About this calculator
The joint diagram (also called the joint force–deflection diagram) is the central mental model of bolted-joint design under dynamic loading. A preloaded bolt and the plates it clamps behave as two springs sharing a common interface: when an external working load P acts on the joint, the bolt is stretched a little further while the clamped parts are relieved of some compression. This calculator quantifies that split — it computes the preload Fi, the bolt load fluctuation Pb, the clamped-part fluctuation Pm, and the resulting bolt stress amplitude, and draws the force–deflection diagram to scale from your inputs.
The reason this matters: the bolt only feels a fraction (typically ~25%) of the external load variation, which is what makes preloaded joints so resistant to fatigue. Getting the preload and stiffness ratio right is usually more effective than choosing a bigger bolt.
Theory and equations
The load sharing is governed by the joint stiffness factor, the ratio of bolt stiffness kb to total stiffness:
Working backward from the requirement that the clamped parts must retain a minimum residual clamping force Fm at the peak of the load cycle:
P is the largest external load the joint can carry while keeping the clamping force at or above Fm. The bolt’s share Pb drives the fatigue check via the alternating stress:
where σe is the endurance strength of the threaded fastener. The diagram drawn below the results shows all of these quantities geometrically: the bolt line rising with slope kb, the clamped-part line falling with slope km, and the external load opening the gap between them.
Worked example: M16, class 8.8
Inputs: M16 coarse (At = 156.67 mm²), class 8.8 (Rp0.2 = 640 MPa), initial stress 70% of yield, required minimum clamping force Fm = 30 kN, stiffness factor Cm = 0.25, endurance strength σe = 43 MPa.
- Preload: Fi = 0.70 × 640 × 156.67 = 70.2 kN.
- Clamped-part fluctuation: Pm = 70.2 − 30 = 40.2 kN.
- Bolt fluctuation: Pb = 40.2 × 0.25/0.75 = 13.4 kN — the bolt sees only a quarter of the total.
- Total admissible external load: P = 13.4 + 40.2 = 53.6 kN.
- Fatigue check: σat = 13 400/(2 × 156.67) = 42.8 MPa — just inside the 43 MPa endurance limit, so the joint passes for infinite life.
Try raising the preload slider to 90%: the admissible external load grows to over 70 kN, but the stress amplitude grows with it and exceeds the endurance limit — the calculator flags it immediately. This trade-off between clamping capacity and bolt fatigue is exactly what the joint diagram is for.
Assumptions and limitations
- Linear-elastic springs; the external load is assumed to act at the bolt head and nut planes (load introduction factor n = 1). Loads introduced inside the clamped stack reduce the bolt’s share further.
- Concentric loading and clamping — no prying or bending effects.
- Embedding losses (preload relaxation from surface flattening) are not deducted; allow for them when specifying the assembly preload.
- Cm must be estimated separately — use the joint stiffness factor calculator or the common default of 0.25.
Frequently asked questions
Why does only part of the external load reach the bolt?
Bolt and clamped parts share the load in proportion to their stiffnesses. Because the clamped plates are usually about three times stiffer than the bolt, the bolt typically sees only ~25% of the load fluctuation — the rest shows up as relief of the clamping pressure.
What is the joint stiffness factor Cm?
Cm = kb/(kb + km) — the bolt’s share of an external load fluctuation. The default 0.25 corresponds to clamped parts three times stiffer than the bolt; compute your actual value with the stiffness calculator.
What if Fm exceeds Fi?
The joint is infeasible — the clamped parts cannot give back more compression than the preload put in. Increase the preload (higher fraction, bigger bolt, stronger class) or relax the minimum clamping requirement.
Stress amplitude vs. stress range — which one matters?
The range Δσ = Pb/At is peak-to-peak; the amplitude σat = Pb/(2At) is the variation about the mean. Bolt endurance limits are conventionally quoted as amplitudes, so the check uses σat.
Where does the endurance strength value come from?
For rolled-thread steel bolts a common size-dependent estimate is σe = 0.75(180/d + 52) MPa — around 43–50 MPa for common sizes. The fatigue strength calculator applies it automatically.