Influence of the Elastic Modulus of the Soil and Concrete Foundation on the Displacements of a Mat Foundation

doi:10.4236/ojce.2013.34027

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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

Email: *[email protected]

Received November 20, 2013; revised December 12, 2013; accepted December 19, 2013

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

www

Dxxyx

Tkw

























qxy





(1)

where D is the flexural rigidity of the plate and is given

by:



12 1



 (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:

0.65

EEB

kEI



 (3)

Vesic [12] improved (3) by:



0.108

0.95

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

0.651

EEB

kEI









(5)

and Equation (4) by:

0.108 0.108

0.95 11

EEB

kEI



 

 



 

(6)

Generally for foundations, Poisson’s ratio is between

0.15 and 0.4 [13], and the term

0.108











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

0.951

EEB

kEI









(7)

Thus by combining (3) and (6), k is expressed by the

following equation:

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



Biot (1937) [11] 0.65 1/12

0.651

EEB

kEI









Vesic (1963) [12] 0.69 0.0868

0.0868

0.691

EEB

kEI









Liu (2000) [14] 0.74 0.0903

0.0903

0.741

EEB

kEI









Daloglu et al. (2000) [15] 0.78 0.0938

0.0938

0.781

EEB

kEI









Fischer et al. (2000) [16] 0.82 0.0973

0.0973

0.821

EEB

kEI









Yang (2006) [17] 0.95 0.108

0.108

0.951

EEB

kEI









Henry (2007) [18] 0.91 0.1043

0.1043

0.911

EEB

kEI









Arul et al. (2008) [19] 0.87 0.1008

0.1008

0.871

EEB

kEI











20d

411 1

sss







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):















(10a)

 

sinh

















 





(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:



12 111

sss







 (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:



,sins

mx ny

qxy QaLB

 

 





 π







with

(13)

Open Access OJCE

O. A. SALL ET AL. 231

4ππ

sinsind d

mx ny

aQ x

LBL B











(14)

4ππ

cos cos

ππ

LmxBny

LB mLnB





 

























(15)

It result that for m and n impair:

16Q

aπ

mn mn (16)

For the calculation of the displacem

ents, we assume

at w(x, y) can also be decomposed into Fourier series:



ππ

,sinsin

mx ny

wxyb

 



 (17

mn LB



 )

Thus:

2w

211

πππ

sin sin

mmxny

LLB



 



 

 

 (18)

211

ππ

sin sin

wnmx



 

 

 

 



πny







(19)

πππ π

sin sin

m nmxny

LBL B







 



 

 



411

ππ

sin sin

wmmx



 



 



 πny







(21)

411

ππ

sin sin

wnmx



 



 

 

 πny

(22)

By replacing the differential equation governing the

behavior of the system we have:

ππ

πππ π

2sin

ππ

sin sin

bD LB

mn mxn

Tksin

mx ny

aLB



























 

































(23)

According to (23), the expression bmn can be given by

the following relation:

22 22

ππ ππ

mn mn

LB LB



 k























(20)







(24)

The axial deflection is:



11 22 22

ππ

sin sin

,πππ ππ

mx ny

QLB

wxy

mn mn

DmnTmn kmn

LB LB













  

 



  



  



 (25)

The total displacement is obtained by summing the

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

1.2

1.4

1.6

1.8

2x 10

B/e

Mod ulus of subgrade react i on k (N/ m

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. 2

1. 4

1. 6

1.8 x 10

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. 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. 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

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

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.

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

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)