As a boy, I remember seeing a dilapidated farm building that had a set of rafter ties in its attic. I noticed that the rafter ties had pulled away from the rafters they were nailed to. The rafter ties were clearly pulling away from their rafters because of the tension forces they were experiencing when under snow load. I noticed that steel cables has been hidden behind the rafter ties to try to reinforce them, but the screws attaching the cables to the rafters were also pulling out. This also was a sign of tremendous tension.
I had forgotten about this old farm building until I saw an article in the Journal of Light Construction (JLC) on the problems associated with using rafter ties on a roof. I became curious where the equation used in this article came from and I decided to derive it myself. I also noticed that the same equation is also built into an online calculator. Note that this post does not presume to train anyone on how to do structural engineering. Instead, I am reviewing the basic mathematics and physics used in doing this sort of work. See a structural engineer if you have a real problem that needs to be solved.
For those who are not familiar with rafter ties, I have included Figure 1 as an illustration of what they are and how they are connected to the rafters. The number of ties required per rafter will vary by situation. The rafter ties can generate tremendous force on the rafter to which it is connected. Purlins are often used in conjunction with rafter ties (see Appendix below for a figure).
|Figure 1: Illustration and Implementation Examples of Rafter and Collar Ties|
Mathematically, rafter and collar ties are handled the same way. However, each type of tie has a different purpose. Unfortunately, the terms collar and rafter tie are often conflated. I found a good set of definitions from a construction forum that I will use here.
- Tension Tie
- A structural member that is subject to net tension.
- Collar Tie
- A tension tie in the upper third of opposing gable rafters that is intended to resist rafter separation from the ridge because of wind or unbalanced roof loads. Collar tie is a colloquial term for collar beam.
- Rafter Tie
- A tension tie in the lower third of opposing gable rafters that is intended to resist the outward thrust of the rafter under load.
Figure 2 is an illustration of a roof constructed using rafter ties.
Calculating Roof Loading
Figure 3 shows how the loading on an individual rafter is defined. Observe that all the loads are specified as if they were projected onto the horizontal plane.
The total load on a rafter is given by W, which is calculated using Equation 1.
where L is the rafter length projected onto the horizontal plane, d is the rafter tributory width , and σ is the roof loading projected onto the horizontal plane. The use of projections onto the horizontal plane seems to be the most commonly used approach.
Rafter and Collar Tie Tension Equation
To establish the relationship discussed in the article, we need to observe that the sum of the angular momentums must be 0. Consider the following free body diagram (Figure 4).
For a static (i.e. non-rotating) situation, the moment about ridge beam connection exerted by the tension tie must equal the moment exerted by the support reaction and the distributed load (Equation 2).
This form is correct, but a more useful form for this discussion substitutes the roof slope and height for L.
Equation 3 is presented in the JLC article. This equation can be modified to produce Equation 4, another commonly seen result.
where w is the unit loading (lbf/ft) along the rafter, expressed as force projected horizontally per unit length.
I plugged the JLC example into Mathcad to verify that I get the result that JLC obtained (Figure 5). Note that this example assumed that the dead load had already been projected onto the horizontal plane.
Using basic statics, I have derived the equations used in in determining the tension in rafter and collar ties. I have duplicated a worked example from JLC magazine. This exercise has shown me that rafter and collar ties are subject to enormous tension forces. These forces make securing the ties to the rafters an engineering challenge. The tension is so high that a large number of nails/screws would be needed for each joint. In fact, so many fasteners are needed that the joint may weaken because of having so many fasteners in a relatively small area.
I also learned that rafter and collar ties have different purposes: collar ties help secure the rafters to the ridge beam during periods of unbalanced loads (i.e. from wind or asymmetrical snow load); rafter ties are used to counteract the outward thrust of the roof.
Note that this post reviews the mathematics performed by a licensed structural engineer in a construction industry publication. It is not intended to be a tutorial on designing rafter or collar ties. That is a job for a professional.
- 18th Annual Eastern Conference of the Timber Framers Guild (Article: Pity the Poor Rafter Pair)
- Interesting forum discussion
- The mechanics of architecture -- free Google Ebook that actually works this problem on pg 121.
Calculation Detail on Moment from a Distributed Load
There is a bit of calculus work associated with modeling the moment due to the distributed load along roof. Figure 6 illustrates this computation.
Illustration Showing Purlins and Collar Ties
The following figure is from this source.