"... there is only one application of which I am aware where the curvature of the earth plays any significant role in the alighnment of vertical structures. That unique situation is the ROTHR (Relocatable Over The Horizon Radar) antennas.. . ."
Fresnel Zone clearance for microwave radio paths.
Bob Swinney
The deal here is that to phase the scanning beam correctly, the vertical transmission towers spaced over a distance measure in miles must be precisely parallel to each other. Not vertical, but precisely parallel.
Erecting a tower vetically is a trivial effort requiring only a plumb line or its modern, more precise instrumentation equivalent. Erecting an array of tower space over miles is a challenging engineering effort due to the curvature of the earth -- you cannot use a plumb line reference.
What is important to realize is that the ROTHR towers are located a considerable distance apart, and not in the 50-yard proximity of the footprint of a very large skyscraper. So, who cares if the four structural members of the building are not precisly parallel. They're certainly sufficiently parallel and vertical enough for both government work and civilian construction.
I would say that buildings where this comes into play need to be tolerant of all of this $#!+. Something as tall as a WTC tower and twice as wide needing to be 1/2 inch wider at the top than the bottom - it better not be so touchy as to easily topple if it ain't!
Just consider wind problems! NYC can expect to get hit by a hurricane badly enough to get peak gusts around 90-100 MPH at "treetop level" at least once per century. Expect that to translate to about 110 MPH on the top half of a building that tall - even though hurricanes have their worst winds in lower parts of the atmosphere! 110 MPH of 74 F air at barometric pressure 28 inches works out to 27.9 pounds per square foot of wind-hit side area assuming drag coefficient 1 if I calculated right! The building's base would have stress a few times that of tension on the upwind side and similar compression stress on the downwind side! In terms of force per building footprint area, multiplied by reciprocal of fraction of the building footprint that has structural members that withstand this force.
Hurricanes ain't the only problem! Consider the Nor'easters of March
1932, March 1962, March 1984, October 1991, December 1992 and March 1993! Boston might also complain about a February 1978 storm and one in early 2005! Along with Nor'easters being worse 1,000 feet up than tropical storms with same treetop level winds are! As for Midwest - how about Ohio-Indiana around January 25 1978?
Also consider the downburst, microburst, gustnado and (less common but often worse) tornado wind effects that severe thunderstorms (some of which have deceptively little lightning and thunder, especially in off-season) have!
LA is a minor improvement, being in a minor tornado hotspot of "western USA" bad enough to be only a minor improvement over the USA east of the Rockies! Worse still - irregularity of tornadoes and other severe thunderstorm wind effects (often there with minor uptick from the unimpressive low lightning/thunder incidence that is normal there) - mostly during an El Nini winter rainy season! Along with bad windstorms other than tornadoes heavily occurring during El Nino winters, and with high incidence of bad winds worse-than-average being worse near/above
1,000 feet than at treetop level!
If a big building is so touchy that 1/2 inch off out of 400 feet is a big structural problem, then the building design has a big structural problem on the drawing board!
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
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.
The Earth's curvature is 8' per mile from what I remember... But taking the diameter of the planet and using a CAD program, you could figure out the difference based on height I suppose...
I always argue with a friend of mine that a 1 mile long "flat" runway would actually rise well above ground level (4' on each end) unless it wasn't truly flat - If it was built to the Earth's curve, it would actually be flat to a level (which works on gravity) but not flat to a perfectionists rules.
Well, see this page for backup on my calculations :
formatting link
Now, using their same equation ...
Ahh, this is too hysterical to do in miles, so I converted everything to feet.
So, A^2 = (3963 * 5280) ^2 + 100 ^2 (for a 100 foot length)
Then, you have to subtract the radius of the earth, in feet (3963 *
5280), and convert feet to inches ...
I get .0029" of curvature over 100 feet. If I haven't made a mistake here, then it does come up smaller than what I was computing before. This isn't the only way to compute this, and takes a couple of shortcuts.
Running the same computation for 1000' gives .287" 1500' gives .645"
Even at these incredibly small angles, the non-linearity of the function shows up clearly in the last 2 numbers.
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.
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