5.4 Mechanical Design: Shafts, Bearings, Gears, Springs, and Fasteners
Key Takeaways
- Mechanical Design & Analysis is one of the largest FE Mechanical domains; component formulas deserve timed practice.
- Shaft stress combines bending and torsion: τ = Tr/J for torsion, σ = Mc/I for bending, with τ = 16T/(πd³) for a solid shaft.
- Rolling-bearing life follows L = (C/P)^a, with exponent a = 3 for ball bearings and 10/3 for roller bearings.
- Spur-gear speed ratio is the inverse tooth ratio; transmitted load Wt = T/r drives bending (Lewis) and contact stress.
- Helical spring rate k = Gd⁴/(8D³N), and bolt/weld joints are sized on tensile-stress area, preload, and shear throat.
Component design is applied mechanics
FE Mechanical design questions are not separate from statics, dynamics, and mechanics of materials — they embed those topics inside hardware: shafts, bearings, gears, springs, fasteners, couplings, and welded or pressure-containing parts. Start every problem by identifying the component, the load path, and the governing failure mode (yield, fatigue, buckling, wear, fracture), then pull the matching relation from the Machine Design section of the FE Reference Handbook.
Shafts carry torque and bending. For a solid circular shaft of diameter d, torsional shear stress is τ = Tr/J = 16T/(πd³), where J = πd⁴/32 is the polar moment of area; bending stress is σ = Mc/I = 32M/(πd³). The angle of twist is φ = TL/(GJ). Shafts almost always face combined bending and torsion, so a ductile shaft is checked with the maximum-shear-stress or distortion-energy theory, and cyclic loading pulls in fatigue. Stress concentration at fillets, keyways, and shoulders raises local stress by the factor Kt (use σmax = Kt·σnom).
Bearings and gears
Rolling-element bearing life is dominated by load. The rated life in millions of revolutions is:
L = (C/P)^a
where C is the basic dynamic load rating, P is the equivalent radial load, and the exponent a = 3 for ball bearings and a = 10/3 for roller bearings. Because of the cube law, doubling the load on a ball bearing cuts life by a factor of 2³ = 8. This sensitivity is the most-tested bearing idea on the FE.
Spur gears transmit motion through meshing teeth. Speed and torque relations:
| Relationship | Formula |
|---|---|
| Speed ratio | n₂/n₁ = N₁/N₂ (inverse of tooth counts) |
| Pitch-line velocity | V = πdn (d = pitch diameter) |
| Transmitted (tangential) load | Wt = T/r = (power)/V |
| Bending stress (Lewis) | σ = Wt·Pd /(F·Y) |
The smaller gear (pinion) spins faster; the larger gear delivers more torque. Gear teeth fail in bending fatigue at the root (Lewis equation) and surface contact/pitting (Hertzian contact stress). A worked check: a pinion with 20 teeth driving a 60-tooth gear gives a 3:1 reduction, so output speed is one-third of input and output torque is roughly tripled (minus friction).
Springs, bolts, and welds
Helical compression springs store energy elastically. The spring rate is:
k = Gd⁴ / (8D³N)
where G is the shear modulus, d the wire diameter, D the mean coil diameter, and N the number of active coils. Note the strong dependence: rate scales with d⁴ and inversely with D³, so a small wire-diameter change moves stiffness dramatically. Maximum shear stress in the wire is τ = 8FD·Ks/(πd³) (Ks corrects for direct shear/curvature). Deflection is δ = F/k.
Bolted joints carry load by preload in tension or by shear across the shank. A bolt's tensile capacity uses the tensile-stress area At, not the nominal diameter: σ = F/At. Proper preload (often 75–90% of proof strength) keeps a tension joint from separating and improves fatigue life by carrying most of the external load through stiffness sharing. 707h·L)**. For eccentric weld groups, add the primary (direct) and secondary (torsional, τ = Tr/J of the weld group) shear vectorially. Identify whether a fastener sees tension, single shear, or double shear before choosing the area.
Kinematics of mechanisms and design judgment
Mechanical Design & Analysis also folds in the kinematics of mechanisms — linkages, cams, and gear trains that convert motion. A foundational tool is Gruebler's (Kutzbach) equation for planar mechanism mobility (degrees of freedom):
M = 3(L − 1) − 2J₁ − J₂
where L is the number of links (including ground), J₁ is the number of one-DOF (pin/slider) joints, and J₂ is the number of two-DOF joints. A standard four-bar linkage has L = 4 and four pin joints, giving M = 3(3) − 2(4) = 1 — a single input fully determines the motion. The Grashof condition then tells whether a link can fully rotate: if the sum of the shortest and longest links is less than or equal to the sum of the other two (S + L ≤ P + Q), at least one link rotates continuously (a crank).
Gear trains multiply ratios in series: the overall speed ratio is the product of the individual mesh ratios, and a compound train lets a small package reach a large reduction. For cams, the follower's displacement, velocity, and acceleration come from differentiating the cam profile — abrupt acceleration changes cause shock, so smooth (e.g., cycloidal) profiles are preferred.
** A shaft that whirls is a critical-speed problem, not a static-stress problem; a bolt that loosens is a preload problem; a gear that pits is a contact-stress problem. The FE rewards selecting the right relationship over grinding a long calculation, so train yourself to name the failure mode before touching the calculator. When a problem gives more data than one formula needs, that surplus is usually a cue to the failure mode being tested — torque plus speed plus a life rating points to bearings; alternating plus mean stress points to fatigue; pressure plus thin wall points to hoop stress.
A solid circular shaft of diameter d = 40 mm transmits a torque of 500 N·m. What is the maximum torsional shear stress?
The equivalent load on a ball bearing is doubled. By approximately what factor does its rated L10 life change?
A helical spring's wire diameter d is increased by 19% (factor 1.19) while all else stays the same. Roughly how does the spring rate change?