How to calculate rolling pressure?

Rolling pressure is calculated to determine how much force is applied over the contact area during a rolling process. It depends on the rolling force and the area of contact between the rollers and the material, and it helps assess material deformation and equipment performance.

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What is Rolling Pressure?

Rolling pressure (p) is the compressive pressure acting between the rolls and the metal strip during rolling.

It depends on:

• Material strength
• Amount of thickness reduction
• Friction between roll and strip
• Roll diameter

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Common Assumptions in Rolling Analysis

To simplify calculations, we assume:

• Plane strain condition
• Rolls are rigid
• Strip width remains constant
• Homogeneous, isotropic material
• Coulomb friction law applies
• No elastic recovery during rolling

These assumptions lead to approximate but useful formulas.

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1. Basic idea of rolling pressure

When metal passes between two rotating rolls, it is plastically deformed. To cause this deformation, the rolls must apply a pressure greater than the flow stress of the material.

Because:

• Pressure is not uniform along the contact length
• Friction affects deformation

we usually calculate an average rolling pressure.

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2. Key parameters involved

• $\sigma_f$σf​ = flow stress of the material
• $\mu$μ = coefficient of friction
• $R$R = roll radius
• $h_1$h1​ = initial thickness
• $h_2$h2​ = final thickness
• $\Delta h = h_1 – h_2$Δh=h1​−h2​ = draft
• $h_{avg} = \frac{h_1 + h_2}{2}$havg​=2h1​+h2​​ = average thickness
• $L$L = contact length

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3. Contact length in rolling

The contact length between the roll and strip is approximated as:$$\boxed{L = \sqrt{R\,\Delta h}}$$L=RΔh​​

This comes from roll geometry and small-angle approximation.

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4. Flow stress of the material

For strain-hardening materials, flow stress is commonly calculated using:$$\boxed{\sigma_f = K\,\varepsilon^n}$$σf​=Kεn​

where:

• $K$K = strength coefficient
• $n$n = strain-hardening exponent
• $\varepsilon = \ln \left(\frac{h_1}{h_2}\right)$ε=ln(h2​h1​​) = true strain

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5. Ideal rolling pressure (no friction)

If friction is ignored, the average rolling pressure is approximately:$$\boxed{p_{avg} = \sigma_f}$$pavg​=σf​​

This is a simplified assumption.

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6. Rolling pressure with friction (practical case)

When friction is included, the average rolling pressure becomes:$$\boxed{ p_{avg} = \sigma_f \left(1 + \frac{\mu L}{h_{avg}}\right) }$$pavg​=σf​(1+havg​μL​)​

Interpretation:

• $\sigma_f$σf​: pressure required for plastic flow
• $\frac{\mu L}{h_{avg}}$havg​μL​: extra pressure due to friction
• Higher friction → higher rolling pressure
• Larger strip thickness → lower friction effect

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7. Rolling force calculation

Once the rolling pressure is known, the rolling force is:$$\boxed{ F = p_{avg} \times b \times L }$$F=pavg​×b×L​

where:

• $b$b = width of the strip

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8. Summary of steps

1. Calculate draft: $\Delta h = h_1 – h_2$Δh=h1​−h2​
2. Find contact length: $L = \sqrt{R\Delta h}$L=RΔh​
3. Calculate strain: $\varepsilon = \ln(h_1/h_2)$ε=ln(h1​/h2​)
4. Find flow stress: $\sigma_f = K\varepsilon^n$σf​=Kεn
5. Compute rolling pressure: $$p_{avg} = \sigma_f \left(1 + \frac{\mu L}{h_{avg}}\right)$$pavg​=σf​(1+havg​μL​)
6. Compute rolling force: $F = p_{avg} b L$F=pavg​bL

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9. Important assumptions

• Plane strain condition
• Uniform deformation
• Coulomb friction
• No roll flattening (simplified)

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