a timoshenko beam element

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Journal of Sound and Vibration (1972) 22 (4), 475-487

A T I M O SH E N K O B E A M E L E M E N T ?R. DAVIS.R. D. HENSHELLANDG. B. WARBURTON

Department of Mechanical Engineeri ng,University of Nottingham, Nottingham NG7 2RD, England

(Received 20 March 1972)A Timoshenko beam finite element wh ich is based upon the exact differential equationsof an infinitesimal element in static equilibrium is presented. Stiffness and consistent m assmatrices are derived. Convergence tests are performed for a simply-supported beam anda cantilever. The effect of the shear coefficient on frequencies is discussed and a study ismade of the accuracy obtained when analysing frameworks with beams.

1. INTRODUCTIONIn the Bernoulli-Euler theory of flexural vibrations of beam s only the transverse inertiaand elastic forces due to bending deflections are considered. As the ratio of the depth ofthe beam to the wavelength of vibration increases the Bernoulli-Euler equation tends tooverestimate the frequency. The applicability of this equation can be extended by includingthe effects of the shear deformation and rotary inertia of the beam . The equation whichincludes these secondary effects was derived by Timoshenko [I, 21 and solutions of it forvarious boundary conditions have been published [3-51. The dynamic stiffness matrixmethod has been used [6 , 71 for studying the dynamic behaviour of two- and three-dimen-sional framew orks comp rised of Bernoulli-Euler or Timoshenko beam s.

The stiffness and mass matrices for the simple beam finite element were published in1963 by Leckie an d Lindberg [8]. Their analysis was based on the exact differential equationsof an infXtesima1 element in static equilibrium. In the present pap er Leckie and Lindbergswork is extended to include shear deformation and rotary inertia in the analysis. W henthese secondary effects are taken into accoun t, it is important to be clear which rotation isto be used at the ends of the finite element model. The authors have chosen this rotationto be that of the cross section of the beam, i.e. the rotation of the neutral axis plus the shearingangle. With this it is possible to impo se the correct boundary conditions at say a clamped orfree end of a beam.

Bending and shearing deformations were considered separately by Kapu r [9] for derivingstiffness and consistent mass matrices for a Timoshenko beam . A cubic displacemen t functionwas assum ed for the bending deformation (the displacements at each node being onetranslation and one rotation). A similar assum ption was made for shear deformations, alsoallowing a translational and rotational displacemen t at each node. The resulting elementmatrices were of order 8 x 8 and no stiffness coupling was permitted between bending andshear deformations. The results presented by Kapur agreed very well with exact frequenciesfor simply-supported beam s and cantilevers. How ever, in general structures, w here allbeam s are not collinear, comp lications arise with Kapurs element in coupling up forcesand displacements, and each node must be treated specially. Kapurs frequencies converged

t Presentedat the British Acoustical Societymeetingon Finite element echniques n structuralvibrations,at the Institute of Sound and Vibration Research, University of Southampton, England, on 24 to 25 March1971.30 475


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476 R. DAVIS, R. D. HENSHELLAND G. B. WARBURTONmuch m ore rapidly than some other answ ers obtained with a finite element T imoshenkobeam published by Archer [lo]. Kapur wrote that Archers element could not representthe exact boundary conditions at a clamped or free end of a beam . Mid-side nodes were usedin the work reported in reference [l l] to improve the rate of convergence of a beam withshear deformation and rotary inertia. The respective authors of both reference [ll] andreference [12] (who have also published stiffness and ma ss matrices for a Timoshenko beam )do not state which rotations were used in formulating their matrices.

In the present paper the matrices for a Timoshenko beam are derived. A section is includeddescribing how the boundary conditions are satisfied and convergence tests are carried o uton a simply-supported beam and a cantilever. The significance of the K factor, which mu stbe used when shear deformation is allowed, is then discussed . F inally, the accuracy whichcan be obtained when analysing a two-dimensional structure with beam s is studied.

2. DERIVATION OF ELEMENT MA TRICES2.1. DIFFERENTIALEQUATIONS

Consider an infinitesimal element of beam of length 6x and flexural rigidity El. Theelement is in static equilibrium under the forces sho wn in Figure 1 t

Figure 1. Forces and displacemen ts on inkitesimal element of beam.The shear an gle, #, is measured as positive in an anticlockwise direction from the normal

to the midsurface to the outer face of the beam.Now F/A = GZ@. (1)The static equilibrium relations are dhf_Fdx- dF 0= ,dx

The rotation of the cross section in an anticlockwise direction (see Figure 1) ise=g+g.The stress-strain relation in bending is

(4)

(5)d*w d$ M&T+&=FZ-t A list of notation used is given in the Appendix.


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A TIMOSHENKO BEAM ELEMENT 477Solution of equations (I), (2), (3) and (5) yields

F= a,, (6)*2!_,

GKA(7)

M=a,x+a2, (8 ).

2.2. ELEMENT STIFFNESS MATRIXThe above four equations can now be applied to the finite element by substituting the end

conditions (see Figure 2) in them.

Figure 2. Generalized forces and displacements on finite element.The rotations at the ends of the beam , a 2 and a.,, can be expressed as rotations of the

cross section by using equation (4). The displacem ents 6, to S4 can be related to the constantsccl to aq by

where

t s i l = { w 1 e 1 w 2 @Z>T and /I = EI/GKA. Similarly the forces can be related to the constants bywhere (11)

and the elements of (PI} are defined in Figure 2 . Substituting for {at> from equation (10)in equation (11) gives(Pt] = W V [xl- @,I


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47 8or

R. DAVIS, R. D. HE NSHE LL AND G. B. WARBU RTON

lpi>= IsI si19where [S] is the elemen t stiffness ma trix

[S]= EIL(L2 + 128)[

12 6L -12 6L6L 4L2 + 12/3 -6L 2L2 - 128-12 -6L 12 -6L 6L 2L2- 128 -6L 4L2 + 128 1

If /3 = 0 the above matrix reduces to the element stiffness matrix derived by Leckie andLindberg [8] for a thin beam.2.3. ELEMENT MASS MATR IX

The kinetic energy of a finite element, T, vibrating at circular frequency, w, including theeffects of shear deformation and rotary inertia, can be written d own as

T= ~ j[pdr'+pl(P+~~]dr.0

Substituting for w from equation (9), # from equation (7) and {al} from equation (10) andintegrating with respect to x then gives

T= ~PAw~{SJ= PI-= [HI [xl- @iIwhere [H] may be found readily as

WI=

Here y = I/A. The element ma ss matrix is thenPOW -= WI [Xl-

which can be set up easily on a computer.

SYMMETR ICAL

3. BOUNDARY CONDITIONSThe two types of boundary condition which give rise to some difficulty are clamped or

free ends.In the case of the c lamped end of a beam 6 i = a2 = 0 (see Figure 2) and some shear strainmust be possible. In our case this shear strain (#J) is provided by the reaction at the wall(Pi) to give #i = PJGKA from equation (1).For a free end of a beam with no point load at the end the shear strain must be zero. Sinceat the free end of a beam the vertical reaction is zero, then tha t part of the rotation of thecross section, by using equations (1) and (4), which gives the shear strain, will also be zero.


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ATIMOSHENKOBEAMELEMENT 4794. CONVE RGEN CE TE ST

In this section the finite element derived in section 2 is tested for convergence on a simplysupported beam and a cantilever. The frequencies are compared with exact values compu tedby using references [3] and [6] for the simply-supported beam and cantilever respectively.In reference [9] a similar test was carried out to compare the performance of a Timoshenkobeam element with that given in reference [IO]. Insufficient information was given in reference[9] to enable the present authors to make a direct comparison. However, the value of k / lused by Kapu r and Archer was used to compute our frequencies. The exact frequencies aregiven in non-dimensional form as C in Table 1, with

C = w ,( p14 /Ek2 ) i 2 ,where w, is the nth natural frequency.

TABLE 1Percen t age er r o r in f r equenc ies betw een f i n i t e elemen t a nd exac t T i m oshenko th eo r y fo r a

sim p l y - s uppo r t ed beam and can t i l e v erDegrees of freedom on full beamSimply-supported beam CantileverF .

M ode 2 4 8 16 2 4 8 16 at bt1 20.11 1.79 0.39 0.09 1.01 0.25 0.06 0.01 11.5 5.772 69.13 37.36 3.99 0.98 67.10 3.63 1.23 0.31 38 35.33 - 56.69 12.08 3.19 - 50.86 5.68 1.54 72 684 - 42.07 32.23 6.81 - 64.19 10.78 3.69 108 107t Columns a and b give the percentage difference between the exact Timoshenko and Bernoulli-Eulertheories for the simply-supp orted beam and cantileve r respectively. Y= @3, K = 0.85, k / l = @08. Exactvalues of C computed by the authors for the first four modes a re 8.8397,28*4 61, 51.49 8 and 7 5.364 for thesimply-supported beam an d 3.3241 , 16.28 9, 36.70 8 and 58.2 79 for the cantilever, where C = u& Z4/Ek2)12.Table 1 shows the results for four idealizations and a lso gives an indication of the effect

of shear deformation and rotary inertia on the first few mod es by giving the percentagedifference in frequencies between the Timoshenko and Bernoulli-Euler theories. The elementpresented shows itself to converge from above onto exact answ ers given by a solution ofTimoshenkos equation. The rate of convergence of the element dep ends on the depth towavelength ratio of the beam u nder consideration, i.e. on the relative importance of bendingto shear deformation. In the dynamic case the deflection curve for an element is still thestatic curve. Consequen tly the shear force is constant along the length so that the strainenergy stored by shear deformation is also constant. During vibration the shear force on anelement is not constant and so the element gives a poorer representation of shear deforma-tion than bending deformation.

5. THE SHEAR COEF F IC IE NTThe finite element presented converges from above onto frequencies which are obtained

from a solution of Timoshenkos equation. The question arises as to the accuracy of thesefrequencies compared with exact answers obtained from the elasticity equations. Cowper[13] has obtained an exact solution for a simply-supported beam . In his paper values offundam ental frequency comp uted by using Timoshenkos equation for two different values


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48 0 R. DAV IS, R. D. HENSH ELLAND G. B. WARB URTONof the shear coefficient are compared with exact values for a variation of beam d epth tolength ratio up to one. The discrepancy between frequencies was shown to depend on thevalue of K used and to increase with increasing depth to length ratio of the beam. Thediscrepancy was greatest when the value K= 3 was used; here K was regarded as the ratioof the average shear stress on a cross-section to the shear stress at the centroid, a parabolicdistribution of shear stress being assumed. Far better accuracy was achieved by using

K= 10(1+ v)12+ llv -For v = 0.3, K = 0.85. This formula was derived by Cowper in an earlier paper [14], and itis a by-product of an analysis whereby Timoshenkos beamintegration of the three-dimensional elasticity equations. equations were derived by

TABLE 2Error s in the Timoshenko heory or t he i rst na tural requency of a simpl y- support ed eam

Percentage error b etween exact [13] and finite elementh/ l

K 0.1 0.2 @3 0.4 0.5 O -6 0.7 0.8 0.9 1.0O-66 -0.350 -1.247 -2.385 -3.540 -4606 -5552 -6.377 -7.092 -7.711 -8.245O-85 -0.020 -0.079 -0.162 -0.258 -0.358 -0.456 -0.549 -0.636 -0.714 -0.7830.86 -OXI -0.028 -0.063 -0.111 -0.165 -0.223 -0.279 -0.334 -0.384 -0.429

Finite element frequencies were obtained from a ten element idealization of a half simply-supported beampercentage error = ( wr~~,~,mmat 1 x 100

Material properties for the analysis were E = 30 x 1 0 lb f/h?, v = @ 3, pg - 0.283 lb f/in.(E = 2w8 GN/m*, p = 7.83 x lo3 kg/m3)Table 2 gives this comp arison between the exact and finite element frequencies for the

two values of K used by Cow per and also for a value obtained using a formula derived byGoodman 1151 n which K must satisfy

160 - P) _ 8(3 - 2~) + 8 _ I = oK3 K2 K where p = (1 - 2v)/2(1 - v). For v = 0.3, K = O-86 . The above expression ensures that thefrequency given by the Timoshenko equation is correct in the limit of zero wavelength.

It appea rs from these results that exact frequencies can be approached provided theappropriate value of K is used.

6. ACCURAC Y OBTAINED W HEN ANALYSING A TWO-DIMENSIONAL STRUCTUREWITH BEAMSThe accuracy of frequencies when analysing two-dimensional structures with beam s does

not appear to have been reported in the literature. The object of this section is to carry outsuch a study on the natural frequencies of in-plane vibration of a single bay portal frame.There have been no exact frequencies obtained by using two-dimensional theory publishedfor a moderately thick portal frame; therefore the finite element m ethod was used to providesome very accurate frequencies.


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A TIMOSHENKO EAMELEMENT 48 16.1. SIMPLY-SUPPORTED EAM USING PLANE STRESSELEMENTS

In order to assess the validity of using finite elements to idealize a portal frame a preliminarystudy was carried out to compare the fundam ental frequency of a simply-supported beamidealized with a 17-node plane stress finite element [16] with exact frequencies obtained byusing Cowpers equation [14]. One half of a beam of total length 100 in was idealized byusing a mesh of two elements through the depth of the beam and ten along h alf the length

TABLE 3Percentage err or in un & menta l f r equency between a plane stress f i n i t e element an alysis of a

sim ply-supported beam and exact requenciesh/l 0.1 O-2 O-3 0.4 0.5 0.6 0.7 O-8 0.9 1.0% error OXI03 0 .020 oQ65 0.145 O-268 0.435 0.643 0.892 l-177 l-495

Exact frequencies Hz) for simply-supportedbeam o btained by using equation (5) of reference [13] (whichalso apply to Table 2) were 90*308, 172772, 243.083, 300-707, 347.135, 384.418, 414467,438*854, 458%06,and 475,266 with E = 30 x 106 lbf/in, v = 0.3 and pg = 0283 lbf/i$.of the beam.? The depth of the beam was varied from 10 to 100 in with the length held constant.With boundary conditions similar to those impo sed in Cow pers theory the number ofdegrees of freedom on the half beam was 399. Continuou s reduction [ 171was used to reducethe eigenvalue problem solved to 62. Table 3 shows the percentage error be tween the finiteelement frequencies and those from Cowpers equation. The sma ll percentage errors obtainedbetween the frequencies suggested that the method could be used to analyse at least one-dimen sional structures and that reducing the eigenvalue problem did not effect the funda-mental frequency very much.6.2. PORTAL FRAM E ANALYSIS6.2.1. Pl ane str ess elements

The 17-node element used for the simply-supported beam an alysis was also used for theportal frame. Half the portal frame w as idealized, and by suitably changing the boundaryconditions, anti-symmetrical and symmetrical modes of vibration for a full portal framewere simulated . Two different meshes of elements were used (see Figure 3) and use wasagain m ade of the front solution to reduce the size of the eigenvalue problem solved from316 to 3 9 and 696 to 69 for the coarse and fine mesh es, respectively.

The leg length of the three sides of the portal frame was kept con stant at 10 in and thebreadth of the legs was varied from 1 to 9 in. The length of the legs was taken as the lengthto the centre of the comer joints as shown in Figure 4.6.2.2. Beams

The portal frame was analysed by using both Bernoulli-Euler and Timoshenko beamtheories. In each case the length of the legs of the portal frame was as defined in Figure 4.The stretching energy of the legs was taken into account in both B ernoulli-Euler andTimoshenko cases. The portal frame frequencies were compu ted by using the Timoshenkobeam elements derived in section 2 of this paper and also by using the dynamic stiffnessmatrix method for the Bernoulli-Euler and Timoshenko beam theories. Fo r the beam finiteelement analysis a half portal frame was idealized w ith 15 equal leng th elements giving 44and 43 degrees of freedom for the antisymmetrical and sym metrical modes of vibration,

tlin-254cm.


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482 R. DAVIS, R. D. HENSHELL AND G. B. WARBURTON

Figure 3. Plane stress idealizations of portal frames.

respectively. In Table 4 percentage errors between finite element and dynamic stiffnessmatrix frequencies are given for the first six modes of vibration for portal frame leg thicknessfrom 1 to 9 in.6.2.3. Com pari son between beam and p l a n e str ess r equencies

I n Figure 5 the results described in sections 6 .2.1 and 6 .2.2 have been plotted for the firstanti-symmetrical and symmetrical modes of vibration for the single bay portal frame.


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A T I M O SH E N K O B E A ME L E M E N T 483The graph shows that refining the mesh has changed the frequency of the anti-symm etrical

mode to a greater extent than that of the symmetrical mode especially as the leg thicknessis increased. This is because the deformation in the region of the joint is much m ore com-plicated in the first antisymmetrical than in the first symm etrical mode. In the case of thecoarser mesh idealization this region of the structure did not contain a sufficient number ofelements to represent the deformation properly.

F igure 4. Portal frame dimensions.T A B L E 4

Percen t age er r o r s in requency betw een T im oshenk oJin i t e elemen t and dyn am ic s t i f f ness m a t r i xm et hods o r i n - p l a ne Y i b r a t i o n s o f a po r t a l r am e w i t h a l eg l engt h o f 10 i n

Percentage error in frequency between Timoshenko finite element and dynamic stiffnessmatrix methodsWMode O-1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.089 OX90 0.0913 0.036t 0.101 0.138 0.167 0*190 0*125t 0.123f 0.121t 0.118t4 0.044 0*106t 0*14Ot 0*127t O-126? O-207 0.219 0,229 0.235t5 0.1327 0*129t 0.158f 0*219t O-253? 0.274t 0.287+ 0*292t o-2936 0.219 0,163 O-200 0.227 O-245 0,256 O -262 0.266 0.267t Antisymmetricd modes; others symmetrical. H alf portal frame idealized by using 15 equal lengthelements giving 44 and 43 degrees of freedom for antisymmetricaland symmetrical modes of vibrati on,respectively.E = 30 x lo6 I bf/in2, Y= O-3,K = 0.85 and pg = 0.283 I bf/in .In fact the plane stress Unite element frequencies are not exact for the following reasons.

(i) The element used was a conforming one which allows a cubic variation in strain a longeach of its sides. It is certain that if more elements had been used, an d the sam e size eigenvalueproblem solved, lower frequencies would have been obtained. (ii) The eigenvalue reductiontechnique used, which reduces freedoms out on a static basis by using certain master freedoms:will have raised the frequencies slightly from their true value for the model taken. The reasonswhy the beam frequencies can be expected to differ from the plane stress results are as follows.(iii) In both beam theories a linear axial and bending stress distribution across the thicknessof the beam is implied. (iv) In the Bernoulli-Euler theory shear deformation and rotaryinertia are not allowed. (v) For the Timoshenko beam it was not certain wha t value of theshear coefficient (K) was the correct one to use (K= 0.85 was used). (vi) The comer of theportal frame was assum ed to be rigid. (vii) When two beams are joined at right anglesthere is a geometric misrepresentation of the structure at this point.


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48 4 R. DAV IS , R . D . HE NSHE LL AN D G. B . WARBU RTON

4oc

20C

0I 1 1 I I I I I I

0.2 0.4 0.6 0.8h/L

Figure 5 . N atural frequencies of portal frames obtained by using plane stress finite element and dynam icstiffness matrix idealizations. -, Fine me& plain stress; ---, coarse me& plain stress; --, Bernoulli-Euler; ----, Bernoulli-Euler (Barkan); -----, Timoshenko; ---, Timoshenko (Barkan).Figure 5 show s that the Timoshenko beam theory gives a better estima tion of the first

antisymmetrical mode of the portal frame than the Bernoulli-Euler theory and vi ce versafor the symm etrical mode. It is not certain why this is so. It can only be said that it seemsthat beam theory is inadequate for the general ana lysis of frameworks in which the depthsof the beams are of the same order a s the distance between joints. It is suggested that reasons(vi) and (vii) are the most imp ortant contributors to the error. Barkan [18] has suggestedthat an allowance can be mad e for joint flexibility by using a modified length for the beams.He has published a graph which can be used to give the modified values. T his graph wa sused by the present authors to obtain new, shorter, leg lengths for portal frame leg thicknessesfrom 1 to 5 in. The new lengths were then used for computing frequencies using the dynamic


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A TIMOSHENKO BEAM ELEMENT 485stiffness matrix method for both Bernoulli-Euler and Timoshenko beam theories. Bark ansbook does not indicate if his theory is meant to be applied to static analyses only, or whetherit may be used in dynamic problems, i.e. the mass of the missing portion of the frame mu stbe allowed for. The authors assum ed that the theory could only be used for static problemsand app lied a simple correction mu ltiplier to the frequencies to allow for this. The resultsobtained are plotted on Figure 5. The effect of Barkans theory on the answers is that neither

Figure 6. Idealization of joints.the Bernoulli-Euler nor Timoshenko theories give a good estimation of the first anti-sym-metrical mode and that the Timoshenko beam theory is now superior to the Bem oulli-Euler theory for the Grst symm etrical mode. It seems that B arkans theory over-correctsthe frequencies and that modified leg lengths shou ld depend on the mode of vibration beingexamined.

The geometric misrepresentation which is incurred when two beam s are joined at rightangles (see Figure 6(a)) is perhaps the largest contributor to the error in frequencies. Thisproblem could be overcome by formulating beam finite elements w ith ends similar to thoseshown in Figure 6(b) and (c). This would be satisfactory for the type of joints analysed inthis paper, but for the general analysis of frameworks a whole series of beam s with specialends would be needed.

7. CONCLUSIONSA beam element w hich includes the effects of shear deformation and rotary inertia has been

presented which converges from above onto a solution of Timoshenkos equation. It hasbeen show n that the element will converge onto an exact solution of the elasticity equationsfor a simply-supported beam provided that the correct value of the shear coefficient is used.By using the rotation of the cross section as the angular displacemen t of the beam it hasbeen possible to satisfy all types of boundary conditions. Accurate frequencies were comp utedfor a single bay portal frame using a plane stress Unite element. These results showed that theBernoulli-Euler and Timoshenko beam theories were generally unsatisfactory when thedepth of the portal frame legs were of the same order as the distance between the joints.

ACKNOWLEDGMENTSThe authors would like to acknowledge mem bers of the PAFE C [19] group who havecontributed routines which have been used in comp uting results for this paper. One of the

authors (R. Davis) would like to acknowledge the financial support of the Science ResearchCouncil.


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486 R. DAV IS , R . D . HEN SHEL L AN D G. B . WARBU RTON

S. P. TIMOSHENKO92 1 Phi losophical Magazine 41,744-746. On the correction for shear o f thedifferential equation for transverse vibrations of prismatic bars.S. P. TIMOSHENK O 922 Phi losophical Magazine 43, 125-131. On the transverse vibrations ofbars of uniform cross-section.

5.

R. A. ANDE RSON 953 Jour nal of Appl iedMechanics 20,504510. Flexural vibrations in uniformbeams according to the Timoshenko theory.T. C. HUANG 1961 Journal of Applied M echanics 28,579-584. The effect of rotatory inertia andof shear deformation on the frequency and normal mod e equations of uniform beams w ithsimple end conditions.R. W. TRAILL-NASH and A. R. COLLAR 1953 Quarterly Jour nal of Mechanics and AppliedMathematics 6, 186-22 2. The effects of shear flexibility and rotatory inertia on the bendingvibrations of beams.

6. R. D. HENSH ELL, . BENNETT,H. MCCA LLION nd M . MILNER1965 Proceedings of he nstitutionof Electrical Engineers 112, 2133-2139. Natural frequencies and mod e shapes of transformercores.

7.8.9.

10 .11.12 .

13 .14.15 .

16 .17 .

R. D. HENSH ELL nd G. B. WAR BURTON 969 I nternational Journal of Numeri cal Methods inEngineeri ng 1,47-66. Transmission of vibration in beam systems.F. A. LECIUE and G. M. LINDB ERG 1963 Aeronautical Quarterl y 14, 224240. The effect oflumped pa rameters on beam frequencies.K. K . KAPU R 1966 Jour nal of the Acoustical Society of Ameri ca 40, 1058-1063. Vibrations of aTimoshenko beam, using finite element appro ach.J. S. AR CHER 1965 American I nstitute of Aeronautics and Astronautics Jour nal 3, 1910-1918.Consistent matrix formulations for structural analysis using finite element techniques.W. CARNEGIE,J. THOMASand E. DOKUM ACI 1969 Aeronautical Quarterly 20, 321-332. Animproved metho d of matrix displacement analysis in vibration problems.R. ALI, J. L. HEDGES,B. MILLS, C. C . NORV ILLE nd 0. GURD~GAN 1971 Proceedings of theI nstitution of Mechanical Engineers, Automobile Division 185,665-690. The application offinite element techniques to the analysis of an automobile structure.G. R. COPPER 1968 Proceedings of the Ameri can Society of Civil Engineers 94, EM6,1447-1453.On the accuracy of Timoshenkos beam theory.G. R. COWPER1966 Jour nal of Appl ied Mechanics 33, 335-340. The shear coefficient in Timo-shenkos beam theory.L. E. GOODMANnd J. G. SUTHERLAND951 Journalof Appli edMechanics l&217-218, discussionof paper, Natural frequencies of continuous beams of uniform span length by R. S. Ayreand L. S. Jacobsen.I. ERGATO UDIS, . M. IRONS and 0. C. ZIENKIEWICZ1968 I nternational Journal of Solids andStructures 4, 3142. Curved isoparam etric quadrilateral elements for finite element analysis.B. M. IRONS 1965 American I nstitute of Aeronauti cs and Astronautics Journal 3, 961-962.Structural eigenvalue problems: elimination of unwanted variables.18 . D. D. BARKAN1962 Dynamics of Bases and Foundations. New York: McGraw-Hill Book CO.19 . PAFEC 70 User s Manual. University of Nottingham: Departm ent of Mechan ical Engineering.

APPENDIXNO TA T IO N

AEFGf:ZKkL1

cross-sectional area of beamelastic modulusshear forceshear modulusfrequency of vibrationacceleration due to gravity (3 86.4 in/s2 in all numerical work )total de pth of beamsecond m oment of area of cross section about a principal axis normal to the plane o f bendingshear coefficientradius of gyration of cross section abo ut a principal axis normal to the plane of bendinglength of beam elementtotal length of beam or length of portal frame leg

REFERENCES


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MPlWXaf61iis

w

A TI M O SH J ZN K O B E A M E L E M E N Tbending momenti th generalized forcetransverse deflection of beamdistance along length of beamconstan ts of integrationith generalized displacementPoissons ratiorotation of cross-sectionmass densityshear straincircular frequency

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a timoshenko beam element

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