Load paths and safety factors: every link in the chain

The link nobody checked

• Most machine failures happen at a link nobody checked: a key, a weld, a bolt. Tracing the load path from source to ground lists every link that needs a calculation.
• For the pump drive: motor torque, gear mesh, shaft, bearings, housing, flange bolts, base frame, floor. Each arrow is an interface, and each interface can fail.

A factor of safety you can explain

• The factor of safety is $n = \text{strength}/\text{stress}$. Build it from named uncertainties: load 20% high, material 15% weak, model 10% optimistic.
• They multiply: $n = 1.2 \times 1.15 \times 1.1 = \mathbf{1.52}$. A single "use 2" hides what it covers.

A duty cycle becomes one load

Forces along the path

• The motor gives $T = 300/8 = \mathbf{37.5}$ N·m at the 51 mm pinion, so the tooth force is $W_t = 2T/d = \mathbf{1471}$ N and $W_r = W_t\tan20° = \mathbf{535}$ N.
• The total, 1565 N, is reacted by the bearings: 1.02 kN and 0.55 kN for a gear 70 mm from one bearing on a 200 mm span.

Torque multiplies, speed divides

• Power is conserved along the path, less losses: $P = T\omega$. The pinion's 37.5 N·m at 1440 rpm and the output's 300 N·m at 180 rpm both carry $\mathbf{5.65}$ kW.
• So each link after the gear sees eight times the torque. The output key, not the motor key, is the one that needs checking.

Worked example: the key on the output shaft

• 300 N·m on the 30 mm shaft pushes the key sideways with $F = 2T/d = 20$ kN. An 8 × 7 mm key, 40 mm long, shears over $8 \times 40$ mm²: $\tau = \mathbf{62.5}$ MPa.
• It crushes against half its height, $40 \times 3.5$ mm²: $\sigma = \mathbf{142.9}$ MPa. Key steel at 350 MPa yield and $n = 1.52$ allows 132.9 in shear and 230 in crushing: it passes both.

Three load-path slips

Recap