Transcript GCSE Mathematics

Pressure Vessel
A simulated response in ANSYS
By Terence James Haydock
1
Overview
• Introduction
• Required Knowledge
• Key Equations
• Tutorial / Methodology
2

Engineering Data

Geometry

Model

Setup

Solution

Results
• Summary
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.
3
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.
4
Key Equation’s for Design
𝑡=
𝐷𝑝
2𝑆
• Thick walled cylinder (brittle) open or
closed: Lame’s equation is applied.
𝑡=
𝐷
2
(
𝑆+𝑝
𝑆 −𝑝
• Thick walled cylinder (ductile) closed
ends: Clavarino’s equation is applied.
𝑡=
𝐷
2
[
𝑆+ 1 −2𝑢 𝑝
𝑆 − 1+𝑢 𝑝
• Thick walled cylinder (ductile) open
ends: Birnie’s equation is applied.
𝑡=
𝐷
2
[
𝑆+ 1 −𝑢 𝑝
𝑆 − 1+𝑢 𝑝
• Thin walled cylinders:
− 1)
− 1]
− 1]
do = external diameter of tube or cylinder (mm)
σa = stress in axial direction (MPa)
pi = internal pressure in the cylinder (MPa) t = wall thickness (mm)
po = external pressure in the cylinder (MPa) u = poisson’s ratio (typically 0.3)
S = allowable stress (MPa)
di = internal diameter of cylinder (mm)
5
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.
6
ANSYS Methodology
Engineering Data
Geometry/
Sketch
7
Generate 3D Model
• 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
bottom of the vessel!
8
or
Simplify were possible
•
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 90⁰, instead of the previous 360⁰
9
Mesh and Refine your Part
Mesh with the default
values i.e. Advanced
size function OFF
Change to FIXED
Change to On:
Proximity
10
Apply Boundary Conditions
Provide a frictionless
support to three
surfaces
Provide a pressure to
the inside surfaces
with a magnitude of
100MPa
11
Check Your Results in Theory
σa = stress in axial direction (MPa)
pi = internal pressure in the tube or cylinder (MPa)
po = external pressure in the tube or cylinder (MPa)
ri = internal radius of tube or cylinder (mm)
ro = external radius of tube or cylinder (mm)
1)
1) Stress in Circumferential Direction (Hoop Stress) (x-axis)
3)
σc = [(pi ri2 - po ro2) / (ro2 - ri2)] - [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))]
2) Stress in Axial Direction (y-axis)
σa = (pi ri2 - po ro2 )/(ro2 - ri2)
3) Stress in Radial Direction (z-axis)
σr = [(pi ri2 - po ro2) / (ro2 - ri2)] + [ri2 ro2 (po - pi) / r2 (ro2 - ri2)]
12
2)
Generate your Results
Hoop Stress
(x-direction)
Axial Stress
(y-direction)
Radial Stress
(z-direction)
13
Compare Results
Theory
Hoop (x) MPa
Axial (y) MPa
Radial (z) MPa
14
ANSYS
% error
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
15
References
1. Oberg. E. et al. Machinery’s Handbook 28th Edition, 2008
Industrial Press INC, New York
2. Lawrence. K. ANSYS Workbench Tutorial, 2007
SDC Publications
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