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.
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)