Moment Resisting Bolted Connections

In this article I will discuss about the design of bolted connections that resist moments. In this article you will find key design techniques, tips and best practices in this field. Answers will be provided to the questions like, How to calculate the required bolt size for a moment resisting bolted connection? Structural bolts are primarily used in shearing, tension, and their various combinations, which are commonly encountered in bolted connections designed to resist out-of-plane moments. I will provide explanations in a clear, straightforward and simplified manner to ensure that even individuals with limited knowledge can benefit from this article. Engineering students, Engineers seeking to refresh their knowledge, and those interested in learning about structural engineering will benefit from this article.

Before looking into the design principles of moment resisting bolted connections, it is essential to understand the difference between two types of such connections: In-plane and Out-of-plane moments. The following figures illustrate the distinction between these two types.

Fig-1: Difference between connections resisting Out-of-Plane and In-Plane Moments

Fig-2: Splicing of a beam- Connection resisting In-Plane Moment

Connections resisting Out-of-Plane Moment

As illustrated in the Fig-1 above, in this type of connection the bolts are utilised in tension as well as in shearing. The vertical arrows on left hand side (vertical reactions) shall be used to determine shearing in the bolts. Similarly, the horizontal arrow on the left hand side (horizontal reactions, T1 and T2) shall be used to determine the bolt tension in the bolts at two different levels. The lowest horizontal arrow pointing toward opposite direction is compression horizontal reaction. Magnitude of this compression must be equal to summation of tensions in the bolts (T1 and T2). 

Fig-3: Determination of shear force and tension in bolts due to out-of-plane moment

It can be seen from the Figure-3 that shear in the bolts is equal to half the total vertical force,V. However, determination of tension in the bolts depends on magnitude of  moment and spacing of the bolts.

Let's assume a rotation point is at point O. We have only two unknowns, i.e., T1 and T2. Therefore, let's write two equations. Using similarity of the triangle shown in the Fig-3,

T1/(d1+d2)= T2/(d1)

T2 = (T1/(d1+d2))xd1 

Then, T2 = d1xT1/(d1+d2) ................(1)

Moment must be resisted by T1 and T2. 

Therefore, writing  moment equation about rotation point O,

T1x(d1+d2)+T2xd1=M

T1x(d1+d2)+(d1xT1/(d1+d2))) x d1=M ............... (2)

T1x(d1+d2)+d1^2 x T1/(d1+d2)=M

T1x(d1+d2)^2+d1^2xT1= Mx(d1+d2)

T1x((d1+d2)^2+d1^2)= Mx(d1+d2)

T1= Mx(d1+d2)/((d1+d2)^2+d1^2)

Find out the value of T1 from this equation and substitute in equation (1) to find the value of T2.

If we have more than two rows of bolts in a section then more equations like equation (1) shall be written in function of a common tension T1 or tension in any other single bolt. Then equation similar to (2) shall be written involving tension in all bolts, i.e., T1,T2,T3,.... Then value of common tension can be calculated using equation like (2). After that, tensions on other bolts can be calculated using equations similar to (1).

The topmost bolt has the maximum tension, therefore, all bolts are sized for the tension in the topmost bolt. Possible prying caused extra force needs to be added to the maximum tension.

Let's see the following example for completeness:

Say total shear force, V=200 kN

Max. out-of-plane moment, M=10 kNm

d1= 35 mm, d2= 100 mm

Let's say we have two bolts each in top and bottom rows, 4 bolts in total. We will find maximum forces in the bolts.

Total vertical shear in each bolt, v= 200/4=50 kN

After entering the values of M, d1 and d2, in the last form of equation-(2), we get T1=69.41 kN

Entering the value of T1 in the equation-(1), we get, T2= 18 kN

It's always a good practice to check the calculation by applying these values in the equations used,

69.41x(0.035+0.1)+18x0.035= 10 kNm. This satisfies the original equation of moment. Therefore, the calculation of T1 and T2 is correct. 

Therefore, the bolts shall be designed for tension= 69.41/2= 34.7 kN and shear=50 kN (Two bolts are assumed to be at T1 level)

The connection plate shall be designed to resist these forces. A good guideline is, thickness of the connection plate shall be about 1.25 times diameter of the bolts.

Some national regulations permit assuming the rotation point at the lowest row of the bolts. In this particular case, we only need to determine the tension in the top bolt. Consequently, there is no requirement to utilize equation (1) as the T2 equals zero. Similarly, in this example, the moment is solely resisted by T1. Therefore, equation (2) only contains one unknown variable, T1, which can be directly calculated. If there are more than two rows of bolts, equations similar to (1) and (2) should be formulated without considering the bolts in the lowest row. Nevertheless, when calculating the vertical shear, it is necessary to include the bolts from all rows.

Connections resisting In-Plane Moments

In the same Fig-1 the connection resisting in-plane moment is shown on the right hand side. In this type of connections the bolts are utilised only in shearing. Using the reaction moment and vertical reactions on left hand side are used to determine the shearing in the bolts. Determination of shearing due to the vertical reaction is simple,i.e., total vertical reaction divided by number of bolts. Determination of shear due to reaction moment is carried out in the follows steps:

Fig-4: Determination of shear force in bolts due to in-plane moment

-Determine centroid of bolted connection

-Determine value of Polar Moment of Inertia , Ip =∑(x2+y2)

Here, x and y are horizontal and vertical distance of the bolt measured from centroid of connection

-Determine length of "r" for each bolt

-Determine shear, "v" for each bolt which will be acting in the direction perpendicular to line "r",

v =M.r/Ip

Here, M is reaction moment

-Determine the horizontal and vertical components of shear "v", say vx = v.sin a and vy = v.cos a respectively. Similarly, all vx and vy can be calculated for each bolt.

Once vertical component, vy for each bolt has been calculated the shear due to vertical reaction can be added to find the total shear in each bolt.

Therefore, total shear in a bolt, vb= vy+V'/n  here, n is total number of bolts and V' is total vertical reaction. All bolts are sized for the maximum shear calculated this way. The maximum shear will occur at the most distant bolt from the centroid. Number of plates shall be two in  the best arrangement. The plates shall be designed for bending moment equal to the applied point load multiplied by the distance of point load from centroid of  the bolted connection. The plates shall also be designed for bearing and possible tear-outs. 

Calculation is best prepared in a table. Following example describes the process of calculation in detail:

Fig-4A: Resolution of resultant shear

Fig-5 Computation of bolt shear-  In-Plane Moment

Explanation of the calculation for these figures is as follows:

With this we know, what forces shall be used for design of the bolts and plate.

Splicing of "I" Beam

The Fig-2 shows a special case of moment resisting connection that is employed to splice a beam. This is a very common connection which resists a in-plane moment in combination with axial force in the beam. Similar connection is used in the splicing of columns as well.

Because of orientation of its components, design of connection for flanges and web are carried out in different ways. Web is designed for Shear and Bending Moment whereas the flanges are designed only for Tension and Compression. Depending on individual Moment of Inertia of the web and flanges only part of bending moments is used to design their connections. Since the web contributes to major portion of moment of inertia than the flanges do, the greater portion of bending moment is used to design the connection of the web. The remaining portion of the bending moment is used to design the connection of the flanges. The horizontal two plates of an "I' section are called flanges and the vertical plate is called a web.

Similarly, since the flanges are ineffective in carrying the vertical shear, connection of the web is designed for full shear force. Similar to distribution of bending moment, the axial force is distributed according to individual cross sectional area of the flanges and the web. 

Fig-6:Forces acting on beam connection

Forces that act on a beam splice is shown in the Fig-6 above. M*, V* and N* are bending moment, shear force and axial force respectively. These forces are applicable at the cross section where the splicing is being made. Similarly, Ntf* and Ncf* are the associated forces acting on the tension and compression flanges respectively. Here, "*" denotes the factored value of these forces.

The splice plates and the associated bolts visible in Fig-2 at the web are designed for major portion of the bending moment, shear force and some portion of the axial force. The moment acting on this connection is resisted by in-plane moment strength. The connection is designed for the related portion of shear and axial force.

Similarly, the splice plates and the associated bolts visible at top of the flanges in the same figure are designed for tension and compression caused by the remaining bending moment and the related part of the axial force.

Below is an explanation of how moment and forces are distributed to the beam.

Let's  say,

at and ac are cross sectional area of flange at tension and compression respectively.

Similarly, aw and A are cross sectional area of web and total beam respectively

I and Iw are moment of inertia of total beam section and web section respectively.

Nc1*, Nt1*: Compression and tension force on the compression and tension flanges respectively

Nc2*, Nt2*: Compression and tension force on the web respectively

fy is the yield stress which is a maximum allowed stress in the extreme fibre of the flange

Fig-7:Stress Distribution in beam cross section

Now let's see what portion of forces are distributed to different components of the beam

Portion of total axial force to be applied in two flanges, Nf* = N* x (ac+at)/A

Portion of total bending moment to be applied in two flanges, Mf* = M* x (I-Iw)/I

Portion of total vertical shear force to be applied in two flanges, Vf* = 0

Portion of axial force to be applied in the web, Nw* = N* x aw/A

Portion of bending moment to be applied in the web, Mw* = M* x Iw/I

Portion of vertical shear force to be applied in the web, Vw* = V*

In structural engineering it's not a practice to show a multiplication symbol in any formula like this. However, considering the readers with less practice in structural engineering I am showing the mark "x" to indicate a multiplication symbol.

Using the forces shown in Fig-6, the connections must be designed for the following forces:

Flange Splice

Total force to be used for the design of compression flange connection,

Nc1* = Mf*/(hc+dc+dt+ht) - Nf*/2

Here "-" appears brcause the axial force is tension, hence the portion of axial force is deductible from compression force Nc1* 

Similarly, total force to be used for the design of tension flange connection,Nt1* = Mf*/(hc+dc+dt+ht) + Nf*/2

The tension and compression flange splice plates shall be designed for tension and compression respectively. Refer Design of Compression Members and Method of Structural Design for the design concept of compression and Tension plates respectively. additionally the plates shall  be checked for bearing and tear-out as well.

Web Splice

Total axial force to be used for the design of web connection = Nw*

Total shear force to be used for the design of web connection = V*

Total bending moment to be used for the design of web connection = Mw*+V* x e

Following, Fig-8 explains what "e" is.

Fig-8: Connection of web

In practice only two rows of bolts are not used in the web splicing. However, for the sake of simplicity, the Fig-8 shows so.

Maximum shear in the bolts shall be calculated using the procedure shown in Connections resisting In-Plane Moments above. In order to add the bolt shear due to vertical shear and the bolt shear due to axial force the bolt shear caused by in-plane moment shall be resolved into vertical and horizontal direction.

Total vertical shear in a bolt, Vbv* = v.cos a + V*/(no. of bolts, 4 in this case)

Total horizontal shear in a bolt, Vbh* = v.sin a + Nw*/(no. of bolts, 4 in this case)

Total resultant shear on a bolt, Vbr* = ((vbv*)^2 + (vbh*)^2)^(0.5)

Bolts shall be designed for Vbr*. Whereas the web connection plates shall be designed for bending moment = (Mw*+V* x e) and orthogonal forces Vbv* and Vbh*. The plates shall be designed also for the bearing and tear-out. Refer Fig-2 for arrangement of the web connection plates. In this case the bolts shall be designed for double shearing.

This shows how the forces are determined which shall be used to design the individual connections of flanges and web of a beam.

In this article I have explained how the forces in the bolts are determined in the in-plane and out-of plain moment resisting connections. I have also explained how the forces in the bolts of a beam splice shall be calculated. Similar approach shall be used to determine the bolted connections of the columns. Refer Design of bolts and Plates for basic process of designing bolts and plates.

In the next article I will explain how the connections at footings are designed.

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