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)
4 Advances in Acoustics and Vibration
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
Advances in Acoustics and Vibration 5
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 2 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31425 314133 3141332 628319 62908 628106 6281063 942478 94503 941761 9417644 125664 126271 125494 1254965 157080 158267 156749 1567546 188496 190552 187926 1879377 219911 223186 219011 2190348 251327 256231 249988 2500349 282743 289749 280845 28092710 314159 323806 311568 311705
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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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Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
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)
4 Advances in Acoustics and Vibration
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
Advances in Acoustics and Vibration 5
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 2 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31425 314133 3141332 628319 62908 628106 6281063 942478 94503 941761 9417644 125664 126271 125494 1254965 157080 158267 156749 1567546 188496 190552 187926 1879377 219911 223186 219011 2190348 251327 256231 249988 2500349 282743 289749 280845 28092710 314159 323806 311568 311705
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
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)
4 Advances in Acoustics and Vibration
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
Advances in Acoustics and Vibration 5
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 2 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31425 314133 3141332 628319 62908 628106 6281063 942478 94503 941761 9417644 125664 126271 125494 1254965 157080 158267 156749 1567546 188496 190552 187926 1879377 219911 223186 219011 2190348 251327 256231 249988 2500349 282743 289749 280845 28092710 314159 323806 311568 311705
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
4 Advances in Acoustics and Vibration
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
Advances in Acoustics and Vibration 5
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 2 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31425 314133 3141332 628319 62908 628106 6281063 942478 94503 941761 9417644 125664 126271 125494 1254965 157080 158267 156749 1567546 188496 190552 187926 1879377 219911 223186 219011 2190348 251327 256231 249988 2500349 282743 289749 280845 28092710 314159 323806 311568 311705
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Acoustics and Vibration 5
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 2 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31425 314133 3141332 628319 62908 628106 6281063 942478 94503 941761 9417644 125664 126271 125494 1254965 157080 158267 156749 1567546 188496 190552 187926 1879377 219911 223186 219011 2190348 251327 256231 249988 2500349 282743 289749 280845 28092710 314159 323806 311568 311705
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Advances in Acoustics and Vibration
Table 3 Dimensionless frequency parameter 120582119894for the simply supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
Ferreira [10] Lee and Schultz [12] Proposed element1 314159 31169 311568 3115692 628319 60993 609066 6090943 942478 88668 884052 8842294 125664 113984 113431 1134925 157080 137089 136132 1362826 188496 158266 156790 1570937 219911 177811 175705 1762398 251327 195991 193142 1939979 282743 213030 209325 21060610 314159 229117 224441 226257
Table 4 Dimensionless frequency parameter 120582119894for the clamped supported Timoshenko beam
Mode Euler theory ℎ119897 = 0002
Ferreira [10] Lee and Schultz [12] Proposed element1 473004 47345 472998 4729982 785320 78736 785295 7852963 109956 110504 109950 1099504 141372 142526 141359 1413605 172788 174888 172766 1727686 204204 207670 204168 2041747 235619 240955 235567 2355788 267035 274833 266960 2669809 298451 309398 298348 29838210 329867 344748 329729 329786
Table 5 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 001
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
Table 6 Dimensionless frequency parameter 120582119894for the clamped
supported Timoshenko beam
Mode Euler theory ℎ119897 = 01
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
Advances in Acoustics and Vibration 7
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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Acoustics and Vibration 7
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
International Journal of
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
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