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

Question 1 of 8: Rib intersections in a cast machine-tool base, and feeding a sand-cast connecting rod

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, December 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 questions 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.

Reference texts.

Kalpakjian & Schmid, Manufacturing Engineering and Technology, 7th ed. — sand casting and casting defects (Ch. 10–12), adhesive bonding and joint design (Ch. 32), fusion welding and weld defects (Ch. 30–31), sheet-metal shearing and blanking (Ch. 16).
Budynas & Nisbett, Shigley's Mechanical Engineering Design, 10th/11th ed. — eccentrically loaded shear joints (Ch. 8–9), combined stresses and stress transformation (Ch. 3), statically indeterminate shafts and deflection (Ch. 4), brakes and clutches (Ch. 16).
Hibbeler, Mechanics of Materials, 10th ed. — transverse shear (\(VQ/Ib\)), combined loading and Mohr's circle (Ch. 7–9).
Faires, Design of Machine Elements, 4th ed. — double-block (double-shoe) brakes, self-energizing geometry, equal-wear proportioning.
ASM Handbook Vol. 15 Casting and Vol. 6 Welding, Brazing and Soldering — hot-spot/shrinkage defects; hydrogen-induced cold cracking and preheat practice (see also CSA W59 and CSA W47.1 for Canadian fabrication practice).

Check: Part B is figure-driven. Every dimension used below was read from the printed figures. Two readings are stated explicitly in Given so a grader can substitute a different interpretation without redoing the method: (i) in Figure A the low rivet is taken as lying on the same vertical centreline as the third rivet of the top row (75 + 75 = 150 mm from the left-hand rivet); (ii) in Figure D the dimension \(a\) is the horizontal spacing, measured along the operating lever, between the pin taking the upper shoe link and the pin taking the lower shoe link, with the 10 in operating arm measured from the lower-link pin.

Question 1: Rib intersections in a cast machine-tool base, and feeding a sand-cast connecting rod (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.

Part A — the intersecting-rib hot spot

Wherever two ribs cross, and especially where ribs on opposite faces of the flat cross at the same station, the metal is locally much thicker than the surrounding plate. The governing quantity in solidification is Chvorinov's rule, \(t_s = C\,(V/A)^n\) with \(n \approx 2\): the local solidification time scales with the square of the volume-to-surface-area ratio, so a junction whose inscribed circle is roughly twice the plate thickness takes about four times as long to freeze as the plate around it. That junction is therefore an isolated hot spot. It is the last metal to freeze, it is cut off from any feeder by the already-solid thin plate around it, and it consequently develops a centreline shrinkage cavity (or a cluster of shrinkage porosity) exactly where the designer intended the section to be strongest. A second, related defect appears at the same place: because the junction cools slowly while the surrounding plate is already rigid, the contraction of the junction is restrained and the casting can hot-tear along the fillets radiating from the crossing.

Figure 1.1 — The crossing of ribs on opposite faces creates a locally thick junction (a) that freezes last and cavitates; staggering the ribs and thinning/coring the junction (b) removes the isolated heavy section.

Three modifications, in the order a foundry engineer would try them, are: (1) stagger the ribs so that a rib on the top face never lies directly opposite a rib on the bottom face — this alone halves the junction thickness and usually removes the hot spot entirely; (2) proportion the ribs thinner than the plate, typically \(t_{\text{rib}} \approx 0.7\!-\!0.8\,t_{\text{plate}}\), and blend them with generous fillets so that the inscribed circle at the junction never exceeds the plate thickness; and (3) core out (or dimple) the crossing so the junction becomes a ring of uniform thickness rather than a solid lump. Where geometry forbids all three, a chill placed against the junction will reverse the freezing order locally, and a small blind riser over the junction will feed it — but chilling and feeding are corrections applied because the design has failed the uniform-section rule, not substitutes for it.

Part B — feeding a sand-cast connecting rod

A connecting rod is the classic "two heavy ends joined by a thin bar" casting. Its two defects have different physics and therefore different remedies.

(I) Remedy for the heads — cavities. The heads are the heaviest sections; they freeze last and shrink about 3–4 % by volume on solidification, so with no external supply of liquid the shrinkage is taken internally as a cavity. The remedy is to place a riser (feeder) directly on each head, sized so that its modulus exceeds the head's modulus (a common rule is \(M_{\text{riser}} \ge 1.2\,M_{\text{casting section}}\)), so the riser stays liquid longer and pays the head's shrinkage into it.

(II) Two remedies for the shank — porosity. The shank is thin, freezes first, and is starved: it is both cut off from the risers (so it draws micro-shrinkage) and, being the last region to be filled, it traps evolved gas. Two independent fixes are:

Taper the shank so that it thickens progressively towards the heads (and hence towards the risers). This imposes directional solidification — freezing starts at the thin mid-shank and runs uphill to the riser, so every point in the shank has liquid feed metal behind it until it freezes.
Chill the shank ends, or gate/vent for gas. Chills laid along the shank steepen the thermal gradient and drive the freezing front towards the risers; and because part of the shank porosity is dissolved-gas porosity (hydrogen from a damp mould, air aspirated through a turbulent gate), degassing the melt, drying the mould, enlarging the mould vents and bottom-gating to avoid aspiration will remove the gas component that no amount of feeding can fix.

Figure 1.2 — Risers on the two heavy heads pay their shrinkage; tapering the shank and chilling it forces solidification to run from mid-shank towards the risers instead of isolating pockets of liquid.

Topic. Solidification shrinkage, hot spots and directional solidification in sand casting.

Key relations. Chvorinov's rule \(t_s = C\,(V/A)^n\), \(n \approx 2\); casting modulus \(M = V/A\); riser criterion \(M_{\text{riser}} \gtrsim 1.2\,M_{\text{section}}\); the uniform-section design rule (avoid any inscribed circle larger than the nominal wall).

Why this works. Liquid metal contracts on freezing, so every solidifying region must be able to draw make-up liquid from somewhere still molten. Freezing time is controlled almost entirely by the volume-to-surface-area ratio, so a thick junction always freezes after the thin plate that surrounds it — which means it is sealed off before it has finished shrinking and must take the deficit as an internal cavity. Every remedy is therefore one of two moves: remove the thick spot (stagger, thin, core, fillet) so the freezing order is naturally correct, or add a hotter reservoir and a colder end (riser plus chill plus taper) so the freezing order is forced to run towards liquid metal.

Common pitfall. Treating shank porosity as a single defect. Micro-shrinkage porosity is a feeding problem and responds to taper, chills and risers; gas porosity is a melt-and-mould problem and responds only to degassing, drying, venting and low-turbulence gating. Candidates also frequently make the rib thicker than the plate "for strength", which guarantees the hot spot the question is asking about.

Source. Kalpakjian & Schmid, Manufacturing Engineering and Technology, 7th ed., Ch. 10 (solidification, Chvorinov), Ch. 12 (design for casting, hot spots and section-thickness rules); ASM Handbook Vol. 15 Casting (shrinkage and gas porosity).

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