Finite Timoshenko-Type Beam Element with a Crack
The paper presents a method of constructing the stiffness matrix of a Timoshenko-type finite beam element with a single nonpropagating transversal one-edge crack located in the middle of its length. The crack was modelled by adding an additional flexibility matrix to the flexibility matrix of the uncracked element. The terms of the additional matrix were evaluated according to the laws of fracture mechanics. The element was used to perform several numerical tests, the results of which were compared with results of the analytical solutions available in literature. Very good agreement between the presented model and the analytical solution was obtained. The element presented in the paper may he applied to the static and dynamic analysis of many types of structural elements with faults in form of fatigue cracks. The method of formation of the stiffness matrix described in the paper, allows to create finite elements of a beam with various types of cracks (double-edge, circumferential, internal, etc), provided their stress intensity factors are known.
G.R.IRWIN, Analysis of stresses and strains near the end of a crack traversing a plate, Trans. ASME, J. of Appl. Mech., 24, 361-364, 1956.
J .WAUER, On the dynamics of cracked rotors: a literature survey, Appl. Mech. Rev., 43, 13-17, 1990.
A.D.DIMAROGONAS and C.A. PAPADOPOULOS, Vibration of cracked shafts in bending, J. of Sound and Vihr., 91, 583-593, 1983.
C.A.PAPADOPOULOS and A.D.DIMAROGONAS, Coupled longitudinal and bending vibrations of rotating shaft with an open crack, J. of Sound and Vibr., 117, 81-93, 1987.
N.ANIFANTIS and A.D.DIMAROGONAS, Stability of columns with a single crack subjected to follower and vertical loads, Int. J. of Sol. Struct., 19, 281-291, 1983.
C.A.PAPADOPOULOS, A.D.DIMAROGONAS, Coupling of bending and torsional vibration of a cracked Timoshenko shaft, Ing. Archiv., 57, 495-505 1987.
M.D.R.AJAB, A.AL-SABEEH, Vibrational characteristics of cracked shafts, J. of Sound and Vibr., 147, 465-173, 1991.
W.0sTACHOWICZ, M.KRAWCZUK, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, J. of Sound and Vibr., 150, 191-201, 1991.
S.CHRISTIDIS and A.D.S. BARR, One-dimensional theory of cracked BernoulliEuler beams, Int. J. of Mech. Sci., 26, 639-648, 1984.
T.A.HENRY and B.E.OKAH-AVAE, Vibrations in cracked shafts, IME London, Vibration of Rotating Machinery, 15-19, 1976.
H.OKAMURA, W.W.LIU, C-S.CHU and H.LIEBOWITZ, A cracked column under compression, Eng. Fract. Mech., 1, 547-564, 1969.
I.W.MAYERS, and W.G.R.DAVIES, The vibrational behaviour of a rotating shaft system containing a transverse crack, IME London, Vibration of Rotating Machinery, 53-65, 1976.
B.O.DIRR and B.K.SCHMALHORST, Crack depth analysis of a rotating shaft by vibration measurement, Proc, of 11-th Biennial Conference on Mechanical Vibration and Noise, 2, 607-614, 1987.
W.OSTACHOWICZ and M.KRAWCZUK, Vibration analysis of a cracked beam, Comp. and Struct., 36, 245-250, 1990.
W.OSTACHOWICZ and M.KRAWCZUK, Vibration analysis of cracked turbine and compressor blades, ASME 35-th International Gas Turbine and Aeroengine Congress and Exposition, 90-GT-5, 1990.
M.H.H.SHEN and C.PIERRE, Natural modes of Bernoulli-Euler beams with symametric cracks, J. of Sound and Vibr., 138, 115-134, 1990.
B.S.HAISTY and W.T.SPRINGER, A general beam element for use in damage assessment of complex structures, Trans. ASME, J. of Vibr., Acoust. Stress and Rel. in Design, 110, 389-394, 1988.
G.L.QIAN, S.N.GU and J.S.JIANG, The dynamic behaviour and crack detection of a beam with crack, J. of Sound and Vibr., 138, 233-243, 1990.
G.GOUNARJS and A.D.DIMAROGONAS, A finite element of a cracked prismatic beam for structural analysis, Comp. and Struct., 28, 309-313, 1988.
G.L.QIAN, S.N.GU and J.S.JIANG, A finite element model of cracked plates and application to vibration problems, Comp, and Struct,, 39, 483-487, 1991.
G.R.COWPER, The shear coefficient in Timoshenko's beam theory, Trans. ASME, J. of Appl. Mech., 335-340, 1966.
J.F.KNOTT, Fundamentals of fracture mechanics, Butterworths, London 1973.
J.S.PRZEMIENIECKI, Theory of matrix structural analysis, Mc Graw-Hill, New York 1968.
M.KRAWCZUK, Dynamics of one-dimensional media with materials faults, [in Polish], Doctoral Thesis, IFFM-PAS, Gdańsk 1991.
T.HUBER, Elasticity theory, [in Polish], WNT, Warszawa 1956.
O.C.ZIENKIEWICZ, The Finite Element Method in Engineering, [Polish trans.], Arkady, Warszawa 1972.
IBM Application program, Techn. Publ. Depart., New York 1970.
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