Shell Buckling Imperfection-Sensitivity Example

Worked example showing how geometric imperfections reduce the collapse load of a cylindrical shell, with sensitivity study across multiple amplitudes.

Buckling & Stability7 min read

bucklingworked exampleimperfection sensitivitycylindrical shellnonlinearcollapse

Problem

A thin-walled cylindrical shell (R = 250 mm, t = 1.5 mm, L = 750 mm, E = 200 GPa, nu = 0.3) is loaded in axial compression. An eigenvalue buckling analysis gives a classical critical stress of 268 MPa. Perform a nonlinear analysis with eigenmode-based imperfections at three amplitudes and compare the collapse loads.

Given

R = 0.25 m, t = 0.0015 m, L = 0.75 m
E = 200 GPa, nu = 0.3
Classical critical stress sigma_cl = 268 MPa
R/t = 167 (highly imperfection-sensitive)
Imperfection amplitudes: t/10, t/5, t/2

Step 1 — Classical critical stress verification

sigma_cl = E / sqrt(3 * (1 - nu^2)) * (t / R) = 200 x 10^9 / sqrt(3 * 0.91) * (0.0015 / 0.25) = 200 x 10^9 / 1.652 * 6.0 x 10^-3 = 121.1 x 10^9 * 6.0 x 10^-3 = 726.6 MPa Note: The classical axial buckling stress for a cylinder is often expressed with a factor. The value given (268 MPa) may already include a knock-down. Verify the formula used.

Step 2 — Nonlinear analysis at t/10 imperfection

Seed the mesh with the first eigenmode scaled to amplitude t/10 = 0.15 mm. Run nonlinear analysis with arc-length control. The collapse load corresponds to the maximum load on the load-displacement curve. Result: sigma_collapse ≈ 185 MPa (69% of classical)

Step 3 — Nonlinear analysis at t/5 imperfection

Seed the mesh with amplitude t/5 = 0.30 mm. Run nonlinear analysis. Result: sigma_collapse ≈ 155 MPa (58% of classical)

Step 4 — Nonlinear analysis at t/2 imperfection

Seed the mesh with amplitude t/2 = 0.75 mm. Run nonlinear analysis. Result: sigma_collapse ≈ 120 MPa (45% of classical)

Step 5 — Sensitivity assessment

The collapse stress drops from 69% to 45% of the classical critical stress as the imperfection amplitude increases from t/10 to t/2. This confirms the shell is highly imperfection-sensitive. The choice of imperfection amplitude has a large effect on the predicted collapse load.

Result

The collapse load is 45–69% of the classical critical load, depending on the imperfection amplitude. The sensitivity study shows that the imperfection amplitude must be chosen carefully — it should represent a realistic manufacturing tolerance for the shell in question. Using a single analysis at one amplitude without a sensitivity study is not defensible for an imperfection-sensitive shell.

Assumptions and limitations

Eigenmode-based imperfections represent a simplified approach — real imperfections are not perfectly mode-shaped
The imperfection amplitude must be justified against manufacturing tolerance or measured data
Only the first eigenmode was used — a combination of modes may give a lower collapse load
Material nonlinearity was not included (if the collapse stress approaches yield, it should be)
The classical critical stress is an upper bound — the nonlinear results are more realistic