Springs Calculator — Rate, Wahl Stress, Buckling & Fatigue (EN 13906 / DIN 2089)

The spring rate is derived from the stored-energy integral over the coil geometry. The shear modulus G comes from the selected wire grade: patented carbon wire, general spring steel and chrome-vanadium (51CrV4) all use G = 81 500 MPa, and stainless steel (1.4310) uses G = 73 000 MPa. The Wahl correction factor K combines the torsion-curvature effect (the Wahl term) with the direct-shear component, replacing the simpler Ks factor used in early textbooks.

The corrected shear stress τK at BOTH load positions is then compared against the wire-diameter-dependent allowable τ_zul = 0.5·Rm(d), where Rm(d) is the minimum tensile strength for the selected wire grade at that diameter. EN 10270-1 is read as a sampled curve and interpolated; EN 10270-2 and -3 publish Rm in banded diameter ranges and are read as steps, so the value used is the one the standard specifies for the band containing your wire. The verdict takes the smaller of the two positions and also refuses any design whose length at either position falls to the solid length.

Lateral buckling is not a geometry limit — it is a deflection limit. A slender spring stands up perfectly well until it is compressed past a critical deflection, and how far that is depends on how the spring is held. Shigley's end-condition constant α covers the four cases: 0.5 seated flat between parallel plates, 0.707 with one end flat and one pivoted, 1.0 with both ends pivoted, and 2.0 with one end clamped and the other free. A spring is absolutely stable while L0 < π·√(2(E−G)/(2G+E))·D/α — 2.58·D/α for the carbon-steel wires here, the 2.63 usually quoted being that expression evaluated for Shigley's own steel constants. Past that it buckles at y_cr = L0·C₁·[1 − √(1 − C₂/λ²)] with λ = α·L0/D, C₁ = E/(2(E−G)) and C₂ = 2π²(E−G)/(2G+E). The calculator computes that deflection for your seating and compares it against the deflection your loads actually produce. A spring running on a guide rod or in a guide bore is laterally restrained and the check is skipped.

The fundamental surge frequency is the speed at which the spring resonates axially, exciting a standing wave along its length. Operating at or near this frequency causes coil-clash, fatigue damage and, in the limit, spring surge. The formula below applies to a spring seated between two parallel plates (both ends fixed — the standard design case).

Zimmerli's data gives a single TEST POINT that a spring survives indefinitely: (Ssa, Ssm) = (241, 379) MPa unpeened, or (398, 534) MPa shot-peened, both largely independent of the wire alloy. That point is not the failure line — it is one point on it. The Goodman line through it is anchored by the ultimate shear strength Ssu = 0.67·Sut, which gives the fully-reversed shear endurance Sse = Ssa / (1 − Ssm/Ssu); the fatigue factor is then measured along a load line through the origin, because the alternating and mean stresses both scale with the applied force.

The first-cycle yield check must be carried alongside it. Shot peening raises the endurance point but not the shear yield strength Ssy — 0.45·Sut for carbon and alloy spring steel (the Shigley Table 10-6 cold-drawn value; alloy wire is held below the table's 0.50) and 0.35·Sut for stainless — so a peened or high-mean-stress spring is frequently limited by yielding on its very first compression rather than by fatigue. The calculator reports both factors and takes the smaller as the governing one. A governing factor ≥ 1.5 is the usual design target; below 1.0 the spring fails.