Rolling reduces the thickness of a long workpiece between rotating rolls.

• Flat rolling: plates/sheets. Shape rolling: non-flat (I-beams, rails, pipes).
• Plates > 6 mm (ship hulls, boilers, nuclear vessels); sheets < 6 mm (car/aircraft bodies, appliances, cans).

Flat-Rolling Mechanics

Strip enters at $h_0$, leaves at $h_f$; roll surface speed $V_r$. Strip speed rises through the gap and is highest at exit. At the neutral (no-slip) point strip and roll speeds match; the roll is faster before it, the strip faster after it. Friction is necessary to draw the strip in, but excess friction raises forces/power and damages the surface.

Draft = $h_0 - h_f$. Maximum draft:

$(h_0 - h_f)_{\max} = \mu^2 R$

($mu$ = friction coefficient, $R$ = roll radius) — higher friction and larger rolls allow a greater draft.

Roll Force & Power

$F = L\,w\,Y_{\text{avg}}$

$L$ = roll–strip contact length, $w$ = strip width, $Y_{\text{avg}}$ = average true stress (mean of entry yield stress and exit true stress).

$\text{Power} = \frac{F L N}{60000}\ \text{kW}$

with $N$ in rpm (Power = torque × angular speed, torque $= Fcdot L/2$). Recall $1 text{hp} = 0.7457 text{kW}$.

Reduce roll force by: lowering friction, smaller-diameter rolls, smaller reductions per pass, or rolling hot.

Geometric Effects

• Roll bending makes the strip thicker at its center → grind rolls with a camber (larger center diameter); a given camber suits one load/width.
• Rolls also flatten elastically (like a tire).
• Spreading: width grows after rolling, more so when width/thickness is small.

Flat-Rolling Practice

• Hot rolling breaks down ingots/slabs → wrought structure (finer grains, more ductile).
• Cold rolling → better surface finish, tolerances, properties; smaller roughness.
• Pack rolling rolls 2+ layers together (Al foil: matte foil-to-foil side, shiny roll side).
• Sheet passed through leveling rolls to improve flatness.

Defects

Wavy edges (roll bending), cracks (poor ductility), residual stresses (non-uniform deformation). Small-diameter rolls → compressive surface residual stresses (good for fatigue).

Mills & Other Operations

• Tandem rolling: continuous through several stands; volume flow is conserved (roll speed must sync with thickness).
• Roll materials need strength & wear resistance (cast iron, cast/forged steel). Hot rolling of steel uses no lubricant (water cools rolls, breaks up scale).
• Shape rolling: bars, channels, I-beams, rails through specially shaped rolls.
• Seamless pipe/tube: a rotating compressed round bar develops a central cavity (induced tensile stresses), expanded over a mandrel.