Bolted Joint Stiffness Calculation Example

The bolted joint stiffness calculation decides how an external load is shared between the bolt and the clamped members. Get the joint stiffness factor wrong and your fatigue and separation checks are wrong with it. Here is the standard derivation as a reviewable worksheet.

1. Model the bolt as two springs in series

The bolt within the grip is an unthreaded shank plus a threaded length, so its stiffness is the series combination:

k_s := A_b · E_b / L_s (unthreaded shank)
k_t := A_s · E_b / L_t (threaded portion)
k_b := 1 / (1/k_s + 1/k_t) (combined bolt stiffness)

2. Member stiffness — Shigley frustum model

The clamped members are modeled as pressure cones with a 30° half-angle. The Shigley closed-form for a member of the frustum stack is:

k_m := (π · E_m · d · tanα) / ln( ((2·L·tanα + D − d)·(D + d)) / ((2·L·tanα + D + d)·(D − d)) )

where d is the hole diameter, D the bearing diameter, L the grip length, and α = 30°. Carrying units through this expression by hand is exactly where mistakes appear—see unit-aware engineering calculations.

3. Joint stiffness factor

C := k_b / (k_b + k_m)

C is the fraction of an external tensile load taken by the bolt; (1 − C) is relieved from the clamped members. A lower C means a more efficient, more fatigue-friendly joint.

4. Share the external load

With preload F_i and external load P:

ΔF_b := C · P — added bolt tension
ΔF_m := (1 − C) · P — lost clamp force
F_b := F_i + ΔF_b — total bolt tension
F_m := F_i − ΔF_m — remaining clamp force (must stay > 0 to avoid separation)
σ_b := F_b / A_s — bolt stress under load

Worked numbers

For an M16 steel bolt (E = 200 GPa) clamping aluminium members (E = 70 GPa), grip 50 mm, preload 40 kN, external load 20 kN, a typical result is C ≈ 0.3: the bolt sees ~30% of the external load and the members are relieved of ~70%. The bolt tensile stress under load comes out near 320 MPa.

Make it reviewable

Because every term depends on the last, this is a poor fit for a spreadsheet and a strong fit for a worksheet where the derivation, units, and assumptions sit together (why).

Open Calculeaf and build this bolted joint check with real units in a few minutes.