5 ms·

A building is say 500 feet tall. The top moves say 2 feet. In order for this to happen every 10 feet of the building doesn't shift sideways a certain amount.

A building is say 500 feet tall. The top moves say 2 feet. In order for this to happen every 10 feet of the building doesn't shift sideways a certain amount. Every 10 feet of the building bends a certain amount. A VERY small amount. Tiny fractions of a degree. So the bottom 10 feet of the building bends say 0.001 degrees. The bottom is flat, the top is tilted 0.001 degrees. The next 10 feet bends an additional 0.001 degrees, but its base was already tilted 0.001 degrees so its top is tilted 0.002 degrees. The next 10 feet bends an additional 0.001 degrees, but its base was already tilted 0.002 degrees so its top is tilted 0.003 degrees. Repeat this 50 times (for 500 total feet) and you've got 0.050 degrees of tilt at the top which might be noticeable. Further, you have to add up displacements the whole way from the bottom to the top as well. If you took a picture of the building sideways it would look like this: http://www.codecogs.com/users/23287/Cantilever-Beams-101.png http://www.codecogs.com/users/23287/Cantilever-Beams-101.png

mturmon 11y ago
I like your reasoning, but I wonder about magnitudes. Since: 10 feet/story * sin(0.001 degree) = .00017 feet/story then: 50 stories * .00017 feet/story = .0085 feet [building] If things added up this way, your magnitudes would be way off. In fact, since sin(x) is linear for small x, you'd need about 100x more angle to get 100x more deflection. 100x more deflection would be about 0.85 feet one way, or 1.70 feet peak-to-peak. Two feet peak-to-peak for 50 stories matches some of the figures quoted in the article for buildings of this scale ("A hundred-story building, by comparison, may move on the order of two-and-a-half to three feet to each side..."). However, on reflection, I think the offsets cumulate quadratically. Because: * \ \ [story 2] \ | | [story 1] | The first story has an angular offset, but the angular offset of the second story adds to that of the first, etc. This must be common knowledge among structural engineers. In that case, since we can just add all the displacements, the appropriate quadratic multiplier would be 50 * 49/2 = 1225, not just 50, and the total displacement is: 1225 * .00017 feet/story = 0.21 feet. This would mean your original, very tiny, angles were only about 5x too low. Nice work!
- msandford 11y ago
  Yeah it was a total guess, not informed by doing any actual math. Thanks for adding some real numbers to the discussion!

mturmon 11y ago

I like your reasoning, but I wonder about magnitudes. Since: 10 feet/story * sin(0.001 degree) = .00017 feet/story then: 50 stories * .00017 feet/story = .0085 feet [building] If things added up this way, your magnitudes would be way off. In fact, since sin(x) is linear for small x, you'd need about 100x more angle to get 100x more deflection. 100x more deflection would be about 0.85 feet one way, or 1.70 feet peak-to-peak. Two feet peak-to-peak for 50 stories matches some of the figures quoted in the article for buildings of this scale ("A hundred-story building, by comparison, may move on the order of two-and-a-half to three feet to each side..."). However, on reflection, I think the offsets cumulate quadratically. Because: * \ \ [story 2] \ | | [story 1] | The first story has an angular offset, but the angular offset of the second story adds to that of the first, etc. This must be common knowledge among structural engineers. In that case, since we can just add all the displacements, the appropriate quadratic multiplier would be 50 * 49/2 = 1225, not just 50, and the total displacement is: 1225 * .00017 feet/story = 0.21 feet. This would mean your original, very tiny, angles were only about 5x too low. Nice work!

msandford 11y ago
Yeah it was a total guess, not informed by doing any actual math. Thanks for adding some real numbers to the discussion!