Bolt Preload – Joint Stiffness, Tightening Torque and Clamp Force

Bolt preload is the initial tensile force introduced into a fastener during tightening. The corresponding compressive force clamps the joint members together and is fundamental to the performance of most highly loaded bolted joints.

This page provides engineering methods for estimating bolt preload, bolt and joint stiffness, the distribution of external separating loads, thermal effects, tightening torque and methods of establishing or measuring bolt tension.

See also: Fastener Engineering Tables and Design Data.

Page sections:
Bolt preload | Bolt stiffness | Joint stiffness | External load distribution | Thermal effects | Tightening torque | Combined stress | Preload methods | Preload accuracy

Nomenclature

Symbol	Description	Typical Units
Ab	Effective bolt cross-sectional area	mm2 or m2
At	Bolt tensile stress area	mm2 or m2
d	Nominal bolt diameter	mm or m
Fp	Initial bolt preload	N
Fe	External separating load applied to the joint	N
ΔFb	Increase in bolt load caused by external load	N
ΔFj	Reduction in joint clamping force caused by external load	N
Fb	Total tensile force in bolt	N
kb	Bolt stiffness	N/mm or N/m
kj	Effective stiffness of clamped joint members	N/mm or N/m
C	Bolt load fraction or joint stiffness factor	dimensionless
E	Young's modulus	N/mm2 or N/m2
L	Effective length	mm or m
δ	Elastic deflection	mm or m
T	Tightening torque	N·m
K	Nut factor used in simplified torque-preload equation	dimensionless
μ	Thread coefficient of friction	dimensionless
μc	Bearing-face coefficient of friction	dimensionless
α	Coefficient of thermal expansion when used in thermal equations	K-1
ΔT	Temperature change	K or °C

Bolt Preload

A tightened bolt is stretched elastically while the joint members are compressed. The tensile force in the bolt and the compressive clamping force in the joint are approximately equal and opposite before an external load is applied.

A sufficiently high and controlled preload is generally desirable because it helps maintain joint contact, reduces the alternating component of bolt stress under fluctuating external loads, improves resistance to loosening and can prevent relative movement between the joint faces.

However, preload must remain within the permissible capacity of the fastener, threads and joint. Excessive preload can cause bolt yielding, thread stripping, bearing damage or damage to the clamped components.

Preload Relative to Proof Load

The original RoyMech guidance used a preload of approximately 80% of bolt proof load as a typical value. This is a useful traditional engineering reference point for some non-critical steel bolted joints, but it is not a universal preload requirement.

$$F_p\approx 0.8F_{\mathrm{proof}}$$

Bolt Stiffness

Within the elastic range, a bolt behaves approximately as a tension spring. For a uniform section:

$$\delta =\frac{FL}{EA}$$

Therefore the axial stiffness is:

$$k=\frac{F}{\delta }=\frac{EA}{L}$$

A real bolt normally consists of several sections having different cross-sectional areas. The unthreaded shank, threaded portion and the elastic contribution of the head and engaged threads can therefore be treated as springs connected in series.

$$\frac{1}{k_b}=\sum _i\frac{L_i}{E_i{A}_i}$$

For sections of the same material:

$$\frac{1}{k_b}=\frac{1}{E}\sum _i\frac{L_i}{A_i}$$

The original RoyMech method allows approximately for elasticity in the bolt head and engaged nut/thread region by using effective lengths:

$$L_{\mathrm{se}}=L_s+0.4d_s$$ $$L_{\mathrm{te}}=L_t+0.4d_r$$

where L_s is the shank length, L_t is the threaded length within the effective bolt spring, d_s is the shank diameter and d_r is the thread root diameter.

Clamped Joint Stiffness

Determining the stiffness of the clamped members is generally more difficult than determining bolt stiffness because the compressive load spreads through the material beneath the bolt head, nut and washers.

For a simple uniform compressed section:

$$k_j=\frac{EA}{L}$$

For several joint components in series:

$$\frac{1}{k_j}=\sum _i\frac{L_i}{E_i{A}_i}$$

The original RoyMech notes use a rough preliminary approximation in which the compressed region is treated as an annulus having an outside diameter of approximately 2.5 times the bolt diameter and an inside diameter approximately equal to the bolt diameter.

External Load Distribution Between Bolt and Joint

When an external tensile force tends to separate a preloaded joint, the whole external load is not normally added directly to the existing bolt preload.

Before joint separation, part of the external load increases bolt tension and the remainder reduces compression in the clamped members.

Define the bolt load fraction:

$$C=\frac{k_b}{k_b+k_j}$$

The increase in bolt load caused by an external separating load F_e is:

$$\Delta F_b=C{F}_e$$

The corresponding reduction in joint compression is:

$$\Delta F_j=(1-C)F_e$$

The total bolt force while the joint remains closed is therefore:

$$F_b=F_p+C{F}_e$$

The remaining joint compression is:

$$F_j=F_p-(1-C)F_e$$

Joint Separation Load

Separation begins when the remaining compressive force in the joint reaches zero:

$$F_j=0$$

Therefore the idealised external separation load is:

$$F_{e,\mathrm{sep}}=\frac{F_p}{1-C}$$

Thermal Effects on Bolt Preload

If the bolt and clamped components have similar coefficients of thermal expansion and experience approximately the same temperature change, thermal effects on preload may be small.

If their coefficients of thermal expansion or temperatures differ, differential expansion can increase or decrease bolt preload.

For an idealised bolt and joint having the same effective length L and uniform temperature change ΔT:

$$\Delta L_j={\alpha }_{j}L\Delta T$$ $$\Delta L_b={\alpha }_{b}L\Delta T$$

The differential free expansion is:

$$\Delta L_{\mathrm{diff}}=({\alpha }_j-{\alpha }_b)L\Delta T$$

The equivalent stiffness of the bolt and joint acting in series is:

$$k_{\mathrm{eq}}=\frac{k_b{k}_j}{k_b+k_j}$$

For this simple model, the change in bolt preload is:

$$\Delta F_T=k_{\mathrm{eq}}({\alpha }_j-{\alpha }_b)L\Delta T$$

and:

$$F_{p,\mathrm{hot}}=F_p+\Delta F_T$$

Setting Preload by Tightening Torque

The most widely used approximate relationship between tightening torque and bolt preload is:

$$T=K{F}_{p}d$$

where:

• T = tightening torque;
• K = empirical nut factor;
• F_p = desired preload;
• d = nominal bolt diameter.

Legacy Nut-Factor Reference Values

The following values are retained from the original RoyMech page as indicative historical engineering values.

Steel Thread / Surface Condition	Indicative K
As received – stainless on mild or alloy steel	0.30
As received – mild or alloy steel on similar material	0.20
Cadmium plated	0.16
Molybdenum-disulphide grease	0.14
PTFE lubrication	0.12

Example – Simplified Torque Estimate

For:

• Target preload F_p = 20 kN
• Nominal diameter d = 10 mm = 0.010 m
• Nut factor K = 0.20

$$T=0.20\times 20000\times 0.010=40N·m$$

This is an estimate, not a guarantee that tightening to 40 N·m will produce exactly 20 kN preload.

Thread and Bearing Friction Torque

A more detailed tightening model separates the torque required to advance the thread from the torque lost through friction in the thread and beneath the rotating nut or bolt head.

The original equation uses:

• F_p = desired bolt preload;
• p = thread pitch;
• d_m = mean thread diameter;
• μ = coefficient of thread friction;
• μ_c = coefficient of bearing-face or collar friction;
• α = half included thread angle, 30° for ISO metric threads;
• r_c = effective collar friction radius.

Friction normally consumes most of the applied tightening torque. Consequently relatively small changes in thread or bearing-face friction can produce substantial changes in achieved preload.

This is the principal limitation of torque-controlled tightening.

Legacy Simplified Friction Equation

The original RoyMech page gives a further approximation for a standard 60° thread by assuming:

• μ_c = μ;
• d_m ≈ 0.92d;
• α = 30°;
• r_c ≈ 0.625d.

This reduces to:

$$T\approx F_p(0.159p+1.156\mu d)$$

Friction reference data are available on the Coefficient of Friction page.

Bolt Stress During Tightening

During torque tightening the bolt is subjected simultaneously to axial tension from the preload and torsion generated by thread friction. Consequently the stress state during tightening is more severe than that produced by axial preload alone.

The original RoyMech expression uses:

• F = bolt tensile force;
• T_f = thread torque;
• A_r = thread-root area;
• d_r = thread-root diameter.

For some calculations the tensile stress area A_t can be used with an equivalent stress diameter:

$$d_t=\sqrt{\frac{4A_t}{\pi }}$$

See Screw Thread Stress Areas and Thread Stripping for tensile stress-area calculations.

Methods of Establishing Bolt Preload

There are several ways of establishing or measuring preload. Selection depends on the required accuracy, fastener size, access, joint geometry, number of fasteners and installation conditions.

Torque-Controlled Tightening

Torque-controlled tightening is convenient and inexpensive but the achieved preload is strongly affected by friction. Where accurate preload is required, the torque procedure should be based on representative torque/clamp-force testing or validated installation data.

Turn-of-Nut / Angle Tightening

After the joint has reached a defined snug condition, the nut or bolt is rotated through a specified additional angle. This controls fastener extension more directly than torque alone.

The required rotation depends on the bolting system, grip length and joint configuration. Compressible coatings, settlement and an inconsistent snug condition can affect the result.

Direct Tension Indicators

Direct tension indicators provide an indication of achieved bolt preload rather than relying solely on tightening torque. Structural DTI systems use specially formed washers whose projections compress as bolt tension increases.

The original RoyMech notes described checking the remaining gap using a feeler gauge. That remains useful historical information, but the installation and acceptance procedure should be taken from the specification applicable to the actual DTI system.

Bolt Elongation Measurement

If the effective bolt stiffness is known, preload can be inferred from measured elastic elongation:

$$F_p=k_b\delta$$

For a uniform bolt section this becomes:

$$F_p=\frac{EA\delta }{L}$$

Ultrasonic Measurement

Ultrasonic instruments can infer bolt extension from changes in acoustic time-of-flight. They are particularly useful where bolt tension must be measured without direct access to both ends of the fastener.

Accuracy depends on calibration, bolt material, geometry, temperature, surface condition and operator procedure.

Hydraulic Bolt Tensioning

A hydraulic tensioner applies a direct axial tensile force to the bolt. While the bolt is stretched, the nut is run down; hydraulic pressure is then released and part of the applied tension remains as bolt preload.

Hydraulic tensioning avoids the high torsional stress associated with torque tightening and is particularly useful for large studs and heavily loaded flange joints.

Allowance must be made for load transfer and preload loss when hydraulic pressure is released.

Multi-Jackbolt Tensioners

Multi-jackbolt tensioners use a number of smaller jackbolts around a main nut body. The individual jackbolts can be tightened using much lower torque than would be required to tighten a conventional large nut directly.

The original RoyMech page specifically identified Superbolt-type systems as a practical option for large bolts requiring otherwise very high tightening torques.

Strain Gauges and Instrumented Fasteners

Strain gauges, load-indicating bolts and other instrumented fasteners can provide direct or indirect measurement of bolt tension. These methods are useful where preload verification is more important than installation simplicity.

Thermal Tensioning

The original RoyMech notes also describe thermal tensioning. The bolt is heated, installed and the nut brought into position. Cooling contracts the bolt and generates tensile preload.

For a simple free thermal-strain approximation:

$$\epsilon =\frac{\sigma }{E}$$

and:

$$\epsilon =\alpha \Delta T$$

giving:

$$\Delta T=\frac{\sigma }{E\alpha }$$

This is an idealised relationship. In practice, heat transfer to the joint, temperature gradients, installation time, relaxation and uncertainty in the final grip condition make thermal preload control difficult.

Legacy Structural Bolting Information

Under the historical BS 4604 approach, high-strength friction-grip bolts were installed to specified minimum tensions using methods including:

• calibrated torque tightening;
• part-turn / turn-of-nut tightening;
• direct tension measurement.

The original RoyMech notes emphasised that torque-controlled results could show substantial variation even when a defined installation procedure was followed. They also noted the importance of establishing the correct snug condition before turn-of-nut tightening.

Those observations remain useful engineering principles even though the referenced structural-bolting standard is now legacy.

Indicative Bolt Preload Accuracy

The following figures are retained from the original RoyMech page. They are useful for comparing the approximate relative capability of different preload-setting methods, but they should not be interpreted as guaranteed accuracy limits.

Preload Method	Original RoyMech Indicative Accuracy
Operator feel	±35%
Torque wrench	±25%
Angle tightening / turn-of-nut	±15%
Load-indicating washer	±10%
Measured bolt elongation	±5%
Hydraulic bolt tensioning	approximately ±1% to ±10%
Strain gauges / ultrasonics	approximately ±1%
Multi-jackbolt tensioner	approximately ±5%

Preload Loss After Tightening

The preload immediately after tightening is not necessarily the preload that remains in service. Possible causes of preload loss include:

• embedding and flattening of surface asperities;
• joint settlement;
• gasket relaxation;
• creep of soft joint materials;
• paint or coating compression;
• temperature changes;
• plastic deformation beneath the bolt head, nut or washer;
• thread or bearing-surface yielding;
• vibration and relative movement where the joint is inadequately clamped.

These effects should be considered where maintaining a minimum service clamp force is important.

Practical Bolt Preload Design Considerations

• Establish the clamp force required to prevent separation, slip, leakage or other unacceptable joint behaviour.
• Select a fastener and property class capable of providing the required preload with adequate margin.
• Check bolt tensile stress, thread capacity and bearing stresses beneath the head, nut and washers.
• Determine bolt and joint stiffness where external separating loads are significant.
• Allow for preload scatter when specifying the target installation preload.
• Account for settlement, embedding, gasket relaxation and other expected preload losses.
• Consider fatigue using the fluctuating bolt load rather than simply applying the full fluctuating external joint load to the bolt.
• Use controlled friction conditions when tightening by torque.
• Where preload is critical, consider direct tension measurement, elongation measurement, hydraulic tensioning or another method less dependent on friction.

Torque / Clamp-Force Testing

ISO 16047:2005 with Amendment 1:2012 specifies conditions for torque/clamp-force testing of threaded fasteners and related parts. It provides a controlled test framework for determining torque and clamp-force characteristics rather than assuming a universal nut factor.

This is particularly relevant when a tightening torque is to be established for a specific fastener, coating and lubrication system.

Related RoyMech Bolt Preload Calculators

Use these calculators for bolt preload, tightening torque, proof load and clamp-force checks.

Calculator	Use
Bolt Preload Calculator	Estimate bolt preload and clamp force from tightening torque.
Bolt Tightening Torque Calculator	Calculate tightening torque from target preload, diameter and nut factor.
Bolt Proof Load Calculator	Calculate proof load and compare it with the proposed preload.
Washer Bearing Pressure Calculator	Calculate bearing pressure beneath a washer, nut face or bolt head from clamp force.
Bolt Tensile Stress Calculator	Calculate tensile stress from axial bolt load.
Bolted Joint and Fastener Calculators	View all RoyMech bolt and fastener calculators.

Related Threaded Fastener References

• Fastener Engineering Tables and Design Data – main RoyMech threaded-fastener reference.
• Bolt Proof Load, Tensile Load and Tightening Torque – ISO fastener loads and approximate tightening torque data.
• Metric Bolt, Screw and Nut Strength – fastener property classes, tensile strength, yield strength and proof stress.
• Screw Thread Stress Areas, Thread Stripping and Engagement – tensile stress areas and thread stripping calculations.
• Bolted Joint Design – joint behaviour under tensile and shear loading.
• Bolted Joint Stiffness – additional methods for estimating clamped-member stiffness.
• ISO Metric Screw Thread Dimensions – ISO metric thread dimensions and reference data.
• Screw Thread Tolerances – equations and tables for thread tolerance calculations.
• Screw Thread Locking Systems – methods of resisting unwanted fastener loosening.

External Engineering References

1. ExcelCalcs – Detailed Bolted Joint Calculations – legacy spreadsheet-based bolted-joint calculation reference.
2. Bolt Science – specialist technical information on bolted joints and fastener tightening.
3. Superbolt – multi-jackbolt tensioning systems and technical information.
4. Rotabolt – bolt tension monitoring systems.
5. Futek Bolt Torque Calculator – external bolt torque calculation resource.
6. SmartBolts – visual direct-tension-indicating fasteners.
7. Applied Bolting – direct tension indicators and structural bolting information.
8. SureBolt – ultrasonic bolt tension measurement information.
9. MITCalc – engineering calculation software including bolted-joint calculations.
10. University of Western Australia – Loads in an Elastic Bolted Assembly – detailed engineering notes on bolted-joint loading.

Standards References

• ISO 16047:2005 + Amd 1:2012 – Fasteners – Torque/clamp force testing.
• ISO 898-1:2013 – Mechanical properties of fasteners made of carbon steel and alloy steel – Part 1: Bolts, screws and studs with specified property classes.
• BS EN 14399 series – High-strength structural bolting assemblies for preloading.
• BS EN 14399-9:2018 – High-strength structural bolting assemblies for preloading – System HR or HV – Direct tension indicators for bolt and nut assemblies.
• BS 4604 – legacy British Standard referenced by the original RoyMech page for high-strength friction-grip structural bolting. Retained here for historical engineering reference rather than current specification.

ISO 16047 remains the published ISO torque/clamp-force testing standard and includes Amendment 1:2012. For critical tightening specifications, use the applicable current standard together with fastener-manufacturer data and, where appropriate, testing of the actual fastener, lubricant and coating combination.

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