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Exact Bernoulli‐Euler static stiffness matrix for a range of tapered beam‐columns
Author(s) -
Banerjee J. R.,
Williams F. W.
Publication year - 1986
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.1620230904
Subject(s) - beam (structure) , mathematics , buckling , bessel function , extrapolation , stiffness , mathematical analysis , timoshenko beam theory , bending stiffness , axial symmetry , flexural rigidity , stiffness matrix , geometry , structural engineering , engineering
Bernoulli‐Euler theory and Bessel functions are used to obtain explicit expressions for the exact static stiffnesses for axial, torsional and flexural deformation of an axially loaded beam which is tapered such that A varies as y l , GJ as y m +2 , and I 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, x is the distance from one end of the beam, l and m can have any value and n is 1,2 or − 1. The work complements the similar dynamic stiffness derivations of Reference 2. Numerical results for a beam with substantial taper ( c = 1.0) 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 critical buckling loads of a clamped/clamped member that are exceeded by any trial load so that an existing algorithm can be used to obtain the exact critical buckling loads of structures which contain tapered members.