Compression Spring Design: Spring Rate and Shear Stress (DIN 2089)

A helical compression spring is defined by three geometric parameters: the wire diameter d (the cross-section of the wire itself), the mean coil diameter D (measured to the centreline of the wire, not the outside or inside), and the number of active coils n (the coils that actually deflect under load — dead coils at each end do not count). The ratio c = D/d is called the spring index and is the single most important non-dimensional parameter in spring design: a low index (tight coils, c < 4) makes a stiff spring that is hard to coil and prone to stress concentration; a high index (c > 20) produces a floppy, unstable spring. Practical designs target c = 6–12.

End conditions affect total coil count but not active coils: squared-and-ground ends (the standard for precision springs) add two inactive coils, giving a total Nt = n + 2. Solid (block) length is then Lc = Nt · d for ground ends, or (Nt + 1) · d for unground — the calculator applies these per Shigley Table 10-1 / DIN 2089-1 conventions.

The spring rate k (N/mm) is the load required to deflect the spring by one millimetre. It follows directly from the stored-elastic-energy integral over the coil geometry and depends only on the shear modulus of the wire, the geometry, and the number of active coils. Patented carbon wire and spring steel use G = 81 500 MPa; stainless steel (1.4310) uses G = 73 000 MPa; chrome-vanadium (51CrV4) uses G = 81 500 MPa.

Because the rate depends on D³ and d⁴, small changes in diameter have a large effect on stiffness. Doubling the wire diameter (at constant D and n) increases the rate 16-fold; doubling the coil diameter reduces the rate 8-fold. This means wire diameter is the primary tuning knob for stiffness, while coil diameter and active coils offer secondary adjustment.

When a helical spring is loaded axially, the wire is in torsion — but not uniform torsion. The inner surface of each coil sees a higher stress than the outer surface because of the curvature of the wire (a stress-concentration effect), and a direct transverse shear component acts across the wire cross-section. The Wahl correction factor K combines both effects into a single multiplier that depends only on the spring index c. It replaces the simpler direct-shear factor Ks = 1 + 0.5/c used in some older texts, which can underestimate the peak stress by 10–20 % for springs with c < 8.

The corrected shear stress τK is compared against the admissible shear stress τ_zul = 0.5 · Rm(d), where Rm(d) is the wire's minimum tensile strength at the actual wire diameter d, read from the EN 10270 tabulated curves for the selected grade. The safety factor SF = τ_zul / τK should be at least 1.3 for a well-designed spring in quasi-static loading. Note that τ_zul increases as wire diameter decreases because thinner wire has a higher tensile strength — so a smaller-diameter wire is not inherently weaker; the allowable scales with it.

DIN 2089-1 and EN 13906-1 specify the allowable shear stress for a helical compression spring as τ_zul = 0.5 · Rm(d), where Rm(d) is the minimum tensile strength of the wire at diameter d, taken from the EN 10270 material tables for the selected wire grade: EN 10270-1 for patented carbon wire (grade SH) and spring steel, EN 10270-2 for chrome-vanadium (51CrV4, grade FDCrV), and EN 10270-3 for stainless steel (1.4310, grade NS). The tensile strength decreases as wire diameter increases — for example, patented wire at d = 2 mm has Rm ≈ 1 960 MPa, falling to about 1 580 MPa at d = 8 mm — so the allowable shear stress is wire-size dependent and must be read from the table rather than assumed constant.

A static safety factor SF = τ_zul / τK ≥ 1.3 is the typical DIN 2089 target for quasi-static or slowly varying loads. For cyclic loading, DIN 2089 supplements the static check with the Zimmerli–Goodman fatigue analysis, which computes a separate fatigue safety factor based on the alternating and mean stress components mapped onto the Haigh diagram. The /springs calculator performs both checks.