Discussion of “Asymptotic Derivation of Shear Beam Theory from Timoshenko Theory” by Dewey H. Hodges

摘要

This technical note presents an asymptotic derivation of shear beam theory from Timoshenko theory and a comparison with a similar derivation of Euler-Bernoulli theory using standard energy principles. The author concluded: “Neither Euler-Bernoulli nor shear beam theory predicts frequencies that correspond to the high-frequency branch of Timoshenko theory, when it exists. As is known, the agreement between Euler-Bernoulli theory and the low-frequency spectrum of Timoshenko theory is good when E and G are of the same order, when the mode number is small, and the beam is slender. This agreement improves as the beam becomes slenderer, as the mode number decreases, or as E/G decreases. On the other hand, shear beam theory is shown to be valid only when two conditions hold: E G and rnL where r =cross-sectional radius of gyration, n=mode number, and L =beam length. The agreement between shear and Timoshenko theories is excellent in this regime and improves as E/G increases,as the mode number increases, and as the beam becomes fatter. Results indeed confirm that when these two conditions are enforced, results from shear beam theory and the low-frequency branch of Timoshenko theory are practically identical.”