research article finite element formulation for stability

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013 Article ID 841215 7 pageshttpdxdoiorg1011552013841215

Research ArticleFinite Element Formulation for Stability andFree Vibration Analysis of Timoshenko Beam

Abbas Moallemi-Oreh1 and Mohammad Karkon2

1 Department of Mechanical Engineering Shahreza Branch Islamic Azad University Shahreza Iran2Department of Civil Engineering Ferdowsi University of Mashhad Mashhad Iran

Correspondence should be addressed to Mohammad Karkon karkon443gmailcom

Received 18 February 2013 Revised 1 April 2013 Accepted 1 April 2013

Academic Editor K M Liew

Copyright copy 2013 A Moallemi-Oreh and M Karkon This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beamCubic displacement polynomialand quadratic rotational fields are selected for this element Moreover it is assumed that shear strain of the element has the constantvalue Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energyand its stationing to shear strain By exploiting these interpolation functions beam elementsrsquo stiffness matrix is also examinedFurthermore geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation onstability and beam free vibration At last accuracy and efficiency of proposed element are evaluated through numerical testsThesetests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis

1 Introduction

Two versions of theories have been developed for analysis ofbeams In Euler-Bernoulli theory the displacement of beam isconsideredwithout shear effectsThismethod gives appropri-ate and acceptable response in thin beam inwhich shear effectis insignificant However in this approach by increasing thethickness of beam and shear effect deformation the error ofresponse is increasing [1] Correspondingly the effect of sheartransformation is formulated in Timoshenko theory There-fore this method has a better result especially in deep beamsin which shear effect is impressive Although the rotationalinertia of thick beams was investigated by Rayleigh for thefirst time Timoshenko has developed this theory and formu-lated shear effect Due to the complexity of the governingequations of the free vibration and stability of beams ingeneral numerical methods such as finite element have beendeveloped profoundly Up to now many elements have beenpresented based on Timoshenko theory These elements areclassified into two groups which are simple and higher orderelements Some researchers used simple two-node elementswith four degrees of freedom [2ndash4] Thomas et al haveexamined the elements proposed by other researchers [3]

The first high-order element was proposed by Kapur witheight degrees of freedom [5] Lees and Thomas formulated acomplex element by applying independent polynomial seriesfor displacement and rotation fields [6 7] Also this methodhas been used byWebster [8] Rao and Gupta have examinedfree vibration of rotating beams [9] In some methods likeisoparametric formulation displacement and rotation fieldsare assumed dependently with the same order [10] Basedon Euler-Bernoulli theory Goncalves et al have presentedfrequency equation and vibration modes for classical bound-ary conditions such as clamped free pinned and slidingsupports [11] Lee and Schultz have considered free vibrationof Timoshenko beam through pseudospectral method [12]

So far very little research has been done on buckling ofTimoshenko beam with respect to free vibration analysisExploiting an approximate method based on finite elementsformulation Wieckowski and Golubiewski examined beamstability of Euler-Bernoulli and Timoshenko theories [13]Also Kosmatka has proposed a two-node element for stabil-ity and free vibration analysis of Timoshenko beam [14]

In this study a new beam element is proposed for freevibration and stability analysis of Timoshenko beams Inorder to compute shape function of the beam element cubic

2 Advances in Acoustics and Vibration

119908119895

120579119895

119895

ℎ119904

119897119908119894

120579119894

119894

119909

Figure 1 Timoshenko beam element

displacement polynomial and quadratic rotational fields areselected In addition shear strain of the element is assumedconstant Then by exploiting the bending and shear strainenergy of the beam and stationary with respect to constantshear strain interpolation function of this element has beencarefully calculated In the following by using these shapefunctions the stiffness matrix geometric stiffness matrixand mass matrix of the proposed element have been exactlydetermined Finally several numerical tests are performed toinvestigate the robustness of this element for free vibrationand stability analysis of beams with different boundaryconditionsThe findings prove that the suggested element hashigh level of accuracy and free of shear locking

2 Finite Element Formulation

In the finite elementmethod displacement and rotation fieldsof the element are associated with interpolation functions tonodal degrees of freedom Figure 1 shows proposed elementwith two nodes The shape functions of the Timoshenkobeam are calculated based on Figure 1 In order to computeshape function of the beam in Figure 1 cubic displacementpolynomial and quadratic rotational fields are selected Addi-tionally it is assumed that shear strain has the constant valueof 1205740 Based on these assumptions (1) can be written as

follows

119908 =119908119894

2(1 minus 119904) +

119908119895

2(1 + 119904)

+ 1205730119897 (1 minus 119904

2) + 1205731119897119904 (1 minus 119904

2)

120579 =120579119894

2(1 minus 119904) +

120579119895

2(1 + 119904) + 120572

0(1 minus 119904

2)

120574 = 1205740

119904 =2119909

119897minus 1

(1)

In these equations 1205731 1205730 1205720 and 120574

0are unknown parame-

ters In order to determine their values at first the equation ofshear strain for Timoshenko beam is established By utilizingthe shear strain value equal to 120574

0 the subsequent equations

will be available

120574 =119889119908

119889119909minus 120579 =

2

119897sdot119889119908

119889119904minus 120579

1205740=

2

119897(minus

119908119894

2+119908119895

2minus 21205730119897119904 + 1205731119897 minus 3120573

11198971199042)

minus 120579119894(1 minus 119904

2) minus 120579119895(1 + 119904

2) minus 1205720(1 minus 119904

2)

(2)

In the present formula the coefficients of the terms 119904 and 1199042

are equivalent to zero Therefore in the succeeding lines 1205720

1205731are determined in terms of the unknown parameter 120574

0

Γ =2

119897(119908119895minus 119908119894) minus (120579

119894+ 120579119895)

1205730=

1

8(120579119894minus 120579119895)

1205720= minus

3

2(1205740minus1

2Γ)

1205731=

1

61205720

(3)

At this stage there is only one unknown constant 1205740 which

can be discovered through the condition of minimum strainenergy It should be added that the structural strain energy isthe sum of bending and shear strain energy Bending strainenergy is calculated in the following way

119880119887=

119864119868

2int

119897

0

1205812119889119909 =

119864119868119897

4int

1

minus1

1205812119889119904 (4)

In (4) 120581 represents curvature which is determined as follows

120581 = minus2

119897sdot119889120579

119889119904= 1205810minus 6

1199041205740

119897

1205810=

1

119897(120579119894minus 120579119895+ 3119904Γ)

(5)

Substituting (5) into (4) leads to the following bending strainenergy

119880119887= 1198800+ 6119864119868(minus

Γ1205740

119897+1205742

0

119897)

1198800=

119864119868119897

4int

1

minus1

1205812

0119889119904

(6)

Besides (7) shows the energy of shear strain

119880119904=

119866119860

2119891119904

int

1

0

1205742119889119909

=119866119860119897

4119891119904

int

1

minus1

1205742

0119889119904

=119866119860119897

2119891119904

1205742

0

(7)

In this equation 119891119904is shear correction factor which depends

on cross section shape of beam This coefficient for rectan-gular section is 56 By adding the bending and shear strainenergy together total strain energy is found as follows

119880 = 119880119887+ 119880119904= 1198800minus6119864119868Γ

1198971205740+6119864119868

1198971205742

0+119866119860119897

2119891119904

1205742

0 (8)

Implementing 1205971198801205971205740= 0 will give the following results

1205740=

6119864119868Γ

1198661198601198972119891119904+ 12119864119868

= 120575Γ

120575 =6120582

1198972 + 12120582 120582 =

119891119904119864119868

119866119860

(9)

Advances in Acoustics and Vibration 3

Substituting 1205731 1205730 1205720 and 120574

0into (1) the succeeding shape

functions for Timoshenko beam can be as follows

119908

120579119899

= [1198731

1198735

1198732

1198736

1198733

1198737

1198734

1198738

]

119908119894

120579119899119894

119908119895

120579119899119895

1198731=

1

4[2 + 119904

3(1 minus 2120575) + 119904 (minus3 + 2120575)]

1198732=

119897

4[05 (1 minus 119904

2) + (119904

3minus 119904) (05 minus 120575)]

1198733=

1

4[2 minus 119904

3(1 minus 2120575) minus 119904 (minus3 + 2120575)]

1198734=

119897

4[minus05 (1 minus 119904

2) + (119904

3minus 119904) (05 minus 120575)]

1198735=

1

4119897[6 (1 minus 119904

2) (minus1 + 2120575)]

1198736=

1

4[minus1 + 119904 (minus2 + 3119904) + 6 (1 minus 119904

2) 120575]

1198737=

1

4119897[minus6 (1 minus 119904

2) (minus1 + 2120575)]

1198738=

1

4[minus1 + 119904 (2 + 3119904) + 6 (1 minus 119904

2) 120575]

(10)

In finite elementmethod displacements and strain field of theelement can be related to the nodal degrees of freedom withinterpolation functions Hence equations of finite elementformulation can be written as follows

119910

120579 = [119873] 119863119864

120576 =

119889119910

119889119909

minus119889119910

119889119909+ 120579

= [119861] 119863119864

(11)

where

[119861] =[[

[

0119889

119889119909119889

119889119909minus1

]]

]

[119873] (12)

and [119873] is the matrix of interpolation function Also 119863119864

is nodal displacement and [119861] is strain matrix Consequentlystiffness matrix of Timoshenko element can be obtained asfollows

[1198700] = int [119861]

119879[119863119898] [119861] 119889119909 (13)

where [119863119898] is the elasticitymatrix for Timoshenko beam ele-

ment

[119863119898] = [

[

119864119868 0

0119866119860

119891119904

]

]

(14)

By calculating (13) the stiffness matrix of the proposed ele-ment is obtained in the succeeding forms

[1198700] =

119864119868

1198973 + 12119897120582

times[[[

[

12 6119897 minus12 6119897

6119897 41198972+ 12120582 minus6119897 2119897

2minus 12120582

minus12 minus6119897 12 minus6119897

6119897 21198972minus 12120582 minus6119897 4119897

2+ 12120582

]]]

]

(15)

3 Mass Matrix

The kinetic energy 119879 of an elemental length 119897 of a uniformTimoshenko beam is given as follows [1]

119879 =1

2int

1198972

minus1198972

120588119860 1199102119889119909 +

1

2int

1198972

minus1198972

120588119868 1205792119889119909 (16)

In this equation 120588 is the mass density of the material of thebeam and 119868 is the second moment of area of cross sectionTherefore the mass matrix of the element has been two partsone related to translations and the other related to rotationsin the form of

[119872] = [1198721] + [119872

2]

=119897

2int

1

1

120588119860[119873119908]119879[119873119908] 119889119904

+119897

2int

1198972

minus1198972

120588119868[119873120579]119879[119873120579]119879

119889119904

(17)

The translation mass matrix [1198721] is achieved as follows

[1198721] =

1205881198601198975

210(12120582 + 1198972)2

times[[[

[

11989811

11989812

11989813

11989814

11989812

11989815

minus11989814

11989816

11989813

minus11989814

11989811

minus11989812

11989814

11989816

minus11989812

11989815

]]]

]

(18)

In the following the entries of this matrix are introduced

11989811

=6

1198974(1680120582

2+ 294119897

2120582 + 13119897

4)

11989812

=1

1198973(1260120582

2+ 231119897

2120582 + 11119897

4)

11989813

=9

1198974(560120582

2+ 841198972120582 + 3119897

4)

11989814

= minus1

21198973(2520120582

2+ 378119897

2120582 + 13119897

4)

11989815

=2

1198972(126120582

2+ 211198972120582 + 13119897

4)

11989816

= minus3

21198972(168120582

2+ 281198972120582 + 13119897

4)

(19)


In addition the rotation mass matrix [1198722] can be obtained

as follows

[1198722] =

1205881198681198972

30(12120582 + 1198972)2

times[[[

[

11989821

11989822

minus11989821

11989822

11989822

11989823

minus11989822

11989824

minus11989821

minus11989822

11989821

minus11989812

11989822

11989824

minus11989812

11989823

]]]

]

(20)

The entries of this matrix are defined in the below form

11989821

= 36119897

11989822

= minus3 (60120582 minus 1198972)

11989823

=4

119897(360120582

2+ 151198972120582 + 3119897

4)

11989824

=1

119897(720120582

2minus 601198972120582 minus 1198974)

(21)

4 Geometric Stiffness Matrix

The concept of the neutral state of equilibrium is used forbuckling analysis of beam In the mathematical formulationof elastic stability of beam the neutral equilibrium is usedassuming a bifurcation of the deformations That is at thecritical load of the possible two paths of deformations (oneassociatedwith the stable equilibrium and the other pertinentto the unstable equilibrium condition as shown in Figure 2)the beam always takes the buckled form In addition to theexistence of this bifurcation of equilibrium paths the elasticstability analysis of plates assumes the validity of Hookersquoslaw By considering Figure 2 at the buckled form the axialshortening of beam can be acquired as follows

119889119904 = radic(1198891199092 + 1198891199102)

≃ 119889119909 +1

2(119889119910

119889119909)

2

119889119909

997904rArr 119889120575 =1

2(119889119910

119889119909)

2

119889119909

(22)

As a result the strain energy of axial load 119875 can be expressedas follows

Δ119882 =119875

2int

119897

0

(119889119910

119889119909)

2

119889119909 (23)

Based on (23) geometric stiffness matrix of the element canbe calculated as follows

[119870119892] = int

119897

0

[119889119873119908

119889119909]

119879

119875[119889119873119908

119889119909] 119889119909 (24)

119889119910

119889119909

119889119904

119889120575

Figure 2 Straight and buckled form

By calculating this equation the geometric stiffness matrixbecomes as follows

[119870119892] =

119875

60119897

times[[[

[

1198961198921

1198961198922

1198961198923

1198961198922

1198961198922

1198961198924

minus1198961198922

1198961198925

1198961198923

minus1198961198922

1198961198921

minus1198961198922

1198961198922

1198961198925

minus1198961198922

1198961198924

]]]

]

(25)

where

1198961198921

= 60 + 12120573

1198961198922

= 6120573119897

1198961198923

= minus60 minus 12120573

1198961198924

= 51198972+ 31205731198972

1198961198925

= 31205731198972minus 51198972

120573 =1198974

(1198972 + 12120582)2

(26)

For stability analysis and determination of themagnitude of astatic compressive axial load thatwill produce beambucklingthe following eigenvalue will be achieved

det ([119870]) = det ([1198700] minus 119875 [119870

119892]) = 0 997904rArr 119875cr (27)

The lowest positive eigenvalue of this equation is the magni-tude of buckling load and the corresponding eigenvector isthe deformed shape of the buckled beam The exact solutionof beam-column buckling load with shear deformation effectis obtained as succeeding form [15]

119875cr =1205872119864119868

1198712eff

times (1

1 + (1205872119891119904119864119868) (1198712eff119866119860)

)

(28)

where119871eff is the effective beam length inwhich (119871eff = 119871) and(119871eff = 1198712) are used for pinned-pinned beams and fixed-fixed beams respectively

41 First Example (Vibration Analysis) The efficiency of pro-posed element is evaluated by analyzing the free vibrationof beam with simply and clamped supports for various


Table 1 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam

Mode Euler theory ℎ119897 = 0002

Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31428 314158 3141582 628319 62928 628310 6283103 942478 94573 942449 9424504 125664 126437 125657 1256575 157080 158596 157066 1570686 188496 191127 188473 1884767 219911 224113 219875 2198838 251327 257638 251273 2512889 282743 291793 282666 28269210 314159 326672 314053 314098

0 050

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus02 0 020

01

02

03

04

05

06

07

08

09

1No3

(a)

minus04minus02 00

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus05 0 050

01

02

03

04

05

06

07

08

09

1No3

(b)

Figure 3 The first three modes of buckling for beam-column (a) simply supported and (b) clamped supported








Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam



Table 5 Dimensionless frequency parameter 120582119894for the clamped

supported Timoshenko beam


Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687

lengths to thickness ratio The Poissonrsquos ratio of this beamis ] = 03 and shear correction factor is taken 56




Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598

To compare other researchersrsquo results frequency dimen-sionless parameter 120582

119894 defined in (29) has been shown in


Table 7 Critical load of the simply and clamped supported beam-column

119897ℎSimply supported Clamped supported

Analytical solution Ferreira [10] Proposed element Analytical solution Ferreira [10] Proposed element10 80138 80218 801386 29766 29877 29770100 8223 8231 82225 32864 32999 328641000 00082 00082 000822 00329 00330 00329

(Tables 1 2 3 4 5 and 6) for 10 first frequencies of this beamby exploiting 40 elements

1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)

Tables 1 2 3 and 4 5 6 show dimensionless parametersof natural frequency of beam-free vibration for simply andclamped supports respectively considering three ratios ofthickness to length Results have been compared to otherresearchersrsquo studies The mentioned tables demonstrate thatthe accuracy of the proposed element is very high in analyz-ing free vibration of beam

42 Second Example (Buckling Analysis) The efficiency ofthe proposed element is determined in buckling analysis Forthis propose critical load of a simply and clamped supportedbeam-column is computed The modulus of elasticity Pois-sonrsquos ratio and length of this beam-column are 119864 = 101198907 ] =

13 and 119897 = 1 respectively The buckling loads of simplyand clamped supported beam-columns are listed in Table 7It demonstrates that the proposed element has high accuracyin buckling analysis Figure 3 shows the first three modes ofbuckling for simply and clamped supported beam-columnrespectively

5 Conclusion

This study has proposed a new beam finite element formu-lation for the stability and free vibration analysis of beamswith shear effect deformation For this purpose displacementfield of the element has been selected from the third degreerotation field has been selected from the second degree andshear strain is assumed constant value By employing thebending and shear strain energy of the element and stationaryrespect to unknown shear strain this value is obtainedIn the following using the shear strain the interpolationfunctions for displacement and rotation fields of element hasbeen exactly calculated Then these interpolation functionshave been used and stiffness matrix geometric stiffnessmatrix and mass matrix of the proposed element have beenclearly obtained Evaluating the efficiency and accuracy of theelement for free vibration and stability analysis of beam withsimply and clamped supports desirable results are obtainedThe results show high accuracy and efficiency of the proposedelement in calculating natural frequencies and critical load ofbeam with different boundary conditions

References

[1] M Petyt Introduction of Finite Element Vibration AnalysisCambridge University Press 2nd edition 2010

[2] R E Nickel and G A Secor ldquoConvergence of consistentlyderived Timoshenko beam finite elements rdquo International Jour-nal for Numerical Methods in Engineering vol 5 no 2 pp 243ndash252 1972

[3] D L Thomas J M Wilson and R R Wilson ldquoTimoshenkobeam finite elementsrdquo Journal of Sound and Vibration vol 31no 3 pp 315ndash330 1973

[4] D J Dawe ldquoA finite element for the vibration analysis of Timo-shenko beamsrdquo Journal of Sound and Vibration vol 60 no 1pp 11ndash20 1978

[5] K K Kapur ldquoVibrations of a Timoshenko beam using finiteelement approachrdquo Journal of the Acoustical Society of Americavol 40 pp 1058ndash1063 1966

[6] AW Lees and D LThomas ldquoUnified Timoshenko beam finiteelementrdquo Journal of Sound and Vibration vol 80 no 3 pp 355ndash366 1982

[7] A W Lees and D LThomas ldquoModal hierarchical Timoshenkobeam finite elementsrdquo Journal of Sound and Vibration vol 99no 4 pp 455ndash461 1985

[8] J J Webster ldquoFree vibrations of shells of revolution using ringfinite elementsrdquo International Journal of Mechanical Sciencesvol 9 no 8 pp 559ndash570 1967

[9] S S Rao and R S Gupta ldquoFinite element vibration analysis ofrotating timoshenko beamsrdquo Journal of Sound and Vibrationvol 242 no 1 pp 103ndash124 2001

[10] A J M Ferreira MATLAB Codes for Finite Element AnalysisSpringer 2008

[11] P J P Goncalves M J Brennan and S J Elliott ldquoNumericalevaluation of high-order modes of vibration in uniform Euler-Bernoulli beamsrdquo Journal of Sound and Vibration vol 301 pp1035ndash1039 2007

[12] J Lee and W W Schultz ldquoEigenvalue analysis of Timoshenkobeams and axisymmetric Mindlin plates by the pseudospectralmethodrdquo Journal of Sound and Vibration vol 269 no 3ndash5 pp609ndash621 2004

[13] ZWieckowski andM Golubiewski ldquoImprovement in accuracyof the finite element method in analysis of stability of Euler-Bernoulli and Timoshenko beamsrdquoThin-Walled Structures vol45 no 10-11 pp 950ndash954 2007

[14] J B Kosmatka ldquoAn improved two-node finite element forstability and natural frequencies of axial-loaded Timoshenkobeamsrdquo Computers and Structures vol 57 no 1 pp 141ndash1491995

[15] B Z P Bazant and L Cedolin Stability of Structures OxfordUniversity Press New York NY USA 1991

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119908119895

120579119895

119895

ℎ119904

119897119908119894

120579119894

119894

119909

Figure 1 Timoshenko beam element

displacement polynomial and quadratic rotational fields areselected In addition shear strain of the element is assumedconstant Then by exploiting the bending and shear strainenergy of the beam and stationary with respect to constantshear strain interpolation function of this element has beencarefully calculated In the following by using these shapefunctions the stiffness matrix geometric stiffness matrixand mass matrix of the proposed element have been exactlydetermined Finally several numerical tests are performed toinvestigate the robustness of this element for free vibrationand stability analysis of beams with different boundaryconditionsThe findings prove that the suggested element hashigh level of accuracy and free of shear locking

2 Finite Element Formulation

In the finite elementmethod displacement and rotation fieldsof the element are associated with interpolation functions tonodal degrees of freedom Figure 1 shows proposed elementwith two nodes The shape functions of the Timoshenkobeam are calculated based on Figure 1 In order to computeshape function of the beam in Figure 1 cubic displacementpolynomial and quadratic rotational fields are selected Addi-tionally it is assumed that shear strain has the constant valueof 1205740 Based on these assumptions (1) can be written as

follows

119908 =119908119894

2(1 minus 119904) +

119908119895

2(1 + 119904)

+ 1205730119897 (1 minus 119904

2) + 1205731119897119904 (1 minus 119904

2)

120579 =120579119894

2(1 minus 119904) +

120579119895

2(1 + 119904) + 120572

0(1 minus 119904

2)

120574 = 1205740

119904 =2119909

119897minus 1

(1)

In these equations 1205731 1205730 1205720 and 120574

0are unknown parame-

ters In order to determine their values at first the equation ofshear strain for Timoshenko beam is established By utilizingthe shear strain value equal to 120574

0 the subsequent equations

will be available

120574 =119889119908

119889119909minus 120579 =

2

119897sdot119889119908

119889119904minus 120579

1205740=

2

119897(minus

119908119894

2+119908119895

2minus 21205730119897119904 + 1205731119897 minus 3120573

11198971199042)

minus 120579119894(1 minus 119904

2) minus 120579119895(1 + 119904

2) minus 1205720(1 minus 119904

2)

(2)

In the present formula the coefficients of the terms 119904 and 1199042

are equivalent to zero Therefore in the succeeding lines 1205720

1205731are determined in terms of the unknown parameter 120574

0

Γ =2

119897(119908119895minus 119908119894) minus (120579

119894+ 120579119895)

1205730=

1

8(120579119894minus 120579119895)

1205720= minus

3

2(1205740minus1

2Γ)

1205731=

1

61205720

(3)

At this stage there is only one unknown constant 1205740 which

can be discovered through the condition of minimum strainenergy It should be added that the structural strain energy isthe sum of bending and shear strain energy Bending strainenergy is calculated in the following way

119880119887=

119864119868

2int

119897

0

1205812119889119909 =

119864119868119897

4int

1

minus1

1205812119889119904 (4)

In (4) 120581 represents curvature which is determined as follows

120581 = minus2

119897sdot119889120579

119889119904= 1205810minus 6

1199041205740

119897

1205810=

1

119897(120579119894minus 120579119895+ 3119904Γ)

(5)

Substituting (5) into (4) leads to the following bending strainenergy

119880119887= 1198800+ 6119864119868(minus

Γ1205740

119897+1205742

0

119897)

1198800=

119864119868119897

4int

1

minus1

1205812

0119889119904

(6)

Besides (7) shows the energy of shear strain

119880119904=

119866119860

2119891119904

int

1

0

1205742119889119909

=119866119860119897

4119891119904

int

1

minus1

1205742

0119889119904

=119866119860119897

2119891119904

1205742

0

(7)

In this equation 119891119904is shear correction factor which depends

on cross section shape of beam This coefficient for rectan-gular section is 56 By adding the bending and shear strainenergy together total strain energy is found as follows

119880 = 119880119887+ 119880119904= 1198800minus6119864119868Γ

1198971205740+6119864119868

1198971205742

0+119866119860119897

2119891119904

1205742

0 (8)

Implementing 1205971198801205971205740= 0 will give the following results

1205740=

6119864119868Γ

1198661198601198972119891119904+ 12119864119868

= 120575Γ

120575 =6120582

1198972 + 12120582 120582 =

119891119904119864119868

119866119860

(9)





119908

120579119899

= [1198731

1198735

1198732

1198736

1198733

1198737

1198734

1198738

]

119908119894

120579119899119894

119908119895

120579119899119895

1198731=

1

4[2 + 119904

3(1 minus 2120575) + 119904 (minus3 + 2120575)]

1198732=

119897

4[05 (1 minus 119904

2) + (119904

3minus 119904) (05 minus 120575)]

1198733=

1

4[2 minus 119904


1198734=

119897


2) + (119904

3minus 119904) (05 minus 120575)]

1198735=

1

4119897[6 (1 minus 119904

2) (minus1 + 2120575)]

1198736=

1


2) 120575]

1198737=

1

4119897[minus6 (1 minus 119904

2) (minus1 + 2120575)]

1198738=

1

4[minus1 + 119904 (2 + 3119904) + 6 (1 minus 119904

2) 120575]

(10)


119910

120579 = [119873] 119863119864

120576 =

119889119910

119889119909

minus119889119910

119889119909+ 120579

= [119861] 119863119864

(11)

where

[119861] =[[

[

0119889

119889119909119889

119889119909minus1

]]

]

[119873] (12)



[1198700] = int [119861]

119879[119863119898] [119861] 119889119909 (13)


ment

[119863119898] = [

[

119864119868 0

0119866119860

119891119904

]

]

(14)


[1198700] =

119864119868

1198973 + 12119897120582

times[[[

[

12 6119897 minus12 6119897

6119897 41198972+ 12120582 minus6119897 2119897

2minus 12120582


6119897 21198972minus 12120582 minus6119897 4119897

2+ 12120582

]]]

]

(15)

3 Mass Matrix


119879 =1

2int

1198972

minus1198972

120588119860 1199102119889119909 +

1

2int

1198972

minus1198972

120588119868 1205792119889119909 (16)


[119872] = [1198721] + [119872

2]

=119897

2int

1

1

120588119860[119873119908]119879[119873119908] 119889119904

+119897

2int

1198972

minus1198972

120588119868[119873120579]119879[119873120579]119879

119889119904

(17)


[1198721] =

1205881198601198975

210(12120582 + 1198972)2

times[[[

[

11989811

11989812

11989813

11989814

11989812

11989815

minus11989814

11989816

11989813

minus11989814

11989811

minus11989812

11989814

11989816

minus11989812

11989815

]]]

]

(18)


11989811

=6

1198974(1680120582

2+ 294119897

2120582 + 13119897

4)

11989812

=1

1198973(1260120582

2+ 231119897

2120582 + 11119897

4)

11989813

=9

1198974(560120582

2+ 841198972120582 + 3119897

4)

11989814

= minus1

21198973(2520120582

2+ 378119897

2120582 + 13119897

4)

11989815

=2

1198972(126120582

2+ 211198972120582 + 13119897

4)

11989816

= minus3

21198972(168120582

2+ 281198972120582 + 13119897

4)

(19)



as follows

[1198722] =

1205881198681198972

30(12120582 + 1198972)2

times[[[

[

11989821

11989822

minus11989821

11989822

11989822

11989823

minus11989822

11989824

minus11989821

minus11989822

11989821

minus11989812

11989822

11989824

minus11989812

11989823

]]]

]

(20)


11989821

= 36119897

11989822

= minus3 (60120582 minus 1198972)

11989823

=4

119897(360120582

2+ 151198972120582 + 3119897

4)

11989824

=1

119897(720120582

2minus 601198972120582 minus 1198974)

(21)



119889119904 = radic(1198891199092 + 1198891199102)

≃ 119889119909 +1

2(119889119910

119889119909)

2

119889119909

997904rArr 119889120575 =1

2(119889119910

119889119909)

2

119889119909

(22)


Δ119882 =119875

2int

119897

0

(119889119910

119889119909)

2

119889119909 (23)


[119870119892] = int

119897

0

[119889119873119908

119889119909]

119879

119875[119889119873119908

119889119909] 119889119909 (24)

119889119910

119889119909

119889119904

119889120575



[119870119892] =

119875

60119897

times[[[

[

1198961198921

1198961198922

1198961198923

1198961198922

1198961198922

1198961198924

minus1198961198922

1198961198925

1198961198923

minus1198961198922

1198961198921

minus1198961198922

1198961198922

1198961198925

minus1198961198922

1198961198924

]]]

]

(25)

where

1198961198921

= 60 + 12120573

1198961198922

= 6120573119897

1198961198923


1198961198924

= 51198972+ 31205731198972

1198961198925

= 31205731198972minus 51198972

120573 =1198974

(1198972 + 12120582)2

(26)


det ([119870]) = det ([1198700] minus 119875 [119870

119892]) = 0 997904rArr 119875cr (27)


119875cr =1205872119864119868

1198712eff

times (1

1 + (1205872119891119904119864119868) (1198712eff119866119860)

)

(28)







0 050

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus02 0 020

01

02

03

04

05

06

07

08

09

1No3

(a)

minus04minus02 00

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus05 0 050

01

02

03

04

05

06

07

08

09

1No3

(b)















Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687





Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598








1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119908

120579119899

= [1198731

1198735

1198732

1198736

1198733

1198737

1198734

1198738

]

119908119894

120579119899119894

119908119895

120579119899119895

1198731=

1

4[2 + 119904

3(1 minus 2120575) + 119904 (minus3 + 2120575)]

1198732=

119897

4[05 (1 minus 119904

2) + (119904

3minus 119904) (05 minus 120575)]

1198733=

1

4[2 minus 119904


1198734=

119897


2) + (119904

3minus 119904) (05 minus 120575)]

1198735=

1

4119897[6 (1 minus 119904

2) (minus1 + 2120575)]

1198736=

1


2) 120575]

1198737=

1

4119897[minus6 (1 minus 119904

2) (minus1 + 2120575)]

1198738=

1

4[minus1 + 119904 (2 + 3119904) + 6 (1 minus 119904

2) 120575]

(10)


119910

120579 = [119873] 119863119864

120576 =

119889119910

119889119909

minus119889119910

119889119909+ 120579

= [119861] 119863119864

(11)

where

[119861] =[[

[

0119889

119889119909119889

119889119909minus1

]]

]

[119873] (12)



[1198700] = int [119861]

119879[119863119898] [119861] 119889119909 (13)


ment

[119863119898] = [

[

119864119868 0

0119866119860

119891119904

]

]

(14)


[1198700] =

119864119868

1198973 + 12119897120582

times[[[

[

12 6119897 minus12 6119897

6119897 41198972+ 12120582 minus6119897 2119897

2minus 12120582


6119897 21198972minus 12120582 minus6119897 4119897

2+ 12120582

]]]

]

(15)

3 Mass Matrix


119879 =1

2int

1198972

minus1198972

120588119860 1199102119889119909 +

1

2int

1198972

minus1198972

120588119868 1205792119889119909 (16)


[119872] = [1198721] + [119872

2]

=119897

2int

1

1

120588119860[119873119908]119879[119873119908] 119889119904

+119897

2int

1198972

minus1198972

120588119868[119873120579]119879[119873120579]119879

119889119904

(17)


[1198721] =

1205881198601198975

210(12120582 + 1198972)2

times[[[

[

11989811

11989812

11989813

11989814

11989812

11989815

minus11989814

11989816

11989813

minus11989814

11989811

minus11989812

11989814

11989816

minus11989812

11989815

]]]

]

(18)


11989811

=6

1198974(1680120582

2+ 294119897

2120582 + 13119897

4)

11989812

=1

1198973(1260120582

2+ 231119897

2120582 + 11119897

4)

11989813

=9

1198974(560120582

2+ 841198972120582 + 3119897

4)

11989814

= minus1

21198973(2520120582

2+ 378119897

2120582 + 13119897

4)

11989815

=2

1198972(126120582

2+ 211198972120582 + 13119897

4)

11989816

= minus3

21198972(168120582

2+ 281198972120582 + 13119897

4)

(19)



as follows

[1198722] =

1205881198681198972

30(12120582 + 1198972)2

times[[[

[

11989821

11989822

minus11989821

11989822

11989822

11989823

minus11989822

11989824

minus11989821

minus11989822

11989821

minus11989812

11989822

11989824

minus11989812

11989823

]]]

]

(20)


11989821

= 36119897

11989822

= minus3 (60120582 minus 1198972)

11989823

=4

119897(360120582

2+ 151198972120582 + 3119897

4)

11989824

=1

119897(720120582

2minus 601198972120582 minus 1198974)

(21)



119889119904 = radic(1198891199092 + 1198891199102)

≃ 119889119909 +1

2(119889119910

119889119909)

2

119889119909

997904rArr 119889120575 =1

2(119889119910

119889119909)

2

119889119909

(22)


Δ119882 =119875

2int

119897

0

(119889119910

119889119909)

2

119889119909 (23)


[119870119892] = int

119897

0

[119889119873119908

119889119909]

119879

119875[119889119873119908

119889119909] 119889119909 (24)

119889119910

119889119909

119889119904

119889120575



[119870119892] =

119875

60119897

times[[[

[

1198961198921

1198961198922

1198961198923

1198961198922

1198961198922

1198961198924

minus1198961198922

1198961198925

1198961198923

minus1198961198922

1198961198921

minus1198961198922

1198961198922

1198961198925

minus1198961198922

1198961198924

]]]

]

(25)

where

1198961198921

= 60 + 12120573

1198961198922

= 6120573119897

1198961198923


1198961198924

= 51198972+ 31205731198972

1198961198925

= 31205731198972minus 51198972

120573 =1198974

(1198972 + 12120582)2

(26)


det ([119870]) = det ([1198700] minus 119875 [119870

119892]) = 0 997904rArr 119875cr (27)


119875cr =1205872119864119868

1198712eff

times (1

1 + (1205872119891119904119864119868) (1198712eff119866119860)

)

(28)







0 050

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus02 0 020

01

02

03

04

05

06

07

08

09

1No3

(a)

minus04minus02 00

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus05 0 050

01

02

03

04

05

06

07

08

09

1No3

(b)















Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687





Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598








1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











as follows

[1198722] =

1205881198681198972

30(12120582 + 1198972)2

times[[[

[

11989821

11989822

minus11989821

11989822

11989822

11989823

minus11989822

11989824

minus11989821

minus11989822

11989821

minus11989812

11989822

11989824

minus11989812

11989823

]]]

]

(20)


11989821

= 36119897

11989822

= minus3 (60120582 minus 1198972)

11989823

=4

119897(360120582

2+ 151198972120582 + 3119897

4)

11989824

=1

119897(720120582

2minus 601198972120582 minus 1198974)

(21)



119889119904 = radic(1198891199092 + 1198891199102)

≃ 119889119909 +1

2(119889119910

119889119909)

2

119889119909

997904rArr 119889120575 =1

2(119889119910

119889119909)

2

119889119909

(22)


Δ119882 =119875

2int

119897

0

(119889119910

119889119909)

2

119889119909 (23)


[119870119892] = int

119897

0

[119889119873119908

119889119909]

119879

119875[119889119873119908

119889119909] 119889119909 (24)

119889119910

119889119909

119889119904

119889120575



[119870119892] =

119875

60119897

times[[[

[

1198961198921

1198961198922

1198961198923

1198961198922

1198961198922

1198961198924

minus1198961198922

1198961198925

1198961198923

minus1198961198922

1198961198921

minus1198961198922

1198961198922

1198961198925

minus1198961198922

1198961198924

]]]

]

(25)

where

1198961198921

= 60 + 12120573

1198961198922

= 6120573119897

1198961198923


1198961198924

= 51198972+ 31205731198972

1198961198925

= 31205731198972minus 51198972

120573 =1198974

(1198972 + 12120582)2

(26)


det ([119870]) = det ([1198700] minus 119875 [119870

119892]) = 0 997904rArr 119875cr (27)


119875cr =1205872119864119868

1198712eff

times (1

1 + (1205872119891119904119864119868) (1198712eff119866119860)

)

(28)







0 050

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus02 0 020

01

02

03

04

05

06

07

08

09

1No3

(a)

minus04minus02 00

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus05 0 050

01

02

03

04

05

06

07

08

09

1No3

(b)















Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687





Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598








1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0 050

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus02 0 020

01

02

03

04

05

06

07

08

09

1No3

(a)

minus04minus02 00

01

02

03

04

05

06

07

08

09

1No1

minus05 0 050

01

02

03

04

05

06

07

08

09

1No2

minus05 0 050

01

02

03

04

05

06

07

08

09

1No3

(b)















Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687





Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598








1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 47330 472840 4728402 785320 78675 784690 7846923 109956 110351 109800 1098014 141372 142218 141062 1410645 172788 174342 172246 1722536 204204 206783 203338 2033557 235619 239600 234325 2343588 267035 272857 265192 2652539 298451 306616 295926 29603210 329867 340944 326514 326687





Ferreira[10]

Lee andSchultz[12]

Proposedelement

1 473004 45835 457955 4579622 785320 73468 733122 7331933 109956 98924 985611 9859184 141372 122118 121454 1215405 172788 143386 142324 1425136 204204 163046 161487 1618417 235619 181375 179215 1798078 267035 198593 195723 1966419 298451 214875 211185 21252310 329867 230358 225735 227598








1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1205822

119894= 1205961198941198972radic

120588119860

119864119868

(29)




5 Conclusion


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article finite element formulation for stability

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