1 By Terence James Haydock Pressure Vessel A simulated response in ANSYS

2 Overview Introduction Required Knowledge Key Equations Tutorial / Methodology  Engineering Data  Geometry  Model  Setup  Solution  Results Summary

3 Introduction Aim: To design a steel pressure vessel and subject it to an internal pressure of 100MPa. A finite element analysis is to be carried out in ANSYS and will show the stress responses of the simulation.

4 Required Knowledge Strength of cylinders subjected to internal pressure: When designing a cylinder to withstand internal pressure there are three key questions that need to be answered so as to choose the right formula, these are – 1.The kind of material i.e. brittle or ductile? (Cast iron, steel etc. are brittle. Brass, bronze etc. are ductile) 2. Open or closed cylinder ends/caps? 3.Is the cylinder classed as a thin or thick walled cylinder? (ratio of wall thickness to inside dia. Is <=0.1 for thin walled, >0.1 for thick walled cylinder.

5 Key Equation’s for Design Thin walled cylinders: σ a = stress in axial direction (MPa) p i = internal pressure in the cylinder (MPa) p o = external pressure in the cylinder (MPa) d i = internal diameter of cylinder (mm) Thick walled cylinder (brittle) open or closed: Lame’s equation is applied. d o = external diameter of tube or cylinder (mm) t = wall thickness (mm) u = poisson’s ratio (typically 0.3) S = allowable stress (MPa) Thick walled cylinder (ductile) closed ends: Clavarino’s equation is applied. Thick walled cylinder (ductile) open ends: Birnie’s equation is applied.

6 Tutorial The vessel in question is subjected to an internal pressure of 100MPa. It has an outer diameter of 500mm, a total height of 700mm and a wall thickness of 25mm. The interior wall surface has a 25mm fillet radius to provide a smooth transition to the end caps.

7 ANSYS Methodology Engineering Data Geometry/ Sketch

8 Sketch and revolve What do you notice? The vessel has planes of symmetry above and below its mid-point. As such, you need only analyze the top or bottom of the vessel! Generate 3D Model

9 We can take our simplification even further. Once you have split the vessel, the same method can be applied again, only this time we can section into quarters. As shown, we scale down the dimensions, though this time we revolve at 90 ⁰, instead of the previous 360⁰ Simplify were possible

10 Mesh and Refine your Part Mesh with the default values i.e. Advanced size function OFF Change to FIXED Change to On: Proximity

11 Provide a frictionless support to three surfaces Provide a pressure to the inside surfaces with a magnitude of 100MPa Apply Boundary Conditions

12 σ c = [(p i r i 2 - p o r o 2 ) / (r o 2 - r i 2 )] - [r i 2 r o 2 (p o - p i ) / (r 2 (r o 2 - r i 2 ))] 1) Stress in Circumferential Direction (Hoop Stress) (x-axis) 3) Stress in Radial Direction (z-axis) σ r = [(p i r i 2 - p o r o 2 ) / (r o 2 - r i 2 )] + [r i 2 r o 2 (p o - p i ) / r 2 (r o 2 - r i 2 )] σ a = (p i r i 2 - p o r o 2 )/(r o 2 - r i 2 ) 2) Stress in Axial Direction (y-axis) σ a = stress in axial direction (MPa) p i = internal pressure in the tube or cylinder (MPa) p o = external pressure in the tube or cylinder (MPa) r i = internal radius of tube or cylinder (mm) r o = external radius of tube or cylinder (mm) Check Your Results in Theory 1) 2) 3)

13 Hoop Stress (x-direction) Axial Stress (y-direction) Radial Stress (z-direction) Generate your Results

14 Compare Results TheoryANSYS% error Hoop (x) MPa Axial (y) MPa Radial (z) MPa

15 Summary Through this tutorial a number of key issues have been addressed with regards to the design and analysis of pressure vessels. 1.Materials 2.Construction 3.Thick or Thin walled 4.Various formula have been presented to help determine the right course of action in designing for wall thickness 5.Various formula have been presented to help determine the stress generated in the x, y, and z direction 6.Numerical (ANSYS) and analytical methods for results

16 References 1.Oberg. E. et al. Machinery’s Handbook 28 th Edition, 2008 Industrial Press INC, New York 2. Lawrence. K. ANSYS Workbench Tutorial, 2007 SDC Publications