Plasticity Correction

The problem

Classical elastic shell-buckling theory — the lineage from von Kármán and Donnell through Timoshenko — derives critical loads from a bifurcation analysis that assumes infinitely elastic material behaviour. For slender shells whose critical stress is well below the proportional limit, this assumption is justified and the theory predicts instability accurately (modulo imperfection sensitivity). For stockier shells, however, the elastic critical stress can approach or exceed the material yield stress, and the theory breaks down.

Once local yielding begins, the shell’s effective bending and membrane stiffness drop in a way that the elastic eigenvalue problem cannot capture. In the limiting case of very thick walls, elastic instability is irrelevant and the shell fails by plastic collapse — a mechanism governed by yield-surface interaction, not by bifurcation. Between pure elastic instability at one extreme and pure plastic collapse at the other lies an interaction region where neither description applies cleanly. The governing failure mode in this region is a coupled elastic-plastic instability whose capacity is lower than either theory predicts in isolation.

Traditional design codes handle this interaction with empirical plasticity reduction factors: a curve, typically a function of the ratio σ_cr,el / f_y, that discounts the elastic critical stress to account for yielding. These factors are calibrated conservatively and serve their purpose as lower-bound allowables, but they obscure the underlying physics. Worse, they treat plasticity and geometric imperfection sensitivity as independent effects, when in reality the two are coupled: real shells yield locally at imperfection crests long before the nominal membrane stress reaches f_y, so plasticity amplifies the imperfection response and vice versa. Ignoring this coupling forces the code curve to carry extra conservatism that penalises structural efficiency.

The MDC approach

MDC replaces empirical plasticity reduction factors with a physics-based correction that is continuous across the elastic, transition, and plastic regimes and respects the coupling between plasticity and imperfection sensitivity. Rather than applying an independent knockdown after the elastic analysis, the correction operates within the capacity prediction itself, so that regime transitions are handled smoothly and the governing failure mechanism is identified at every analysis point. The correction is calibrated against numerical simulation campaigns and cross-checked against published experimental data. The full derivation and validation for cylindrical shells under axial compression is given in Wagner et al. (2025), Proc. R. Soc. A 481(2321), DOI 10.1098/rspa.2025.0196. Extensions of the methodology to further geometries and load cases are the subject of ongoing research.

Scope and limitations

The MDC plasticity treatment is a pre-design and verification tool. It captures the global interaction between elastic instability, imperfection sensitivity, and material yielding for shell segments under membrane-dominated stress states. It does not replace geometrically and materially nonlinear analysis with imperfections (GMNIA) for local effects such as cutouts, concentrated loads, welded joints with residual stress patterns, or strain-rate-dependent behaviour.

For the cylindrical-shell case under axial compression, the full methodology — derivation, calibration, and experimental validation — is documented in Wagner et al. (2025), Proc. R. Soc. A 481(2321). For other geometries and load cases, the tool provides MDC allowables where the underlying methodology has passed internal validation; results from public standards (NASA SP-8007, EN 1993-1-6, DNV RP-C202, ECSS-HB-32-24, GOST 34233) are always available alongside for comparison.