RE-HOBI362 Rachel Hoffman-Bice, PhD
Seek & Geek

Seek & Geek 6: The Fencing Foil

A Rayleigh-Ritz buckling load for a tapered blade — and why it has to buckle upward

Period Week 6

This past weekend I went fencing for the first time, and became fascinated by the sword — a foil. When one fencer hits another, the foil buckles and deflects upward at the middle.

That direction is critical. If the foil deflects the wrong way — downward — the tip can slide up under the opponent’s protective mask.

So this week: the buckling of a fencing foil, and how its cross-sectional geometry might guarantee it buckles in the right direction.

Three annotated photographs of the foil: the spring-loaded button at the tip that registers a hit, the wires running along the length of the blade, and a close-up showing the semi-circular cross section

Buckling is the point

It’s worth noting that the sword buckling is desirable. It takes roughly 7.5 N on the button at the tip to register a hit, and fencers are taught to impart more than the minimum to make sure the hit lands. That extra force is what buckles the blade.

Looking closely at the foil afterwards, I noticed the semi-circular cross section and the taper along its length, and my instinct said both were deliberate.

Handwritten setup. The semi-circular cross section is characterised with centroid at 4r over 3 pi and I-xc of approximately 0.11 r to the fourth. The tip button is sketched with its 0.5 inch and 0.25 inch diameters, noted as spring loaded and requiring 750 g, about 7.5 N, to register a hit per Bay State Fencers. A sketch of a hit shows the blade bowing. The problem is then framed as a fixed, simply-supported tapered beam 90 cm long, to be solved by Rayleigh-Ritz, with four boundary conditions requiring a fourth-order shape function

Four boundary conditions — zero slope and displacement at the fixed end, zero displacement and moment at the tip — mean a fourth-order trial function.

Handwritten solution for the shape function coefficients, working through the boundary conditions to a-1 equals minus 5 L over 2 and a-2 equals 3 L squared over 2, boxing the result phi equals x to the fourth minus five halves L x cubed plus three halves L squared x squared, along with its first and second derivatives, and the Rayleigh-Ritz quotient for the critical load

The taper makes it messy

Handwritten treatment of the varying moment of inertia. The radius is written as a linear function of position along the blade, giving I-xc as 0.11 times that function to the fourth power, which when substituted into the Rayleigh-Ritz integral produces an expression annotated "this is a total mess to figure out!" For simplicity on a first pass, a constant average moment of inertia of 1.11 by 10 to the minus 11 is used with a Young's modulus of 200 GPa for tempered annealed low carbon steel, giving EI of about 2.23

The radius varies along the blade, so the moment of inertia does too — and carrying that through the Rayleigh-Ritz integral gets ugly fast. For a first pass I took a constant average moment of inertia instead, giving EI ≈ 2.23.

Handwritten evaluation of the Rayleigh-Ritz quotient, expanding and integrating both polynomials over the blade length and dividing, arriving at a critical load of 55 N. The note reads that a thrust force of 55 N, about 5.5 kg, is required to buckle the fencing foil, and that this is a bit greater than experiments found online showing about 37.8 N of force bends the foil to shorten it by 17.5 cm

   
Predicted buckling load 55 N (≈ 5.5 kg)
Published experiment ~37.8 N to shorten the foil by 17.5 cm

Higher than the measured value, and there are good reasons why.

Why the prediction runs high

Handwritten discussion noting that the Rayleigh-Ritz calculation is not an exact solution, which accounts for some of the difference, and — importantly — that the analysis does not include the eccentricity of the sword or its initial deflection under gravity, sketched as an initial imperfection w-nought. Below, two sketches contrast a "good" upward bend against a bend that leaves a gap where the tip can slide under someone's mask and hurt them, with the conclusion that the orientation of the cross section and the initial displacement force it to bend a certain way

Rayleigh-Ritz gives an upper bound rather than an exact answer, so some overshoot is expected. More importantly, the analysis leaves out the eccentricity of the blade and its initial deflection under gravity — a real foil is never the perfectly straight column the idealisation assumes.

And that imperfection is exactly what makes the mechanism safe.

Handwritten explanation of eccentric loading: a cross-section sketch shows the force F applied at a point beneath the neutral axis, so the load is eccentric and produces a bending moment that bows the blade upward. Below, a second pair of sketches shows that gravity acting on the sword pre-inclines it, adding an initial deflection w-nought that biases which way it buckles

The force lands beneath the neutral axis of the semi-circular section, so the loading is eccentric — it carries a moment that bows the blade in one particular direction. Gravity adds an initial deflection that biases it the same way.

Between the cross-section orientation and that initial displacement, the foil is made to bend upward, away from the opponent’s mask. A safety property falling out of the section geometry rather than a separate mechanism.