ORIGINAL PAPER

Assessment of empirical equations for the compression indexof fine-grained soils in Missouri

Site Onyejekwe • Xin Kang • Louis Ge

Received: 10 February 2014 / Accepted: 7 August 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract The use of correlations and empirical relation-

ships in geotechnical engineering provides a fast, cost-

effective means of predicting the value of a parameter

based on the values of certain other, possibly more easily

determined, parameters. The correlation between two or

more soil properties has been found to be dependent in

varying degrees on soil type, the testing method used to

obtain the numerical value of the parameter itself and the

homogeneity of the soil. Many empirical correlations

among soil properties have been published. These corre-

lations, based on widely sourced data, may not be appro-

priate for local situations. Hence, there is a need for

correlations that are based on local data. This paper eval-

uated the validity of published empirical equations for the

index of fine-grained soils in Missouri, USA. Four indices

were used in the assessment including the root mean square

error, the ratio of the estimated to laboratory-determined

compression index, the ranking index and the ranking

distance. Results reveal the overall best correlations for the

Southeast Region and ‘‘Other Regions’’ of Missouri are

given by Azzouz et al. (Soils Found 16:19–29, 1976).

Keywords Fine-grained soils � Compression Index �Index properties � Correlations � Ranking distance �Ranking Index � RMSE

Introduction

Correlations and empirical relationships are used exten-

sively in geotechnical engineering. The use of correla-

tions and empirical relationships provides a fast, cost-

effective means of predicting the value of a parameter

based on the values of some other, possibly more easily

determined, parameters provided that the appropriate

correlations are employed. Generally, the more easily

obtained parameters are correlated to the difficult-to-

obtain parameters. The correlation between two or more

soil properties has been found to be dependent in varying

degrees on soil type, the testing method used to obtain

the numerical value of the parameter itself and the

homogeneity of the soil (Uzielli et al. 2007). Many cor-

relations between soil properties have been published. A

publication by Kulhawy and Mayne (1990) presents over

50 such correlations.

Knowledge of the consolidation properties of a soil is

important in geotechnical design, particularly as they relate

to settlement of structures. These properties are determined

from the standard consolidation test, i.e., incremental

loading test (ASTM D2435-11 2011) or constant rate of

strain test (ASTM D4186-12 2012). The incremental

loading test is the predominantly test method used. Using

either test method, consolidation properties are usually

determined in terms of the compression index, Cc and the

coefficient of consolidation, Cv. Cc is used to predict the

magnitude of settlement and Cv, a rate parameter, is used to

predict the rate of settlement (i.e., time required for an

S. Onyejekwe

Road Sector Development Team, Federal Ministry of Works,

Abuja, Nigeria

e-mail: sonyejekwe@yahoo.com

X. Kang

Department of Civil, Architectural, and Environmental

Engineering, Missouri University of Science and Technology,

Rolla, MO 65409, USA

e-mail: xkb4c@mst.edu

L. Ge (&)

Department of Civil Engineering, National Taiwan University,

Taipei, Taiwan

e-mail: louisge@ntu.edu.tw

123

Bull Eng Geol Environ

DOI 10.1007/s10064-014-0659-8

amount of settlement to occur). However, the cost of

consolidation testing is relatively high compared to other

common engineering tests and may be considered cost-

prohibitive for very small projects. Hence it is cost effec-

tive to develop correlations between consolidation prop-

erties and other easily obtained properties, like the index

properties.

In view of the cost implications of the consolidation test,

investigators have correlated the consolidation properties

of soils and other easily and cheaply obtained properties,

like the index properties. Skempton (1944) presented the

first well-known correlation between Cc and liquid limit,

LL for remolded soils:

Cc ¼ 0:007 ðLL� 10Þ ð1Þ

Many correlations between consolidation properties and

index properties have been developed by several investi-

gators based upon this first correlation (e.g., Cozzolino

Table 1 Some correlations

between index properties and

consolidation parameters

NC normally consolidated

* Identifier for correlationsreviewed in this paper

Correlation * Applicability References

Cc = 0.01wn A All clays Koppula (1981)

Cc = 0.01(wn - 7.549) B Clays Rendon-Herrero (1983)

Cc = 0.0115wn C Organic silt and clays Bowles (1979)

Cc = 0.01(wn - 5) D All clays Azzouz et al. (1976)

Cc = 0.006(LL - 9) E All clays with LL \ 100 % Azzouz et al. (1976)

Cc = 0.008(LL - 12) F All clays Sridharan and Nagaraj (2000)

Cc = 0.009(LL - 10) G All clays Terzaghi and Peck (1967)

Cc = 0.063(LL - 10) Egyptian clay Mahmoud and Abdrabbo(1990)

Cc = 0.048(LL - 10) Brazilian clays Bowles (1979)

Cc = 0.007(LL - 10) Remoulded clays Skempton (1944)

Cc = 0.0046(LL - 9) Brazilian clays Cozzolino (1961)

Cc = 0.009(LL - 10) NC clays Terzaghi and Peck (1967)

Cc = 0.009(LL - 8) Osaka Bay clay Tsuchida (1991)

Cc = 0.009LL Tokio Bay clay Tsuchida (1991)

Cc = 0.008(LL - 12) All clays Sridharan and Nagaraj (2000)

Cc = 0.007(SI ? 18) All clays Sridharan and Nagaraj (2000)

Cc = 0.014(PI ? 3.6) H All clays Sridharan and Nagaraj (2000)

Cc = PI/74 I All Clays Wroth and Wood (1978)

Cc = 0.29(e0 - 0.27) J Inorganic soils Hough (1957)

Cc = 0.35(e0 - 0.5) K Organic soils Hough (1957)

Cc = 0.156e0 ? 0.0107 L All clays Bowles (1979)

Cc = 1.15(e - e0) All clays Nishida (1956)

Cc = 0.29(e0 - 0.27) Inorganic soils Hough (1957)

Cc = 0.35 (e0 - 0.5) Organic soils Hough (1957)

Cc = 0.246 ? 0.43 (e0 - 0.25) Motley clays: Sao Paulo,Brazil

Cozzolino (1961)

Cc = 1.21 ? 1.055 (e0 ? 1.87) Lowland of Santos, Brazil Cozzolino (1961)

Cc = 0.75(e0 - 0.5) Soils with low plasticity Sowers (1970)

Cc = 0.208e0 ? 0.0083 Chicago clays Bowles (1979)

Cc = 0.156e0 ? 0.0107 All clays Bowles (1979)

Cc = 0.2343(LL/100) Gs M All clays Nagaraj and Murty (1985)

Cc = 0.2926(LL/100) Gs N All clays Nagaraj and Murty (1985)

Cc = 0.5 Gs(PI/100) O All Clays Wroth and Wood (1978)

Cc = 0.009wn ? 0.005LL P All Clays Koppula (1981)

Cc = 0.037(e0 ? 0.003LL - 0.34) Q Clays: Greece, parts of USA Azzouz et al. (1976)

Cc = -0.156 ? 0.411e0

? 0.00058LLR All Clays Al-Khafaji and Andersland

(1992)

Cc = 0.048(e0 ? 0.001wn - 0.25) Clays: Greece, parts of USA Azzouz et al. (1976)

Cc = 0.37(e0 ? 0.003LL? 0.0004wn - 0.34)

Clays: Greece, parts of USA Azzouz et al. (1976)

S. Onyejekwe et al.

123

1961; Terzaghi and Peck 1967; Azzouz et al. 1976; Ogawa

and Matsumoto 1978; Bowles 1979, 1996; Mahmoud and

Abdrabbo 1990; Carrier 1985; Sridharan and Nagaraj

2000). The compressibility characteristics of a soil can be

correlated to other soil properties such as natural water

content (wn), dry unit weight of soil (d), natural void ratio

(e0), liquid limit (LL), plastic limit (PL), plasticity index

(PI), liquidity index (LI), activity (ACT), the passing per-

centage of 2 lm fines and the specific gravity of soil par-

ticles (Gs), among others; hence, numerous correlations

between consolidation properties and these and other

properties have been developed. The choice of the property

for correlation seems to be dependent on the test soil, test

conditions (including the data availability) and type of

analyses executed. Azzouz et al. (1976) presented several

correlations between Cc and Cr and index properties.

Investigators such as Giasi et al. (2003), Sridharan and

Nagaraj (2000), Ozer et al. (2008) and others present

several correlations between Cc and index properties. Some

of these correlations are presented in Table 1.

The rather large number of correlations and the

equally large number of different parameters considered

in these correlations indicates variability exists when

these correlations are used. This may be due to corre-

lations being developed based on site-specific data and/or

widely sourced data. Hence, the use of these correlations

may not result in efficient, cost-effective outcomes in a

local situation. Consequently, this paper assesses the

validity of some of these correlations between the

compression index and index properties for fine-grained

soils in Missouri.

Data

Data for this study were obtained principally from the

Missouri Transportation Geotechnics Research Program

(MTGRP) and Missouri Department of Transportation

(MoDOT) geotechnical database. Data was obtained from

laboratory tests on undisturbed samples from five loca-

tions: North Kansas City (Clay County); Warrensburg

(Johnson County); New Florence (Montgomery County);

St. Charles (St. Charles County); and Hayti (Pemiscot

County). The geology of these locations is described as

follows. The Warrensburg site (located in the Western

Plains geological region) is generally composed of soft,

silty soils overlying shale, sandstone or limestone bed-

rock. The St. Charles site, located in the Glaciated Plains

geological region, is an alluvial site located within the

floodplain between the Mississippi and Missouri Rivers

near the confluence of those rivers. The subsurface is

composed of approximately 11 m of soft, fat clay over-

lying alluvial sands. The New Florence site, located in

the Glaciated Plains geological region, is an ‘‘upland

soil’’ site composed of relatively stiff clay overlying

sandstone, limestone and/or shale bedrock. The Pemiscot

site, located in the Southeast Lowlands geological region,

is an alluvial site composed predominantly of soft, fat

clays and located within the flood plain of the Missis-

sippi River. The subsurface is composed of approxi-

mately of 8–21 m of soft clay overlying alluvial sands.

Locations for field exploration are shown in Fig. 1.

Index test and compression test data used in this study is

a subset of a larger laboratory test database that includes

strength test data. All tests were conducted in accordance

with the provisions of the applicable ASTM standards:

Natural moisture content (ASTM D2216-10 2010); liquid

limit, plastic limit, and plasticity index (ASTM D4318-10

2010); specific gravity (ASTM D854-10 2010); and con-

solidation—plurality, incremental loading test (ASTM

D2435-11 2011); others, constant rate of strain test (ASTM

D4186-12 2012). Data was collated and validated. Data

validation entailed the investigation of identified outliers so

as to exclude only true outliers from the analysis.

Considering the very different characteristics of the

Pemiscot site which, due to the high compressibility of the

soils in the area, is considered atypical soil in Missouri, the

geotechnical parameters from Pemiscot (Southeast Region)

site were analyzed separately while data from the other

sites (other regions) were grouped together.

Descriptive statistics of the parameters in this paper and

their frequency distributions are presented in Table 2 and

Figs. 2 and 3, respectively.

Fig. 1 Major geologic regions of Missouri (Saville and Davis 1962)

Assessment of empirical equations for the compression index

123

Analysis

Existing correlations developed from widely sourced

data may not be appropriate for local conditions.

Uncritical adoption of these correlations could possibly

introduce a great deal of error and subsequent uncer-

tainty in the estimated values of undrained shear

strength would result either in inadequate, unsafe

designs or overly-conservative, inefficient designs

(Figs. 4, 5).

Fig. 2 Histogram of data—Other regions

Fig. 3 Histogram of data—Southeast region

Table 2 Descriptive statistics

of variables used in the study

wn natural water content, e0

initial void ratio, LL liquid limit,

PL plastic limit, PI plasticity

index, Cc compression index,

Min minimum, Max maximum,

SD standard deviation

Parameter Others Southeast

Min Max Mean SD Count Min Max Mean SD Count

wn (%) 11.4 67.2 30.7 9.96 189 25.6 81.4 48.0 14.47 113

LL (%) 23.8 108 50.2 21.4 159 23.0 115.0 66.7 25.59 90

PL (%) 13.0 47 21.2 5.8 158 5 42.2 27.5 6.51 90

PI 1.0 80 28.5 17.8 157 0 82.0 38.8 21.81 91

e0 0.429 1.801 0.901 0.277 184 0.655 2.287 1.422 0.398 83

Cc 0.073 0.821 0.265 0.131 180 0.085 1.078 0.484 0.238 103

S. Onyejekwe et al.

123

Fig. 4 Estimated (Cccal) versus measured (Cclab

) Compression index—Other regions

Assessment of empirical equations for the compression index

123

Analysis in this paper is focused on assessing the

uncertainty associated with applying the existing correla-

tions to the data on fine-grained soils from Missouri.

Uncertainty in geotechnical properties can be formally

grouped into two classes: aleatory and epistemic uncer-

tainty (Lacasse and Nadim 1996; Whitman 1996; DNV

2007). Aleatory uncertainty represents the natural ran-

domness of a property and hence is a function of the spatial

variability of the soil property. Epistemic uncertainty

results from lack of information and shortcomings in

measurement and/or calculation; for example, systematic

error resulting from factors such as the methods of property

measurement, quantity of available data and modeling

errors. Analysis in this case is aimed at model-related

epistemic uncertainty.

Table 1 shows a preponderance of linear correlations

based on the value of a single index property. Some of the

most commonly used correlations are those based on liquid

limit, one of the most easy-to-obtain index properties.

Compressibility and liquid limit of a soil can be correlated

in that they both relate to the nature and mineralogical

composition of the soil. LL, the moisture content at which a

soil becomes a semi-solid, is a measure of the quantity of

water attracted by the soil particles. Compressibility is

related to the size of its constituent particles and hence its

mineralogy.

The reliability of LL-based correlations has been ques-

tioned by some investigators (Cherubini 1991; Al-Khafaji

and Andersland 1992; Sridharan and Nagaraj 2000). Soils

having the same liquid limit but different plastic and

shrinkage limits have been shown to have a different

compressibility (Sridharan and Nagaraj 2000). It is

expected that the liquid limit does not fully represent the

plasticity characteristics of soil. In order to use a more

representative plasticity parameter and improve the reli-

ability, investigators have developed correlations based on

the plasticity (Wroth and Wood 1978; Sridharan and

Nagaraj 2000) and shrinkage (Sridharan and Nagaraj 2000)

indices. The shrinkage index (SI) is the difference between

the shrinkage limit and the liquid limit (SI = SL–LL).

Sridharan and Nagaraj (2000) showed that the shrinkage

index yields the best correlation with compressibility

characteristics. However, the SI was not determined in this

study.

Table 1 also shows a number of correlations based on

initial void ratio (e0). The initial void ratio is generally used

to define the compressibility of a soil. The initial void ratio

is representative of the in situ state of the soil and also

describes the structure of the soil through its geologic

history. Unlike the void ratio, liquid limit and plasticity

properties do not take into account the stress history of the

soil. Hence, an ideal correlation between Cc and index

properties will include parameters that account for both the

soils’ initial state and soil type; i.e., the ideal correlation is

a multi-parameter correlation. This is particularly true for

undisturbed samples.

Several multi-parameter correlations are presented in

Table 1. Correlations that are based on multiple regressions

are more accurate in estimating the compression index

(Azzouz et al. 1976; Sridharan and Nagaraj 2000; Giasi

et al. 2003). However, in order to obtain reliable multiple

parameter regression-based correlations the variables con-

sidered must be absolutely independent of each other and

the physical significance of the mathematical operation

must be taken into consideration (Li and White 1993;

Cherubini and Giasi 2000; Giasi et al. 2003). Hence, cor-

relations such as those that contain e0 and cd, e0 and wn are

unreliable as those parameters are related and not

independent.

Considering the foregoing, a number of single-parame-

ter (wn, LL, PL, e0) and multi-parameter (LL, Gs; PI, Gs;

wn, LL; e0, LL) correlations were selected for review.

Single-parameter correlations were selected on the basis of

their being in popular use. Multi-parameter correlations

were selected on the basis of the independence of their

constituent parameters. Correlations for review were also

selected on the basis of their applicability. Most of the

selected correlations are applicable to all clays. Two cor-

relations applicable to organic clays were selected due to

the organic nature of the soils in the southeast region of

Missouri.

The validity of the selected correlations was assessed on

the basis of four criteria: root mean square error (RMSE);

the first (mean) and second moment (standard deviation)

statistics of the ratio of the estimated to laboratory-deter-

mined compression index, K; ranking index; and ranking

distance. Descriptions of these criteria are listed as follows.

RMSE is the square root of the average the squared

difference between the values calculated using a correla-

tion and the corresponding observed values determined

from laboratory tests. Errors in RMSE are squared before

they are averaged; consequently, relatively high weight is

given to large errors. This means the RMSE is most useful

when large errors are particularly undesirable. The RMSE

has been used by investigators (e.g., Alvarez Grima and

Babuska 1999; Finol et al. 2001; Gokceoglu 2002; Yılmaz

2006; Ozer et al. 2008) to evaluate the performance of

empirical equations. RMSE is determined using the fol-

lowing formulae:

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

n

X

n

1

ðCcest� Cclab

Þ2s

ð2Þ

where n is the number of data points, Cccalis the com-

pression index calculated from empirical equations, Cclabis

the compression index determined directly from laboratory

S. Onyejekwe et al.

123

Fig. 5 Estimated (Cccal) versus measured (Cclab

) Compression index—Southeast region

Assessment of empirical equations for the compression index

123

tests. The lower the RMSE value, the better the model

performance.

The first (mean) and second moment (standard devia-

tion) statistics of the ratio of the estimated compression

index to the laboratory-determined compression index is

denoted by K, which is determined using the formulae:

K ¼ Cccal

Cclab

ð3Þ

where Cccalis the predicted compression index and Cclab

is

the laboratory-determined compression index. The accu-

racy of a method is its ability to predict the laboratory-

determined compression index; it is represented by the

mean of K (Briaud and Tucker 1988; Cherubini and Orr

2000; Giasi et al. 2003). The precision of a method refers to

the scatter around the mean; it is quantified by the standard

deviation of K (Briaud and Tucker 1988; Cherubini and Orr

2000; Giasi et al. 2003). Theoretically, K ranges from a

minimum of 0 to a maximum of infinity with an optimum

value of one. This results in a nonsymmetric distribution of

K around the mean, which does not give an equal weight of

underprediction and overprediction (Briaud and Tucker

1988). The first and second moment statistics of K have

been used by several investigators (e.g., Titi and Abu-

Farsakh 1999; Abu-Farsakh and Titi 2004; Cherubini and

Orr 2000; Orr and Cherubini 2003; Cherubini and Giasi

2000; Giasi et al. 2003) to evaluate the performance of

empirical equations.

The ranking index (RI) is one of the two methods pro-

posed by Briaud and Tucker (1988) to alleviate the prob-

lem of nonsymmetrical distribution of K data. The other

being presenting the results for all methods in the form of

log normal distribution curves. Hence, with RI it is possible

to express an overall judgment on the quality of a corre-

lation while taking into consideration the mean value and

the standard deviation of all the K data. The ranking index

is determined using the formulae (Briaud and Tucker

1988):

RI ¼ lðln½K�Þ

�

�

�

�

�

�þ sðln½K�Þ ð4Þ

where l and s represent the mean and standard deviation of

the series of analyzed data, respectively. RI has been used

by several investigators (e.g., Briaud and Tucker 1988;

Cherubini and Orr 2000; Orr and Cherubini 2003; Cheru-

bini and Giasi 2000; Giasi et al. 2003) to evaluate the

performance of empirical equations.

Ranking distance (RD), proposed by Cherubini and Orr

(2000), is another method of expressing an overall judg-

ment on the quality of a calculation method that takes into

consideration the mean value and the standard deviation of

all the K data. RD represents, on a plot with mean (l)

values on the x axis and standard deviation (s) on the y axis,

the distance of the point representing a computation using a

particular correlation from the point representing the

optimum condition (l = 1 and s = 0). RD is determined

using the formulae (Cherubini and Orr 2000):

RD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1� lðKÞÞ2 þ ðsðKÞÞ2q

ð5Þ

where l and s represent the mean and standard deviation of

the series of analyzed data respectively. RD and RI provide

a different evaluation of the suitability of a given correla-

tion equation to fit a measured value (Cherubini and Orr

2000). For correlation equations where the precision,

indicated by the standard deviation, mean value and the

accuracy are similar, RD gives a better result than RI, while

for those that are either very accurate or very precise RI

gives the best result. RD gives equal weight to accuracy

and precision. RD has been used by several investigators

(Cherubini and Orr 2000; Cherubini and Giasi 2000; Orr

and Cherubini 2003; Giasi et al. 2003) to evaluate the

performance of empirical equations.

Results and discussion

A summary of the results of the RMSE, K, RI and RD

analysis for the ‘‘Other Regions’’ and Southeast Region of

Missouri are presented in Tables 3 and 4, respectively. The

results of the RMSE, RI and RD analysis for the Other

Regions and Southeast Region are presented graphically in

Figs. 6 and 7, respectively.

In terms of RMSE, for the single-parameter correlations,

correlation D, proposed by Azzouz et al. (1976), with wn as

variables gave the best performance with RMSE of 0.107

and 0.213 for the Other Regions and Southeast Region, as

shown by the K values in Tables 3 and 4, respectively.

Among the double-parameter correlations, correlation Q,

proposed by Azzouz et al. (1976), with e0 and LL as

variables gave the best performance with RMSE of 0.159

for Other Regions. Correlation M proposed by Nagaraj and

Murty (1985) with LL and Gs as variables gave the best

performance with RMSE of 0.317 for the Southeast

Region. The overall best performing correlation was cor-

relation D for Other Regions (RMSE = 0.107) and the

Southeast Region (RMSE = 0.213).

There was a general overestimation (K [ 1) of Cc par-

ticularly for the Other Regions; eight out of twelve single

parameter correlations and four out of six double parameter

correlations had over 50 % of the K values greater than 1.

Correlation K had the highest mean K value (over 90 %

were larger than 1) while over 70 % of the K values (mean)

in correlations A, C, J, N and P were greater than 1. A

similar overestimation, but of a lower magnitude, was

observed for the Southeast Region where four out of twelve

S. Onyejekwe et al.

123

single parameter correlations and three out of six double

parameter correlations, over 50 % of the K values were

greater than 1. Over 74.8 % of the K values for correlation

P were greater than 1 (a mean K of 1.94); the mean K value

of correlation C was 1.35, however, which had the same

percentage of K values larger than 1.

In terms of accuracy (in terms of nearness of the mean

of K to 1), for the single parameter correlations, correlation

D, proposed by Azzouz et al. (1976), with wn as variable

gave the best prediction with a K mean of 0.94 and 1.03 for

the Other Regions and the Southeast Region, respectively.

For the double-parameter correlations, correlation Q,

Table 3 Results of the RMSE,

K, RI and RD analyses—Other

Regions

# Correlation RMSE K RI RD

% [ 1 Mean SD

A Cc = 0.01wn 0.120 77.8 1.11 0.303 0.357 0.322

B Cc = 0.01(wn - 7.549) 0.109 34.4 0.86 0.225 0.490 0.265

C Cc = 0.0115wn 0.148 90.0 1.28 0.348 0.496 0.444

D Cc = 0.01(wn - 5) 0.107 48.3 0.94 0.247 0.390 0.253

E Cc = 0.006(LL - 9) 0.163 37.8 1.21 0.496 0.550 0.540

F Cc = 0.008(LL - 12) 0.184 50.6 1.54 0.628 0.792 0.827

G Cc = 0.009(LL - 10) 0.213 60.0 1.79 0.731 0.941 1.077

H Cc = 0.014(PI ? 3.6) 0.288 67.8 2.29 1.044 1.208 1.659

I Cc = PI/74 0.251 58.9 2.05 0.937 1.106 1.404

J Cc = 0.29(e0 - 0.27) 0.140 86.7 1.19 0.390 0.449 0.434

K Cc = 0.35(e0 - 0.5) 0.258 93.9 1.70 0.598 0.811 0.924

L Cc = 0.156e0 ? 0.0107 0.157 6.1 0.54 0.163 0.991 0.491

M Cc = 0.2343(LL/100)Gs 0.181 57.2 1.50 0.618 0.755 0.798

N Cc = 0.2926(LL/100)Gs 0.229 70.6 1.88 0.772 0.977 1.169

O Cc = 0.5Gs(PI/100) 0.380 68.9 2.82 1.293 1.428 2.235

P Cc = 0.009wn ? 0.005LL 0.305 80.6 2.15 0.749 1.069 1.376

Q Cc = 0.37(e0 ? 0.003LL - 0.34) 0.159 47.2 1.02 0.306 0.432 0.306

R Cc = -0.156 ? 0.411e0 ? 0.00058LL 0.161 33.3 0.92 0.277 0.570 0.288

Table 4 Results of the RMSE,

K, RI and RD analyses—

Southeast Region

# Correlation RMSE K RI RD

% [ 1 Mean SD

A Cc = 0.01wn 0.216 48.5 1.17 0.59 0.449 0.615

B Cc = 0.01(wn - 7.549) 0.216 28.2 0.96 0.45 0.463 0.454

C Cc = 0.0115wn 0.240 70.9 1.35 0.68 0.589 0.763

D Cc = 0.01(wn - 5) 0.213 32.0 1.03 0.50 0.402 0.499

E Cc = 0.006(LL - 9) 0.323 19.4 0.83 0.30 0.593 0.342

F Cc = 0.008(LL - 12) 0.318 33.0 1.04 0.37 0.378 0.373

G Cc = 0.009(LL - 10) 0.331 48.5 1.22 0.44 0.490 0.490

H Cc = 0.014(PI ? 3.6) 0.368 60.2 1.45 0.77 0.709 0.891

I Cc = PI/74 0.343 50.5 1.25 0.69 0.632 0.738

J Cc = 0.29(e0 - 0.27) 0.324 36.9 1.19 0.64 0.477 0.664

K Cc = 0.35(e0 - 0.5) 0.374 63.1 1.65 0.93 0.818 1.139

L Cc = 0.156e0 ? 0.0107 0.374 3.9 0.55 0.28 1.065 0.529

M Cc = 0.2343(LL/100)Gs 0.317 36.9 1.07 0.41 0.362 0.412

N Cc = 0.2926(LL/100)Gs 0.336 53.4 1.34 0.51 0.584 0.609

O Cc = 0.5Gs(PI/100) 0.461 62.1 1.73 0.96 0.954 1.202

P Cc = 0.009wn ? 0.005LL 0.439 74.8 1.94 0.86 0.954 1.270

Q Cc = 0.37(e0 ? 0.003LL - 0.34) 0.329 24.3 1.03 0.45 0.405 0.456

R Cc = -0.156 ? 0.411e0 ? 0.00058LL 0.328 24.3 1.02 0.43 0.393 0.433

Assessment of empirical equations for the compression index

123

proposed by Azzouz et al. (1976), with e0 and LL as

variables gave the most accurate prediction with a mean

K of 1.02 for the Other Regions, while correlation R pro-

posed by Al-Khafaji and Andersland (1992) with e0 and LL

as variables gave the most accurate prediction for the

Southeast Region with a mean K value of 1.02. Comparing

the K values obtained using the single parameter and

double-parameter correlations, the double-parameter cor-

relations were found to be the best overall with correlation

Q the best for the Other Regions and correlation R the best

for the Southeast Region, both with mean K values of 1.02.

In terms of precision (determined in terms of smallness

SD of K) for the single-parameter correlations, correlation

L, proposed by Bowles (1979), with e0 as variable gave the

most precise prediction of Cc with a mean SD of 0.163 and

0.28 for the Other Regions and Southeast Region, respec-

tively. For the double-parameter correlations, correlation R,

proposed by Al-Khafaji and Andersland (1992), with e0

and LL as variables gave the best prediction with a SD

mean of 0.277 for the Other Regions. Correlation M pro-

posed by Nagaraj and Murty (1985) with LL and Gs as

variables gave the best prediction with a SD mean of 0.41

for the Southeast Region.

The best single parameter correlations in terms of RI for

the Other Regions and Southeast Region are correlation

A (RI = 0.357) proposed by Koppula (1981) and correla-

tion F (RI = 0.378) proposed by Sridharan and Nagaraj

(2000), respectively. For double parameter correlations it

was correlation Q (RI = 0.432) proposed by Azzouz et al.

(1976) and correlation M (RI = 0.362) proposed by Nag-

araj and Murty (1985) for the Other Regions and the

Southeast Region, respectively. The overall best correla-

tions were correlation A (RI = 0.357) and correlation

M (RI = 0.362) for the Other Regions and Southeast

Region, respectively. From Figs. 6 and 7 the RI curve

seems like an exaggeration of the RMSE curve where the

RI curve generally follows the profile of the RMSE curve

but plots above it.

For RD the most efficient single parameter correlations

for the Other Regions and Southeast Region are correlation

E (RD = 0.342) proposed by Azzouz et al. (1976) and

correlation D (RD = 0.253) proposed by Azzouz et al.

(1976), respectively. For double parameter correlations it

was correlation R (RD = 0.288) proposed by Al-Khafaji

and Andersland (1992) and correlation M (RD = 0.412)

proposed by Nagaraj and Murty (1985) for the Other

Regions and Southeast Region, respectively. The overall

best correlations were correlation D (RD = 0.253) and

correlation E (RD = 0.342) for the Other Regions and

Southeast Region, respectively.

The difference in value of RI and RD for each corre-

lation is shown Figs. 6 and 7 for the Other Regions and

Southeast Region, respectively. Larger differences

correspond to cases where the accuracy of the calculated Cc

value (mean of K is not close to 1.0) is poor, while smaller

differences correspond to higher accuracy prediction levels

of the Cc. Given the shortcomings of RMSE and RI, RMSE

highlights large errors (low accuracy) while RI considers

both accuracy and precision but does not assign equal

weight to them. RD, which gives equal weight to accuracy

and precision, is a better parameter for comparing the

suitability of the different correlation equations.

Conclusions

The suitability of some published correlations between the

compression index and index properties as they relate to

fine-grained Missouri soils were assessed in this paper.

Relevant data on fine-grained Missouri soils were grouped,

largely according to geology, into two regions (the ‘‘Other

Regions’’ and the Southeast Region) for this study. The

validity of these published correlations was assessed in

terms of four indices: the RMSE, the ratio of the estimated

to laboratory-determined compression index (K), the

ranking index (RI) and the ranking distance (RD). How-

ever, the ultimate/final assessment was made by means of

the RD index that assigns equal weight to the accuracy and

Fig. 6 RMSE, RI, and RD—Other Regions

Fig. 7 RMSE, RI, and RD—Southeast Region

S. Onyejekwe et al.

123

the precision of prediction. The RMSE does not consider

precision while the ranking index does not assign equal

weight to the accuracy and the precision of prediction.

Based on the analyses conducted in this study, the fol-

lowing conclusions can be reached:

• There was a general overestimation of Cc as evidenced

by the percentage of K values greater than 1.00.

• The overestimation of Cc was more pronounced in the

case of the Other Regions where, for example, the

percentage of K values greater than 1.00 as high as 90%

was recorded for correlation K proposed by Hough

(1957).

• Compared with double parameters correlation, the

single parameter correlation performed best for the

data studied in this research.

• The best single-parameter correlations for the Other

Regions and Southeast Region are correlation

E (RD = 0.342) proposed by Azzouz et al. (1976),

Cc = 0.006 (LL - 9) and correlation D (RD = 0.253)

proposed by Azzouz et al. (1976), Cc = 0.01 (wn - 5),

respectively.

• The best two-parameter correlations for the Other

Regions and Southeast Region are correlation

R (RD = 0.288) proposed by Al-Khafaji and Anders-

land (1992), Cc = -0.156 ? 0.411e0 ? 0.00058LL,

and correlation M (RD = 0.412) proposed by Nagaraj

and Murty (1985), Cc = 0.2343 (LL/100) Gs,

respectively.

• The overall best correlations for the Other Regions and

Southeast Region are correlation D (RD = 0.253)

proposed by Azzouz et al. (1976), Cc = 0.01 (wn -

5), and correlation E (RD = 0.342) proposed by

Azzouz et al. (1976), Cc = 0.006 (LL - 9),

respectively.

Acknowledgments The Missouri Department of Transportation

(MoDOT)/Missouri Transportation Institute (MTI) Transportation

Geotechnics Research Program (MoDOT/MTI-TGRP) was jointly

executed by MoDOT, the Geotechnical Engineering programs of the

Department of Civil and Environmental Engineering at the University

of Missouri-Columbia (MU) and the Department of Civil, Architec-

tural and Environmental Engineering at the Missouri University of

Science and Technology (S&T) and the Geological Engineering

program of the Department of Geological Sciences and Engineering

at the Missouri University of Science and Technology (S&T).

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