Bending, deep drawing and spinning

Why it matters

Brackets, enclosures, kitchen sinks, beverage cans, fuel tanks and lamp reflectors are all formed from flat sheet without cutting. The press-shop engineer has to get the flat blank size right before forming, choose a bend radius the sheet can survive, allow for springback, and decide whether a cup can be drawn in one stroke. Each of these is a short calculation.

Key ideas

Bending. A sheet bent over a radius stretches on the outside and compresses on the inside; the neutral axis in between keeps its length. For sharp bends the neutral axis shifts toward the inside, so its position is given as K·t from the inner surface (K ≈ 0.33 when R < 2t, ≈ 0.5 when R ≥ 2t; take from your data book). The length of the neutral axis through the bend is the bend allowance, used to compute the flat blank length.

• Minimum bend radius — the outer-fibre tensile strain must not exceed what the metal can stand; less ductile and thicker sheets need larger radii. Bending across the rolling direction is safer than along it.
• Springback — when the load is removed the elastic part of the bend recovers, so the final angle is smaller and the radius larger than in the die. It increases with yield strength, with R/t (thinner sheet or larger radius) and with lower elastic modulus. It is compensated by overbending, bottoming (coining) the bend, stretch bending, or warm bending.
• Operations — V-bending in a V-die (press brakes), edge (wiping) bending, roll bending of plates into cylinders, roll forming of long profiles, flanging, hemming, seaming.
• Bending force — estimated from the tensile strength, width, thickness squared and die opening; a V-die needs about four times the force of a wiping die for the same sheet.

Deep drawing. A flat blank, held by a blank holder, is pushed by a punch through a die to form a cup. The flange is drawn radially inward, so it is compressed circumferentially (and tends to wrinkle); the cup wall carries the punch force in tension (and tends to tear, usually near the punch nose). The blank holder must press hard enough to stop wrinkling but not so hard that friction makes the wall tear.

• Drawing ratio DR = D_blank / d_punch. The limiting drawing ratio (LDR) is about 2.0–2.2 for most ductile sheets; larger ratios need several draws (redrawing) with intermediate annealing if required. Equivalently, the reduction r = (D − d)/D should be below about 0.5 in the first draw.
• Anisotropy — sheets with high normal anisotropy (R̄, the ratio of width to thickness strain in a tensile test) resist thinning and can be drawn deeper; planar anisotropy causes earing (wavy rims).
• Ironing — thinning the cup wall by a die clearance smaller than the wall thickness; used on beverage cans to get thin, uniform walls.
• Blank size — by constant surface area (thickness assumed unchanged), a flat-bottomed cup of diameter d and height h without flange needs D = √(d² + 4dh), ignoring corner radii.
• Defects — wrinkling (flange or wall), tearing, earing, surface scratches.

Spinning. A disc is rotated on a lathe-like machine and pressed over a mandrel by a roller or tool.

• Conventional spinning — the blank is folded over the mandrel; thickness stays about the same and diameter shrinks. Low tooling cost, for small quantities: reflectors, cookware, bells.
• Shear spinning (flow turning) — diameter stays the same and the wall thins following the sine law, t_f = t₀·sin α (α = cone half-angle). Large strains, for rocket nose cones and similar.
• Tube spinning — reduces tube wall thickness over a mandrel.

Formulas

BA = α × (R + K·t) — bend allowance (mm); α = bend angle (rad), R = inner bend radius (mm), t = thickness (mm), K = neutral-axis factor.

e_outer = 1 / (1 + 2R/t) — engineering strain of the outer fibre.

R_min = t × (50/r − 1) — minimum bend radius (mm); r = percentage reduction of area in a tensile test (empirical rule).

R_i / R_f = 4(R_i·Y/(E·t))³ − 3(R_i·Y/(E·t)) + 1 — springback; R_i = radius before release, R_f = after, Y = yield strength, E = elastic modulus.

F_b = k_b × σ_UTS × w × t² / D — bending force (N); w = bend length (mm), D = die opening (mm), σ_UTS in MPa, k_b ≈ 1.33 for V-bending and 0.33 for wiping (edge) bending.

D = √(d² + 4·d·h) — blank diameter for a cylindrical cup without flange (mm).

DR = D / d_p ≤ LDR ≈ 2, r = (D − d_p)/D — drawing ratio and reduction.

F_d ≈ π × d_p × t × σ_UTS × (D/d_p − 0.7) — empirical maximum drawing force (N).

t_f = t₀ × sin α — shear spinning wall thickness.

Worked examples

Example 1 (standard — bending). A 3 mm steel sheet (σ_UTS = 400 MPa), 500 mm long, is bent through 90° with an inside radius of 6 mm in a V-die of opening 24 mm. Take K = 0.5. Find the bend allowance, the outer-fibre strain and the bending force.

1. BA = (π/2) × (6 + 0.5 × 3) = 1.5708 × 7.5 = 11.78 mm.
2. e = 1 / (1 + 2 × 6/3) = 1/5 = 0.20 (20%).
3. F_b = 1.33 × 400 × 500 × 3² / 24 = 99 750 N.

Answer: BA ≈ 11.8 mm, e = 20%, F ≈ 99.8 kN.

Example 2 (GATE level — deep drawing). A cylindrical cup 60 mm in diameter and 40 mm high (no flange) is drawn from 1 mm sheet with σ_UTS = 350 MPa. Find the blank diameter, check whether one draw is enough (LDR = 2), and estimate the drawing force.

1. D = √(60² + 4 × 60 × 40) = √(3600 + 9600) = √13 200 = 114.9 mm.
2. DR = 114.9 / 60 = 1.91 < 2 → one draw is possible.
3. F_d = π × 60 × 1 × 350 × (1.915 − 0.7) = 65 973 × 1.215 = 8.01 × 10⁴ N.

Answer: D ≈ 115 mm, single draw, F ≈ 80 kN.

Example 3 (shear spinning). A 4 mm disc is shear-spun into a cone of half-angle 30°. t_f = 4 × sin 30° = 2 mm.

Common mistakes

• Using the inner radius alone for bend allowance; the neutral axis lies at R + K·t.
• Using degrees for α in the bend-allowance formula — it must be in radians.
• Believing thicker sheet springs back more for the same bend radius; springback grows with R/t, so thin sheet springs back more.
• Using the cup's outside height or including the flange incorrectly in blank-size problems — read what the dimensions refer to.
• Comparing blank diameter with die diameter instead of punch diameter for the drawing ratio.
• Thinking a higher blank-holder force always helps; too much causes tearing.

For GATE PI

Numericals ask for bend allowance and flat length, bending force, minimum bend radius, blank diameter for a cup, drawing ratio and number of draws, drawing force, and wall thickness in shear spinning. One-mark items test springback factors, the role of the blank holder, wrinkling vs tearing, earing and anisotropy, and ironing. Practise the cup blank-size formula and the LDR check together.

Quick check

1. Blank diameter for a cup d = 50 mm, h = 30 mm?
2. Does springback increase or decrease with yield strength?
3. Bend allowance for a 60° bend, R = 10 mm, t = 2 mm, K = 0.5?
4. What defect does insufficient blank-holder force cause?
5. Why do sheets with high normal anisotropy draw deeper?

Answers: 1. √(2500 + 6000) ≈ 92.2 mm. 2. Increases. 3. (π/3) × 11 ≈ 11.5 mm. 4. Wrinkling of the flange. 5. They resist thinning, so the cup wall does not tear as easily.