Bolt Tightening Torque & Preload (K-factor)
Estimates the required installation assembly torque to generate target thread preload tension.
Primary Mathematical Expression
Design Schematic
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| T | Tightening torque | N·m | in·lb | The turning moment applied to tighten the bolt. |
| K | Nut factor | dimensionless | dimensionless | The global friction and thread coefficient (torque coefficient). |
| F_i | Bolt preload | N | lbf | The initial tension clamping force generated in the bolt. |
| d | Nominal diameter | mm | in | The major thread outer diameter. |
Step-by-Step Derivation
- 1
Tightening torque T is required to overcome thread friction, head friction, and stretch the bolt to generate clamp force (preload Fi).
- 2
Rigorous screw thread mechanics (from VDI 2230) state: T = Fi * [ (d2 / 2) * tan(\psi + \rho') + (d_w / 2) * \mu_w ].
- 3
This rigorous form combines thread helix angle (\psi), thread friction angle (\rho'), and washer bearing radius/friction coefficient (d_w, \mu_w).
- 4
To simplify, engineers introduce the dimensionless nut factor K (or torque coefficient) that maps this entire complex geometry.
- 5
Dividing the torque by (Fi * d) defines K: K = T / (Fi * d), which rearranges to: T = K * F_i * d.
Worked Example Calculation
An M12 structural bolt (d = 12 mm = 0.012 m) requires a clamp preload force of 40 kN (40,000 N). Calculate the recommended assembly tightening torque under lightly oiled conditions (K = 0.15).
- •Identify input parameters: d = 12 mm = 0.012 m, F_i = 40,000 N, K = 0.15.
- •Apply the K-factor torque formula: T = K * F_i * d.
- •Compute: T = 0.15 * 40,000 * 0.012 = 72 N·m.
Engineering Assumptions
- •Thread profile matches standard ISO metric or Unified Inch layouts.
- •Nut factor K represents friction conditions accurately.
Design Limitations
- •Nut factor K varies significantly with surface finish, thread tolerances, plating, and lubrication conditions. Using torque alone has a scattering preload variation tolerance of up to \pm 25%.
Academic References & Standards
Systematic Calculation of Bolted Joints
Standard defining complete thread friction and torque mechanics.
Shigley's Mechanical Engineering Design - Fasteners
Textbook outlining nut factors, friction coefficients, and proof strengths.