4‑Hi Roll Stack & Flatness Lab
Elastic roll bending · Mechanical crown · DS/NDS tilt · Rolling-force coupling · Flatness modes · Buckling tendency
4‑Hi mill · live stress and buckling map
Awaiting calculation
Live stress / buckle map · static steel rolls
Live strip color map
The steel rolls remain the approved reference illustration. The strip between the rolls uses calculated transverse stress for its color. Indicative waves are drawn only where the local buckling screen predicts onset; their amplitude and wavelength are exaggerated and not physical predictions. The approved steel-roll image remains unchanged.
Example buttons change incoming crown and recalculate.
After tension release · wave onset estimate
Elastic screening, not a measured flatness resultCalculate to assess unloaded-strip buckling.
Checks each compressed width band against an estimated plate-buckling threshold. Below threshold means residual stress remains but visible waves are not predicted by this simplified check. Coefficients and effective strip restraints are assumptions; calibration is needed.
Operating point and flatness indicators
Results · current / optimized preview
Main outcomes and actuator solutionElastic roll centerline deflection
Relative to straight chord between barrel shoulders; positive means outward gap opening.
Transverse strip thickness / roll gap
Exit shape after average-gauge regulation; incoming profile shown on secondary normalized axis.
Longitudinal residual stress distribution
Sign + tensile (red), − compressive (blue); applied exit tension shown separately.
Estimated axial roll bending stress
Elastic-beam bending stress estimate. It does not include Hertz contact stress, notch concentration or fatigue limits.
Actuator solution and load balance
HGC forces are calculated equilibrium reactions at the BUR chocks; positive values are separating-load magnitudes.
Flatness mode decomposition · before / after correction
I-units (1 I = 10⁻⁵ strain)P₁ = wedge/asymmetry · P₂ = center/edge (first symmetric shape order) · P₃ = asymmetric higher-order · P₄ = quarter-region (second symmetric shape order). Minimizing the RMS alone can conceal individual modes, so all four are reported.
Rolling mechanics · Von Kármán, yielding and buckling
Von Kármán slab equilibrium (schematic):
d(σₓ · h)/dx = 2p [tan(θ) ± μ]
Plane-strain compressive yield condition:
p − σₓ = σᵧ,plane strain
Von Mises: σᵧ,ps ≈ 1.155 · YS
Tresca: σᵧ,ps ≈ 1.000 · YS
Normal contact p(x) and local draft determine separating force. Friction reverses direction at the neutral plane.
Out-of-plane buckling is a DIFFERENT criterion:
σcompressive > σcritical(h, W, mode, boundary)
Zipf / Shohet–Townsend crown-ratio deadband:
−80 (h/W)² ≤ [C_in/h_in − C_out/h_out] ≤ 40 (h/W)²
Negative outside bound: edge-wave tendency.
Positive outside bound: center-buckle tendency.
Crown-ratio limits indicate center/edge tendency only. A separate compressed-band plate-buckling screen checks whether waves may appear after exit tension is removed.
Model interpretation and validation limits
Enter parameters and calculate.
Equations and buckling thresholds draw on Mark E. Zipf, Advanced Cold Rolling 201: Shape / Flatness (particularly p. 17 and pp. 140–145), and Mechanics of the Roll Bite / Dissecting Roll Bite Forces. The slab solver, two-beam contact assumption, wear law, transferred strain and optimizer are engineering approximations implemented for this prototype, not equations validated by those notes.