# Bolted Joint Design

Bolted Joints are detachable connections which are primarily used to facilitate assembly or when there is a requirement for components to be freely disassembled as part of a maintenance and inspection schedule. They are also utilised when permanent joining methods, such as welding, are unsuitable. The purpose of the bolt is to clamp two or more components together by threading into either a nut or tapped hole in the final clamped component in the stack. The bolt itself only experiences a fraction of the load applied to the clamped components, assuming it is designed correctly. The reason for this will be explained in detailed in this article as it is not necessarily intuitive at first glance. Bolted joints are used extensively in the Automotive, Aerospace, Civil Construction, Oil and Gas, Defence, Nuclear, Rail and Materials Handling sectors.

### 1.0 Bolted Joint Applications

This article considers a bolted joint to be a structural connection which is designed to resist forces and moments but also must not be susceptible to prying, fatigue, fretting and corrosion. The designer must consider the application and determine which failure mechanisms apply.

For example, badly designed joints can induce significant prying forces which must be added to the applied bolt tensile forces. These prying forces can be sufficient for the joint to separate and if separation occurs, the bolt forces will rise proportionally with the load applied to the joint. Consequently, the benefit of pre-loading the joint, in the first place, is lost. Pre-loaded joints should be designed to remain clamped throughout their design life. However, edge separation resulting from an applied bending moment may be tolerated and this will be explained in more detail in subsequent sections.

When designing machine components which experience variable loading the stress concentrations at the thread root and the radius under the bolt head need to be considered and techniques such as the Goodman Relation can be employed to ensure that the bolts survive the design life [aw. cit.].

When elastic micro-slip gives rise to a phenomenon known as fretting, tribological aspects need consideration such as surface finish and heat treatment processes which alter the surface properties [aw. cit.].

In the Oil and Gas sector hydrogen induced stress corrosion cracking or HISC is a major design concern due to the chemical composition of produced fluids and other environmental factors, such as cathodic protection requirements. To mitigate this phenomenon the bolt material hardness needs restricting to prevent any embrittlement caused when external hydrogen atoms diffuse into the crystalline matrix of the low alloy steel and reduce ductility around surface micro cracks [aw. cit.].

These failure mechanisms are beyond the scope of this article. However, they will be covered succinctly to inform the reader that simply designing for forces and moments alone may not be sufficient to prove a bolted joint is fit for purpose.

### 2.0 Calculating Bolt Group Forces and Stresses

One of the first tasks when designing a bolted joint is to establish the joint configuration and subsequently the forces experienced by each bolt. Initially the bolt size is usually estimated and then checked by calculation. It is usually the clamped component geometry which defines the bolt pattern and number of bolts. For the purposes of calculation, it can be assumed that there are four types of loading that a bolted joint can experience. These four types are as follows:-

1) Direct Tension Loading acting through the centre of the bolt group.

2) In Plane Shear Loading acting through the centre of the bolt group.

A bolted joint can experience one or all of these types of loading concurrently. If we assume the principle of linear superpositioning, which is considered conservative [aw. cit.], then the bolt forces can be calculated for each type of loading independently and then summed to give a total force. It is important to note that externally applied forces do not necessarily translate into forces experienced by the bolt. If a bolt is torque tightened in such a way that a pre-load force is generated in the bolt shank then the bolt usually only experiences a fraction of the applied load. Also, a pre-loaded bolted joint will resist shearing forces by friction grip and subsequently external shearing forces will not be translated to the bolt shank. This concept is explained in detail in the following section.

If a moment is applied directly to the clamped plates of a single bolt joint or prying action is significant in a multi-bolt joint, then an additional calculation may be required. It may be necessary to check that any reduction in plate stiffness caused by the clamped parts bending does not adversely affect the performance of the joint. This is covered thoroughly in the German design standard VDI 2230 and the designer should use the approach in this standard. This concept is not covered in this article.

#### 2.1 Direct Tension Loading acting through the centre of the bolt group

When a tensile force acts through the centroid or centre of a bolt group then it is generally assumed that the force is distributed equally over the total number of bolts in the group. However, this simplification assumes that the bolted joint clamped components are infinitely stiff and the load is transferred directly through the clamped components to each bolt. The designer should be aware that quite often this is not the case and prying can arise if the clamped components deform significantly. When the clamped components deform enough for prying to occur the joint material which is outstanding from the furthest bolt from the centroid is placed into compression which gives rise to a prying force. This prying force must be added to the calculated axial bolt force to maintain equilibrium. Alternatively, if the joint clamped components are designed using single curvature bending i.e. neglect the outstanding material compressive forces, then the prying forces can be ignored but at the expense of thicker joint plates [aw. cit.].

The equation below defines the bolt force for this type of loading:-

$$F = F_t/n - Eq. 2.1.0$$

Where :-

$$F_t$$ = Externally applied tensile force acting on bolted joint.
$$n$$ = Total number of bolts in bolted joint.

#### 2.2 In Plane Shear Loading acting through the centre of the bolt group

This type of loading occurs when external forces act in plane to the faying surfaces and gives rise to shear through the bolt shank or threaded cross section. As with direct tension loading the external applied force is shared evenly over the total number of bolts in the group.

The equation below defines the bolt force for this type of loading:-

$$F = F_s/n - Eq. 2.2.0$$

Where :-

$$F_s$$ = Externally applied shear force acting on bolted joint.
$$n$$ = Total number of bolts in bolted joint.

#### 2.3 Eccentric Loading applied out of plane to the bolt group

This type of loading occurs when external forces act in plane to the faying surfaces and gives rise to shear through the bolt shank or threaded cross section. As with direct tension loading the external applied force is shared evenly over the total number of bolts in the group.

Sometimes bolt patterns are unavoidably arbitrary and taking moments to resolve bolt forces becomes impractical. Two methods are employed by various steel construction institutions around the world. These are the Elastic method, which assumes a triangular distribution of bolt forces [aw. cit.] and the Plastic Method which assumes that each row of bolts achieves its full design strength. The latter method relies on the ductility of the joint components and is more complex to use for the none structural engineer. Subsequently, it is beyond the scope of this article but if the reader is interested, further explanation is given in reference [aw. cit.].

#### 2.3.1 Elastic Method

In the UK it is common practice to assume a triangular distribution of bolt forces which are all in tension. They react against a compression zone which sits somewhere near the lower bolts in the joint. It is common practice when designing beam to column connections in buildings to assume the compression zone is in line with the compression flange of the beam [aw. cit.].

In this article, we will assume the centroid or neutral axis of the bolt group lays in the centre of the group, contrary to the above. This implies that there are as many bolts in tension as there are in compression. This can be shown to be a conservative assumption as the bolt which experiences the greatest tension is further away from the compression zone than the centre of the joint. As the resistance to an applied moment is defined by the bolt group moment of inertia, the further the bolt is away from the neutral axis then the greater its contribution to the inertia value by the square relationship to distance.

Using this assumption, it can be shown that a bolted joint behaves in a similar manner to a beam in bending or a shaft in torsion when external moments are applied. Just as with a beam or shaft the bolts furthest from the centroid or neutral axis of the joint experience the largest force when a moment is applied. If we extend this analogy and postulate that the beam bending and shaft torsion formulas can be rewritten to give bolt forces then the following expressions are derived :-

$$M/I = \sigma/y - Eq. 2.3.1.0$$

Which rearranges to :-

$$\sigma = My/I - Eq. 2.3.1.1$$

Where :-

$$M$$ = Externally applied moment
$$I$$ = Second Moment of Area
$$\sigma$$ = Bending Stress
$$y$$ = Vertical distance from neutral axis

However, we are interested in finding the force at each bolt which is analogous to finding the force at a given distance from the neutral axis of a beam in bending. As Stress is defined as force per unit area, equation 2.3.1.1 can be expressed in terms of F or Force as:-

$$F = MAy/I - Eq. 2.3.1.2$$

Where :-

$$F$$ = Force at bolt position y from the neutral axis
$$A$$ = Bolt cross-sectional area

As the moment and bolt position values are prescribed at the design stage we need to find the bolt group moment of inertia, I to enable us to analytically determine the applied force at each bolt. If we assume that the bolt group moment of Inertia is analogous to the 2nd moment of area used in beam bending theory then we should be able to use the parallel axis theorem, which is expressed for beam bending as:-

$$I_{xx} = I_{gg} + Ay^2 - Eq. 2.3.1.3$$ [aw. cit.] Where :-

$$I_{xx}$$ = Second moment of area taken about bolt group neutral axis
$$I_{gg}$$ = Second moment of area taken about one bolt cross sectional area neutral axis

We can think of the bolt group moment of inertia as a quantity that defines how a set of points or infinitesimally small areas resist an applied moment. These points are coincident to the centre of the bolt positions. The more points (or bolts) that are scattered further away from the neutral axis, the larger the moment of inertia value and the more resistance the bolted joint gives to the applied moment. Hence, the larger the inertia the lower the forces are at each bolt in the joint. If we draw parallels to a beam, the material is concentrated in the flange area of a beam which is the furthest away from the neutral axis. This gives the greatest 2nd moment of area and subsequently greatest resistance to bending. We can generate an expression for the bolt group moment of inertia by assuming the radius of this small area approaches 0 and all the bolts are of the same nominal size. See the derivation below :-

The second moment of area of any circle is defined by :-

$$\pi r^4/4 - Eq. 2.3.1.4$$

Equation 2.3.1.3 can now be rewritten as:-

$$I_{xx} = A(r^2/4 + y^2) - Eq. 2.3.1.5$$

If we let r tend to 0 as we are considering a point, then the $$r^2/4$$ term can be removed from the expression above and thus can be rewritten as:-

$$I_{xx} = Ay^2 - Eq. 2.3.1.6$$

This expression only considers one remote point from the neutral axis. The total moment of inertia for the bolt group is the sum of all $$Ay^2$$ terms or:-

$$A\displaystyle\sum_{i=1}^n y_i^2 - Eq. 2.3.1.7$$

If we substitute equation 2.3.1.7 into 2.3.1.2 then we get:-

$$F = My/\displaystyle\sum_{i=1}^n y_i^2 - Eq. 2.3.1.8$$

Now the bolt area A is no longer part of the expression for bolt force.

Up to now we have only considered a uniaxial state of bending, however a bolted joint can experience bending about two axes. An expression can be derived for a state of multiaxial bending based upon the above theory and as we have assumed the principle of linear superpositioning then we can simply sum the two values to achieve a total force. See below :-

$$F_{Combined} = M_x x/\displaystyle\sum_{i=1}^n x_{i}^2 + M_y y/\displaystyle\sum_{i=1}^n y_{i}^2 - Eq. 2.3.1.9$$

The designer would need to evaluate expression 2.3.1.9 for all bolt instances to determine the largest bolt force produced by multiaxial bending. As mentioned previously this can be added to forces produced by other types of loading to obtain a total maximum bolt force. If the position of the neutral axis is not obvious by inspection, which is usually the case for an arbitrary set of points, then the position can be calculated from a given datum. The position is found by taking moments about the datum line. The following expression can be used to find the distance to the neutral axis from a given datum.

$$X_{na} = \displaystyle\sum_{i=1}^n (x_{i}-x_{datum})/n - Eq. 2.3.1.10$$

$$Y_{na} = \displaystyle\sum_{i=1}^n (y_{i}-y_{datum})/n - Eq. 2.3.1.11$$

Where:-

$$n$$ = Number of bolts
$$x_{datum}$$ = Arbitary datum position to measure all x bolt positions from.
$$y_{datum}$$ = Arbitary datum position to measure all y bolt positions from.

Remote forces which do not act through the centroid of the bolt group and which act in the same plane as the faying surface tend to induce a torsion moment on the group. There are two methods used to evaluate the bolt forces for this type of loading. It is common practice in the UK to assume that the forces are distributed such that the shear is proportional to the distance from the centroid of the group. This method is analogous to torsion in a shaft, i.e. the extreme fibres or bolts, in our case, experience the largest forces. The alternative method is known as the Instantaneous Centre method and is commonly used in the US. It assumes that the centre of rotation is continually adjusted until the three basic equations of equilibrium are satisfied. This is an iterative approach and is beyond the scope of this article. The elastic method is found to be very conservative when compared to test results [aw. cit.]. This is because it is assumed that the translational and rotational components of the forces are independent and superimposed to calculate the bolts forces, when in reality they act together. We will describe the elastic method below.

#### 2.4.1 Elastic Method

This method assumes that the bolt group rotates around its elastic centroid when an external moment is applied. In much the same way as the bolt group moment of Inertia is calculated in section 1.3.1 the polar moment of Inertia of the bolt group requires evaluating. From the theory of the torsion of a shaft, the polar moment of inertia is defined as the sum of the two perpendicular axis moment of inertias of the cross section.

As in section 2.3.1 we can utilise the shaft in torsion equations to derive an expression for bolt forces. See below:-

$$T/J = \tau/R - Eq. 2.4.1.0$$ [aw. cit.]

Where:-

$$T$$ = Torque
$$J$$ = Polar moment of inertia
$$\tau$$ = Shear Stress
$$R$$ = Radial dimension from centre of shaft to position of calculation

As $$\tau$$ is defined as force per unit area we can rewrite equation 2.4.1.0 as :-

$$F = TAR/J - Eq. 2.4.1.1$$

Where:-

$$F$$ = Force
$$A$$ = Cross Sectional Area

If we recall the definition of the polar moment of inertia from above, the following expression is derived, making use of equation 1.3.10.

$$J = E(A*rx^2) + E(A*ry^2) - Eq. 2.4.1.2$$

If we substitute equation 2.4.1.2 into 2.4.1.1 we get:-

$$F = T*R/( E(rx^2) + E(ry^2)) - Eq. 2.4.1.3$$

The expression gives the force at each bolt due to an eccentric load acting in plane to the faying surfaces. Again, if the positions of the two neutral axes are not obvious by inspection the equations 2.3.1.10 and 2.3.1.11 can be used to find these dimensions.