Thin-Walled Pressure Vessels (5.4)

5.4 Thin Walled Pressure Vessels Lo assume there are no "through thickness stresses -ie we use a membrane model with the stresses only acting within the wall of the pressure vessel cylindrical vessels \[\sigma_{1}\] \[\sigma_{1}\] \[\sigma_{2}\] \[\sigma_{2}\] \[\sigma_{2}\] \[\sigma_{2}\] \[s_{1}\] \[\sigma_{1}\] \[\sigma_{2}\] \[\sigma_{1}\] P \[\Delta x\] \[P2r~\Delta x=\sigma_{1}t~ax~2\] \[\sigma_{1}=\frac{\rho_{r}}{t}\] \[P2\pi c^{3}=\sigma_{2}2\pi rt\] \[\sigma_{1}=\frac{\rho_{r}}{2\pi}(=\frac{\sigma_{1}}{t})\] spherical vessels spherical pressure vessels are the most effective means of containing something under pressure \[P2\pi r^{2}=\sigma2\pi r~t\] \[\sigma=\frac{Pr}{2t}\] P note: no shear stresses in spherical vessels bo all directions are principal stress directions 55 Generalised Plane Stress Lo if you don't ignore the z axis & conduct the analysis for \(\sigma_{3}\) , you just do the same ها thing & add 2 more Mohr's circles based on the rotation around the other 2 prinupal axes ها find that the absolute max shearing in-plane shearing stress. Stress isnt necessarily the same as the max 4 plastic deformation is controlled by the absolute max shearing stress (which doesn't always occur in the 1-2 plane) 4 hence, plastic deformation will often begin in an out-of-plane direction rotated by 45" from one of the in-plane principal directions => 4 this situation is called a state of generalised plane stress 4 if only ơ is acting on the third plane but no shearing stresses (Fxz or Tyz), then the 2-direction is a principal direction & ơ is the 3rd principal stress (ơ3) 4 can determine ơ of using results from the stress transformation analysis & draw the 3 Mohr's circles.