Thick cylinders and shells:
Thick Walled Cylinders,
Internal Pressure Only.

**Deformation in
thin cylindrical and spherical shells**

Thick cylinders and shells

**Thick Walled Cylinders**

Under the action of radial
Presssures at the surfaces, the three Principal Str esses 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 fix ing.

Let u be the radial shift at a raadius 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 (43)

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)

The maximum Shear Stress is half the
"Stress difference" at the inside. Thus using equation (45)

From which as before, D = 13.43 in.

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