# Thick Walled cylinders and Spheres

**Contents**

## Thick Walled Cylinders

**cylinder**is the central working part of a reciprocating engine or pump, the space in which a piston travels.

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0001.png** cannot be found in /users/23287/ThickWalled-CnS-0001.png. Please contact the submission author.

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0002.png** cannot be found in /users/23287/ThickWalled-CnS-0002.png. Please contact the submission author.

## Internal Pressure Only

Pressure Vessels are found in all sorts of engineering applications. If it assumed that the Internal Pressure is at a diameter of , and that the external pressure is zero (Atmospheric) at a diameter , then using equation (9)**Pressure**is the force per unit area applied in a direction perpendicular to the surface of an object.

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0003.png** cannot be found in /users/23287/ThickWalled-CnS-0003.png. Please contact the submission author.

## The Error In The

If the thickness of the cylinder walls is , then and this can be substituted into equation (11)**mean**diameter is used in the thin cylinder formula, then the error is minimal.

##### Example - Example 1

**thickness**required to withstand an internal pressure of The maximum Tensile Stress is limited to and the maximum Shear Stress to

- The
**Thickness**=

## The Plastic Yielding Of Thick Tubes.

**yield strength**or

**yield point**of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically.

**Assumptions made in the Theory of Plastic Yielding.**

- Yield takes place when the maximum Stress difference (or Shear Stress) reaches the value corresponding the yield in simple tension. This assumption has been found to be in good agreement with experimental results for ductile material.
- The Material exhibits a constant yield Stress in tension and there is NO Strain hardening; i.e., it is an ideal elastic material.
- The longitudinal Stress in the tube is either zero or lies algebraically between the Hoop and Radial Stresses. From this it follows that the maximum Stress difference is determined by the Hoop and Radial Stresses only.

**Hoop and Radial Stresses in the Plastic Zone**The equilibrium equation (7) must apply and the yield criterion based on the assumptions stated above, provided that and are stresses of the opposite type, is:

**Partially Plastic Wall**Consider a thick tube of internal radius and external radius , subjected to an internal pressure only of such a magnitude that the material below a radius of is in the plastic state (i.e. is the boundary between the inner plastic region and the outer elastic region). If is the radial Stress at , it is stated by the elastic theory for internal pressure only (Equation (12))that the maximum Stress difference is (i.e. just reaching the yield conditions at ).

##### Example - Example 5

**pressure**and

**show**the variation of Stresses across the wall. What are the

**pressures required**for initial yield and complete yield? Assume that yield occurs due to maximum shear stress and neglect Strain Hardening. In simple Tension equals

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0005.png** cannot be found in /users/23287/ThickWalled-CnS-0005.png. Please contact the submission author.

- The pressure for the
**initial yield**is - The pressure for a
**complete yield**is

## Compound Tubes.

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0003.png** cannot be found in /users/23287/ThickWalled-CnS-0003.png. Please contact the submission author.

##### Example - Example 6

**stress**and

**show**in a diagram the variations of Hoop Stress in the two tubes. What is the

**initial difference**of diameters prior to assembly ?

- For the inner tube :

- Similarly for the outer tube:

- Stresses due to Internal Pressure:

##### MISSING IMAGE!

**13108/img_cyl11_0006.jpg** cannot be found in /users/13108/img_cyl11_0006.jpg. Please contact the submission author.

- The initial difference of diameters at the common surface

## A Hub Shrunk Onto A Solid Shaft

The Shaft will be subjected to an external pressure , and if and are the Hoop and Radial Stresses at a radius , the equilibrium equation (7) will be obtained as for a "Thick Cylinder".##### Example - Example 7

**necessary force**fit allowance and the maximum circumferential

**Stress**in the hub. and Poisson's Ratio = for both. The coefficient of friction between the surfaces is . If after assembly the shaft is subjected to an axial compressive Stress of , what is the resulting

**increase**in the maximum circumferential Hub Stress ?

- For the shaft

- For the Hub

- Force fit

## Thick Spherical Shells

At any radius let the circumferential or Hoop Stress be Tensile and the Radial Stress Compressive. If is the radial shift then it was shown earlier that the Hoop Strain is given by and the radial Strain by .**spherical shell**is a generalization of an annulus to three dimensions. A spherical shell is the region between two concentric spheres of differing radii.

##### MISSING IMAGE!

**23287/ThickWalled-CnS-0007.png** cannot be found in /users/23287/ThickWalled-CnS-0007.png. Please contact the submission author.

**If there is internal pressure only**:- The Maximum Stress is the value of at the inside i.e. And the maximum Shear Stress,