Reference · Methodology

Bolt Torque & Preload Analysis

ISO 898-1 · VDI 2230 · Shigley's 11th Ed. Ch. 8

Complete methodology for calculating tightening torque, preload force, and safety factor for metric and imperial threaded fasteners. All equations derived from first principles with cited standards.

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[01]Nomenclature

Symbol	Description	Unit (SI)	Unit (Imperial)
d	Nominal bolt diameter	mm	in
p	Thread pitch	mm	—
n	Threads per inch	—	TPI
K	Nut factor	—	—
A_s	Tensile stress area	mm²	in²
S_p	Proof strength	MPa	psi
F_p	Proof load	N	lbf
F_i	Target preload	N	lbf
T	Tightening torque	N·m	ft·lbf
σ_t	Tensile stress	MPa	psi
SF	Safety factor	—	—

[02]Equations

Tensile Stress Area

The tensile stress area accounts for the reduced cross-section at the thread root. The metric formula per ISO 898-1 uses the pitch diameter approximation. The imperial equivalent uses threads per inch with a constant derived from UNS 60° thread form geometry.

Metric (ISO 898-1)

A_s = \frac{\pi}{4}(d - 0.9382 \cdot p)^2

Imperial (UNS 60°)

A_s = \frac{\pi}{4}\left(d - \frac{0.9743}{n}\right)^2

Proof Load

Proof load is the maximum tensile force the bolt can sustain without permanent deformation. It is the product of proof strength (from the bolt grade specification) and tensile stress area.

F_{\text{proof}} = S_{\text{proof}} \times A_s

Target Preload

Target preload is a fraction of proof load, typically 60–90%. The percentage depends on the application, joint criticality, and whether torque control or turn-of-nut method is used.

F_i = \text{target\%} \times F_{\text{proof}}

Tightening Torque

The short-form torque equation per VDI 2230. K is the nut factor (dimensionless torque coefficient) that accounts for thread friction, under-head friction, and thread geometry.

T = K \times F_i \times d

Tensile Stress

Actual tensile stress in the bolt at the target preload.

\sigma_t = \frac{F_i}{A_s}

Safety Factor

Ratio of proof strength to actual tensile stress. Values above 1.0 indicate the bolt is within proof load. Values above 1.5 are typically considered adequate for non-critical joints.

SF = \frac{S_{\text{proof}}}{\sigma_t}

Nut Factor (K) Values

Condition	K Range	Typical K
Dry (unlubricated)	0.18–0.22	0.20
Oiled	0.14–0.18	0.15
Anti-seize compound	0.10–0.14	0.12
Waxed	0.15–0.19	0.17
Cadmium plated	0.11–0.15	0.13
Zinc plated	0.17–0.22	0.20
Molybdenum disulfide (MoS₂)	0.08–0.12	0.10
PTFE (Teflon)	0.09–0.13	0.11

[03]Bolt Grade Data

ISO 898-1 (Metric)

Grade	Proof Strength (MPa)	Yield Strength (MPa)	Tensile Strength (MPa)
4.6	225	240	400
5.8	380	420	520
8.8	580	640	800
10.9	830	940	1,040
12.9	970	1,100	1,220

SAE J429 (Imperial)

Grade	Proof Strength (psi)	Yield Strength (psi)	Tensile Strength (psi)
Grade 2	55,000	57,000	74,000
Grade 5	85,000	92,000	120,000
Grade 8	120,000	130,000	150,000

[04]Worked Example

M10 × 1.5, Class 8.8, Dry Assembly (K = 0.20), 75% of proof.

Step 1: A_s = (π/4) × (10 − 0.9382 × 1.5)² = 57.99 mm²

Step 2: F_proof = 580 × 57.99 = 33,634 N

Step 3: F_i = 0.75 × 33,634 = 25,226 N

Step 4: T = 0.20 × 25,226 × 0.010 = 50.45 N·m

Step 5: σ_t = 25,226 / 57.99 = 435.0 MPa

Step 6: SF = 580 / 435.0 = 1.33

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[05]Assumptions & Limitations

Assumes elastic behavior only; bolt is not torqued beyond proof load
Nut factor K is an empirical approximation; actual value depends on surface finish, lubrication, plating, and thread fit class
Does not account for joint relaxation, embedment loss, or thermal effects
Torque-angle method or stretch-based methods may be more accurate for critical joints
Imperial tensile stress area formula assumes UNS 60° thread form
Proof strength values are for bolt diameters ≤ 16mm (ISO) or ≤ 1" (SAE); larger diameters have reduced values

[06]References

[1]VDI 2230 Part 1 (2015) — Systematic Calculation of Highly Stressed Bolted Joints

[2]ISO 898-1:2013 — Mechanical Properties of Fasteners Made of Carbon Steel and Alloy Steel

[3]Budynas, R.G. & Nisbett, J.K. — Shigley's Mechanical Engineering Design, 11th Ed., Ch. 8

[4]Bickford, J.H. — Introduction to the Design and Behavior of Bolted Joints, 4th Ed.

[5]SAE J429 — Mechanical and Material Requirements for Externally Threaded Fasteners

Related References

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