How this calculation works
A thick cylinder does not carry the same hoop stress at every radius. Lamé theory resolves radial and circumferential stress across the wall. That distribution is useful when a thin-wall average would hide a higher inner-surface value.
A = (pi a² − po b²)/(b² − a²); B = a²b²(pi − po)/(b² − a²); σr = A − B/r²; σθ = A + B/r²A worked example
With a = 50 mm, b = 100 mm and 10 MPa internal pressure, A is 3.33333 MPa. Inner hoop stress is 16.66667 MPa and outer hoop stress is 6.66667 MPa. Radial stress runs from −10 MPa at the bore to zero outside.
Good to know
Axisymmetric elastic stress away from end disturbances only. External-pressure collapse, plasticity, thermal gradients, openings and fatigue need different checks. This is not a pressure-vessel code calculation or a permitted working pressure.
Reference: BCcampus — pressure vessel stress context (opens in a new tab) ↗
What this model answers
A thick cylinder does not carry the same hoop stress at every radius.
Lamé theory resolves radial and circumferential stress across the wall. That distribution is useful when a thin-wall average would hide a higher inner-surface value.
The Plane stress principal stresses and Mohr circle calculator can explain two-dimensional stress transformations, but the thick cylinder’s radial, hoop and axial stresses form a three-dimensional state.
Keep the input conventions consistent
Both geometry fields are radii.
Pressures are entered as positive compressive magnitudes, with the same pressure reference on each boundary. Outputs use tensile-positive stress: the radial boundary stress is therefore the negative of the pressure applied there.
Read the result carefully
The worked table checks the radial boundary conditions directly.
At the inner surface σr equals minus internal pressure; at the outer surface it equals minus external pressure. Closed-end axial stress is an additional result, not a third pressure input.
Radial stress at each pressure boundary provides a direct consistency check on the sign convention. The bore radial stress should match negative internal pressure and the outside value negative external pressure. Hoop stress varies through the wall, so evaluating only one nominal average misses the stated thick-wall distribution.
Work through the numbers
With a = 50 mm, b = 100 mm and 10 MPa internal pressure, A is 3.33333 MPa.
Inner hoop stress is 16.66667 MPa and outer hoop stress is 6.66667 MPa. Radial stress runs from −10 MPa at the bore to zero outside.
Where the calculation stops
Axisymmetric elastic stress away from end disturbances only.
External-pressure collapse, plasticity, thermal gradients, openings and fatigue need different checks. This is not a pressure-vessel code calculation or a permitted working pressure.
A few questions, answered
Is hoop stress the same as radial stress?
No. They act in perpendicular directions and satisfy different equations and boundary conditions.
What should I change in the comparison?
Add external pressure and inspect how both boundary conditions change.