## Back-tie rigging – tension and compression forces

The other day I had a question come through about how I worked out the tension and compression forces for a post, pole or tree when you back-tie for support.

### The context?

Before I get to the question, we need to set the context:

- The front anchor is in the right location; however, it needs a back anchor to make it acceptable with sufficient margin.
- The back anchor is some distance from the front anchor.
- Multiple strands are tensioned to reduce stretch in the rope or cord over the back-tie distance.
- The front anchor can sustain any forces applied in compression, but the post needs support for a bending (moment) force as the rigging is off the ground.
- Having the anchor rigging higher is to get it off the ground, so it’s easier and safer to operate. For example, it has less friction for a rescue raise, or if you are abseiling/rappelling over an edge, it makes the transition easier.

### Now to the Question

I wondered what trig functions you are using on page 194 of the guide (Rope Rescue and Rigging Field Guide)? I understand the “factors”, I’d like to run the numbers to better my understanding.

### Answer

Good question. I had my workings scribbled on a bit of paper. I have put it into a 2-page summary for you to work through (see the download at the end of this post). The following is the process I used to work out the factors in the book.

#### Notes

- Orientating the drawing vertically made it easier to see what’s happening. Imagine having the load hanging on the end of the post. But it can be in any orientation.
- C1 is a solid object subject to (and can handle) compression such as a post (tree, pole or similar).
- T1 is the rigging that supports the post and subject to tension such as a rope (wire, cable or equivalent).
- It’s a 2 stage process. First, work out the tension (T1), then the compression (C1).
- Use the three basic trig ratios – SOH CAH TOA – to solve a missing side in a right-angled triangle.
- Check out this link if you need a reminder of how it works. (https://calcworkshop.com/triangle-trig/sohcahtoa/).

#### Stage 1

- Start with the tension on the rigging T1. Break this down into x and y vectors T1x and T1y.
- For every action, there is an equal and opposite reaction. Whatever the load is also on the T1y vector.
- The angle formed by T1x-T1 is the same as the angle formed by T1-C1 (alternate angles inside a transversal). I have put an equal sign on the drawing.
- Now use SOH (Sin angle = Opposite/Hypotenuse) with the formed right-angle triangle (Tx-Ty) to work out the tension on T1 (which is the hypotenuse).
- Manipulate the equation to get T1 = L/Sin angle.
- Plugin the values you know (the load(L) and the angle) to determine the answer.

Check out the download worksheet below for the workings and examples.

#### Stage 2

- Now use CAH (Cos angle = Adjacent/Hypotenuse) with the formed right-angle triangle (C1-T1) to work out the compression force on C1 (which is the adjacent).
- Manipulate the equation to get C1 = T1 x Cos angle.
- Plugin the values you know (which is T1 from stage 1 and the angle) to determine the answer.

Check out the download worksheet below for the workings and examples.