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. 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 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 upon, are used because the tools of answer for short-time dynamic methods. the instance of utilizing the ray expansions for studying the effect reaction of spatially curved thin-walled beams of open profile is tested via fixing the matter concerning 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 effect of the preliminary stresses at the dynamic fields has been investigated.
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Extra resources for Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile
144), Eqs. 17. 143) coincide, ð3:145Þ respectively, Appendix 2 fiðkÀ1Þ ki ¼ d2 xðkÀ1Þ d À cos u À &h sin u &g ðkÀ1Þ ðkÀ1Þ ds2 & ds ' dhðkÀ1Þ þ & sin u þ gðkÀ1Þ ðK þ sÞ À xðkÀ1Þ & sin u ds with Appendix 2 49 & ' dgðkÀ1Þ À hðkÀ1Þ ðK þ sÞ þ xðkÀ1Þ & cos u À & cos u ds e 1y e 1y þx ðkÞ & sin u À x ðkÞ & cos u; ð3:146Þ & ' dgðkÀ1Þ d2 hðkÀ1Þ À h þ ðK þ sÞ ðK þ sÞ þ x & cos u ðkÀ1Þ ðkÀ1Þ ds2 ds & ' dxðkÀ1Þ À & sin u þ hðkÀ1Þ & sin u À gðkÀ1Þ & cos u ds d ð3:147Þ xðkÀ1Þ & sin u À gðkÀ1Þ ðK þ sÞ þ x1k À ðkÀ1Þ & cos u; ds & ' d2 gðkÀ1Þ dhðkÀ1Þ fiðkÀ1Þ si ¼ þ gðkÀ1Þ ðK þ sÞ À xðkÀ1Þ & sin u À ðK þ sÞ ds2 ds & ' dxðkÀ1Þ þ & cos u þ hðkÀ1Þ & sin u À gðkÀ1Þ & cos u ds d hðkÀ1Þ ðK þ sÞ À xðkÀ1Þ & cos u þ x1k À ð3:148Þ ðkÀ1Þ & sin u; ds ( 0 dxðkÀ1Þ 2d À& g0ðkÀ1Þ cosuÀh0ðkÀ1Þ sinu F1ðkÀ1Þ ¼qG1 ds ds ) À Á 1k þ& ax cosuþay sinu xðkÀ1Þ fiðkÀ1Þ ki ¼ þ2&qG22 ( À Ádx1k ðkÀ1Þ ax cosuþay sinu þ2qG22 ðK þsÞx1k ðkÀ1Þ ( þ2&qG22 cosu x1x ðkÀ1Þ À ds dg0ðkÀ1Þ ds À Á þðK þsÞ ay cosuÀax sinu !
Mech. 189, 87–121 (2007) 54 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams 9. D. P. Reddy, Note on wave propagation in linearly viscoelastic media. ZAMP 18, 141–144 (1967) 10. A. V. Shitikova, Ray method for solving dynamic problems connected with propagation of wave surfaces of strong and weak discontinuities. Appl. Mech. Rev. 48, 1–39 (1995) Chapter 4 Impact Response of Thin-Walled Beams of Open Profile Abstract The dynamic theory of thin-walled beams of generic open section with a spatially curved and twisted longitudinal axes proposed in the previous chapter with due account for the axial precompression is utilized here for the analysis of the impact response, since the derived hyperbolic system of recurrent equations together with the ray expansions allow one to describe the short-time processes, in particular, the processes of shock interaction, because the convergence of the ray series essentially depends on the rapidity of the duration of the process under consideration.
79 within the accuracy of arbitrary constants, while the discontinuities h0ðkÞ ; g0ðkÞ ; and x1k ðkÞ are found from the algebraic Eqs. 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.
Dynamic Response of Pre-Stressed Spatially Curved Thin-Walled Beams of Open Profile by Yury A. Rossikhin, Marina Shitikova