# Thin Walled Cylinders and Spheres

**Contents**

## Thin Walled Cylinders Under Pressure.

**Circumferential**or

**Hoop Stress**, the

**Longitudinal Stress**, and the

**Radial Stress**.

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

## Thin Spherical Shells Under Internal Pressure.

As in the previous section the radial Stress will be neglected and the circumferential or hoop Stress is assumed to be constant.**spherical shell**is a generalization of an annulus to three dimensions. A spherical shell is the region between two concentric spheres of differing radii.

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## Cylindrical Shells With Hemispherical Ends.

**Pressure**is an effect which occurs when a force is applied on a surface. Pressure is the amount of force acting on a unit area. The symbol of pressure is .

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## Volumetric Strain On The Capacity Of A Cylinder

The capacity of a cylinder is , so if the dimensions increase by and there will be an increase in volume and:**Volume**is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

##### Example - Example 1

**additional water**will be pumped in after an initial filling at atmospheric pressure? Assume that the circumferential Strain at the junction of cylinder and hemisphere is the same for both. For the drum material For water

- The
**additional water**must be or at atmospheric pressure.

## A Tube Under A Combined Loading.

##### Example - Example 4

**maximum and minimum principle Stresses**and the

**maximum shearing Stress**.

- The
**maximum and minimum principle Stresses**are and - The
**maximum shearing Stress**is

## Wire Winding Of Thin Walled Cylinders.

One way to strengthen a thin walled tube against an Internal Pressure is to wind the outside with wire under tension. This puts the tube into compression and consequently reduces the Hoop Stress. In many applications the maximum Stress will be in the wire, which must be made of high-tensile material.##### MISSING IMAGE!

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- Consider the wire to be replaced by an equivalent cylindrical shell of thickness . Note that this gives the longitudinal cross sectional area, i.e.

- Let the Initial Tensile Stress in the wire
- Let the initial Compressive Stress in the cylinder be . Then for equilibrium:

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- When an internal pressure is applied, let the Stresses be tension in the wire, and tensile Hoop Stress in the cylinder.

- The final longitudinal Stress in the Cylinder is

- As the wire and cylinder remain in contact, the change in Hoop Strain due to the application of the internal pressure must be the same for both. i.e.,

##### Example - Example 5

**tension**at which the wire must have been been wound if an internal pressure of produces a tensile circumferential stress of in the tube.

- The equivalent Wire thickness. (see equation (2)

- If is the winding stress in the wire, the initial Hoop Stress in the tube ( see equation (3)

- If the final stresses are and the equilibrium equation (4) gives:

- Equating the change in the Hoop Strain for the wire and tube and
**neglecting**longitudinal stress in the tube.

- The winding Tension

- The
**tension**is

## Rotational Stresses In Thin Cylinders.

When a Cylinder rotates about its axis a centrifugal force will occur in its walls, which will produce a Hoop Stress . This stress may be assumed to remain constant at any given angular velocity. Let the cylinder have a mean radius , a wall thickness of and made of material with a density of . Assume that the angular velocity of revolution is . The centrifugal force on an element of unit length which subtends an angle at the centre of the cylinder ( see diagram) equals:##### MISSING IMAGE!

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##### Example - Example 6

**thickness**of the rim. The Density is

- The
**thickness**is