Mechanical Design Failure Mode Cases

Name the part and the failure mode

Mechanical-design questions are easiest as failure-mode selection problems. A shaft, gear, spring, bearing, bolt, weld, key, or pressure vessel does not have one generic strength — it can yield, buckle, fatigue, wear, loosen, fracture, deflect too far, or fail by contact stress. The equation follows the failure mode. * A rotating shaft carrying a pulley sees torque plus bending, so fatigue and combined stress may govern. A slender compression member is a buckling problem even with a high-strength alloy. A bolt group under eccentric load is a load-distribution problem before it is a shear calculation.

Worked problem: bearing life

A ball bearing has a basic dynamic load rating C = 26 kN and operates at equivalent load P = 5.2 kN. The rating-life model is L10 = (C/P)^3 (in millions of revolutions). So L10 = (26/5.2)^3 = (5.0)^3 = 125 million revolutions. If the prompt then doubles the load to P = 10.4 kN, L10 = (26/10.4)^3 = (2.5)^3 = 15.6 million — exactly one-eighth of the original, because life scales with the inverse cube of load. This sensitivity is a favorite FE concept: small load increases cause large life reductions.

Worked problem: gear tangential force

A spur gear transmits P = 8 kW at N = 900 rpm on a pitch diameter D = 150 mm (pitch radius r = 0.075 m). Torque: omega = 2pi900/60 = 94.25 rad/s, so T = P/omega = 8000/94.25 = 84.9 N·m. The tangential (pitch-line) force is W_t = T/r = 84.9 / 0.075 = 1132 N. The classic miss is dividing by the diameter instead of the radius, which halves the force to 566 N. If the answer choices differ by a factor of two, recheck radius versus diameter.

Worked problem: thin-wall pressure vessel

A closed cylindrical tank has internal gauge pressure p = 1.5 MPa, inside radius r = 0.40 m, and wall thickness t = 8 mm. For a thin wall (r/t = 50 > 10), hoop stress sigma_h = pr/t = 1.5e6 * 0.40 / 0.008 = 75 MPa, and longitudinal stress sigma_l = pr/(2t) = 37.5 MPa. Hoop is exactly twice longitudinal, which is why cylindrical vessels split along their length, not around their circumference. Use the pressure difference across the wall as the driver when an external pressure is also given.

Springs, fasteners, and review style

Spring rate is geometry-sensitive: for a helical compression spring, k = Gd^4 / (8D^3*N), so wire diameter d enters to the fourth power. A deflection answer is not done if the prompt also asks for safety — check the shear stress with the Wahl factor. Fasteners need a load path: a centered shear connection may divide load among bolts equally, but an eccentric bracket adds secondary shear from the moment, and a preloaded joint behaves differently from a pin.

When you miss a design item, log the earliest wrong assumption — static yielding chosen over fatigue, stress concentration ignored, wrong equivalent bearing load — because the next similar problem is recognized by failure mode before any lookup.

Worked problem: fatigue with an endurance limit

A rotating shaft carries a steady transverse load, so each point sees fully reversed bending — the mean stress is zero and the alternating stress equals the bending amplitude. The endurance limit of the steel is Se = 220 MPa and the alternating stress is sigma_a = 110 MPa. 0**. If the prompt instead added a mean stress, you would need a Goodman or Soderberg line combining sigma_a/Se and sigma_m/Su — not a plain yield check. The defining FE trap is comparing the alternating stress to yield strength and getting a falsely high safety factor, because a part can survive a static load yet still fail in fatigue.

Recognizing 'rotating' or 'cyclic' or 'reversed' in the prompt should immediately switch you from a static model to an endurance model.

Worked problem: eccentric bolt group

A bracket bolt pattern carries a downward load P applied at an eccentricity e from the bolt-group centroid. Each bolt sees two contributions: a primary shear P/n shared equally among the n bolts, and a secondary shear from the moment M = Pe, distributed in proportion to each bolt's distance r from the centroid (F = Mr / sum(r^2)). The two vectors add geometrically, and the most heavily loaded bolt is the one where primary and secondary shear most nearly align. Dividing the load equally among all bolts — ignoring the moment entirely — understates the critical bolt force and is the standard distractor.

The lesson: whenever a load is offset from the fastener-group centroid, it is a moment-distribution problem first and a shear calculation second.