22-Mec-A4 Design and Manufacture of Machine Elements · May 2013

Question 2 of 8: Fracture of a stretch-formed pressing and the forming-limit diagram

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examination, 07-Mec-A4 Design and Manufacture of Machine Elements, May 2013 — 3 hours, open book, any non-communicating calculator permitted. Eight questions on six pages, divided into Part A (manufacturing processes, Q1–Q4) and Part B (machine-element design, Q5–Q8). The rubric asks for three from Part A and two from Part B, five questions constituting a complete paper, all of equal value (20 % each). All eight questions are solved here, because this document is a study resource rather than an examination script.

Reference texts.

Kalpakjian & Schmid, Manufacturing Engineering and Technology, 7th ed. — grinding (Ch. 26), sheet-metal forming and the forming-limit diagram (Ch. 16), sand casting and casting defects (Ch. 11–12).
Budynas & Nisbett, Shigley's Mechanical Engineering Design, 10th/11th ed. — combined stresses and static failure theories (Ch. 3, 5), fatigue (Ch. 6), shaft design (Ch. 7), bolted joints in shear (Ch. 8), welded joints (Ch. 9).
Hibbeler, Mechanics of Materials, 10th ed. — stress transformation and combined loading.
ASM Handbook Vol. 15, Casting — gas, penetration and mould-wall-movement defects in no-bake sand systems.

Check: Part B is figure-driven. Every dimension used below was read from the printed figures (Fig. S4–S7). Where the drawing dimensions a distance from a face rather than from a bolt centre (Q7), the reading is stated explicitly in Given so a grader can substitute a different interpretation without redoing the method.

Question 2: Fracture of a stretch-formed pressing and the forming-limit diagram (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) How to analyse the problem

The diagnostic tool for a sheet-forming failure is circle-grid analysis, and it should be the first action. A regular array of small circles (typically 2.5 mm diameter) is electrochemically etched onto blanks before forming. After forming, each circle has become an ellipse whose major and minor axes give the two principal surface strains directly:

$$\varepsilon_1 = \ln\!\left(\frac{d_{\text{major}}}{d_0}\right), \qquad \varepsilon_2 = \ln\!\left(\frac{d_{\text{minor}}}{d_0}\right)$$

Measuring the ellipses immediately adjacent to the crack, and along a path running from a safe region into the crack, gives the strain path that the material followed, not just its end point. Those pairs are then plotted on the forming-limit diagram for that grade and thickness, and the answer to "why did it fail" becomes visible: either the point lies above the forming-limit curve, or it lies below the curve but the material was substandard.

The circle-grid work is supported by four other checks that a production engineer would run in parallel. First, examine the fracture: a fracture preceded by visible local necking is a formability limit, while a fracture with no necking and a shear-textured surface points at edge damage or a very sharp die radius. Second, verify the material against the certificate — strain-hardening exponent $n$, normal anisotropy $\bar r$, thickness and the yield point — because an incoming-material change is the most common cause of a press that "suddenly" starts cracking. Third, audit the tooling: drawbead height and radius, die and punch radii for wear or pickup, and blank-holder pressure and its uniformity around the perimeter. Fourth, audit the lubricant: type, amount, coverage and whether it is actually reaching the drawbead.

(b) Likely strain state at fracture

The question states that the part is formed by almost pure stretching, using drawbeads. Drawbeads exist to restrain the flange so that metal cannot draw inward; all of the shape change must then come from thinning the sheet already inside the die opening. That restraint means the minor strain is close to zero. The strain state at the fracture is therefore plane strain:

$$\varepsilon_2 \approx 0, \qquad \varepsilon_1 > 0, \qquad \varepsilon_3 = -\varepsilon_1 \ \text{(volume constancy)}$$

This is the worst possible place to be. The forming-limit curve is V-shaped with its minimum at $\varepsilon_2 = 0$, the value conventionally called $\mathrm{FLC}_0$; the curve rises to the left into the drawing quadrant and rises to the right into biaxial stretching. A pressing operated with locked drawbeads sits in the trough of the curve, so it has the least major-strain capability the material can offer. The fracture point plots at or just above $\mathrm{FLC}_0$, on the vertical axis.

(c) Two remedies shown on the FLD, shape unchanged

Because the part shape is fixed, the total amount of surface area that must be created is fixed. What can still be changed is where that area comes from and how evenly it is distributed. Both remedies are drawn on the FLD as arrows that move the critical point from above the curve to below it.

Remedy 1 — move the strain path to the left, into the drawing quadrant. Reduce the drawbead restraint (lower the bead, increase its radius, or reduce the blank-holder force locally on the side that is cracking). Metal is then allowed to draw in from the flange, the minor strain becomes negative, and the operating point slides down and to the left along a steeply rising branch of the FLC. On the diagram this is an arrow from $(0,\,\varepsilon_1)$ toward $(-\varepsilon_2,\,\varepsilon_1')$ with $\varepsilon_1' < \varepsilon_1$, comfortably under the curve. This is the single most effective change available.
Remedy 2 — move the strain path to the right, into balanced biaxial stretch, by improving lubrication and reducing local friction hot spots. Where friction pins the sheet against the punch nose, deformation localises: one small region reaches $\mathrm{FLC}_0$ while the surrounding metal is barely strained. A better lubricant (or a locally applied dry film) lets the strain spread over a much larger area, so the peak major strain falls even though the same shape is produced. On the FLD this appears as an arrow pointing right and downward from the fracture point toward the safe zone beneath the right-hand branch of the curve.

Both arrows terminate below the FLC with a working margin; industry practice is to require a margin of roughly 10 % major strain between the highest measured point and the curve, and to treat the band between as the "marginal" zone.

(d) If none of that works

The escalation runs from cheapest to most disruptive, and each step raises the curve itself rather than moving the point under it:

Upgrade the steel grade. Move from a commercial-quality grade to a drawing-quality or interstitial-free grade. A higher strain-hardening exponent $n$ raises the whole FLC (for low-carbon steel, $\mathrm{FLC}_0$ scales roughly with $n$), and a higher normal anisotropy $\bar r$ improves resistance to thinning.
Increase the blank thickness. $\mathrm{FLC}_0$ rises approximately linearly with sheet thickness up to about 2.5 mm, so a step from 0.8 mm to 1.0 mm buys real margin at a modest mass penalty.
Split the operation into two stages — a preform that redistributes the metal, then a restrike to final shape. If work hardening becomes the limit, an intermediate anneal restores ductility.
Change the blank shape or add a tailor-welded blank, placing a thicker or more formable coupon exactly where the crack occurs while keeping the finished part identical.
Change the process route: warm forming (raising formability substantially for aluminium alloys in particular) or hydroforming, in which pressurised fluid replaces a rigid punch and produces a far more uniform strain distribution.

Topic: Forming-limit diagrams, circle-grid strain analysis and the role of drawbeads in sheet-metal pressing.

Key relations: True principal surface strains $\varepsilon_1=\ln(d_{\text{maj}}/d_0)$, $\varepsilon_2=\ln(d_{\text{min}}/d_0)$; volume constancy $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$; the FLC minimum lies at $\varepsilon_2=0$ (plane strain); $\mathrm{FLC}_0$ increases with the strain-hardening exponent $n$ and with sheet thickness $t$.

Why this works: Necking in sheet metal is a geometric instability: it begins when the rate at which the material work-hardens can no longer compensate for the rate at which the load-carrying section shrinks. In plane strain there is no minor-direction contraction to feed the deforming region, so the instability arrives at the lowest major strain of any loading path. Moving the strain path off the $\varepsilon_2=0$ axis in either direction delays that instability, which is exactly why the FLC is V-shaped and why the whole discipline of press-shop troubleshooting reduces to steering the strain path away from the trough.

Common pitfall: Assuming that more blank-holder force or a taller drawbead always improves a pressing. Restraint prevents wrinkling, but on a stretch-dominated part it drives the strain path straight into plane strain and causes the very split it was meant to prevent. The second pitfall is measuring the deformed circles only at the crack, where the strain has already localised; the useful information is in the gradient of strain approaching the crack.

Source: Kalpakjian & Schmid, Manufacturing Engineering and Technology, 7th ed., Ch. 16 (Sheet-Metal Forming Processes), §16.3 on forming-limit diagrams and §16.7 on drawbeads and blank-holder control.

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