Spring design for the PE Mechanical exam

Helical compression springs are the exam's standard spring problem: given the geometry and material, find the rate, the stress, the deflection, or how springs combine. Four relations carry almost all of it — and one correction factor people forget.

Spring rate

For a round-wire helical spring:

k = G·d⁴ / (8·D³·N)

• G — shear modulus, d — wire diameter, D — mean coil diameter, N — number of active coils.

Two details bite: D is the mean coil diameter (not OD or ID), and N is active coils (not total — end coils don't deflect). Deflection then follows from δ = F/k.

Stress — don't skip the correction factor

The handbook gives the shear stress in the wire as:

τ = K_s · 8·F·D / (π·d³), with K_s = (4C + 2) / (4C − 3)

where C = D/d is the spring index. That correction accounts for the wire's curvature plus the direct shear, and it's always > 1 — so using the uncorrected 8FD/πd³ under-predicts the stress. A tighter index (small C) means a larger correction.

You'll also see this as the Wahl factor in textbooks — a slightly different expression that lands within about 1%. Either way the point is the same, but work from the form the handbook prints: it's the one you'll be able to find on exam day.

Combining springs — series is softer

This is the one most people get backwards because it's the opposite of the resistor intuition:

• Series (springs end-to-end): compliances add → softer, 1/k_eq = Σ 1/k_i.
• Parallel (side by side, sharing the load): stiffer, k_eq = Σ k_i.

A long stack of identical springs in series is more flexible, not less.

Energy

A spring stores U = ½·k·δ² = ½·F·δ — useful when a problem routes through energy (e.g., impact or a spring-mass system feeding a natural-frequency question).

Where people lose points

• Skipping the correction factor — the uncorrected formula under-predicts the real wire stress.
• Series vs. parallel backwards — series springs are softer (compliances add), the opposite of the reflex guess.
• OD/ID instead of mean D — the rate and stress both use the mean coil diameter.
• Total coils vs. active coils — only active coils contribute to the rate.

Drill it

Practice these on real problems — the tutor walks any you miss, grounded in the worked solution. Where this lives: MD&M Module 5 — Components & Assemblies (this area); springs feed the natural-frequency work in Mechanical vibrations.