25-Nav-A6 Advanced Strength of Materials (25-Mec-A6) · May 2016

Question 3 of 6: Sheet-Metal Forming — Forming Limit Diagram and Bendability

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

Notes on this paper

Reference texts: Kalpakjian & Schmid, Manufacturing Engineering and Technology, 8th ed.; Groover, Fundamentals of Modern Manufacturing, 7th ed.; Hibbeler, Mechanics of Materials, 10th ed.; Hibbeler, Engineering Mechanics: Statics, 14th ed.; Shigley's Mechanical Engineering Design, 11th ed.

Check: the exam is headed "98-Mar-A6, Design and Manufacture of Machine Elements" and Part A (Q1–Q3) is entirely machining/casting/forming content with zero naval-architecture material. Solved here as the paper actually printed; reference texts above are chosen for the real content. Part B (Q4–Q6) is genuine strength-of-materials/machine-design content.

Check: Q5 states the shaft modulus as "E = 30 ksi," which is off by three orders of magnitude for steel (actual E ≈ 29,000–30,000 ksi = 30×10⁶ psi); the shaft's computed self-weight, w = γA = 0.283 × (π/4)(3²) = 2.00 lb/in, comes out to a clean round number confirming the 3-in-diameter reading, and the printed "30 ksi" is treated as a dropped exponent (E = 30×10⁶ psi used throughout).

Question 3: Sheet-Metal Forming — Forming Limit Diagram and Bendability (equal value, Part A)

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.

(i)(a) How to analyze the failure. Perform a circle-grid (or laser-etched grid) strain analysis on the failed part: a grid of circles etched on the blank before forming deforms into ellipses, and measuring each ellipse's major and minor axis gives the major strain $\varepsilon_1$ and minor strain $\varepsilon_2$ at any location, including the fracture site. Plotting $(\varepsilon_2,\varepsilon_1)$ against the material's Forming Limit Diagram (FLD) shows directly whether the local strain state exceeded the Forming Limit Curve (FLC), and where relative to the curve the split occurred.

(i)(b) Likely strain state at fracture. Drawbeads deliberately lock the flange so essentially no material can draw in from outside the die cavity; the part must therefore accommodate the punch stroke entirely by thinning in place. This locked-flange condition is the classic route to a plane-strain state (minor strain $\varepsilon_2 \approx 0$) along much of the part, and plane strain sits at the lowest, most restrictive point of the FLC — both equal-biaxial stretching ($\varepsilon_1\approx\varepsilon_2$) and drawing ($\varepsilon_2<0$) tolerate more major strain before splitting. A drawbead-locked stretch failure is therefore most likely to have occurred at or very near plane strain.

(i)(c) Two FLD-based remedies (shape unchanged). (1) Reduce the drawbead restraining force slightly (a shallower bead or lower bead lock-up force) so a controlled, small amount of material is allowed to draw in from the flange; this shifts the local strain path away from plane strain toward the safer, negative-minor-strain (drawing) side of the FLD without changing the part's final geometry. (2) Redistribute strain by changing the friction/lubrication condition over the die and punch radius (e.g., a lower-friction die lubricant or a lower blank-holder pressure where a holder is also used) so that strain that was concentrating at one plane-strain "hot spot" is spread over a larger area of the part, lowering the peak local strain below the FLC without altering the die shape.

(i)(d) If neither remedy works. Change the blank material to a higher-formability grade (higher strain-hardening exponent $n$ and higher total elongation, which raises the FLC itself); introduce an intermediate process anneal between forming stages if the part is formed in more than one hit, restoring ductility consumed by strain hardening; or move to warm forming, which raises ductility directly, or to a fundamentally different process (e.g., hydroforming) that can distribute strain more favourably than a rigid punch and die.

(ii)(a) Bending without orange peel. Specify a fine, uniform grain size. Orange peel (a visibly rough, pebbly surface after forming) results from individual coarse surface grains deforming somewhat independently and becoming visible; a fine grain size keeps the surface smooth because many small grains average out any grain-to-grain deformation difference.

(ii)(b) Bending to zero radius. Specify high ductility — high total elongation, a high strain-hardening exponent $n$, and (for the same alloy) the softest available temper. Bending to a radius approaching zero imposes very large local strain at the outer fibre, and only a highly ductile sheet can accommodate that strain without cracking.

(ii)(c) Greatest resistance to permanent deformation in service. Specify high yield strength $S_y$. Resistance to permanent (plastic) set is governed directly by yield strength — the higher $S_y$ is, the higher the applied stress must be before permanent deformation begins. This directly trades off against (b): a temper hardened for high $S_y$ is generally less ductile and harder to bend tightly, which is why formed parts are often bent soft and then heat-treated to raise strength afterward.

Topic: Forming limit diagrams and sheet-property selection for forming versus service performance.

Key relations: FLD axes are major strain $\varepsilon_1$ (vertical) vs. minor strain $\varepsilon_2$ (horizontal); the FLC is lowest (most restrictive) near plane strain ($\varepsilon_2=0$) and rises toward both drawing ($\varepsilon_2<0$) and equal-biaxial stretching ($\varepsilon_2>0$).

Why this works: A locked flange (drawbeads fully engaged) removes the draw-in degree of freedom that would otherwise let the part relieve strain by pulling material inward, forcing the strain path toward the FLC's weakest point; every remedy that keeps the shape fixed works by giving some of that degree of freedom back, either through the bead itself or through friction/lubrication.

Common pitfall: assuming equal-biaxial stretching (not plane strain) is always the critical condition for a "pure stretch" part — drawbead-locked stretch forming specifically pushes the strain state toward the FLC's minimum, which is plane strain, not equal-biaxial.

Source: Kalpakjian & Schmid, Manufacturing Engineering and Technology, Ch.16 (sheet-metal forming, forming limit diagrams) and Ch.7 (mechanical behaviour, bendability).

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