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Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams
Author(s) -
Banerjee J. R.,
Williams F. W.
Publication year - 1985
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620211212
Subject(s) - bessel function , extrapolation , beam (structure) , mathematics , vibration , mathematical analysis , bernoulli's principle , stiffness , range (aeronautics) , natural frequency , constant (computer programming) , timoshenko beam theory , euler's formula , geometry , physics , structural engineering , materials science , optics , acoustics , engineering , computer science , composite material , thermodynamics , programming language
Bernoulli‐Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as y n and GJ and I both vary as y ( n + 2 ) , where A , GJ and I have their usual meanings; y = ( cx / L ) + 1; c is a constant such that c > − 1 ; L is the length of the beam; and x is the distance from one end of the beam. Numerical checks give better than seven‐figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.