experimental study of estimating the subgrade reaction modulus
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ORIGINAL PAPER
Experimental Study of Estimating the Subgrade ReactionModulus on Jointed Rock Foundations
Jaehwan Lee1 • Sangseom Jeong1
Received: 10 August 2015 / Accepted: 22 December 2015 / Published online: 23 January 2016
� Springer-Verlag Wien 2016
Abstract The subgrade reaction modulus for rock foun-
dations under axial loading is investigated by model foot-
ing tests. This study focuses on quantifying a new subgrade
reaction modulus by considering rock discontinuities. A
series of model-scale footing tests are performed to
investigate the effects of the unconfined compressive
strength, discontinuity spacing and inclination of the rock
joint. Based on the experimental results, it is observed that
the subgrade reaction modulus of the rock with disconti-
nuities decreases by up to approximately 60 % of intact
rock. In addition, it is found that the modulus of subgrade
reaction is proportional to the discontinuity spacing, and it
decreases gradually within the range of 0�–30� and tends to
increase within the range of 30�–90�.
Keywords Subgrade reaction modulus � Rockfoundation � Rock discontinuity � Model-scale footing tests
1 Introduction
Soil–foundation interaction is a challenging problem in
geotechnical engineering. Because of the complex behavior
of soil, the subgrade in soil–foundation interaction prob-
lems is replaced by a simpler system called a subgrade
model. In the practical design of mat foundations, struc-
tural engineers prefer to model the soil mass as a series of
elastic springs, known as the Winkler foundation. In other
words, the modulus of subgrade reaction is assumed to be
the elastic constant of the springs.
Over 90 % of the shallow foundations constructed in
South Korea are constructed on weathered rocks. The
weathered rocks, which occupy two-thirds of the total land
area of the Korean peninsula, are generally the result of the
physical weathering of granite-gneiss of varying thick-
nesses up to 40 m. It is clear that most rocks cannot be
accurately represented as isotropic linear elastic materials
because of the presence of joints and discontinuities.
Discontinuities in rock masses often result in strengths that
are less than that of the intact rock. Therefore, the presence
of these discontinuities creates weakness planes along
which failures may initiate and propagate. The overall
behavior of the rock mass is affected by the mechanical
properties of the intact rock and by the condition of the
discontinuities.
There are several studies for determining the elastic
modulus of the rock mass using empirical correlations
with the rock properties. Heuze (1980) stated that the
modulus of deformation of rock masses ranges between
20 and 60 % of the modulus measured on intact rock
specimens in the laboratory. Hoek and Brown (1980a, b)
proposed an empirical failure criterion for rock masses
containing two parameters that are related to the degree of
rock mass fracturing. Empirical expressions have also
been proposed between those parameters and the rock
quality designation (RQD), the RMR, and the Q ratings.
However, previous studies have focused on the fact that
deformation modulus of rock mass with the discontinuity
is smaller than that of the intact rock. Because the sub-
grade reaction modulus is proportional to deformation
modulus, the subgrade reaction modulus of the rock mass
with discontinuity is expected to be smaller than that of
the intact rock.
& Sangseom Jeong
soj9081@yonsei.ac.kr
1 Department of Civil and Environmental Engineering, Yonsei
University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749,
Republic of Korea
123
Rock Mech Rock Eng (2016) 49:2055–2064
DOI 10.1007/s00603-015-0905-9
Author copy
ve
e rock
served thatt
with disconti-
ely 60 % of intactof intact
e modulus of subgradeof subgrade
discontinuity spacing, and itdiscontinuity spacin
the range of 0range of 0��–30–30� and tends tond tends to
ge of 30of ��–90–90��..
ubgrade reaction modulusreaction modulus � RockRock discontinuitycontinuity �� Model-scale footing tesModel-sca
1 IntroductionIntroduction
Soil–foundation interaction is a chil–foundation interactio
geotechnical engineering. Becautechnical engineeri
f soil, the subgrade in sooil,
ms is replaced by ais r
l. In the prac. In
gineers
i
dulus of subgrade reaction is assumed to besubgrade reaction is assumed to be
constant of the springs.f the springs.
r 90 % of the shallow foundations constructshallow foundations
uth Korea are constructed on weathered roa are constructed on weathered
weathered rocks, which occupy two-thirds ofrocks, which occupy two-thirds of
area of the Korean peninsula, are generalKorean penins
physical weathering of granite-gnephysical weathering of
nesses up to 40 m. It is clear tnesses up to 40 m. It
accurately represented as isa
because of the presenbec
Discontinuities in roDisco
are less than thae le
of these d
which
beh
Since the 1930s, many studies have been performed by
many researchers on the subgrade reaction modulus (ks) of
soil (Biot 1937; Terzaghi 1955; Vesic 1961; Meyerhof and
Baikie 1963; Vlassov and Leontiev 1966; Kloppel and
Glock 1979; Selvadurai 1985; Horvath 1989; Daloglu and
Vallabhan 2000; Elachachi et al. 2004; Moayed and Naeini
2006). In addition, several empirical methods for soil have
been proposed. However, there are very few available
methods of ks for the rock mass compared with that of the
soil because the available loading test data for the rock
mass were insufficient.
The predicted results for the existing subgrade reaction
modulus of soil differ from those of rock masses, although
precise analysis is performed by empirical methods for soil.
Less is known about the subgrade reaction modulus in
weathered rock, which occupies two-thirds of the total land
area of the Korean peninsula. The need for more research
on the subgrade reaction modulus of rock masses has been
emphasized.
This paper is intended to evaluate the subgrade reaction
modulus of jointed weathered rocks. A series of model-
scale footing tests are performed to take into account var-
ious factors influencing the subgrade reaction modulus, i.e.,
the rock discontinuity spacing and inclination. Based on the
obtained results, an appropriate and simple subgrade
reaction modulus (kj) is proposed, particularly for jointed
rock foundations.
2 Available Methods for Determiningthe Subgrade Reaction Modulus
The subgrade reaction modulus (ks) is a mathematical
constant that represents the foundation’s stiffness; it is
defined as the ratio of the pressure (q) against the mat to the
settlement (d) at a given point,
ks ¼ q
dð1Þ
where q is the soil pressure at a given point and d is the
settlement of the mat at the same point.
The subgrade reaction modulus is not a constant for a
given soil; it depends upon a number of factors, such as the
width and the shape of the foundation in addition to the
depth of embedment of the foundation. There are numerous
semi-empirical models that can be used to determine the
subgrade reaction modulus as a function of the elastic
modulus (E), the Poisson’s ratio (m) of the soil, and the
footing width (B). Previous authors have each suggested a
different but suitable expression. The empirical methods
are summarized in Table 1. Many studies (Biot 1937;
Terzaghi 1955; Vesic 1961; Meyerhof and Baikie 1963;
Vlassov and Leontiev 1966; Kloppel and Glock 1979;
Selvadurai 1985; Horvath 1989; Daloglu and Vallabhan
2000) have investigated effective factors and approaches
for determining ks.
Terzaghi (1955) suggested values of ks for a
30 cm 9 30 cm rigid slab placed on a soil medium. His
work showed that the value of ks depends on the dimen-
sions of the area acted upon by the subgrade reaction, and
he incorporated size effects in his equations.
For footings on sand,
ks ¼ k0:3ðkN=m3Þ � B ðmÞ þ 0:3 ðmÞ2B ðmÞ
� �2ð2aÞ
For footings on clay,
ks ¼ k0:3 ðkN=m3Þ � 0:3 ðmÞB ðmÞ
� �ð2bÞ
where ks is the desired value of the modulus of subgrade
reaction for full-sized footings, k0.3 is value of k from a
plate load test, B is the footing width.
Table 1 Empirical methods of subgrade reaction modulus (ks)
Proposer Empirical method Application
Biot (1937)ks ¼ 0:95
B� Es
1�m2s
h i� EsB
4
EI
� �0:108 Infinite beams resting on an elastic soil continuum
Terzaghi (1955) Sand
ks ¼ k0:3ðkN=m3Þ � B ðmÞþ0:3 ðmÞ2B ðmÞ
h i2Clay
Rigid plate placed on a soil medium
Vesic (1961)ks ¼ 0:65
B� Es
1�v2s
h i�
ffiffiffiffiffiffiffiffiffiffiEs�B4
EI
12
qBeams resting on elastic half space
Meyerhof and Baikie (1963) ks ¼ Es
Bð1�v2s ÞBuried circular conduits
Vlassov (1966) ks ¼ Esð1�vsÞð1þvsÞð1�2vsÞ �
l2B
� �Beams and plates resting on elastic half space
Kloppel and Glock (1979) ks ¼ 2Es
Bð1þvsÞ Buried circular conduits
Selvadurai (1985) ks ¼ 0:65B
� Es
ð1�v2s ÞBuried circular conduits
2056 J. Lee, S. Jeong
123
Author copy
ble
t of the
or the rockk
g subgrade reactionde reaction
f rock masses, althoughes, although
y empirical methods for soil.y empirical methods
subgrade reaction modulus inbgrade reaction modulus in
occupies two-thirds of the total landcupies two-thirds of the total land
n peninsula. The need for more researchsula. The need for more r
de reaction modulus of rock masses has beenon modulus of rock masses h
ed.
is paper is intended to evaluate the subgrade res intended to evaluate the s
modulus of jointed weathered rocks. A seriesmodulus of jointed weathered rocks.
scale footing tests are performed to take incale footing tests are performed
ious factors influencing the subgrade reaus factors influencing the su
the rock discontinuity spacing and irock discontinuity
obtained results, an approprained results, an
eaction modulus (ion kjk ) is pr
k foundations.foun
ðð11ÞÞ
he soil pressure at a given point andessure at a given point an d is thethe
nt of the mat at the same point.at the same point.
he subgrade reaction modulus is not a constanade reaction modulus is not a c
given soil; it depends upon a number of factors,it depends upon a number of factors
width and the shape of the foundation inthe shape of the foun
depth of embedment of the foundation.mbedment of th
semi-empirical models that can besemi-empirical models t
subgrade reaction modulus a
modulus (m E), the Poisson
footing width (foot B). Pre
different but suitiffere
are summariz
Terzaghi
Vlass
S
yalf spa
Vesic (1961) showed that ks depends upon the stiffness
of the soil, as well as the stiffness of the structure, so that
similarly sized structures of different stiffnesses will yield
different values of ks for the same applied load. He
extended Biot’s solution by providing the distributions of
deflection, moment, shear and pressure along the beam. He
found that the continuum solution correlated with the
Winkler model
ks ¼ 0:65
B� Es
1� v2s
� ��
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEs � B4
EI
12
r; ð3Þ
where Esis the elastic modulus of the soil, ms is Poisson’sratio of the soil, E is elastic modulus of the beam, I is
the moment of inertia of the beam.
Vesic (1961) suggested an equation for ks to use in the
Winkler model. For practical purposes, Vesic’s equation
reduces to
ks ¼ Es
B 1� v2s ð4Þ
Vlassov and Leontiev (1966) introduced an equation for
beams and plates resting on elastic half-space, but the
ambiguities of estimating l in Table 1 (a non-dimensional
parameter) make the problem more complex (Sadrekarimi
and Akbarzad 2009). The equations given by Meyerhof and
Baikie (1963), Kloppel and Glock (1979), and Selvadurai
(1985) were proposed for computing the horizontal sub-
grade reaction modulus in buried circular conduits.
3 Model-Scale Footing Tests
In this study, the subgrade reaction modulus and the
bearing capacity of the mat foundations considering rock
discontinuities (or joints) were investigated by performing
model-scale footing tests.
UTM Controller
UTM (Universal Test Machine)
FootingSpecimen
SteelTesting
Box
Fig. 1 Test set-up for model-scale footing tests
Fig. 2 Simple preparation
process of jointed rock
specimens: a preparing
gypsum–sand–water mix,
b aluminum mold, c pouring
gypsum–sand–water mix into
the mold and d jointed rock
specimen
Experimental Study of Estimating the Subgrade Reaction Modulus on Jointed Rock Foundations 2057
123
hor copyed an equation fion
c half-space, but theut t
able 1 (a non-dimensionalon-dimension
m more complex (Sadrekarimicomplex (Sadrekarim
e equations given by Meyerhof andns given by Meyerhof and
ppel and Glock (1979), and Selvaduraick (1979), and Selvadurai
oposed for computing the horizontal sub-for computing the horizontal sub-
ion modulus in buried circular conduits.us in buried circular condu
3 Model-Scale Footing Tests-Scale Footing Tests
In this study, the subgrade reactiIn this study, the subg
bearing capacity of the mat foubearing capacity of the
discontinuities (or joints) wed
model-scale footing testmoorAAuthck
3.1 Testing Apparatus and Specimen Preparation
Model-scale footing tests were performed in a
48 cm 9 48 cm 9 28 cm steel box (Fig. 1). The square
mat foundation, 8 cm wide (B), was made of an aluminum
plate 4 cm thick (t). Tests were performed with the foot-
ings located at the rock surface. The mat foundation was
loaded vertically at a constant rate of 2 9 10-5 m/s until a
settlement of at least 0.1B occurred. The ultimate bearing
capacity was defined as the bearing stress that produced a
relative settlement of 0.1B. Although choosing to define
qult at a relative settlement of s/B is arbitrary, the 0.1B
method is convenient and easy to remember, and it may
actually be close to the average soil strain at failure (Cerato
and Lutenegger 2007).
Natural rock blocks with regular discontinuity patterns
are required for model-scale footing tests. It is impossible
to perform a number of tests under various boundary
conditions because it is difficult to prepare large rock block
samples and to make regular discontinuity patterns with the
rock block. In this study, therefore, two industrial gypsum
plasters were used to make rock-block specimens with
regular discontinuity patterns by using a developed
experimental apparatus. These plasters can be molded into
any shape when mixed with water and sand. The photos for
the process of preparing artificial jointed rock specimens
are shown in Fig. 2.
The properties of the artificial rock are similar to those
of typical weathered rocks (Indraratna et al. 1998; Yang
and Chiang 2000; Jiang et al. 2004; Seol et al. 2008).
Table 2 summarizes the gypsum–sand–water ratio, the
unconfined compressive strength (qu), and the Young’s
modulus (Es) of the rock specimens that were used in this
study.
3.2 Test Boundary Conditions
To study the factors that influence the subgrade reaction
modulus and the bearing capacity of rock foundations, a
total of 21 model footing tests were conducted on rock
specimens under various conditions in consideration of the
rock mass discontinuity inclination and spacing. The
effects of the joint roughness, filling material, and dis-
continuous friction angle itself were not taken into account.
In this study, the unconfined compressive strength (qu), the
discontinuity inclination (Id, where the index d refers to the
discontinuity or joint), and the discontinuity apparent
Table 2 Material properties of test samples
Parameters Artificial rock A Artificial rock B
gypsum–sand–water ratio 1.5:1.5:1 2:1:1
UCS (MPa) 15 24
Es (MPa) 860 1520
Table 3 Summary of test
boundary conditionsVariable Values
Artificial rock A Artificial rock B
UCS (MPa) 15 24
Discontinuity spacing, Sd (cm) Intact, 4, 8, 12, 16 Intact, 4, 8, 12
Discontinuity inclination, Id (�) 0�, 30�, 60�, 90� 90�
Fig. 3 Testing devices and boundary conditions: a plan view and
b front view
2058 J. Lee, S. Jeong
123
yer
uity appa
Author copy
to
tos for
specimenss
are similar to thoser to those
atna et al. 1998; Yang1998; Yang
. 2004; Seol et al. 2008).. 2004; Seol et al
gypsum–sand–water ratio, thepsum–sand–water ratio, the
ve strength (strength (qquu), and the Young’s), and the Young’s
e rock specimens that were used in thisspecimens that were used
AuAAATable 2able 2 Material properties of test samplesMaterial properties of test s
Parameters Artificial roameters A
gypsum–sand–water ratio 1.5:1um–sand–water rat
CS (MPa)(MP
MPa)Pa)
or copy
spacing (hereafter Sd) were determined to represent the test
boundary conditions. Two unconfined compressive
strengths (15 and 24 MPa), four discontinuity apparent
spacings (0.5, 1.0, 1.5, and 2.0 B) and four discontinuity
inclinations (0�, 30�, 60�, and 90�) were used as the test
conditions. A summary of the test boundary conditions is
given in Table 3.
Figure 3 shows the testing devices and the boundary
conditions. The overall dimensions of the boundaries com-
prise a width of 3.0 times the mat width (B) from the mat
center and a height equal to 3.5 times the mat width (B).
These dimensions were considered adequate to eliminate the
influence of boundary effects on the mat performance based
on a review of the literature (Boussinesq 1883). The mat is
made of an aluminum plate, which allows for rigorous
analysis of the behavior under only rock mass conditions,
regardless of the mat foundation properties.
3.3 Test Results and Discussion
A total of 21 individual tests were conducted under various
boundary conditions described in the previous sections.
The rock specimens with joints at various conditions are
shown in Fig. 4. In this study, only a selection of typical
test results is presented. The modulus of subgrade reaction
was considered as an initial tangent modulus (around
s = 2 mm) over the estimated working range of bearing
pressure. From the results obtained from the model tests,
the modulus of subgrade reaction (ks) and the ultimate
bearing capacity (qult) decreased compared with intact rock
due to the rock discontinuities.
3.3.1 Effect of the Unconfined Compressive Strength (qu)
Figure 5a shows the stress-settlement curves from typical
tests on artificial rock specimens with two unconfined
compressive strengths (qu = 15 and 24 MPa) under the
same discontinuity spacing [Sd = 12 cm (1.5B)] and
inclination (Id = 90�). As shown in Fig. 5b, c, with an
increase in the unconfined compressive strength (qu), the
modulus of subgrade reaction (ks) increases. In addition,
the ultimate bearing capacity (qult) of the stress–settlement
curves tends to increase as qu increases.
3.3.2 Effect of the Rock Discontinuity Spacing (Sd)
Figure 6 shows the stress–settlement curves with different
discontinuity spacings [Sd = 4 cm (0.5B), 8 cm (1.0B),
12 cm (1.5B), 16 cm (2.0B), and intact rock] under two
discontinuity inclinations, i.e., 60� and 90�. The stress–
settlement curve tends to be similar to the curve of intact
rock as the discontinuity spacing (Sd) increases. As shown
in Fig. 7, it is noted that the modulus of subgrade reaction
(ks) is proportional to the discontinuity spacing (Sd). In
addition, qult tends to increase as the discontinuity spacing
(Sd) increases.
Sd = 16 cm (2.0B) Sd = 12 cm (1.5B) Sd = 8 cm (1.0B) Sd = 4 cm (0.5B)
Id = 0
48cm
16cm
28cm
48cm
2cm
12cm
48cm
28cm
2cm
12cm
48cm
8cm
8cm
8cm
28cm
2cm
48cm
28cm
4cm2cm
4cm4cm4cm4cm
Id = 30
16cm
30º
28cm
48cm
30º
12cm
48cm
28cm
30º
8cm8cm 8cm
48cm
28cm
30º
4cm
48cm
28cm
4cm4cm
Id = 60
60º
16cm
28cm
48cm
12cm
48cm
28cm
60º 60º
48cm
28cm
8cm8cm 8cm
60º
4cm
48cm
28cm
4cm4cm
Id = 90
16cm
28cm
48cm 48cm
28cm
12cm
48cm
28cm
8cm8cm 8cm 4cm
48cm
28cm
4cm4cm
Fig. 4 Rock specimens with joints at various conditions (qu = 15 MPa)
Experimental Study of Estimating the Subgrade Reaction Modulus on Jointed Rock Foundations 2059
123
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the test
ompressivee
inuity apparentp
d four discontinuityscontinuity
) were used as the testas the test
test boundary conditions istest boundary condi
e testing devices and the boundaryesting devices and the boundary
erall dimensions of the boundaries com-mensions of the boundari
of 3.0 times the mat width (mes the mat width (B) from the matm
d a height equal to 3.5 times the mat width (qual to 3.5 times the
e dimensions were considered adequate to eliminons were considered adequ
nfluence of boundary effects on the mat performnfluence of boundary effects on the m
on a review of the literature (Boussinesq 18n a review of the literature (Bou
made of an aluminum plate, which aade of an aluminum plate
analysis of the behavior under onllysis of the behavi
regardless of the mat foundatiordless of the mat
Test Results andTest
f 21 i
the modulus of subgrade reactionthe modulus of subgrad
bearing capacity (bearing capacity (qqult) decrease)
due to the rock discontinuid
3.3.1 Effect of the3.3.1
Figure 5a
tests
co
copyooooooopyycooopy
3.3.3 Effect of the Rock Discontinuity Inclination (Id)
Figure 8 shows the stress–settlement curves with varying
discontinuity inclinations (Id = 0�, 30�, 60�, and 90�)under the same discontinuity spacing (Sd = 12 cm). As
shown in Fig. 9, the modulus of subgrade reaction (ks) and
the ultimate bearing capacity (qult) of a rock foundation
with vertical discontinuity (90�) are greater than that of a
rock with inclined discontinuities. For the condition in
which the joint inclination is 0�, there is no failure along
discontinuity planes, and failure occurs in rock material. In
addition, the values of ks and qult decrease gradually within
the range of 0�\ Id\ 30� and tend to increase within the
range of 30�\ Id\ 90�. In the case of 30� and 60�, therock mass fails along the plane of rock discontinuities. It is
observed that the least value of ks and qult is in the vicinity
of joint inclination of 30�. When the discontinuity incli-
nation is 90�, each rock block behaves as an individual
column, and the bearing capacity increases. This trend is in
general agreement with previous research regarding the
bearing capacity of the footing (Roy 1993; Sutcliffe et al.
2004; Maghous et al. 2008).
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
qu=24MPa
qu=15MPa
qu=24MPa
qu=15MPa
0
200
400
600
800
1000
1200
1400
1600
12 14 16 18 20 22 24 26 28Mod
ulus
of s
ubgr
ade
reac
tion
(ks)
(MN
/m3 )
Unconfined compressive strength (qu) (MPa)
intactsd/B=1.5sd/B=1.0sd/B=0.5
IntactSd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
0
10
20
30
40
50
60
12 14 16 18 20 22 24 26 28
Ulti
mat
e be
arin
g ca
paci
ty (q
ult)
(MPa
)
Unconfined compressive strength (qu) (MPa)
intactsd/B=1.5sd/B=1.0sd/B=0.5
IntactSd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
(a)
(b)
(c)
Fig. 5 Effect of unconfined compressive strength (qu): a stress–
settlement curves with qu = 15 and 24 MPa [Sd = 12 cm (1.5B),
Id = 90�], b modulus of subgrade reaction, c ultimate bearing
capacity
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact
16
12
8
4
Sd=16cm (2.0B)Sd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
Intact
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact161284
Sd=16cm (2.0B)Sd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
Intact
(a)
(b)
Fig. 6 Stress–settlement curves with different discontinuity spacing:
a discontinuity inclination (Id) = 60�, b discontinuity inclination
(Id) = 90�
2060 J. Lee, S. Jeong
123
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4 Proposed Modulus of Subgrade Reactionfor Jointed Rock
The modulus of subgrade reaction can be obtained from
field and laboratory tests, and semi-empirical equation.
Among the semi-empirical methods, Vesic’s model is
widely applied to mat foundations in the literature (Bowles
1996). However, these empirical methods were suggested
based on the soil; therefore, they do not consider the
influence of rock mass discontinuities.
A rock foundation is supported by the jointed rock mass,
and not by intact rocks. The behavior of foundations on
rock is largely dependent on the strength of the rock mass.
The rock mass consists of intact rock and discontinuities
(joints or fractures, faults, and possibly bedding planes).
Discontinuities usually have a lower resistance, higher
deformability, and conductivity than the intact rock and, in
most cases, govern the behavior of the rock mass. Clearly
rock mass discontinuities affect the bearing behavior.
Discontinuities may significantly influence the strength of
0
200
400
600
800
1000
1200
0 0.5 1 1.5 2 2.5Mod
ulus
of s
ubgr
ade
reac
tion
(ks)
(MN
/m3 )
Discontinuity spacing/mat width ratio (Sd/B)
90
0
60
30
Id=90
Id=0
Id=60
Id=30
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5
Ulti
mat
e be
arin
g ca
paci
ty (q
ult)
(MPa
)
Discontinuity spacing/mat width ratio (Sd/B)
90
0
60
30
Id=90
Id=0
Id=60
Id=30
(a)
(b)
Fig. 7 Effect of discontinuity spacing (Sd): a modulus of subgrade
reaction, b ultimate bearing capacity
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
9006030
Id=90Id=0Id=60Id=30
Fig. 8 Stress–settlement curves with different discontinuities
0
200
400
600
800
1000
1200
0 15 30 45 60 75 90Mod
ulus
of s
ubgr
ade
reac
tion
(ks)
(MN
/m3 )
Discontinuity Inclination (º)
S/B=2.0S/B=1.5S/B=1.0S/B=0.5
Sd=16cm (2.0B)Sd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
0
5
10
15
20
25
30
35
40
0 15 30 45 60 75 90
Ulti
mat
e be
arin
g ca
paci
ty (q
ult)
(MPa
)
Discontinuity Inclination (º)
S/B=2.0S/B=1.5S/B=1.0S/B=0.5
Sd=16cm (2.0B)Sd=12cm (1.5B)Sd=8cm (1.0B)Sd=4cm (0.5B)
(a)
(b)
Fig. 9 Effect of discontinuity inclination (Id): a modulus of subgrade
reaction, b ultimate bearing capacity
Experimental Study of Estimating the Subgrade Reaction Modulus on Jointed Rock Foundations 2061
123
Author copy
thorhohohoo2
ing/mat width ratio (Sd/ooy spacing (Sd): dulus of subgrade
a g capac typac tyapacity
AutAutAAAAAAA025
30
3
40
opyor co
pyyyyy9(º) y
r copyr crr cr cccrr opyyyopooppyyopyopycopycocoppyycopycocoppyycocococoppppyyyyo
reaction
the rock mass, depending on their inclination and the nat-
ure of the filling material in the discontinuities (Pells and
Turner 1980; Zhang and Einstein 1998; Zhang 2010; Jeong
et al. 2010; Lee et al. 2013). Thus, besides the intact rock
properties, the influence of rock mass discontinuities must
be taken into account.
The results obtained from the model tests show that the
modulus of subgrade reaction of rock foundation is mainly
affected by the rock strength and the rock mass disconti-
nuities. The model-scale footing test results are shown in
Fig. 10. The modulus of subgrade reaction is considered as
an initial tangent modulus, and the results show that a
reduction in the modulus of subgrade reaction (ks) occurred
due to rock discontinuities. In this study, a joint reduction
factor (Jf) is proposed considering rock discontinuities
based on the test results. Joint reduction factor (Jf) is
defined as the ratio of the subgrade reaction modulus of the
jointed rock and that of intact rock. Figure 11 shows a joint
reduction factor (Jf) chart as a function of the discontinuity
inclination (Id) and the discontinuity spacing/mat width
ratio (Sd/B). The modulus of the subgrade reaction, con-
sidering rock discontinuities, decreased by up to approxi-
mately 60 %, compared to the intact rock. Additionally, the
modulus of subgrade reaction is proportional to the dis-
continuity spacing, and it decreases gradually within the
range of 0�–30� and tends to increase within the range of
30�–90�. Table 4 summarizes the values of the joint
reduction factor (Jf) based on the obtained results.
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact9006030
Id=90Id=0Id=60Id=30
Intact
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact9006030
Id=90Id=0Id=60Id=30
Intact
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact9006030
Id=90
Id=60Id=30
Intact
Id=0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8
Stre
ss (M
Pa)
Settlement (mm)
Intact9006030
Id=90
Id=60Id=30
Intact
Id=0
(a) (c)
(b) (d)
Fig. 10 Stress–settlement curves with varying discontinuity inclination: a Sd = 16 cm (2.0B), b Sd = 12 cm (1.5B), c Sd = 8 cm (1.0B) and
d Sd = 4 cm (0.5B)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 15 30 45 60 75 90
Join
t Red
uctio
n Fa
ctor
Major Discontinuity Inclination (º)
S/B=2.0S/B=1.5S/B=1.0S/B=0.5
Sd /B=2.0Sd /B=1.5Sd /B=1.0Sd /B=0.5
Fig. 11 Joint reduction factor (Jf) chart
2062 J. Lee, S. Jeong
123
Author copy
on their inclination and the nat-their inclination and the nat-
rial in the discontinuities (Pells andin the discontinuities (Pells and
g and Einstein 1998; Zhang 2010; JeongEinstein 1998; Zhang 2010
e et al. 2013). Thus, besides the intact rock013). Thus, besides the inta
the influence of rock mass discontinuities muce of rock mass disco
en into account.c
The results obtained from the model tests shoThe results obtained from the model
modulus of subgrade reaction of rock foundamodulus of subgrade reaction of
affected by the rock strength and the rofected by the rock strength
nuities. The model-scale footing testies. The model-scale f
Fig. 10. The modulus of subgrad10. The modulus
n initial tangent modulusnitia
uction in the modulution
o rock discontro
JfJ ) isf
hor copy
r opyccc ppypycop
2 3opopopoopId=pI
Id=0
pity inclination: a Sdo
Finally, a subgrade reaction modulus (kj) for jointed
rock is proposed. In particular, by considering the discon-
tinuity spacing and inclination, a simple and improved kj is
introduced in this study. The kj can be expressed by the
joint reduction factor (Jf) and subgrade reaction modulus.
As a result, kj is suggested as follows:
kj ¼ Jf � ks ¼ Jf � E
Bð1� m2Þ� �
ð5Þ
where Jf is the joint reduction factor (Table 4), the other
parameters were defined previously.
5 Conclusions
The main objective of this study was to propose the sub-
grade reaction modulus of a rock mass that can consider the
rock discontinuity conditions. Due to the limitations
inherent in field measurements in Korean rock, a series of
experimental studies were performed to investigate the
factors that influence the subgrade reaction modulus for
rock foundations. The conclusions of this study are as
follows:
1. Based on the results of the model footing tests, the
influential factors for predicting the subgrade reaction
modulus (ks) and the ultimate bearing capacity (qult)
for rock foundations are introduced. The results
showed that the values of ks and qult are highly
dependent on factors, i.e., the strength of the rock
mass, the spacing, and the inclination of the rock
discontinuity. With the increase in both the rock
strength and discontinuity spacing, ks and qult increase.
2. In particular, the rock mass discontinuity is found to
affect the subgrade reaction modulus of the rock
foundation. Consequently, the discontinuity in the rock
mass causes the reduction of the modulus of subgrade
reaction (ks) by up to approximately 60 % compared to
the intact rocks. In addition, the value of ks decreases
gradually within the range of 0�–30� and then tends to
increase within the range of 30�–90�. It is observed
that the least value of ks is in the vicinity of joint
inclination of 30�.3. As a result, the joint reduction factor (Jf) is proposed
by varying the major factors, i.e., the discontinuity
inclination (Id) and the discontinuity spacing/mat width
(Sd/B). By taking into account the rock mass discon-
tinuity, the proposed subgrade reaction modulus (kj)
can be used for jointed weathered rocks.
Acknowledgments This work was supported by the National
Research Foundation of Korea (NRF) Grant funded by the Korean
government (MSIP) (No. 2011-0030040).
References
Biot MA (1937) Bending of infinite beams on an elastic foundation.
J Appl Mech Am Soc Mech Eng 59:A1–A7
Boussinesq J (1883) Application des potentials a l’etude de
l’equilibre et due movement des solides elastiques. Gauthier-
Villars, Paris
Bowles JE (1996) Foundation analysis and design, 5th edn. McGraw-
Hill, New York
Cerato AB, Lutenegger AJ (2007) Scale effects of shallow foundation
bearing capacity on granular material. J Geotech Geoenviron
Eng 133(10):1192–1202
Daloglu AT, Vallabhan CVG (2000) Values of K for slab on winkler
foundation. J Geotech Geoenviron Eng 126(5):463-471
Elachachi SM, Breysse D, Houy L (2004) Longitudinal variability of
soils and structural response of sewer networks. Comput Geotech
31:625–641
Heuze FE (1980) Scale effects in the determination of rock mass
strength and deformability. Rock Mech 12:167–192
Hoek E, Brown ET (1980) Empirical strength criterion for rock
masses. ASCE J Geotech Eng 106:1013–1035
Hoek E, Brown ET (1997) Practical estimates of rock mass strength.
Int J Rock Mech Min Sci 34(8):1165–1186
Horvath JS (1989) Subgrade models for soil–structure interaction
analysis. J Found Eng Curr Princ Pract Proc ASCE 20:599–612
Indraratna B, Haque A, Aziz N (1998) Laboratory modeling of shear
behaviour of soft joints under constant normal stiffness condi-
tions. Geotech Geol Eng 16:17–44
Jeong SS, Cho HY, Cho JY, Seol HI, Lee DS (2010) Point bearing
stiffness and strength of socketed drilled shafts in Korean rocks.
Int J Rock Mech Min Sci 47:983–995
Jiang Y, Xiao J, Tanabashi Y, Mizokami T (2004) Development of an
automated servo-controlled direct shear apparatus applying a
constant normal stiffness condition. Int J Rock Mech Min Sci
41:275–286
Kloppel K, Glock D (1979) Theoretische und experimentelle
untersuchungen zu den traglastproblemen beigewiecher, in die
erde eingebetteter rohre. Veroffentlichung des Instituts Statik
und Stahlbau der Technischen Hochschule Darmstadt, H-10
Lee JH, You KH, Jeong SS, Kim JY (2013) Proposed point bearing
load transfer function in jointed rock-socketed drilled shafts. Soil
Found 53(4):596–606
Maghous S, Bernaud D, Freard J, Garnier D (2008) Elastoplastic
behavior of jointed rock masses as homogenized media and finite
element analysis. Int J Rock Mech Min Sci 45(8):1273–1286
Meyerhof GG, Baikie LD (1963) Strength of steel sheets bearing
against compacted sand backfill. Highway research board
proceedings 30
Table 4 Joint reduction factor
Sd/Ba Id
b
0 30 60 90
0.5 0.63 0.43 0.58 0.71
1.0 0.73 0.58 0.64 0.78
1.5 0.83 0.63 0.71 0.86
2.0 0.85 0.67 0.73 0.91
a Discontinuity spacing/mat widthb Major discontinuity inclination
Experimental Study of Estimating the Subgrade Reaction Modulus on Jointed Rock Foundations 2063
123
Author copy
o propose the sub-e the sub-
ss that can consider theconsider the
Due to the limitationsDue to the lim
nts in Korean rock, a series ofs in Korean rock, a series of
were performed to investigate there performed to investigate the
nce the subgrade reaction modulus fore subgrade reaction mod
ns. The conclusions of this study are asonclusions of this study
Based on the results of the model footing tethe results of the model
influential factors for predicting the subgrainfluential factors for predic
modulus (modulus ks) and the ultimate bearingand the ultima
for rock foundations are introdfor rock foundations
showed that the values ofshowed that the
dependent on factors, i.edependent on f
mass, the spacing, amass
discontinuity. Wisco
ength and d
articu
the
nfinite beams on an elastic foundation.ms on an elastic foundation
c Mech Eng 59:A1–A7g 59:A
Application des potentials a l’etude dedes potentials a l’etude de
due movement des solides elastiques. Gauthier-ement des solides elastiques. Gauthier-
aris
(1996) Foundation analysis and design, 5th edn. McGration analysis and design, 5th
ill, New York
rato AB, Lutenegger AJ (2007) Scale effects of shallowutenegger AJ (2007) Scale effects of shal
bearing capacity on granular material. J Geotecapacity on granular material. J Geot
Eng 133(10):1192–12023(10):1192–1202
Daloglu AT, Vallabhan CVG (2000) Values oVallabhan CVG
foundation. J Geotech Geoenviron En. J Geotech G
Elachachi SM, Breysse D, Houy L (2Elachachi SM, Breysse D,
soils and structural response
31:625–641
Heuze FE (1980) ScaleHeuz
strength and defstr
Hoek E, Brownek
masses.
Hoek E, B
I
H
Pells PJN, Turner RM (1980) End-bearing on rock with particular
reference to sandstone. Proc Int Conf Struct Found Rock,
Balkema, Rotterdam 1: 181–190
Roy N (1993) Engineering behavior of rock masses through study of
jointed models. Ph.D thesis, Indian Institute of Technology,
Delhi, India
Sadrekarimi J, Akbarzad M (2009) Comparative study of methods of
determination of coefficient of subgrade reaction. Electron J
Geotech Eng 14:1–14
Selvadurai APS (1985) Soil–pipeline interaction during ground
movement. In: Bennett FL, Machemehl JL (eds) Arctic, civil
engineering in the Arctic offshore. ASCE speciality conference,
San Francisco, pp 763–773
Seol HI, Jeong SS, You KH (2008) Shear load transfer for rock-
socketed drilled shafts based on borehole roughness and
geological strength index (GSI). Int J Rock Mech Min Sci
45(6):848–861
Sutcliffe D, Yu HS, Sloan SW (2004) Lower bound solutions for
bearing capacity of jointed rock. Comput Geotech 31(1):23–36
Terzaghi KV (1955) Evaluation of coefficient of subgrade reaction.
Geotechnique 5(4):297–326
Vesic AB (1961) Beams on elastic subgrade and Winkler’s hypoth-
esis. Proc 5th Int Conf Soil Mech Found Eng 845–850
Vlassov VZ, Leontiev NN (1966) Beams, plates, and shells on elastic
foundations. Translated from Russian, Israel Program for
Scientific Translations, Jerusalem
Yang ZY, Chiang DY (2000) An experimental study on the
progressive shear behavior of rock joints with tooth-shaped
asperities. Int J Rock Mech Min Sci 37:1247–1250
Zhang L (2010) Method for estimating the deformability of heavily
jointed rock masses. J Geotech Geoenviron Eng 136:1242–1250
Zhang L, Einstein HH (1998) End bearing capacity of drilled shafts in
rock. J Geotech Geoenviron Eng 124:574–584
Ziaie Moayed R, Naeini SA (2006) Evaluation of modulus of
subgrade reaction in gravely soils based on standard penetration
test (SPT). Proceedings of the sixth international conference on
physical modelling in geotechnics, 6th ICPMG, Hong Kong,
Chapter 115
2064 J. Lee, S. Jeong
123
yof modulu
standard penetratinetra
ernational conference onence
Author copy, 6th ICPMG, Hong Kong,MG, Hong Kon
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