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Re: big dig structural failure - epoxy anchors overhead supporting gravity

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On Jul 15, 2006, at 1:26 PM, S. Gordin wrote:

This is not how I understand it. With bridges, for example, there are two criteria to limit vibrations - one based on rigidity (natural frequency) and the other one based on the mass of the structure.
It's first principles--frequency = (1/2pi)sqrt(k/m). Increase m and the frequency decreases. The system gets relatively (stiffness relative to mass) softer. With frequency dependent excitation, adding mass might move the system frequency away from the excitation frequency, but that won't happen with suddenly applied loads, like the pressure wave. Unless the system is extremely stiff, a suddenly applied load always produces twice the stress and deflection as the same load statically applied, and the system goes through a couple of cycles of vibration at a lower or higher frequency depending on how stiff it is. The system frequency affects the displacement and how long it takes the response to die out.

Those two bridge criteria are related because mass and frequency are related, but there are counter tendencies. What usually happens with steel frameworks is that it's difficult to increase the stiffness without also adding mass--moreover the square root dependency means that a 20% stiffness increase only increases the frequency by 10%--such a small change gets lost in the white noise. Sometimes you can play tricks like changing from girders to trusswork, but only in special circumstances. I've played around with trying to increase first mode frequencies for years, and with steel frames like crane structures, once you've set the span, you're pretty much stuck with the frequency you get.

I expect with concrete supported by steel, most of the mass is concrete and most of the stiffness is steel, so you have a little more latitude to make meaningful changes. I don't do concrete so there's probably more to it than just that, especially with bridges.

Christopher Wright P.E. |"They couldn't hit an elephant at
chrisw(--nospam--at)   | this distance" (last words of Gen.
.......................................| John Sedgwick, Spotsylvania 1864)

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