vibration of timoshenko beams on three-parameter elastic foundation

Computers and Structures 88 (2010) 294–308

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Computers and Structures

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Vibration of Timoshenko beams on three-parameter elastic foundation

K. MorfidisDepartment of Civil Engineering, Division of Structural Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

a r t i c l e i n f o

Article history:Received 1 September 2009Accepted 4 November 2009Available online 6 December 2009

Keywords:Beams on elastic foundationThree-parameter elastic foundationEigenvalue analysisVibrationsTimoshenko beamFinite element method

0045-7949/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruc.2009.11.001

E-mail address: [email protected]

a b s t r a c t

In the present paper, the process of the formulation of the equations of dynamic equilibrium and of therespective equations of natural vibration of Timoshenko beams on three-parameter elastic foundationwithin the framework of the second order theory is presented. The corresponding mass matrices are alsoformulated. In order to define the values of the parameters of the three-parameter model, an analyticalmethod is applied. With the aid of two numerical examples, it is proved that the three-parameter modelis capable of yielding natural period values that sufficiently approximate those derived by the solutionusing 2D finite element soil models.

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1. Introduction

The problem of beams resting on elastic foundation is oftenencountered in the analysis of the foundations of buildings,highway and railroad structures, and of geotechnical structuresin general. The solution of this problem demands the modelingof the mechanical behavior of the beam and the soil, as well asthe modeling of the interaction between them. Another necessaryprerequisite, however, is the correct modeling of the inertial char-acteristics of the structure and the soil so that the behavior of thesoil–structure under dynamic (e.g. seismic) load system can berendered.

The process of modeling the mechanical behavior of elasticallysupported structural elements under static or/and dynamic loadswith the use of the well-known beam/column and shell elementsis straightforward. As far as the soil is concerned, assuming thatit is homogenous, isotropic and has a linear elastic behavior, twomajor categories of soil models are encountered in the literature:(a) the continuous medium models and (b) the so-called ‘‘mechan-ical” models.

The continuous medium models are based on the equations ofelastic semi-infinite space and display the highest level of accu-racy. Nevertheless, the exact solution of their equations is a veryarduous task, feasible only in certain simple cases (see e.g. [1,2]).The solution of the problem of the continuous medium may alsobe achieved numerically, with the help of the finite element meth-od, for any case of geometric configuration and static or dynamicload. However, in order to employ 2D finite element models, the

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engineer must possess a solid theoretical grounding as well as am-ple experience, so as to be in a position to evaluate and use the re-sults of the analysis that are obtained in terms of stresses.

With respect to mechanical models, their accuracy is consider-ably smaller as they are founded in various simplifying assump-tions. Consequently, they are simpler and easier to use in thestudy of most problems. Analytical descriptions of these modelsand of their assumptions may be found in a relatively large numberof bibliographical references (see e.g. [1,3–5]). The simplest andmost frequently employed mechanical model is that of Winkler[6]. This model can be described in terms of a simple mathematicalformulation, which enables the finding of closed analytical solu-tions for problems of static or dynamic analysis. As a result, manyformulations of the equations of dynamic equilibrium and naturalvibration of beams on Winkler-type elastic foundation are to beencountered in the literature (see e.g. [7–9]). References to the for-mulation of the respective mass matrices are also numerous (seee.g. [10–12]). The weakness of the Winkler model in the productionof realistic results in a lot of cases lies chiefly in the fact that itoverlooks the influence of the soil on either side of the structure.It was for this specific reason that a number of mechanical modelswhich do not feature this drawback were proposed. These are di-vided into two categories: the two-parameter (see e.g. [13–15])and the three-parameter models (see e.g. [16,17]).

Two-parameter models are characterized by their relativelysimple mathematical formulation, which enables the analyticalsolution of the equations governing the behavior of beams restingon elastic foundation under static or dynamic load. A variety ofpublished papers on the equations of dynamic equilibrium andthe solution of the problem of natural vibration, as well as on theformation of mass matrices (see e.g. [18–21]), is thus available.

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Nomenclature

F cross-sectional area (m2)Fs effective shear cross-sectional area (m2)b width of the cross-section of the beam (m)c constant of the upper spring layer of the Kerr model

(kN m�3)C = c � b (kN m�2)T modulus of elasticity of the beam (kN m�2)Ts modulus of elasticity of the soil (kN m�2)g constant of the shear layer of the Kerr model (kN m�1)G = g � b (kN)Gb shear modulus of the beam (kN m�2)H depth of the elastic subgrade (m)I moment of inertia (m4)k constant of the lower spring layer of the Kerr model

(kN m�3)K = k � b (kN m�2)L length of the beam (m)

m mass of the beam/unit length (t m�1)M bending moment (kN m)N axial force (kN)Q shear force (kN)r2 = I/F radius of gyration (m2)T = 2p/x natural period (s)eG geometrical axial strainV vertical force (kN)vs Poisson’s ratio of the soilwc displacement of the upper spring layer of the Kerr

model (m)wk displacement of the lower spring layer of the Kerr model

(m)w = wk + wc total displacement (m)b shear rotation of the beam’s cross-sectionw bending rotation of the beam’s cross-sectionx natural frequency (–)

K. Morfidis / Computers and Structures 88 (2010) 294–308 295

Three-parameter models constitute a generalization of two-parameter models, as they incorporate the influence of the soilon either side of the structure and also take into account the ‘‘dis-continuity” of the deformations of the soil surface at the bound-aries between its loaded and unloaded regions. The bibliographyconcerning the dynamics of beams on three-parameter elasticfoundation is considerably limited. The only published papersfocusing on this specific problem are those of Rades [22], andMorfidis and Avramidis [23]. Rades [22] dealt with the dynamicbehavior of a rigid beam on Kerr-type three-parameter elasticfoundation. Meanwhile, Morfidis and Avramidis [23] were the firstto formulate and solve the dynamic problem of a Timoshenkobeam resting on Kerr-type three-parameter elastic foundationwithin the context of the first order theory. The aforementionedpaper led to the conclusion that the Kerr-type three-parametermodel allows the calculation of the natural periods of elasticallysupported beams, yielding values that are very close to those ob-tained by the solution with 2D finite elements.

In the present paper, the process of formulating the equations ofdynamic equilibrium and natural vibration of Timoshenko beamson Kerr-type three-parameter elastic foundation within the con-text of the second order theory is presented. These equations are

Fig. 1. Timoshenko beam on Kerr-type three-param

analytically solved and all the possible solution cases are laidout. Furthermore, the mass matrices that are necessary for thesolution of the problem of the natural vibration of elastically sup-ported structures using the finite element method are formulatedon the basis of the application of the Lagrange equations [24]and the ‘‘exact” interpolation functions. By means of two numericalexamples, it is demonstrated that the three-parameter model iscapable of producing results relating to the natural periods of sin-gle beams as well as of plane frames that satisfactorily converge tothe respective results derived from 2D finite element solutions,which are considered as the reference solutions. With regard tothe values of the parameters of the Kerr model, the analyticalmethod proposed by the author (see [5,25–27]) is applied.

2. Formulation of the differential equations of dynamicequilibrium

Consider a Timoshenko beam element of length L resting on aKerr-type three-parameter elastic foundation (Fig. 1), loaded by adynamic load q(x, t) (for a more detailed description of the Kerr-type three-parameter elastic foundation see [16,25,26]). In order

eter elastic foundation – boundary conditions.

Fig. 2. Infinitesimal Timoshenko beam element resting on Kerr-type three-parameter elastic foundation.

296 K. Morfidis / Computers and Structures 88 (2010) 294–308

to derive the equations of dynamic equilibrium of the beam, theprinciple of virtual work will be applied on the basis of the follow-ing assumptions: (a) the behavior of the beam material is linearelastic; (b) the cross-section is rigid and constant throughout thelength of the beam and has one plane of symmetry; (c) shear defor-mations of the cross-section of the beam are taken into accountwhile elastic axial deformations are ignored; (d) the equationsare derived bearing in mind the geometric axial deformations(Fig. 2); (e) the axial forces N acting on the ends of the beam arenot changed with time and are conservative and (f) the inertialbehavior of the soil and the damping forces are ignored. Planestrain conditions are valid for the soil–beam system. This meansthat a foundation strip of infinite length and finite width b, equalto the width of the cross-section of the beam, is considered.

In general, during the vibration of the beam work is producedby the external dynamic load, the inertial forces, the dampingforces and the internal cross-sectional forces (see Fig. 2). Accordingto the principle of virtual work, when a system is in a state of equi-librium, the virtual work of all the forces acting on the system isequal to zero for every virtual displacement.

In the case under analysis the mathematical formulation of theprinciple of virtual work is:

dW tot ¼ dWbeam þ dW ðIÞsoil þ dW ðIIÞ

soil þ dW ðIIIÞsoil þ dWext load ¼ 0 ð1Þ

where

dWbeam ¼ dWM þ dWQ þ dWN þ dW Inertial ð2aÞis the virtual work of the internal forces of the beam (that is, of thebending moments M, the shear forces Q and the axial forces N) andof the inertial forces. The inertial forces are of two types: forces dueto the translational acceleration of the beam’s cross-sections andmoments due to the angular acceleration of the beam’s cross-sec-tions (see Fig. 2)

dW ðIIÞsoil ¼ dW ðIIÞ

c þ dW ðIIÞG þ dW ðIIÞ

k ð2bÞ

is the virtual work of the interaction forces acting between soil andbeam (i.e. the forces developing due to the deformation of the twospring layers and the shear layer of the model) in field II of the soil–beam system (Fig. 1). For fields I and III, the following applyrespectively:

dW ðIÞsoil ¼ dW ðIÞ

G þ dW ðIÞk ð2cÞ

dW ðIIIÞsoil ¼ dW ðIIIÞ

G þ dW ðIIIÞk ð2dÞ

It is noted here that in fields I and III, on both sides of the beam, theupper spring layer is not deformed and thus the correspondingterms in relations (2c) and (2d) do not exist.

dWext load is the virtual work of the external dynamic forces act-ing on the beam.All above expressions constitute functions of thebasic parameters of the problem (w), (wk) and (w) (see Figs. 1and 2). These functions are not determined in the same way inthe three fields of the soil–beam system. Consequently, the calcu-lation of the virtual work for these fields is carried out separately,by applying the well-known rules of the calculus of variation in thefollowing way (see e.g. [25]):

� Field I

dW ðIÞtot ¼ dW ðIÞ

soil ¼ dW ðIÞG þ dW ðIÞ

k

¼Z 0

�1FGIðx; tÞ � d

@wkI

@x

� �� dxþ

Z 0

�1FkIðx; tÞ � dwkIðx; tÞ½ �dx

¼Z 0

�1�K �wkI þ G � @2wkI

@x2

!" #� dwkI � dx

� G � @wkI

@x� dwkI

� �0

�1ð3aÞ

� Field II

dW ðIIÞtot ¼ dWbeam þ dWext load þ dW ðIIÞ

soil

¼ ðdWM þ dWQ þ dWN þ dW InertialÞ þ dWext load

þ dW ðIIÞc þ dW ðIIÞ

G þ dW ðIIÞk

� �¼ �

Z L

0Mðx; tÞ � d @wðx; tÞ

@x

� �dx�

Z L

0½Qðx; tÞ � dbðx; tÞ�dx

þ N �Z L

0½deGðx; tÞ�dxþ

Z L

0½qðx; tÞ � dwIIðx; tÞ�dx

þZ L

0½Fcðx; tÞ � dwcIIðx; tÞ�dxþ

Z L

0FGðx; tÞ � d

@wkII

@x

� �� dx

þZ L

0½Fkðx; tÞ � dwkIIðx; tÞ�dxþ

Z L

0½FZ;inertðx; tÞ � dwII�dx

þZ L

0½Minertðx; tÞ � dw�dx


¼Z L

0GbAs �

@2wII

@x2 �@w@x

!� N � @2wII

@x2

!� C � ðwII �wkIIÞ

"

þ qðx; tÞ �m � @2wII

@t2

#� dwIIdx

þZ L

0EI � @2w

@x2

!þ GbAs �

@wII

@x� w

� ��m � r2 � @

2w

@t2

" #� dwdx

þZ L

0C � ðwII �wkIIÞ þ G � @2wkII

@x2

!� K �wkII

" #� dwkIIdx

� EI � @w@x

� �� dw

� �L

0

� GbAs �@wII

@x� w

� �� N � @wII

@x

� �� dwII

� �L

0

� G � @wkII

@x� dwkII

� �L

0ð3bÞ

� Field III

dW ðIIIÞtot ¼ dW ðIIIÞ

soil ¼ dW ðIIIÞG þ dW ðIIIÞ

k

¼Z þ1

LFGIIIðx; tÞ � d

@wkIII

@x

� �� dx

þZ þ1

L½FkIIIðx; tÞ � dwkIIIðx; tÞ�dx

¼Z þ1

L�K �wkIII þ G � @2wkIII

@x2

!" #� dwkIII � dx

� G � @wkIII

@x� dwkIII

� �þ1L

ð3cÞ

The differential equations of dynamic equilibrium in the threefields that comprise the soil–beam system, as well as the respec-tive boundary conditions that are valid for x = 0, x = L and x = ±1,are obtained from the above the equations and take the followingform:

� Fields I and III

� K �wkI þ G � @2wkI

@x2

!¼ 0 ð�1 < x < 0Þ ð4aÞ

� K �wkIII þ G � @2wkIII

@x2

!¼ 0 ðL < x < þ1Þ ð4bÞ

� Field II

GbAs �@2wII

@x2 �@w@x

!� N � @2wII

@x2

!� C � ðwII �wkIIÞ þ qðx; tÞ

�m � @2wII

@t2 ¼ 0 ð5aÞ

EI � @2w@x2

!þ GbAs �

@wII

@x� w

� ��m � r2 � @

2w

@t2 ¼ 0 ð5bÞ

C � ðwII �wkIIÞ þ G � @2wkII

@x2

!� K �wkII ¼ 0 ð5cÞ

Eqs. (5a), (5b), and (5c) constitute a system of three differentialequations. The unknowns are the three basic parameters of theproblem: wII, wkII and w in field II. All of them are coupled to eachother. However, appropriate transformations may lead to thedecoupling of parameters wkII and w. As a result, the following dif-ferential equations are derived:

k1 �@6wkII

@x6 þ k2 �@4wkII

@x4 þ k3 �@2wkII

@x2 þ K �wkII � q

"

þ EIU

� �� @

2q@x2

#þ @4

@x2@t2 k4 �@2wkII

@x2 þ k5 �@2wkII

@t2 þ k6 �wkII

" #

þ @2

@t2 k7 �@2wkII

@t2 þ k8 �wkII �m � r2

U� q

" #¼ 0 ð6aÞ

k1 �@6w@x6 þ k2 �

@4w@x4 þ k3 �

@2w@x2 þ K � w� 1þ K

C

� �� @q@xþ G

C

� �� @

3q@x3

" #

þ @4

@x2@t2 k4 �@2w@x2 þ k5 �

@2w

@t2 þ k6 � w" #

þ @2

@t2 k7 �@2w

@t2 þ k8 � w" #

¼ 0 ð6bÞ

where

k1 ¼ �EI � G

C� h; k2 ¼ EI � h � 1þ K

C

� �þ G � EI

U� N

C

� �;

k3 ¼ ðN � GÞ � K � EIU� N

C

� �; k4 ¼

m � GC� EI

Uþ r2 � h

� �

k5 ¼ �m2 � r2 � G

U � C ;

k6 ¼ �m � 1þ KC

� �� EI

Uþ r2 � h

� �þ G � 1

Cþ r2

U

� �� ;

k7 ¼m2 � r2

U� 1þ K

C

� �

k8 ¼ m � 1þ KC

� �þ r2 � K

U

� �; h ¼ 1� N

U; U ¼ Gb � As

Thus, the values of the basic parameters wkII and w can be calcu-lated from Eqs. (6a) and (6b), while the third parameter wII can bedetermined from Eq. (5c). The boundary conditions that apply forx = 0, x = L and x = ±1 are supplied in Fig. 1.

3. Formulation of the differential equations of natural vibration

Eqs. (4a), (4b), (5a), (5b), and (5c) form the starting point for thederivation of the equations of natural vibration. For q(x, t) = 0, weassume that the solutions of these equations take the form (Fou-rier’s method of the separation of variables, see e.g. [20]):

wkIðx; tÞ ¼WkIðxÞ � ft; wkIIIðx; tÞ ¼WkIIIðxÞ � ftðtÞ ð7a;bÞ

wIIðx; tÞ ¼W IIðxÞ � ftðtÞ; wkIIðx; tÞ ¼WkIIðxÞ � ftðtÞ;wðx; tÞ ¼ WðxÞ � ftðtÞ ð7c;d; eÞ

and after some appropriate transformations the equations of natu-ral vibration are derived:

U � d2W II

dx2 �dWdx

!� N � d2W II

dx2

!� C � ðW �WkIIÞ

þx2 �m �W II ¼ 0 ð0 < x < LÞ ð8aÞ

EI � d2W

dx2 þU � dW II

dx�W

� �þx2 �m � r2 �W ¼ 0 ð0 < x < LÞ ð8bÞ

C � ðW II �WkIIÞ þ G � d2WkII

dx2

!� K �WkII ¼ 0 ð0 < x < LÞ ð8cÞ

� K �WkI þ G � d2WkI

dx2 ¼ 0 ð�1 < x < 0Þ ð8dÞ

Fig. 3. Degrees of freedom of a finite element of a Timoshenko beam on Kerr-typeelastic foundation.


� K �WkIII þ G � d2WkIII

dx2 ¼ 0 ðL < x < þ1Þ ð8eÞ

With respect to Eqs. (8a), (8b), and (8c), it is possible to decouplethe unknown functions, as was the case with Eqs. (5a), (5b), and(5c). Therefore, the following decoupled equations of natural vibra-tion are derived:

k1 � d6Wdx6

� �þ k2 � d4W

dx4

� �þ k3 � d2W

dx2

� �þ k4 �W ¼ 0 ðaÞ

k1 � d6WkII

dx6

� �þ k2 � d4WkII

dx4

� �þ k3 � d2WkII

dx2

� �þk4 �WkII ¼ 0 ðbÞ

W II ¼ g4 �WkII þ g1 �d2WkII

dx2

� �ðcÞ

9>>>>>>>=>>>>>>>;! ð0 < x < LÞ

ð9a;b; cÞ

� K �WkI þ G � d2WkI

dx2 ¼ 0 ð�1 < x < 0Þ ð9dÞ

� K �WkIII þ G � d2WkIII

dx2 ¼ 0 ðL < x < þ1Þ ð9eÞ

where

k1 ¼ EI � h � g1; k2 ¼ ½U� h � ðU�x2 �m � r2Þ� � g1 þ EI � g2;

k3 ¼ �ðU�x2 �m � r2Þ � g2 þ EI � g3 þU � g4;

k4 ¼ �ðU�x2 �m � r2Þ � g3; g1 ¼ �GC;

g2 ¼ �C �x2 �m

U

� �� g1 þ h � g4;

g3 ¼CU� C �x2 �m

U

� �� g4; g4 ¼ 1þ K

C

The boundary conditions required for the solution of Eqs. (9a)–(9e)are identical with those that are valid in the equations of dynamicequilibrium, and are given in Fig. 1.

4. Solution of the differential equations of natural vibration

4.1. Equations of fields I and III

Eqs. (9d) and (9e) are linear differential equations of second or-der. These are solved in fields –1 < x < 0 and L < x < +1 respec-tively with the use of familiar mathematical methods. If theboundary conditions of Fig. 1 are taken into account, the followingis obtained:

Region I ðx � 0Þ : WkIðxÞ ¼Wk0 � eðk=GÞx ðx � 0Þ ð10ÞRegion III ðx > LÞ : WkIIIðxÞ ¼WkL � e�ðk=GÞðx�LÞ ðx � LÞ ð11Þ

The above solutions are dependent upon the values of the displace-ments of the lower layer of springs of the soil model (Fig. 1) at theends of the beam (i.e. for x = 0 and x = L), Wk0 and WkL. These valueswill be calculated after solving the differential equations of field II.

4.2. Equations of field II

The differential equations of field II (9a) and (9b) are homoge-nous linear differential equations of the sixth order. Their solutionstake the following general form:

WkIIðxÞ ¼X6

i¼1

ðCi � fiÞ ¼ fc and WðxÞ ¼X6

i¼1

ðCðIIÞi � fiÞ ¼ fcII ð12a;bÞ

The third parameter of displacement WII is calculated by Eq.((9a)–(c)), combined with the solutions of (9a) and (9b), i.e. Eqs.(12a) and (12b):

W IIðxÞ ¼X6

i¼1

ðCðIIIÞi � fiÞ ¼ fcIII ð13Þ

In the above relations:

f ¼ ½f1 . . . f6�; c ¼ fC1 . . . :C6g;cII ¼ fCðIIÞ1 . . . CðIIÞ6 g; cIII ¼ fCðIIIÞ1 . . . CðIIIÞ6 g ð14a;b; c;dÞ

Vector f consists of the real functions fi (of the spatial coordinatex) whereas the vectors c, cII and cIII consist of the integration con-stants of differential Eqs. (9b), (9a) and (9c), respectively. Theform of fi depends on the coefficients of Eqs. (9a) and (9b) and,at the same time, on a series of auxiliary quantities formed whilesolving the aforementioned equations (see e.g. [25,27]). Conse-quently, the form of functions fi is ultimately dependent uponthe values of the three parameters of the soil model, the constantsthat govern the mechanical behavior of the beam (bending andshear stiffness), the value of axial force N, as well as the valueof the natural frequency x. This means that during the iterativeprocess of defining the natural frequencies, and while x takeson different trial values, the form of fi may be modified, despitethe fact that the values of the coefficients determining themechanical behavior of the soil and the beam, as well as axialforce N, are constant. Therefore, all potential solution cases ofEqs. (9a) and (9b) must be incorporated in the solution algorithmof the problem. As is indicated by mathematics, equations of theform of (9a) and (9b) have 10 different solution forms, set asidethe virtually impossible cases in which conditions of equality be-tween the auxiliary quantities (formed while solving these specificequations) are required. The solution algorithm of the problemmust thus incorporate these 10 solution forms, which are allavailable in the bibliography (see e.g. [5,27]).

5. Derivation of mass matrices

In order to form the mass matrices, the Lagrange equations willbe applied (see e.g. [12]). The process starts by defining the degreesof freedom of a Timoshenko beam element resting on Kerr-typethree-parameter elastic foundation, which are demonstrated inFig. 3.

On the basis of this figure, the vector of the generalized coordi-nates according to Lagrange is:

uT ¼ uðtÞT ¼ w1 w1 wk1 w2 w2 wk2½ �¼ u1ðtÞ u2ðtÞ u3ðtÞ u4ðtÞ u5ðtÞ u6ðtÞ½ � ð15Þ

The displacement parameters w, wk and w in the interior of the ele-ment may thus be expressed as functions of the generalized coordi-nates in the following way:


wðx; tÞ ¼X6

i¼1

ðuiðtÞ � /iwðxÞÞ ¼ /xu;

wkðx; tÞ ¼X6

i¼1

ðuiðtÞ � /iðxÞÞ ¼ /u;

wðx; tÞ ¼X6

i¼1

ðuiðtÞ � /icðxÞÞ ¼ /c; ð16a;b; cÞ

where

/x ¼ /1w . . . /6w½ �; / ¼ /1 . . . /6½ �; /c ¼ /1c . . . /6c½ �

are vectors which contain the interpolation functions of the ele-ment that are formed (see Appendix A) on the basis of the exactsolutions of the differential equations governing the problem ofthe bending of Timoshenko beams resting on Kerr-type three-parameter elastic foundation under static load (see e.g. [25,26]). Itmust be stressed that these equations may also be derived by theequations of natural vibration (9a) and (9b), provided that the valueof x is set to 0 in the calculation of parameters k1–k4 and g1–g4. Inorder to develop the Lagrange equations:

@

@t@T@ _u

� �þ @U@u¼ pðtÞ ð17Þ

it is necessary to calculate the potential energy U and the kinetic en-ergy T of the element. The stiffness matrix is obtained from the for-mer (see [26]), while the mass matrix that is demanded here isobtained by the latter. As far as the kinetic energy of a Timoshenkobeam element is concerned, the following equation is valid:

TðtÞ ¼ 12�Z L

0m � @w

@t

� �2

dxþ 12�Z L

0ðm � r2Þ � @w

@t

� �2

dx ð18Þ

From Eqs. (16a) and (16c) respectively, the following can bederived:

@w@t

� �2

¼ @w@t� @w@t¼ ð/w � _uÞT � ð/w � _uÞ ¼ _uT � /T

w � /w � _u ð19aÞ

@w@t

� �2

¼ @w@t� @w@t¼ ð/c � _uÞT � ð/c � _uÞ ¼ _uT � /T

c � /c � _u ð19bÞ

The introduction of the previous two equations into (18) yields thefollowing:

TðtÞ ¼ 12�Z L

0m � @w

@t

� �2

dxþ 12�Z L

0ðm � r2Þ � @w

@t

� �2

dx

¼ 12� _uT �

Z L

0m � /T

w � /wdx�

� _uþ 12� _uT

�Z L

0ðm � r2Þ � /T

c � /cdx�

� _u ð20Þ

This may be simplified as follows:

TðtÞ ¼ 12� _uT �Mt � _uþ 1

2� _uT �Mr � _u ð21Þ

where

Mt ¼Z L

0m � /T

w � /w dx; Mr ¼Z L

0ðm � r2Þ � /T

c � /c dx ð22a;bÞ

are the mass matrix for translational inertia and rotatory inertia,respectively, while the mass matrix expressing the total kinetic en-ergy of the element is:

M ¼Mt þMr ð23Þ

The above are 6 � 6 square matrices. By combining (A.4a), (Appen-dix A) and (22a), we obtain:

Mt ¼Z L

0m � /T

w � /w dx ¼ JTRw �m �

Z L

0fT � f dx

� �� JRw ðJRw ¼ JT

w � R�1Þ

ð24Þ

Similarly, a combination of equations ((A.4a), Appendix A) and(22b) gives:

Mr ¼Z L

0ðm � r2Þ � /T

c � /c dx ¼ JTRc � ðm � r2Þ

�Z L

0fT � f dx

� �� JRc JRc ¼ JT

c � R�1

� �ð25Þ

A conclusion that may be drawn from Eqs. (24) and (25) is thatthe formulation of the mass matrix M necessitates the priorformulation of matrices Jw, Jc, R, as well as the calculation of thematrix I:

I ¼Z L

0fT � f dx ð26Þ

These matrices depend on the form of the functions that make upvector f. Since this vector consists of functions fi, which constitutethe exact solutions of the problem of static equilibrium of Timo-shenko beams resting on three-parameter elastic foundation, it issufficient to calculate matrices Jw, Jc, R, I, only for those solutioncases of the differential equations which may come up on the basisof the usual values acquired by the mechanical characteristics of thesoil and the beam, as well as by axial force N. An analytical investi-gation carried out in [5] aided in identifying all three specific solu-tion cases. The mass matrices for these cases are presented inAppendix B.

To conclude, it should be stressed that the 6 � 6 dimensions ofthe mass (and stiffness) matrices of Timoshenko beam elements onKerr-type elastic foundation create the problem of forming thetotal mass (and stiffness) matrices of plane frames which alsoinclude conventional planar beam/column elements, whose stiff-ness and mass matrices are 4 � 4. It is for this reason that the ‘‘sta-tic condensation” of the mass matrix of Eq. (23), as well as of therespective stiffness matrix, is absolutely essential. This process ofstatic condensation is one of matrix transformations, described inmany references (see e.g. [28]), and may easily be incorporatedinto algorithms of structural analysis.

6. Estimation of the values of the soil parameters

The problem of specifying realistic values of the soil parametersinvolved in two- and three-parameter models has already been ad-dressed in the bibliography (see e.g. [1,29]). The lack of appropriatelaboratory tests or in situ measurement methodologies, throughwhich it would be possible to overcome this problem, has alsobeen acknowledged [1]. Concerning the simpler two-parametermodels, the sole method of defining the two soil parameters re-ported in the bibliography is the analytical method of Vallabhanand Das [30,31], which is founded on a quite different formulationof the equations of the two-parameter Vlasov model (‘‘modified”Vlasov model). With regard to three-parameter models, it is possi-ble to relate their parameters with the modulus of elasticity Ts, theshear modulus Gs, and the depth H of the elastic subgrade, throughReissner’s relations [17]. In addition, a relation between the param-eters of the Reissner model and the respective parameters of theKerr model can be derived [5].

(a) (b)

. . . . . . . .

Fig. 4. Relation of the parameters nck and k to polynomial functions on the basis of symmetric (a) and anti-symmetric and (b) static loading.


In the present paper, the estimation of the values of the threeparameters of the Kerr model is achieved through the applicationof the analytical method that has been proposed by the author[5] and used successfully in the case of beams under static loads[25,26]. It is reminded here that this method is founded on the iter-ative process of the modified Vlasov model [30] for the determina-tion of two of the three parameters (parameters K, G). It is to benoted that parameter K (constant of the lower spring layer) andparameter G (constant of the shear layer) correspond with theparameters of the Vlasov model. As regards the third parameterof the Kerr model (parameter C, constant of the upper spring layer),an assumption is made, within the context of the method, that it isrelated to parameter K through the relation:

C ¼ nck � K ð27Þ

Coefficient nck is termed as ‘‘relation factor” of parameters C and K(see [25,26]). The steps to be followed for its calculation are:

1. Compilation of a database consisting of a series of beams ofvarious lengths, resting on soils with appropriately selectedmechanical characteristics, so as to cover a wide range of val-ues of the index of relative stiffness of the soil–beam systemk, as dictated by Vlasov (Fig. 4b).

2. Analysis of the beams of the database using the modified Vla-sov model and determination of the respective parameters Kand G. The analysis is carried out for symmetric loading (con-centrated vertical load at the middle of the beam) and foranti-symmetric loading (concentrated moment at the middleof the beam).

3. Analysis of the beams of the database for the load cases ofstep 2, this time using a 2D finite element model in the mod-eling of the soil. It is through this analysis that the stress anddeformation quantities to be used as target-values (referencevalues) for the next step of the procedure are obtained.

4. Iterative process of defining the optimum factor nck, for whichthe results using the Kerr model approximate those derivedwhen employing the reference model, for all the beams ofthe database.

5. The process described above yields pairs of values (k-nck),which aid in the unique matching of each value of the indexk with a value of the optimum factor nck. With the aid ofnon-linear regression analyses, these pairs lead to the deter-mination of polynomial functions, which in turn assist inthe calculation of the optimum factor nck not only for thosevalues of k for which the aforementioned analyses weremade, but for every single value of k. These functions for

the symmetric and anti-symmetric loading of the beams aresupplied in Fig. 4. (High order polynomial functions havebeen chosen in order to achieve the highest possible approx-imation. However, choice of lower order functions is possibleand does not induce any problems.)

After studying the above figure, the following may beconcluded:

(a) The approximation of pairs (k-nck) by the polynomial func-tions is closer for values of k that are greater than 1.00 inthe case of loading with a symmetric static load.

(b) The method does not converge in the determination ofspecific values of nck in the case of loading with an anti-symmetric static load, for values of k that are smallerthan 1.50. However, as will be clarified by the numericalexamples that follow, the application of the method ispossible even in cases in which k < 1.50, with very satisfac-tory results. In these cases, the value of nck for k = 1.50 isemployed.

The curves of Fig. 4 are incorporated into the solution algo-rithms of the problem of the natural vibration of beams or planeframes on Kerr-type elastic foundation. In combination with theadded incorporation of the iterative process of the modified Vlasovmodel, the determination of the values of the three parameters in-volved in the Kerr-type three-parameter model is achieved forevery case. Finally, it must be underlined that the curves ofFig. 4a and those of Fig. 4b are used for the determination of ‘‘sym-metric” and ‘‘anti-symmetric” mode shapes respectively.

7. Numerical examples

Two numerical applications are laid out in the present para-graph. The first relates to the calculation of the natural periods ofbeams with free ends (Fig. 5), while the second concerns the calcu-lation of the natural periods of three-storey plane frames (Fig. 7).The main objectives of these two examples are:

1. The application of the solution of Eqs. (9a)–(9e) in the desig-nation of the natural periods of beams on Kerr-type three-parameter elastic foundation, as well as the comparativeevaluation of this model, on the one hand, and one andtwo-parameter models, on the other hand (Fig. 5).

2. The application of the mass matrix (Eqs. (22a), (22b) and (23))in the calculation of the natural periods of plane frames

Fig. 5. Beam and soil data used in the analyses of the first example.


resting on Kerr-type three-parameter elastic foundation. Thematrix formulation of the problem of natural vibration isemployed so as to make use of the mass matrices, while thisexample is solved with the aid of the finite element method.

In order to comparatively evaluate one-, two- and three-parameter models, the 2D finite element model, by far the mostprecise model available, is used as a reference model in bothexamples (see [26]). Of course, the best choice concerning thereference model would be the model which yields results fromthe analytical solution of the equations of the elastic half-space.Nevertheless, analytical solutions are obtainable only for verysimple loading cases of academic character. Within the frame-work of the applications examined in the present paper, thesolution using the 2D Finite Element Model is the most precisesolution available.

A number of assumptions were necessary in the solution pro-cess: (a) assumption of plane stress conditions, (b) assumption oflinear elastic behavior of the beam and the subgrade, and (c)assumption of conditions of bilateral contact between the beamand the soil. The depth of the elastic subgrade was set at 60 m.

Two program codes in Fortran 90/95 were developed for theanalyses. The first one is linked with the application of the analyt-ical solutions of the problem of the natural vibration of beams onelastic foundation modeled by one-, two- and three-parameter

Table 1Solution steps of the problem of the natural vibration of beams on three-parameterelastic foundation.

Step 1: Trial value for xo: x = xi,Step 2: Formation of vector f (Eq. 14a): f = [f1....f6] (Depends on the solution

case of Eqs. (9a) and (9b))Step 3: Formation of matrices Jw, Jc (see Appendix B)Step 4: Formation of the matrix equation for the calculation of the integration

constants: F(xi) � ce = 0

FðxiÞ ¼ð0Þ ¼ d

dx

�h � f 00 � J

Tw � f0 � JT

c 0 0f 00 � J

Tc 0 0

f0 �1 0f 00

ffiffiffiffiffiffiffiffiffiffiK � Gp

0h � f0L � J

Tw � fL � JT

c 0 0f 0L � J

Tc 0 0

fL 0 �1f 0L 0

ffiffiffiffiffiffiffiffiffiffiK � Gp

2666666666664

3777777777775

ce ¼cd

� �c ¼ fC1 . . . :C6g ðsee Eq:ð14bÞÞd ¼ Wk0

WkL

� �ðsee Eqs:ð10Þandð11ÞÞ

The matrix F is formed on the basis of the boundary conditions of Fig. 1Step 5: Check of the value of the determinant of matrix F(xi)If det[F(xi)] = 0 ? xi is a natural frequency of the soil–beam systemIf det[F(xi)] – 0 ? xi is not a natural frequency of the soil–beam systemStep 6: New trial value for x: x = xi+1

models (Table 1). The second one is related with the programmingof the matrix formulation of the problem of natural vibration (seee.g. [32]), with the aim of calculating the natural periods of planeframes resting on elastic foundation modeled by one, two andthree-parameter models. The determination of the natural periodsin this code was effected by the incorporation of the ‘‘subspaceiteration method” and the ‘‘simultaneous iteration method” (see[32]).

The analytical method of defining the parameters involved insoil models that was presented in the previous paragraph isapplied in both algorithms in question. In conclusion, it is notedthat the well-known program SAP2000 [33] was used in bothnumerical applications, for the solutions with the 2D-FEM refer-ence model.

7.1. Example 1: calculation of the natural periods of free-end beams

In this example, the first three natural periods of beams, withcharacteristics that may be found in Fig. 5, were calculated. Thesame figure demonstrates the mechanical characteristics of thesoils employed in the calculations.

The results of the analyses comparing one, two and three-parameter models with the 2D-FEM reference model are illustratedin the diagrams of Fig. 6. The presentation of these results is basedin the value of the index k (Fig. 4b), which corresponds to the beamand soil data used.

More specifically, the above figure shows the divergences of thevalues of the first three natural periods, which are obtained fromthe analyses with one, two and three-parameter models, fromthe respective values derived from the analyses using the 2D-FEM reference model. A study of the diagrams leads to the follow-ing conclusions:

� The Kerr model proves to be the most efficient in terms of theapproximation of the reference value of the first natural per-iod (corresponds with a ‘‘symmetric” mode shape). Its effi-ciency increases when the values acquired by k are great.More particularly, the divergences from the reference valuesdo not exceed 5% for values of k greater than 1.00. Wherethe values of k are smaller, the divergences are greater butdo not exceed 13.5% in any of the cases. The Vlasov modelranks second, as its efficiency does not alter substantially inrelation to the value of k, with divergences ranging between20% and 25%. In contrast, the Pasternak and Winkler modelsare not sufficient in approximating the reference values. Thelowest divergence of the former is approximately 73%, whilethat of the latter is approximately 66%.

-100%

-80%

-60%

-40%

-20%

0%

20%

0.45 0.65 0.85 1.05 1.25 1.45 1.65 1.85 2.05

T2(b)

λ

-100%

-80%

-60%

-40%

-20%

0%

0.45 0.65 0.85 1.05 1.25 1.45 1.65 1.85 2.05

(Vlasov) (Kerr) (Winkler) (Pasternak)

T1

(a)

λ

-100%

-80%

-60%

-40%

-20%

0%

20%

40%

0.45 0.65 0.85 1.05 1.25 1.45 1.65 1.85 2.05

T3(c)

λ

MODE 1

MODE 2

MODE 3

Fig. 6. Deviations (in %) of the first three natural periods of the four foundation models under investigation from the respective reference (2D-FEM model) values.


� With regard to the second natural period (corresponds withan ‘‘anti-symmetric” mode shape), the Kerr model appearsto be the most precise for values of k greater than 1.40. It isworth noting that the efficiently of this model in fields(0 < k < 1.4) and (k > 1.4) is widely different. To illustrate,when k > 1.4, the maximum divergence from the referencevalue is 3% whereas, when k < 1.4, it may even mount to30%. This discrepancy may be clarified by studying Fig. 4b,where the second mode shape is ‘‘anti-symmetric”. Themethod employed for the determination of nck does not con-verge for values of k smaller than 1.50. Therefore, the factornck corresponds with the value k = 1.50 in such cases,unavoidably leading to greater divergences for the specificfield of k. In spite of this, the Kerr model is not inferior thanthe rest of the models in terms of efficiency, since its diver-gences are of the same order as the respective divergencesof the Vlasov model for k < 1.50. Finally, the Pasternak andWinkler models are the least precise, featuring divergencesthat exceed 50% in every case.

� With reference to the third natural period (corresponds witha ‘‘symmetric” mode shape), the Kerr model once again estab-lishes itself as the most efficient. The difference to beobserved in comparison with the equally ‘‘symmetric” firstmode shape is that the efficiency of the model in questiondecreases with an increasing value of k. Nevertheless, itsdivergences are never greater than 22%. The Vlasov model isagain more precise than the other two models, but its effi-ciency is not uniform in all the range of the values acquiredby k. When these values are great, the divergences of thismodel approximate those of the Winkler and Pasternak mod-els, which are still the least efficient.

7.2. Example 2: calculation of the natural periods of three-storey planeframes

The present paragraph constitutes a presentation of the resultsderived from the calculation of the first two natural periods of thethree three-storey plane frames given in Fig. 7.

The reason for the calculation of the first two natural periods isthat these periods correspond with a natural vibration of theframes along their horizontal and vertical axis (Fig. 8). On the basisof analysis made on frames with less storeys, it was concluded thatthe greatest of the two periods corresponds with a natural vibra-tion along the vertical axis, in the case of a combination of longfoundation beams and soils of low stiffness. Furthermore, it wasobserved that as the number of storeys of the frame increases,the greatest period corresponds, as a rule, with a natural vibrationalong the horizontal axis. More specifically, with respect to thethree-storey frames examined within the context of this example,the greatest natural period corresponds with a natural vibrationalong the horizontal axis in all cases.

Another point to be highlighted is that the auxiliary static load-ing used to determine parameters K and G (step 2 of the process ofdetermining the parameters, see paragraph 6) is composed of theuniformly distributed vertical loads on the beams (Fig. 7) and thehorizontal concentrated forces at the storey levels which are line-arly distributed along the height. The rationale behind this choicewas the association of all the results of the example with resultsof analyses demanded by the seismic codes. As regards the choiceof the curve from which the value of nck is to be taken (Eq. (27)), itwas deemed that the curve of Fig. 4a should be employed both forthe calculation of the natural period corresponding with a vibra-tion along the vertical axis and for the calculation of the natural

Fig. 8. The mode shapes that correspond with the first two natural periods of the frames.

Fig. 7. Frame and soil data used in the analyses of the second example.


period corresponding with a vibration along the horizontal axis. Athorough application of the method (as laid out in paragraph 6)would necessitate the use of the curve of Fig. 4a in order to deter-mine the natural period that corresponds with a vibration alongthe vertical axis (‘‘symmetric” mode shape) and the use of thecurve of Fig. 4b for the determination of the natural period that

corresponds with a vibration along the horizontal axis (‘‘anti-sym-metric” mode shape). The sole purpose for selecting the curve ofFig. 4a only was to simplify the algorithm employed, as it was con-sidered that it should be relatively simple within the framework ofthe example. This decision ultimately contributed, to some extent,to a decrease in the efficiency of the Kerr model, as will be

Vlasov Kerr Pasternak Winkler

2 0.54 -53.8% 9.4% 3.9% 345.9%

1 0.58 -52.4% 6.6% -2.1% 320.0%

5 0.82 -45.0% 15.2% -4.3% 311.6%

3 1.07 -38.1% 20.3% -11.3% 282.8%

4 1.17 -30.2% 22.5% -17.0% 261.5%


2 0.66 -36.4% 3.6% 14.9% 254.8%

1 0.71 -39.6% 2.4% 12.6% 246.8%

5 1.00 -22.2% 7.7% -7.0% 191.5%

3 1.31 -14.7% 9.3% -13.6% 145.6%

4 1.44 -9.0% 7.9% -8.6% 108.1%


2 1.40 -35.4% 13.9% 29.9% 201.6%

1 1.51 -34.8% 13.8% 27.8% 194.0%

5 2.12 -23.7% 18.0% 3.5% 149.1%

3 2.77 -17.7% 14.9% -16.0% 112.2%

4 3.04 -13.3% 9.6% -12.5% 84.2%

-40%

-30%

-20%

-10%

0%

10%

20%

0.66 0.71 1.00 1.31 1.44

Vlasov Kerr Pasternak

FRAME F3-B

-60%

-40%

-20%

0%

20%

0.54 0.58 0.82 1.07 1.17


FRAME F3-A

-40%

-20%

0%

20%

40%

1.40 1.51 2.12 2.77 3.04


FRAME F3-C

Fig. 9. Deviations (in %) of the natural period which corresponds to the vibration along the horizontal axis of the frames of the four foundation models under investigationfrom the respective reference (2D-FEM model) values.


demonstrated by the results that follow. All the same, the results ofthe specific model are sufficiently close to the reference results.Still, the recommended process for the formulation of an algorithmfor general use must necessarily coincide the one described in par-agraph 6.

Figs. 9 and 10 illustrate the natural periods corresponding witha natural vibration of the frames along their horizontal and verticalaxis, respectively.

After a careful study of Fig. 9, the following conclusions arise:

� The Kerr model appears to be the most precise in the case offrames F3-B and F3-C. Especially in the case of F3-B, its diver-gences from the reference values do not exceed 10%. In thecase of F3-A, which is shorter in length than the other twoframes, the Kerr model lacks in precision when compared tothe Pasternak model, but still excels in comparison with theremaining two models. Even in this case, however, its diver-gences are not greater than 22%.

� The Pasternak model yields satisfactory results in the case offrames F3-A and F3-B. In contrast, in the case of F3-C, whichhas a larger span, its results are not as accurate, especially forthe ‘‘soft” soils E1 and E2 (divergences of the order of 30%).

� The Vlasov model proves to be less accurate than the Kerr andPasternak models in all cases. Its greatest divergences appear,as a rule, in the case of the ‘‘soft” soils E1 and E2, and espe-cially in the case of the frame F3-A, which is the shortest inlength.

� The divergences of the Winkler model are quite substantial,mainly in the case of ‘‘soft” soils. In general, in none of thecases studied do these divergences remain on a satisfactorylevel (they normally exceed 150%).

A more general assumption to be made is that, when calculatingthe first natural period that corresponds with a vibration along thehorizontal axis of the frame, the influence of the soil on either sideof the frame does not play a vital role in the convergence of the ref-erence values. On the contrary, the consideration of the shearstresses of the soil is of vital importance. This conclusion isgrounded in the fact that the Pasternak model, which takes theseshear stresses into account but does not account for the influenceof the soil on either side of the frames, is more precise than the Vla-sov model, which takes into consideration both of these influences.Fig. 9 proves that the combination of the two influences leads tothe consideration of the soil as stiffer than it actually is. Therefore,the Kerr model, with the added consideration of the local deforma-tions under the loaded surface of the soil (see [25,26]), leads to acloser approximation of the actual stiffness of the soil.

Having examined Fig. 10, we may conclude the following:

� The Kerr model is more precise that all the other models, withthe exception of the cases of ‘‘soft” soils E1 and E2 for framesF3-A and F3-B, where the Pasternak model seems to excel.Nevertheless, its divergences do not exceed 10% in any ofthe cases analysed. This model is thus more efficient herethan in the calculation of the natural period correspondingwith a vibration along the horizontal axis of the frame. Thismay easily be explained by taking into consideration the fol-lowing facts: (a) the first mode shape along the vertical axis is‘‘symmetric” and (b) the curve of Fig. 4a, which is obtained bythe static symmetric loading of beams (paragraph 6), wasused for the calculation of the factor nck.

� The Pasternak model is considerably precise in the case of‘‘soft” soils E1 and E2. However, that does not apply to all

SOIL Vlasov Kerr Pasternak Winkler

2 0.54 -24.0% -10.6% -5.5% 100.7%

1 0.58 -22.3% -9.7% 0.1% 90.6%

5 0.82 -22.5% -7.1% 17.6% 95.5%

3 1.07 -22.7% -4.7% 38.6% 92.4%

4 1.17 -24.5% -3.2% 67.6% 97.8%


2 0.66 -22.3% -9.6% 6.2% 85.1%

1 0.71 -20.6% -8.5% 0.4% 76.1%

5 1.00 -21.0% -5.8% 52.7% 79.3%

3 1.31 -21.5% -3.3% 73.8% 75.1%

4 1.44 -20.7% -2.2% 72.8% 76.9%


2 1.40 -21.5% -3.0% -13.5% 66.5%

1 1.51 -22.2% -2.2% -14.4% 58.8%

5 2.12 -22.6% 3.1% 17.8% 61.9%

3 2.77 -24.9% 5.1% 45.8% 60.0%

4 3.04 -27.6% 3.2% 40.7% 65.1%

-40%-20%

0%20%

40%60%

80%

0.54 0.58 0.82 1.07 1.17


FRAME F3-A

-40%

-20%

0%

20%

40%

60%

80%

0.66 0.71 1.00 1.31 1.44


FRAME F3-B

-30%

-15%

0%

15%

30%

45%

1.40 1.51 2.12 2.77 3.04


FRAME F3-C

Fig. 10. Deviations (in %) of the natural period which corresponds to the vibration along the vertical axis of the frames of the four foundation models under investigation fromthe respective reference (2D-FEM model) values.


cases, since this model appears to be rather inaccurate whenthe soils are ‘‘hard” (divergences greater than 40%).

� The Vlasov model does not vary in terms of efficiency, as itsdivergences range between 20.6% and 27.6%.

� Once again, the Winkler model is shown to be the least pre-cise. All the same, it produces divergences that are signifi-cantly smaller than in the case of the calculation of thenatural period that corresponds with a vibration along thehorizontal axis.

8. Summary and conclusions

In the present paper the equations of dynamic equilibrium andthe equations of natural vibration of Timoshenko beams on Kerr-type three-parameter elastic foundation are formulated. In orderto derive these equations, the influence of constant axial forcesat the ends of the beam (second order theory) as well as the influ-ence of the soil on either side of the beam, have been taken into ac-count. Apart from that, the respective mass matrices are formed byapplying the Lagrange equations.

Two numerical examples are presented so as to evaluate the re-sults derived from application of the equations of natural vibrationand of the mass matrices. The first example refers to the calcula-tion of natural periods of beams with free ends, while the secondrelates to the calculation of the natural periods of three-storeyplane frames. In both examples, the reliability of the one, twoand three-parameter soil models were also evaluated. This wasaccomplished by using as reference the model in which the soilis modeled with the aid of 2D shell finite elements. In order to esti-mate the values for the soil model parameters, an analytical meth-od proposed in previous papers is applied that is based on the

modified Vlasov model and on the appropriate correlation of theparameters of the Kerr model. It is explicitly described here andis quite easy to program.

Evaluation of the results of the first numerical application led tothe following conclusions:

� The Kerr model proves to be the most accurate among allmodels examined in the calculation of the first three naturalperiods. The Vlasov, Pasternak and Winkler models then fol-low in terms of accuracy.

� The Pasternak and Winkler models display divergence andare therefore insufficient in approximating the referenceresults. This weakness is attributed to the inability of the spe-cific models to take into account the influence of the soil oneither side of the beams.

� Evaluation of the results of the second numerical example ledto the following conclusions:

� The factors affecting the level of accuracy with which the ref-erence values of the natural periods are approximated are: (a)consideration of the influence of soil on either side of theframes; (b) consideration of the shear stresses developing inthe soil and (c) consideration of the local deformations underthe loaded surface of the soil.

� The Kerr model, which is capable of taking into account allthree factors mentioned above, achieves the optimal conver-gence with respect to the reference results in almost all casesstudied. The Vlasov model, which takes into consideration theinfluence of the soil on either side of the frames and the shearstresses in the soil, yields natural period values that are smal-ler than the reference values. This indicates that the specificmodel overestimates the stiffness of the soil. The Pasternak


model, which only takes into consideration the stresses underthe loaded surface of the soil and not the influence of the soilon either side of the frames, appears to be more precise thanthe Vlasov model in the case where calculation of the naturalperiod corresponds with a natural vibration along the hori-zontal axis of the frames. However, the same is not alwaystrue in the case of calculation of the natural period that cor-responds to a natural vibration along the vertical axis of theframes. Finally, the Winkler model, which does not considerany of the aforementioned factors, is the least precise of allinvestigated models.

Acknowledgement

The results presented in this paper are based on a postdoctoralresearch investigation financially supported by the Greek StateScholarships Foundation (IKY).

Appendix A. Formulation of the interpolation functions

The displacement parameters in the interior (i.e., betweennodes) of the element are expressed through Eqs. (12a), (12b)and (13). The specification of these parameters requires the calcu-lation of 18 integration constants (see Eqs. (14b), (14c) and(14d)). Nevertheless, these constants are not independent. Withthe help of Eq. (8c), it is possible to relate vector cIII to vectorc, while Eq. (8b) aids in the relation of vector cII to vector c. Itmay thus be demonstrated that the following relations are valid(see [26]):

wkðxÞ ¼ f � c; wðxÞ ¼ f � JTw � c; wðxÞ ¼ f � JT

c � c ðA:1a;b; cÞ

where Jw, Jc are 6 � 6 square matrices whose form depends on func-tions fi(x) which in turn depend on the form of the solution of thedifferential equations that govern the bending of Timoshenkobeams on Kerr-type three-parameter elastic foundation under staticloading (these matrices are derived from the equations presented inAppendix B, by setting x = 0).

The first step towards the formulation of the interpolation func-tions consists in the expression of the vector of the generalized La-grange coordinates u (Eq. (15) and Fig. 3) as a function of the vectorof the integration constants c (Eq. (14b)). By inserting the Eqs.(F.1a), (F.1b), (F.1c) for x = 0 and x = L into (15), and after appropri-ate transformations, we obtain:

u ¼ R � c) uðtÞ ¼ Rðx ¼ 0; x ¼ LÞ � c ðA:2Þ

where R is a 6 � 6 square matrix supplied in Appendix B.From the relation (F.2) the following is obtained:

c ¼ R�1 � u ðA:3Þ

The second and final step in the process of formulation of the inter-polation functions is the introduction of (A.3) in Eqs. (F.1a), (F.1b),(F.1c). As a result:

wðx; tÞ ¼ fðxÞ � JTw � R

�1 � uðtÞwkðx; tÞ ¼ fðxÞ � R�1 � uðtÞ wðx; tÞ ¼ fðxÞ � JT

c � R�1 � uðtÞ

)

)wðx; tÞ ¼ /wðxÞ � uðtÞwkðx; tÞ ¼ /ðxÞ � uðtÞ wðx; tÞ ¼ /cðxÞ � uðtÞ

)

)/wðxÞ ¼ fðxÞ � JT

w � R�1

/ðxÞ ¼ fðxÞ � R�1

/cðxÞ ¼ fðxÞ � JTc � R

�1

ðA:4a;b; cÞ

Appendix B. Mass matrices

M ¼ m � ðJTw � R

�1ÞT �Z L

0ðfT � fÞdx

� �� JT

w � R�1

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Mt

þ ðm � r2Þ � ðJTc � R

�1ÞT �Z L

0ðfT � fÞdx

� �� JT

c � R�1

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Mr

Jw ¼

Jw11 0 0 0 0 0

0 Jw22 0 0 0 0

0 0 Jw33 Jw34 0 0

0 0 Jw43 Jw44 0 0

0 0 0 0 Jw55 Jw56

0 0 0 0 Jw65 Jw66

266666666664

377777777775;

Jc ¼

Jc11 0 0 0 0 0

0 Jc22 0 0 0 0

0 0 Jc33 Jc34 0 0

0 0 Jc43 Jc44 0 0

0 0 0 0 Jc55 Jc56

0 0 0 0 Jc65 Jc66

266666666664

377777777775;

R ¼

f0 � JTw

f0 � JTc

f0

fL � JTw

fL � JTc

fL

266666666664

377777777775

f0 ¼ fx¼0

fL ¼ fx¼L

� Case 1

f ¼ eR1xje�R1xjeRx � cosðQxÞjeRx � sinðQxÞje�Rx � cosðQxÞje�Rx � sinðQxÞ �

R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�bþ

ffiffiffiffiDp3

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b�

ffiffiffiffiDp3

q� J1

3

r; R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2p

þm2

s;

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2p

�m2

s

m ¼ �12�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�bþ

ffiffiffiffiDp3

qþ


ffiffiffiffiDp3

qþ 2 � J1

3

� �;

n ¼ffiffiffi3p

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�bþ

ffiffiffiffiDp3

q�


ffiffiffiffiDp3

q� �D ¼ a3 þ b2

a ¼ 13� � J2

1

3þ J2

" #; b ¼ 1

2� 2 � J3

1

27� J1 � J2

3þ J3

" #;

J1 ¼k2

k1; J2 ¼

k3

k1; J3 ¼

k4

k1

The relations from which parameters k1, k2, k3, k4 are obtainedare given in paragraph 3, setting x = 0.

Jw11 ¼ Jw22 ¼ A1; Jw34 ¼ �Jw43 ¼ Jw65 ¼ �Jw56 ¼ A2;

Jw33 ¼ Jw44 ¼ Jw55 ¼ Jw66 ¼ A3; Jc11 ¼ �Jc22 ¼ A4

Jc43 ¼ �Jc34 ¼ Jc65 ¼ �Jc56 ¼ A6; Jc33 ¼ Jc44 ¼ �Jc55 ¼ �Jc66 ¼ A5


A1 ¼ 1þ KC

� �� G

C� R2

1; A2 ¼ 2 � R � Q � GC;

A3 ¼ 1þ KC

� �� G

C� ðR2 � Q 2Þ

A4 ¼U � R1

�EI � R21 þU

" #� A1; A5 ¼

a1 � b1 þ a2 � b2

b21 þ b2

2

!�U;

A6 ¼a1 � b2 � a2 � b1

b21 þ b2

2

!�U; U ¼ Gb � As

a1 ¼ A3 � Rþ A2 � Q ; a2 ¼ A2 � R� A3 � Q ;b1 ¼ �EI � ðR2 � Q 2Þ þU; b2 ¼ 2 � EI � R � Q

� Case 2

f ¼ eRxje�RxjeR1xje�R1xjeR2xje�R2x �

R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi�ap

� cos/3

� �� J1

3

s;

R1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi�ap

� cosð/þ 2pÞ

3

� �� J1

3

s;

R2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi�ap

� cosð/þ 4pÞ

3

� �� J1

3

s

/ ¼ cos�1 � bffiffiffiffiffiffiffiffiffi�a3p

� �; a ¼ 1

3� � J2

1

3þ J2

" #;

b ¼ 12� 2 � J3

1

27� J1 � J2

3þ J3

" #; J1 ¼

k2

k1; J2 ¼

k3

k1J3 ¼

k4

k1


Jw11 ¼ Jw22 ¼ A1; Jw33 ¼ Jw44 ¼ A2; Jw55 ¼ Jw66 ¼ A3;

Jw34 ¼ Jw43 ¼ Jw65 ¼ Jw56 ¼ 0

Jc11 ¼ �Jc22 ¼ A4; Jc33 ¼ �Jc44 ¼ A5; Jc55 ¼ �Jc66 ¼ A6;

Jc34 ¼ Jc43 ¼ Jc65 ¼ Jc56 ¼ 0

A1 ¼ 1þ KC

� �� G

C� R2; A2 ¼ 1þ K

C

� �� G

C� R2

1;

A3 ¼ 1þ KC

� �� G

C� R2

2

A4 ¼U � R

�EI � R2 þU

� �� A1; A5 ¼

U � R1

�EI � R21 þU

" #� A2;

A6 ¼U � R2

�EI � R22 þU

" #� A3; U ¼ Gb � As

� Case 3

f ¼ eRxje�Rxj cosðQxÞj sinðQxÞj cosðQ 1xÞj sinðQ1xÞ �

R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi�ap

� cos/3

� �� J1

3

s;

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

ffiffiffiffiffiffiffi�ap

� cosð/þ 2pÞ

3

� �þ J1

3

s;

Q1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

ffiffiffiffiffiffiffi�ap

� cosð/þ 4pÞ

3

� �þ J1

3

s

/ ¼ cos�1 � bffiffiffiffiffiffiffiffiffi�a3p

� �; a ¼ 1

3� � J2

1

3þ J2

" #;

b ¼ 12� 2 � J3

1

27� J1 � J2

3þ J3

" #; J1 ¼

k2

k1; J2 ¼

k3

k1; J3 ¼

k4

k1


Jw11 ¼ Jw22 ¼ A1; Jw33 ¼ Jw44 ¼ A2; Jw55 ¼ Jw66 ¼ A3;

Jw34 ¼ Jw43 ¼ Jw65 ¼ Jw56 ¼ 0

Jc11 ¼ �Jc22 ¼ A4; Jc43 ¼ �Jc34 ¼ A5;

Jc65 ¼ �Jc56 ¼ A6 Jc33 ¼ Jc44 ¼ Jc55 ¼ Jc66 ¼ 0

A1 ¼ 1þ KC

� �� G

C� R2; A2 ¼ 1þ K

C

� �þ G

C� Q 2;

A3 ¼ 1þ KC

� �þ G

C� Q 2

1

A4 ¼U � R

�EI � R2 þU

� �� A1; A5 ¼

U � QEI � Q 2 þU

� �� A2;

A6 ¼U � Q 1

EI � Q 21 þU

" #� A3; U ¼ Gb � As

References

[1] Selvadurai APS. Elastic analysis of soil–foundation interaction. Amsterdam:Elsevier; 1979.

[2] Poulos HG, Davis EH. Elastic solutions for soil and rock mechanics. NewYork: John Wiley & sons Inc.; 1974.

[3] Kerr AD. Elastic and viscoelastic foundation models. J Appl Mech1964;31:491–8.

[4] Dutta SC, Roy R. A critical review on idealization and modeling for interactionamong soil foundation–structure system. Comput Struct 2002;80:1579–94.

[5] Morfidis K. Research and development of methods for the modelling offoundation structural elements and soil (in Greek). PhD thesis, Dep. of CivilEngineering, Aristotle University of Thessaloniki, 2003.

[6] Winkler E. Die Lehre von der Elastizitat und Festigkeit. Dominicus: Prague;1867.

[7] Timoshenko S, Young DH. Vibration problems in engineering. New York: D.Van Nostrad Company Inc.; 1955.

[8] Mathews PM. Vibrations of a beam on elastic foundation. ZAMM1958;38:105–15.

[9] Cheng FY, Pantelides CP. Dynamic Timoshenko beam–column on elastic media.J Struct Eng 1988;114(7):1524–50.

[10] Eisemberger M, Yankelevski DZ, Adin MA. Vibration of beams fully or partiallysupported on elastic foundation. Earthquake Eng Struct Dynam1985;13:651–60.

[11] Eisemberger M, Clastornik J. Vibrations and buckling of a beam on a variableWinkler elastic foundation. J Sound Vib 1987;115:233–41.

[12] Lai YC, Ting BY, Lee WS, Becker BR. Dynamic response of beams on elasticfoundation. J Struct Eng 1992;118(3):853–8.

[13] Filonenko-Borodich MM. Some approximate theories of elastic foundation (inRussian). Uchenyie Zapiski Moskovskogo Gosudarstvennogo UniversitetaMekhanica 1940;46:3–18.

[14] Pasternak PL. On a new method of analysis of an elastic foundation by meansof two constants (in Russian). Moscow, USSR: Gosudarstvennoe IzdatelstvoLiteraturi po Stroitelstvu I Arkhitekture; 1954.

[15] Vlasov VZ, Leont’ev UN. Beams, plates and shells on elastic foundations.Gosudarstvennoe Izdatel’stvo, Fiziko-Matematicheskoi Literatury, Moskva,1960. [(Translated from Russian), Israel Program for Scientific Translations1966, Jerusalem].

[16] Kerr AD. A study of a new foundation model. Acta Mech 1965;I/2:135–47.

[17] Reissner E. Note on the formulation of the problem of the plate on an elasticfoundation. Acta Mech 1967;4:88–91.

[18] Valsangkar AJ, Pradhanang R. Vibrations of beam-columns on two-parameterelastic foundation. Earthquake Eng Struct Dynam 1988;16:217–25.


[19] Franciosi C, Masi A. Free vibrations of foundation beams on two-parameterelastic soil. Comput Struct 1993;47(3):419–26.

[20] Wang TM, Stephens JE. Natural frequencies of Timoshenko beams onpasternak foundation. J Sound Vib 1977;51(2):149–55.

[21] Yokoyama T. Vibrations of Timoshenko beam-column on two-parameterelastic foundations. Earthquake Eng Struct Dynam 1991;20:355–70.

[22] Rades M. Forced vibrations of a rigid body on a three-parameter foundation.Int J Mech Sci 1971;13:573–83.

[23] Morfidis K, Avramidis IE. Exact eigenvalue solutions for and comparativeperformance of the Kerr-type three-parameter model of a beam on elasticfoundation. In: Proceedings of PACAM X 2008. Cancun, Mexico, p. 349–52.

[24] Lagrange J. Mecanique Analytique. t. I, 1788.[25] Avramidis IE, Morfidis K. Bending of beams on three-parameter elastic

foundation. Int J Solids Struct 2006;43:357–75.[26] Morfidis K. Exact matrices for beams on three-parameter elastic foundation.

Comput Struct 2007;85:1243–56.

[27] Morfidis K. Investigation of the influence of the soil flexibility and itsmodelling on the structural response within the contexts of the responsespectrum method and the static pushover analysis (in Greek). Research Reportfunded by Greek State Scholarships Foundation (I.K.Y.), Athens, 2008.

[28] Bathe KJ. Finite element procedures in engineering analysis. Englewood Cliffs,New Jersey: Prentice-Hall; 1982.

[29] Kerr AD. On the determination of foundation model parameters. J Geotech Eng1985;111(11):1334–40.

[30] Vallabhan CVG, Das YC. Parametric study of beams on elastic foundations. JEng Mech 1988;114(12):2072–82.

[31] Vallabhan CVG, Das YC. Modified Vlasov model for beams on elasticfoundations. J Geotech Eng 1991;117(6):956–66.

[32] Bhatt P. Programming the dynamic analysis of structures. London: Spon Press;2002.

[33] SAP NonLinear Version 7.44. Integrated finite element analysis and design ofstructures. Computers and Structures Inc, Berkeley, CA, USA, 2000.

vibration of timoshenko beams on three-parameter elastic foundation

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