Wire and Tube Drawing: Calculate Drawing Force
Compute drawing stress and drawing force in wire and tube drawing using Siebel's approximation. Enter the mean flow stress, the initial and final diameter (or cross-sections), the friction coefficient and the half die angle – the calculator returns the true strain, area reduction, the ideal-deformation, friction and shear shares, the drawing force and the drawing-limit check with a traffic-light rating, live on every input.
Drawing Calculator (Drawing Force by Siebel)
Model: Siebel's drawing-force equation for the cylindrical conical pass without back tension (wire and tube drawing). It accounts for the ideal, friction and shear shares, but not for back tension, deformation heat, speed and lubrication effects or the exact flow curve. Guide values for design, not experimentally validated.
Results
Calculating …
Formulas and fundamentals
The basis is Siebel's drawing-force equation: σ_z = kf_m·φ·(1 + µ/α) + (2/3)·kf_m·α. Here kf_m is the mean flow stress of the material over the pass, φ the true strain, µ the friction coefficient in the die zone and α the half die angle in radians. The first term combines the ideal deformation work kf_m·φ and the friction share kf_m·φ·µ/α, the second term (2/3)·kf_m·α captures the additional shear work from redirecting the material flow at the cone. The drawing stress acts on the final cross-section, so the drawing force is F = σ_z·A1.
The true strain describes the logarithmic change of area: φ = ln(A0/A1) = 2·ln(d0/d1) for round wire. It must not be confused with the engineering area reduction ε = (A0 − A1)/A0 – for small reductions both are close, for large reductions φ runs noticeably higher (at 36 percent reduction φ ≈ 0.45). Because φ is logarithmic, the true strains of several passes simply add up, whereas the reductions do not.
The drawing stress has a minimum over the die angle: too small an angle enlarges the friction area and thus the friction share µ/α, too large an angle enlarges the shear share (2/3)·α. Between them lies an optimum angle. Decisive is the drawing limit: the drawing stress σ_z must stay below the flow stress of the already work-hardened wire at the die exit kf_m,end, otherwise the wire yields behind the die instead of inside it and breaks. In practice this limits the area reduction per pass (round wire usually to about 20 to 35 percent).
Worked example
A wire is drawn from d0 = 10 mm to d1 = 8 mm. The mean flow stress is kf_m = 500 N/mm², the friction coefficient µ = 0.05 and the half die angle α = 6° (0.10472 rad). The cross-sections are A0 = 78.54 mm² and A1 = 50.27 mm², so the area reduction is 36 percent and the true strain φ = ln(78.54/50.27) = 0.4463.
The shares of the drawing stress are: ideal deformation share kf_m·φ = 223.1 N/mm², friction share kf_m·φ·µ/α = 106.5 N/mm² and shear share (2/3)·kf_m·α = 34.9 N/mm². In total σ_z = 364.6 N/mm².
The drawing force is F = σ_z·A1 = 364.6·50.27 ≈ 18,328 N, i.e. about 18.3 kN. Since σ_z at 364.6 N/mm² is well below the flow stress of 500 N/mm², the pass is feasible – the drawing-limit utilisation is about 73 percent.
Frequently asked questions
How do true strain φ and area reduction differ?
The area reduction ε = (A0 − A1)/A0 is the intuitive percentage decrease, the true strain φ = ln(A0/A1) the logarithmic quantity that enters the deformation work. For small reductions both are almost equal, for large ones φ runs higher. True strains of several passes add up, reductions do not.
What is the drawing limit and when does the wire break?
The drawing stress σ_z must be smaller than the flow stress of the already work-hardened wire at the die exit kf_m,end. If σ_z reaches this value, the wire yields behind the die and necks down instead of being formed inside it – it breaks. This limits the area reduction per pass to typically about 20 to 35 percent.
Why is there an optimum die angle?
The drawing stress consists of a friction share that falls with µ/α and a shear share that rises with (2/3)·α. Too small an angle enlarges the friction area, too large an angle the shear work. In between lies a minimum; typical half die angles are around 5 to 8 degrees.
How do I get the mean flow stress kf_m?
kf_m is the mean of the flow curve between the initial and final true strain of the pass, approximately the arithmetic mean of kf at the entry and kf,end at the exit. For strongly hardening materials the exit flow stress is much higher than at entry; for the drawing-limit check kf_m,end is the decisive value.
What does Siebel's equation not account for?
It is an approximation for the cylindrical conical pass without back tension. It ignores back tension, deformation and friction heat, speed-dependent effects, lubricant film conditions, the exact flow curve and the die geometry with bearing zone. For reliable designs the values serve as a guide and are validated experimentally.
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