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It is well-known that the Timoshenko quotient always gives better results than the corresponding Rayleigh quotient, but its implementation is not straightforward, at least for non-uniform redundant beams. Quite recently a modified approach has been proposed [1], in which the main difficulty is overcome, and some preliminary results were given for a tapered beam. In this paper an iterative procedure is suggested, which leads to closer approximations to the true results, and to dramatic improvements in the Rayleigh quotient performances. Consequently, narrow lower-upper bounds can be deduced. Clamped beams and clamped-supported beams with rectangular cross-sections and linearly varying height are thoroughly investigated, providing some interesting comparisons with the results given in [1]. The exact differential equations have been solved for this particular cross-section variation law, in terms of Bessel function, so that exact critical loads and free frequencies can be used to illustrate the performance of the proposed approach. A small program was written, by using the symbolic package Mathematica, so that a large sample of numerical examples could be offered.
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