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Which failure arrives first — page 17

Essays 385 to 386 of 386 on this thread, in the same order.

A crossed hanger swings through nearly all of its force. The largest and smallest force in each crossed hanger of a tied arch of 100 m span and 17 m rise on 32 hangers, with 60 kN/m of dead load and a live load of 60 kN/m running on from either end, each bar running from the one to the other. The worst range is 678 kN, from 49 to 727, in hanger 11; the largest force anywhere is 756 kN. 6 of the 32 go slack at some stage, and the worst range is 90% of the largest force anywhere: the live load's shear passes through the crossed hangers as through a truss's web, and swings them from nearly nothing to nearly their most.

The hangers that carry the shear twice

Crossing a tied arch's hangers turns them into the web of a truss, which is why the tie stops bending. It also means a live load's shear passes through them, so as a load runs on from either end each crossed hanger swings from nearly nothing to nearly its most: on a 100 m arch the worst swings through 678 kN, 90% of the largest force in any hanger, where a vertical hanger swings through 194, half of its own. More dead load raises the force and leaves the range alone. The arrangement that is best for strength is the one that hands its hangers to fatigue.

A damper too large stops damping. The damping of a structure with 2.0% of its own, against the size of a viscous damper held by a brace — the damper's force over the brace's at the structure's own frequency, on a logarithmic scale — for braces a quarter as stiff as the structure, as stiff, and four times as stiff. With the brace at 0.25 the damping peaks at 7.9% for a damper of 0.86; with the brace at 1.00 the damping peaks at 22.7% for a damper of 0.58; with the brace at 4.00 the damping peaks at 63.9% for a damper of 0.29. Either side of the peak it falls: a small damper barely moves, and a large one barely strokes — it locks, and the brace carries the motion as a spring.

Dynamics

The damper is only as good as its brace

A viscous damper added to a frame is held by a brace, and the brace is a spring in series with it. That pair has a ceiling: the most damping it can add is (√(1+κ) − 1)/2, κ being the brace's stiffness over the frame's, reached at one damper size and lost on either side of it. A brace as stiff as the frame allows 20.7%; a quarter as stiff, 5.9%. A damper larger than the best one is not better — it strokes less, locks, and turns the brace into a strut that moves the resonance instead of flattening it.

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