Re: Can a structure be built which is tens or even hundreds of miles high?

938120564.Eg

An interesting question, and one that has been posed by others for a 
similar reason.

To answer the first part of the question, yes, there are physical limits to 
the height of such a structure.  Basically, it boils down to the ratio of 
the strength of the structural material to the density.

If you started with just a balloon with a wire hanging down to the earth's 
surface, the weight of the entire wire will be carried by the very top 
strand.  The limit the wire can carry is described by its breaking 
strength, usually given in terms of physical load divided by its cross 
sectional area.  For a round wire, this load, P will be: 

P/ (pi*r*r) = breaking stress  where pi = 3.14159 and r = radius of wire

The weight of a length of L wire below it can be calculated from the 
density, the radius, and the length.

P = pi*r*r*(density)*L

setting the above two loads, P equal and solving for the length gives

L = breaking stress/density

For a high strength steel like spring wire, the breaking strength can 
exceed 200,000 pounds per square inch and the density is = .29 lb/cubic 
inch.  So the wire can be about 200,000/.29 inches long, or about 10 miles.


If we built a truss like structure like the Eifel Tower, we can minimize 
the wind loads and other factors that will tend to topple it, but you can't 
get around the weight effect.

I can't explain the problem in 3D so easily, but imagine building a 
truss structure in 2 dimensions made of triangles like the one below.  Each 
triangle is made up of 3 legs or segments.  Adjacent triangles share a 
segment.  Such a design can put all of the load uniformly into the 
triangles below. (I don't show the horizontal segments.)

      /\             1 triangle of 3 segments (two sides and the bottom)
     /\/\            3 triangles of a total of 6 segments
    /\/\/\           5 triangles of a total of 9 segments
   /\/\/\/\          7 triangles of a total of 12 segments


If you add up the weight of the segments above each layer, you will find 
that the 2nd layer has to hold the weight of 3 segments, the 3rd layer has 
to hold up (3+6)=9 segments, the 4th layer has to hold up (3+6+9)=18 
segments, etc.

If each segment of the truss is carrying an equal share of the weight above 
it, then the table below shows the weight carried by each segment as you go 
up down the structure:

Layer      segments    total segments above         weight above           
                        that layer                  per segment in layer
  2           6         3                                     3/6 = 0.5
  3           9         9                                     9/9 = 1.0
  4           12        18                                    18/12 = 1.5
  5           15        30                                    30/15 = 2.0

As you can see, the trend is increasing the load per segment due to the 
weight above.

In reality, the structure is 3D, so the number of trusses below can be 
greater or less than that shown, but no matter what, the weight becomes a 
factor eventually.  If we could fabricate a metal bar with helium trapped 
inside, perhaps the bouyancy of the helium would counteract the weight of 
the bar.  Otherwise, even lightweight materials like composite plastics 
will eventually succomb to the weight limit.

Even a mountain is presumably limited by the compressive strength of the 
soil or rock from which it is made.

The second part of the question asks whether such a structure would benefit 
a space launch.  Although rockets tend to use less fuel if launched at a 
higher altitude, this is usually done by carrying the rocket on an airplane 
and dropping it before ignition.  Presumably, your tower would have a 
similar benefit.  

The velocity effect of the earth's radius would be minimal.  We launch 
rockets near the equator to take advantage of the speed gained by the 
earth's rotation, as opposed to the north pole, where the velocity is pure 
rotation. I think I recall that the earth is roughly 8000 miles in 
diameter, so the velocity on the ground is:

radius * rotational velocity of earth =
4000 miles * (2 pi radians)/(24 hours) = 1047 mph

Even if you build a 200 mile high tower, the radius will be only 4200 
miles, or about 5% more.
The velocity will increase only 5% as well.

I seem to recall scientists posing the possibility of putting satelites in 
stationary orbit and then hanging thin wires from them down to the earth's 
surface.  Small payloads could presumably be run up the wire to orbit.  Not 
sure how serious they were about the topic, but the practical limits are 
severe, as you can see.

Moderator's Note: The excellent novel "The Fountains of Paradise" by Arthur
C. Clarke deals with such a "space elevator." If you could build one, it 
would lower the cost of transfer to orbit to pennies per pound. The catch
is that you would need superstrong materials, stronger than anything we
have now...