Sheet Metal Bending

What happens inside the bend

Clamp a flat strip between a punch and a V-die and press. The metal directly under the punch tip wraps to a radius, and across the thickness of that wrapped zone the strain varies linearly: the outer surface is stretched in tension, the inner surface is compressed, and somewhere in between sits a fibre that changes length not at all — the neutral axis. As long as the strain at the surfaces stays below the elastic limit, the metal springs flat the instant you lift the punch. To make a permanent angle you must push the surface strain past yield so that part of the cross-section deforms plastically.

But yielding the surface does not yield the whole thickness. The strain falls to zero at the neutral axis, so a band of material around the centre never exceeds its elastic limit. When the punch retracts, that elastic core unloads like a released spring and partially straightens the bend. The bend angle opens up; the inside radius grows. This is springback, and it is unavoidable in any room-temperature bend — the only question is how much, and how to plan for it.

The neutral axis and the K-factor

The neutral axis does not sit at the geometric centre of the thickness. Because tension and compression are not symmetric — the outer fibres stretch farther than the inner fibres compress — the neutral axis migrates toward the inside of the bend. Its position is expressed as the K-factor: the distance from the inside surface to the neutral axis, as a fraction of thickness.

K = d_na / t

  d_na = distance from inside surface to neutral axis (mm)
  t    = material thickness (mm)
  K    ≈ 0.33  for tight radii (R/t < 1)
  K    ≈ 0.50  for generous radii (R/t > 5)

K matters because it sets the bend allowance — the length of neutral-axis arc swallowed by the bend. Get K wrong and your flat blank comes out the wrong length, the legs end up short or long, and every hole punched in the flat lands in the wrong place after forming.

Bend allowance (arc length of neutral axis):
  BA = (π / 180) · A · (R + K·t)

Bend deduction (how much shorter the flat is than leg-to-leg):
  BD = 2·(R + t)·tan(A/2) − BA

Flat length:
  L_flat = leg_1 + leg_2 − BD

  A = bend angle (degrees)
  R = inside bend radius (mm)
  t = thickness (mm)

Springback, quantified

Springback can be derived from the elastic unloading of the bent section. A common closed-form estimate relates the final radius R_f to the loaded radius R_i:

R_i / R_f = 4 (R_i σ_y / E t)³ − 3 (R_i σ_y / E t) + 1

  R_i = inside radius under load (mm)
  R_f = inside radius after release (mm)
  σ_y = yield strength (MPa)
  E   = Young's modulus (MPa)
  t   = thickness (mm)

The key grouping is σ_y / E. Springback grows when yield strength is high and modulus is low, because a stiff, low-yield material stores more recoverable elastic strain per unit of plastic strain. That single ratio explains the workshop folklore: aluminium springs back far more than steel even though it is "softer." Aluminium 6061-T6 has σ_y ≈ 275 MPa and E ≈ 69 GPa (σ_y/E ≈ 0.0040); mild steel has σ_y ≈ 250 MPa and E ≈ 200 GPa (σ_y/E ≈ 0.00125). The aluminium ratio is about three times larger, and so is its springback.

Overbending — the practical fix

Because springback opens the angle, the cure is to close it further during forming. To finish a 90° included angle with 2° of springback, the punch drives the metal to an 88° included angle; on release it relaxes the 2° back to 90°. The overbend angle is whatever cancels the measured springback for that exact material, thickness, radius and tooling combination — which is why production shops keep empirical bend tables rather than trusting a formula. A typical CNC press brake controller stores a springback offset per tool and per material and adjusts the ram depth automatically.

Method	Air bending	Bottoming	Coining
Punch reaches die bottom?	No — stops short	Yes — touches die faces	Yes — squeezes bend zone
Angle set by	Ram depth	Die angle	Die angle + pressure
Inside radius set by	V-die opening (~0.16·V)	Punch nose radius	Punch nose radius
Springback	Highest (1–3°+)	Low (<1°)	Near zero
Relative force	1× (baseline)	3–5×	5–8×
Tooling cost	Low, flexible	Medium	High, matched dies
Best for	Prototyping, varied angles	Repeatable production angles	Tight tolerance, thin gauge

Worked example: bending force on a press brake

How much tonnage to air-bend a 1 m long, 2 mm mild-steel part in a 16 mm V-die? Use the standard air-bend force formula:

P = (650 · UTS · t²) / V        (force per metre, in N/mm of bend length)

  UTS = 450 MPa   (mild steel, ~S275)
  t   = 2 mm   → t² = 4 mm²
  V   = 16 mm   (die opening, ~8× t)

P = (650 × 450 × 4) / 16
  = 1,170,000 / 16
  = 73,125 N per metre
  ≈ 73 kN/m
  ≈ 7.5 tonne over the 1 m bend

Widening the die to a 20 mm opening drops the force to about 59 kN/m, but the inside radius grows from roughly 0.16·16 ≈ 2.6 mm to 0.16·20 ≈ 3.2 mm. Force, radius and springback all trade against one another, which is why die selection is the first decision a brake operator makes.

Worked example: flat-pattern length

An L-bracket has two 50 mm legs, a 90° bend, an inside radius R = 3 mm in t = 2 mm steel, with K = 0.42. What length of flat stock do you need?

Bend allowance:
  BA = (π/180) · 90 · (3 + 0.42 × 2)
     = 1.5708 × (3 + 0.84)
     = 1.5708 × 3.84
     = 6.03 mm

Bend deduction:
  BD = 2·(R + t)·tan(A/2) − BA
     = 2·(3 + 2)·tan(45°) − 6.03
     = 2 × 5 × 1.0 − 6.03
     = 10 − 6.03
     = 3.97 mm

Flat length:
  L_flat = 50 + 50 − 3.97 = 96.03 mm

Cut the blank at 96 mm, not 100 mm. Use K = 0.50 by mistake and you would predict a 96.2 mm blank — only 0.2 mm off here, but the error compounds across a part with a dozen bends and turns a tolerance-perfect drawing into a pile of scrap.

Bend radius, grain and cracking

• Minimum inside radius. The outer fibre strain is roughly t / (2R + t). Push R too small and that strain exceeds the material's elongation limit, producing orange-peel texture and then cracks. Ductile mild steel and dead-soft aluminium bend to about 0.5–1.0× thickness; high-strength low-alloy steels need 1.5–3× and hardened alloys 3–4×.
• Grain direction. Rolled sheet has elongated grains. Bending across (perpendicular to) the rolling grain tolerates a tighter radius; bending along the grain cracks more readily because the stretched outer fibres run lengthwise through weak grain boundaries. Drawings often specify "bend across grain."
• Bend relief. A bend that ends at an edge tends to tear at the corner. A small relief notch at each end of the bend line stops the crack before it starts.

Failure modes and trade-offs

• Springback drift. Incoming material varies — a coil with 10% higher yield strength springs back more. Without per-coil offset, a batch that was dead-on at the start of the day creeps out of angle tolerance by the end.
• Outer-fibre cracking. Radius too tight, grain wrong, or material too hard. Anneal, increase the radius, or reorient the bend line.
• Die-mark scoring. The die shoulders drag across the part as it forms, leaving lines on the outside surface. Polished or urethane-insert dies, or a film between part and die, prevent this on cosmetic parts.
• Multi-bend interference. A later bend can collide with the flange formed by an earlier one. Bend sequence planning — usually working from the inside out — avoids the trap.
• Wrinkling near holes. A hole too close to the bend line distorts into an oval as the surrounding metal stretches. Keep holes at least 2.5× thickness plus the radius away from the bend.
• Over-coining cracks. Coining eliminates springback but the high local pressure can crack brittle or work-hardened material and accelerates tool wear; reserve it for thin, ductile, tight-tolerance parts.