and 10. The K-S test is relatively strict, so most p-values are less than

0.05. Nonetheless, nearly all histograms of ε are bell-shaped and skewed to the left, suggesting that

the lognormal hypothesis is more plausible than non-bell-shaped (e.g., uniform distribution) or

symmetric (e.g., normal distribution) distributions. As a result, the actual value of σc is still modeled

as a lognormal random variable, with mean = unbiased prediction = 1.18×(14.63×Is50) and COV =

0.445.

Consider again the Is50-σc model. Let the site-specific Is50 value for the new design site be

denoted by Is50,new and the actual value of the site-specific σc be denoted by σc,new. The point

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estimate (median value) for σc,new is simply b×(predicted target value) = 1.18×(14.63×Is50,new). The

95% confidence interval for σc,new is

( ) ( )( )( ) ( )( )

2

2

s50,new 2

2

b predicted target valueexp 1.96 ln 1

1

1.18 14.63 Iexp 1.96 ln 1 0.445

1 0.445

×× ± × + δ

+ δ

× ×= × ± × +

+

(7)

It is also possible to represent σc,new using a standard normal random variable Z for the purpose of

the first-order reliability method (Hasofer and Lind 1974; Ditlevsen and Madsen 1996):

( ) ( ) ( )s50 2c s502

b 14.63 IActual = exp ln + ln 1 Z = exp ln 15.77 I +0.425 Z

1

× ×σ + δ × × ×

+ δ (8)

The use of the calibrated transformation models in Table 11 is demonstrated by 24 igneous

intact rock cases (Bukit Timah granite; e.g., Wu et al. 2000) collected at a site at Dairy Farm Road

in Singapore (Cai 2017). These cases are summarized in Table 12. Their weathering grades (based

on GSE-GWPR 1995) range from Grade I to Grade III. These cases are not in ROCK/9/4069. These

cases contain information on n, Is50, σbt, σc, and E. Each of the input parameters (n, Is50, or σbt) is

used to predict the median value and 95% confidence interval for σc or E. Care is taken to ensure

that the application range given in Table 11 is applicable. The actual value of σc or E is then

compared with the prediction result. Figure 17 shows the predicted median values and 95%

confidence intervals. The horizontal coordinate of a data point indicates the actual value of σc or E,

whereas the vertical coordinate indicates the predicted value. The vertical bar indicates the 95%

confidence interval. If the vertical bar covers the 1:1 line (dashed lines), it implies that the 95%

confidence interval covers the actual value of σc or E. The length of the vertical bar depends on the

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precision of the transformation model (δ). It is not the purpose of this paper to improve the

precision of any model, but to ensure that a meaningful probabilistic statement can be made on the

target parameter of interest (σc or E). In this example, the statement is that there is a 95% chance of

the actual value of σc or E falling within the vertical bar. It is clear that the vertical bars cover the

1:1 line for the vast majority of the cases (theoretically, one expects 24×0.05 ≈ 1.2 case to fall

outside the vertical bar on the average).

CONCLUSIONS

In this paper, a global intact rock database (ROCK/9/4069) is developed, and existing

transformation models for intact rock properties in the literature are investigated. This global

database contains igneous, sedimentary, and metamorphic intact rocks with wide range of

characteristics and from a wide range of geographical locales. The vast majority (> 95%) of intact

rocks in the database are in their natural moisture contents. Two types of transformation models are

considered: models that predict the uniaxial compressive strength (σc models) and models that

predict the Young’s modulus (E models).

In general, there is no strong evidence indicating that the transformation relationships among

intact rock properties exhibited by ROCK/9/4069 data points depend on rock classes (igneous,

sedimentary, and metamorphic). Furthermore, there is no strong evidence that weathered rocks

exhibit significantly different transformation relationships. There is no strong evidence that foliated

metamorphic rocks exhibit significantly different transformation relationships, either.

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It is found that the existing transformation models are mostly data-specific or site-specific in the

sense that they fit well to their own calibration databases but do not necessarily fit well to the

ROCK/9/4069 database. In fact, many transformation models do not fit the general trend of the

ROCK/9/4069 database. It is also evident that the ROCK/9/4069 database has coverage wider than

most existing transformation models. The ROCK/9/4069 database is further used to calibrate the

bias and variability for the existing transformation models (see Tables 9 and 10 for the calibration

results), and models with relatively large application ranges and with relative small transformation

uncertainties are selected as generic transformation models. It is further found that the

transformation models for E larger transformation uncertainties compared with those for σc. The Is50

information is the most effective in terms of estimating σc, and the Sh and Vp information is the

most effective in terms of estimating E. It is also evident that the transformation uncertainties for

intact rock models are significantly higher than those for soil (clay and sand) models. This study

specifies whether a property is corrected for sample size or other effects explicitly. For uncorrected

properties such as σc, σbt, and E, the associated transformation models may carry additional

uncertainty due to this lack of correction. In short, the transformation uncertainties presented in this

study are conservative.

The main objective of this paper is to calibrate existing transformation models, not to produce

new models. In the companion paper (Ching et al. 2017a), the ROCK/9/4069 database will be

further adopted to develop the multivariate probability distribution for the nine parameters of intact

rocks. New probabilistic transformation models that admit multiple input parameters will be

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developed in this companion paper.

ACKNOWLEDGMENTS

The authors are grateful to Dr Jun Gang Cai from Tritech Consultants Pte Ltd for sharing the

Singapore Bukit Timah granite dataset.

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Miocene limestone. Rock Mechanics and Rock Engineering, 38(1), 69-76.

Wang, Y., and Aladejare, A.E. 2015. Selection of site-specific regression model for characterization

of uniaxial compressive strength of rock. International Journal of Rock Mechanics and

Mining Sciences, 75, 73-81.

Wang, Y., and Aladejare, A.E. 2016. Bayesian characterization of correlation between uniaxial

compressive strength and Young's modulus of rock. International Journal of Rock Mechanics

and Mining Sciences, 85, 10-19.

Wu, C., Hao, H., and Zhou, Y. 2000. Statistical properties of the Bukit Timah granite in Singapore.

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Yagiz, S. 2009. Predicting uniaxial compressive strength, modulus of elasticity and index properties

of rocks using the Schmidt hammer. Bulletin of Engineering Geology and the

Environment, 68(1), 55-63.

Yaşar, E., and Erdogan, Y. 2004. Correlating sound velocity with the density, compressive strength

and Young's modulus of carbonate rocks. International Journal of Rock Mechanics and

Mining Sciences, 41(5), 871-875.

Yilmaz, I. 2009. A new testing method for indirect determination of the unconfined compressive

strength of rocks. International Journal of Rock Mechanics and Mining Sciences, 46(8),

1349-1357.

Zhang, L. 2016. Engineering Properties of Rocks, 2nd Edition. Elsevier Ltd., Cambridge, MA, USA.

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Zhang, L. 2017. Evaluation of rock mass deformability using empirical methods – A review.

Underground Space, 2(1), 1-15.

Zorlu, K., Ulusay, R., Ocakoglu, F., Gokceoglu, C., and Sonmez, H. 2004. Predicting intact rock

properties of selected sandstones using petrographic thin-section data. International Journal of

Rock Mechanics and Mining Sciences, 41, 93–98.

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Figure Captions

Figure 1 Two Is50-σc models and their calibration databases.

Figure 2 Is50-σc models for different rock classes and data points in ROCK/9/4069.

Figure 3 Transformation uncertainty (data scatter) resulting from pairwise correlation between a

measured property and a desired design property (after Ching et al. 2017b).

Figure 4 Data points in ROCK/9/4069 with simultaneous information for RN and RL.

Figure 5 σc-Et50 and σc-Eav data points in ROCK/9/4069.

Figure 6 Correlation behaviors among intact rock properties for rocks with known and unknown

weathering grades.

Figure 7 Correlation behaviors among intact rock properties for foliated and non-foliated rocks.

Figure 8 n-σc models for different rock classes and data points in ROCK/9/4069.

Figure 9 RL-σc models for different rock classes and data points in ROCK/9/4069.

Figure 10 Sh-σc models for different rock classes and data points in ROCK/9/4069.

Figure 11 σbt-σc models for different rock classes and data points in ROCK/9/4069.

Figure 12 Vp-σc models for different rock classes and data points in ROCK/9/4069.

Figure 13 RL-E models for different rock classes and data points in ROCK/9/4069.

Figure 14 Sh-E models for different rock classes and data points in ROCK/9/4069.

Figure 15 σc-E models for different rock classes and data points in ROCK/9/4069.

Figure 16 Vp-E models for different rock classes and data points in ROCK/9/4069.

Figure 17 Prediction results for the 24 cases in Singapore.

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Table 1 Transformation models for σc in the literature.

Model Rock class Literature Transformation model Figure #

[curve #] Application range Similar models

n-σc

Igneous Tuğrul and Zarif (1999) σc ≈ -16.55×n+183 Fig. 8a [1] n: 0.2~4.3

Sedimentary

Lashkaripour (2002) σc ≈ 210.1×n-0.821 Fig. 8b [2] n: 1~35

Sabatakis et al. (2008) σc ≈ 123×e(-0.12n) Fig. 8b [3] n: 0.04~32.8

Rabbani et al. (2012) σc ≈ 135.9×e-0.048n Fig. 8b [4] n: 0.01~20

σc ≈ 143.8×e-0.0695n Fig. 8b [5] n:5~20

Mixed Tuğrul (2004) σc ≈ 195×e-0.21n Fig. 8d [6] n: 2.12~12

Kiliç and Teymen (2008) σc ≈ 147.16×e-0.0835n Fig. 8d [7] n: 0.16~37.81

RL-σc

Igneous

Dinçer et al. (2004) σc ≈ 2.75×RL-35.83 Fig. 9a [1] RL: 25.01~53.4

Aydin and Basu (2005) σc ≈ 1.4459×e0.0706RL Fig. 9a [2] RL: 20~64.67

Ng et al. (2015) σc ≈ 8.79×e0.0386RL Fig. 9a [3] RL: 21.66~56.51

Sedimentary

Cargill and Shakoor (1990) σc ≈ e0.018ρ×RL+2.9 ** RL: 27~49

Torabi et al. (2008) σc ≈ 4.1077×RL-61.96 Fig. 9b [4] RL: 19.96~65

Torabi et al. (2010) σc ≈ 0.047×RL2-0.176×RL+27.682 Fig. 9b [5] RL: 16~66.89

Minaeian and Ahangari (2013) σc ≈ 0.678×RL Fig. 9b [6] RL: 5.4~48.67

Bruno et al. (2013) σc ≈ e-4.04+2.28×ln(RL) Fig. 9b [7] RL: 16.53~60

Mixed

Deere and Miller (1966) σc ≈ 8.59×RL-240.6 Fig. 9d [8] RL: 32.03~58

σc ≈ 6.9×100.0087ρ×RL +0.16 ** RL: 21.5~58

Aufmuth (1973) σc ≈ 6.9×101.348×log(ρ×RL)-1.325 **

Sachpazis (1990) σc ≈ 4.29×RL-67.5 Fig. 9d [9] RL: 20.87~60

Katz et al. (2000) σc ≈ 2.208×e0.067RL Fig. 9d [10] RL: 23.97~71.11

Kulhawy and Prakoso (2003) σc/pa ≈143.1×e0.048RL Fig. 9d [11] RL: 15~70

Morales et al. (2004) σc ≈ e0.053RL+1.332 Fig. 9d [12] RL: 10.04~63.8

Kılıç and Teymen (2008) σc ≈ 0.0137×RL2.2721 Fig. 9d [13] RL: 17~63

Karaman and Kesimal (2015) σc ≈ 0.1383×RL1.743 Fig. 9d [14] RL: 11.82~59.59

Sh-σc

Sedimentary Koncagul and Santi (1999) σc ≈ 0.895×Sh+41.98 Fig. 10b [1] Sh: 14.93~47.62

Mixed Deere and Miller (1966)

σc ≈ 3.54×Sh-42.8 Fig. 10d [2] Sh: 17.94~105

σc ≈ 6.9×100.0041ρ×Sh+0.62 ** Sh: 11~105

Altindag and Guney (2010) σc ≈ 0.1821×Sh1.5833 Fig. 10d [3] Sh: 9~100 Kiliç and Teymen (2008): σc ≈ 0.159×Sh

1.6269

σbt-σc

Igneous Tuğrul and Zarif (1999) σc ≈ 6.667×σbt+4.867 Fig. 11a [1] σbt: 13.99~30.78

Sedimentary Nazir et al. (2013) σc ≈ 9.25×σbt0.947 Fig. 11b [2] σbt: 3.02~12.44

Mixed

Altindag and Guney (2010) σc ≈ 12.38×σbt1.0725 Fig. 11d [3] σbt: 0.49~29.34

Prakoso and Kulhawy (2011) σc ≈ 7.8×σbt Fig. 11d [4] σbt: 5~25.64

Karaman et al. (2015) σc ≈ 4.874×σbt+24.301 Fig. 11d [5] σbt: 4.4~34.4

Is50-σc Igneous

Irfan and Dearman (1978) σc ≈ 30×Is50 Fig. 2a [1] Is50: 0.7~9.2

Chau and Wong (1996) σc ≈ 12.5×Is50 Fig. 2a [2] Is50: 1.2~10

Basu and Aydin (2006) σc ≈ 21×Is50 Fig. 2a [3] Is50: 0.35~7.85

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Ng et al. (2015) σc ≈ 7.86×Is50+13.82 Fig. 2a [4] Is50: 1.08~8.19

Sedimentary

Bieniawski (1975) σc ≈ 23×Is50 Fig. 2b [5] Is50: 0.28~14

Rusnak and Mark (1999): σc ≈ 21×Is50

Lashkaripour (2002): σc ≈ 21.4×Is50

Kahraman (2001): σc ≈ 23.62×Is50-2.69

Cargill and Shakoor (1990) σc ≈ 23×Is50+13 Fig. 2b [6] Is50: 1.48~13.8 Akram and Bakar (2007): σc ≈ 22.792×Is50+13.295

Tsiambaos and Sabatakakis (2004) σc ≈ 7.3×Is501.71 Fig. 2b [7] Is50: 0.47~7

Zorlu et al. (2004) σc ≈ 10.3×Is50+28.1 Fig. 2b [8] Is50: 0.2~13 Kahraman and Alber (2006): σc ≈ 17.91×Is50+7.93

Akram and Bakar (2007) σc ≈ 11.076×Is50 Fig. 2b [9] Is50: 0.8~5.24

Çobanoğlu and Çelik (2008) σc ≈ 8.66×Is50+10.85 Fig. 2b [10] Is50: 2.44~11.36

Metamorphic Kahramam and Gunaydin (2009) σc ≈ 18.45×Is50-13.63 Fig. 2c [11] Is50: 2.2~13.3

Basu and Kamran (2010) σc ≈ 11.103×Is50+37.66 Fig. 2c [12] Is50: 1.08~5.93

Mixed

Deere and Miller (1966) σc ≈ 20.7×Is50+29.6 Fig. 2d [13] Is50: 0.69~13.79

Broch and Franklin (1972) σc ≈ 24×Is50 Fig. 2d [14] Is50: 0.42~14 Prakoso and Kulhaway (2011): σc ≈ 23.3×Is50

Kahraman (2001) σc ≈ 8.41×Is50+9.51 Fig. 2d [15] Is50: 1.32~16.21

Kahraman et al. (2005) σc ≈ 10.91×Is50+27.41 Fig. 2d [16] Is50: 1.6~13.3

Kiliç and Teymen (2008) σc ≈ 100×ln(Is50)+13.9 Fig. 2d [17] Is50: 0.9~8.6

Mishra and Basu (2013) σc ≈ 14.63×Is50 Fig. 2d [18] Is50: 1.15~14.13 Yilmaz (2009): σc ≈ 13.33×Is50+7.44

VP-σc

Igneous Sousa et al. (2005) σc ≈ 22.03×VP

1.247 Fig. 12a [1] VP: 2.34~5.75

Ng et al. (2015) σc ≈ 8.45×e0.4Vp Fig. 12a [2] VP: 2.19~5.66

Sedimentary Altindag (2012) σc ≈ 12.746×VP

1.194 Fig. 12b [3] VP: 1~6.75

Minaeian and Ahangari (2013) σc ≈ 5×VP Fig. 12b [4] VP: 0.90~6.43

Mixed

Kahraman (2001) σc ≈ 9.95×VP1.21 Fig. 12d [5] VP: 1.02~6.3

Ersoy and Atici (2006) σc ≈ 22.73×VP-67.73 Fig. 12d [6] VP: 3.27~6.44

Kiliç and Teymen (2008) σc ≈ 2.304×VP2.4315 Fig. 12d [7] VP: 1.51~6.74

* σc = uniaxial compressive strength (in MPa); n = porosity (in %); ρ = density (in g/cm3); RL = L-type Schmidt hammer hardness; Sh = Shore scleroscope hardness; σbt = Brazilian tensile strength (in MPa); Is50 = point load strength index (in MPa); Vp = P-wave velocity (in km/s); pa = one atmosphere pressure = 101.3 kPa. ** Transformation model cannot be plotted in a two-dimensional figure.

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Table 2 Transformation models for E in the literature.

Model Rock class Literature Transformation model Model No. Application range Similar models

RL-E

Igneous Dinçer et al. (2004) E ≈ 0.47×RL-6.25 Fig. 13a [1] RL: 24.8~53.4

Aydin and Basu (2005) E ≈ 1.0405×e0.058RL Fig. 13a [2] RL: 25.1~65.76

Mixed

Deere and Miller (1966) E ≈ 0.601×ρ×RL-20.27 ** RL: 21.5~58

Aufmuth (1973) E ≈ 0.0069×101.061×log(ρ×RL)+1.861 **

Beverly et al. (1979) E ≈ 0.192×ρ2×RL -12.71 **

Sachpazis (1990) E ≈ 1.94×RL-33.92 Fig. 13d [3] RL: 22.64~53.56

Katz et al. (2000) E ≈ 0.00013×RL3.09074 Fig. 13d [4] RL: 24.01~73.3

Yagiz (2009) E ≈ 0.0987×RL1.5545 Fig. 13d [5] RL: 39~59

Sh-E

Sedimentary Shalabi et al. (2007) E ≈ 0.971×Sh-26.91 Fig. 14b [1] Sh: 38.02~55


E ≈ 0.739×Sh+11.51 Fig. 14d [2] Sh: 11~105

E ≈ 0.268×ρ×Sh+12.62 ** Sh: 11~105

σc-E

Igneous

Aggistalis et al. (1980) E ≈ σc2.2248 Fig. 15a [1] σc: 22.31~62.79

Tuğrul and Zarif (1999) E ≈ 0.35×σc-12 Fig. 15a [2] σc: 120~193

Begonha and Braga (2002) E ≈ 0.0183×σc1.3646 Fig. 15a [3] σc: 20~157

Dinçer et al. (2004) E ≈ 0.17×σc+0.28 Fig. 15a [4] σc: 32.93~112.7

Sonmez et al. (2004) E ≈ 0.4358×σc0.6759 Fig. 15a [5] σc: 9.53~48.41

Sedimentary

Lashkaripour (2002) E ≈ 0.1033×σc1.0863 Fig. 15b [6] σc: 35.57~120 Pappalardo (2015): E ≈ 0.199×σc-3.97

Gokceoglu and Zorlu (2004)* E ≈ 0.456×σc+11.6 Fig. 15b [7] σc: 17~152.19 Shalabi et al. (2007): E ≈ 0.531×σc+9.567

Vásárhelyi (2005) E ≈ 0.374×σc Fig. 15b [8] σc: 1.07~39

Ocak (2008) E ≈ 0.5342×σc0.7672 Fig. 15b [9] σc: 2~152

Mixed

Deere and Miller (1966) E ≈ 0.303×σc-0.8745 Fig. 15d [10] σc: 21~329.61

Sachpazis (1990) E ≈ 0.3752×σc+4.4279 Fig. 15d [11] σc: 22~177.43

Prakoso (2002) E/pa ≈ 5280×(σc/pa)0.62 Fig. 15d [12] σc: 0.1~500

Shalabi et al. (2007) E ≈ 0.531×σc+9.567 Fig. 15d [13] σc: 16.83~127

VP-E

Igneous Begonha and Braga (2002) E ≈ 11.011×ln(VP)-78.923 Fig. 16a [1] VP: 1.43~6.42

Sedimentary

Moradian and Behnia (2009) E ≈ 2.06×VP2.78 Fig. 16b [2] VP: 1.83~6.54

Altindag (2012) E ≈ 0.919×VP1.9122 Fig. 16b [3] VP: 1~2.88

Pappalardo (2015) E ≈ 5.076×VP-15.72 Fig. 16b [4] VP: 3.57~6.25

Mixed Yaşar and Erdogan (2004) E ≈ 10.67×VP-18.71 Fig. 16d [5] VP: 3.11~5.6

Ersoy and Atici (2007) E ≈ 8.06×VP-28.33 Fig. 16d [6] VP: 3.68~6.44

* E = Young’s modulus (in GPa); σc in MPa, ρ in g/cm3; Vp in km/s; pa = one atmosphere pressure = 101.3 kPa.

** Transformation model cannot be plotted in a two-dimensional figure.

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Table 3 Clay/sand/rock parameter databases.

Database Reference Parameters of interest # data

points

# sites/

studies U

niv

aria

te CLAY/16 Phoon and Kulhawy (1999b)

γ, γd, wn, PL, LL, PI, LI, φ′, su, suFV

, qc, qt, SPT-N,

DMT (A, B), PMT pL a

SAND/11 Phoon and Kulhawy (1999b) φ′, Dr, qc, SPT-N, DMT (A, B, ID, KD, ED), PMT

(pL, EPMT) b

ROCK/8 Prakoso (2002) γ (or γd), n, R, Sh, σbt, Is, σc, E c

ROCK/13 Aladejare and Wang (2017) ρ, Gs, Id2, n, wc, γ, RL, Sh, σbt, Is50, σc, E, ν d

Mult

ivar

iate

CLAY/5/345 Ching and Phoon (2012a) LI, su, sure, σ

’p, σ

’v 345 37 sites

CLAY/7/6310 Ching and Phoon (2013, 2015) su from 7 different test procedures 6310 164 studies

CLAY/6/535 Ching et al. (2014) su/σ'v, OCR, qtc, qtu, (u2−u0)/σ

'v, Bq 535 40 sites

CLAY/10/7490 Ching and Phoon (2014a,b) LL, PI, LI, σ'v/Pa, σ

'p/Pa, su/σ

'v, St, qtc, qtu, Bq 7490 251 studies

F-CLAY/7/216 D’Ignazio et al. (2016) suFV

, σ′v, σ′p, wn, LL, PL, St 216 24 sites

J-CLAY/5/124 Liu et al. (2016) Mr, qc, fs, wn, γd 124 16 sites

SAND/7/2794 Ching et al. (2017b) D50, Cu, Dr, σ'v/Pa, φ′, qt1, (N1)60 2794 176 studies

EMI-

ROCK/8/26000

+

Kim and Hunt (2017) σc, σbt, ρ, CAI, PPI, cohesion, direction shear,

triaxial confining 26000+ -

ROCK/9/4069 This study γ, n, RL, Sh, σbt, Is50, Vp, σc, E 4069 184 studies

Note: ρ = density; Gs = specific gravity; Id2 = slake durability index; n = porosity; wn = water content; γ = unit weight; γd = dry unit weight; CAI = Cerchar abrasivity index; punch penetration = punch penetration index; R = Schmidt hammer hardness (RL = L-type Schmidt hammer hardness); Sh = Shore scleroscope hardness; σbt = Brazilian tensile strength; Is = point load strength index (Is50 = Is for diameter 50 mm); Vp = P-wave velocity; σc = uniaxial compressive strength; E = Young’s modulus; ν = Poisson ratio; LL = liquid limit; PI = plasticity index; LI = liquidity index; σ

’v = vertical effective

stress; σ’p = preconsolidation stress; wn = water content; su = undrained shear strength for clay; su

FV = field vane su; su

re

= remoulded su; St = sensitivity; OCR = overconsolidation ratio, qc = cone tip resistance; fs = sleeve frictional resistance; qt = corrected cone tip resistance, qtc = (qt-σv)/σ

'v = normalized cone tip resistance, qtu = (qt-u2)/σ

'v = effective cone tip

resistance; u0 = hydrostatic pore pressure; (u2-u0)/σ'v = normalized excess pore pressure; Bq = pore pressure ratio = (u2-

u0)/(qt-σv); Pa = atmospheric pressure = 101.3 kPa; σ'v = vertical effective stress; qt1 = (qt/Pa)/(σ'v/Pa)

0.5; SPT-N =

standard penetration test blow count; N60 = corrected SPT-N; (N1)60 = N60/(σ'v/Pa)0.5

; DMT (A, B, ID, KD, ED) = dilatometer A & B readings, material index, horizontal stress index, modulus; PMT (pL, EPMT) = pressuremeter limit stress, modulus; D50 = median grain size; Cu = coefficient of uniformity; Dr = relative density; φ′ = effective friction angle; Mr = subgrade resilience modulus. a - The no. of data groups varies between 2 and 42 depending on the clay parameter. Statistics are calculated at the data group level. The average no. of data points/data group varies between 16 and 564. Details given in Tables 1-3, Phoon and Kulhawy (1999b). b - The no. of data groups varies between 5 and 57 depending on the sand parameter. Statistics are calculated at the data group level. The average no. of data points/data group varies between 15 and 123. Details given in Tables 1-3, Phoon and Kulhawy (1999b). c – The no. of data groups varies between 30 and 174 depending on the rock parameter with no differentiation of rock type [igneous (intrusive, extrusive, pyroclastic), sedimentary (clastic, chemical), metamorphic (foliated, non-foliated)]. Statistics are calculated at the data group level. The average no. of data points/data group varies between 3 and 161 for σc (Prakoso, 2017). Details given in Table 4.4, Prakoso (2002). d – The no. of data groups varies between 2 and 47 depending on the rock parameter and rock type (igneous, sedimentary, or metamorphic). Statistics are calculated at the data group level. The average no. of data points/data group varies between 7 and 92. Details given in Tables 2-4, Aladejare and Wang (2017).

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Table 4 Statistics of the nine parameters in the ROCK/9/4069 database at the global level.

Parameter N

(Igneous + Sedimentary + Metamorphic)

Mean COV Min Max

γ (kN/m3) 1288 (238+879+171) 24.6 0.12 15.0 34.7

n (%) 1371 (289+957+125) 11.0 1.02 0.01 55.0

RL 812 (356+390+066) 41.4 0.29 8.1 81.6

Sh 333 (62+233+038) 47.6 0.44 8.4 100.0

σbt (MPa) 854 (196+512+146) 8.2 0.61 0.07 34.4

Is50 (MPa) 1303 (420+659+224) 4.2 0.69 0.05 17.4

σc (MPa) 3226 (857+1971+398) 72.3 0.77 0.68 379.0

E (GPa) 1495 (477+895+123) 26.5 0.99 0.03 116.3

VP (km/s) 1858 (619+989+250) 4.0 0.39 0.44 8.0

Table 5 Summary statistics of the nine parameters in the ROCK/9/4069 database at the site level

(igneous intact rocks).

Property

No. of

data

groups

No. of tests/group Range of

data

Property mean value Property COV (%)

Range Mean Range Mean Range Mean*

γ (kN/m3) 17 4-20 9 16.68-31.02 16.98-30.07 24.55 0.21-10.18 3.36 (4.9)

n (%) 9 4-60 25 0.06-31.02 0.22-30.31 7.11 15.16-73.25 45.61 (69.7)

RL 14 4-145 23 16.8-65.76 29.24-62.57 45.76 2.03-26.17 12.86

Sh 2 4-4 4 19-80.5 25.25-73.7 49.48 7.04-19.23 13.14 (24.2)

σbt (MPa) 10 4-23 11 1.2-34.4 1.9-16.75 11.59 6.39-51.11 22.96

Is50 (MPa) 11 5-145 35 0.1-14.16 1.85-10.78 5.22 14.14-84.80 41.30

σc (MPa) 33 4-145 21 2.2-294 17.63-246 112.42 0.95-89.46 33.16

E (GPa) 22 4-92 18 0.96-109.3 1.68-85.95 34.26 2.87-75.10 30.16

VP (km/s) 22 4-150 25 1.16-7.98 2.88-7.55 4.53 0.20-37.41 13.70

* value in parenthesis is reproduced from Table 2, Aladejare and Wang (2017)


(sedimentary intact rocks).

Property

No. of

data

groups


data



γ (kN/m3) 42 4-126 18 16.25-34.68 18.17-29.04 24.04 0.34-15.81 5.14 (6.2)

n (%) 43 4-126 19 0.2-55 0.43-38.40 13.51 1.51-141.48 33.55 (53.4)

RL 19 4-44 15 12-67 20.40-57.35 33.31 4.05-31.75 16.43

Sh 10 4-31 15 8.4-98 13.36-76.11 40.65 6.18-35.31 21.73 (20.3)

σbt (MPa) 21 4-45 16 0.07-17.99 1.25-14.85 7.03 1.77-81.06 28.02

Is50 (MPa) 34 4-44 16 0.05-13.35 0.17-9.69 3.47 4.07-75.50 30.52

σc (MPa) 112 4-66 14 0.68-203 1.92-159.13 61.39 1.16-88.70 32.08

E (GPa) 55 4-66 13 0.03-99.9 0.14-62.47 25.88 1.47-120.85 40.16

VP (km/s) 56 4-80 16 0.44-6.76 0.81-6.27 3.41 1.04-44.69 13.72


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(metamorphic intact rocks).

Property

No. of

data

groups


data



γ (kN/m3) 10 5-20 11 23.74-32.92 25.70-30.69 27.18 0.96-5.82 2.57 (1.8)

n (%) 7 4-24 13 0.03-9 0.39-6.67 2.25 13.61-113.60 54.13 (61.8)

RL 3 4-8 7 39.9-59.7 47.01-51.60 49.78 3.29-16.04 9.81

Sh - - - - - - - - (20.8)

σbt (MPa) 8 5-20 8 3.45-26.3 6.67-18.12 10.89 3.06-53.74 27.19

Is50 (MPa) 12 5-41 16 1.03-11.08 1.27-6.92 4.04 11.37-40.87 30.37

σc (MPa) 21 4-41 13 3-272.83 4.88-200.43 79.62 2.48-74.22 28.96

E (GPa) 7 5-20 9 1-116.3 4.49-102.33 54.59 4.03-82.12 21.85

VP (km/s) 16 4-32 12 1.48-6.73 2.06-6.27 4.32 0.66-31.48 10.42


Table 8 Coefficient of variation of intact rock parameters with a mix of rock types [igneous

(intrusive, extrusive, pyroclastic), sedimentary (clastic, chemical), metamorphic (foliated, non-

foliated)] represented in each parameter (Source: Table 4.4, Prakoso (2002); Table 3.9, Phoon et al.

2016)

Test type Property Number of

data groups

Coefficient of Variation (%)

Mean S.D. Range

Index γ, γd 79 1.0 1.2 0.1 – 8.6

n 30 24.2 18.6 3.0 – 71

R 54 8.7 5.4 1.4 – 26

Sh 59 11.1 8.5 1.4 – 38

Strength σc 174 14.0 11.7 0.8 – 61

σbt 54 19.4 12.9 3.8 – 61

Is 66 20.5 14.3 2.8 – 59

Stiffness Et50 72 20.5 16.9 1.4 – 69

Note: Clastic and chemical sedimentary rocks dominate the data base, followed by metamorphic non-foliated, intrusive

igneous, and extrusive igneous rocks. The databases for metamorphic foliated and igneous pyroclastic rocks are small.

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Table 9 Bias and variability for various σc transformation models when benchmarked against ROCK/9/4069.

Model Rock class Literature Transformation model Application range Calibration results

N b δ p-value

n-σc

Igneous Tugrul and Zarif (1999) σc ≈ -16.55×n+183 n: 0.2~4.3 193 0.56 0.720 < 0.01

Sedimentary

Lashkaripour (2002) σc ≈ 210.1×n-0.821 n: 1~35 559 1.66 0.752 < 0.01

Sabatakis et al. (2008) σc ≈ 123×e(-0.12n) n: 0.04~32.8 635 1.88 1.062 < 0.01

Rabbani et al. (2012) σc ≈ 135.9×e-0.048n n: 0.01~20 635 0.59 0.817 < 0.01

σc ≈ 143.8×e-0.0695n n:5~20 635 0.65 0.883 < 0.01

Mixed Tugrul (2004) σc ≈ 195×e-0.21n n: 2.12~12 828 0.88 1.335 < 0.01

Kilic and Teymen (2008) σc ≈ 147.16×e-0.0835n n: 0.16~37.81 911 0.91 0.747 < 0.01

RL-σc

Igneous

Dincer et al. (2004) σc ≈ 2.75×RL-35.83 RL: 25.01~53.4 226 0.80 0.464 0.18

Aydin and Basu (2005) σc ≈ 1.4459×e0.0706RL RL: 20~64.67 319 1.80 0.577 0.60

Ng et al. (2015) σc ≈ 8.79×e0.0386RL RL: 21.66~56.51 319 1.28 0.446 0.35

Sedimentary

Cargill and Shakoor (1990) σc ≈ e0.018ρ×RL+2.9 RL: 27~49 129 0.55 0.584 0.01

Torabi et al. (2008) σc ≈ 4.1077×RL-61.96 RL: 19.96~65 338 0.84 1.018 < 0.01

Torabi et al. (2010) σc ≈ 0.047×RL2-0.176×RL+27.682 RL: 16~66.89 360 0.71 0.543 < 0.01

Behnaz Minaeian (2013) σc ≈ 0.678×RL RL: 5.4~48.67 291 2.52 0.566 < 0.01

Bruno et al. (2013) σc ≈ e-4.04+2.28×ln(RL) RL: 16.53~60 344 1.04 0.628 0.09

Mixed

Deere and Miller (1966) σc ≈ 8.59×RL-240.6 RL: 32.03~58 493 0.83 1.805 < 0.01

σc ≈ 6.9×100.0087ρ×RL +0.16 RL: 21.5~58 171 0.84 0.577 < 0.01

Aufmuth (1973) σc ≈ 6.9×101.348×log(ρ×RL)-1.325 166 0.40 0.504 0.01

Sachpazis (1990) σc ≈ 4.29×RL-67.5 RL: 20.87~60 646 0.79 2.284 < 0.01

Katz et al. (2000) σc ≈ 2.208×e0.067RL RL: 23.97~71.11 735 1.92 0.637 0.02

Kulhawy and Prakoso (2003) σc/pa ≈143.1×e0.048RL RL: 15~70 735 0.62 0.545 0.03

Morales et al. (2004) σc ≈ e0.053RL+1.332 RL: 10.04~63.8 735 1.94 0.562 0.01

Kilic and Teymen (2008) σc ≈ 0.0137×RL2.2721 RL: 17~63 700 1.15 0.632 0.60

Karaman and Kesimal (2015) σc ≈ 0.1383×RL1.743 RL: 11.82~59.59 664 0.78 0.531 0.02

Sh-σc

Sedimentary Koncagul and Santi (1999) σc ≈ 0.895×Sh+41.98 Sh: 14.93~47.62 225 1.47 0.908 0.06


σc ≈ 3.54×Sh-42.8 Sh: 17.94~105 288 1.03 3.211 < 0.01

σc ≈ 6.9×100.0041ρ×Sh+0.62 Sh: 11~105 111 0.64 0.565 < 0.01

Altindag and Guney (2010) σc ≈ 0.1821×Sh1.5833 Sh: 9~100 297 1.15 0.650 0.08

σbt-σc

Igneous Tugrul and Zarif (1999) σc ≈ 6.667×σbt+4.867 σbt: 13.99~30.78 144 17.36 0.834 < 0.01

Sedimentary Nazir et al. (2013) σc ≈ 9.25×σbt0.947 σbt: 3.02~12.44 334 1.19 0.459 0.06

Mixed

Altindag and Guney (2010) σc ≈ 12.38×σbt1.0725 σbt: 0.49~29.34 741 0.76 0.522 0.57

Prakoso and Kulhawy (2011) σc ≈ 7.8×σbt σbt: 5~25.64 525 1.31 0.496 0.04

Karaman et al. (2015) σc ≈ 4.874×σbt+24.301 σbt: 4.4~34.4 751 1.31 0.585 0.08

Is50-σc Igneous

Irfan and Dearman (1978) σc ≈ 30×Is50 Is50: 0.7~9.2 368 0.59 0.497 0.25

Chau and Wong (1996) σc ≈ 12.5×Is50 Is50: 1.2~10 341 1.32 0.444 0.28

Basu and Aydin (2006) σc ≈ 21×Is50 Is50: 0.35~7.85 353 0.89 0.589 0.07

Ng et al. (2015) σc ≈ 7.86×Is50+13.82 Is50: 1.08~8.19 420 2.53 1.337 < 0.01

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Sedimentary

Bieniawski (1975) σc ≈ 23×Is50 Is50: 0.28~14 601 0.81 0.636 < 0.01

Cargill and Shakoor (1990) σc ≈ 23×Is50+13 Is50: 1.48~13.8 624 0.92 1.740 < 0.01

Tsiambaos and Sabatakakis (2004) σc ≈ 7.3×Is501.71 Is50: 0.47~7 511 1.39 0.897 < 0.01

Zorlu et al. (2004) σc ≈ 10.3×Is50+28.1 Is50: 0.2~13 624 0.94 0.813 < 0.01

Akram and Bakar (2007) σc ≈ 11.076×Is50 Is50: 0.8~5.24 436 1.67 0.464 0.01

Cobanoglu (2008) σc ≈ 8.66×Is50+10.85 Is50: 2.44~11.36 624 2.24 1.211 < 0.01

Metamorphic Kahramam (2009) σc ≈ 18.45×Is50-13.63 Is50: 2.2~13.3 140 1.17 0.383 0.92

Basu and Kamran (2010) σc ≈ 11.103×Is50+37.66 Is50: 1.08~5.93 182 1.39 1.046 < 0.01

Mixed

Deere and Miller (1966) σc ≈ 20.7×Is50+29.6 Is50: 0.69~13.79 1226 0.65 1.206 < 0.01

Broch and Franklin (1972) σc ≈ 24×Is50 Is50: 0.42~14 1171 0.75 0.499 0.07

Kahraman (2001) σc ≈ 8.41×Is50+9.51 Is50: 1.32~16.21 1226 1.71 0.908 < 0.01

Kahraman et al. (2005) σc ≈ 10.91×Is50+27.41 Is50: 1.6~13.3 1226 1.01 0.900 < 0.01

Kilic and Teymen (2008) σc ≈ 100×ln(Is50)+13.9 Is50: 0.9~8.6 1019 0.55 1.207 < 0.01

Mishra and Basu (2013) σc ≈ 14.63×Is50 Is50: 1.15~14.13 1074 1.18 0.445 0.29

VP-σc

Igneous Sousa et al. (2005) σc ≈ 22.03×VP

1.247 VP: 2.34~5.75 328 0.66 0.568 < 0.01

Ng et al. (2015) σc ≈ 8.45×e0.4Vp VP: 2.19~5.66 422 4.76 1.463 < 0.01

Sedimentary Altindag (2012) σc ≈ 12.746×VP

1.194 VP: 1~6.75 717 0.85 0.550 < 0.01

Minaeian (2013) σc ≈ 5×VP VP: 0.90~6.43 711 2.74 0.587 < 0.01

Mixed

Kahraman (2001) σc ≈ 9.95×VP1.21 VP: 1.02~6.3 1247 1.26 0.632 < 0.01

Ersoy and Atici (2006) σc ≈ 22.73×VP-67.73 VP: 3.27~6.44 858 2.51 0.846 < 0.01

Kilic and Teymen (2008) σc ≈ 2.304×VP2.4315 VP: 1.51~6.74 1234 1.15 0.829 < 0.01

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Table 10 Bias and variability for various E transformation models when benchmarked against ROCK/9/4069.

Model Rock class Literature Transformation model Application range Calibration results

N b δ p-value

RL-E

Igneous Dincer et al. (2004) E ≈ 0.47×RL-6.25 RL: 24.8~53.4 63 1.47 0.583 0.30

Aydin (2005) E ≈ 1.0405×e0.058RL RL: 25.1~65.76 134 1.17 0.658 0.02

Mixed

Deere and Miller (1966) E ≈ 0.601×ρ×RL-20.27 RL: 21.5~58 82 0.35 0.809 0.06

Aufmuth (1973) E ≈ 0.0069×101.061×log(ρ×RL)+1.861 78 0.19 0.965 < 0.01

Beverly et al. (1979) E ≈ 0.192×ρ2×RL -12.71 78 0.45 0.850 0.18

Sachpazis (1990) E ≈ 1.94×RL-33.92 RL: 22.64~53.56 208 0.43 0.909 < 0.01

Katz et al.(2000) E ≈ 0.00013×RL3.09074 RL: 24.01~73.3 289 1.47 0.985 0.02

Yagiz (2009) E ≈ 0.0987×RL1.5545 RL: 39~59 123 0.56 0.914 < 0.01

Sh-E

Sedimentary Shalabi et al. (2007) E ≈ 0.971×Sh-26.91 Sh: 38.02~55 38 1.11 0.454 0.99

Mixed Deere and Miller (1966) E ≈ 0.739×Sh+11.51 Sh: 11~105 197 0.61 0.712 0.01

E ≈ 0.268×ρ×Sh+12.62 Sh: 11~105 107 0.54 0.826 < 0.01

σc-E

Igneous

Aggistalis et al. (1980) E ≈ σc2.2248 σc: 22.31~62.79 121 3.32 1.263 0.02

Tugrul and Zarif (1999) E ≈ 0.35×σc-12 σc: 120~193 97 1.09 0.513 < 0.01

Begonha and Sequeira Braga (2002) E ≈ 0.0183×σc1.3646 σc: 20~157 314 3.04 1.087 0.02

Dincer et al. (2004) E ≈ 0.17×σc+0.28 σc: 32.93~112.7 428 70.10 1.401 < 0.01

Sonmez et al. (2004) E ≈ 0.4358×σc0.6759 σc: 9.53~48.41 112 2.01 1.140 0.08

Sedimentary

Lashkaripour (2002) E ≈ 0.1033×σc1.0863 σc: 35.57~120 437 2.69 0.855 0.13

Gokceoglu and Zorlu (2004) E ≈ 0.456×σc+11.6 σc: 17~152.19 873 0.82 1.656 < 0.01

Vásárhelyi (2005) E ≈ 0.374×σc σc: 1.07~39 349 1.02 1.279 < 0.01

Ocak (2008) E ≈ 0.5342×σc0.7672 σc: 2~152 779 1.77 0.948 < 0.01

Mixed

Deere and Miller (1966) E ≈ 0.303×σc-0.8745 σc: 21~329.61 1152 1.23 0.941 < 0.01

Sachpazis (1990) E ≈ 0.3752×σc+4.4279 σc: 22~177.43 977 1.43 2.183 < 0.01

Prakoso (2002) E/pa ≈ 5280×(σc/pa)0.62 σc: 0.1~500 1405 2.04 1.879 < 0.01

Shalabi et al. (2007) E ≈ 0.531×σc+9.567 σc: 16.83~127 977 1.21 1.749 < 0.01

VP-E

Igneous Begonha (2002) E ≈ 11.011×ln(VP)-78.923 VP: 1.43~6.42 160 2.51 0.657 < 0.01

Sedimentary

Moradian and Behnia (2009) E ≈ 2.06×VP2.78 VP: 1.83~6.54 303 0.25 0.857 < 0.01

Altindag (2012) E ≈ 0.919×VP1.9122 VP: 1~2.88 114 1.51 0.971 < 0.01

Pappalardo (2015) E ≈ 5.076×VP-15.72 VP: 3.57~6.25 185 3.76 0.710 < 0.01

Mixed Yasar and Erdogan (2004) E ≈ 10.67×VP-18.71 VP: 3.11~5.6 192 0.90 0.724 < 0.01

Ersoy and Atici (2007) E ≈ 8.06×VP-28.33 VP: 3.68~6.44 340 3.61 0.877 < 0.01

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Table 11 Selected generic transformation models with bias corrected (bias correction in bold).

Model Literature

Unbiased transformation model in

the form of (actual target value) ≈

b×(predicted target value)

δ Application range

σc models

n-σc Kılıç and Teymen (2008) σc ≈ 0.91×(147.16×e-0.0835n

) 0.747 n: 0.16~37.81

RL-σc Karaman and Kesimal (2015) σc ≈ 0.78×(0.1383×RL1.743

) 0.531 RL: 11.82~59.59

Sh-σc Altindag and Guney (2010) σc ≈ 1.15×(0.1821×Sh1.5833

) 0.650 Sh: 9~100

σbt-σc Prakoso and Kulhawy (2011) σc ≈ 1.31×(7.8×σbt) 0.496 σbt: 5~25.64

Is50-σc Mishra and Basu (2013) σc ≈ 1.18×(14.63×Is50) 0.445 Is50: 1.15~14.13

VP-σc Kahraman (2001) σc ≈ 1.26×(9.95×VP1.21

) 0.632 VP: 1.02~6.3

E models

RL-E Katz et al. (2000) E ≈ 1.47×(0.00013×RL3.09074

) 0.985 RL: 24.01~73.3

Sh-E Deere and Miller (1966) E ≈ 0.61×(0.739×Sh+11.51) 0.712 Sh: 11~105

σc-E Deere and Miller (1966) E ≈ 1.23×(0.303×σc-0.8745) 0.941 σc: 21~329.61

VP-E Yaşar and Erdoğan (2004) E ≈ 0.90×(10.67×VP-18.71) 0.724 VP: 3.11~5.6

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Table 12. Intact rock data collected in Singapore (Cai 2017).

# Rock

type

Weathering grade

based on GSE-

GWPR (1995)

Input parameters Parameters to be

estimated

γ

(kN/m3)

n

(%)

σbt

(MPa)

Is50

(MPa)

σc

(MPa)

E

(GPa)

1 Granite Grade II 25.9 0.6 5.24 166.2 66.9

2 Granite Grade II 25.9 13.41 6.04 184.6 66.6

3 Granite Grade II 25.9 5.72 144.3 71.3

4 Granite Grade I 26.0 0.3 12.33 14.96 209.1 72.3

5 Granite Grade III 25.9 10.69 101.5 76.2

6 Granite Grade III 26.0 10.73 9.20 108.0 72.3

7 Granite Grade II 26.0 0.4 8.81 199.0 74.0

8 Granite Grade II 26.1 12.43 14.45 189.2 73.1

9 Granite Grade II 26.1 10.07 158.6 79.8

10 Granite Grade III 25.8 1.2 11.33 99.3 91.4

11 Granite Grade I 26.1 11.80 242.1 75.5

12 Granite Grade I 26.0 10.04 13.07 231.1 77.5

13 Granite Grade III 25.8 11.39 133.0 98.4

14 Granite Grade I 26.1 12.13 5.47 224.1 71.9

15 Granite Grade II 26.0 8.75 196.0 76.5

16 Granite Grade II 25.9 0.4 14.16 16.87 143.2 68.6

17 Granite Grade I 26.0 11.74 217.7 72.1

18 Granite Grade I 26.0 12.15 11.49 198.6 74.7

19 Granite Grade I 26.0 8.42 226.2 64.2

20 Granite Grade II 26.0 0.2 12.67 9.31 186.1 104.0

21 Granite Grade II 26.0 0.3 8.90 173.5 64.2

22 Granite Grade II 26.0 11.93 13.05 208.5 66.7

23 Granite Grade I 26.1 0.5 13.38 209.0 69.7

24 Granite Grade I 26.0 14.21 13.13 222.7 88.9

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APPENDIX BASIC INFORMATION FOR ROCK/9/4069 DATABASE

Table A1 Basic information for the ROCK/9/4069 database.

# study Reference Site # data points

Rock type γ (kN/m3) n (%) σc (MPa) E (GPa) Vp (km/s)

1 Aggistalis et al. (1996) Greece 30 Weathered Basalt (Grade I~II; GSE-GWPR) 17.1~91.2 1.2~12.1 Greece 62 Weathered Gabbro (Grade I~II; GSE-GWPR) 6.3~107.5 1.0~9.8

2, 3 Aydin and Basu (2005), Basu and Aydin (2006) Hong Kong 40 Weathered Granite (Grade I~IV) 1.3~21.0 6.3~196.5 4.5~53.2 4 Adebayo and Umeh (2007) Nigeria 30 Limestone 59.8~119.2

5 Agustawijaya (2007) Australia 5 Weathered Sandstone (unknown grade) 3.1~10.5

Victoria, Australia 14 Weathered Siltstone (unknown grade) 3.4~11.1 6 Arman et al. (2007) Turkey 43 Limestone 33~52

7 Aoki and Matsukura (2008)

Indonesia 1 Andesite 12 63.8

Japan

1 Gabbro 5.4 152.7 2 Granite 1.1 162.7~175.1 1 Limestone 20 24.8 2 Sandstone 5.8~6.9 72.2~101.5 2 Tuff 17~39.5 15.5~59.7

8 Altindag and Guney (2010) Unknown

1 Andesite 86.1 1 Diorite 193 1 Gabbro 210 5 Granite 145.2~188 8 Limestone 68.9~139.4 4 Marble 48.8~115.8

22 Sandstone 7~156 1 Siltstone 69.5

9 Anikoh and Olaleye (2013) Nigeria 5 Shale 22.1~26.2 26~40 30.7~36.7 10 Aydin et al. (2013) Turkey 9 Granite 25.9~27.2 94~145 3.6~5.6

11 Anikoh et al. (2015) Nigeria 5 Granite 24.8~26.5 1~2 146.6~197 5 Limestone 25.2~27.0 2.2~3.3 74.4~133.6

12 Arslan et al. (2015) Pakistan 18 Limestone 22.9~27.1 42~108 13 Azimian and Ajalloeian (2015) Fars, Iran 40 Marl 20.5~25.7 6.3~33.7 15.3~88.9 5.6~30.7 1.2~4.0 14 Abdullah et al. (2016) Malaysia 2 Weathered Granite (Grade II~III) 27~28 23~60 15 Aono et al. (2016) Japan 1 Granite 25.8 0.7 5.3

16 Aqla et al. (2016) Indonesia 1 Andesite 25.5 0.8 71.5 9.5 5.4 1 Limestone 26.3 1.9 28.9 7.3 6.0

17 Bell (1978) United Kingdom 29 Sandstone 10.2~20.5 33.2~112.4 19.7~46.2 18 Bell et al. (1997) United Kingdom 10 Mudstone 23.8~25.1 25.7~45.4

19 Bearman (1999) United Kingdom

1 Andesite 139.2 63.6 2 Diorite 128.8~274.8 54.5~64.2 1 Granite 132.4 39.1 2 Greywacke 165.4~226.3 49.6~57.0 3 Limestone 47.8~156.7 25.5~57.2 1 Quartzite 138.6 36.4

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2 Sandstone 76.3~162.2 15.9~39.1 20 Bell and Lindsay (1999) South Africa 27 Sandstone 5.6~10.1 77~214 10.9~99.9 21 Begonhaa and Braga (2002) Portugal 163 Weathered Granite (Grade I~IV; ISRM) 0.7~3.9 1.3~6.4

22 Balci et al. (2004) Turkey

1 Claystone 27.1 58 1 Limestone 26.7 121 57 3 Sandstone 26~26.2 87~174 17~33.3 1 Siltstone 26 58 30 4 Tuff 16.7~17.7 11~27 1.3~2.4

23 Basarira and Karpuz (2004) Turkey 6 Marl 15.0~23.6 2.7~24.9 0.9~2.9 24 Buyuksagis and Goktan (2005) Turkey 7 Marble 39.2~87.0 25 Buyuksagis (2007) Turkey 2 Granite 25.9~29.9 0.3~0.8 168~292 26 Basem K. Moh’d (2009) France 15 Limestone 16.9~23.5 11.7~36.1 10.4~80.2 2.1~4.6 27 Basu et al. (2009) Sao Paulo, Brazil 40 Weathered Granite (Grade I~IV; GSE-GWPR) 73~214 41.1~70.2 1.9~5.5 28 Basu and Kamran (2010) Jharkhand, India 15 Schist (foliated) 40.1~107.9 29 Bastola and Chugh (2015) Illinois, US 44 Coal 17.5~25.9

30 Cargill and Shakoor (1990)

New York, US 1 Dolomite 95.7 Ontario, Canada 1 Gneiss (foliated) 288.2

Indiana, US 1 Limestone 56.9 Ohio, US 2 Limestone 122.3~169.4

Ontario, Canada 1 Marble 119.1 New York, US 1 Sandstone 208

Ohio, US 4 Sandstone 34.8~64 Pennsylvania, US 2 Sandstone 199~270.7

31 Christaras et al. (1994) France

1 Andesite 21.6 28.7 3.8 3 Basalt 29.4~29.9 101.8~114.4 6.4~6.5 1 Granite 26.2 64.0 5.19 1 Limestone 21.6 19.9 3.5 1 Phonolite 25.3 56.5 5.3

32 Chitty et al. (1994)

Vermont, US 5 Granite 216.9~222 54.2~58.0 3.5~3.6 Massachusetts, US 9 Limestone 16.2~17.1 58.2~61.5 29.5~31.6 4.3~4.4

Tennessee, US 5 Marble 126.6~154.5 68.5~85.7 6.0~6.4 Massachusetts, US 14 Mudstone 16.6~17.2 46~59.3 25.2~29.6

Vermont, US 5 Sandstone 64.7~66.7 17.1~17.7 2.0~2.2 33 Chen and Hsu (2001) Taiwan 12 Marble

34 Chatterjee and Mukhopadhyay (2002) Andhra, India

1 Gneiss (foliated) 1.7 52 1 Limestone 4.2 41.8 7 Sandstone 3~22 5~48 14~38 3 Siltstone 6~18 15.9~35 15~22

35 Ceryan et al. (2008) Turkey 20 Granite 1.3~18.5 2.8~200.3 2.1~5.3

36 Çobanoĝlu and Çelik (2008) Turkey 25 Limestone 40.9~91.3 4.6~5.1 25 Sandstone 71.6~146.8 4.6~5.0

37 Cai (2010) Manitoba, Canada

1 Amphibolite 110 1 Pegmatite 169 2 Peridotite 67~155 1 Quartzite 152 3 Schist (foliated) 65~73

38 Ceryan et al. (2013) Turkey 55 Carbonate rock 4.9~33.9 7.3~24.1 1.3~4.3

39 Çopur et al. (2013) Turkey 1 Harzburgite 26 58 2.1 1 Serpentine 24.4 38 2.3

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40 D'Andrea et al. (1965) Bureau of Mines, US

2 Basalt 149.1~292.0 62.3~100.5 5.2~6.6 2 Chalk 15.0~19.1 3.1~5.2 2.0~2.5 2 Dolomite 54.8~81.3 28.3~40.5 4.0~5 1 Gabbro 200.3 101.1 7.1 9 Granite 132.1~224.4 19.1~81.0 3.7~6.0

12 Limestone 35.2~179.4 17.9~91.6 3.4~6.4 1 Marble 251.0 106.1 6.7 1 Peridotite 122.4 109.7 6.6 2 Quartzite 214.9~311.5 42.3~53.4 4.4~4.9 1 Rhyolite 96.1 35.7 4.1 5 Sandstone 10.6~40.9 1.9~5.9 1.3~2.1 2 Schist (foliated) 142.2~165.5 69.4~76.9 5.5~5.6 1 Serpentine 112.7 53.2 5.4 2 Slate (foliated) 85.4~172.3 65.9~71.4 5.2~5.5 1 Syenite 129.7 75.6 5.9 3 Taconite 233.7~304.8 61.6~97.0 5.0~6.1

41 Dinçer et al. (2004) Turkey 14 Andesite 18.3~25.1 38.5~112.7 7.8~18.3 4 Basalt 25.4~26.5 65~108 11.6~21.2 6 Tuff 17.5~19.6 32.9~52 5.1~9.1

42 Dinçer et al. (2008) Turkey 18 Caliche 17.3~22.9 16.2~32.5 2.7~10.4 0.2~1.4 0.4~1.6 43 Diamantis et al. (2009) Greece 32 Serpentine 19.2~125.7 4.8~5.8 44 Dehghan et al. (2010) Afghanistan 30 Travertine 1.0~10.3 22.7~71.5 3.1~11.5 4.8~5.8 45 Diamantis et al. (2011) Greece 35 Peridotite 65.2~241.6 26.4~69.3 7.1~8.0

46 Ersoy and Waller (1995)

South Africa 1 Diorite 375.2 37.4 United Kingdom 1 Granite 106.2 19.8

Italy 1 Limestone 28.2 9.7

United Kingdom 1 Sandstone 37.5 9.9 1 Siltstone 90.5 17.7

47 Ersoy et al. (2005) Turkey 1 Gabbro 29.9 0.01 172 22 4 Granite 25.3~25.7 0.2~0.5 134.1~161.5 14.8~18.5 1 Syenite 25.9 0.4 168 19.9

48, 49 Ersoy and Atici (2007) Atici and Ersoy (2009)

Turkey

2 Andesite 21.7~22.4 12.5~15.8 53~57.8 7.5~7.9 5.1~5.4 1 Breccia 25.3 1.2 48.4 15.2 5.6 1 Dacite 25.7 6.4 65.3 14.3 5.2 1 Gabbro 292

Spain 3 Granite 134~162 Turkey 4 Limestone 26.1~26.8 0.3~0.5 49.7~87.2 19.6~24.2 6.3~6.4

United Kingdom 1 Limestone 60 Turkey 1 Marble 26.6 0.7 85.4 28.6 6.4

United Kingdom 1 Mudstone 43 2 Sandstone 85~175 2 Siltstone 91

Turkey 1 Tuff 15.9 42.0 6.4 1.3 2.3

50 Efimov (2009) Siberian, Russia

1 Dolerite 29.4 379 112 6.0 2 Gabbro 26.5~29.4 189.5~290 62~80 4.8~5.5 3 Granite 25.5~25.7 129.9~176.6 40~70 3.8~5.6 1 Marble 26.5 80 18 6

51 Erguler and Ulusay (2009) Turkey 8 Marl 6.1~53.9

14 Mudstone 2.1~48.1

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6 Siltstone 5.3~42.1 3 Tuff 26.1~33.6

52 Endait and Juneja (2014) Maharashtra, India 41 Basalt 1.3~19.6 2.2~181.7

53 Franklin and Hoek (1970)

United Kingdom 1 Granite 0.4 190.9 Oregon, US 2 Limestone 12.1 59.3

Italy 1 Marble 0.1 92.4 United Kingdom 2 Sandstone 12.6~16.1 52.7~80.1

Alaska, US 1 Sandstone 0.2 197.0

54 Fener et al. (2007) Turkey 3 Limestone 128.8~175 6~6.1 5 Travertine 45.4~83.3 3.7~5.4

55 Fereidooni (2016) Hamadan, Iran 8 Hornfels 0.4~2.8 95.9~272.8 3.0~5.5 56 Ghosh and Srivastava (1991) Himachal, India 11 Granite 25~119

57 Güney et al. (2005) Turkey 5 Limestone 52.5~138.2 4.6~6.2 1 Marble 48.7 5.8 1 Travertine 58.8 5.4

58 Gorski et al. (2007) Sweden 20 Granite 25.8~26.1 4.9~5.2

59 Graue et al. (2011) Germany

1 Basalt 24.7 63.1 1 Latite 24.1 126.4 1 Muschelkalk 22.1 48.4 3 Sandstone 20.6~21.3 45.1~86.7 2 Trachyte 22.9~23.1 65.5~75.5

60 Gupta and Sharma (2012) Himalaya, India 18 Quartzite 25.5~27.7 0.3~1.5 46~141 1.5~5.5

61 ~67 Hatzor and Palchik (1997, 2000, 2002)

Palchik (2006, 2007, 2010, 2011) Israel

3 Chalk 23.7~30 31.9~37.4 9.5~11.7 29 Dolomite 2~29.3 32~274 6.1~82.3 40 Limestone 5.7~37.9 14~187.2 6.2~60.5

68 Huang and Wang (1997) US

1 Diorite 26.7 163.4 27.5 1 Granite 25.5 130.0 38.5 1 Granodiorite 25.6 153.2 36.2 1 Limestone 26.8 83.7 15.3 1 Sandstone 26.6 102.9 48.7 1 Schist (foliated) 26.5 73.1 9.8

69 Hawkins (1998) United Kingdom 8 Bath stone 9.8~14.1

24 Limestone 48.8~150.6 24 Sandstone 18.6~185.5

70 Horsrud et al. (1998) North Sea 11 Shale 3~55 0.8~12.2

71 Hecht et al. (2005) Germany 4 Conglomerate 23.2~25.4 21~135 3.3~4.6 2 Sandstone 23.1~23.7 54~88 3.1~3.5

72 Hoseinie et al. (2012) Iran 6 Granite, Limestone, Monzonite, Sandstone,

Syenite, Travertine

73 Heidari et al. (2013) Hamadan, Iran 20 Sandstone 3.4~17.5 44.9~92.5 6.1~12.4

74 Hoseinie et al. (2014) East Azerbaijan, Iran

1 Granite 26 87.5 60.8 2 Limestone 25.4~27.4 40~51 44.1~58.8 1 Monzonite 25.3 57 51.2 1 Sandstone 22.0 14 9.3 2 Travertine 24.1~25.0 50.5~53 48.8~70.2

75 Hasancebi (2016) Turkey 32 Basalt 3.6~6.1 24.0~172.8

76 Hedtmann and Alber(2016) Germany 1 Andesite 26.5 1 156 33.9 1 Rhyolite 25.8 0.9 216 36.5 1 Sandstone 23.8 8.8 90 15.9

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1 Slate (foliated) 27.0 1.2 164 23.3 77 Ng et al. (2015) Macao, China 145 Weathered Granite (Grade III; ISRM) 20.3~112.9 1.2~5.9

78 Jaeger (1966) Italy 1 Marble

NSW, Australia 1 Sandstone 1 Trachyte

79 Jizba (1991) Texas, US 126 Sandstone, Shale 23.9~27.6 0.7~20.0

80 Jeng et al. (2004) Taiwan 12 Graywacke 11.5~20.7 19.9~86 5~14 1 Quartzwacke 24.6 14.5 2.6

81 Jeong and Jeon (2016) Shandong, China 1 Sandstone 23.1 8.2 64 10.5 2.3

82 Kasim and Shakoor (1996)

Pennsylvania, US 1 Diabase 181 Ontario, Canada 2 Dolomite 49.1~179.5

Indiana, US 2 Dolomite 34.2~134.8 Virginia, US 2 Granite 133.5~136.5

Delaware, US 1 Limestone 152 New York, US 1 Limestone 209.6

Connecticut, US 1 Limestone 156.2 Indiana, US 1 Limestone 108.4

New York, US 1 Marble 184.2 Maryland, US 1 Marble 104.9 Alabama, US 1 Marble 91.1 Kentucky, US 1 Sandstone 57.3 Illinois, US 2 Sandstone 136.8~144.2

Massachusetts, US 1 Sandstone 34 New York, US 2 Sandstone 129~162.4 Maryland, US 1 Schist (foliated) 137.9

Ontario, Canada 1 Syenite 151 83 Khaksar et al. (1999) Australia 22 Sandstone 20.9~25.1 2.6~16.6 84 Koncagül and Santi (1999) Kentucky, US 31 Shale 30.7~99.5

85 ~87 Kahraman et al. (2000)

Kahraman (2001) Kahraman et al. (2003)

Turkey

1 Diabase 29.0 110.9 10.9 5.2 3 Dolomite 28.7~29.2 68~189.8 6.8~30.2 5.1~6.3

16 Limestone 18.3~29.0 15.7~152.7 0.8~22.4 2.2~5.6 21 Marl 16.3~24.1 4.4~82.4 0.2~4.8 1~3.4 1 Metasandstone 26.8 25.7 10.6 5.2 5 Sandstone 25.0~29.4 20.1~149.2 1.6~13.9 2~5.2 2 Serpentine 25.8~28.3 54.3~69.1 20.2~21.1 2.9~5 1 Tuff 18.2 10.1 0.2 1.2

88 Koçkar and Akgün (2003) Turkey 12 Limestone 25.8~26.6 28~117 23~61 15 Schist (foliated) 26.2~27.1 3~104 1~88

89 Kahraman et al. (2004) Turkey 5 Limestone 79.5~175 6~6.2 8 Travertine 45.4~112.3 3.7~5.5

90 ~95

Kahraman et al. (2005) Kahraman and Gunaydin (2007)

Kahraman (2007) Kahraman and Gunaydin (2008) Kahraman and Gunaydin (2009) Kahraman and Toraman (2016)

Turkey

1 Amphibolite 27.4 1.05 3.7 6 Andesite 25.3 1.2~10.7 65.8~164.1 5.1 2 Basalt 25.4 2.1~5.5 202.9 4.4 1 Gneiss (foliated) 26.3 0.5 4.0

10 Granite 25.0~26.4 0.5~1.2 84.9~133.2 4.4~5.6 Spain 6 Granite 25.2~26.1 0.4~3.6 90~120.3 3.9~4.8

Turkey 1 Granodiorite 2.5 109.2 9 Limestone 26.1~26.4 0.2~2.6 60.8~147.1 5.1~5.6

19 Marble 25.1~26.8 0.1~0.4 24.1~90.5 3.4~5.6

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1 Metagabbro 0.7 115.4 2 Migmatite 27.1 0.7~1.3 203.6 5.5 2 Quartzite 26.5 0.4~0.9 111.5 5.2 4 Sandstone 23.8~24.5 1.4~3.6 120.3~168.6 4.7~4.9 3 Schist (foliated) 26.4 1.0~2.0 70.9~186.5 3.4 2 Serpentine 26.6 0.3~0.9 210.6 5.5

12 Travertine 18.0~24.2 1.2~15.5 39.8~87.8 4.2~5.2 2 Volcanic bomb 23.7 3.2~3.8 50.2 4.1

96 Kahraman and Alber (2006) Germany 24 Breccia 9.8~86.6 3~16.8

97 Karakus and Tutmez (2006) Turkey

1 Amphibolite 89.1 6.2 1 Dacite 90.2 3.5 5 Limestone 11.5~58.0 3.5~5.8 1 Marble 75.1 4.9

98 Khandelwal and Singh (2009)

Chhattisgarh, India 1 Coal 18.1 16.4 2.0 1.8 Jharkhand, India 1 Coal 18.9 20.6 2.4 1.9

Maharashtra, India 1 Coal 21.1 54.7 3.2 2.1 Madhya, India 1 Coal 17.2 8.2 1.5 1.7

Chhattisgarh, India 1 Sandstone 18.3 17.3 2.2 1.9 Jharkhand, India 1 Sandstone 19.4 32.1 2.8 2.0

Maharashtra, India 1 Sandstone 21.1 58.3 3.4 2.1 Madhya, India 1 Sandstone 17.7 11.4 1.7 1.8

Chhattisgarh, India 1 Shale 17.9 15.8 1.8 1.9 Jharkhand, India 1 Shale 18.5 25.3 2.3 1.9

Maharashtra, India 1 Shale 21.0 57.2 3.4 2.1 Madhya, India 1 Shale 17.5 10.7 1.5 1.8

99 Khandelwal and Ranjith (2010)

Rajasthan, India 1 Dolomite 3.3 1 Granite 4.4

Gujarat, India 1 Limestone 3.0 Rajasthan, India 1 Limestone 6.2 Madhya, India 1 Limestone 3.2

Rajasthan, India 3 Marble 2.4~3.7 Uttar, India 1 Quartzite 5.2

Rajasthan, India 3 Sandstone 2.7~3.0 Jharkhand, India 1 Shale 1.5

100 Klanphumeesri (2010) Thailand 10 Limestone 10 Marble 10 Sandstone

101 Khanlari and Abdilor (2011) Hamadan, Iran 40 Limestone 25.4~27.1 56.2~104.0 3.5~6.7

102 Kumar et al. (2011) Karnataka, India

1 Chalk 3.3 21.3 1 Ironstone 0.3 83.2 2 Limestone 0.7~4.6 17.2~71.8 1 Marl 1.2 58.3 1 Sandstone 1.1 62.2 1 Shale 5.5 15.2

103 Kurtulus et al. (2011) Turkey 20 Serpentine 32.7~114.3 3.4~5.4 4.1~5.3 104 Khalily et al. (2013) Kopet Dagh, Iran 16 Limestone

104 ~106

Karaman et al. (2015) Karaman and Kesimal (2015a) Karaman and Kesimal (2015b)

Turkey 2 Andesite 134.5~173 4.7~5

13 Basalt 34~197 3.0~5.9 1 Breccia 41 4.2

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12 Dacite 56~132 3.9~5.0 1 Granite 215

10 Limestone 7.7~120 1.4~5.4 8 Metabasalt 66~158 3.9~5.0 1 Volcanic Breccia 41

107 Kassab and Weller (2015) Egypt 42 Sandstone 20.1~34.7 2.3~3.4 108 Kurtulus et al. (2015) Turkey 32 Limestone 1.9~2.9 19~46 38.3~69.1 3.5~5.9

109 Kahraman et al. (2016) Turkey 3 Claystone 14.1~34.3 1 Limestone 16.4

110 Kumari et al. (2016) Victoria, Australia 11 Granite 76.3~143.3 6.3~9.1

111 Larson et al. (1987) Hawaii, US 1 Phosphorite 18.7 32 32.6 10.1 3.5 1 Tuff 15.5 38 3.7 0.3 1.0

112 Liu et al. (2004) Sweden 5 Granite 217~260 57~68 113 Liu et al. (2014) Gansu, China 6 Granite 26.5~26.6

114 Manghnani and Woollard (1965) Hawaii, US

1 Amphibolite 28.9 96.3 6.8 5 Basalt 19.6~25.5 34.5~55.6 4.4~5.5 1 Eclogite 27.6 64.5 5.8 1 Hawaiite 25.4 40 4.2 1 Trachyte 25.5 54 5.2

115 Merriam et al. (1970)

California, US 1 Diorite 202.7 20.7 4 Gabbro 156.5~258.6 17.2~82.7

NSW, Australia 1 Granite 126.0 57.2

California, US 1 Granodiorite 104.1 62.1 1 Granodiorite 124.1 41.4 1 Monzonite 209.6 37.9

116 McVay et al. (1992) Florida, US 14 Limestone 1.1~9.8

117 Meulenkamp and Grima (1999) Netherlands

3 Dolomite 75.2~119.8 4 Granite 186.8~262 3 Granodiorite 256.6~274.6

14 Limestone 62~168.8 1 Mudstone 6.6 8 Sandstone 35.4~91.7

118 Moradianl and Behnia (2009)

Gilan, Iran 27 Limestone 25.7~27.2 40.7~143.1 13.7~90.5 1.8~6.5 Harasan Dam, Iran 2 Limestone 23.3~25.1 39.8~64.8 22.7~23.3 3.4~4.1 Meymeh Dam, Iran 6 Limestone 23.6~26.8 46.1~99.1 6.6~79.5 2.8~6.2

Tang Sorkh Dam, Iran 9 Limestone 20.6~28.7 10.2~53.4 1.1~53.3 1.8~5.7 Mamlo Dam, Iran 8 Marl 22.9~23.9 17.3~48.7 4.0~7.8 1.8~2.7 Sarni Dam, Iran 4 Marl 21.8~23.1 11.3~23.9 3.2~5.0 2.4~2.5

Siabisheh Dam, Iran 8 Sandstone 20.0~22.8 10.4~44.2 0.8~9.3 2.5~3.9 119 Manouchehrian et al. (2012) Nepal 44 Sandstone 1~12 1.3~51.6

120, 121 Mishra and Basu (2012) Mishra and Basu (2013)

Madhya, India 20 Granite 26.5~27.2 0.1~0.4 91.5~201.7 5.4~6.3 20 Sandstone 21.3~25.5 2.9~15.5 12.8~172.0 2.7~5.0 20 Schist (foliated) 26.9~28.6 0.2~0.5 21.4~95.1 5.1~6.3

122 Mahmoud (2013) Saudi Arabia 30 Sandstone 26.0~36.2 123 Marques et al. (2013) Minas Gerais, Brazil 82 Metagranite 13.9~243.1

124 Mikaeil and Ataei (2014) West Azerbaijan, Iran 5 Granite 125~173 4 Marble 68~74.5 3 Travertine 53~63

125 Mohamad et al. (2015) Malaysia 24 Weathered Shale (Grade II~IV; ISRM) 20.5~34.7 5.5~61.1 1.3~2.9

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126 Mustafa et al. (2015) Pakistan 5 Granite 17~82 5 Marble 45~66 5 Schist (foliated) 17~112

127 Mendive et al. (2016) Uruguay

1 Breccia 27.6 41 1 Carbonate rock 27.9 80 1 Diorite 27.4 99.5 3 Granite 25.8~27.5 63.2~108.8 3 Monzonite 26.5~27.0 24.5~73.9 1 Mylonite 27.2 139.3 1 Trachyte 26.6 19.2

128 Nicksiar and Martin (2012) Sweden 10 Diorite 171~294 72~80 129 Nazir et al. (2013a) Malaysia 20 Limestone 52.2~85.6 130 Nazir et al. (2013b) Malaysia 20 Limestone 21.2~100.7 131 Nefeslioglu (2013) Turkey 66 Claystone, Mudstone 16.3~20.1 0.7~4.1 0.03~1.0 0.5~1.4

132 Ozcelik et al. (2004) Turkey 2 Limestone 26.3~26.4 59.4~108.1 19.1~21.5 6 Marble 26.4~26.9 28.7~65.6 9.9~21.1

133 Okewale and Olaleye (2013) Nigeria 10 Limestone 25.9~27.0 59.4~77.2

134 Ozturk and Nasuf (2013) Turkey

1 Diabase 110.9 2 Granite 106.3~114.5

12 Limestone 27.9~175 1 Marble 134.2 2 Marl 21.4~39.5 1 Mudstone 146.5 3 Sandstone 70.5~154.1 1 Serpentine 54.3 1 Shale 82.9 2 Siltstone 55.1~104.1 7 Travertine 50.3~112.3

135 Palchik (1999) Turkey 15 Sandstone 27.5~47.2 7.1~19.8 1.4~2.5 136 Prikryl (2001) Germany 13 Granite 0.9~4.5

137 Prakoso (2002)

Unknown 2 Andesite 10.4~10.4 Massachusetts, US 4 Argillite (foliated) 4.5~33.2

Hong Kong 1 Granodiorite 6.5 Florida, US 3 Limestone 4.4~25.1

Ontario, Canada 1 Limestone 74 Hong Kong 1 Metasandstone 26

Victoria, Australia 38 Mudstone 0.4~8.6 Japan 1 Mudstone 0.6

South Africa 1 Mudstone 1.1 United Kingdom 1 Mudstone 6.3

South Africa 1 Quartzite 231 NSW, Australia 5 Sandstone 6~30

South Africa 2 Sandstone 2.5~18 Unknown 2 Sandstone 18~142

Colorado, US 3 Shale 0.8~3.8 Connecticut, US 1 Shale 68.6 New York, US 1 Shale 0.8 NSW, Australia 3 Shale 6.8~32

Japan 2 Shale 0.6~1.4

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NSW, Australia 1 Shale 33.3 South Africa 1 Shale 182

Unknown 7 Shale 0.5~20.7 Northern Territory, Australia 1 Siltstone 1

Unknown 1 Siltstone 18.0 Naples, Italy 1 Tuff 4.0

138 Palchik and Hatzor (2004) Israel 50 Chalk 18.9~43.9 20.9~63.3 9.3~20.5 139 Pappalardo (2015) Italy 25 Dolomite 25.2~27.5 2.4~10.4 10.2~112 2.38~18.8 3.3~6.3

140 Pittino et al. (2016) Austria 13 Gneiss (foliated) 2.3~3.5 3 Schist (foliated) 1.2~1.4

141 Qu and Zheng (2015) Shanxi, China 1 Coal 15.0 60.6 6.1

142 Ramana and Venkatanarayana (1973) Mysore, India

2 Basalt 26.2~30.0 7~9 62.4~104.7 5.0~6.6 4 Diabase 29.4~31.0 3 59.1~105.2 6.0~6.7 6 Dolerite 29.4~30.8 4~8 62.5~109.3 5.2~6.5 5 Granulite 29.8~32.9 6~7 82.9~102.5 5.4~6.7 2 Hornblendite 31.9~34.1 3~10 97.9~99.2 5.6~6.4 1 Quartzite 27.4 5 58 5.2 9 Schist (foliated) 26.9~31.4 4~8 92.7~116.3 5.8~6.7

143 Reddish and Yasar (1996) United Kingdom

2 Limestone 39~63.2 7.3 1 Mudstone 19 6.6 2 Sandstone 72~80 10.5 1 Siltstone 70 9.0

144 Rajabzadeh et al. (2012) Fars, Iran 25 Limestone 0.2~9.8 32.9~138.6 7.3~17.4 24 Marble 0.2~2.9 43.1~101.8 5.9~15.9

145 Rahmouni et al. (2013) Morocco 6 Calcarenite 25.7~35.8 3.6~3.8

146 Schmidt (1972)

Wisconsin, US 1 Argillite (foliated) 26.8 216.5 52.4 5.9 1 Basalt 29.3 281.3 90.3 6.7

Minnesota, US 1 Diabase 29.3 367.5 80.7 6.0 1 Dolomite 25.7 95.2 5.3

Minnesota, US 2 Gabbro 28.0~28.6 182.7~204.1 63.4~88.9 5.9~6.7 Wisconsin, US 1 Gabbro 28.7 172.7 102.0 6.4

Minnesota, US 3 Granite 26~26.1 151.7~199.6 64.1~67.6 5.4~5.6 1 Limestone 24.3 97.9 42.8 5.1

Wisconsin, US 1 Marble 28.0 125.1 80.0 5.8 Michigan, US 1 Pegmatite 25.8 87.9 40.7 5.0 Minnesota, US 2 Quartzite 25.6~25.8 153.4~301.3 40.0~64.8 5.1~5.3 Wisconsin, US 1 Quartzite 25.9 218.2 72.4 5.3 Michigan, US 1 Schist (foliated) 29.3 204.1 100.7 6.3

Minnesota, US 3 Taconite 30.1~33.0 354.0~361.3 89.6~108.3 5.8~6.1 1 Trap Rock 26.3 67.6 58.6 5.7

147 Sachpazis (1990) Greece

2 Dolomite 105~188 49~71 15 Limestone 22~157 8~67 12 Marble 78~121 29~61

United Kingdom 3 Limestone 81~311 34~71 1 Marble 88 57

148 Sousa et al. (2005) Portugal 9 Weathered Granite (Grade I~IV; GSE-GWPR) 62.4~158.5 2.3~5.8

149 Shalabi et al. (2007) Illinois, US 21 Dolomite 19.5~23.9 21.4~96.6

Michigan, US 8 Dolomite 23.9~26.7 37.2~127 19.1~80.7

14 Limestone 20.4~25.9 19.9~109.9

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New York, US 1 Limestone 27.6 109 64.8 1 Marble 28.4 59.3 50.3

Puerto Rico, US 11 Shale 24.5~26.6 11.2~55.1 16.2~41.1

150 Sharma and Singh (2008)

Deccan Trap, India 4 Weathered Basalt (unknown grade) 58~72 2.7~3.0 Madhya, India 5 Coal 8.2~9.5 2.0~2.1

Rampur-Wantung Series, India 8 Phyllite (foliated) 4.4~5.2 1.9~2.0 Lower Siwalik, India 17 Sandstone 22~52 2.1~2.6

Rampur-Wantung Series, India 4 Schist (foliated) 19.6~24.8 2.2~2.3 Jharkhand, India 10 Shale 22~28 2.1~2.3

151 Sonmez and Tunusluoglu (2008) Turkey

3 Andesite 64.5~77.8 1 Conglomerate 28.8 1 Diabase 105.6 1 Granodiorite 135.4 2 Greywacke 58.5~67.3 6 Limestone 22~94.4 1 Marble 22.2 6 Marl 4.1~94.3 2 Mudstone 43.7~48.4 1 Peridotite 64.6 2 Sandstone 30.6~46.3 3 Schist (foliated) 5.2~65.0 1 Serpentine 54.5 1 Syenite 115.7 1 Travertine 35.8 2 Tuff 7.3~10.0

152 Sarkar et al. (2010) Himachal, India

10 Limestone 24.5~26.5 68.4~84.5 2.6~3.9 10 Quartzite 25.8~26.5 93.2~112.3 3.5~4.2 10 Schist (foliated) 25.7~26.4 20.3~28.5 2.1~2.5 10 Slate (foliated) 25.9~27.6 24.2~49.3 3.6~4.3

152 Sarpün et al. (2010) Turkey 14 Volcanic rock 23.3~29.1 13.2~57.1 18.4~47.1 3.3~4.7

153 Sharma et al. (2011) India

5 Andesite 5.4~5.5 3 Basalt 5.4~5.8 4 Conglomerate 2.1~2.2 4 Gneiss (foliated) 3.6 6 Granite 5.0 3 Quartzite 3.6~3.8 2 Sandstone 2.2~2.5

15 Weathered Sandstone (unknown grade) 2.1~2.3 3 Schist (foliated) 2.4~2.6 6 Siltstone 2.2~2.3

154 Sarkar et al. (2012) India

11 Basalt 21.9~29.3 50.3~165.2 2.2~5.1 5 Weathered Basalt (unknown grade) 22.4~23.6 65.3~73.2 2.8~3.0 5 Coal 18.5~19.8 12.5~14.5 1.5~1.6 5 Dolomite 28.9~29.1 127.1~132.4 5.0~5.1 8 Gneiss (foliated) 23.7~28.1 70.8~143.8 2.6~4.9 3 Granite 27.0~29.0 120.7~141.2 4.2~4.8

17 Limestone 23.3~27.3 68.4~117.0 2.6~4.3 10 Quartzite 25.8~26.5 93.2~112.3 3.5~4.2 10 Sandstone 20.0~23.3 20.1~48.6 2.1~2.5

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10 Schist (foliated) 22.6~23.0 20.3~28.5 2.1~2.5 10 Shale 22.5~23.6 39.0~45.5 2.4~2.6

155 Singh et al. (2013) India 15 Limestone 20.8~31.3 29~61.8 2.2~3.7

156 Su et al. (2016) Turkey

1 Andesite 68 10.7 2 Limestone 56~70 7.5~22.9 2 Marble 68~71 21.5~24.4 3 Sandstone 57~75 8.52~20 1 Siltstone 81 14.8

157 Tamrakar et al. (2007) Nepal 44 Sandstone 1.3~51.6 0.1~1.1

158 ~ 160

Tiryaki (2006) Tiryaki and Dikmen (2006)

Tiryaki (2008) United Kingdom

2 Grit 37~58 9.3~18.7 8 Limestone 91.6~192.9 43.2~65.7 4 Mudstone 18~23 15.7~25.2

19 Sandstone 7~156 5.6~39.8

161 Tumac et al. (2007) Turkey

1 Limestone 121 57 3 Sandstone 87.4~173.6 17~33.3 1 Serpentine 38.1 2.3 1 Siltstone 57.9 30 5 Tuff 5.7~26.6 0.4~2.4

162 Torabi et al. (2010) Semnan, Iran 41 Sandstone, Siltstone, Shale 25~224 163 Török and Vásárhelyi (2010) Hungary 40 Travertine 22.0~25.4 30.3~124.3 3.8~5.5 164 Tahir et al. (2011) Pakistan 30 Limestone 26.6~61.8

165, 166 Tumac et al. (2016a,b) Turkey 7 Marble 26.2~26.5 63.8~81.5 23.5~31.7 5.0~7.0 167 Ulusay et al. (1994) Turkey 15 Sandstone 55~96 6.5~9.1

168 Unlu and Yilmaz (2014) Turkey

1 Andesite 1 Basalt 1 Limestone 1 Sandstone

169 Ündül et al. (2016) Turkey 23 Limestone 53.3~151.5 39.3~76.6 5.6~6.8

170 Verwaal and Mulder (1993) Netherlands

3 Dolomite 21.9~23.9 10.7~19.7 39~67 32~40 1 Granite 25.2 155 49

15 Limestone 19.0~26.6 0.5~37.9 22~203 9~80 1 Marble 26.5 0.4 94 49 5 Sandstone 19.9~26.5 0.8~27 31~198 9~54

172 Vásárhelyi (2005) Italy 45 Limestone 11.4~52.2 0.9~38.8 0.4~21.1 173 Vasconcelos et al. (2007) Portugal 19 Granite 26~159.8 11.0~63.8 1.9~4.8

174 Wagner and Schumann (1970) South Africa

1 Marble 178 43.7 1 Norite 274 100 1 Quartzite 205 75.5 1 Sandstone 103 30.8 1 Shale 185 70

175 Wijk (1989) Sweden 1 Granite 239 Norway 1 Marble 214 Sweden 1 Sandstone 81.7

176 Wang et al. (2015)

Anhui, China 8 Coal 3.2~8.5 1.6~2.1 Shanxi, China 4 Coal 2.2~2.4 1.7~2.1 Jiangxi, China 7 Coal 0.2~2.8 1.9~2.5 Xuecun, China 5 Coal 2~5.1 1.9~2.1

177 Xeisakis and Samaras (1996) Greek 13 Marble 26.2~27.4 0.03~1.0 45.3~111.4 178 Xue et al. (2015) China 7 Gneiss (foliated) 191~206 51.7~57.3

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179 Yagiz (2009)

Oregon, US 1 Andesite 25.8 178 Massachusetts, US 1 Argillite (foliated) 24.6 52

Maryland, US 1 Argillite (foliated) 25.7 95 Mexico 2 Basalt 22.9~26.5 73~129

Idaho, US 1 Basalt 23.6 57 Oregon, US 1 Basalt 26.9 205

New Zealand 1 Calc-slicate rock 28.3 177 Oregon, US 1 Coal 20.5 9.5

New York, US 1 Diorite 25.5 47 California, US 1 Gneiss (foliated) 27.8 227 Georgia, US 2 Gneiss (foliated) 26.2~26.9 151~190

New York, US 1 Gneiss (foliated) 27.6 68 Massachusetts, US 1 Granite 26.3 75

California, US 1 Granite 25.7 165 Colorado, US 1 Granite 26.4 165

New Jersey, US 2 Granite 26.8~27.4 166~327 South Korea 3 Granite 25.6~26.5 177~315 Colorado, US 2 Limestone 24.8~25.7 142~159 Indiana, US 1 Limestone 22.2 24 Nevada, US 1 Limestone 24.5 106

New York, US 1 Limestone 25 155 Ontario, Canda 1 Limestone 22.5 64

Pennsylvania, US 1 Limestone 24.6 78 Tennessee, US 1 Limestone 26.5 143

Texas, US 1 Limestone 23.8 71 Colorado, US 1 Marble 26.2 180

New Zealand 1 Metaandesite 27.5 129 1 Metadolorite 28.7 76

Colorado, US 1 Mudstone 26 150

New Zealand 2 Orthogneiss (foliated) 27.6~28.9 58~163 1 Paragneiss (foliated) 27.4 110

Portland, Oregon, US 1 Quartzite 26.4 182 Colorado, US 3 Sandstone 21.4~23.5 21~120 New York, US 1 Schist (foliated) 27.9 82

Colorado, US 2 Shale 20.7~23.3 57~109 1 Siltstone 22.1 82

New York, US 1 Syenite 26.3 188

180 Yu et al. (2015) Qinghai, China 2 Limestone 24.5~25.5 20.3~21.6 4.4~4.7 8 Sandstone 20.4~25 8.1~18.0 2.3~4.7

181 Yilmaz and Citiroglu (2016) Turkey 1 Gneiss (foliated) 39.25 6.4 1 Limestone 46.25 7.1

182 Zamora et al. (1994) Italy 2 Latite 25.8~26.7 3.7~4.5 7 Tuff 19.6~25.2 3.1~4.7

183 Zhao and Li (2000) Singapore 23 Granite 25.7~56.1 184 Zarif and Tuĝrul (2003) Turkey 20 Limestone 1.1~4.1 74.2~138.1 16.8~46.8 3.2~6.8

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Figure 1 Two Is50-σc models and their calibration databases.

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Figure 2 Is50-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 3 Transformation uncertainty (data scatter) resulting from pairwise correlation between a measured property and a desired design property (after Ching et al. 2017b).

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Figure 4 Data points in ROCK/9/4069 with simultaneous information for RN and RL.

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Figure 5 σc-Et50 and σc-Eav data points in ROCK/9/4069.

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Figure 6 Correlation behaviors among intact rock properties for rocks with known and unknown weathering grades.

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Figure 7 Correlation behaviors among intact rock properties for foliated and non-foliated rocks.

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Figure 8 n-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 9 RL-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 10 Sh-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 11 σbt-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 12 Vp-σc models for different rock classes and data points in ROCK/9/4069.

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Figure 13 RL-E models for different rock classes and data points in ROCK/9/4069.

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Figure 14 Sh-E models for different rock classes and data points in ROCK/9/4069.

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Figure 15 σc-E models for different rock classes and data points in ROCK/9/4069.

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Figure 16 Vp-E models for different rock classes and data points in ROCK/9/4069.

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Figure 17 Prediction results for the 24 cases in Singapore.

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Documents

Generic transformation models for some intact rock properties