Compression spring calculator per DIN EN 13906-1
Calculate cold-formed cylindrical helical compression springs per DIN EN 13906-1: wire diameter, coil diameter, number of coils and two working points yield the spring rate, spring forces, shear stresses with correction factor, solid length and buckling safety.
Compression spring calculator (DIN EN 13906-1)
Rm is interpolated from the reference table for EN 10270-1 SH/DH by d (0.07 … 17 mm).
| Point | s [mm] | L [mm] | F [N] | τ [N/mm²] | τk [N/mm²] |
|---|---|---|---|---|---|
| P1 | 32.04 | 87.96 | 300 | 382 | 447.8 |
| P2 | 53.4 | 66.6 | 500 | 636.6 | 746.4 |
Checks
Geometry and solid dimensions
- Total coils nt
- 10.5
- Solid length Lc
- 42 mm
- Minimum gaps Sa
- 6.66 mm
- Minimum usable length Ln
- 48.66 mm
- Usable deflection sn / Fn
- 71.34 mm / 668 N
- Deflection to solid sc / Fc
- 78 mm / 730.4 N
- Remaining travel from s2 to Ln
- 17.94 mm
- Outer dia. De / inner dia. Di
- 36 / 28 mm
Static assessment per DIN EN 13906-1 with uncorrected shear stress against 0.5·Rm (operation) and 0.56·Rm (solid); stress correction per Bergsträßer. Cold-formed springs, linear characteristic, centric axial loading.
Spring characteristic
Formulas and fundamentals
The spring rate of a cylindrical compression spring with a linear characteristic follows from the geometry: R = G·d⁴ / (8·Dm³·n) with shear modulus G, wire diameter d, mean coil diameter Dm and the number of active coils n. Force and deflection are linked by F = R·s; the two working points can be entered as deflection s, force F or spring length L and are converted internally (s = L0 − L).
The shear stress in the wire is τ = 8·F·Dm / (π·d³). Due to the wire curvature the stress is higher at the inner edge of the coil; this is captured by the Bergsträßer correction factor k = (w + 0.5) / (w − 0.75) with the spring index w = Dm/d. The corrected stress τk = k·τ governs the fatigue assessment; following the logic of the standard, the static assessment uses the uncorrected stress because the local stress peak relaxes by setting.
For cold-formed springs: total coils nt = n + 2, solid length Lc = nt·d for closed and ground ends or Lc = (nt + 1.5)·d for closed, unground ends. To prevent coil contact in service, the standard requires the sum of minimum gaps Sa = n·(0.0015·Dm²/d + 0.1·d); for dynamic loading Sa is increased by the factor 1.5. The minimum usable length is Ln = Lc + Sa, the maximum usable deflection sn = L0 − Ln.
The static strength assessment compares the operating stress at point 2 with τperm = 0.5·Rm and the stress of the fully compressed spring (solid force Fc = R·(L0 − Lc)) with τc,perm = 0.56·Rm. The minimum tensile strength Rm depends on the wire diameter and is interpolated linearly from a built-in reference table for spring steel wire EN 10270-1 SH/DH; for other materials Rm is entered directly. The solid-length check ensures the spring does not suffer inadmissible setting when compressed solid during assembly.
The buckling check uses the closed-form approximation of the standard: with a = 1 − G/E and b = 0.5 + G/E, the buckling deflection is sK = L0·(0.5/a)·(1 − √(1 − (a/b)·(π·Dm/(ν·L0))²)). The seating coefficient ν describes how the spring ends are guided (ν = 0.5 for both ends fixed up to ν = 2 for one free end). If the expression under the root becomes negative, the spring is buckling-proof for any deflection. The spring index should stay within the manufacturable range 4 ≤ w ≤ 16 (exceptions up to 18, preferably 7 ≤ w ≤ 10); the tool warns outside it.
Worked example
Given: a cold-formed compression spring with d = 4 mm, Dm = 32 mm, n = 8.5 active coils and L0 = 120 mm made of spring steel wire EN 10270-1 SH/DH (G = 81,500 N/mm²). The working points are given as forces: F1 = 300 N and F2 = 500 N. The spring rate is R = 81,500 · 4⁴ / (8 · 32³ · 8.5) = 9.3635 N/mm; this yields the deflections s1 = 32.04 mm and s2 = 53.40 mm, i.e. the lengths L1 = 87.96 mm and L2 = 66.60 mm.
With w = 32/4 = 8, k = 1.1724. The shear stress at point 2 is τ2 = 8 · 500 · 32 / (π · 4³) = 636.6 N/mm², corrected τk2 = 746.4 N/mm². Solid dimensions: nt = 10.5, Lc = 42 mm, Sa = 6.66 mm, Ln = 48.66 mm. At d = 4 mm the Rm table interpolates to 1745 N/mm², i.e. τperm = 873 N/mm² and τc,perm = 977 N/mm². The solid stress τc = 929.9 N/mm² stays below; both checks are fulfilled.
The buckling check shows the importance of the end seating: with both ends hinged (ν = 1) the buckling deflection is only sK = 27.25 mm, so the spring would buckle at s2 = 53.4 mm. Only with both ends fixed and guided in parallel (ν = 0.5) does the root expression become negative and the spring is buckling-proof for any deflection.
Frequently asked questions
Why are there an uncorrected and a corrected shear stress?
The wire curvature increases the stress at the inner edge of the coil. The Bergsträßer factor k captures this local peak. For the static assessment the standard uses the uncorrected stress because the peak relaxes by setting; for the fatigue assessment the corrected stress τk governs.
What is the solid length Lc and why is it checked?
Lc is the length of the fully compressed spring where all coils touch. During assembly or misuse the spring can be pressed solid; to avoid permanent setting, the solid stress must stay below 0.56·Rm.
What role does the seating coefficient ν play in the buckling check?
ν describes how the spring ends are guided: both ends fixed gives ν = 0.5 (most favourable case), both ends hinged ν = 1, one free end ν = 2. The smaller ν, the larger the admissible deflection before buckling. Buckling-prone springs can be guided on a rod or in a sleeve.
Where does the tensile strength Rm come from and why does it depend on the wire diameter?
Patented cold-drawn spring steel wire gets stronger with decreasing diameter. The tool interpolates reference values of the minimum tensile strength for EN 10270-1 SH/DH between 0.07 and 17 mm; for other wire grades and materials Rm is entered directly. Binding ranges are given in EN 10270.
What is the sum of minimum gaps Sa?
Sa is the minimum gap between the coils at the largest operating deflection required by the standard, so the coils do not touch and the characteristic stays linear. For dynamic loading Sa is increased by the factor 1.5. The minimum usable length is Ln = Lc + Sa.
Does the tool cover the fatigue strength assessment?
No. For dynamic loading the tool calculates the corrected stress range τkh = τk2 − τk1 as the input quantity, but the full fatigue assessment requires the fatigue strength diagrams of DIN EN 13906-1 (depending on material, wire grade and shot peening). Above about 10,000 load cycles such an assessment is required.
Why does it use τperm = 0.5·Rm and not 0.45·Rm?
The tool uses the permissible operating stress τperm = 0.5·Rm (and the solid stress τc,perm = 0.56·Rm) per DIN EN 13906-1; this value is documented in the technical literature (Metallfedern, Table 4.18) with an explicit reference to the standard. Older and general literature such as the Roloff/Matek formula collection uses the more conservative τperm = 0.45·Rm without reference to EN 13906-1. The standard-backed value 0.5·Rm governs here; anyone who deliberately wants a more conservative design can allow for roughly 10 % lower utilisation.