# OT: curvature of the earth and buildings--was g-dElson right?

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• posted on November 18, 2006, 9:10 pm

Awl,
(See the original thread on alt.machines.cnc, where Elson dissed me and I thusly started the usual web-bristling, w/ the usual huff, puff, and attendant dance. But really, he shouldna dissed me, man, shouldna dissed me....)
But moving onwerd and forwerd:
Even tho I had it on good authority that no goddamm builder on earth uses the earth's curvature ceptin builders of long-assed bridges, the image of oh-so slowly diverging radii of the earth did bring this to mind:
For a tall building building to be "truly plumb", *each wall* would have to be plumb, ie, lie exactly along the *line of the earth's radius*, from the point where sed wall touches the ground.
*Necessarily*, then, the walls of a tall building must **diverge**, just as the earth's radii diverge, from the earth's center to the base of the building.
Rough calcs indicate that a 1500 ft tall building, w/ a 400' x 400' base, would have its top floor about 1/2" longer on an edge than the ground floor. (!!!) Iow, a "properly" constructed building will be a very long truncated pyramid, point-side down, base up.
If a skyscraper is constructed w/ strictly parallel walls, as far as gravity is concerned there is actually a tipping "moment", sorta like the Leaning Tower of Pizza. Not significant, of course, but hey, 1/2" is 1/2". Which gives the top floor an area of 160,400 sq ft, vs the ground floor's 160,000 sq ft. Which is, hey, but another lawyer's/stockbroker's office, which is, hey, but even mo' rent.
Or, inversely/conversely/obversely/reversely/obtusely, A tall building might have to be built pyramidally, *narrow end up*, to *counteract* the slight tipping moment of one (or both) of the walls not lying strictly in line w/ the earth's radii. Uh oh, there goes the lawyer's office....
I can see Elson now, w/ his gadget at the Sears tower....
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Mr. P.V.\'d (formerly Droll Troll), Yonkers, NY
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• posted on November 18, 2006, 9:46 pm
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1/2" on an Empire-State-sized building? I trust your calcs, but do you really think there are builders working at that scale to *that* accuracy? (Other than just interesting "measuring" exercises, that is)
Interesting concept, (I've often pondered it myself), but swallowed up entirely in the tolerance (or margin for error).
You could argue, f'r'instance, that any one "vertical" strut would indeed have the earth-centre-pointing vector contained entirely within itself, unless it was considerably less than 1/2" thick.
-- Jeff R. (pass me another beer)
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• posted on November 18, 2006, 9:50 pm
You're preaching to the choir here. :) Read the claims in the original thread. goodgawd.... 1/2" is positively *gargantuan* compared to the other stuff. :)
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Mr. P.V.\'d (formerly Droll Troll), Yonkers, NY
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• posted on November 20, 2006, 3:57 am

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• posted on November 20, 2006, 7:52 am
this all makes several invalid assumptions, one being that the earth is nearly spherical, the other that gravity is uniform - see WGS-84 for example for datums for earth parametrics

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• posted on November 18, 2006, 11:15 pm

I would ASSume that how they are actually constructed would be that the imaginary centerline of the building's core would be constructed plumb and the building would be squared to that. Therefore there would be no "tipping moment" as WRT the earth's gravity all outside walls would be leaning in equally. However, I haven't personally built anything bigger than a large shed, so I dunno.
IRL it seems that most tall buildings have terraces and therefore the upper floors are smaller in areas than the lower floors discussion above notwithstanding.
nate
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• posted on November 19, 2006, 12:05 am
You are correct. I guess ahm in my alarmist mode. :)
And you are correct about terraced buildings. This is actually required in some areas by the urban planners, as well as setbacks for these tall buildings. The reason can be plainly seen ( or not seen!) by parts of very lower Manhattan, where walking on the sidewalk is much like being in the bottom of a goddamm well--you see the sun only when it is directly overhead! Ergo, the terracing. Oh, and the area of the top floor would be 16,040, not 16,400: 400.05 x 400.05, approx.
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Mr. P.V.\'d (formerly Droll Troll), Yonkers, NY
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• posted on November 18, 2006, 11:27 pm
Parallel lines averaged to the center plumb line should work fine.
Some folks think a plumb bob won't work next to mountain.
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• posted on November 18, 2006, 11:59 pm
Technically, they might be correct. Would have to calculate relative masses, tho. Main earth no doubt swamps the mountain. But....
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Mr. P.V.\'d (formerly Droll Troll), Yonkers, NY
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• posted on November 19, 2006, 10:53 pm
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If you really are interested, go to: http://www.amse.net/The%20Geological%20Concept/2-A%20 (Modern%20Geological).htm
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• posted on November 20, 2006, 5:34 pm
On Sat, 18 Nov 2006 18:59:44 -0500, "Proctologically Violated©®"
[on plumb bobs not working next to mountains]

Nope. Plumb-bobs work no matter what's around them, they always show you which way down in. Granted, that might not always be towards the center of the planet, if there's a big enough mass nearby, sideways, but you don't care about that.
You care about what direction the building is going to try to collapse in.
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• posted on November 18, 2006, 11:40 pm

Ya know , there's something to this . I read once of a (oblong , and big !)tank/fountain/whatever , that was supposed to have an even volume of water falling over the edge all the way around . Kind of a water curtain effect . They forgot to include the curvature in their calcs , the thing was only overflowing in the middle ! Don't recall how they solved it , but I recall pics of it "after mods" . T'was wayyyy cool .
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• posted on November 19, 2006, 12:00 am
Coriolis force?
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Mr. P.V.\'d (formerly Droll Troll), Yonkers, NY
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• posted on November 19, 2006, 2:07 am

No , the water surface actually was curved to match that of the earth . The tank (for lack of a better word) lip was straight ... therefor it was only flowing over the edge in the middle .
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Snag aka OSG #1
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• posted on November 19, 2006, 12:20 am

I don't know whether I don't know whether these buildings are actually designed this way, but I'll bet they are BUILT this way. The reason is, they use LEVELS when putting the walls up! So, the walls have to diverge by that tiny amount. You'd never be able to measure it on one floor, but if you were to carefully measure the walls over a few floors, I think they might show the divergence.

Ahh, I can't measure that one, because it is too far off the local vertical!
Jon
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• posted on November 19, 2006, 1:18 am
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The Boeing plant in Everett, WA is about .6 miles long according to my odometer. You can see the curvature of the earth in the ramp in the front of the building, and if you look just right at the building from an approaching highway you can see the curvature of the roof to match. Haven't been on the roof but once, and I wasn't looking for that when I was up there. Since the building is actually many separately constructed building joined together by semi-flexible joints and floor plates between buildings for earthquakes and natural ground movement, you can't see the curve inside the building unless you put your head to the floor and look from one end to the other in the main aisleway inside. It's .3 miles deep, and arranged so that there are two buildings front to back and six or so longways. I imagine that if you could somehow measure the centerline of the posts that hold up the walls and roof, you might be able to measure the difference in span between the floor and the roof structure.
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• posted on November 20, 2006, 9:43 pm
Carl McIver wrote:

He he! Now that I've got the formula figured out, a 6 mile span has a divergence from a straight line of roughly 24 feet. Of couse if you consider both ends "level" then the rise in the middle appears to be 12', I think.
Jon
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• posted on November 29, 2006, 3:37 am

I shoulda been more clear. I wrote ".6" when I oughta written "0.6"
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• posted on November 19, 2006, 2:03 am

Properly constructed buildings are built with parallel lines, not truly "plumb" lines in the strictest sense. The alignment is done with optical instruments calibrated to set benchmarks which which ignore the divergence that you describe. If tall buildings were built with plumb bobs, perhaps the divergence would be more of a problem, but then, where would you hang the plumb bob from?. Large civil structures, such as bridges, are often subject to this divergence due to their large size. Bridge towers will indeed be further apart at the top than the bottom due to the earth's curvature.
Sorry, can't comment on tipping moments.
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• posted on November 19, 2006, 3:23 am

The problem with the Sears Tower was its asymmetric construction. There were multiple rectangular structural elements all bolted together, but each section terminated at a different height. Thus, when they got to the highest center section, they found that the very top of the building was cockeyed (leaning in one particular direction).
A similar analogy would be a man coming home from work while each of his say 5 children jump on him and hang on at different levels. If the children weigh enough, the man is going to have difficulty standing straight.
I briefly worked in the Sears Tower earlier in my lifetime and up above the 100th floor was not a good place to be during the big storms that occasional hit Chicago with 40 - 50+ mph winds. The building swayed so much that hanging light fixtures would swing swing back and forth.
Beachcomber