Bead Roller and Panel Stiffness

What is modelled

A bead roller squeezes a strip between a die with a rib and a die with a matching groove, so the strip is stretched into a circular-arc bead as it feeds through. Both dies are linked by a one-to-one pair of 40-tooth gears, so they turn at the same speed in opposite senses and drive the strip at the roll surface speed. The bead raises the second moment of the cross-section, so a panel with a bead bends much less under a given load.

Equations

• Bead arc of chord w and depth h: radius ρ = (w²/4 + h²)/(2h), half-angle φ = asin(w/2ρ), length 2ρφ, strain 2ρφ/w − 1.
• Section second moment I = ∫ (y − yc)² t ds along the centre line; for the arc alone ρ³ t (φ + sinφ cosφ − 2 sin²φ/φ).
• Simply supported panel under a uniform line load q: δ = 5 q L&sup4; / (384 E I).
• Roll surface speed v = 2π R n / 60; gear mesh rate 40 n / 60 teeth per second.

Checks run in the page

• Arc length and chord of the bead by polyline against the closed form.
• Second moment of the arc by numerical integration against the closed form.
• Finite-difference beam deflection against 5qL&sup4;/(384EI).
• Feed distance by integrating the crank speed against 2πR per turn.

Honest simplifications

The bead is a perfect circular arc, the sheet keeps its thickness, the panel is a simply supported strip of linear steel (E = 200 GPa) and no forming force, springback or die contact is modelled. The 3D page shows the machine; the numbers here illustrate the mechanics and are not a forming-process design.

• Press Deep narrow bead and compare the large stiffness gain with the high strain across the bead.
• Press Shallow wide bead to see that depth matters far more than width for stiffness.
• Press Thin sheet, long span and see how deflection grows with the fourth power of the span.