tailieunhanh - MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 11

Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 378 Structural elements subsection bending is highly confined in the vicinity of the interface and outside this zone of accommodation the radial displacements are very close to the asymptotic values given here. Pressurized toroidal shell As sketched in Figure we consider a toroidal shell of circular cross-section loaded by an internal pressure p assumed to be uniform. An approximate solution may be reasonably proposed according to which the displacement field is the superposition of two distinct radial dilatations one concerning the cross sectional meridian circles radius a and the other the parallel circles of radius lying between R a and R a. The resulting displacement field is thus written as Í 71 i 72 . Furthermore if the aspect ratio R a of the torus is sufficiently large it can be assumed that 71 and 72 are essentially constant. The problem can be then solved by using the Rayleigh-Ritz or Galerkin procedure based again on the principle of minimum potential energy. However as the trial function is constant in the local frame I n it may be found more expedient to write directly the equilibrium equations in terms of 71 and 72. Both methods are successively worked out below. First the metrics of the shell is given by ds2 g2 do2 gp dp2 r2 do2 a2 dip2 go r R a sin p gp a Figure . Toroidal shell Arches and shells string and membrane forces 379 The principal curvature radii are 1 dp Rp dsm 1 Rp a a 1 sin p R R a sin p Ro r sin p The displacement field is Ệ Un Wt q2n q 1 i i cos pt sin pn U q2 q1 sin p W q1 cos p The small strains are given by the relations in which the following variables are in correspondence a p Ra Rp a Xa W p o Ro R a p Xp V 0 sin p Xĩ U V is a free rotation around the torus axis and may be discarded. Then q2 q1 q2 sin p npp 7oo D a R a sin p The elastic stresses are Npp Noo Eh 1 V2 Npp Eh q2 1 2 i a hpp _noo q1 q2 sin p V z----2__----- R a sin p Eh i q1 q2 sin p q2 1 V2 I R a sin p a 1 V V 1 Noo 380 Structural .

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