Thick-Walled Cylinder Worked Example
Lame equation calculation for a thick-walled cylinder under internal pressure, showing inner and outer surface stresses and comparison with the thin-wall prediction.
Problem
A thick-walled cylinder has an inner radius of 100 mm, an outer radius of 150 mm and an internal pressure of 20 MPa. Calculate the hoop stress at the inner and outer surfaces using the Lame equations and compare with the thin-wall prediction.
Given
- a = 100 mm
- b = 150 mm
- p_i = 20 MPa
- p_o = 0
- r/t = 100/50 = 2.0, so thick-wall theory is required
Step 1 — Lame constants
For p_o = 0: C1 = a^2 * p_i / (b^2 - a^2) = 10000 * 20 / (22500 - 10000) = 200000 / 12500 = 16.0 MPa C2 = a^2 * b^2 * p_i / (b^2 - a^2) = 10000 * 22500 * 20 / 12500 = 360000 MPa*mm^2
Step 2 — Hoop stress at inner surface (r = a = 100)
sigma_theta(a) = C1 + C2/a^2 = 16.0 + 360000/10000 = 16.0 + 36.0 = 52.0 MPa
Step 3 — Hoop stress at outer surface (r = b = 150)
sigma_theta(b) = C1 + C2/b^2 = 16.0 + 360000/22500 = 16.0 + 16.0 = 32.0 MPa
Step 4 — Thin-wall prediction
sigma_h = p * r / t = 20 * 125 / 50 = 50.0 MPa (using mean radius r = 125 mm, t = 50 mm). The thin-wall prediction is 50.0 MPa, which is between the Lame inner (52.0) and outer (32.0) values but does not capture the 62% variation through the wall.
Result
The inner surface hoop stress is 52.0 MPa and the outer surface hoop stress is 32.0 MPa. The thin-wall prediction of 50.0 MPa is close to the inner surface value but misses the significant through-thickness variation that is characteristic of thick-walled cylinders.