Nonlinear Collapse Example
Worked example of a nonlinear buckling analysis for a stiffened panel, showing load-displacement response, limit point, collapse load and comparison with eigenvalue.
Problem
A stiffened panel (aluminium, 1.0 m x 0.5 m, t = 2 mm, two blade stiffeners) is loaded in axial compression. An eigenvalue analysis gives a first load factor of 2.3 (P_cr = 115 kN for a 50 kN reference load). Perform a nonlinear analysis with imperfections and determine the collapse load.
Given
- Panel: 1.0 m x 0.5 m, skin t = 2 mm, two stiffeners
- Material: aluminium, E = 70 GPa, sigma_y = 280 MPa
- Eigenvalue load factor: lambda = 2.3 (P_cr = 115 kN)
- Reference load: 50 kN
- Imperfection: first eigenmode at amplitude t/5 = 0.4 mm
Step 1 — Setup the nonlinear model
Activate nonlinear geometry (large deformations). Use an elastic-plastic material model with yield at 280 MPa. Seed the mesh with the first eigenmode at 0.4 mm amplitude. Apply the compressive load using arc-length control to trace the post-buckling path.
Step 2 — Run the analysis
The load increases along a stable path. At approximately 78 kN, the skin begins to buckle (skin buckling event). The load continues to increase as load redistributes to the stiffeners. At approximately 108 kN, the stiffeners begin to deform laterally. The load reaches a maximum (limit point) at approximately 125 kN, then decreases as the panel collapses.
Step 3 — Extract the load-displacement curve
Plot the applied load against the end shortening. The curve shows: (1) an initial linear region, (2) a stiffness reduction at skin buckling (78 kN), (3) continued load increase with reduced stiffness, (4) a limit point at 125 kN, (5) post-collapse unloading. The collapse load is 125 kN.
Step 4 — Compare with eigenvalue
The eigenvalue load was 115 kN. The nonlinear collapse load is 125 kN — higher than the eigenvalue. This is because the stiffened panel has post-buckling reserve: after skin buckling, the stiffeners carry additional load and the panel exceeds the eigenvalue prediction. This is typical for stiffened panels with stable post-buckling behaviour.
Step 5 — Check stress at collapse
Extract the stress at the collapse load. The stiffener stress at collapse is approximately 260 MPa — close to the yield stress of 280 MPa. The collapse involves both geometric nonlinearity (post-buckling) and material nonlinearity (approaching yield). The collapse mode is stiffener column buckling combined with skin post-buckling.
Result
The nonlinear collapse load is 125 kN, which is higher than the eigenvalue load of 115 kN due to post-buckling reserve. The panel collapses by a combination of skin post-buckling and stiffener instability. The collapse load is the defensible prediction of the ultimate load capacity. The eigenvalue analysis correctly identified the buckling mode but underestimated the collapse load.
Assumptions and limitations
- Eigenmode-based imperfection at a single amplitude — a sensitivity study is recommended
- Elastic-plastic material model with bilinear hardening — more complex material models may be required
- Arc-length control captures the post-buckling path but may miss local snap-through events
- The result is for a single load case — combined loading may produce a lower collapse load
- The analysis does not include residual stress from manufacturing