A Finite Element for the Vibration Analysis of Timeshenko Beams

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Journal of Sound and Vibrat i on 1978) 60(l), 1 -20

A FINITE ELEMENT FOR THE VIBRATION ANALYSIS OFTIMOSHENKO BEAMS

D. J. DAWEDepartment of Civil Engineeri ng, The Uni versity of Birmingham,

Birmingham B15 2TT, England

(Received 11 February 1978)

A Timoshenko beam finite element is presented which has three nodes and two degreesof freedo m per node, namely the values of the lateral deflection and the cross-sectionalrotation. The element proper ties are based on a coupled displacement field; the lateraldeflection is interpolated as a quintic polynomial function and the cross-sectional rotationis linked to the deflection by specifying satisfaction of the governing differential equationof mom ent equilibrium in the absence o f the rotary inertia term . Numerical results con6t-mthat this proce dure does not preclude convergence to true Timoshe nko theory solutionssince rotary inertia is included in lumped form at element ends. The new Timoshe nkobeam element has good convergence characteristics and where comparison can be madein numerical studies it is shown to be generally mor e etkient than previous elements.

1. INTRODUCTIONThe classical Bernouilli-Euler theory pred icts the frequencies of flexural vibration of thelower modes of slender beams with adequate precision. However, because in this theorythe effects of transverse shear deformation and rotary inertia are neglected the errors a sso-ciated with it become increasingly large as the beam depth increases and as the wavelength ofvibration decreases.

The effect of rotary inertia on beam frequencies was first considered by Rayleigh [l] andlater Timoshenko [2,3] extended this to include the effects of transverse shear deformation.In Timoshenko theory the rotation of a cross-section, 6, is the sum of the shearing ang le,JI, and the rotation of the neutral axis, dw/dx (w here w is the transverse displacement andx is the longitudinal axis). The problem is thus governed by two variables, w and 6 say,rather than by w alone as in Bemouilli-Euler theory. Exact solutions of Timoshenkosequations are available for simply supported beams [4,5] and for beams with other boundaryconditions [6,7]. Results obtained through use of Timoshenko equations have been shownto be in good agreement with results obtained by using the classical equations of theoreticalelasticity [3, 81.

A variety of finite elements h as been presented [9-201 for the dynamic analysis of beamsin which shear and rotary inertia effects are significant. The basic Timoshenko beam elementhas a total of four degrees of freedom (W and 8 at both ends) and can be based on a coupleddisplacement field which exactly satisfies the governing equations of a beam which is unloadedbetween its ends. Higher order elements are based on independent polynomial assumptionsfor the bending and shear deformation; these various models have more than two degreesof freedom per node and/or more than two nodes. The existing Timoshenko beam elementsare discussed in the next section.In the present work a Timoshenko beam element having a total of six degrees of freedomis presented. The element properties are based on the representations of w and 8 by quintic

110022460X/78/0908-001 1 SOZ.OO /O 0 1978 Academic Press Inc. (London) Limited


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12 D. J. DAWEand quartic polynomial functions respectively, but the polynomial coefficients are coupledthrough consideration of the differential equations of equilibrium. The element degr ees offreedom are the values of w and 0 at the two end points and the centre point of the element.Numerical studies are presented to show the way in which the performance of the newelement compares with that of existing elements.

2. RE VIEW OF EXISTING EL EME NTSThe four degree of freedom basic Timoshenko beam element has appeared in a variety

of guises [9, 10, 12, 14-17, 191 and, as noted by D. L. Thomas, Wilson and Wilson [18],som e confusion has arisen both d ue to error s of detail in published wor k and in som e in-stances, to misconceptions as to which rotational degrees of freedom should be used at thenodes. The correct nodal degrees of freedom are values of the transverse displacement, w, andthe cross-sectional rotation, 8, but in som e cases [14 , 171 dw/d x has been us ed in place of 8;this does not allow the correc t represen tation of the clamp ed boundary condition or thecorrec t coupling of adjacent elements in general situations.Davis, Henshell and Warburton [16] and Narayanaswami and Adelman [19] have shownthat the stiffness matrix of the basic element can be based upon the exac t solution of thedifferential equations of static equilibrium of a beam unloaded along its length except atits ends. Under these conditions the shear forc e is constant along th e length of the elementand is equal to the first derivative of bending mom ent. Hence bending and shear deformationscan be coupled and no extra de gree s of freed om are involved in the beam analysis; the lateraldeflection is a cubic polynomial function of the longitudinal co-ordinate and the cross-sectional rotation is a (dependen t) quadratic function. The philosophy has something incomm on with that adopted by Egle [21] in deriving a simplified version of the Timosh enkotheory for the beam as a whole.

The general use of the basic element stiffness ma trix and associated consistent massmatrix invibrationapplications has beencriticized by J. Thom as and Abbas [20] who argue thatit is invalid to use this Timosh enko beam element in vibration wor k where the shear for cedue to dynamic loading is clearly of variable intensity. How ever, although the governingequilibrium equations cannot be satisfied at the micros copic level within each element thisdoes not mean that equilibrium conditions are not satisfied at the mac roscop ic level (at thenodes) for an assembly of elements, and that convergence to true Timoshenko theory energylevels will not occur w ith use of an increasing number of elements. The situation is notunlike that which exists in finite element vibration analysis based on Bernouilli-Euler theorywhe re the well-proven basic element [22], with a cubic displacemen t function, is also implicitlybased on the exac t satisfaction of the governing differential equation of the beam elem entcarrying no load between its ends.

The earliest higher order element was presented by Kapur [l 1 ] and has a total of eightdegrees of freedom. The element is based on considering the total transverse displacementas the summation of parts due to bending and due to shear. The variation of each of thesedisplacement parts along the element is represe nted by a cubic polynomial function andthe degrees of freedom at each end node are the values of the bending and shear displacementsand their first derivatives. Nickel and Secor [17] describe a seven degree of freedom elementbased on a cubic function for w and a quartic for 8; the degrees of freedom are w, dw/dxand 0 at the two end nodes plus 0 at a mid-point node. Carnegie, J. Thomas and Dokum aci[13] base an eight degree of freedom element on cubic functions for both w and 0, withvalues of these quantities as degrees of freedom at four node points located at each end atthe one-third points. An element due to D. L. Thom as, Wilson and Wilson [18] has sixdegrees of freedom with values of w, 0 and $ as degrees of freedom at the two end points;


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FINITE ELEMENT FOR TIMOSHENKO BEAMS 13these degrees of freedom are associated with an assumed cubic variation of w and a linearvariation of $. Finally J. Thomas and Abbas [20] have described an element having eightdegrees of freedom, comprising values of w, dw/dx, 8 and d0/dx at the element ends; thesedegrees of freedom are selected so that all standard displacement and force boundary condi-tions can be imposed at beam ends. Since this element is based on assumed independent cubicfunctions for w and 8 it is, as noted by D. L. Thomas [23], simply a new form of the elementof Carnegie, J. Thomas and Dokumaci [13], obtained by a suitable transformation of nodaldisplacements. However, because of the different conditions applied at element junctionsand beam bo undaries the numerical results obtained by using the two forms can be quitedifferent [23, 241 ; on an element basis the earlier version is superior but on a degree-of-freedom basis the latter version is more accurate.Of the existing higher order elem ents all but that of Carnegie, Thom as and Doku maci[13] involve use of quantities at the end nodes of the element other than those w hich arestrictly required by the variational procedure. For such elemen ts difficulties are introducedin analysing assemblies of beam elements which are not unidirectional or in analysingcomplex structures which combine beam an d other types of finite element.

Some comparisons of the numerical performance of the available Timoshenko beamelements in vibration analyses of single beams are recorded elsewhere [18,20,23,24] and arealso provided in section 4 of this paper.

3. TIMOSHENKO BEAM ELEMENT PROPERTIESAn infinitesimal length 6x of a uniform beam element vibrating with circular frequency

p is shown in Figure 1 in dynamic equilibrium under the action of the shear forces and b endingmom ents and the inertia force and couple. T imoshenko beam theory is based on the followingset of equations (see the Appendix for a list of notation) :

dF/dx = pAwp2, dM/dx - F = -pZl.lZ?, (132)M = EZdB/dx, F= AGK t,b, 8 = dw/dx + $. (3,495)

Of these equations the first two are the dynamic equilibrium equations, the next two arethe elasticity or stress/strain relations, and the last is a comp atibility relation linking thegeometric deform ations. Normally, in applying the finite element displacement methodonly the elasticity and compatibility relations are explicitly satisfied within an elemen t, of

M

Iv

T xFigure 1. Equilibrium of an infinitesimal length of beam.


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14 D. J. DAWEcourse , and the micro scopic equilibrium equations are approx imated by the variationalprocedure.

The derivations of the basic, four degr ee of freedo m element by Davis, Henshell andWarburton [16] and Narayanaswani and Adelman [ 191 are based on explicitly satisfyingequations (l)-(5) with the right-hand sides of equations (11 and (2) set to zero. This resultsin an element with constant shea r force along its length, linear variation of mom ent, quadraticvariation of cross-sectional rotation and cubic variation of transverse displacement.

The element mode l derived here is based on explicit satisfaction of the homogeneousform of equation (2), that is

dM/dx - F= 0, (6 )so that on using eq uations (3), (4) and (5) a relationship is obtained linking w and 0 in theform

EId2 9/dx2 = GKA(tl - dwldx). (7 )This proce dure means that the rotary inertia term is ignored in the mom ent equilibriumequation within the element but the effec t of rotary inertia will nevertheless be included in lumped form at the nodes.

During the vibration the spatial variation of w and 0 along the beam element is initiallyassumed to be

w=A,+A,x+A2x2+A,~3+A4x4+As~s,8=Bo+B,x+B2x2+B,x3+B4x4. (8 )

By using equation (7) the coefficients B, . . . B, can be expressed in terms of the coefficientsA o... As by equating coefficients of powers of x. This procedure yields

B, = 120e2A, + 6&A3 + A,, B, = 24&A, + 2A,,B, = 60&A, + 3A,, B, = 4A,, B4 = 5A,, (9 )

where E = EI/GKA.It is convenient to expre ss the full list of coefficients A. . . . A,, B, . . . B, in terms of

A o... As alone in matrix form asf=Rb, (10)

wheref=(A,. . . As,B,,. . . B4}, b={A,,. . . As } and R is an 11 x 6 rectangular matrix.The nodal displacements of the element are the values of w and 0 at the ends 1 and 2 of

the element and at the element centre point 3. Applying the boundary conditions gives

whered = Cb, b= C-d,

d = {w,, e,, ~2902, ~3963).

(11)

The strain energy, U, and kinetic energy, T, of the uniform Timoshenko beam element are

.=ii,:[($~dx+KGA _si-3dx),

Tw2dx+I-I/2

(12)


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F I N I T E L E M E N T O R T I M O SH E N K O E A M S 15By using equations (8)-(11) it is a simple matter to express these energies in the usualquadratic forms

U=+dkd and T=$p2dTmd,where

il/2 I/2

k = (C-l ) RT EZ j - DTD1dx+KGA j D;DJD2dx R C-1 (13)-II2 -f/2and

il/2 l/2

m=(C-l)TRT A 1 DfD ,dx+Z I D:D,dx RC -1

(14)-l/2 -l/2

are the required element stiffness and mass matrices. T he matrices D, to D, are defined asD, = [0, 0, 0, 0, 0, 0, 0, 1, 2x, 3x2, 491,

D, = [l, x, , x3, 4, 5, , , , 0, 1,Ds = [0, , , , , , , x, x2, x3, a+] (15)

4. N U M E R I C A L R E S U L T SResults of the application of the Timoshenko beam element to the calculation of the firstfour natural frequencies of three simply supported beams are given in Table 1. If the beams areassumed to have a solid rectangular cross-section the three quoted values of r /L of 0.008,

0.04 and 0.08 correspond to beam depth/length ratios of 0*02771 28,0.138564 and 0.277128,respectively. In each case the effect of shear deformation and rotary inertia on any frequencyT A B L E 1

F in i t e elemen t requency resu l t s o r th ree s im p l y suppo r t ed beam s (K = 0 .85, v = 0 .3)values of 1

-bValues of % error in finite element frequencies for the E xactfollowing number of elements (in complete beam) Exact BernouillirP M ode 7

A ---- T imoshenko -E uler2 3 4 6 8 16 32 theory theory

0.008 1 0.03 o*Oo o+I O o+I O 0.00 - - -2 0.47 0.04 0.00 0.00 o*O o: 50.620.88 0.37.58 0.05.20 0.00.06 o*Oo.00

0.04 1 0.08 0.00 o*OO OGO 003 0.00 0.00 0.002 1.19 0.19 0.02 0.01 OG I o$IO o*OO 0.003 9445 140 0.33 0.09 0.04 0.02 0.01 0.004 166.26 2.19 1.08 0.45 0.13 0.08 0.02 O-010.08 1 0.19 o*OO 0.00 OGO 0.00 0.00 0.00 0.002 2.20 0.46 0.13 0.08 0.04 0.03 0.01 0.003 108.10 246 062 0.29 0.14 009 0.02 o*OO4 66.5 3.08 1.72 0.69 0.27 0.16 0.05 0.01

9.8750 9.869639.278 39.47887.823 88.826154.79 157.919.5710 9.869635.359 39.47871.657 88.826113.85 157.918.8397 9.869628461 39.47851.498 88.82675,365 157.91


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16 D. J. DAWE

0 4 6 I2 16 0 4 8 I2 I6Number of degrees of freedom

Figure 2. Percentage errors in the calculated frequencies of a simply supported beam with K = 0 4 5 ,r / L = 0 .0 8. (a) Mode 1, I.* = 8.8397; (b) mode 2, ,I= 28,461; (c) mode 3, I.= 51.498; (d) mode 4,1t* = 73.365. -o--, Present element; --;--, basic element [16]; -a--, element of Thomas and AbbasWI.is clear on comp arison of the quoted values based on the Timosh enko and the Bernouilli-Euler theories.

The re sults of Table 1confirm that the finite element answe rs a lways converge, from above,onto the exact results as given by the Timosh enko theory. To demo nstrate this conclusivelyvery many mor e elements have been used than are needed to yield results of sufficient accuracy

Number of degrees of freedomFigure 3. Percentage errors in the calculated frequencies of a cantilever beam with K = 2/3, r / L = 0.02.(a) Mode I, ,l12= 3.500; (b) mode 2,1 = 21.35; (c) mode 3, ,I1/Z = 57.47. -0-, Present element; --O--,

basic element [16]; -z--, element of J. Thomas and Abbas [20]; -.-A-.-, element of D. L. Thomas ef al.[18];----0------,elementofCarnegieelnl. [13].


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FINITE ELEMENTFOR TIMOSHENKO EAMS 17for all practical purposes. Convergence to good practical values of accuracy is in fact rapidfor all the beam s; even for the very deep beam the use of four elements (with 16 degrees offreedom in the whole beam ) is suflicient to predict the first four frequencies to within O-7%.As the exact solution is closely approximated, within half a per cent say, a reduction in therate of convergence becomes evident for some modes. This behaviour is presumably linkedwith the constraint imposed by the specific exclusion of the effect of rotary inertia in theequilibrium equations at the microscopic level within each element and its inclusion onlyat at the macroscopic level as a lump ed quantity. Be that as it may the overall effect onconvergence rate is very small.

Numbar o f degrws o f f reedomFigure 4. Percentage errors in the calculated frequencies of a cantilever beam with K = 0.65, r/L @05.(a) Mode 1, A1/z= 3.419; (b) mode 2, A112 18.61; (c) mode 3, A12= 44.62. Key as for Figure 3.

For the very deep simply supported beam, w ith r/L = O-08, results for the Timoshenkobeam elements of Davis, Henshell and Warburton [16] and Thomas and Abbas [20] areavailable and these are compared graphically with the results for the present element inFigure 2. For a given number of degrees of freedom the present element clearly com paresvery well in this problem with the earlier elements. (In Figures 2-5 the quoted number ofdegrees of freedom are those in the complete beam after application of the boundaryconditions.)

Results are available in the literature of the application of a number of Timoshenko beamelement mod els to the solution of cantilever beam dynamic problems. Three cantilevershave been considered in references [23] and [24]; these are a slender beam A with r/L = 0.02an d K = 213, a beam B of intermediate thickness with r/L = 0.05 and K = O-65, and a deepbeam C with r/L = 0.08 and K = 2/3. The models for which results are documented are thebasic four degree of freedom model, the eight degree of freedom models of Carnegie, J.Thomas and Dokumaci [131 and J. Thomas and Abba s [20] and the six degree of freedom


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18 D. J. DAWE

Number of degrees of freedomFigure 5. Percentage errors in the calculated frequencies of a cantilever beam w ith K = 2/3, r/L = 0.08.(a) Mode 1, l.2/2 3.284; (b) mode 2,1 = 15.49; (c) mode 3,J. liz = 34.30 . Key as for Figure 3.

model of D. L. Thomas, Wilson and Wilson [18]. These results are shown together withresults based on the present model in Figures 3-5. The present model appears to be themost efficient for all three cantilever beam s.

5. DISCUSSION AND CONCLUSIONSThe new six degree of freedom Timoshenko beam element presented here is based on a

displacemen t field which couples the primary variables w and 6 by specifically satisfyingwithin the element the differential equation of mom ent equilibrium with the rotary inertiaterm ignored. Numerical results demonstrate that convergence to exact Timoshenko theorysolutions does take place since the effect of rotary inertia is included at the end points ofelements. The new finite element mode l is based on the assumption of a quintic variation ofw along the element length with, correspondingly, a coupled quartic variation of 8. Thenew model is clearly related in its philosophy to the four degree of freedom basic model aspresented by Davis, Henshell and Warburton [16] and Narayanaswami and Adelman [19].

As with the basic mode l the new mode l ha s the considerable advantage of using onlyvalues of the primary variables as nodal degre es of freedo m. This facilitates the use of thiselement in the analysis of fram ed structures and of general structures involving other typesof finite element.

Comparison of the performance of the new model, where possible, with that of the basicmodel and of previous higher ord er models shows that for practical levels of accuracy thenew model is generally superior on a total degree of freedom basis, whatever the relativedepth of the beam. F urthermore, with the exception of the model of Carnegie, Thomas andDokumaci [13], previous higher order models have the disadvantage of using values of


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FINIlZEL.EMENTFO R TIMOSHENKO BEAMS 19other than the primary variables a s nodal degrees of freedom. The higher order m odels are,of course, based on independent assumptions of the spatial variation of the primary variablesand consequen tly considerably more element deg rees of freedom are involved for given ordersof interpolation. If independent interpolation is to be used then it might be mentioned thatrecent work [25] on the vibration of plates using Mindlin theory (the equivalent theory inplate analysis to that of Timoshenko in beam analysis) sugg ests that an increase in efficiencywould result if orders of interpolation higher than those used in previous beam m odels wereemployed.

REFERENCES1. LORD RA~HGH 1877 Theory of Sound. London: Macmillan and Co.2. S. P. TIMOSHENKO92 1 Phi losophical Magazine 41, 744-746. On the correction for shear of thedifferential equation for transverse vibrations of prismatic bars.

7.

8.

9.10 .11 .12 .13 .14 .15 .

16 .17 .18 .19 .

20 .21 .22 .

S. P. TIMOSI-&NKO 1922 Phil osophical Magazine~43, 125-131. On the transverse vibration ofbars of uniform cross-section.R. A. AND ERSON 953 Jour nal of Appl ied Mechanics 20,504-510. Flexural vibrations in uniformbeams according to the Timoshenko theory.C. DOLP H 1954 Quarterly of Appl ied Mathematics 12, 175-187. On the Timoshenko theory oftransverse beam vibrations.T. C. HUNG 1961 Jour nal of Appl ied Mechanics 28,579584. The eff ect of rotary inertia and ofshear deformation on the frequency and normal mode equations of uniform beams with simpleend conditions.T. C. HUANG and C. S. KUNG 1962 Developments in Theoretical and Appl ied Mechanics 1,59-71. New tables of eigenfunctions representing normal mod es of vibration of Timos henkobeams.R. M. DAVIES 1948 Phi losophical Transactions of the Royal Society A240, 375-457. A criticalstudy of the Hopkinson pressure bar.R. B. MCCALLEY 1963 General Electr ic Knol ls Atomic Power Laboratory, Schenectady, NewYork, Report No. DIGISA 63-73. Rotary inertia correction for mass matrices.J. S. ARC HER 1965 American Znstitute of Aeronautics and Astronautics Journal 3, 1910-1918.Consistent matrix formu lations for structural analysis using finite element techniques.K. K. KAPUR 1966 Jour nalof the Acoustical Society of America 40, 1058-1063. Vibrations of aTimoshe nko beam using a finite element ap proac h.J. S. PRZEMIENLECKI96 8 Theory of Matrix Structural Analysis. New York: McGraw-Hill.W. CARNEGIE, . THO MAS nd E. DOKUMACI 1969 Aeronautical Quarterly 20, 331-332. An im-proved meth od of matrix displacem ent analysis in vibration problem s.R. T. SE VERN 970 Journal of Strain Analysis 5,239241. Inclusion of shear deformation in thestiffness matrix for a beam element.R. ALI, J. L. HEDGES,B. MILLS, C. C. N ORVILLE nd 0. GURXIGAN 1971 Proceedings of theI nstitution of Mechanical Engineers, Automobile Division 185, 665690. The application offinite element tech niques to the analysis o f an automobile structure.R. DAVIS, R. D. HE NSHE LL nd G. B. WARBURTON 972 Journal of Sound and Vibration 22,475-487. A Timoshenko beam element.R. E. NICKEL and G. A. SE COR1972 I nternational Journal of Numerical Methoa!s n Engineeri ng5,243-253. Convergen ce of consistently derived Timoshe nko beam finite elements.D. L. THOMA S, . M. WILSON nd R. R. Wnso~ 1973 Journalof Soundand Vibration 31,315-330.Timos henko beam finite elements.R. NARAYANASWAMInd H. M. ADELMAN1974 Ameri can I nstitute of Aeronautics and Astro-nautics Journal 12, 1613- 1614. Inclusion of transverse shear deformation in finite elementdisplacement formulations.J. Tr-rohus and B. A. H. ABBAS1975 Journal of Sound and Vibration 41,291-299. Finite elementmodel for dynamic analysis of Timoshenko beam.D. M. EO LE 1969 NASA C& 1317. An approximate theory for transverse shear deformationand rotary inertia e ffects in vibrating beam s.F. A. LECKIE nd G. M . LINDBERG 963 Aeronautical Quarter ly 14,224-240. The eff ect of lumpedparameters on beam frequencies.


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20 D. J. DAWE23. D. L. THOMAS 976 Journal ojsound and Vibration 46,285-288. Com ments on Finite elementmodel for dynamic analysis of Timoshenko beam.24. J. THOMAS and B. A. H. ABBAS 1976 Journal of Sound and Vibration 46,288-290. Authors reply.25. D. J. DAWE 1978 Journal ofSound and Vibration 59, 441452. Finite strip models for vibrationof Mindlin plates.AEFGI

KL

IvlTukmPrWX8I

APPENDIX : NOTATIONcross-sectional area of beamYoungs modulus of elasticityshear forceshear modulussecond m oment of area of beam cross-sectionshear coefficientbeam lengthbending momentkinetic energystrain energyelement stiffness matrixelement lengthelement mass matrixradian frequencyradius of gyration of beam cros s-section (r = Z/A)lateral deflectionco-ordinate mea sured along element axis, origin at centrerotation of cross-section= pAL4p2/EI, frequency parameter

p mass densityv shear strain

A Finite Element for the Vibration Analysis of Timeshenko Beams

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