Crippling Stress⚠ unverified

Understanding the Parameters

• Width-to-thickness ratio $b/t$ — the master variable. A wide, thin flange ($b/t$ large) buckles locally at a low stress and cripples early; a stubby, thick element ($b/t$ small) resists local buckling and cripples near yield. The negative exponent captures this: as $b/t$ grows, $\sigma_{cc}$ falls. Design charts and codes are all built around $b/t$ (and the analogous plate-buckling parameter).

• Yield stress $\sigma_y$ — unlike global column buckling (which ignores strength), crippling does scale with material strength, because a crippled element is one that has yielded locally in its buckled corners. But it can never exceed $\sigma_y$ — the physical cap the simplified formula omits.

• Crippling coefficient $K_c$ — bundles the edge-support condition of the element: a flange with one free edge (a "one-edge-free" element, like an outstanding leg of an angle) cripples earlier than a web supported on both edges. Real methods use different constants for one-edge-free vs no-edge-free elements; $K_c=0.3$ here is a generic value.

• The output $\sigma_{cc}$ — the average stress at which the element crumples. A built-up stiffener's crippling is the area-weighted average of its individual flats' crippling stresses, and that value then acts as the local "yield" ceiling in the column-buckling check of the whole stringer.

Derivation (Approaching a Proof)

Crippling has no closed-form derivation — it is an inherently nonlinear, post-buckling, partly-plastic phenomenon — so engineering practice uses a semi-empirical approach anchored to plate-buckling theory. The elastic buckling stress of a flat plate under edge compression is

$$\sigma_{cr} = k\,\frac{\pi^2 E}{12(1-\nu^2)}\left(\frac{t}{b}\right)^2,$$

i.e. $\sigma_{cr} \propto (b/t)^{-2}$. If crippling merely equalled plate buckling it would follow this $-2$ power. But a buckled plate keeps carrying load in its supported edge strips (the "effective width" effect), so it collapses at a stress higher than initial buckling, with a weaker dependence on $b/t$. Gerard correlated extensive test data into non-dimensional forms of the type

$$\frac{\sigma_{cc}}{\sigma_y} = \beta\left(\frac{b}{t}\sqrt{\frac{\sigma_y}{E}}\right)^{-m},$$

with $m \approx 0.4$–$0.85$ depending on the element/section type. The registry expression $\sigma_{cc} = K_c\,\sigma_y\,(b/t)^{-0.4}$ is a stripped-down version of this correlation — the $-0.4$ exponent and coefficient $K_c$ standing in for Gerard's fitted constants, and the $\sqrt{\sigma_y/E}$ grouping absorbed into $K_c$. $\blacksquare$ It should always be truncated at $\sigma_{cc} \le \sigma_y$.

Dimensional check. $\sigma_y$ is a stress; $b/t$ and $K_c$ are dimensionless, so $\sigma_{cc}$ is $\text{Pa}$. $\checkmark$

History and Development

• Needham and Gerard. Practical crippling methods were developed in the 1940s–50s by Joseph Needham (the "angle method") and George Gerard, who reduced large aircraft-industry test programmes to non-dimensional curves. Their correlations, tabulated in references like Bruhn and Niu, remain the standard for metal stiffened structure.

• Effective width. The reason crippling exceeds buckling is the effective-width concept (von Kármán): once a plate buckles, load sheds to the stiff material near the supported edges, so the panel keeps taking load. Effective width and crippling are two faces of post-buckled panel behaviour.

• Stringer design. In a skin–stringer panel, each stringer's crippling stress caps the stress it can reach before its flats crumple; this becomes the cutoff on the column-buckling curve for the stringer-plus-effective-skin column (Stringer Crippling Load). Crippling, column buckling, and skin buckling together set the panel's compression allowable.

Related Concepts: Stringer Crippling Load, Buckling Stress, Buckling Load Factor, Euler Buckling Load, Slenderness Ratio, Margin Of Safety

Notes: Registry calculator crippling-stress (unverified). Simplified empirical power law — real methods (Gerard/Needham) use element-and-edge-specific coefficients, sum area-weighted flats, and cap at $\sigma_y$ (the formula does not). $K_c = 0.3$ and exponent $-0.4$ are generic stand-ins. Crippling is local post-buckled collapse, distinct from global column buckling. Defaults $1.0$ (except $K_c$).