Metallurgical Rolling Mills, Hot/Cold Strip Pass Schedules & Roll Separating Force
Rolling Mill Draft, Max Friction Bite & Roll Force Formula
Calculate Pass Thickness Draft Delta_h (mm), Friction-Limited Maximum Self-Entry Draft mu^2*R (mm), Projected Roll-Bite Contact Length L_p (mm), Plane-Strain Roll Separating Force (kN), and Bite Status.
Command a pass draft Delta_h larger than mu^2 * R and the work rolls skid helplessly without pulling the slab into the bite!
Reducing flat metal strip of width w (mm) from entry thickness h_0 (mm) to exit thickness h_f (mm) between work rolls of radius R (mm) takes Thickness Draft Delta_h = h_0 - h_f over Projected Contact Arc Length L_p = SQRT(R * Delta_h) mm! For the rotating work rolls to pull the slab unaided into the roll gap upon initial contact, horizontal tangential friction component mu * F_n * COS(alpha) must overcome horizontal normal rejection F_n * SIN(alpha) (i.e. TAN(alpha) <= mu), which via small-angle geometry L_p / R ~= SQRT(Delta_h / R) establishes the famous Maximum Unaided Bite Draft Law: Delta_h_max = (mu^2) * R! Under Von Mises plane-strain yield ((2 / SQRT(3)) * Y_avg ~= 1.1547 * Y_avg), Ideal Roll Separating Force across the stand housings reaches F_roll = ((2 / SQRT(3)) * Y_avg * w * L_p) / 1,000 kN!
Verify on a 1,000 mm Wide Hot/Warm Strip Mill Stand—Entry_Thickness h_0 = 12.00 mm (B2), Exit_Thickness h_f = 9.20 mm (B3 -> Pass Draft Delta_h = 2.80 mm in E2), Work_Roll_Radius R = 320.0 mm (B4), Roll_Bite_Friction mu = 0.11 (B5 -> Max Friction Bite Delta_h_max = 3.87 mm in E3), Strip_Width w = 1,000.0 mm (B6), and Mean_Flow_Stress Y_avg = 185.0 MPa (B7): Contact Arc L_p = SQRT(320 * 2.80) = 29.93 mm produces Plane-Strain Roll Separating Force of 6,394.33 kN (~652 metric tons, E4), and Bite Check confirms "SELF-ENTERING BITE OK" (E5)!
| Strip Rolling Pass KPI (h0=12.0 -> hf=9.2 mm, R=320 mm, mu=0.11, w=1,000, Y=185) | Formula | Verified Output |
|---|---|---|
| Pass Thickness Draft dh (mm) | Entry_h0_mm - Exit_hf_mm | 2.80 mm (4.50 mm @ hf = 7.50 mm) |
| Max Unaided Friction Draft (mm) | (Friction_Mu ^ 2) * Roll_Radius_R | 3.87 mm |
| Plane-Strain Roll Force (kN) | ((2/SQRT(3)) * Yavg * w * SQRT(R*dh))/1000 | 6,394.33 kN (8,106.29 @ hf = 7.50) |
| Roll-Gap Entry Bite Feasibility | Draft_dh <= (Mu^2) * R | SELF-ENTERING BITE OK (SKID @ 7.5!) |
Von Mises Plane-Strain Roll Force & Friction-Limit Bite Formula
Validate h_0 > h_f > 0, Roll_Radius R > 20 mm, Friction mu in [0.02, 0.50], Width w > 0, and Mean_Flow_Stress Y_avg > 0!
=LET(ha_m, B2, hb_m, B3, rr_m, B4, mu_f, B5, wd_m, B6, yv_m, B7, IF(OR(hb_m <= 0, ha_m <= hb_m, rr_m <= 20, mu_f < 0.02, mu_f > 0.5, wd_m <= 0, yv_m <= 0), "Invalid Rolling Input", LET(dh_m, ha_m - hb_m, dm_x, (mu_f ^ 2) * rr_m, IF(dh_m <= dm_x, "SELF-ENTERING BITE OK", "ROLL SKIDDING REFUSAL"))))Over-Greedy 4.50 mm Pass Roll-Skidding Refusal Check (Set Exit Thickness h_f = 7.50 mm)
Attempt to screw down Exit_Thickness B3 to 7.50 mm in a single pass (Draft E2 = 4.50 mm > Max Friction Bite 3.87 mm in E3): E5 immediately flags "ROLL SKIDDING REFUSAL"—warning mill operators that lubricated rolls cannot grip the nose of the strip without ragging or pusher bars!
=IF(ISTEXT(E2), E2, LET(lp_m, SQRT(B4 * (B2 - B3)), ROUND(((2 / SQRT(3)) * B7 * B6 * lp_m) / 1000, 2)))Try the example
Use the original values or change them to test your understanding.