Estimate flat rolling — squeezing a metal strip thinner between two rotating rolls. Adjust the strip width, entry and exit thickness, roll radius and flow stress to see the draft, contact arc length, roll force and roll torque update in real time, and grasp the enormous force on a rolling mill with a simplified plane-strain model.
Parameters
Strip width w
mm
Entry thickness h₀
mm
Strip thickness before entering the rolls
Exit thickness h₁
mm
Strip thickness after the rolls (must be less than h₀)
Roll radius R
mm
Average flow stress Y
MPa
How hard the material resists plastic flow; higher for cold or hard stock
Results
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Draft Δh (mm)
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Reduction (%)
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Contact arc length (mm)
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Roll force (kN)
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Roll torque (both rolls) (kN·m)
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Reduction verdict
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Flat-rolling cross-section — rolling animation
A strip of thickness h₀ is gripped by two counter-rotating rolls and squeezed down to exit thickness h₁ (and made correspondingly longer). The arrows show the roll force pushing the rolls apart.
L is the projected contact arc length (mm), 1.15Y the plane-strain mean pressure (Y is the average flow stress, MPa). The roll force F grows with strip width w, flow stress Y and contact length L. Δh: draft, R: roll radius.
$$\Delta h = h_0 - h_1,\qquad r = \frac{\Delta h}{h_0}\times 100\,[\%]$$
Draft Δh and reduction r. h₀: entry thickness, h₁: exit thickness. The reduction shows how hard the strip was squeezed.
$$M = F\cdot\frac{L}{1000}\quad[\text{N·m}]$$
Roll torque M (total for both rolls), estimated by treating the roll force F as acting at an effective lever arm of about the contact length L.
What is roll force in rolling?
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"Rolling" is that process where you squeeze a metal sheet thinner between two rollers, right? It's so everyday I've never really thought about it.
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Exactly. And by tonnage, rolling is the single most important of all metal-forming processes. The great majority of the steel and aluminium in the world passes between the rolls of a rolling mill on its way to becoming sheet, plate, strip, bar or section. In flat rolling, the strip is fed into a gap set slightly closer together than the strip is thick. The rolls grip it by friction, drag it through, and squeeze it thinner — that is flat rolling.
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So if it gets thinner, the material has to go somewhere — it doesn't just disappear, right?
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Good point. Metal conserves its volume, so whatever it loses in thickness it gains in length. With narrow stock it also spreads a little wider, but in flat rolling the width barely changes and almost all of it turns into extra length. The thickness removed in one pass is the "draft Δh", and that draft as a fraction of the entry thickness is the "reduction". Move the entry h₀ and exit h₁ sliders on the left and you will see both numbers change.
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I see. So the "roll force" is the force with which the rolls push on the strip?
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More precisely, it is the force with which the two rolls are pushed apart as they squeeze the strip — the roll separating force. It is enormous, often thousands of kilonewtons. That is why the mill's frame, bearings and screw-down system must all be built to withstand it. The rolls themselves bend elastically under the load, so they are deliberately ground with a slight barrel-shaped "camber" to roll a flat strip. The roll force decides the motor power and the thickness tolerance — it is the heart of the design.
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What makes that roll force grow? The numbers jump quite a bit when I move the sliders.
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The formula F = w·L·1.15Y makes it clear. The wider the strip w, and the higher the material's flow stress Y — that is, the thinner and more work-hardened the cold material — the larger the force. And the contact arc length L = √(R·Δh) grows with both the roll radius R and the draft Δh. So heavy reductions are taken on mills with large, powerful rolls. Conversely, reducing a long product gradually through many stands instead of all at once keeps the force per stand under control.
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A red warning appears when the reduction gets too big. Is taking a lot off in one pass really a problem?
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Yes. Above a 35% reduction this tool flags a caution. A heavy single pass sends the force and torque soaring, and it also makes "bite failure" — where the rolls cannot grip and pull the strip in — more likely. So on the shop floor engineers keep a feel for light, standard and heavy drafts, and hold the per-pass reduction to a sensible range. If that is not enough, you simply add more passes — that is the basic strategy of rolling.
Frequently Asked Questions
It uses a simplified plane-strain model. First the draft is found as Δh = h0 − h1, then the projected contact arc length as L = √(R·Δh), where R is the roll radius. The roll separating force is estimated as F = w·L·1.15Y, treating the mean pressure as 1.15 times the average flow stress Y acting over the projected contact area w·L, where w is the strip width. On a real mill, friction and elastic flattening push the force higher, but this formula is enough for an early design estimate.
The draft Δh is the thickness removed in one pass itself (Δh = h0 − h1, in mm). The reduction is the draft expressed as a fraction of the entry thickness (reduction = Δh/h0 × 100, in %) and shows how hard the strip was squeezed. For example, rolling a 10 mm entry strip down to 8 mm gives a draft of 2 mm and a reduction of 20%. Because the roll force grows with the draft through the contact arc, taking the whole reduction in one heavy pass gives a larger force than spreading it out.
The roll force F = w·L·1.15Y is set by three things: the strip width w, the average flow stress Y, and the contact arc length L. The contact arc L = √(R·Δh) itself grows with both the roll radius R and the draft Δh. So a wider strip, a harder or work-hardened material, a larger roll radius, and a heavier draft per pass all increase the force. This is why cold rolling of thin, work-hardened material needs enormous forces, and why a long product is reduced gradually through many stands to spread the load.
Because the roll force reaches hundreds to thousands of kilonewtons and governs every part of a rolling mill. The frame, bearings and screw-down system must all be built to withstand this enormous force. The rolls themselves bend elastically under it, so they are deliberately ground with a slight barrel-shaped camber to roll a flat strip. The roll force also decides the motor power and the achievable thickness tolerance. Get it wrong and the strip waves, the gauge drifts, and in the worst case the rolls or housing fail — so an early estimate is essential.
Real-World Applications
Steelworks hot and cold rolling lines: Mills that reduce slabs and billets into sheet, plate and strip are the main arena for roll-force calculation. In hot rolling the material is soft at high temperature with a low flow stress, so heavy drafts are possible; in cold rolling the material is hard and work-hardens, so the force is far larger. On a tandem mill (a multi-stand continuous mill), the draft is distributed between stands by balancing force, torque and power, with the final stand finishing the gauge.
Non-ferrous sheet and foil production: Aluminium can stock, copper and stainless strip, and even micron-thick aluminium foil — rolling is the workhorse of thin-product manufacture. For ultra-thin foil the material work-hardens severely and its flow stress becomes very high, so cluster mills and Sendzimir mills support small work rolls with large back-up rolls, withstanding the enormous force while taking precise, tiny drafts.
Designing and upgrading rolling mills: When designing a new mill or stand, the roll force is estimated from the intended materials, dimensions and draft schedule to set the housing (frame) stiffness, bearing capacity, screw-down cylinder thrust and drive-motor rating. A simple calculation like this tool is useful as a first read before going into detailed elastic-plastic FEM or rolling-theory analysis, and it lets you compare roll diameters and draft distributions quickly.
Troubleshooting rolling problems: When a strip waves, the gauge drifts along the length, or rolls wear or spall early, evaluating the roll force and roll deflection gives clues. If the force is higher than expected, the roll camber or draft distribution is reviewed; if bite failure occurs, the per-pass reduction is lowered. Knowing the force level is the starting point for shop-floor improvement.
Common Misconceptions and Pitfalls
The biggest misconception is assuming this simple formula gives the real mill roll force directly. The F = w·L·1.15Y used here is the simplest model, taking the mean pressure as 1.15 times the flow stress (the plane-strain Mises factor). In real rolling, friction between roll and strip makes the contact pressure rise toward the middle — the "friction hill" — so the force is larger than the simple estimate. On top of that, the rolls flatten elastically under the reaction from the strip, lengthening the contact arc and pushing the force higher still. In cold rolling of thin stock, friction and flattening together can more than double the force. Treat this tool as an early design estimate, and finish with Bland-Ford or Hill rolling theory, or with FEM analysis.
Next, the belief that the flow stress Y is a single fixed value for each material. Flow stress varies strongly with temperature, strain and strain rate. In hot rolling, the higher the temperature the lower Y, so the same draft gives a smaller force. In cold rolling, the more you squeeze the more the material work-hardens and the higher Y becomes, so the force rises pass after pass. The "average flow stress" entered here is a representative mid-pass value between the entry and exit conditions. In practice, choose that average from the material's stress-strain curve or hot deformation-resistance data to match the conditions of the pass.
Finally, the oversimplification that a bigger draft is more efficient because it cuts the number of passes. Increasing the draft per pass does reduce the pass count, but it also lengthens the contact arc L = √(R·Δh), so both force and torque rise sharply. If the force exceeds the mill's capacity, the housing deflects excessively, the gauge is not held, and in the worst case the rolls or spindles fail. Push the reduction too far and the rolls cannot draw the strip in — "bite failure". The maximum draft the rolls can bite is set by the friction coefficient and the roll radius, so in reality the per-pass reduction must stay within the machine's capacity and the bite condition. A draft schedule is distributed wisely across several passes while watching force, torque, power, bite and gauge accuracy together.
How to Use
Enter strip width (wNum, typically 500–2000 mm for industrial mills) and adjust via wRange slider
Set entry thickness h0Num (initial gauge, 10–100 mm) and exit thickness h1Num (final gauge, must be less than entry) using respective range sliders
Input roll radius rNum (100–300 mm for laboratory mills, 400–800 mm for hot strip mills) to establish contact arc geometry
Execute simulation to calculate draft Δh, contact arc length via Hertzian contact, roll separating force, and torque requirements for both rolls
Worked Example
Aluminum 5083-H19 strip: width 800 mm, entry thickness h0=25 mm, exit thickness h1=18 mm, roll radius r=250 mm. Draft Δh = 7 mm; reduction = 28%. Contact arc length ≈ 59 mm (calculated from Δh and roll geometry). Roll separating force ≈ 1850 kN assuming mean flow stress of 180 MPa and coefficient of friction μ=0.15. Torque per roll ≈ 54.6 kN·m. Reduction verdict: acceptable for single-stand operation without intermediate annealing.
Practical Notes
Higher reductions (>40% per pass) in aluminum and copper require lower speeds and cooling; steel typically tolerates 30–50% drafts cold and 60–75% hot
Contact arc length is the critical dimension—exceeding mill roll diameter by too much causes excessive force and energy consumption; verify against available motor horsepower
Torque output scales with roll radius and friction coefficient; wet rolling (μ≈0.08) demands 30–40% less torque than dry rolling (μ≈0.20)
Exit thickness precision depends on roll gap control tolerance; typical ±0.1 mm in modern mills requires load-cell feedback systems