Bolt Tensile and Shear Stress Area
Tensile stress area, shear area, and stiffness relationships for threaded fasteners under axial and shear loading.
Definition
The tensile stress area of a threaded fastener is the effective cross-sectional area at the threaded section, accounting for the thread geometry. It is used to calculate axial stress in the bolt under preload and external load.
Equations
Tensile stress area (ISO metric):
A_t = (π/4) · (d₂ + d₃)² / 4
Simplified (approximate):
A_t ≈ 0.7854 · (d − 0.9382·p)²
where d₂ = pitch diameter, d₃ = minor diameter
d = nominal diameter, p = pitch
Bolt axial stress:
σ_bolt = F / A_t
Bolt shear stress (threads in shear plane):
τ = V / A_s
Torque-preload relationship:
T = K_t · F_pre · d
(K_t ≈ 0.20 for dry, 0.15 for lubricated)
Bolt stiffness:
k_b = A_t · E / L_grip
Joint stiffness factor (n):
n = k_b / (k_b + k_m)
(k_m = clamped member stiffness)Variables
| Symbol | Definition | Units |
|---|---|---|
| A_t | Tensile stress area | mm² |
| A_s | Shear area at thread | mm² |
| d | Nominal bolt diameter | mm |
| p | Thread pitch | mm |
| F | Axial force (preload or external) | N |
| V | Shear force | N |
| σ_bolt | Bolt axial stress | MPa |
| τ | Bolt shear stress | MPa |
| T | Applied torque | N·mm |
| K_t | Torque coefficient (friction) | — |
| F_pre | Preload force | N |
| k_b | Bolt stiffness | N/mm |
| k_m | Clamped member stiffness | N/mm |
| n | Joint stiffness factor | — |
| L_grip | Clamped grip length | mm |
| E | Young's modulus | GPa |
Common Tensile Stress Areas (Metric)
| Size | Pitch (mm) | A_t (mm²) | Notes |
|---|---|---|---|
| M6 | 1.0 | 20.1 | |
| M8 | 1.25 | 36.6 | |
| M10 | 1.5 | 58.0 | |
| M12 | 1.75 | 84.3 | |
| M14 | 2.0 | 115 | |
| M16 | 2.0 | 157 | |
| M20 | 2.5 | 245 | |
| M24 | 3.0 | 353 | |
| M30 | 3.5 | 561 |
Notes and Limitations
- The torque-preload relationship has significant scatter (±20-30%) due to friction variability
- For critical joints, specify preload directly and verify by bolt elongation or ultrasonic measurement
- The joint stiffness factor n determines what fraction of external load the bolt carries before separation
- Member stiffness k_m depends on the clamped geometry — use the frustum (cone) model or FEA
Related Knowledge
See the Joints & Interfaces Knowledge category for bolted joint theory, preload, separation and the Huth fastener flexibility model. See How to Check a Bolted Joint for the practical workflow.