Consider the figure below, which shows a bolt under pre-tension.

The bolt is pre-loaded to a tension , and the plate(s) is pre-loaded to a compression .

The lengthening of the bolt under this pre-tension is:

The shortening of the plate (or in many cases, multiple plates) under this pre-compression is:

The figure below shows the loads on the bolt and plate after an external force is applied. In the figure, the example that is shown is a moment connection that uses angles instead of welds. Under application of a beam moment, each leg of each angle is pulled in a way that causes some additional tension in the bolt and some reduction in compression of the plate. This force that’s applied to the connection assembly is denoted in the figure. What we’ll prove is that only a small fraction of this force, , causes additional tension in the bolt, so long as the bolt has been sufficiently pre-tensioned.

First, let’s look at the additional lengthening of the bolt, in terms of the force , which is so far unknown.

(1)

Now, let’s look at the lengthening of the plate(s), which is written in terms of unknown variables and .

(2)

We’d like to find an expression that relates to and . In other words, we’d like to know how the pre-load in the bolt, , affects the final tension in the bolt, , when the connection assembly is subjected to external tensile loads, . We currently have two equations and many “unknowns” (, , , , ). We need to consider force equilibrium and displacement compatibility in order to solve.

Since the bolts and the plates “lengthen” by the same amount:

(3)

From equilibrium of our two free-body diagrams (prev. figures that contain , , and , , ), we have two additional equations:

(4)

(5)

We can see in the above figure that:

Additionally, we can assume that the bolt and the plate have the same (they are both steel), so that:

We now have equations (1 2 3 4 5) and unknowns (, , , , ), so we can solve for the desired result ( expressed in terms of and ):

(6)

From eq. 6, we can conclude that only a small fraction of the load “” is actually “added” to the bolt, so long as a) the plate is big and b) the bolt has been sufficiently pre-tensioned so that the bolt and plate remain in contact when “” is applied. Eq. 6 would not be applicable if the bolt is not sufficiently pre-tensioned (e.x. would simply be equal to , if is zero).

**Pre-tensioned bolts have a relatively constant tension, even if the external loading ( i.e. “”) is cyclical. This relatively constant tension is necessary to avoid bolt failure due to fatigue. **