Bolted joints are commonly used in machinery to securely connect two or more components together. Compared to other types of fasteners, they are favored because of their strength in supporting loads, ease of disassembly, and reliability.

Bolt clamping force is an essential factor in the design of bolted joints. Read more to learn about clamping force calculations, methods of indication, and the factors affecting its value.

## Clamping Force Calculation

A bolted joint connects two structural elements using a bolt and a nut. The bolt is inserted through a hole in both elements and tightened by turning the nut along the threads. The tightening process generates a compressive force that squeezes the elements together and secures the joint.

This compressive force that the structural elements experience is called the clamping force or clamping load. In a way, it is a measure of the tightness of the bolted joint.

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The figure below shows the clamping force acting on the two materials being clamped together by a bolt and nut assembly. Calculating the value of the clamping force depends on the loading condition on the joint.

### Preload Calculation

At zero external load condition, the clamping force is equal to the preload or the initial load of the bolted joint. In a properly designed joint, the value of the preload should be enough to keep the joint securely fastened and prevent it from loosening or separating under the expected loading conditions it will be subjected to.

When deciding on the preload, there is not a one-size-fits-all solution. Several factors must be considered to make an accurate estimate of the required value

However, as a general rule, the preload is 75% of the proof strength for removable fasteners and 90% of the proof strength for permanent fasteners. If the proof strength value is unknown, it can be estimated to be equal to 85% of the yield strength.

### Clamping Force Under Compressive External Load

Under a compressive external load, the members are assumed to have infinite stiffness. Hence, no load is carried by the fastener and the clamping force that the structural elements experience is simply equal to the preload plus the applied compressive load. This can be mathematically expressed as:

Where:

- F
_{c}= clamping force that the structural elements experience [N] - F
_{i}= preload [N] - P = applied load [N]

In general, compressive external loads do not cause bolt failure since the force in the bolt is unaffected by the compressive load. That is, the compressive load is equal to the preload.

### Clamping Force Under Tensile External Load

Under tensile load, the bolt and the members are assumed to have finite stiffness. Following Hookeâ€™s Law, the clamping force on the members can be calculated using the formula:

Where:

- k
_{m}= stiffness of the structural members [N/m] - k
_{b}= stiffness of the bolt [N/m]

In addition, the tensile force on the bolt can be calculated using the formula:

Where:

- F
_{b}= tensile force on the bolt [N]

Note that the formulas above assume that there is no separation happening between the joints. Once the joint separates, then the bolt will carry all the external load, that is, F_{b}=P.

#### Factor Of Safety Against Joint Separation

Bolted joints subjected to tension can typically fail due to joint separation or permanent elongation of the bolt.

The tensile external load required to cause joint separation can be calculated using the formula:

Where:

- P
_{0}= magnitude of the tensile force that would cause joint separation [N]

From the formula above, it follows that the factor of safety against joint separation can be calculated using the formula:

Where:

- n
_{sep}= factor of safety against joint separation [unitless]

#### Factor Of Safety Against Bolt Elongation

For a threaded fastener under tensile load, the axial stress in the threaded region can be calculated using the formula:

Where:

- Ïƒ
_{b}= tensile stress in the boltâ€™s threaded region [N/m^{2}] - A
_{t}= tensile area of the bolt [m^{2}]

From the formula above, it follows that the factor of safety for exceeding the proof strength of the bolt can be calculated using the formula:

Where:

- n
_{p}= factor of safety against elongation or yielding [unitless] - S
_{p}= proof strength of the bolt [Pa]

## Clamping Force Indication

Applying the correct amount of preload clamping force to a bolted joint is crucial to prevent joint failure or damage to the parts being joined or the bolt itself. Over-tightened joints may stress structural elements and lead to breakage, while under-tightened joints may loosen the connection between elements and cause structure separation or fatigue. Therefore, a good tightness indicator is necessary to ensure that the joint is tightened enough.

Because the clamping force of a bolt depends on various factors, such as the diameter and thread pitch of the bolt, the type and thickness of the materials being joined, the torque applied during tightening, and the coefficient of friction between the bolt threads and the mating surfaces, estimating the actual clamping force can be tricky.

In common practice, torque is commonly used as an indicator of the tightness of a bolted joint, which is measured directly from a torque wrench. It’s assumed that the tightening torque is linearly proportional to the bolt preload generated at the joint. This is expressed in the equation:

Where:

- T = tightening torque [N-m]
- K = nut factor [unitless]
- d
_{b}= bolt diameter [m]

The nut factor depends on the material and lubricant used, and is normally provided in ranges. The nut factors of the most common lubricants are shown in the table below.

Standards have also been developed to standardize the suggested tightening torque values to produce the needed bolt clamping force. For SAE bolts, the suggested tightening torque values are shown in the table below.

### SAE Grade 5 Bolt Recommended Tightening Torque

### SAE Grade 8 Bolt Recommended Tightening Torque

Although the torque method is the most common method in practice, the assumption that torque is linearly proportional to the clamping force doesn’t hold in reality since the preload generated under a single torque value varies with friction. Hence, various other methods have been developed to indicate clamping force more accurately.

A piezoelectric load-sensing washer has been developed to directly measure the clamping force exerted by the bolt it is on. It utilizes the capacity of piezoelectric materials to generate voltage when they are subjected to mechanical stress in order to read the electrical output of the filaments and relate it to the clamping force being experienced real-time.

In addition, some other methods that are used to indicate clamping force include the use of ultrasonic sensing, computer-controlled wrench, and strain gages. These methods have different levels of accuracies as shown in the table below.