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Physics Pillar

Regime Analysis

Short, intermediate, long — a shell’s slenderness Z decides whether it buckles over its whole length or in one local band. MDC reads the knockdown at your Z instead of one factor for every length.

This page is part of the Skalnav technical documentation series. For the full peer-reviewed methodology, see Wagner et al. (2025), Proc. R. Soc. A, DOI 10.1098/rspa.2025.0196.

Schematic: three cylinders with the same radius and wall but increasing length. The short one buckles in lobes that span its whole height, the intermediate one in rows of diamond-shaped dimples, the long one in a single local band while the rest of its length stays smooth.
Short · low ZIntermediateLong · high Z

Why this matters

Classical handbooks give one knockdown factor for a given radius-to-thickness ratio, whatever the length. Yet a shell’s length changes how it buckles. The Batdorf parameter Z = L² / (R t) · √(1−ν²) captures this: it grows with the square of the length and falls as radius and wall thickness grow.

  • Short shells (low Z) — the buckle spans the whole length between the end rings, which hold the shell round and take part in carrying it.
  • Intermediate Z — the transition: the buckle no longer reaches both ends, and rows of diamond-shaped dimples form.
  • Long shells (high Z) — buckling is local. The dimples take a size set by radius and wall, not by length, so beyond this point the length no longer matters.

A single factor that ignores Z cannot be right at both ends of this range.

How MDC handles it

Skalnav computes Z from your geometry. The MDC knockdown is a surface over Z and the imperfection amplitude, fitted to Skalnav FE Monte-Carlo campaigns, so a short and a long shell with the same wall each get their own knockdown. The knockdown chart in the app plots the curve over Z and marks your design on it.

Whether the wall also yields is a separate question — see Plasticity Correction.

See where your shell sits on Z.

Open Skalnav, change the length, and watch your design move along the knockdown curve over Z.