# Stresses with thickwalled tube theory

In summary, the conversation discusses the mounting of a ring on a solid shaft and calculating the stress of the ring. The ring has an inner diameter of 90mm - 50 micrometers and the shaft has a diameter of 90mm + 20 micrometers, resulting in a difference of 110 micrometers. The stress can be calculated using the thickwalled tube theory with plane strain. The conversation also mentions material data for the ring and shaft, including their respective E-moduli and Poisson's ratios. Further help can be found in thread 289475.
Hello

A ring should be mounted on a solid shaft. The rings inner diameter is smaller than than the shaft diamater so it will be forced on.
for my problem the inner diamater for the ring is 90mm -50 $$\mu$$m
the shaft is 90mm + 20$$\mu$$m

so there can be a difference of 110 $$\mu$$m.
This should give a certain stress after mounting the ring.

That stress is what I need to calculate

Im using thickwalled tube theory. plane strain.
I have attached a figure to make it easier...

If the radial stress in the ring starts with
A, B are constants as we know.

with BC:

so A=B/(OR^2)

and at the inner radius the stress is equal to the inner pressure: -Pi
So A=-Pi/(1/OR^2 -1/IR^2)/)

So the radial stress for the ring is :
SigmaRing=[-Pi/(1/OR^2 -1/IR^2)]/OR^2 + [Pi/(1/OR^2 -1/IR^2)]*1/r^2

If I do equlibrium for the solid shaft:
Witch BC:

The constants A=0 and B=Pi*(IR^2)
So that stress is Sigmar=-Pi(IR^2)/r^2

What should I do to introduce the radial mismatch of 110 micro meters?
And after that how do I get the stress in the ring with my material data below ?

E-modulus ring =540 GPa
Poissons ring = 0.24
E-modulus shaft=205 GPa
Poissons shaft = 0.3

Thanks for any help

#### Attachments

• mounting.jpg
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## 1. What is thickwalled tube theory?

Thickwalled tube theory is a mathematical model used to analyze stresses and strains in cylindrical structures with a large difference between the inner and outer radii. It takes into account both the internal and external pressures acting on the tube, as well as any external forces or moments.

## 2. How is thickwalled tube theory different from thinwalled tube theory?

The main difference between thickwalled and thinwalled tube theory is the ratio of the inner and outer radii of the tube. Thickwalled tube theory is used when this ratio is large, while thinwalled tube theory is used when the ratio is small. Thickwalled tube theory also considers the effects of radial and circumferential stresses, while thinwalled tube theory only considers axial stresses.

## 3. What types of structures can be analyzed using thickwalled tube theory?

Thickwalled tube theory can be applied to a variety of cylindrical structures, such as pipes, pressure vessels, and storage tanks. It is also commonly used in the design of mechanical components, such as bearings and gears.

## 4. What are the assumptions made in thickwalled tube theory?

Thickwalled tube theory makes several assumptions, including: the material is homogeneous and isotropic, the tube is perfectly cylindrical, and the stress distribution is linearly elastic. It also assumes that the internal and external pressures act uniformly on the tube.

## 5. How is thickwalled tube theory used in practical applications?

Thickwalled tube theory is used in engineering and design to predict the stresses and strains in cylindrical structures, which helps determine the appropriate material and geometry to ensure structural integrity. It is also used in failure analysis and quality control to identify potential weak points in a structure.

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