Guy cables for extreme towers

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.

Mass of Extremely Tall Truss Towers

Well, that took an embarrassingly long time. For background, I started off trying to determine the limits for building compression towers high into the atmosphere. The ground rules were that the tower had to be stable with only passive structure, withstand realistic wind loads, and be built using conventional, current materials. With this goal in mind, I developed a Mathematica package called ttas.m (Truss Tower Analysis System) that would automatically generate a finite element model, run an analysis, and iteratively size the truss elements to satisfy local failure criteria (Euler buckling of the element, local wall buckling, and strength).

I wanted to actually use ttas package to do parametric studies of towers up to about 50 km in height. I quickly discovered a lot of inadequacies in the approach. One big problem was that in my first few examples global buckling was easily satisfied, so I assumed that with minor tweaks to the geometry a user could set a reasonable slenderness ratio and buckling would not be an issue. Not true. I eventually designed a system that simultaneously satisfies the global buckling criterion, and local strength criteria.  My initial set of parametric studies is up on the web site now. Pretty satisfying that I got this far, and the results are definitely interesting. In the process, the analysis package got a major upgrade, and the new version has been posted, along with new user notes.See the main web site page on tall towers for links.

The results can pretty much be summarized with the log-log plot of mass versus height for various materials, shown below. We have assumed a 70 m/sec wind over the entire height of the tower, where air density is a function of height. We also assumed a 10^5 kg mass at the top of the tower (although another result is that total mass is a weak function of the top mass). The tower mass forms a straight line in log-log space, and the curves for different materials are nearly parallel. A typical model is also shown with the truss tubes drawn to scale.

I don’t know where I’m going next. In the tower study, it became obvious that I need to look at multi-level trusses (truss elements are made from additional trusses), but the amount of work required may not be justified. I feel like I left sub-orbital refueling incomplete, and would like to get back to that subject. I’ll take a few days off and see what moves me. The joys of hobby engineering.

Tower Mass versus Height. 6-leg truss, 70 m/sec wind

Typical Truss Model. Six-Leg configuration

 

 

Tall Towers

I’ve had a major side diversion into the idea of extreme compression towers. A couple of years ago I wrote some basic Mathematica tools for looking at truss towers using the finite element method combined with some slick automatic modeling functions. For space applications, it is always assumed that compression towers were impractical and the focus has been on tensile structures such as the space elevator. While compression structures are certainly challenging because of stability issues, I had never seen a complete treatment that covered all the major structural requirements. So before creeping old age made me loss the earlier work I spent some time working on the analysis tools. It turned out to be a significant slug of work, at least for a retired guy, and thus the two months since my last post.The main website page for extreme tower work is here.

Of course, a tower can only get you above the atmosphere. Climbing to the top provides only a tiny portion of the total energy needed to travel in space. Still, there could be some useful applications. Getting above the bulk of the atmosphere does reduce the total velocity change needed to reach orbit by reducing or eliminating atmospheric drag (see, for example Landis, 2003). As discussed elsewhere one the website, vertical accelerators can be used for deep space applications (geosynchronous or beyond). One novel proposal combines a rotating tether with a tall tower to launch payloads (www.fisherspacesystems.com). A tower into the jet-stream might be a great place to put a windmill. And then there’s all the conventional uses of towers such as transmitting and observation.

We’ll start by acknowledging that in all likelihood, extreme towers are not going to be cost effective for any application. But I think it will be a fascinating project to see if one can be designed that meets the most basic criteria for strength and stability with a reasonable total mass.

There are a number of speculative papers that start to address the physical possibility of extreme tall towers. For our purposes, “extreme” means getting above a major part of the atmosphere, say starting at 10 km. Alexander Bolonkin has written about conventional compression towers, gas-filled towers, and structures based on electrostatic repulsion. These papers focus on the compression strength aspect and gloss over global stability and wind issues. Finally, there are a whole host of what I call kinematic structures. These use the momentum of a moving mass stream to keep a structure in tension. I’d like to study kinematic structures elsewhere on the site in the future.

I’m wary of assuming an active control system can be used to manage structural buckling. In tower discussions, one sometimes hears buckling dismissed by saying there will be some active system that keeps the tower perfectly aligned. While an active system can certainly balance a broomstick or a structure with a finite number of degrees-of-freedom, it is hard to imagine how it would work in a continuously flexible tower. In addition, the restoration forces involved for a structure with a mass of a few million kilograms would be huge. Therefore, we will demand that our tower have natural stability. I’m sure that active controls will be necessary to damp vibrations, and possibly improve alignment, but a control system should not be required to simply stand up.

We will also start the studies with free-standing towers only. Intuitively, it may be hard to scale guy-wire supported structures to heights of 10-100 km. This is also a limitation in what can be done easily with analysis. I’d like to keep the analysis linear, but extremely long guy-wires will have a great deal of sag and therefore a nonlinear force-deflection response. Guy-wires to help resist wind loads are more promising than wires to provide stability. Finally, we’ll limit the studies to conventional structures and materials. I don’t dismiss pneumatic structures, but let’s try to obvious first.

So far, only the analysis package and a user’s guide have been written. The guide walks through the design of a 10 km tall tower using a conventional, intermediate strength composite material. The examples are meant to demonstrate the system and show a preliminary feasibility of an extremely tall tower. Using a material with specific strength of 2.25E5 (m/sec)^2, and supporting a 10,000 kg mass at the top, the total tower mass was 6.4E8 kg. For comparison, that’s about 1.4 times the mass of one of the World Trade Center Towers. The design withstands a hurricane force wind (90 m/sec) over applied to the entire height. More complete studies and tradeoffs using the system are planned. I need to complete the tools to allow for breaking down the structure into finer level trusses. For now, there is a basic truss structure in which the truss elements are assumed to be cylindrical tubes. The tubes are weight optimized with strength, wall stability, and Euler buckling stability constraints. However, for a structure of this scale, the elements of the main truss structure should also be trusses. And so on as in a fractal geometry. I’m currently working on automatically breaking down the structure to finer level trusses. One motivation is to reduce the area presented to a wind load.

It will also be fun to look at some of the geometric systems that have been devised by civil engineers of earlier times. One promising concept is to use hyperboloid arrangements.

Hyperboloid Pylon Structures. Shukhov Oka Towers. photo by Igor Kazus