How do we analyse the effects of loading on cylinders?
Continuing on in our series looking at the various calculations used in Civil Structural Analysis, we’re going to dive into analysing the effects of loading on cylinders.
What is a cylinder?
Firstly, what is a cylinder? Many items like steam boilers or large pipes are subjected to an internal fluid/gas pressure uniformly distributed over the internal surfaces. This internal pressure gives rise to various stress distributions in the cylinder walls.
Cylinders can be classified as either ‘thick’ or ‘thin’ cylinders according to the following relations:
A vessel is thin walled when the ratio of internal diameter D to wall thickness t is greater than 20:1.
D / t > 20
Aircraft fuselages are subjected to an internal gas pressure, and a typical fuselage structure is shown below.
At the normal stratospheric cruising altitudes of 30,000 to 38,000 feet the outside pressure is 0.3 to 0.2 atmospheres, respectively, while the internal pressure is maintained at a level equal to that found at 8,000 feet, or about 0.7 atmospheres. Therefore, the pressure difference across the thin skin of the cabin ranges from 0.4 to as much as 0.5 atmospheres, or 6 to 7 psi (40 to 50 kPa).
Hoop (Circumferential) Stress
Let’s look at a thin cylinder of internal diameter D subjected to an internal pressure p.
This will give rise to a circumferential or hoop stress σθ and a longitudinal stress σL.
Consider the equilibrium of a half cylinder of length L sectioned across a diameter.
Pressure force = Force in cylinder wall
Longitudinal Stress
Looking at the equilibrium of a half cylinder cut by a transverse plane
Pressure force = Force in cylinder wall
Note: If D/t is greater than 20, then the radial stress σr (equal to -p on the inner surface and zero on the outer surface) can be neglected.
Strains and Changes in Volume
Let the length and diameter of a thin-walled cylinder with closed ends increase by amounts δL and δD, respectively, when the internal (gauge) pressure is increased from zero to p.
Therefore, if the original internal volume is V, the change of volume,
Effects of end plates and joints
So far, we have assumed uniform material properties throughout the vessel without considering the effects of endplates and joints, which are necessary requirements for vessel production. In general, the strength of the vessel will be reduced by the presence of, for example, riveted joints or welds, and should be considered by the introduction of a joint efficiency factor η.
Keep an eye out for more exciting articles diving into Civil Structural Analysis
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