In my previous post, I described calculations for free-standing truss towers that could be on the order of 40 km tall. When I did the study, I stated that I was dubious about the use of guys (or guy wires, guy cables) for extreme towers. The problem I was thinking about was the degree of sag in guys that went up to 20 or 40 km. The sag would make the cable ineffective for resisting wind loads and for aiding in the buckling performance of the tower.
When I was working as a structural engineer, I used to be paid well for my engineering intuition. What I learned over the years is that intuition is not worth much once you leave the realm of the familiar. One always must do calculations to inform intuition. So, off we go doing calculations on guys for extreme towers. The basic solution for a catenary is well known and part of many introductory calculus courses. However, starting from the classical solution, you can examine many aspects of a guy system, including combining elastic elongation with the geometric effects of the hanging catenary. The study document in the main web site goes into great detail. It’s a bit rough, but all the derivations and code development are included.
The conclusion is that for synthetic high-strength materials such as Kevlar, it appears that using guys with attachment points 6 km above the ground looks practical. If one can use half of the available strength for tightening the cable and supporting its own weight, then 12 km is possible. This is well short of the 40 km towers we were designing, but still may be useful because most of the wind load occurs at low altitude. I have not yet looked at how including guys affects the tower design. These calculations assume that the ground anchor point distance from the tower is equal to the height. In other words, a 45 degree line. The functions available in the document can be used for other variations.
For a bit of fun, I’ve included a Mathematica widget that shows the shape of a cable given the horizontal component of tension force, and the position of the endpoint on the tower (horizontal distance from the anchor point and altitude). The widget will be visible and functional if you have the Wolfram CDF player installed on your browser. Not particularly useful, but a nice demonstration of installing a widget into a WordPress post.
I have a couple of more things I want to do with tall towers before I leave the subject for a while. The calculations are complicated enough that if I leave the topic, it will take a lot of energy to get back into the flow.