About this calculator
The joint stiffness factor Cm decides how much of an external load fluctuation the bolt actually feels — the single most important parameter in the fatigue behaviour of a preloaded joint. This calculator derives it from first principles: the bolt is modelled as an elastic rod, and the clamped plates as the material inside the 30° compression cone that spreads out from the bolt head. Enter the bolt diameter, clamped thickness, and the Young’s moduli of bolt and clamped material, and you get kb, km, Cm, and the load share carried by the bolt.
The value of Cm feeds directly into the joint diagram and fatigue strength calculators, replacing the generic 0.25 assumption with a number for your actual geometry.
Theory and equations
Both springs follow the axial stiffness of an elastic member, k = AE/l. For the bolt, the full nominal cross-section over the clamp length lm is used:
The clamped parts are not compressed uniformly — the head pressure spreads into the material as a cone with a flank angle of about φ = 30°. The model represents this zone as an equivalent hollow cylinder: the cone starts at the washer-face diameter d2 = 1.5 d and widens to d3 = d2 + lm tan 30°, and the effective area uses the mean diameter minus the bolt hole:
The stiffness factor and the bolt’s share of any external load fluctuation follow as:
For standard steel-on-steel proportions Am comes out near 3 Ab, which is where the familiar rule of thumb Cm ≈ 0.25 (“the bolt sees a quarter of the load”) originates.
Worked example: M20 bolt, 30 mm steel stack
Inputs: d = 20 mm, lm = 30 mm, E = 210 GPa for both bolt and plates.
- Bolt stiffness: Ab = (π/4) × 20² = 314.2 mm², so kb = 314.2 × 210 000 / 30 = 2.20 GN/m.
- Cone geometry: d2 = 1.5 × 20 = 30 mm; d3 = 30 + 30 × tan 30° = 47.32 mm; mean 38.66 mm.
- Effective clamped area: Am = (π/4)(38.66² − 20²) = 859.7 mm² — about 2.7 times the bolt area.
- Clamped stiffness: km = 859.7 × 210 000 / 30 = 6.02 GN/m.
- Stiffness factor: Cm = 2.20/(2.20 + 6.02) = 0.268 — the bolt carries 26.8% of any load fluctuation, close to the classical 1/4.
Swap the plates for aluminium (Em = 70 GPa) and km drops to a third: Cm jumps to about 0.52, i.e. the bolt suddenly absorbs half of every load cycle — a dramatic difference for fatigue life that the generic 0.25 assumption would completely miss.
Assumptions and limitations
- Single-cone model with fixed 30° flank angle and washer-face diameter 1.5 d; VDI 2230 uses more refined cone/sleeve combinations for extreme geometries.
- Bolt stiffness uses the nominal shank cross-section; long threaded portions or waisted shanks make the bolt more elastic (lower Cm — conservative to ignore for fatigue).
- No gaskets: a soft gasket dominates the clamped stiffness and invalidates the cone model.
- The cone must fit inside the part — flanges narrower than d3 reduce the effective area.
Frequently asked questions
Why is Cm ≈ 0.25 such a common assumption?
Standard steel-on-steel proportions give clamped parts roughly three times stiffer than the bolt, hence kb/(kb+km) ≈ 1/4. Thin stacks, big clearance holes, or soft materials shift it substantially — compute the real value when fatigue matters.
What exactly is the compression cone?
The head pressure spreads into the clamped material at roughly 30° from the bearing face; only the material inside that cone is effectively compressed. The cone (surface diameter 1.5 d, widening with depth) defines the effective area for km.
Is low Cm good or bad?
Good for fatigue — the bolt sees less of the load fluctuation. Long elastic bolts and stiff clamped parts lower Cm; gaskets raise it sharply.
What if bolt and plates are different materials?
Use the real moduli: steel ~210 GPa, aluminium ~70 GPa, cast iron ~100–120 GPa. An aluminium housing under a steel bolt roughly doubles-to-triples Cm versus all-steel — often decisive in fatigue-critical aluminium assemblies.