Research Paper - ijerst · analysis of isotropic beams are plenty, the literature on static and dynamic analysis of functionally graded beams on elastic foundations have not been

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Int. J. Engg. Res. & Sci. & Tech. 2016 Trilochan Rout et al., 2016

SIGMOID TIMOSHENKO BEAM ON VARIABLEELASTIC FOUNDATION

A sigmoid Timoshenko beam resting on variable elastic foundation is investigated for its dynamicbehavior. Finite element method is used for simply supported end condition of the beam. Thestatic part of the governing differential equation of the beam is solved to determine the shapefunctions to obtain accurate results. Various variable foundation models like linear, parabolicand sinusoidal model are considered for investigation. It is observed that parabolic foundationout of the chosen models renders better dynamic behavior for the beam.

Keywords: Sigmoid, Sinusoidal, Foundation modulus, Power index

INTRODUCTIONConventional composite materials are preferredover alloys for engineering applications as theypossess better properties. But these materialssuffer from the problem of delamination at theinterface resulting from stress mismatch thereon. Functionally Graded Material (FGM), anadvanced composite consists of two or moredissimilar materials and the properties of thismaterial can be designed to vary continuouslywith respect to spatial coordinates by varying thevolume fractions of its constituent phases alongspatial coordinates so as to improve the strength,toughness and high temperature withstandingability. FGMs are regarded as one of the mostpromising candidates for many engineering1 Centurion University of Technology & Management, Parlakhemundi, Odisha.2 Department of Mech Engg., CIT, Bhubaneswar, Odisha.3 Department of Mech Engg., PMEC, Berhampur, Odisha.

Int. J. Engg. Res. & Sci. & Tech. 2016

ISSN 2319-5991 www.ijerst.comVol. 5, No. 2, May 2016

© 2016 IJERST. All Rights Reserved

Research Paper

sectors such as the aerospace, aircraft,automobile, defense industries, electronic andbiomedical sectors. Many machine and structuralcomponents in aforesaid sections can bemodeled as beams. A rigorous literature surveyis carried out and an overview of the survey isgiven below.

The calculation of the natural periods of beamswith free ends and three-storey plane framesindicates that the Kerr model is the most accurateamong all models examined in the calculation ofthe first three natural periods and then follow theVlasov, Pasternak and Winkler models in termsof accuracy based on (a )consideration of theinfluence of soil on either side of the frames; (b)consideration of the shear stresses developing

Surya Narayan Padhi1, Ramesh Chandra Mohanty2 and Trilochan Rout3*

*Corresponding Author: Trilochan Rout [email protected]

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in the soil; and (c) consideration of the localdeformations under the loaded surface of the soil(Morfidis, 2010). The results from the simulationof the transient elastic wave propagation in aninfinite beam on viscoelastic foundation supportsthat the high-order nonreflecting artificial boundarycondition is accurate for solving elastic wavepropagation problems of infinite (or semi-infinite)beams on viscoelastic foundation (Liu and Li,2003). For each load velocity, there is a criticalfrequency in which the maximum deflection andbending moment are in their highest level. Thesetwo critical frequencies are independent of eachother and variation of the maximum bendingmoment and deflection vs. load frequency followdifferent trend (Kargarnovin and Younesian, 2004).The distribution of CNT, foundation stiffness andvolume fraction of CNT have significant effectson the natural frequencies and critical bucklingload of the CNTRC beams. The beams with FG-X distribution have higher fundamental frequencyas well as critical buckling load in comparisonwith other distributions. The natural frequenciesand critical buckling loads decrease withincreasing the slenderness ratio (Yas andSamadi, 2012).

The convergence of the Galerkin truncation forTimoshenko beams on foundations is found tobe slightly slower than the Euler-Bernoulli beam.Furthermore, the vertical deflections of theTimoshenko beam are slightly larger (Ding et al.,2014). In the study of the free vibration behaviorof the FG beams made of Al2O3/Al,Wattanasakulpong and Ungbhakorn (2012) foundthat the trend of frequency results for variousmodes of vibration decreases as the volumefraction indexes increase, except for the case ofthe E-E boundary conditions in which the trendof the first two modes is reversed owning to the

effects of translational and rotational springs atboth ends (Wattanasakulpong and Ungbhakorn,2012).

Rout has studied the dynamic stability ofFunctionally Graded Material (FGM) beamssubjected to parametric excitation using finiteelement method. He used first order sheardeformation theory (Timoshenko beam theory) forthe analysis of the beams (Rout, 2012). Rongying,Shen Hongxing, Hua Xianding, Jin have developedan analytical method for determining the bending–torsion coupled random response of the thin-walled beams with monosymmetrical cross-sections which takes into account the effects ofwarping stiffness, shear deformation and rotatoryinertia on the random response of twoappropriately chosen thin-walled beams (Junet al., 2004). Dong and Xing-jian have investigatedthe vibration characteristics of a cantileverlaminated beam with surface bondedpiezoelectric materials. They developed a steppedbeam model using Timoshenko beam theory topredict analytically the natural frequencies andmode shapes of a stepped laminated compositebeam. In this model, the effect of sheardeformation and rotary inertia of the steppedbeam were taken into account (Dong, 2006). Hou,Shui-hung Yan, Qing-xu Feng and De-xing haveworked on the stabilization problem of a couplednonuniform Timoshenko beam system usingfeedback control and shown that the energy ofthe closed loop system exponentially decays tozero and can be proven to be asymptoticallystable with comparatively fewer feedback controls(Hou et al., 2005). Chakraborty et al. (2003) havedeveloped a beam finite element to study thethermoelastic behaviour of functionally gradedbeam structures with exponential and power lawvariation of material properties along thickness.

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Though the literatures on static and dynamicanalysis of isotropic beams are plenty, theliterature on static and dynamic analysis offunctionally graded beams on elastic foundationshave not been reported to the best of the authors’knowledge. In the present article, SFGO beamhinged at both the ends and resting on variableelastic foundation is considered for dynamicanalysis.

MATERIALS AND METHODSFormulationA functionally graded sigmoid beam with steel andaluminium as its constituent phases is consideredfor analysis as shown in Figure 1a. The beam,hinged at both the ends is subjected to a dynamicaxial load tPPtP ds cos . Where, t is time,

sP is the static component, dP is the amplitudeof the dynamic component and is the

frequency of the applied dynamic load componentof tP . The mid-longitudinal (x-y) plane is chosenas the reference plane for expressing thedisplacements as shown in Figure 1b. Thethickness coordinate is measured as z from thereference plane. The axial displacement, thetransverse displacement, and the rotation of thecross-section are u , w and respectively. Figure1c shows a two nodded beam finite element havingthree degrees of freedom per node.

The generalized displacement vector of theelement can be given as

111ˆ iiiiii wuwuu ...(1)

The equation of motion for the elementsubjected to axial force tP can be expressed interms of nodal degrees of freedom as

0ˆˆ uktPkum gef ...(2)

Figure 1a: Functionally Graded Sandwich Beam Subjected to Dynamic Axial Load

Fig. 1(a) Functionally graded sandwich beam subjected to dynamic axial load.Figure 1b: The Coordinate System with Generalized Forces and Displacements for the FGSWBeam Element

FGM

Z, w, V

Y, M, φ

X, u, N

Fig. 1(b) The coordinate system with generalized forces and

displacements for the FGSW beam element.

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The axial load tP is taken as tPPtP d cos ,, tPPtP d cos ,

so that, PPs and PP dd .. P i is the

critical buckling load of an isotropic beam withsimilar geometrical dimensions and endconditions and , d are called static anddynamic load factors respectively. Consideringthe application of static and dynamic componentof load in the same manner we have Equation(2) in the following form

0ˆcosˆ uktPkum gdef

...(3)

feef kkk

where ek , fk , m and gk are element

elastic stiffness matrix, foundation stiffnessmatrix, mass matrix and geometric stiffnessmatrix respectively. Assembling the elementmatrices as used in Equation (3), the equation inglobal matrix form which is the equation of motionfor the straight beam, can be expressed as

0ˆcosˆ UKtPKUM gdef

...(4)

feef KKK

M , eK , fK , gK are global mass,

elastic stiffness, foundation stiffness and

geometric stiffness matrices respectively and U

is global displacement vector. Equation (4)represents a system of second order differentialequations with periodic coefficients of the Mathieu-Hill type. The periodic solutions for the boundarybetween the dynamic stability and instabilityzones can be obtained from Floquet Theory(BOLOTIN (1964)) as follows. Considering firstorder expansion, a solution with twice the timeperiod which is of practical importance isrepresented by

2

cos2

sinˆ11

td

tctU

...(5)

Substituting Equation (5) into Equation (4) and

comparing the coefficients of2

sint and

2cos

t

terms the condition for existence of theseboundary solutions with twice the time period isgiven by

0ˆ4

2/2

UMKPK gdef

...(6)

Figure 1c: Beam Element Showing Generalized Degrees of Freedom for ith Element

i I+1

wiwi+1

i ui+1ui

Fig. 1(c) Beam element showing generalized degrees of freedom for ith element.

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The solution of Equation (10) will give two sets

of values of

1

for given values of , d ,

P , and 1 . The plot between d and

1

will

will give the regions of dynamic instability.

Element MatricesThe element matrices for the SFG beam elementare derived following the procedure as proposedby Chakraborty et al. (2003).

Shape FunctionsThe displacement fields according to first ordershear deformation beam theory is expressed as

),(),,,(),,(),(),,,( txwtzyxWtxztxutzyxU

...(11)

The cross-sections are assumed to remainplane after the deformation.

The longitudinal and shear strains are

.,x

w

xz

x

uxzxx

...(12)

wherex

w

is the slope of the deformed

longitudinal axis

The stress-strain relation in matrix form canbe given by

xz

xx

xz

xx

zGk

zE

)0

0)(...(13)

where xx is the normal stress in longitudinaldirection and xz is shear stress in zx plane,

)(zE is Young’s modulus and )(zG is shearmodulus and k is shear correction factor..

Equation (6) represents an eigenvalue problemfor known values of , d , and P . Thisequation gives two sets of eigenvalues () bindingthe regions of instability due to the presence ofplus and minus sign. The instability boundary canbe determined from the solution of the equation

04

2/2

MKPK gdef

...(7)

Free Vibration

When =0, d =0, and2

, Equation (7) is

reduced to a problem of free vibration as

02 MK ef ...(8)

The solution of Equation (8) gives the value ofnatural frequencies

Static StabilityWhen =1, d =0, and 0 , Equation (7) isreduced to the problem of static stability as

0 gef KPK ...(9)

The solution of Equation (9) gives the valuesof buckling loads.

Regions of Instability1 ,, the fundamental natural frequency and P

the critical buckling load of an isotropic beam withsame geometrical dimensions and endconditions are calculated from Equation (8) andEquation (9) respectively.

Choosing 11

,, Equation (7) can be

rewritten as

04

2/21

2

1

MKPK gdef

...(10)

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The material properties of the FGM that variesalong the thickness of the beam are assumed tofollow sigmoid distribution given by

2/01)( 11 hzzgRzgRzR bt

02/1)( 22 zhzgRzgRzR bt

...(14)

n

h

zzg

21

2

111

n

h

zzg

21

2

12 ...(15)

where )(zR denotes a material property suchas, E, G, etc., Rt and Rb denote the values ofthe properties at topmost and bottommost layerof the beam respectively, and n is an index.

The kinetic energy T and elastic strain energyS of an element are given respectively as

dAdxt

W

t

UzT

l

A

22

0

)(2

1 ...(16)

dAdxxx

uz

xz

x

uzES

l

A

0

2

2

2

2)(2

1

l

A

dAdxx

w

x

wzG

0

2

2 2)(2

1 ...(17)

The governing differential equations can bederived by applying Hamilton’s principle aspresented below.

0)(

u

ST

0)(

w

ST, and

0)(

ST...(18)

The shape functions for the displacement fieldfor finite element formulation are obtained bysolving the static part of the Equation (18) withthe following consideration.

,2321 xaxaau

,37

2654 xaxaxaaw

.21098 xaxaa ...(19)

The above displacement fields are substitutedin the static part of Equation (18) in order to findout the constants of polynomials. Subsequently,the displacement fields are expressed in termsof the nodal degree of freedoms as follows.

uxwuu T ˆ ...(20)

x , a 3 x 6 matrix is the required shapefunction.

Better convergence can be achieved as theshape functions are obtained from the exactsolution of static part of the governing differentialequation. Now the shape function can also beexpressed as

Twu xxxx ...(21)

where, xu , xw , x are the shapefunctions for the axial, transverse and rotationaldegree of freedom respectively.

Element Elastic Stiffness MatrixThe general force boundary conditions for theelement can be given as

A

xxx xB

x

uAdAN

1111

A

xzx x

wAdAV 55

A

xxx xD

x

uBdAzM

1111 ...(22)

102



where, xN , xV , xM are axial force, shear forceand bending moment respectively acting at theboundary nodes. Similarly substituting Equation(20) into Equation (22), we get

Fuke ˆ ...(23)

where Txxxxxx lMlVlNMVNF )()()()0()0()0(

is the nodal load vector and

ek is the required element elastic stiffness

matrix.

Element Mass MatrixThe element mass matrix is derived bysubstituting Equation (20) into the expression ofkinetic energy Equation (26).

umuTT ˆˆ

2

1 ...(24)

where

][][][][][ uwu mmmmm , and they

can be expressed as

l

uT

uu dxIm0

0

l

wT

ww dxIm0

0

l

T dxIm0

2

l

uTT

uu dxIm0

1

...(25)

where, dAzzzIIIA 2

210 1)(

um , wm , m , represent the contribution of u ,w , degree of freedom to the mass matrix and

um represents the mass matrix arising due tothe coupling between u and . Though the massmatrix is not derived from the exact solution ofthe governing differential equation, it includes theeffect of rotary inertia as seen in Equation (25)which definitely improves the accuracy of thesolution.

Element Geometric Stiffness MatrixWhen an axial load P is applied on the beamelement, the work done by the load can beexpressed as

l

p dxx

wtPW

0

2

)(21

...(26)

Substituting the value of from Equation (20)into Equation (26) the work done can beexpressed as

dxuutP

Wl

w

T

wT

p ˆˆ2

)(

0

''

ukutP

g ˆˆ2

)( ...(27)

where, l

w

T

wg dxk0

'' is called the element

geometric stiffness matrix.

Element Elastic Foundation MatrixThe work done by the foundation is given by theexpression

l

vf dxwxkW0

2

2

1

l

wT

wvT dxuxku

0

ˆˆ2

1

uku fT ˆˆ

2

1 ...(28)

103



where, vk is the foundation stiffness parameter

per unit width of the beam and l

wT

wvf dxxkk0

is element foundation stiffness matrix.

The foundation stiffness that varies along thelength of beam considered in present study is asfollows.

xkk ov 1 for linear variation of foundation

stiffness

21 xkk ov for parabolic variation of

foundation stiffness and

xkk ov sin1 for sinusoidal variation of

foundation stiffness.

The variation of the chosen foundation modulealong length of the beam is depicted in Figure 2.

Validation of the FormulationThe present result is found to be in goodagreement with the result of Li (2008) as given inTable 1.

Natural Frequencies(rad/s)

PresentMethod

Li (2008)Difference

(%)

6431.63 6457.93 -0.4

21702.7 21603.18 0.46

40029.06 40145.42 -0.28

59799.26 59779.01 0.03

80290.11 79686.16 0.75

Table 1: Comparison of Natural FrequencyBetween Present Method and Li’s Result

1

2

3

4

5

RESULTS AND DISCUSSIONThe simulation is carried out for a SigmoidFunctionally Graded Ordinary (SFGO) beam forsimply supported end conditions. The constituentphases chosen are steel and aluminum.

The non-dimensional natural frequencies,

21

4

11

EI

AL and the non-dimensional

foundation modulus,EI

LkK v

4

Frequency AnalysisAn SFGO beam with steel-rich bottom isconsidered for the dynamic analysis. The lengthof the beam is 0.5 m, width is 0.1 m with variousthicknesses. The material properties are asfollows.

Steel: E = 2.1 x 10111 Pa, G = 0.8 x 10111 Pa, =7.85 x 103 kg/m3. The shear correction factor k= 0.8667.

Aluminium: E = 0.7 x 10111 Pa, G = 0.2697 x1011 Pa, = 2.707 x 103 kg/m3.

1 = 6724.9, the fundamental frequency ofan equivalent ordinary steel beam of thickness0.125 m.

Figures 3a and 3b show respectively the effectof slenderness parameter (L/h) on first andsecond mode non-dimensional natural frequency

Figure 2: Variation of Foundation StiffnessAlong Length of Beam for Various Foundation

Models

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11600

1650

1700

1750

1800

1850

1900

1950

2000

length(x)

found

atio

n s

tiff

nes

s(k

)

linear foundationparabolic foundationsinusoidal foundation

104



of the superior material steel decreases withincrease power index in the mixture of steel andaluminum thereby decreasing the stiffness of thebeam.

Figure 5a shows the effect of linear foundationon first mode natural frequency of beam withsteel-rich bottom for property distributionaccording to power index = 1. It is observed thatthe first mode frequency increases with increaseof foundation modulus. The effect of sinusoidal

Figure 3a: Effect of Slenderness Parameteron Fundamental Frequency

2 3 4 5 6 7 8 9 107

7.5

8

8.5

9

9.5

S lenderness para meter L/h

Fu

nd

amen

tal

freq

uen

cy 1

n=1K= 0

Figure 3b: Effect of Slenderness Parameteron Second Mode Frequency

2 3 4 5 6 7 8 9 1010

15

20

25

30

35

40

Slenderness parameter L/h

Sec

on

d m

ode

freq

uen

cy 2

n=1K= 0

of an SFGO beam with steel-rich bottom. It isfound that the natural frequencies of both themodes increase with the non-linearly with theparameter.

Figures 4a and 4b show respectively the effectof power index on first and second mode naturalfrequency of an SFGO beam with steel-richbottom. The frequencies decrease as the powerindex varies from 1 to 3 in case of both themodes. It may be due to the fact that the amount

Figure 4a: Effect of Power Indexon Fundamental Frequency

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 38.3

8.35

8.4

8.45

8.5

8.55

8.6

8.65

Power index n

Fun

dam

enta

l fr

eque

ncy

1

L/h=4K=0

Figure 4b: Effect of Power Index on SecondMode Frequency

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

21

21.05

21.1

21.15

21.2

21.25

21.3

21.35

Power index n

Fun

dam

enta

l fre

que

ncy

1

L/h=4K=0

105



foundation on fundamental frequency is depictedin Figure 5b. The fundamental non-dimensionalfrequency increases with foundation modulus inslightly non-linear manner.

The effect of parabolic foundation on non-dimensional frequency shown in Figure 5creveals a similar result as shown in Figure 5b.

To have a comparison among variousfoundations as regards their effect on dynamicbehavior of the beam the following relations havebeen used.

The above chosen relations ensure the valueof foundation constants at two ends of beam samefor all the foundation models as shown in Figure 3.

Figure 6 shows the comparison among thefoundations. It is observed that the parabolicfoundation model gives the highest non-dimensional frequency, the linear foundationmodel the intermediate frequency while thesinusoidal model rendering the lowest frequency.

CONCLUSIONFinite element method is used to investigate the

Figure 5a: Effect of Linear Foundationon Fundamental Frequency

0 100 200 300 400 500 600 700 800 900 10008.5

8.6

8.7

8.8

8.9

9

9.1

9.2

9.3

9.4

Foundation modulus K

Fun

dam

enta

l fre

quen

cy

1

L/h=4=0.4n=1

Figure 5b: Effect of Sinusoidal Foundationon Fundamental Frequency

0 100 200 300 400 500 600 700 800 900 10008.5

8.6

8.7

8.8

8.9

9

9.1

9.2

9.3

9.4


Fun

dam

enta

l fre

que

ncy

1

L/h=4=0.4n=1

Figure 5c: Effect of Sinusoidal Foundationon Fundamental Frequency

0 100 200 300 400 500 600 700 800 900 10008.5

8.6

8.7

8.8

8.9

9

9.1

9.2

9.3

9.4


Fun

dam

enta

l fre

quen

cy 1

L/h=4=0.4n=1

Figure 6: Comparison of the Effects of VariousFoundations on Fundamental Frequency forthe Foundation Modulus at the Ends of theBeam Remaining Same for All Foundations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 18.8

8.85

8.9

8.95

9

9.05

Linear foundation parameter

Fun

dam

enta

l fre

quen

cy 1

Li near foundationSi nusoidal foundat ionParabolic foundat ion

106



6. Kargarnovin M H and Younesian D (2004),“Dynamics of Timoshenko Beams onPasternak Foundation Under Moving Load”,Vol. 31, pp. 713-723.

7. Li X F (2008), “A Unified Approach forAnalyzing Static and Dynamic Behaviours ofFunctionally Graded Timoshenko and Euler-Bernaulli Beams”, Vol. 318, Nos. 4-5,pp. 121-122.

8. Liu T and Li Q (2003), “Transient ElasticWave Propagation in an Infinite TimoshenkoBeam on Viscoelastic Foundation”, Vol. 40,pp. 3211-3228.

9. Morfidis K (2010), “Vibration of TimoshenkoBeams on Three-Parameter ElasticFoundation”, Comput. Struct., Vol. 88, Nos.5-6, pp. 294-308.

10. Rout T (2012), “On the Dynamic Stability ofFunctionally Graded Material Beams UnderParametric Excitation”, Dep. Mech. Eng.,Ph.D., No. 508.

11. Wattanasakulpong N and Ungbhakorn V(2012), “Free Vibration Analysis ofFunctionally Graded Beams with GeneralElastically End Constraints by DTM”,pp. 297-310.

12. Yas M H and Samadi N (2012), “InternationalJournal of Pressure Vessels and PipingFree Vibrations and Buckling Analysis ofCarbon Nanotube-Reinforced CompositeTimoshenko Beams on Elast icFoundation”, Int. J. Press. Vessel. Pip.,Vol. 98, pp. 119-128.

dynamic behaviour of SFGO beam resting onvariable elastic foundation. The followingconclusions may be drawn from the aboveanalysis:

1. Beam with higher slenderness parametershould be used for better dynamic behavior.

2. Sigmoid distribution with the value power indexas one should be designed to ensure the betterperformance of the beam.

3. The parabolic foundation model should bepreferred over the selected foundations foranalysis.

REFERENCES1. Chakraborty A, Gopalakrishnan S and Reddy

J N (2003), “A New Beam Finite Element forthe Analysis of Functionally GradedMaterials”, International Journal ofMechanical Science, Vol. 45, pp. 519-539.

2. Ding H, Yang Y, Chen L and Yang S (2014),“Vibration of Vehicle – Pavement CoupledSystem Based on a Timoshenko Beam ona Nonlinear Foundation”, J. Sound Vib.,Vol. 333, No. 24, pp. 6623-6636.

3. Dong X (2006), “Vibration Analysis of aStepped Laminated Composite TimoshenkoBeam”, Vol. 32, No. 2005, pp. 572-581.

4. Hou S, Yan Q and Feng D (2005),“Stabilization of Coupled Non-UniformTimoshenko Beam System”, Vol. 63,pp. 2329-2333.

5. Jun L, Rongying S, Hongxing H and XiandingJ (2004), “Response of MonosymmetricThin-Walled Timoshenko Beams to RandomExcitations”, Vol. 41, pp. 6023-6040.

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Research Paper - ijerst · analysis of isotropic beams are plenty, the literature on static and dynamic analysis of functionally graded beams on elastic foundations have not been