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connection for research and other day to day academic activity. Although a router is present for the purpose, faulty LAN cable creating A TIMOSHENKO BEAM ELEMENT?R. DAVIS. R. D. HENSHELL AND G. B. WARBURTONDeto be that of the cross section of the beam, i.e. the rotation of the neutral axisplus the shearingangle. With this it is possible to impose the correct boundary conditions at saya clamped orfree end of a beam.Bending and shearing deformations were considered separately by Kapur [9] for derivingstiffness and consistent mass matrices for a Timoshenko beam. A cubic displacement function

beams are not collinear, complications arise with Kapurs element in coupling up forcesand displacements, and each node must be treated specially. Kapurs frequencies convergedt Presented at theBritish Acoustical Society meeting on Finite element techniques in structural vibrations,at the Institute of Sound and Vibration Research, University of Southampton, Eng

land, on 24 to 25 March1971.30 475476 R. DAVIS, R. D. HENSHELL AND G. B. WARBURTONmuch more rapidly than some other answers obtained with a finite element Timoshenkobeam 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 nodeswere usedin the work reported in reference [l l] to improve the rate of convergence of abeam withshear deformation and rotary inertia. The respective authors of both reference [

ll] andreference [12] (who have also published stiffness and mass matrices for a Timoshenko beam)do not state which rotations were used in formulating their matrices.In the ppartment of Mechanical Engineering,University of Nottingham, Notto of the depth ofthe beam to the wavential equationsof an infinitesimal element in static equilibrium is presented. Stiffness and consistent massmatrices 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 is

made of the accuracy obtained when analysing frameworks with beams.1. INTRODUCTIONIn the Bernoulli-Euler theory of flexural vibrations of beams only the transverse inertiaand elastic forces due to bending deflections are considered. As the ratielengthof 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 which

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includes these secondary effects was derived by Timoshenko [I, 21 and solutionsof 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 frameworks comprised of Bernoulli-Euler or Timoshenko beams.The stiffness and mass matrices for the simple beam finite element were published in1963 by Leckie and Lindberg [8]. Their analysis was based on the exact differential equationsof an infXtesima1 element in static equilibrium. In the present paper Leckie andLindbergswork is extended to include shear deformation and rotary inertia in the analysis. Whenthese secondary effects are taken into account, it is important to be clear which rotation isto be used at the ends of the finite element model. The authors have chosen thisrotationresent paper the matrices for a Timoshenko ingham NG7 2RD, England(Received 20 March 1972)A Timoshenko beam finite element which is based upon the exact differwas assumedfor the bending deformation (the displacements at each node being onetranslation and one rotation). A similar assumption was made for shear deformations, also

allowing a translational and rotational displacement at each node. The resultingelementmatrices 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 beams and cantilevers. However, in general structures, where all various 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 frameworks comprised of Bernoulli-Euler or Timoshenko beams.The stiffness and mass matrices for the simple beam finite element were publishe

d in1963 by Leckie and Lindberg [8]. Their analysis was based on the ex