Structural Engineering & Statics
where the load goes
Concept

Second order analysis — where it appears

Analysis on the deformed structure, so that the loads acting through its deflections add to its moments. For a frame the effect is roughly an amplification of the sway moments by 1/(1 − 1/αcr), and it matters wherever αcr is below about ten.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A milliradian of slack costs the brace more than a milliradian of crookedness. For the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side by braces of 5,691 kN·m/rad — 2.0 times the 2,846 that just reaches 5,627 kN·m — erected with 5.0 mrad of initial twist: the moment in the middle brace when the girder reaches 5,627 kN·m, against each brace's slack (solid), and against the same amount added instead to the girder's initial twist, the braces tight (dashed). With tight braces and no extra twist, 34.1 kN·m; each milliradian of slack adds 8.0, each milliradian of initial twist 6.8. At 10.0 mrad, 114.0 against 102.4.

The braces with slack in their bolts

A cross-frame bolted into clearance holes resists nothing until the girder it braces has twisted far enough to take up the clearance, and then resists fully. It is a two-way brace that starts late. The girder, erected a little crooked, twists freely into its braces while the load is still small, and from then on the braces hold a girder that is more crooked than the one they were fitted to. The slack acts as an imperfection, and a costly one: each milliradian of it adds more to the middle brace's force than a milliradian of extra crookedness would, and no stiffness takes it back out. On a 24 m plate girder braced at twice the stiffness it needs, three milliradians of slack — a pair of ordinary clearances over its depth — raise the middle brace's force by 70 per cent.

stability · Beam bracing Where the knees begin to alternate. The shakedown factor (solid) and collapse factor (thin) of a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, on pinned bases, with 160 kN of gravity that comes and goes and a 1,000 kN roof carried at its knees, against the peak of a wind that reverses, first-order (dashed) and amplified. First-order, the shakedown and collapse factors are one line until the knees alternate, from a wind of 110 kN; amplified, from 90 kN. At 60 kN: shakedown 0.727 first-order and 0.692 amplified, collapse 0.727 and 0.692. At 120 kN: shakedown 0.533 first-order and 0.488 amplified, collapse 0.571 and 0.539. At 200 kN: shakedown 0.348 first-order and 0.326 amplified, collapse 0.400 and 0.366.

The roof that makes the wind worse every storm

A portal frame's shakedown check is made on its elastic moments, and the elastic moments a heavy roof sways are larger than first-order analysis says: each storm's wind is amplified by 1/(1 − N/Ncr), on its way out and on its way back. On pinned bases under a 1,000 kN roof that takes the wind at which the knees start to alternate from 110 kN down to 90, and the shakedown load down by 4.8 per cent where gravity leads and 8.4 where the wind does. Fixed bases are worth more under a heavy roof than without one, but by half again, not twice.

materials · Shakedown

Named alongside it

The objects these essays reach for when they reach for this one.

AmplificationBracingBucklingImperfectionJoint stiffnessLateral-torsional bucklingMechanismMode shapePlastic hingePortal frameRatchetingShakedown

All concepts