22-Mec-A4 Design and Manufacture of Machine Elements · Undated paper
Question 3 of 6: Springback control on a precision bent lever
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations, May 2019 — 16-Mec-A4 Design and Manufacture of Machine Elements. Three hours, open book, any non-communicating calculator. Six questions in two parts: Part A (Q1–Q3, manufacturing processes) and Part B (Q4–Q6, machine-element analysis). The rubric asks for two questions from each part; all six are solved here. All questions carry equal value (25 %).
Reference texts. S. Kalpakjian and S. Schmid, Manufacturing Engineering and Technology, 7th ed. (Part A: Ch. 16 sheet-metal forming, Ch. 26 grinding); M. Groover, Fundamentals of Modern Manufacturing, 6th ed. (Ch. 20, 25); R. Budynas and K. Nisbett, Shigley's Mechanical Engineering Design, 10th ed. (Ch. 3 stress, Ch. 6 fatigue, Ch. 7 shafts and keys, Ch. 16 brakes); R. Hibbeler, Mechanics of Materials, 10th ed. (Ch. 6, 7, 9 transverse shear and stress transformation).
Question 3: Springback control on a precision bent lever (25 marks)
(a) What varies in a production batch of strip. Sheet and strip are sold to tolerances on dimensions and to a specification band on properties, not to a single value, so several quantities scatter from coil to coil and, more slowly, along a single coil:
Yield strength. The dominant variable. A typical commercial temper carries a specification band of roughly ±10 % about nominal, arising from heat chemistry, finishing temperature, cooling rate and skin-pass reduction.
Thickness. Held to a rolling tolerance (commonly a few per cent), and varying across the width as crown and along the length.
Strain-hardening exponent $n$ and plastic strain ratio $\bar r$, which follow the texture developed in rolling and annealing, and which set how the bend zone work-hardens.
Planar anisotropy. Properties differ with the angle between the bend axis and the rolling direction, so a blank nested at a different angle bends differently and has a different minimum bend radius.
Residual coil shape — coil set and crossbow — plus variations in surface finish and residual mill oil, which change friction in the die.
These matter here because springback in bending is governed by the elastic-recovery group $S_y/E$ together with the ratio of bend radius to thickness. Kalpakjian gives the recovery of a bend as
with $R_i$ the initial (die) radius, $R_f$ the radius after release, $T$ the strip thickness, $S_y$ the yield strength and $E$ the modulus. Since $E$ is essentially constant for a given alloy family while $S_y$ and $T$ are precisely the two quantities that scatter, the batch variation in the as-formed angle is inherited directly from the batch variation in the strip.
Given. Low-carbon strip, nominal $S_y = 250\ \text{MPa}$ with a $\pm 10\,\%$ coil-to-coil band, $T = 1.0\ \text{mm}$, die radius $R_i = 5.0\ \text{mm}$, $E = 207\ \text{GPa}$.
Find. The spread in the finished included angle if a single fixed overbend is set from the nominal properties.
Evaluate the recovery group at the nominal condition. With $R_iS_y/(ET) = (5)(250)/[(207000)(1)] = 6.039\times10^{-3}$, the formula gives $R_i/R_f = 0.98188$.
Convert to an overbend. The bend arc length is conserved on release, so the angle scales with the same ratio, $\theta_f = \theta_i\,(R_i/R_f)$. To land on 90° at nominal properties the tool must be set to $\theta_i = 90/0.98188 = 91.66^\circ$ — that is, an overbend of about $1.66^\circ$.
Apply the same fixed tool to the extremes of the property band. At $S_y = 225\ \text{MPa}$ the ratio is $0.98370$ and the part closes to $90.17^\circ$; at $S_y = 275\ \text{MPa}$ it is $0.98007$ and the part opens to $89.83^\circ$.
Quote the residual spread. The finished angle therefore wanders across $$\boxed{\Delta\theta \approx 0.33^\circ \ \text{(about } \pm 0.17^\circ \text{)}}$$ from coil variation alone, before thickness scatter, tool wear or lubrication are counted.
(b) Partly — overbending is necessary but it cannot "always assure" the angle. Overbending is the right first move: it removes the mean springback, and no bending process for a precision part should be set up without it. What it cannot do is remove the variance. The overbend is fixed in the tool steel; the springback it is compensating is a function of each coil's own $S_y$ and $T$. When the property band shifts, the compensation is no longer matched, and the part comes off at the wrong angle in a perfectly repeatable way. The calculation above puts a number on that: roughly a third of a degree of drift, which for a camera linkage held to a tenth of a degree is a scrap-generating amount.
The correct answer is therefore to overbend and to add a mechanism that is insensitive to the strip:
Bottoming or coining the bend. Pressing the punch nose hard into the radius plastically compresses the material through its thickness at the bend. With the elastic core largely eliminated there is very little stored energy to recover, and the finished angle is set by the tool geometry rather than by $S_y$. This is the standard production answer for precision bends and is why coining is used on small, thin parts of exactly this type.
Stretch bending. Superimposing a tensile force along the strip while bending shifts the neutral axis toward the inner surface and puts the whole section into net tension, so far less of the cross-section unloads elastically. Springback falls dramatically and becomes much less sensitive to yield strength.
Closed-loop or adaptive forming. Measure the angle in the press (or the punch force, from which $S_y$ can be inferred) and adjust the stroke part by part. This tracks the property drift instead of assuming it away.
Control the input. Buy strip to a narrowed yield band, keep to a single supplier and temper, restrict the nesting orientation relative to the rolling direction, and sort or segregate by coil so that a fixed overbend can at least be re-set per coil.
The recommendation for this part is to overbend as a first-order correction, bottom the bend to kill the residual sensitivity, and specify strip with a tightened yield range — with in-process angle checks at coil changes to catch anything the first two miss.
Springback of the 90° camera lever with a fixed overbend
