Home | | **Strength of Materials I** | | **Strength of Materials for Mechanical Engineers** | Deformation in thin cylindrical and spherical shells

Under the action of radial Pressures at the surfaces, the three Principal Stresses will be . These Stresses may be expected to vary over any cross-section and equations will be found which give their variation with the radius r.

**Deformation in thin cylindrical and spherical shells**

Thick cylinders and shells

**Thick Walled Cylinders**

Under the action of radial Pressures at the surfaces, the three Principal Stresses will be . These Stresses may be expected to vary over any cross-section and equations will be found which give their variation with the radius r.

It is assumed that the longitudinal Strain e is constant. This implies that the cross-section remains plain after straining and that this will be true for sections remote from any end fixing.

Let u be the radial shift at a radius r. i.e. After Straining the radius r becomes (r + u). and it should be noted that u is small compared to r.

**Internal Pressure Only**

Pressure Vessels are found in all sorts of engineering applications. If it assumed that the Internal and Pressure is at a diameter of that the external pressure is zero ( Atmospheric) at a diameter then using equation (22)

**The Error In The "thin Cylinder" Formula**

If the thickness of the cylinder walls is t then and this can be substituted into equation

Which is 11% higher than the mean value given by And if the ratio is 20 then which is 5% higher than

It can be seen that if the **mean** diameter is used in the thin cylinder formula, then the error is minimal.

**Example 1**

The cylinder of a Hydraulic Ram has a 6 in. internal diameter. Find the thickness required to withstand an internal pressure of 4 tons/sq.in. The maximum Tensile Stress is limited to 6 tons/sq.in. and the maximum Shear Stress to 5 tons/sq.in.

If D is the external diameter, then the maximum tensile Stress is the hoop Stress at the inside.

Using equation (43)

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