Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 418 Structural elements and stress variables W Nzz V . N Mi R U d Mzz 2 9 Mze dz Rde Mzz Either the displacement or the stress variable is set to zero according to whether the degree of freedom concerned is fixed or free. As in the case of plates the transverse shear forces at the boundaries differ from their expression at a current point within the shell V Qzu 9-M V Nze Mr Vzu and Vze are known as the Kirchhoff effective shear stress resultants of the first and of the second kind respectively. Elastic vibrations Small elastic strain and stress fields The membrane components of global strains are _ dW _ dV U _1 dW dV nzz dz nee Rãẽ R nze 2 -Rd0 iz The global strains in flexure and torsion are dfe dfz -1 i dfe dfz Xzz dz Xee -Rãẽ Xze T Rd0 dTJ Flexure angles are related to translation displacements by ýe 1U ýz - V J dz R Rde Whence the following components of the elastic stress field N _ Eh 9W dV U i zz 1 - vM dz Rde R J N _ Eh dV U 9W i 1 - v2 Rde R dz Ị Nze Gh dW dri z Rde dz J Bent and twisted arches and shells 419 M - - o M -DIR - dV vg Ị M _ -D 1 - v d2U 1 dv R d9dZ - 2HĨJ . Eh3 where D 1 Equations of vibrations By substituting the elastic stresses into the Love equations we arrive at the following system of vibration equations 1. Axial equation h Eh d2W 1 - v d2W 1 v d2V du i phW - 1 - v2 I dzr 2 2 vR9z j f e ỡ z t 2. Tangential equation phV - Eh v2 h2 i d2V 1 - vd2V 1 v d2W 12R2 R2d92 2 z 2 du h2 d3U . d3u R290 12R2 R2903 dz2d0 J 1 f e G z t 3. Radial equation 12 fl phU ---------------y 1 - v2 U h2 a4u a4u 2 a4u R2 12R2 R29Ớ4 29z29ớ2 R Rz4-J vRWZ - s RV f. . z t dMI Jr R39 dz 420 Structural elements In these equations the terms proportional to h2 12R2 are due to bending and twisting. More generally to understand the relative importance of the various terms appearing in the coupled system to it is advisable to discuss first a few particular .
