By Yury A. Rossikhin, Marina Shitikova
This brief ebook analyses the dynamic balance with admire to small perturbations, in addition to the neighborhood harm of geometrically nonlinear elastic, spatially curved, open part beams with axial precompression. brief waves, that are the surfaces of robust discontinuity and in which the tension and pressure fields event discontinuities, are used as small perturbations; in so doing the discontinuities are thought of to be of small importance. Such waves are initiated in the course of low-velocity affects upon thin-walled beams. the idea of discontinuities and the tactic of ray expansions which permit one to discover the specified fields at the back of the fronts of the temporary waves when it comes to discontinuities in time-derivatives of the values to be discovered, are used because the equipment of answer for short-time dynamic strategies. the instance of utilizing the ray expansions for interpreting the impression reaction of spatially curved thin-walled beams of open profile is tested via fixing the matter in regards to the basic influence of an elastic hemispherical-nosed rod upon an elastic arch representing itself a channel-beam curved alongside an arc of the circumference. The impact of the preliminary stresses at the dynamic fields has been investigated.
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Additional resources for Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile
That is why, below in Sect. 22) we have q½vi;ðkþ1Þ ¼ ÀGÀ1 ½rij;ðkþ1Þ kj þ þ d½rij;ðkÞ o½rij;ðkÞ o½rij;ðkÞ kj þ kj þ sj ds ox oy d½vi;ðkÞ d2 ½vi;ðkÀ1Þ o½vi;ðkÀ1Þ þ& þ sin u ds ds2 ox ! GÀ2 ½vi;ðkþ1Þ À 2GÀ1 À& o½vi;ðkÀ1Þ cos u r0kk : oy ð3:23Þ To satisfy Eq. 14, it is sufficient to put Â Ã rij;ðkÞ ki kj ¼ 0; Â Ã rij;ðkÞ si sj ¼ 0; Â Ã rij;ðkÞ si kj ¼ 0: ð3:24Þ ð3:25Þ ð3:26Þ 24 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams Fig. 2 Scheme of the cross section of a thin-walled beam with a generic open crosssection Then the boundary surface will be free from the normal and tangential stresses.
130 yields x1k ðkÞ ¼ 0; h0ðkÞ ¼ GI G22 1y c ; G21 À G22 ðkÀ1Þ g0ðkÞ ¼ GI G22 1x c : G21 À G22 ðkÀ1Þ ð3:131Þ From Eqs. 128 it follows that on the quasi-transverse wave for k ! 133) we have À1 3 2 1 x1k IpC ¼ s þ s a À s a ¼ c1k x y ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ¼ const; À1 h0ðkÞ ¼ s1ðkÞ À ay s3ðkÞ þ s2ðkÞ ax À s1ðkÞ ay IpC ¼ chðkÞ ¼ const; ð3:134Þ À1 ¼ cgðkÞ ¼ const: g0ðkÞ ¼ s2ðkÞ þ ax s3ðkÞ þ s2ðkÞ ax À s1ðkÞ ay IpC Thus, for the particular case of a straight untwisted thin-walled beam of open profile, it has been found that the transient transverse wave is a pure sheartorsional mode as it follows from Eq.
L. Gol’denveizer, To the theory of thin-walled beams (in Russian). Prikl. Mat. Mekh. 13, 561–596 (1949) 4. P. Timoshenko, Vibration problems in engineering. (Van Nostrand, New York, 1928) 5. Z. Vlasov, Thin-walled elastic beams. C. by the Israel Program for Scientific Translations, Jerusalem (Eng. trnsl. from the 2d Russian Ed. published in 1959 in Moscow). (1961) 6. A. V. Shitikova, The impact of a sphere on a Timoshenko thin-walled beam of open section with due account for middle surface extension.
Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile by Yury A. Rossikhin, Marina Shitikova