22-Mec-A4 Design and Manufacture of Machine Elements · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examinations, May 2016 — 07-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-process theory) and Part B (Q4–Q6, machine-element analysis). Candidates answer two from Part A and two from Part B; four questions constitute a complete paper and all questions carry equal value (25 %). All six are solved here.
Reference texts. Kalpakjian & Schmid, Manufacturing Engineering and Technology (Part A); Groover, Fundamentals of Modern Manufacturing (Part A); Budynas & Nisbett, Shigley's Mechanical Engineering Design, 11th ed. (Part B); Hibbeler, Mechanics of Materials, 10th ed. (Q4, Q5); Norton, Machine Design: An Integrated Approach, 6th ed. (Q6).
Check: two corrections carried through Part B. (1) Q5 prints the shaft modulus as “E = 30 ksi”; a 30 ksi modulus is physically impossible for steel and would make the shaft a rubber band, so it is read as the standard E = 30 × 106 psi (30 Mpsi). (2) Q6 supplies the linkage dimensions but not the cross-section of the levers, so the stress in members 2 and 3 is worked for an explicitly stated assumed section and the required section is also reported. Both readings are flagged where they are used, in the spirit of Note 1 on the cover page.
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.
The analysis is a strain-based one, and the tool is the circle-grid. Electrochemically etch a grid of small circles (typically 2.5 mm diameter) onto blanks, run them through the press, and measure the ellipses that the circles have become in and around the failure. The major and minor diameters give the major and minor engineering strains directly, and converting to true strain gives a point for every measured location. Plotting those points on the forming limit diagram (FLD) for that sheet grade and thickness shows immediately how close each region is to the forming limit curve (FLC), where the failure sits relative to it, and along which strain path the material travelled.
Alongside the strain map, the investigation must confirm the inputs: check the incoming coil certificate and re-test the material for yield strength, tensile strength, total elongation, the strain-hardening exponent $n$ and the normal anisotropy $\bar{r}$, and compare them against specification and against the coils that ran successfully; check sheet thickness and its variation; check lubricant type, quantity and coverage; check blank-holder force and its uniformity around the periphery; check the drawbead penetration, radius and wear; check die and punch radii for wear, galling or pick-up; and check press ram speed, tonnage trace and blank location. Finally, establish whether the failure is a split, a local neck, or a wrinkle — a distinction that changes the diagnosis completely. A production process that once ran and now does not is nearly always a change in one of these variables, most commonly lubrication, blank-holder force, or an incoming-material batch change.
The part is described as formed by almost pure stretching with drawbeads. Drawbeads exist precisely to lock the flange and prevent metal from being drawn in, so all the strain must come from thinning the sheet under the punch. With the flange locked in both directions the sheet is stretched biaxially, and the strain state at the critical location lies in the right-hand side of the FLD, between plane strain and balanced biaxial stretching — that is, with major strain $\varepsilon_1 > 0$ and minor strain $\varepsilon_2 \geq 0$.
The most probable precise state is plane strain, $\varepsilon_2 \approx 0$, at the bottom of the FLC. This is where the forming limit curve reaches its minimum, so it is the least forgiving strain path available and it is where fully locked-in stretch-formed panels characteristically split — along a line, typically over a punch radius or a character line, running perpendicular to the major strain direction. The split is preceded by a visible local neck.
Remedy 1 — move the strain point to the right, away from the plane-strain minimum. Because plane strain is the worst point on the curve, deliberately admitting a little minor strain moves the operating point toward balanced biaxial stretching, where the FLC is higher, and simultaneously reduces the major strain required for the same shape. Practically this is done by reducing the drawbead restraint (shallower bead, larger bead radius, or removing the bead over part of the periphery), lowering the blank-holder force, and improving lubrication so that metal can flow in from the flange instead of all the strain coming from thinning. Enlarging the blank locally, or opening a lock bead into a draw bead on the sides where the metal should feed, achieves the same thing. Note the trade-off: relax the restraint too far and the panel wrinkles instead of splitting, so the process window is bounded on both sides.
Remedy 2 — raise the forming limit curve by changing the sheet. The height of the FLC scales with the strain-hardening exponent $n$ and with the sheet thickness, and its shape is influenced by the normal anisotropy $\bar{r}$. Specifying a deeper-drawing grade with a higher $n$ and higher $\bar{r}$ — for example moving from a commercial-quality grade to a drawing-quality or interstitial-free grade — lifts the whole curve so the same strain point now lies safely below it. Increasing the sheet gauge does the same, at a weight and cost penalty. Both remedies leave the geometry of the pressing untouched, which is what the question requires.
If the strain state cannot be moved and the sheet cannot be improved enough, the remaining options attack the process route rather than the single operation. In rough order of increasing cost and disruption: redistribute the strain over more of the blank by re-designing the addendum and binder surface, the draw-bar or the punch-radius profile — legitimate because the part shape is unchanged even though the tooling shape is not; split the operation into two or more forming stages with an intermediate transfer die, so no single stage demands a strain near the limit; introduce an intermediate anneal between stages to restore ductility; adopt a tailor-welded or tailor-rolled blank so that a thicker or more formable grade is placed exactly where the split occurs; or move to a fundamentally more capable process — hydroforming, which supports the sheet with pressurised fluid and dramatically improves strain distribution, or warm forming for aluminium and magnesium, where a modest temperature rise raises both $n$ and total elongation. Finally, active blank-holder control (variable force in time and around the periphery, or segmented binders) can hold the process inside a window that a fixed blank-holder force cannot.
(a) Bending without orange peel. Specify a fine grain size — ASTM grain size number 7 or finer, i.e. a mean grain diameter well below about 30 µm, and certainly small compared with the sheet thickness so that many grains lie through the section. Orange peel is a purely metallurgical surface roughening: each grain deforms according to its own crystallographic orientation, so on a free (unsupported) tensile surface each grain moves out of plane by a different amount and the surface acquires a texture at the scale of the grains. Nothing about the bending operation causes it and no lubricant cures it. Halving the grain diameter halves the amplitude of the roughening, so the specification is simply a fine, uniform, recrystallised grain structure — obtained by a controlled cold-reduction and annealing schedule. Avoiding a prior heavy anneal that grows the grains, and avoiding the critical strain range that causes abnormal grain growth in a subsequent anneal, are the practical corollaries.
(b) Bending to zero radius. Bending to a nominally zero inside radius demands the maximum possible tensile ductility on the outer fibre, and the property that controls it is not total elongation but reduction of area at fracture. The minimum bend radius expressed in thicknesses is related to the tensile reduction of area $r$ (as a fraction) by
$$\frac{R}{t} = \frac{1}{2r} - 1$$
so $R/t \to 0$ requires $r \to 0.5$, that is a reduction of area of at least about 50 %. Specify accordingly: a high reduction of area, a low yield-to-tensile ratio, high total elongation, and above all cleanliness — low sulphur and low oxide inclusion content, with inclusion shape control by calcium treatment, because elongated manganese-sulphide stringers are the classic initiation site for a bend crack. Specify the bend axis transverse to the rolling direction where the geometry permits, since ductility across the stringers is far lower than along them. A fully annealed, low-carbon, non-work-hardened temper is required; any prior cold work consumes exactly the ductility that the zero-radius bend needs.
(c) Greatest resistance to permanent deformation in service. Permanent deformation in service means yielding under load, so the property to maximise is the yield strength $S_y$ — and specifically the yield strength in the finished part, which includes the increase produced by the forming operation itself and by any bake-hardening. Specify a high-strength grade: a high-strength low-alloy (HSLA) or a bake-hardenable steel, a dual-phase steel, or a precipitation-hardened or strain-hardened aluminium temper. Where stiffness rather than strength governs the deflection — and for most sheet panels it does — note that the elastic modulus is essentially fixed for a given alloy system, so resistance to elastic deflection must be bought with geometry (gauge, beads, curvature) rather than with material. The trade-off is direct and unavoidable: the properties that resist permanent deformation are the opposite of those wanted in (a) and (b), because a high yield strength implies a high yield-to-tensile ratio, lower ductility, a larger minimum bend radius and greater springback.
