See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/349517111
Subgrade reaction modulus of rock masses under the load of single and
multiple footings
Article in Geomechanics and Geoengineering · February 2021
DOI: 10.1080/17486025.2021.1889687
CITATION
READS
1
5,033
2 authors:
Saeed Shamloo
Meysam Imani
Amirkabir University of Technology
Amirkabir University of Technology
9 PUBLICATIONS 29 CITATIONS
26 PUBLICATIONS 192 CITATIONS
SEE PROFILE
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Earth Pressure View project
Upper bound stability analysis of crushed rock slopes considering the Hoek-Brown non linear failure criterion View project
All content following this page was uploaded by Saeed Shamloo on 25 February 2021.
The user has requested enhancement of the downloaded file.
Geomechanics and Geoengineering
An International Journal
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tgeo20
Subgrade reaction modulus of rock masses under
the load of single and multiple footings
Saeed Shamloo & Meysam Imani
To cite this article: Saeed Shamloo & Meysam Imani (2021): Subgrade reaction modulus of rock
masses under the load of single and multiple footings, Geomechanics and Geoengineering, DOI:
10.1080/17486025.2021.1889687
To link to this article: https://doi.org/10.1080/17486025.2021.1889687
Published online: 22 Feb 2021.
Submit your article to this journal
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
https://www.tandfonline.com/action/journalInformation?journalCode=tgeo20
GEOMECHANICS AND GEOENGINEERING
https://doi.org/10.1080/17486025.2021.1889687
Subgrade reaction modulus of rock masses under the load of single and multiple
footings
Saeed Shamloo and Meysam Imani
Geotechnical Engineering Group, Amirkabir University of Technology, Garmsar, Iran
ABSTRACT
ARTICLE HISTORY
Subgrade reaction modulus is an essential parameter in the structural design of footings.
Determination of this parameter is difficult, especially in rock masses containing an irregular dis­
tribution of discontinuities. In this paper, the subgrade reaction modulus of rock masses obeying
Hoek-Brown failure criterion was investigated. It was assumed that the rock masses are under the
pressure of a single isolated footing and also two adjacent strip footings. The subgrade reaction
modulus was determined from the pressure-settlement curve of the rock masses. This study’s results
were presented in two forms, which include the subgrade reduction factor and the interference
coefficient. The obtained results show that the subgrade reaction modulus of rock masses may be
smaller than 50% of intact rocks. Likewise, based on the distance between the multiple footings, the
subgrade reaction modulus may be in the range of 0.7 to 1.3 times the subgrade reaction modulus in
the case of a single isolated footing.
Received 5 July 2020
Accepted 9 February 2021
1. Introduction
The analysis of the interaction between footings and
subsurface materials is a challenging problem in geo­
technical engineering. In some footing-bedding inter­
action analysis methods, the soil/rock bedding
material is replaced by a simple system that contains
a collection of springs. This approach has been used
extensively in the design of foundations by structural
engineers. The stiffness coefficient of these springs,
which is named as the subgrade reaction modulus
(SRM) in the current study, can be obtained using
Equation (1):
ks ¼ q=δ
(1)
In which q is the footing pressure and δ is the corre­
sponding settlement. After Biot (1937), many studies
were performed on the SRM, i.e., ks, of soils. Among
others, Terzaghi (1955) proposed ks value for sand and
clay as Equations (2) and (3), respectively.
�
�
B þ 0:3 2
ks ¼ k0:3
(2)
2B
ks ¼ k0:3
� �
0:3
B
(3)
In which k0.3 (in the unit of force/volume, like MN/m3)
is the value of ks from a plate load test, and B (m) is the
CONTACT Meysam Imani
imani@aut.ac.ir
© 2021 Informa UK Limited, trading as Taylor & Francis Group
KEYWORDS
Hoek-Brown; interference
coefficient; multiple footing;
rock foundation; subgrade
reaction modulus
footing width. This method was initially proposed for
a 30 cm×30 cm concrete rigid slab rested on the soil
bedding surface. The obtained results show that the ks
value depends on the dimensions of the footing.
Vesic (1961) showed that ks is related to both the soil
and structural stiffness. He proposed a precise relation
for ks, which is presented in Equation (4):
�1=
�
� �
0:65
Es
Es � B4 12
ks ¼
�
�
1 ν2s
EI
B
(4)
In which, Es (in the unit of force/area, like MPa) and νs
are the soil modulus of elasticity and the Poisson’s ratio,
respectively, and EI (in the unit of force multiplied by
area, like kN.m2) is the footing flexural stiffness. For
practical applications, one can use Vesic’s simplified
relation as follows:
ks ¼
Es
B 1
ν2s
�
(5)
Vlassov and Leontiev (1966) also proposed the reaction
coefficient for the beams and plates resting on an elastic
semi-infinite medium as follows:
ks ¼
Es ð1 νs Þ
μ
�
ð1 þ νs Þð1 2νs Þ 2B
(6)
The determination of the dimensionless factor μ is ambig­
uous, which makes the use of this method very complicated
2
S. SHAMLOO AND M. IMANI
(Sadrekarimi and Akbarzad 2009). Likewise, Imanzadeh
et al. (2013) showed that soil stiffness has the most signifi­
cant effect on the uncertainty of the SRM, while other
parameters like structural properties have little importance.
As another research, by comparing the results of finite
element models with those obtained from field plate load
tests, Avci and Gurbuz (2018) showed that the modulus of
subgrade reaction decreases with the increasing settlement
of the soils. Many other kinds of research were performed
to determine soil horizontal SRM in buried circular chan­
nels (Meyerhof and Baikie 1963, Klopple and Glock 1979,
Selvadurai 1985, Ziaie Moayed and Naeini 2006).
Because of the complexity of rock media, rock
masses’ load-displacement behaviour has gained little
attention in past research. Among others, Carter and
Kullhawy (1988) presented the load-displacement
response and the corresponding ultimate bearing
capacity of rock masses subjected to loads of
socketed shafts. Also, Kullhawy and Carter (1992)
investigated different aspects of settlement and bear­
ing capacity analysis of foundations on rock masses.
Using plasticity and finite element models, Alhossein
et al. (1992) proposed solutions for strip footings on
a regularly jointed rock mass with one and two joint
sets. To the authors’ knowledge, the problem of
determining the SRM for rock masses was not ade­
quately investigated by researchers. Very few pub­
lished papers can be found regarding this issue in
the literature. The few available studies show that the
SRM of the rock masses is smaller than that of intact
rocks. Since the rock masses’ elastic modulus is
smaller than that of intact rocks and the SRM has
a positive correlation with the elastic modulus.
The experimental research performed by Lee and
Jeong (2016) can be considered as the most recent
study regarding the SRM of jointed rocks. They
constructed some artificial joints with different orien­
tations and spacings in jointed rock specimens and
finally obtained the joint reduction factor (Jf) for
each case. They suggested that the SRM of the
jointed rocks (kj) can easily be obtained from
Equation (7):
kj ¼ Jf � ks ¼ Jf �
Es
B 1
ν2s
�
(7)
In which ks is the SRM of the intact rock, which can
be obtained using Vesic’s simplified formula, i.e.,
Equation (5). This equation is widely used in prac­
tical applications for soil beddings. Es and νs are the
elastic modulus, and the Poisson’s ratio of the intact
rock, respectively, and B is the footing width.
The method presented by Lee and Jeong (2016) is
applicable only for the jointed rocks containing specific
joints and can not be used in general cases. However, in
many practical projects, engineers are faced with rock
masses that contain irregular fractures with no distinct
joint sets. In such situations, the Hoek-Brown failure
criterion (Hoek and Brown 1980) has widely been
applied by the engineers. To the authors’ knowledge,
no precise method is available for obtaining the SRM of
the rock masses obeying the Hoek-Brown failure criter­
ion, and the present paper seems to be the pioneer in
this field. This issue is of paramount importance since
many massive structures, like a bridge, are usually con­
structed on crushed rock masses, and structural engi­
neers need the SRM of the underlying rock mass for
designing the foundation of the structure.
In this paper, by considering different properties for
the rock masses, detailed sensitivity analyses were per­
formed to obtain the SRM of the rock masses. The
results of this study were presented as a factor named
‘subgrade reduction factor’ that can be used for deter­
mining the SRM of the rock masses. Simple tables were
presented for obtaining the subgrade reduction factor,
which can easily be used in practical applications.
Moreover, as a new subject in rock foundation pro­
blems, the effect of multiple footings on the SRM was
also investigated.
2. Analysis method
The SRM is a parameter that can be obtained by
dividing the footing pressure (q) by the correspond­
ing settlement (δ). This parameter is not constant,
and it depends on the width and depth of the foot­
ing. Two common methods are available for deter­
mining this parameter, which includes the loading
experiment on the base material and the numerical
analysis. The latter was used in the present study by
applying the finite element software, Phase2.
2.1. Boundary conditions
In the numerical models, the side boundaries were kept
fixed in the horizontal direction, while the base of the
models was kept fixed in both the vertical and the
horizontal directions. Model meshing was performed
by 6 nodded triangular elements.
An important issue in numerical modelling is the
distance between the footing edges to the sides and
the bottom boundaries of the model. Fattah et al.
(2014a, 2014b, 2014c) showed that these distances
greatly influence the amount of footing settlement
and the corresponding bearing capacity. Lee and
Jeong (2016) considered these distances approxi­
mately equal to 3.5 times the footing width (i.e.,
3.5B). Lees (2016) proposed a distance approximately
equal to 3B is enough. In the present study, to
minimise the boundary effects on the results, the
distance between the footing edge to the side bound­
aries was considered equal to 5B, while a distance
equal to 3.5B was considered from the footing base
to the bottom boundary of the model. Figure 1
represents the general shape of the numerical models
considered in the current research.
In the numerical models, the footing pressure was
applied incrementally to the top boundary of the
model at a rate equal to 2*10−5m/s. Lee and Jeong
(2016) were also considered the same loading rate in
their experimental tests.
Two sets of models were constructed in this
research, including the verification models and the
sensitivity analysis models. In the verification mod­
els, the width of the footing was considered to be
equal to 8 cm, which is consistent with the models
constructed by Lee and Jeong (2016). However, in
the sensitivity analysis models, the footing width was
set to one metre.
2.2. Determination of the SRM from the
pressure-settlement curve
The pressure-settlement curve of the rock masses can
be drawn by applying an incremental footing pres­
sure and obtaining the corresponding settlements.
This curve can be used to determine the bearing
capacity and the SRM of the rock masses. Most
previous numerically obtained pressure-settlement
curves were used for determining the ultimate bear­
ing capacity of soil/rock masses [(Mabrouki et al.
2010, Javid et al. 2015, Sargazi and Hosseininia
2017, Mansouri et al. 2019). This issue is out of the
scope of this paper, and only determining the SRM
Figure 1. Boundary condition for numerical model.
3
pressure (q)
GEOMECHANICS AND GEOENGINEERING
SRM
1
settlement (δ)
Figure 2. An example of a pressure-settlement curve and the
corresponding SRM.
was focused here. The pressure-settlement curve is
generally nonlinear, but the curvature of its initial
part, before the yield point, is much smaller than the
curvature of its post-yield point. As shown in Figure
2, an approximate line can be fitted to the initial part
of the pressure-settlement curve. The slope of this
line was called the SRM. Lee and Jeong (2016) were
used this approach for obtaining the SRM of their
experimental models. In the present research, the
same approach was used for obtaining the SRM of
rock masses.
3. Verification of the numerical modelling
The results of the present study were verified by the
experimental tests performed by Lee and Jeong (2016).
They constructed 21 jointed rock samples with dimen­
sions equal to 48 cm*48 cm*28 cm. Each sample was
subjected to a distributed incremental load with a width
equal to 8 cm, and the corresponding pressuresettlement curve was obtained. Then, they obtained
the SRM as the slope of the nearly linear part of the
curve occurred at the beginning of it.
4
S. SHAMLOO AND M. IMANI
Table 1. The properties of the jointed rocks considered in the
analyses.
Parameter
σci (MPa)
Sd (cm)
Id (degree)
Es (MPa)
Poisson’s ratio (ν)
ϕj (degree)
cj (MPa)
Kn (GPa/m)
Ks (GPa/m)
γ (MN/m3)
Magnitiude
15
4, 8
0, 30, 60, 90
860
0.3
35
0
100
10
0.027
3.1. The pressure-settlement curve and the
corresponding SRM obtained
For all the 21 samples, the numerical models were
constructed in the current study, and the pressuresettlement curves were compared with those
obtained by Lee and Jeong (2016). The properties
considered for the jointed rock are presented in
Table 1. As an example, for the eight models
shown in Figure 3, the results of this comparison
were shown in Figure 4. As shown in Figure 4, the
overall trend of the curves obtained from the numer­
ical models is approximately similar to that obtained
from the experimental tests. The most similarity of
the curves obtained from these two methods was
seen in the approximately linear part at the begin­
ning of the curves. It means that the slope of this
part of the curves, which corresponds to the SRM, is
very close to each other in numerical and experi­
mental models, which shows the applicability of the
numerical models constructed in the current
research. Table 2 presents a comparison between
the SRM obtained from the present study and the
Lee and Jeong (2016) experimental tests.
Unfortunately, Lee and Jeong (2016) did not men­
tion five parameters of their samples, including the
unit weight of the rock mass, the cohesion and the
friction angle of these joints and also the normal and
shear stiffness along the joints. In the current section,
an approximate logical magnitude was assumed for
these parameters, as presented in Table 1. However,
the sensitivity of the results to these assumed values
was investigated later in section 3.3.
3.2. The joint reduction factor
Determination of the SRM by using the in situ tests is
too difficult, especially in rock masses. So, by using
Equation (7), one can easily obtain the SRM for the
jointed rock masses (kj) by having the SRM of the
intact rocks (ks) and the joint reduction factor, Jf.
According to Lee and Jeong (2016), the SRM of the
intact rock can be obtained using Vesic’s simplified
relation (Equation (5)) that was initially proposed for
soil beddings. The Jf can also be obtained from the
simple chart provided in their study. As another ver­
ification of the present research, the joint reduction
factor, Jf, was obtained using the numerical method
and was compared with the experimental tests (Lee
and Jeong 2016). According to Equation (7), the Jf is
the ratio of SRM of jointed rocks to the intact rocks.
All the 21 specimens constructed by Lee and Jeong
(2016) were analysed numerically in the present
paper, and the obtained Jf were presented in Table 3.
This table shows that the difference between the
numerical analyses and the experimental tests is 2 to
9%, which is usually appropriate in practical engineer­
ing applications. The experimental tests performed by
Lee and Jeong (2016) show that SRM of the jointed
rocks is about 43% to 91% of SRM of the intact rocks,
while the numerical results reveal a magnitude of
about 54% to 86%.
3.3. The effect of the joint parameters on the
verification results
As discussed previously, Lee and Jeong (2016) did
not mention the unit weight of their samples and
also the joints cohesion, friction of angle, and the
normal and shear stiffness. The effect of these para­
meters on the numerical results was checked in this
section for model 6 of Figure 3.
Lee and Jeong (2016) ignored the filling material
in the joints for their samples. Therefore, the cohe­
sion along the joints is low and can be considered
equal to zero. On the other hand, different investiga­
tions show that the unit weight of the rock mass
does not have a considerable effect on the strength
of the rock mass (Yang and Yin 2005, Saada et al.
2008, AlKhafaji et al. 2020). As a result, only the
effects of the joint friction angle and the normal
and shear stiffness were investigated here. Table 4
presents the magnitudes considered for these para­
meters. In all cases, the joint shear stiffness was
taken into account equal to 0.1 times the joint nor­
mal stiffness. Other required parameters were select
from Table 1.
Figures 5 and 6 show the effect of the joint fric­
tion angle and the normal and shear stiffnesses,
respectively. These figures reveal that the mentioned
joint parameters do not have a considerable effect on
GEOMECHANICS AND GEOENGINEERING
5
Figure 3. Configuration of some of the samples tested by .Lee and Jeong (2016)
the SRM and the bearing capacity of the jointed
rocks. As an overall conclusion from the current
section, the numerical results obtained in this paper
are in reasonable conformity with the experimental
results.
4. Effect of the rock mass properties on the SRM
In many practical rock engineering projects, rock
masses contain a considerable amount of randomly
distributed discontinuities. The analysis of such rock
S. SHAMLOO AND M. IMANI
30
30
25
25
20
20
Pressure (MPa)
Pressure (MPa)
6
15
10
Present study
15
10
Present study
5
5
Lee & Jeong (2016)
Lee & Jeong (2016)
0
0
0
2
4
Sattlement (mm)
6
0
8
2
25
30
20
25
15
10
Present study
5
8
20
15
10
Present study
5
Lee & Jeong (2016)
Lee & Jeong (2016)
0
0
0
2
4
Settlement (mm)
6
0
8
2
30
25
25
Pressure (MPa)
30
20
15
10
Present study
5
4
Settlement (mm)
6
8
Model 4
Model 3
Pressure (MPa)
6
Model 2
Pressure (MPa)
Pressure (MPa)
Model 1
4
Sattlement (mm)
20
15
10
Present study
5
Lee & Jeong (2016)
Lee & Jeong (2016)
0
0
0
2
4
Settlement (mm)
6
8
0
2
Model 5
4
6
Settlement (mm)
8
Model 6
30
35
25
30
Pressure (MPa)
Pressure (MPa)
25
20
15
10
Present study
5
20
15
10
Present study
5
Lee & Jeong (2016)
Lee & Jeong (2016)
0
0
0
2
4
Settlement (mm)
Model 7
6
8
0
2
4
Settlement (mm)
Model 8
Figure 4. Comparison between the results of the present study with .Lee and Jeong (2016)
6
8
GEOMECHANICS AND GEOENGINEERING
7
Table 2. Comparison between the SRM obtained from the pre­
sent study and the experimental tests performed by Lee and
Jeong (2016).
Model
1
2
3
4
5
6
7
8
Present study
(Values in MN/m3)
6137
6740
4740
5390
5894
6672
6980
7868
Lee and Jeong (2016)
(Values in MN/m3)
7220
7798
4780
5480
6370
7020
8340
9160
Difference (%)
15
13
1
2
7
5
16
14
masses has usually been performed by considering
a homogeneous isotropic rock mass with the HoekBrown failure criterion. Equation (8) defines the gen­
eral form of the Hoek-Brown failure criterion:
(8)
In which, σ1 and σ3 are the major and minor principal
stresses, respectively, σci is the uniaxial compressive
strength of the intact rock, and mb, s, and a are the
Hoek-Brown parameters that can be obtained using
Equations (9)–(11), respectively.
�
�
GSI 100
mb ¼ mi exp
(9)
28 14D
�
�
GSI 100
s ¼ exp
9 3D
1 1�
a¼ þ e
2 6
GSI
16
20
3
(10)
Figure 5. Effect of joints friction angle on pressure-settlement
curve.
35
30
25
P ressure (MP a)
�
�a
σ3
σ 1 ¼ σ 3 þ σ ci mb þ s
σ1
20
15
Kn=50GPa/m
, Ks=5GPa/m
Present
study
10
Kn=150GPa/m , Ks=15GPa/m
30
�
(11)
5
In which mi is an empirical constant for the intact rock,
GSI is the geological strength index, and D is the dis­
turbance factor.
In the following sections, the SRM of the rock masses
obeying the Hoek-Brown failure criterion, i.e., ks(mass),
0
e
Kn=100GPa/m , Ks=10GPa/m
25
Kn=200GPa/m , Ks=20GPa/m
40
Lee & Jeong (2016)
Lee&Jeong[13]
0
2
4
Settlement (mm)
6
8
Figure 6. Effect of normal and shear stiffness of joints on
pressure-settlement curve.
Table 3. Comparison between the joint reduction factor (Jf) obtained from the present study with Lee and Jeong (2016).
Id = 0°
(Sd/B)
0.5
1
1.5
2
Lee & Jeong
0.63
0.73
0.83
0.85
Present study
0.68
0.75
0.79
0.80
Id = 30°
Difference (%)
5
2
4
5
Lee & Jeong
0.43
0.58
0.63
0.67
Id = 60°
0.5
1
1.5
2
Present study
0.54
0.65
0.69
0.75
Difference (%)
9
7
6
4
Id = 90°
Lee & Jeong
Present study
Difference (%)
Lee & Jeong
Present study
Difference (%)
0.58
0.64
0.71
0.73
0.64
0.71
0.74
0.75
6
7
3
2
0.71
0.78
0.86
0.91
0.66
0.75
0.84
0.86
5
3
2
5
8
S. SHAMLOO AND M. IMANI
Table 4. The properties considered for the rock joints.
Parameter
ϕj (degree)
Kn (GPa/m)
Ks (GPa/m)
Magnitude
25, 30, 35, 40
50, 100, 150, 200
5, 10, 15, 20
Table 5. Rock mass properties considered for sensitivity analyses.
Parameter
σci (MPa)
GSI
mi
D
Magnitude
5, 25, 50, 75, 100
10, 30, 50, 70, 90
5, 10, 15
0, 0.5, 0.8
was investigated considering all combinations of the
parameters presented in Table 5. The ks(mass) is the slope
of the initial approximately linear part of the pressuresettlement curve obtained from numerical models. The
rock mass was considered to be under the pressure of
a strip footing with one-metre width. Due to a large
number of the models, only the results of some of them
were presented in the following sections.
The elastic properties of the rock mass calculated
using the following relations that were proposed by
Hoek and Brown (1980) and Vásárhelyi (2009),
respectively:
�
�rffiffiffiffiffiffiffi
GSI 10
D
σ ci
� 10ð 40 Þ
(12)
Em ðGPaÞ ¼ 1
100
2
νm ¼
0:002GSI
0:003mi þ 0:457
(13)
In which Em and νm are the rock mass deformation
modulus and the Poisson’s ratio, respectively, and the
Hoek-Brown parameters, D, σci (MPa), GSI, and mi
were selected from Table 5.
4.1. Effect of σci
Figure 7 shows the effect of σci on the ks(mass), considering
different Hoek-Brown parameters. It can be seen that
increasing the σci results in increasing the ks(mass). The
Figure 7. Effect of σci on ks,mass assuming: (a) D = 0 and mi = 10 (b) D = 0 and GSI = 30, and (c) mi = 10 and GSI = 30.
GEOMECHANICS AND GEOENGINEERING
9
Figure 8. Effect of GSI on ks,mass assuming: (a) D = 0 and mi = 10, (b) D = 0 and σci = 25 MPa and (c) mi = 10 and σci = 25 MPa.
Table 6. Comparison between the ks(mass) (MN/m3)obtained from
the present study with Lee and Jeong (2016) for the case of D = 0
mi = 10 and GSI = 88.
Present study
Lee and Jeong (2016)
Difference(%)
σci = 24 MPa
21743
20880
4
σci = 15 MPa
12478
11814
5
ks(mass) value is less affected by mi, while D and GSI sig­
nificantly influence the ks(mass).
4.2. Effect of GSI
Figure 8 shows the effect of GSI on the ks(mass), consider­
ing different Hoek-Brown parameters. It is clear that
increasing the GSI results in increasing the ks(mass). The
rate of this increment is not considerable for the case of
GSI<60, while for GSI>60, ks(mass) increases by a high
rate. This conclusion means that for the rock masses
with GSI>60, the ks(mass) is considerably affected by GSI.
Additionally, it can be seen that among the Hoek-Brown
parameters, mi has the lowest effect on the ks(mass).
It is interesting to note that Figure 8(a) shows Another
verification of the current study. As can be seen, the ks
obtained by Lee and Jeong (2016) for the rock masses
with σci = 24MPa and 15 Mpa are equal to 20,880 MN/m3
and 11,814 MN/m3, respectively, which were shown in
Figure 8(a) as two single points. These two points were
obtained from the experimental tests performed by Lee
and Jeong (2016) on the samples, not including the joints.
Naturally, such samples behave as rock masses with large
GSI values, i.e., approximately GSI = 90. The ks for the
mentioned σci and GSI were also obtained from the cur­
rent study, and the results were presented in Table 6. This
comparison shows a good agreement between the current
study and the experimental tests.
10
S. SHAMLOO AND M. IMANI
Figure 9. Effect of mi on ks,mass assuming: (a) D = 0 and GSI = 30 (b) D = 0 and σci = 25 MPa (c) σci = 25 MPa and GSI = 30.
4.3. Effect of mi
Figure 9 presents the effect of mi on the ks(mass). As
described in the previous subsection, mi has not
a significant influence on the ks(mass); only a small
increase in ks(mass) can be seen by increasing mi.
4.4. Effect of D
Figure 10 shows the effect of D on the ks(mass). By
increasing the disturbance of the rock mass, ks(mass)
will decrease. The rate of reduction of ks(mass) is widely
affected by σci. As shown in Figure 10(a), increasing
D results in decreasing the ks; the larger the σci, the
more reduction in the ks.
5. Effect of the footing width on the SRM
In the sensitivity analyses performed in the previous
sections, the width of the footing was taken into account
equal to one metre. In order to check the effect of the
footing width on the ks(mass) of the rock masses, several
B magnitudes that are common in practice were con­
sidered, and the obtained ks(mass) was shown in Figure
11. The figure shows that the footing’s width has
a considerable effect on the ks(mass) of the rock masses.
By increasing B, the rock mass settlement beneath the
footing will increase, which results in the reduction in
the ks(mass) (see Equation (1)). This reduction rate is
paramount in small magnitudes of B, especially in
large values of σci.
6. Suggesting a new coefficient for computing
the SRM of rock masses
6.1. Subgrade reduction factor, Jmass
The Hoek-Brown failure criterion is widely used in
practical rock engineering projects that can implicitly
consider the discontinuities of rock masses. Reviewing
the literature shows that no research has been per­
formed to obtain the SRM of the Hoek-Brown rock
masses subjected to a load of strip footings. Therefore,
it is of paramount importance to have a simple method
GEOMECHANICS AND GEOENGINEERING
11
Figure 10. Effect of D on ks,mass assuming: (a) GSI = 30 and mi = 10, (b) mi = 10 and σci = 25 MPa and (c) GSI = 30 and σci = 25 MPa.
to determine the SRM of rock masses in practical appli­
cations without performing time-consuming and com­
plicated numerical analyses. In this section, a new
coefficient named subgrade reduction factor, Jmass, was
proposed that can be used easily to determine the SRM
of rock masses obeying the Hoek-Brown failure criter­
ion. The Jmass can be computed as follows:
�
q
ksðmassÞ
qB 1 ν2i
δ
Jmass ¼
¼ Ei ¼
(14)
δEi
ks
Bð1 ν2i Þ
In which, ks(mass) is the SRM of rock masses obeying the
Hoek-Brown failure criterion that was obtained using the
numerical analyses performed in the present study. Also, ks
is the SRM of the intact rock that can easily be computed
using Vesic’s simplified formula for soils (Equation (5)).
For using Equation (5) for intact rocks, one should have
the modulus of elasticity and the Poisson’s ratio of the
intact rock (Ei and νi, respectively) and the footing width,
B. The corresponding modulus of deformation of the rock
mass can be obtained by Equation (12), using the Hoek-
Brown parameters of the rock mass. Then, using Equation
(15) proposed by Hoek and Dierichs (2005), one can
obtain the modulus of elasticity of the intact rock, Ei, that
can be used in Vesic’s simplified formula.
Ei ¼
Em
D=2
0:02 þ 1þexp1 60þ15D
ð 11
(15)
GSI
Þ
Using Equation (14), the Jmass factor was obtained for
various properties of the rock masses and were pre­
sented in Tables 7–9. As can be seen from these tables,
by increasing σci, GSI, and mi, the Jmass increases, which
means increasing the SRM of the rock mass. However,
increasing D results in decreasing the Jmass.
In practical engineering projects, having the SRM of
intact rocks ks from Equation (5), one can easily obtain
the SRM of the corresponding rock mass ks(mass) as
follows:
ksðmassÞ ¼ Jmass ks
(16)
12
S. SHAMLOO AND M. IMANI
Figure 11. Effect of B on ks,mass assuming: (a) D = 0, mi = 15 and GSI = 50 (b) σci = 25 MPa, D = 0 and mi = 15 (c) σci = 25 MPa, D = 0 and
GSI = 50.
Table 7. Values of Jmass for the case of mi = 5.
Table 8. Values of Jmass for the case of mi = 10.
σci(MPa)
GSI
10
30
50
70
90
D
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
5
0.004
0.003
0.002
0.020
0.009
0.006
0.071
0.026
0.017
0.129
0.094
0.055
0.271
0.114
0.090
25
0.006
0.005
0.003
0.025
0.013
0.007
0.083
0.038
0.024
0.244
0.150
0.109
0.400
0.286
0.225
50
0.008
0.006
0.004
0.027
0.014
0.009
0.124
0.049
0.034
0.291
0.154
0.110
0.463
0.303
0.226
σci(MPa)
75
0.010
0.008
0.005
0.033
0.015
0.010
0.127
0.056
0.042
0.321
0.171
0.113
0.512
0.318
0.227
100
0.011
0.009
0.006
0.034
0.016
0.011
0.128
0.060
0.044
0.323
0.194
0.121
0.523
0.354
0.229
GSI
10
30
50
70
90
D
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
5
0.007
0.005
0.002
0.023
0.011
0.007
0.078
0.035
0.020
0.158
0.102
0.065
0.290
0.176
0.117
25
0.008
0.006
0.004
0.029
0.014
0.010
0.102
0.046
0.027
0.267
0.164
0.113
0.459
0.306
0.241
50
0.009
0.008
0.005
0.030
0.015
0.012
0.124
0.053
0.040
0.341
0.175
0.119
0.474
0.331
0.243
75
0.011
0.009
0.006
0.034
0.016
0.013
0.128
0.060
0.044
0.343
0.177
0.120
0.518
0.332
0.244
100
0.012
0.010
0.007
0.037
0.017
0.014
0.130
0.061
0.047
0.350
0.226
0.122
0.526
0.362
0.253
GEOMECHANICS AND GEOENGINEERING
13
Table 9. Values of Jmass for the case of mi = 15.
σci(MPa)
GSI
10
30
50
70
90
D
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
0
0.5
0.8
5
0.008
0.006
0.003
0.032
0.013
0.009
0.081
0.038
0.022
0.173
0.107
0.070
0.293
0.179
0.121
25
0.010
0.007
0.005
0.033
0.015
0.012
0.118
0.055
0.034
0.273
0.173
0.109
0.462
0.316
0.244
50
0.011
0.009
0.007
0.034
0.016
0.013
0.134
0.060
0.042
0.348
0.178
0.120
0.508
0.333
0.245
75
0.012
0.009
0.008
0.035
0.017
0.014
0.136
0.061
0.044
0.349
0.180
0.122
0.531
0.340
0.249
100
0.013
0.010
0.009
0.037
0.019
0.015
0.139
0.062
0.049
0.356
0.241
0.133
0.535
0.376
0.260
Figure 12. Comparison between the ks, mass obtained from the
present study and the Wyllie (1999) method.
6.2. Verification of the proposed method
In this section, some practical examples were solved to
show the applicability of the presented method. Wyllie
(1999) proposed that the settlement of rock masses can
be calculated as follows:
δv ¼
Cd qBð1
E
ν2 Þ
(17)
In which Cd is the shape factor, q is the uniformly
distributed bearing pressure, B is the footing width,
and E and ν are the rock mass deformation modulus
and Poisson’s ratio, respectively. Therefore, using
Equation (1), which is the general form of the
SRM, one can obtain the SRM of rock masses as
follows:
ks; Wyllie ð1999Þ ¼
q
E
¼
δv Cd Bð1 ν2 Þ
(18)
In which, E and ν can be calculated using
Equations 12 and 13, respectively, and the average
value of Cd for a strip footing is equal to 2.25
(Wyllie 1999). Assuming B = 1 m, mi = 15 and
D = 0, Figure 12 shows a comparison between the
ks, mass obtained from the present study, i.e.,
Figure 13. Schematic of two adjacent footings.
Equation (16), and that obtained from the Wyllie
(1999) method, i.e., Equation (18). Good confor­
mity between the results of the two methods can
be seen, which implies the applicability of the pre­
sented method.
7. The SRM of the rock masses in the case of
two adjacent footings
Some structures founded on the rock masses, like
bridges, may have multiple footings that have been
constructed close to each other. It is necessary to con­
sider the interactive effects of the two adjacent footings
on the SRM of the underneath rock mass in such cases.
In the scope of the authors’ knowledge, there is no
research in this field. So this paper can be considered
a pioneer. Figure 13 shows a schematic of two adjacent
footings that were examined here by applying numerical
analyses. The width of each footing (B) was considered
equal to one metre, and S is the distance between them.
It was assumed that the pressure of both footings was
exerted on the rock mass simultaneously. Considering
14
S. SHAMLOO AND M. IMANI
the problem’s symmetry, the pressure-settlement curve
of one of the footings was used to determine the SRM.
In order to consider the interference effect of the two
adjacent footings, an interference coefficient, α, was
defined as the ratio of the SRM of the rock mass sub­
jected to a load of a footing in the presence of the
adjacent footing, ks(int), to the SRM of the rock mass
subjected to a load of a single isolated footing, ks(iso):
α¼
ks ðintÞ
ks ðisoÞ
(19)
It should be noted that ks(iso) is equal to the ks(mass) that
was obtained numerically in the past section. In order to
investigate the effect of various rock mass properties on
the α coefficient, several numerical models were carried
out, and the results were presented in the following
subsections.
7.1. Effect of σci
Assuming mi = 10 and D = 0, Figures 14 and 15
show the variation of α versus S/B for different
values of σci considering GSI = 10 and 50, respec­
tively. In S/B = 0, the two footings are in touch, and
they behave as a single footing with a width equal to
2B. In this case, the size of the stress bulb beneath
the multiple footings increases with respect to
a single footing, which results in increasing the set­
tlement and decreasing the corresponding SRM.
Thus, the α coefficient becomes smaller than one.
By increasing the distance between the two footings,
the confinement of the passive zones beneath the two
footings increases, which results in increasing the α
coefficient. This increment will continue until reach­
ing a maximum magnitude. The distance between
Figure 14. Effect of σci on α assuming GSI = 10, mi = 10 and D = 0.
the two footings that corresponds to the maximum
value of α was introduced as the critical spacing, Scr.
If the spacing of the two footings becomes larger
than Scr, the α coefficient decreases until reaching
its final magnitude which is equal to one. In such
a case, the footings distance is large enough to be
considered as a single isolated footing without any
influence from each other. Therefore, as a general
result, for S< Scr, the α coefficient increases by
increasing S, while for S> Scr, a reduction in α occurs
by increasing S. It is interesting to mention that the
bearing capacity of multiple footings on rock masses
also conforms such a trend (Javid et al. 2015,
Shamloo and Imani 2021) since both the SRM and
the bearing capacity have a direct positive correlation
with each other and both of them have been
obtained using the same tool, i.e., the pressuresettlement curve. Figures 14 and 15 show that
increasing σci results in increasing α. Also, the Scr
increases slightly by increasing the σci. As an exam­
ple, by increasing the σci from 5MPa to 100MPa, the
Scr/B ratio changes from about 3 to 4. A maximum
increase approximately equal to 60% can be seen in α
when the spacing between the two footings increases
from 0 to Scr. If the S magnitude exceeds the Scr and
reaches about 8B~10B, the two footings’ interference
effect will be disappeared and the α value
becomes one.
7.2. Effect of GSI
Figure 16 shows the variation of α versus S/B for differ­
ent values of GSI, considering σci = 25MPa, mi = 10, and
D = 0. The overall trends of variation of the curves are
similar to the previous subsection. As can be seen,
GEOMECHANICS AND GEOENGINEERING
15
Figure 15. Effect of σci on α assuming GSI = 50, mi = 10 and D = 0.
Figure 16. Effect of GSI on α assuming σci = 25 MPa, mi = 10 and D = 0.
increasing the GSI results in decreasing α, but the cri­
tical spacing (Scr) is always approximately equal to 4B,
which was not considerably affected by the GSI value.
The effect of interference of the footings on the α is
paramount for the rock masses with small values of GSI.
The interference effect of the two footings will disappear
at S/B > 8 for the rock masses with large values of GSI,
while for low GSI magnitudes, this interference effect
will be diminished at S/B > 10.
7.3. Effect of mi
Assuming σci = 25 MPa, and D = 0, Figures 17–19 show
the variation of α versus S/B for different values of mi
considering GSI = 10, 50, and 90, respectively. It was
found that for S> Scr, α will reduce; the higher the mi, the
larger the rate of this reduction. Moreover, increasing
the mi results in increasing α. For the rock masses with
small values of mi, the interference of the two footings
will be disappeared at S/B > 8, while for larger magni­
tudes of mi, this interference effect will be diminished at
about S/B > 10.
7.4. Effect of D
Figure 20 shows the variation of α versus S/B for different
values of D, considering σci = 25 MPa, GSI = 50, and mi
= 10. As can be seen, increasing the D results in decreasing
α and Scr. For the rock masses with D = 0, the interference
of the two footings will be disappeared at S/B > 9, while for
D = 0.8, this interference effect will be diminished at about
S/B > 6. This conclusion means that by increasing D, the
two adjacent footings’ interference effect will be disap­
peared in a smaller spacing between them.
16
S. SHAMLOO AND M. IMANI
Figure 17. Effect of mi on α assuming σci = 25 MPa, GSI = 10 and D = 0.
Figure 18. Effect of mi on α assuming σci = 25 MPa, GSI = 50 and D = 0.
Figure 19. Effect of mi on α assuming σci = 25 MPa, GSI = 90 and D = 0.
GEOMECHANICS AND GEOENGINEERING
17
Figure 20. Effect of D on α assuming σci = 25 MPa, GSI = 50 and mi = 10.
8. Conclusion
The following results were obtained from the numerical
analyses performed in this research:
The SRM values obtained from the numerical ana­
lyses performed in this research are in good accor­
dance with the laboratory tests performed by Lee
and Jeong (2016), with a range of differences
between 1% to 16%.
● Among the Hoek-Brown parameters, GSI has the
highest, and mi has the lowest influence on the
SRM of rock masses. The effects of the rock mass
Hoek-Brown parameters on the SRM are more
paramount in the case of significant Hoek-Brown
parameters than in the case of low magnitudes.
Moreover, increasing σci, GSI, and mi result in
increasing the SRM, while increasing D results in
decreasing the SRM.
● A new coefficient named the subgrade reduction
factor, Jmass, was introduced in this research,
which was defined as the ratio of the SRM of
the Hoek-Brown rock mass to the SRM of the
intact rock. Depending on the rock mass prop­
erties, this coefficient may vary from 0.2% to
54%.
● To take into account the effect of two adjacent foot­
ings on the SRM of rock masses, the interference
coefficient, α, was introduced as the ratio of the SRM
in the case of two nearby footings to that of one
isolated footing. The magnitude of this coefficient
depends on the rock mass properties and the dis­
tance between the two footings. By increasing σci
and mi, the α coefficient will increase, while increas­
ing GSI results in decreasing α. Moreover, if the
distance between the two footings becomes zero,
●
this coefficient would be less than one. By increasing
S from zero to Scr, the α coefficient increases. For
S> Scr, the α coefficient decreases until reaching
a constant magnitude equal to one.
● The presence of two adjacent footings plays an
essential role in the amount of SRM. Depending
on the distance between the two footings, the α
coefficient is between 0.7 and 1.3. In multiple foot­
ings, the minimum value of the SRM occurs at S/
B = 0, while its maximum magnitude occurs at
2 ≤ S/B ≤ 4. Likewise, depending on the strength
properties of the rock mass, if the distance between
the two footings becomes larger than 8 to 10 times
the footing width, the interference of the footings
does not have any effect on the SRM of the rock
mass, and the multiple footings behave as two
isolated ones.
● The width of the footing has an adverse correlation
with the SRM of rock masses. The obtained results
show that by increasing footing width from 0.5 to 3
metres, the SRM of the rock mass decreases in the
range of 13% to 67%, depending on the rock mass
properties.
Notations
The following symbols were used in this paper.
SRM:
ks:
kj:
ks(mass):
Jf:
Jmass:
Scr:
S:
Subgrade reaction modulus;
SRM of the intact rock;
SRM of the jointed rock;
SRM of the rock mass obeying the HoekBrown failure criterion;
Joint reduction factor;
Subgrade reduction factor;
Critical spacing;
Distance between two footings;
18
S. SHAMLOO AND M. IMANI
α:
ks(int):
ks(iso):
S d:
Id:
ϕj:
cj:
K n:
Ks:
γ:
Interference coefficient;
SRM of the rock mass subjected to the load of
a footing in the presence of the adjacent foot­
ing (obtained from the pressure-settlement
curve);
SRM of the rock mass subjected to the load
of a single isolated footing (obtained from
the pressure-settlement curve);
Spacing of the joints;
Inclination of the joints;
Friction angle of the joints;
Cohesion of the joints;
Normal stiffness of the joints;
Shear stiffness of the joints;
Unit weight of the rock mass
Disclosure statement
No potential conflict of interest was reported by the author(s).
References
Alhossein, H., Carter, J.P., and Booker, J.R., 1992. Finite ele­
ment analysis of rigid footings on jointed rock. In:
Proceedings of the 3rd International Conference on
Computational Plasticity, Vol. 1. Barceona, Spain, 935–945.
AlKhafaji, H., Imani, M., and Fahimifar, A., 2020. Ultimate
bearing capacity of rock mass foundations subjected to
seepage forces using modified Hoek-Brown criterion.
Rock Mechanics and Rock Engineering, 53, 251–268.
doi:10.1007/s00603-019-01905-6
Avci, B. and Gurbuz, A., 2018. Modulus of subgrade reaction
that varies with magnitude of displacement of cohesionless
soil. Arabian Journal of Geosciences, 351 (11), 1–8.
doi:10.1007/s12517-018-3713-1.
Biot, M.A., 1937. Bending of infinite beams on an elastic
foundation. Applied Mechanics American Society of
Mechanical Engineers, 59, A1–A7.
Carter, J.P. and Kullhawy, F.H., 1988. Analysis and design of
drilled shaft foundations socketed into rock. In: Rep. EL-5918.
Palo Alto, California: Electric Power Research Institute.
Fattah, M.Y., Shlash, K.T., and Mohammad, H.A., 2014a.
Bearing capacity of rectangular footing on sandy soil
bounded by a wall. Arabian Journal for Science and
Engineering, 39, 7621–7633. doi:10.1007/s13369-014-1353-7
Fattah, M.Y., Shlash, K.T., and Mohammad, H.A., 2014b.
Experimental study on the behavior of strip footing on sandy
soil bounded by a wall. Arabian Journal of Geosciences, 8 (7),
4779–4790. doi:10.1007/s12517-014-1564-y.
Fattah, M.Y., Shlash, K.T., and Mohammad, H.A., 2014c.
Experimental study on the behavior of bounded square footing
on sandy soil. Engineering and Technology Journal, University
of Technology – Iraq, 32 (Part (A),5), 1083–1105.
Hoek, E. and Brown, E.T., 1980. Empirical strength criterion
for rock masses. ASCE Journal of Geotechnical Engineering,
106, 1013–1035.
Hoek, E. and Dierichs, M.S., 2005. Empirical estimation of
rock mass modulus. International Journal of Rock
Mechanics and Mining Sciences, 43, 203–215. doi:10.1016/
j.ijrmms.2005.06.005
Imanzadeh, S., Denis, A., and Marache, A., 2013. Effect of
uncertainty in soil and structure parameters for buried
pipes. Geotech and Geophys, Site Characterization, 4,
1847–1853.
Javid, A.H., Fahimifar, A., and Imani, M., 2015. Numerical
investigation on the bearing capacity of two interfering
strip footings resting on a rock mass. Computers and
Geotechnics,
69,
514–528.
doi:10.1016/j.
compgeo.2015.06.005
Klopple, K. and Glock, D., 1979. Theoretische und experi­
mentelle untersuchungen zu den traglastproblemen beige­
wiecher, in die erde eingebetteter rohre. Veroffentlichung
des Instituts Statik und Stahlbau der Technischen
Hochschule Darmstadt, H–10.
Kullhawy, F.H. and Carter, J.P., 1992. Settlement and bearing
capacity of foundations on rock masses. In: F.G. Bell, ed.
Engineering in rock masses. Oxford, UK: ButterworthHeinemann, 231–245.
Lee, J. and Jeong, S., 2016. Experimental study of estimating
the subgrade reaction modulus on jointed rock
foundations. Rock Mechanics and Rock Engineering, 49,
2055–2064. doi:10.1007/s00603-015-0905-9
Lees, A., 2016. Geotechnical finite element analysis. 1st ed.
London: ICE Publishing.
Mabrouki, A., Benmeddour, D., and Mellas, M., 2010.
Numerical study of the bearing capacity for two interfering
strip footing on sands. Computers and Geotechnics, 37,
431–439. doi:10.1016/j.compgeo.2009.12.007
Mansouri, M., Imani, M., and Fahimifar, A., 2019. Ultimate
bearing capacity of rock masses under square and rectan­
gular footings. Computers and Geotechnics, 111, 1–9.
doi:10.1016/j.compgeo.2019.03.002
Meyerhof, G.G. and Baikie, L.D., 1963. Strength of steel sheets
bearing against compacted sand backfill. Highway Research
Record, 30, 1–19.
Saada, Z., Maghous, S., and Garnier, D., 2008. Bearing capa­
city of shallow foundations on rocks obeying a modified
Hoek–Brown failure criterion. Computers and Geotechnics,
38, 144–154. doi:10.1016/j.compgeo.2007.06.003
Sadrekarimi, J. and Akbarzad, M., 2009. Comparative study of
methods of determination of coefficient of subgrade
reaction. Electronic Journal of Geotechnical Engineering,
14, 1–14.
Sargazi, O. and Hosseininia, E., 2017. bearing capacity of ring
footing on cohesionless soil under eccentric load.
Computers and Geotechnics, 92, 169–178. doi:10.1016/j.
compgeo.2017.08.003
Selvadurai, A.P.S., 1985. Soil–pipeline interaction during
ground movement. In: F.L. Bennett and J.L. Machemehl,
eds.. Arctic, Civil engineering in the Arctic offshore. ASCE
speciality conference. San Francisco, 763–773.
Shamloo, S. and Imani, M., 2021. Upper bound solution for
the bearing capacity of two adjacent footings on rock
masses. Computers and Geotechnics, 129, 1–14.
doi:10.1016/j.compgeo.2020.103855
Terzaghi, K.V., 1955. Evaluation of coefficient of subgrade
reaction. Geotechnique, 5 (4), 297–326. doi:10.1680/
geot.1955.5.4.297.
Vásárhelyi, B., 2009. A possible method for estimating the
Poisson’s rate values of the rock masses. Acta Geodaetica
Geophys Hungarica, 44 (3), 313–322. doi:10.1556/
AGeod.44.2009.3.4.
GEOMECHANICS AND GEOENGINEERING
Vesic, A.B., 1961. Beams on elastic subgrade and Winkler’s
hypothesis. In: Proceedings of the Fifth International
Conference on Soil Mechanics and Foundation Engineering.
Paris, 845–850.
Vlassov, V.Z. and Leontiev, N.N., 1966. Beams, plates, and
shells on elastic foundations. Translated from Russian.
Jerusalem: Israel Program for Scientific Translations.
Wyllie, D.C., 1999. Foundations on rock. 2nd ed. London, UK:
E & FN Spon.
View publication stats
19
Yang, X. and Yin, J., 2005. Upper bound solution for ultimate
bearing capacity with a modified Hoek–Brown failure cri­
terion. International Journal of Rock Mechanics and Mining
Sciences, 42, 550–560. doi:10.1016/j.ijrmms.2005.03.002
Ziaie Moayed, R. and Naeini, S.A., 2006. Evaluation of mod­
ulus of subgrade reaction in gravely soils based on standard
penetration test (SPT). Chapter 115. In: Proceedings of the
sixth international conference on physical modelling in geo­
technics, Hong Kong.