Open Journal of Civil Engineering, 2013, 3, 228-233
Published Online December 2013 (http://www.scirp.org/journal/ojce)
http://dx.doi.org/10.4236/ojce.2013.34027
Open Access OJCE
Influence of the Elastic Modulus of the Soil and Concrete
Foundation on the Displacements of a Mat Foundation
Oustasse Abdoulaye Sall1*, Meissa Fall1, Yves Berthaud2, Makhaly Ba1
1Département Génie Civil, UFR SI-Université de Thiès, Thiès, Sénégal
2UFR Ingénierie, Université Pierre et Marie Curie, Paris, France
Received November 20, 2013; revised December 12, 2013; accepted December 19, 2013
Copyright © 2013 Oustasse Abdoulaye Sall et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we suggest to study the behavior of a mat foundation on subsoil from the plate theory taking into account
the soil-structure interaction. The objective is to highlight the soil-structure interaction particularly the influence of the
rigidities of the soil and the concrete on the subgrade reaction (k) and the displacements of the mat foundation subjected
to vertical loads. From plate theory and the soil-structure interaction, the general equation is reached. This equation de-
pends more on the subgrade properties than the concrete foundation properties. Consequently, the behavior of the mat
foundation is more influenced by soil properties than the concrete.
Keywords: Mat Foundation; Plate Theory; Soil-Structure Interaction; Mechanical Properties
1. Introduction
The structural and geotechnical calculations of civil en-
gineering works involve the limit state method and re-
quire the determination of characteristic values for resis-
tance and deformation criteria of structures and soils.
However, the geotechnical design is mainly based on the
determination of the displacement caused by the actions
applied to foundation and the determination of stresses
under limit state service. The structural design is strongly
based on the determination of stresses and displacements.
A computational approach that takes into account struc-
tural and geotechnical aspects related to the design of
foundation structures must be developed. It is then ques-
tion of interaction between two bodies of very different
characteristics of deformability. The rupture is often fol-
lowed by the formation of a thin region led in the direc-
tion of contact. This area is called soil-structure interface
and it is the location of the major displacements. This
work focused on the foundation slab and, more particu-
larly, on the characterization of the soil-structure inter-
face. A precise knowledge of moduli characterizes its
deformability and stress paths which should facilitate the
optimization of the structural and geotechnical design of
foundation.
2. Modelisation
A foundation is responsible for transmitting the loads
from the superstructure to the soil; it provides an inter-
face between the upper part of the structure and the soil.
A mat foundation is a continuous reinforced concrete
slab and the study may be governed by the theory of
plates whose behavior can be studied from the Lagrange
equation which take into account the soil-structure inter-
action. The solution of the Lagrange equation is possible
with the use of the methods of Fourier series or finite
differences with well-defined boundary conditions. Cha-
racterization of the interface has also allowed us to see
that the soil-structure interaction is important for the de-
sign of foundation. Selvadurai [1] presented a detailed
analysis of the soil-foundation interaction problem, ex-
plaining the different approaches proposed to model this
interaction. These models recognize that soil reaction is a
linear function of the displacement of the soil-founda-
tion interface layer. Several models have been devel-
oped:
 Winkler model [2],
 Elastic continuum model [1];
 Biparametric model [3];
 Filonenko Borodich model [4,5];
 Hetenyi model model [6];
*Corresponding author. 
Pasternak model [7];

O. A. SALL ET AL. 229
 Reissner model [8];
 Vlazov and Leontiev model [9];
 Vlazov modified model [10].
The system is similar to a concrete slab (Eb, νb) resting
on an elastic soil (Es, νs). The plate is assumed to rest on
a spring assembly infinitely close to each other with k as
the modulus of reaction. These springs are connected by
an elastic membrane of shear modulus (2T). Modeling of
the system is shown in Figure 1.
The problem is governed by the following general eq-
uation:

444
4224
22
22
2
2,
www
Dxxyx
ww
Tkw
xy












qxy


(1)
where D is the flexural rigidity of the plate and is given
by:

3
2
12 1
b
b
Ee
D

 (2)
with:
Eb: elastic modulus of the material constituting the
plate;
e: the thickness of the plate;
νb: Poisson’s ratio of the plate;
k is the modulus of subgrade reaction.
Biot [11] developed an empirical formula for k ex-
pressed as follow:
4
12
2
0.65
1
ss
b s
EEB
kEI

 (3)
Vesic [12] improved (3) by:

0.108
4
22
0.95
11
ss
ssb
EEB
kEI






where:
Es is the modulus of subgrade;
νs is the Poisson’s ratio of the subgrade;
B is the width of the foundation;
Eb is the Young’s modulus of the concrete foundation;
I is the moment of inertia of the cross section of the
concrete.
Equation (2) can be written as:
112
4
2
0.651
ss
b s
EEB
kEI




(5)
and Equation (4) by:
0.108 0.108
4
22
1
0.95 11
ss
b ss
EEB
kEI

 
 

 
(6)
Generally for foundations, Poisson’s ratio is between
0.15 and 0.4 [13], and the term
0.108
2
1
1s 




is between
1.0025 and 1.019 [13] (which leads to ignore this term in
the expression) for k in Equation (6) which can be re-
written as follows:
0.108
4
2
0.951
ss
b s
EEB
kEI




(7)
Thus by combining (3) and (6), k is expressed by the
following equation:
4
2
1
ss
EEB
ka EI







(8)
where a and

are constants according to different au-
thors (Table 1).
It should specify that the vertical modulus of subgrade
reaction can be determined from the results of geotech-
nical testing. T is the horizontal elastic modulus of sub-
grade reaction. Vlasov [9] proposes the following rela-
tion:

(4)
Figure 1. Discretisation of the system.
Open Access OJCE

O. A. SALL ET AL. 230
Table 1. Equations giving k [13].
Authors a

k
Biot (1937) [11] 0.65 1/12
12
2
1
4
0.651
ss
EEB
kEI




Vesic (1963) [12] 0.69 0.0868
0.0868
4
2
0.691
ss
EEB
kEI




Liu (2000) [14] 0.74 0.0903
0.0903
4
2
0.741
ss
EEB
kEI




Daloglu et al. (2000) [15] 0.78 0.0938
0.0938
4
2
0.781
ss
EEB
kEI




Fischer et al. (2000) [16] 0.82 0.0973
0.0973
4
2
0.821
ss
EEB
kEI




Yang (2006) [17] 0.95 0.108
0.108
4
2
0.951
ss
EEB
kEI




Henry (2007) [18] 0.91 0.1043
0.1043
4
2
0.911
ss
EEB
kEI




Arul et al. (2008) [19] 0.87 0.1008
0.1008
4
2
0.871
ss
EEB
kEI







2
20d
411 1
H
s
sss
E
T



z
(9)
To a relatively deep layer of soil where the normal
stress may vary with depth, it is possible to use, for the
function Φ(z), the non-linear continuous variable defined
by Equation 10(a). Φ(z) is a function which describes the
variation of the displacement w(x,y) along the z axis,
such that:



01 0et H 
Selvadurai [1] suggests two expressions of Φ(z):

1z z
H






(10a)
 
sinh
sinh
Hz
L
ZH
L









 


(10b)
H: thickness of the soil layer (depth of the rigid sub-
stratum).
And for a linear variation of Φ(z), the shear parameter
model is given after integration by:

 
2
12 111
s
sss
EH
T
 

 (11)
3. Analytical Solutions
Before the calculation of displacements due to load, it
should be consider that the motion of the interface is a
result of the weight of the slab. This displacement is con-
stant on the entire extension of the interface and is a
function of the thickness of the plate and the modulus of
vertical subgrade reaction. The displacement w0 is given
by:
025000wek

 (12)
In the case of an elastic homogeneous soil, a uniform
distribution of the forces applied to the foundation sys-
tem is assumed. This amounts to admitting that the stress
q(x,y) is constant (Q value) from each point of the foun-
dation. For a foundation of infinite dimension, a zero
displacement at the edges of the plate is imposed. If each
edge is far from one to another, this is true. Although,
this questionable assumption allows an accurate resolu-
tion of the problem using the Fourier series. At first, we
assume a uniform distribution of the applied foundation
system forces. So q(x,y) is constant (Q value) for ana-
lytical solution, and the double Fourier series is used.
q(x,y) can be written as:

11
π
,sins
mn
mx ny
qxy QaLB
 
 


 π
in



with
(13)
Open Access OJCE

O. A. SALL ET AL. 231
00
4ππ
sinsind d
LB
mn
mx ny
aQ x
LBL B





y (14)
00
4ππ
cos cos
ππ
L
B
mn
LmxBny
aQ
LB mLnB


 












(15)
It result that for m and n impair:
2
16Q
aπ
mn mn (16)
For the calculation of the displacem
th
ents, we assume
at w(x, y) can also be decomposed into Fourier series:

11
ππ
,sinsin
mx ny
wxyb
 

 (17 mn LB
 
 )
Thus:
2
2w
211
πππ
sin sin
mn
mmxny
b
LLB x

 

 
 
 (18)
2
2
211
ππ
sin sin
mn
wnmx
b
BL y
  
 
 
 

πny
B



(19)
4
22
22
11
πππ π
sin sin
mn
w
xy
m nmxny
b
LBL B



 

 
 

4
4
411
ππ
sin sin
mn
wmmx
b
LL x
  

 

 πny
B



(21)
4
4
411
ππ
sin sin
mn
wnmx
b
BL
y
  

 
 
 πny
B
(22)
By replacing the differential equation governing the
behavior of the system we have:
2
22
11
22
11
ππ
πππ π
2sin
ππ
sin sin
mn
mn
mn
bD LB
mn mxn
Tksin
y
L
BL
mx ny
aLB














 
 












B



(23)
According to (23), the expression bmn can be given by
the following relation:
2
22 22
ππ ππ
2
mn
mn
a
b
mn mn
DT
LB LB

 k











(20)




(24)
The axial deflection is:

22 11 22 22
ππ
sin sin
16
,πππ ππ
2
mx ny
QLB
wxy
mn mn
DmnTmn kmn
LB LB






  
 

  

  

 (25)
The total displacement is obtained by summing the
i
is study that the elastic modulus and
elastic modulus of the soil. Hence the importance of
d splacements given by Equations (12) and (25).
Figures 2 to 7 show the evolution of k according to the
different parameters of the mechanical behavior model.
Figure 2 shows the increase of k with the increase of Es.
Figures 3-5 show that k is sensitive to the mechanical
properties of the soil foundation. These figures show that
k and the displacements vary slightly with the mechani-
cal properties of concrete foundation and are strongly
dependent on elastic modulus of the soil foundation.
4. Conclusion
It appears from th
the Poisson’s ratio of the subgrade are the most influen-
tial parameters on the displacements of the plate. The
results show that modulus of subgrade reaction and dis-
placements varies slightly with the mechanical properties
of concrete foundation and is more influenced by the
mastering the property of the foundation soil is to better
01020 3040 5060708090100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
7
B/e
Mod ulus of subgrade react i on k (N/ m
3)
Es = 8MPa
Es = 7MPa
Es = 6MPa
Es = 8MPa
Figure 2. Modulus of subgrade reaction versus B/e ratio of
the plate for various values of Es.
Open Access OJCE

O. A. SALL ET AL. 232
010 20 30 40 50 60 7080 90100
0. 4
0. 6
0. 8
1
1. 2
1. 4
1. 6
1.8 x 10
7
B/e
Moduul us of subgrade reacti on (N/ m3)
Eb = 33GP a
Eb = 36GP a
Eb = 39GP a
Eb = 43GP a
Figure 3. Modulus of subgrade reaction versus B/e ratio of
the plate for various values of Eb.
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91
0
0. 005
0. 01
0. 015
0. 02
0. 025
0. 03
0. 035
0. 04
x/L
Déplacemen ts (m)
Es=4MPa
Es=6MPa
Es=7MPa
Es=8MPa
Figure 4. Displacements along the median of the plate for
arious values of Es. v
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0. 005
0.01
0. 015
0.02
0. 025
0.03
0. 035
0.04
x/L
Displacements (m)
Eb=33GPa
Eb=36GPa
Eb=39GPa
Eb=43GPa
Figure 5. Displacements along the median of the plate
r various values of Eb.
understand the behavior of foundation structures for op-
timal sizing of these and especially in order to limit the
displacements, which are the vectors of disorder in the
fo
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91
0
0. 005
0. 01
0. 015
0. 02
0. 025
0. 03
0. 035
0. 04
x/L
Déplacem ents (m )
nus = 0.2
nus = 0.25
nus = 0.3
nus = 0.4
Figure 6. Displacements along the median of the plate fo
various values of νs.
r
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91
0
0. 005
0. 01
0. 015
0. 02
0. 025
0. 04
0. 03
0. 035
x/L
Displacements (m
)
Eb=33GPa
Eb=36GPa
Eb=39GPa
Eb=43GPa
Figure 7. Displacements along the median of the plate for
various values of νb.
structures.
RENCES
[1] A. P. S. Selvadurai, “Elastic Analysis of Soil-Foundation
Interaction,” Developments in Geotechnical Engineering,
Vol. 17, 1979, pp. 7-9.
http://dx.doi.org/10.1016/B978-0-444-41663-6.50005-1
REFE
[2] E. Winkler, “Die Lehhre von der Eiastizitat und Festigkeit.
Dominicus,” Prague, 1867.
[3] C. V. G. Vallabhan and Y. C. Das, “Parametric Study of
Beams on Elastic Foundations,” Journal of the Engineer-
ing Mechanics Division, Vol. 114, No. 12, 1988, pp. 2072-
2082.
http://dx.doi.org/10.1061/(ASCE)0733-9399(1988)114:12
(2072)
[4] M. M. Filonenko-Borodich, “Some Approximate Theo
lonenko-Borodich, “A Very Simple Model of an
oundation Capable of Spreading the Load,” Sb
Tr. Mosk. Elektro. Inst. Inzh. Trans., No. 53, 1945.
-
- ries of the Elastic Foundation,” Uchenyie Zapiski Mosk
ovskogo Gosudarstvennogo Universiteta, Vol. 46, 1940,
pp. 3-18.
[5] M. M. Fi
Elastic F
Open Access OJCE

O. A. SALL ET AL.
Open Access OJCE
233
[6] M. Hetényi, “ation: Theory with
Applications i Mechanical Engi-
-
-
Plates on Viscoelastic Foun-
oi.org/10.1061/(ASCE)0733-9410(1991)117:6(
Beams on Elastic Found
n the Fields of Civil and
neering,” University of Michigan Press, Ann Arbor, 1946.
[7] P. L. Pasternak, “On a New Method of Analysis of an
Elastic Foundation by Means of Two Foundation Con
stants,” Gosudarstvennoe Izdatelstro Liberaturi po Stroi
telstvui Arkhitekture, Moscow, 1954.
[8] E. Reissner, “Deflection of
dation,” Journal of Applied Mechanics, Vol. 80, 1958, pp.
144-145.
[9] V. Z. Vlazov and U. N. Leontiev, “Beams, Plates and
Shells on Elastic Foundations,” Israel Program for Scien-
tific Translations, Jerusalem, 1966.
C. V. G[10] . Vallabhan and Y. C. Das, “Modified Vlasov
Model for Beams on Elastic Foundations,” Journal of
Geotechnical Engineering, Vol. 117, No. 6, 1991, pp.
956-966.
http://dx.d
956)
[11] M. A. Biot, “Bending of an Infinite Beam on an Elastic
Foundation,” Journal of Applied Physics, Vol. 12, No. 2,
1937, pp. 155-164. http://dx.doi.org/10.1063/1.1712886
[12] A. B. Vesic, “Beams on Elastic Subgrade and the Wink-
ler’s Hypothesis,” Proceedings of 5th International Con-
ference of Soil Mechanics, 1963, pp. 845-850.
[13] H. Bund, “An Improved Method for Foundation Modulus
in Highway
[14] F. L. Liu, “Rectangular Thick Plates on Winkler Founda-
tion: D
Engineering,” EJGE, Vol. 14, 2009.
eferential Quadrature Element Solution,” Interna-
tional Journal of Solids and Structures, Vol. 37, No. 12,
2000, pp. 1743-1763.
http://dx.doi.org/10.1016/S0020-7683(98)00306-0
[15] A. T. Daloglu and C. V. G. Vallabhan, “Values of k for
Slab on Winkler Foundation,” Journal of Geotechnical
and Geoenvironmental Engineering, Vol. 126, No. 5,
2000, pp. 463-471.
http://dx.doi.org/10.1061/(ASCE)1090-0241(2000)126:5(
463)
[16] F. D. Fischer and E. Gamsjäger, “Beams on Foundation,
Winkler Bedding or Half-Space—A Comparison,” Tech-
nische Mechanike, Vol. 2, 2008, pp. 152-155.
[17] K. Yang, “Analysis of Laterally Loaded Drilled Shafts in
askatchewan, Canada, 2007.
r the Flexure of Plates on Nonlinear Foundation,”
08)134:1(
Rock,” University of Akron, OH, 2006.
[18] M. T. Henry, “Train-Induced Dynamic Response of Rail-
way Track and Embankments on Soft Peaty Founda-
tions,” University of S
[19] S. Arul, S. Seetharaman and Abraham, “Simple Formula-
tion fo
Journal of Engineering Mechanics, Vol. 134, No. 1, 2008,
pp. 110-115.
http://dx.doi.org/10.1061/(ASCE)0733-9399(20
110)