experimental study of estimating the subgrade reaction modulus

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

[email protected]

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


123

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uity appa

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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)


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

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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


Intact

(a)

(b)

Fig. 6 Stress–settlement curves with different discontinuity spacing:

a discontinuity inclination (Id) = 60�, b discontinuity inclination

(Id) = 90�


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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


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


(a)

(b)

Fig. 9 Effect of discontinuity inclination (Id): a modulus of subgrade

reaction, b ultimate bearing capacity


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


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).

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