By Yury A. Rossikhin, Marina Shitikova
This brief ebook analyses the dynamic balance with recognize to small perturbations, in addition to the neighborhood harm of geometrically nonlinear elastic, spatially curved, open part beams with axial precompression. temporary waves, that are the surfaces of sturdy 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 value. Such waves are initiated in the course of low-velocity affects upon thin-walled beams. the speculation of discontinuities and the strategy of ray expansions which permit one to discover the specified fields in the back of the fronts of the brief waves by way of discontinuities in time-derivatives of the values to be came across, are used because the tools of resolution for short-time dynamic techniques. the instance of utilizing the ray expansions for interpreting the influence reaction of spatially curved thin-walled beams of open profile is validated by way of fixing the matter in regards to the common influence of an elastic hemispherical-nosed rod upon an elastic arch representing itself a channel-beam curved alongside an arc of the circumference. The impression of the preliminary stresses at the dynamic fields has been investigated.
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82, in so doing the discontinuities 1y 0 1x h0ðkÞ ; g0ðkÞ ; and x1k ðkÞ have the higher order than the discontinuities xðkÞ ; xðkÞ ; xðkÞ ; and wðkÞ : For arbitrary magnitudes of k, the set of Eqs. 79 can be rewritten in the form dx0ðkÞ ¼ A0ðkÞ ðsÞ; ð3:83Þ ds d 1x ð3:84Þ xðkÞ þ x0ðkÞ & cos u ¼ A1ðkÞ ðsÞ; ds d 1y ð3:85Þ xðkÞ À x0ðkÞ & sin u ¼ A2ðkÞ ðsÞ; ds d 1y 0 2 & sin u þ x & cos u À x & sin 2u ¼ A3ðkÞ ðsÞ; ð3:86Þ wðkÞ À x1x ðkÞ ðkÞ ðkÞ ds where functions AiðkÞ ðsÞ ði ¼ 0; 1; 2; 3Þ are presented in Appendix 3.
To the Timoshenko beam. It is well known that the 46 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams Timoshenko beam equations produce only two transient waves, longitudinal and pffiffiffiffiffiffiffiffiffi transverse, propagating, respectively, with the velocities GL ¼ E=q and GT ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Kl=q: From the results presented in Sect. 1 it is evident that neglecting warping motions for a bisymmetrical beam, as it is seen from Eqs. 21), the magnitudes of which depend essentially on the geometry of the beams’s crosssection.
However, reference to Eq. 22 shows that the value r0kk is multiplied by the k ? e. e. they exert weak effect only on the jumps. 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  à rij;ðkÞ ki kj ¼ 0;  à rij;ðkÞ si sj ¼ 0;  à 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.