Timoshenko Beams

The natural frequencies and mode shapesBeams, Euler-Bernoullimode shape of Timoshenko beams of constantSingle degree-of-freedom systemconstant force cross section, continuously variable cross sectionBeams, Timoshenkovariable cross section, and cross sections with step changesBeams, Euler-Bernoullivariable cross section in properties for numerous combinations of boundary conditions and boundary and in-span attachments are obtained. The in-span and boundary attachmentsBeams, Euler-Bernoulliboundary attachments include springs, concentrated massesBeams, Euler-Bernoulliconcentrated mass, single degree-of-freedom systemsBeams, Timoshenkosingle degree-of-freedom system, and elastically connectedBeams, Euler-Bernoullielastically connected beams. TheBeams, Euler-Bernoullisingle degree-of-freedom system responses to externally applied forces and to a moving mass are determined. The natural frequencies of functionally graded materials are obtained. For all numerical results, comparisons are made to those obtained from the Euler–Bernoulli theory and regions of applicability are inferred.