Analytical Model for Beams on Elastic FoundationsConsidering the Coupling of Horizontal and

Vertical DisplacementsQijian Liu1 and Jianjun Ma2

Abstract: An analytical method is introduced to deal with the coupling problem for Euler-Bernoulli beams on elastic bidimensional founda-tions by considering the horizontal and vertical displacements of the beam-foundation system. The approach is an extension of the modifiedVlasov model. With separation of the variables, the horizontal and vertical displacements are expressed as the displacement function at theground surface and the attenuation function along the depth of the foundations, respectively. The governing equations and the correspondingboundary conditions of the model are obtained via the variational principle. Then, the differential operator method is used to uncouple the gov-erning equations and boundary conditions. An iterative procedure is executed to accomplish the numerical implementation. A parametric studyis conducted to illustrate the effects of the applied loadings and the physical and geometry properties on the static responses of the beam andfoundations. The numerical results show that the coupling elasticity should be taken into account in cases of flexible beams and high soil Pois-son ratios. Moreover, the horizontal loads on the beam significantly affect the response of the beam-foundation system.DOI: 10.1061/(ASCE)EM.1943-7889.0000635. © 2013 American Society of Civil Engineers.

CE Database subject headings: Beams; Elastic foundations; Coupling; Elasticity; Displacement; Analytical techniques.

Author keywords:Beams on elastic foundations;Variational principle; Foundationmodel; Coupling elasticity;Differential operatormethod.

Introduction

The theory of beams on elastic foundations plays an important rolein the analysis of soil-structure interaction problems, such as in re-lation to applications pertaining to the foundation engineering ofbuildings, railroad tracks, pile foundations, and excavation engi-neering. In the past, research has been undertaken to construct an-alytical models of beams on elastic foundations (Selvadurai 1979).A review of foundation models was performed by Dutta and Roy(2002). Most probably, the oldest model used to simulate beams onelastic foundations is the Winkler (1867) model. Theoretically, themain characteristic of the Winkler model is that the underlyingfoundations are assumed to be simulated as a series of independentlinear springs rested by the beam.

TheWinkler model has been widely used in engineering practicebecause of its simplicity (Hetenyi 1946). However, the Winklermodel cannot consider the continuity of elastic foundations becausethe foundation displacement at a point is dependent only on theforce acting on the point in this model. Meanwhile, it is not easy todetermine the spring constant in theWinkler model because its valueis not unique for a certain type of soil. Therefore, to overcome theweakness of Winkler model, several kinds of generalizations of the

Winkler model have been developed as two-parameter foundationmodels in the past few decades by Filonenko-Borodich (1940),Pasternak (1954), and Vlasov and Leontev (1966). The two-parametermodels have beenwidely used to solve the elastic foundation problem(Feng and Cook 1983; Nogami and O’Neill 1985; Nogami and Lam1987; Morfidis and Avramidis 2002). Moreover, another category ofgeneralized models is the so-called three-parameter models (Hetenyi1950; Reissner 1958; Kerr 1965). As a typical three-parametermodel,the Kerr foundation model has been proved to be identical to theVlasov model based on the choice of the attenuation profile (Jonesand Xenophontos 1976). In terms of the Kerr model, the bending andstiffness of beams on elastic foundations have been investigated(Avramidis and Morfidis 2006; Morfidis 2007). Furthermore, DiPaola et al. (2009) showed that elastic foundation models may bereferred to as gradient models, which involve some nonlocality. Onthe basis of the variational formulation of the Reissner model,Challamel et al. (2010) investigated the buckling of an axially loadedEuler-Bernoulli model supported on an elastic nonlocal medium.

In the generalizations of the Winkler model, the classical Vlasovmodel starts from continuum theory on the basis of the variationalprinciple. Therefore, it has a plausible theoretical basis, and thecharacteristic constants in the model can be expressed by the ma-terial properties. However, to determine the two characteristic para-meters in the Vlasov model, an arbitrary parameter g should beintroduced to characterize the distribution of the vertical displace-ment profile in advance. Based on variational calculus, Jones andXenophontos (1977) investigated the response of a plate on elasticfoundations and established a relationship between parameter g andthe displacement of the resting plate. Vallabhan and Das (1988,1991a, b) developed an iterative procedure to determine the valueof g, and the procedure is referred to the modified Vlasov model.In the modified Vlasov model, attenuation of the vertical displace-ment profile of foundations is coupled with the configuration of thebeam.

1Associate Professor, College of Civil Engineering, Hunan Univ.,Changsha, Hunan 410082, P.R. China (corresponding author). E-mail:Q.Liu@hnu.edu.cn

2Doctoral Candidate, College of Civil Engineering, Hunan Univ.,Changsha, Hunan 410082, P.R. China. E-mail: majianjun@hnu.edu.cn

Note. This manuscript was submitted on July 5, 2012; approved onMarch 18, 2013; published online on March 20, 2013. Discussion periodopen until May 1, 2014; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Engineering Mechan-ics, Vol. 139, No. 12, December 1, 2013. ©ASCE, ISSN 0733-9399/2013/12-1757–1768/$25.00.

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Obviously, only the vertical component of the total displace-ments is considered in the models mentioned previously. Thus, thehorizontal component cannot be fully justified. In practical engi-neering the horizontal displacement should be taken into accountunder many conditions, such as in foundations with a high Poissonratio and beams subjected to horizontal loads. To the best knowledgeof the authors, there have only been a few models that take intoaccount coupling of horizontal and vertical displacements of beam-foundation systems (Rao et al. 1971). However, the method is basedon the classical Vlasov model and the same weakness of the de-termination of the attenuation functions cannot be overcome.

In this study, an analytical solution of beams on elastic founda-tions involving the coupling of horizontal and vertical displacementshas been developed. First, the horizontal and vertical displacementsare expressed as the separation of a displacement function at thesurface and an attenuation function along the depth of foundations,respectively. Then, the governing equations for the displacementand attenuation functions in the model are obtained and coupledusing variational calculus. The differential operator method is cho-sen to decouple the governing equations. Following Vallabhan andDas (1988, 1991a, b), an iterative procedure is executed to ac-complish the numerical implementation. Finally, a parametric studyis performed to investigate the static responses of the beam andfoundations under the combined effects of horizontal and verticalloadings. The present approach may be regarded as an extension ofthe modified Vlasov model in the framework of the elastic soil-structure interaction.

Mathematical Model

Displacement Model

The basic model of beams on elastic foundations is depicted inFig. 1.Attention is focused on the static response of an infinitely longslab of finite width resting on an elastic foundation of thicknessH inplane strain conditions. The elastic foundations are assumed to beisotropic and homogeneous. The Young’s modulus and Poissonratio of the foundations are Es and ns, and its Lamé constants are lsand ms, respectively. Here, a strip of the slab can be considered asa beam of width b, height h, and length L. The Young’s modulus andPoisson ratio of the beam are E and n, respectively. The beam issubjected to distributed horizontal and vertical loads, pðxÞ and qðxÞ.For the foundation, the field displacements are denoted as uðx, zÞ andwðx, zÞ along the x- and z-directions, respectively. A Cartesiancoordinate system o-xyz is chosen, with the origin o placed at the leftend of the beam, as shown in Fig. 1.

In this study, the beam is assumed to be perfectly connected to thefoundations; i.e., there is no slippage or separation along the beam-foundation interface. From the elasticity, the horizontal and verticaldisplacements, uðx, zÞ and wðx, zÞ, can be expressed in the full-basisdiscrete form as (Nogami and O’Neill 1985)

uðx, zÞ ¼ P‘n¼1

unðxÞfnðzÞ (1a)

wðx, zÞ ¼ P‘n¼1

wnðxÞcnðzÞ (1b)

where unðxÞ and wnðxÞ5 nth displacement functions along thex-direction at ground surface z5 0, corresponding to uðx, zÞ andwðx, zÞ, respectively; and fnðzÞ and cnðzÞ are chosen as the nthnondimensional attenuation functions representing the variationsof the foundation displacements of uðx, zÞ and wðx, zÞ along thez-direction, respectively.

Basically, the solution for each n may be superimposed to yieldthe complete results for the displacements. In general, the mainnature of the displacement of the beam-foundation system can beobtained by the first-order truncation (n5 1) as

uðx, zÞ ¼ uðxÞfðzÞ (2a)

wðx, zÞ ¼ wðxÞcðzÞ (2b)

where attenuation functions fðzÞ and cðzÞ should satisfy the fol-lowing boundary conditions:

fð0Þ ¼ cð0Þ ¼ 1 (3a)

fðHÞ ¼ cðHÞ ¼ 0 (3b)

It may be easily concluded from Eqs. (3a) and (3b) that the hori-zontal and vertical displacements at surface z5 0 is assumed to beuðx, 0Þ 5 uðxÞ and wðx, 0Þ5wðxÞ. Based on the assumption of theperfectly bonded beam-foundation interface the displacements ofthe beam should be chosen as uðxÞ and wðxÞ along the region of thebeam length.

Principle of Minimum Potential Energy

To obtain the governing equations of the static response of the beamand foundation, the principle of minimum potential energy is used.The minimum potential energy function,P, of the beam-foundationsystem is

P ¼ Ubeam þ Usoil 2Wload (4)

whereUbeam5potential energy of the beam;Usoil5 potential energyof the foundations; and Wload 5 virtual work done by the conser-vative forces.

Potential energy Ubeam can be expressed as

Ubeam ¼ðL0

EI2

�d2wdx2

�2

dxþðL0

EA2

�dudx

�2

dx (5)Fig. 1. Beam resting on the elastic foundation

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and potential energy Usoil can be also written as

Usoil ¼ b2

ð‘2‘

ðH0

�sijɛij

�dx dz (6)

wheresij and ɛij ði, j5 1, 2Þ5 components of the stresses and strainsof the foundations, which can be expressed by the displacementsaccording to the elasticity.

Virtual work Wload done by the conservative forces is

Wload ¼ðL0

qðxÞwðxÞdxþðL0

pðxÞuðxÞdx (7)

Applying the principle of minimum potential energy to Eq. (4) gives

dP ¼ dUbeam þ dUsoil 2 dWload ¼ 0 (8)

where d5 variation. The term, dUbeam, can be attained by applyingthe variation of Eq. (5) as

dUbeam ¼ðL0

d2

dx2

�EI d

2wdx2

�dw dx2

ðL0

EA d2udx2

du dx

þ�EI d

2wdx2

ddwdx

2 ddx

�EI d

2wdx2

�dw

�L0þ�EA du

dxdu

L0

(9)

Similarly, dWload can be determined from the variation of Eq. (7) as

dWload ¼ðL0

qðxÞdw dxþðL0

pðxÞdu dx (10)

Likewise, substituting Eq. (2) into Eq. (6) and performing someintegrations by parts, dUsoil is given by

dUsoil ¼ðH0

ð‘2‘

lsbn0

�dudx

2

fdf dx dz2ðH0

ð‘2‘

lsbn0

f2d2udx2

du dx dzþðH0

ð‘2‘

lsbfdcdz

dudx

dw dx dzþðH0

ð‘2‘

lsbwdudx

dcdz

df dx dz

2

ðH0

ð‘2‘

lsbwdudx

dfdz

dc dx dz2ðH0

ð‘2‘

lsbfdcdz

dwdx

du dx dzþðH0

ð‘2‘

lsbn0

�dcdz

�2

w dw dx dz2ðH0

ð‘2‘

lsbn0

w2d2c

dz2dc dx dz

þðH0

ð‘2‘

msb

�dwdz

�2

cdc dx dzþðH0

ð‘2‘

msbu

�dwdz

�dfdz

dc dx dzþðH0

ð‘2‘

msbcdfdz

dwdx

du dx dzþðH0

ð‘2‘

msb

�dfdz

�2

u du dx dz

2

ðH0

ð‘2‘

msbc2d2wdx2

dw dx dz2ðH0

ð‘2‘

msbcdfdz

dudx

dW dx dz2ðH0

ð‘2‘

msbu

�dwdz

�dcdz

df dx dz2ðH0

ð‘2‘

msbu2d

2f

dz2df dx dz

þ

� ðH0

lsbn0

f2dz dudx

du

�‘2‘

þ

� ð‘2‘

lsbwdudx

dxfdc

�H0

þ

� ðH0

lsbfdcdz

dz w du

�‘2‘

þ

� ð‘2‘

lsbn0

w2dxdcdz

dc

�H0

þ

� ðH0

msbc2dz dw

dxdw

�‘2‘

þ

� ðH0

msbcdfdz

dz u dw

�‘2‘

þ

� ð‘2‘

msbu

�dwdz

�dxcdf

�H0

þ

� ð‘2‘

msbu2dx

dfdz

df

�H0

(11)

Eq. (11) shows that the variation of the potential energy of soil,dUsoil, is very complex and involves the displacement and attenu-ation functions. For example, the third term in Eq. (11) containsthree parts. The first part is the constant lsb. The second part isthe integral

ÐH0 fðdc=dzÞdz, which can be regarded as the attenua-

tion of the displacement functions. The third part is the integralÐ ‘2‘ðdu=dxÞdw dx, which involves the variation of displacementfunctionwðxÞ and the derivative of horizontal displacement function

uðxÞ. Thus, the second part involves the coupling of attenuationfunctions f and c, whereas the third part involves the coupling ofdisplacement functions u and w. Therefore, Eq. (11) indicates thatthe displacement and attenuation functions are coupled to each other.To further investigate the horizontal and vertical coupling in the soil-foundation system, the displacement and attenuation functions inEq. (11) are separated, in which some coefficients hiði5 1, . . . ,12Þare introduced, as given in Table 1. Thus, Eq. (11) can be rewritten as

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dUsoil ¼ðH0

h1fdf dz2ð‘2‘

h2d2udx2

du dxþð‘2‘

h3dudx

dw dxþðH0

h4dcdz

df dz2ðH0

h4dfdz

dc dz2ð‘2‘

h3dwdx

du dxþð‘2‘

h5w dw dx

2

ðH0

h6d2cdz2

dc dzþðH0

h7cdc dzþðH0

h8dfdz

dc dzþð‘2‘

h9dwdx

du dxþð‘2‘

h10u du dx2ð‘2‘

h11d2wdx2

dw dx2ð‘2‘

h9dudx

dw dx

2

ðH0

h8dcdz

df dz2ðH0

h12d2fdz2

df dzþhh2dudx

dui‘2‘

þ ½h4 fdc�H0 þ ½h3 w du�‘2‘ þ�h6dcdz

dc

�H0þhh11

dwdx

dwi‘2‘

þ ½h9 u dw�‘2‘ þ ½h8 c df�H0 þ�h12

dfdz

df

�H0

(12)

Governing Equations of the Displacement

For convenience of expression, the following nondimensional var-iables are introduced:

xp ¼ xL; up ¼ u

L; wp ¼ w

L(13)

In the subsequent analysis, the asterisk will be implied and extractedfor simplicity. Because u and w are the horizontal and verticaldisplacements along the surface z5 0, uB and wB are used to rep-resent the displacement components of the beam for 0# x# 1,whereas uS and wS represent the ground surface for2‘, x, 0 and1, x,‘. For region 0# x# 1, by substituting Eqs. (9)–(12) intoEq. (8) and collecting the coefficients of du and dw, the governingequations of the displacement functions may be obtained as

d4wB

dx42 r1

d2wB

dx2þ r2wB þ r3

duBdx

¼ Q (14)

d2uBdx2

2 s1uB þ s2dwB

dx¼ P (15)

where ri ði5 1, 2, 3Þ and sj ð j5 1, 2Þ 5 coefficients as follows:

r1 ¼ h11L2

EI; r2 ¼ h5L

4

EI; r3 ¼ ðh32 h9ÞL3

EI; s1 ¼ h10L

2

h2 þ EA;

s2 ¼ ðh32 h9ÞLh2 þ EA

; Q ¼ qðxÞL3EI

; P ¼ pðxÞLh2 þ EA

(16)

Further investigation of governing Eqs. (14) and (15) of wB anduB is of interest. It may be concluded that the proposed model is thegeneralization of some available foundationmodels in the absence ofhorizontal displacement uB. For example, if r1 5 0, Eq. (14) is re-stored to themodel for bending of the beam onWinkler foundations.Meanwhile, if the values of r1 and r2 are not equal to zero, Eq. (14) isunderstood and generalized to be the governing equation of the beamin the Pasternak or classical andmodifiedVlasovmodels.Moreover,Eq. (14) has the same expression as the so-called three-parameterfoundation models (such as the Kerr model), provided that theircoefficients in the differential equations are assumed to be identical.It is also expected that the present model may display the same formas the Reissner model if it is extended to modeling the slab-foundation interface. Of interest is that governing Eq. (15) of thehorizontal direction is also coupled with the vertical displacement.On the other hand, it may be justified that the present model has theadvantage of a theoretical background because it is deduced in termsof the variational principle. Furthermore, Eqs. (14) and (15) showthat the horizontal and vertical displacement functions, uB and wB,are coupled with each other, which is basically different from theaforementioned foundation models.

Similarly, the corresponding boundary conditions at x5 0 and 1from Eq. (8) can also be obtained as

d2wB

dx2¼ 0 (17a)

d3wB

dx32h11L

2

EIdwB

dxþ h11L

2

EIdwS

dx¼ 0 (17b)

duBdx

2h2

h2 þ EAduSdx

¼ 0 (17c)

For region 2‘, x, 0 and 1, x,‘

d2wS

dx22 k1wS þ k2

duSdx

¼ 0 (18)

Table 1. Coefficients in Eq. (12)

Coefficient Expression

h1Ð ‘2‘ðlsb=n0Þðdu=dxÞ2dx

h2ÐH0 ðlsb=n0Þf2dz

h3ÐH0 lsbfðdc=dzÞdz

h4Ð ‘2‘ lsbwðdu=dxÞdx

h5ÐH0 ðlsb=n0Þðdc=dzÞ2dz

h6Ð ‘2‘ðlsb=n0Þw2dx

h7Ð ‘2‘ msbðdw=dzÞ2dx

h8Ð ‘2‘ msbuðdw=dzÞdx

h9ÐH0 msbcðdf=dzÞdz

h10ÐH0 msbðdf=dzÞ2dz

h11ÐH0 msbc

2dz

h12Ð ‘2‘ msbu

2dx

Note: n0 5 ns=ð12 nsÞ.

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d2uSdx2

2 d1uS þ d2dwS

dx¼ 0 (19)

where

k1 ¼ h5L2

h11; k2 ¼ ðh92 h3ÞL

h11; d1 ¼ h10L

2

h2; d2 ¼ ðh32 h9ÞL

h2

(20)

Here, the continuous conditions of the beam-foundation system areobtained as

wBð0Þ ¼ wSð0Þ; wBð1Þ ¼ wSð1Þ; wSð6‘Þ ¼ 0;

uBð0Þ ¼ uSð0Þ; uBð1Þ ¼ uSð1Þ; uSð6‘Þ ¼ 0 (21)

Governing Equations of the Attenuation Functions

Collecting the coefficients of df and dc in Eq. (8), the governingequations of the attenuation functions can be written as

d2fdz2

2 z1f2 z2dcdz

¼ 0 (22)

d2cdz2

2 t1c2 t2dfdz

¼ 0 (23)

where

z1 ¼ h1h12

; z2 ¼ h42 h8h12

; t1 ¼ h7h6

; t2 ¼ h8 2h4h6

(24)

The boundary conditions of the attenuation functions are given inEq. (3), and the following four dimensionless parameters, g12g4,can be introduced as

g1 ¼ Hffiffiffiffiffiz1

p; g2 ¼ Hz2; g3 ¼ H

ffiffiffiffit1

p; g4 ¼ Ht2 (25)

Solving Process

Solutions of the Attenuation Functions

To obtain the solutions of the attenuation functions, the differentialoperatormethod is applied to decompose the coupled equations. Thegoverning equations of attenuation functionsf and c [Eqs. (22) and(23)] can be rewritten as

L11fþ L12c ¼ 0 (26a)

L21fþ L22c ¼ 0 (26b)

where Lij ði, j5 1, 2Þ 5 differential operators and are defined as

L11 ¼ d2

dz22 z1; L12 ¼ 2z2

ddz; L21 ¼ 2t2

ddz;

L22 ¼ d2

dz22 t1 (27)

To uncouple Eq. (26), an auxiliary function F1 is introduced as

f ¼ 2L12F1 ¼ z2dF1

dz(28a)

c ¼ L11F1 ¼ d2F1

dz22 z1F1 (28b)

Substituting Eq. (28) into Eq. (26), Eq. (26a) is automatically sat-isfied and Eq. (26b) yields

d4F1

dz42 2m1

d2F1

dz2þ m2F1 ¼ 0 (29)

where

m1 ¼ z1 þ t1 þ t2z22

; m2 ¼ t1z1 (30)

Then, the general solution of Eq. (29) can be written as

F1 ¼ P4i¼1

c1i f1iðzÞ (31)

where c1iði5 1, 2, 3, 4Þ 5 undetermined coefficients; and f1iðzÞði5 1, 2, 3, 4Þ 5 solution functions listed in Table 2. SubstitutingEq. (31) into Eq. (28) and considering boundary conditions Eq. (3),coefficients c1iði5 1, 2, 3, 4Þ can be determined. Thus, attenuationfunctions fðzÞ and cðzÞ may be solved.

Solutions of the Displacement Functions

For regions2‘, x, 0 and 1, x,‘, by applying the differentialoperator method, Eqs. (18) and (19) can be written as

L11uS þ L12wS ¼ 0 (32a)

L21uS þ L22wS ¼ 0 (32b)

where

L11 ¼ k2ddx; L12 ¼ d2

dx22 k1; L21 ¼ d2

dx22 d1; L22 ¼ d2

ddx(33)

The auxiliary function, F2, is chosen as

uS ¼ 2L12F2 ¼ 2d2F2

dx2þ k1F2 (34a)

wS ¼ L11F2 ¼ k2dF2

dx(34b)

Table 2. Constants and Solution Functions of Eqs. (29) and (35)

Solution function

Constant m21 $m2 m2

1 #m2

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

1 2m2

pq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ffiffiffiffiffiffim2

p=2Þ1 ðm1=4Þ

pb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

1 2m2

pq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ffiffiffiffiffiffim2

p=2Þ2 ðm1=4Þ

pf1ðzÞ eaz eaz sinbzf2ðzÞ e2az eaz cosbzf3ðzÞ ebz e2az sinbzf4ðzÞ e2bz e2az cosbz

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Substituting Eq. (34) into Eq. (32), Eq. (32a) is automatically sat-isfied and Eq. (32b) yields

d4F2

dx42 2m3

d2F2

dx2þ m4F2 ¼ 0 (35)

where

m3 ¼ k1 þ d1 þ k2d22

; m4 ¼ k1d1 (36)

Similar to the solution of auxiliary function F1, the general solutionof F2 can be written as

F2 ¼ P4i¼1

c2i f2iðzÞ (37)

whereconstantsc2iði5 1, 2, 3, 4Þ can be determined by the boundaryconditions; and f2iðzÞði5 1, 2, 3, 4Þ 5 solution functions given inTable 2. Parametersm1 andm2, given in Table 2, should be replacedby m3 and m4 during the determination of solution functions f2iðzÞfor F2. Then, applying the boundary conditions of Eq. (21), unknowncoefficients c2i can be determined and expressed by displacementfunctions uB and wB at the ends of the beam (x5 0, 1).

For region 0# x# 1, the solutions of Eqs. (14) and (15) can bewritten as

uBðxÞ ¼ uhBðxÞ þ upðxÞ (38a)

wBðxÞ ¼ whBðxÞ þ wpðxÞ (38b)

where superscripts h and p5 homogeneous and particular solutionsof the displacements, respectively. From Eqs. (14) and (15), theparticular solutions, up and wp, can be easily found as

up ¼ 2 Ps1

(39a)

wp ¼ Qr2

(39b)

To attain the homogeneous solutions, the homogeneous parts ofEqs. (14) and (15) are written as

L11uhB þ L12w

hB ¼ 0 (40a)

L21uhB þ L22w

hB ¼ 0 (40b)

where

L11 ¼ s3ddx

; L12 ¼ d4

dx42 s1

d2

dx2þ s2; L21 ¼ d2

dx22 r1;

L22 ¼ r2ddx

(41)

For Eq. (40), the following auxiliary function F3 can be chosen

uhB ¼ 2L22F3 ¼ 2r2dF3

dx(42a)

whB ¼ L21F ¼ d2F3

dx22 r1F3 (42b)

Substituting Eq. (42) into Eq. (40b), yields

d6F3

dx6þ k1

d4F3

dx4þ k2

d2F3

dx2þ k3F3 ¼ 0 (43)

where

k1 ¼ 2ðr1 þ s1Þ; k2 ¼ s2 þ r1s12 r2s3; k3 ¼ 2r1s2 (44)

The characteristic equation [Eq. (43)] can be transformed intothe classical Cardano formula and calculated analytically as shownin the Appendix. Correspondingly, the solution of auxiliary functionF3 can be expressed as

F3 ¼ P6i¼1

ai f3iðxÞ (45)

where aiði5 1, 2, . . . , 6Þ 5 constants determined by the boundaryconditions of the beam; and f3iðxÞði5 1, 2, . . . , 6Þ 5 solution func-tions depending on the characteristic roots of Eq. (43). If the char-acteristic roots are one real number 6y1 and two complex conjugates6ðy2 1 iy3Þ and 6ðy2 2 iy3Þ, f3i can be written as

Fig. 2. Iterative procedure

Fig. 3. Vertical displacement of the beam on the elastic foundationunder concentrated load

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f f3igT ¼ fey1x, e2y1x, ey2x sin y3x, ey2x cos y3x, e

2y2x sin y3x,

e2y2x cos y3xg (46)

Meanwhile, if the characteristic equation has six different real rootsin the form of 6y1, 6y2, and 6y3, f3i can be written as

f f3igT ¼ fey1x, e2y1x, ey2x, e2y2x, ey3x, e2y3xg (47)

Based on the displacement functions, the bendingmoment and shearforce of the beam can be obtained.

Iterative Procedure

As discussed previously, displacement functions u and w and at-tenuation functions f and c are coupled to each other. To obtain theresponse of the beam on elastic foundations considering the hori-zontal and vertical displacements, an iterative procedure recom-mended by Vallabhan (Vallabhan and Das 1988, 1991a, b) waschosen, as shown in Fig. 2. The iteration begins with a set of ap-proximate values of parameters g12g4. Using the iterative pro-cedure, the step-by-step process of displacements and stresses of thebeam on the elastic foundation can be obtained.

First, the initial approximate values of g12g4 are assumed, e.g.,g1,2,3,4 5 1:0. The solutions of attenuation functions fðzÞ and cðzÞcan be obtained via Eq. (28). Then, substituting fðzÞ and cðzÞ (seeTable 1), parameters h2, h3, h5, h9, h10, and h11 are obtained. UsingEqs. (34) and (38), the horizontal and vertical displacements, uðxÞand wðxÞ, can be calculated. Then, the next series of attenuationparametersg12g4 can be calculated byEqs. (25).As shown in Fig. 2,the iteration will continue until an enough accuracy of parametersgkðk5 1, 2, 3, 4Þ is satisfied, which means the difference betweenthe ith and ði1 1Þth values of gki is smaller than ɛ as���gkðiþ1Þ 2 gki

���, ɛ (48)

whereɛ5 required accuracy and can be usually assumed to be 0.001.

Numerical Results and Analysis

To compare the displacements and stresses of the beam-foundationsystem using the presentmodelwith the available data, the following

geometry and material parameters were chosen: L5 30:48 m,h5 0:9144 m, b5 0:3048 m, E5 20:685 GPa, and n5 0:2.

Validation

The comparisons of the displacements of the beam on elasticfoundations under concentrated loads using the present modelwere performed with those by the modified Vlasov model and the

Fig. 4. Vertical displacement of the beam on the elastic foundation under uniform vertical load

Fig. 5. Horizontal displacement of the beam on elastic foundationsunder uniform vertical load

Table 3. Properties of the Cases under Uniform Load

Case

Dimensionless parameter 1 2 3 4

Modulus ratio, E=Es 3,000 3,000 300 300H=l 0.50 1.00 0.50 1.00

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FEM. The vertically concentrated load was chosen as q5 250 kN.The physical and geometry parameters of the elastic foundations wereEs 5 15MPa, ns 5 0:1, and H5 30:48 m. In the FEM, the beam-foundation system was modeled by a rather fine finite-element (FE)mesh, which included more than 8,000 solid elements. The com-parison results of the beam displacements are shown in Fig. 3, whichindicate that the vertical displacements obtained by the presentmethod and FEM are both greater than those obtained by themodified Vlasov model. The phenomenon is well understood asa result of neglecting the horizontal displacement component in themodified Vlasov model.

On the other hand, comparisons of the vertical displacementsof the beam between the results obtained by the present method andthe published data by Vallabhan and Das (Vallabhan and Das 1988)were also carried out and are shown in Fig. 4. Four kinds of workingconditions, Cases 1–4, were considered during the comparisons (seeTable 3). The beam was assumed to be subjected only to uniformvertical loads q. As shown in Fig. 4, the displacement curvesobtained by the two models are in good agreement.

Effect of the Modulus and Depth of the Foundation

If horizontal displacement uðx, zÞ is considered, the coupling effectoccurs on vertical displacementwðx, zÞ. As shown in Fig. 4, the effectof depthH of the foundation on the vertical displacement of the beamis significant. For Cases 2 and 4, the vertical displacements of thebeam obtained by the present model were usually greater than thoseobtained by Vallabhan and Das (1988). Thus, the increase of depthH results in greater vertical displacements of the beam. Meanwhile,vertical displacements wðxÞ decrease with an increase of Es.

Fig. 5 shows the horizontal displacements of the beam obtainedby the present method with the absence of the horizontal load. It canbe observed that the horizontal displacements of the beam existbecause of the coupling of the horizontal and vertical displacements.However, in contrast to the vertical displacements, the amplitudes ofthe horizontal displacements of the beam are very small. Moreover, anantisymmetrical nature exists in the distribution curves of the hori-zontal displacement. Furthermore, another interesting phenomenon

is that Young’smodulusEs and depthH intensely affect the horizontaldisplacements.

Fig. 6 shows attenuation functions fðzÞ and cðzÞ for Cases 1–4in Table 3, where it can be seen that attenuation function cðzÞdecreases graduallywith the foundation depth.Moreover, the shapesof cðzÞ obtained by the present model agree with those obtainedby Vallabhan and Das (1988). It is also interesting to observe thatattenuation function fðzÞ has a negative value in Cases 2 and 4. Ingeneral, a negative value of fðzÞ indicates that the horizontal dis-placements of the foundation increase at a certain depth and thenreduce to zero at the rigid base. Furthermore, the slope of the at-tenuation function, fðzÞ, is significantly greater than cðzÞ becauseof the fact that the vertical displacements contribute mainly to thetotal displacements.

Fig. 7 shows the vertical displacements of the beamwith differentPoisson ratios of the foundation for Case 1. It is observed that theeffect of the Poisson ratio of the foundation on the vertical dis-placement of the beam is significant. Obviously, when the foun-dation has a lower Poisson ratio value this will result in greatervertical displacements of the beam. The reason may be that with anincrease of the Poisson ratio of foundations, the total deformation offoundations transfers to the horizontal displacement and smallvertical displacement of the beam is reasonable. Thismay also be thereason why the conventional foundation models result in over-estimation of the vertical displacement. This interesting phenom-enon also indicates the horizontal displacement cannot be neglectedwhen the Poisson ratio value of foundations is high.

Fig. 6. Attenuation function of the beam on elastic foundations underuniform vertical load (q5 29:19 kN=m)

Fig. 7. Effect of the Poisson ratio on the vertical displacement of thebeam under uniform vertical load

Table 4. Properties of the Cases under Concentrated Load

Case

Property A1 A2 A3 A4 B1 B2 B3 B4

E=Es 300 300 300 300 3,000 3,000 3,000 3,000H=l 1.00 1.00 2.00 2.00 1.00 1.00 2.00 2.00ns 0.1 0.4 0.1 0.4 0.1 0.4 0.1 0.4

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Effect of the Nature of the Beam-FoundationSystem

According to the nature of the differential equation [Eq. (49); seeAppendix], the beam on elastic foundations can be classified intotwo groups; i.e.,D. 0 for a stiff beam andD, 0 for a flexible beam.

To investigate the effect of the nature of the beam-foundationsystem, the cases of a concentrated vertical load on a flexiblebeam on the foundations with a high Poisson ratio (for soft soil) arediscussed herein. The parameters of the beam-foundation system areconsidered as eight kinds of cases, as shown in Table 4. Fig. 8 showsthe vertical and horizontal displacement curves of the beam on elastic

Fig. 8. Effect of the nature of the beam-foundation system on the vertical and horizontal displacements of the beam on the elastic foundation underconcentrated load

Fig. 9. Vertical displacements of the beam on elastic foundations with various p

Fig. 10. Horizontal displacements of the beam on elastic foundations with various p

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foundations under concentrated load. It can be observed that thebeam becomes stiffer with an increase of E=Es. Moreover, thevertical displacements of the flexible beam are basically larger thanthose of the stiff beam. However, the relationships between thehorizontal displacements and the beam’s stiffness are very complex.On the other hand, it shows that high Poisson ratio ns of the foundationleads to a high value of the horizontal displacement of the beam.

Effect of the Horizontal Load

In practical engineering, usually a beam on an elastic foundation isoften subjected to not only vertical but also horizontal loads.Therefore, the horizontal component of the total displacements ofthe beam may be not neglected. To investigate the effect of thehorizontal load on the response of the beam-foundation system, fourcases were considered, as shown in Table 3. The horizontal loads onthe beam were assumed to be p5 0, 0:1q, and 0:2q.

Fig. 9 shows that the vertical displacement curves under variouscases exhibit significant differences. For Cases 1 and 3, with the

increase of the subjected horizontal loads, the vertical displacementsdecrease at the left part of beam and increase at the right part;whereas this tendency is not evident for Cases 2 and 4. The hori-zontal displacements of the beam on an elastic foundation with adifferent value for horizontal load p are shown in Fig. 10. It maybe concluded that the horizontal displacements increase rapidly withthe increase of the horizontal load. Moreover, it is of interest toobserve that the horizontal displacement uðxÞ is sensitive to depthHof the elastic foundation when p5 0:1q. Fig. 11 shows the atten-uation functions with various values of p, where the attenuationfunctions of Case 1 are totally different from those of the other cases.Moreover, the effects of the horizontal load andEs on the attenuationfunctions are significant.

Figs. 12 and 13 show the bending moment and shear force of thebeam with various horizontal loads p, respectively. Obviously, thehorizontal load may increase the bending moment of the beam,whereas it contributes little to the shear force of the beam.Moreover,the effect of Young’s modulus Es on the shapes of the bendingmoment of the beam is significant. For example, the shape of thebending moment is saddleback when E=Es 5 300.

Fig. 14 shows the effect of the horizontal load on the syntheticdisplacement vector of the beam-foundation system, where it can beseen that the amplitude of the displacements decreaseswith the depthof the foundation as the length of the arrows decreases. When p5 0,the shape of the displacement vector is symmetric, and almost onlythe vertical displacement vector exists. When p5 0:2q, the ap-pearance of the horizontal displacement of the beam-foundationsystem is evident. In this case, it is not practical to neglect thehorizontal displacement component.

Conclusions

An analytical model to deal with the coupling analysis of theBernoulli-Euler beam on elastic bidimensional foundations is pro-posed. The present approach is an extension of the modified Vlasovmodel. With the separation of the variables, the horizontal andvertical displacements of the beam-foundation system are expressedas a displacement function at the ground surface and an attenuationfunction along the depth of the foundation, respectively. The gov-erning equations of the displacement and attenuation functions havebeen obtained to couple them via the variational principle, and theyare uncoupled in terms of the differential operator method. An itera-tive procedure is chosen to fulfill the numerical implementation. Theparametric study indicates that the response of the beam-foundation

Fig. 11.Attenuation functions of the beam on elastic foundations withvarious p ðq5 29:19 kN=m; subscripts 0, 1, and 2 denote the cases ofthe horizontal load at p5 0, 0:1q, and 0:2q)

Fig. 12. Bending moment of the beam on elastic foundations with various p

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system greatly depends on the coupling of the horizontal and verticaldisplacements, especially for the cases of flexible beams under ver-tically concentrated loads on foundations with higher Poisson ratios.Moreover, the static behavior of beams under combined loads issignificant relative to the horizontal component of the displacementfield.

Appendix. Characteristic Solution of Eq. (43)

The solution of Eq. (43) may be assumed as F3ðxÞ5Fp3e

Dx.Substituting the solution into Eq. (43) with respect to Y 5D2, yields

Y3 þ k1Y2 þ k2Y þ k3 ¼ 0 (49)

Eq. (49) can be transformed to the classical Cardano formula withZ5 Y 2 k1=3 as

Z3 þ k4Z þ k5 ¼ 0 (50)

where k4 5 k2 2 k1=3; and k5 5 2k31=272 k1k2=31 k3. The gen-eral solution of Eq. (50) can be written as

Z1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k52þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

2 þ �k43

3r3

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k522

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

22�k43

3r3

s

Z2 ¼ v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k52þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

2 þ �k43

3r3

sþ v2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k522

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

22

�k43

3r3

s

Z3 ¼ v2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k52þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

2 þ �k43

3r3

sþ v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k522

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k52

22

�k43

3r3

s

(51)

wherev5 ð211 iffiffiffi3

p Þ=2. The general solution of Eq. (50) dependson the sign of term D5 ðk5=2Þ2 1 ðk4=3Þ3.

Case 1: D> 0

Root Z1 is a real number, whereas Z2 and Z3 is a conjugate complexpair. Accordingly, root Y1 of Eq. (49) is also a real number, whereasY2 and Y3 is a conjugate complex pair. Thus, the sign of Y leads totwo subcategories of the general solution. Consequently, the generalsolution functions are listed in Eq. (46).

Case 2: D< 0

Roots Zi and Yi ði5 1, 2, 3Þ are real numbers and unequal to eachother. Moreover, the general solution of Eq. (43) depends on thesign of Yi. In general, there are eight kinds of possible solution cases.Consequently, the general solution functions are listed in Eq. (47).

Case 3: D5 0

Roots Zi and Yiði5 1, 2, 3Þ are real numbers. In practice, this case isextremely unlikely to exist.

Acknowledgments

The authors gratefully acknowledge support from the Science andTechnology Project of Hunan Province under Grant No. 2011FJ3124

Fig. 13. Shear force of the beam on elastic foundations with various p

Fig. 14. Synthetic displacement vector of the beam-foundation system

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and the Research Funds for the Young Teachers of HunanUniversity.

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