Structural Engineering & Statics
where the load goes
Concept

Suspension bridge — where it appears

A bridge whose deck hangs from a cable draped between towers and anchored at its ends, the cable carrying the load in tension along its sagging shape. Where the cable is anchored — to the ground or to the deck itself — decides whether its tension stiffens the deck or is cancelled by it.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The same bridge by the two theories it might have been designed by. Deck moment along a 500 m suspended span with half of it loaded, computed twice. Elastic theory treats the deck as a beam and applies the cable's extra tension as an upward load: 80.1 MN·m. Deflection theory keeps the cable's total tension acting on the deck's own deflected shape — a geometric stiffness — and returns 56.3, which is 30 per cent less. The extra cable tension is very nearly the same in both (6.20 against 6.11 MN), so nothing about the cable is in the difference: it is entirely the H·v″ term the older theory drops.

The tension that was left out

A suspension bridge's deck sits on a cable pulling hard along it, and a member with a large tension in it is stiffened by that tension. Leaving the term out of the deck's own equilibrium is what elastic theory does, and on a long span it asks for fourteen times the girder.

structures · Stiffening girder The same bridge, anchored to the ground and to itself. The deck's bending moment under the live load, for a 500 m suspension span with a 50 m sag, a deck of EI 1.03 × 10¹² N·m², 100 kN/m of dead load and 20 kN/m of live load on half the span. Anchored to the ground, the deck carries at most 56.3 MN·m, because the cable's whole tension works on its deflected shape. Anchored to the deck's own ends, 83.0 MN·m — 1.47 times as much, and a little more than Rankine's elastic theory, 80.1, which leaves the tension out altogether. The deck's compression acts on the same deflected shape as the cable's tension, with the opposite sign, and cancels it.

The bridge that pays for its own anchorage

A suspension bridge's deck is light because the cable's tension works on its deflected shape and stiffens it. Tie the cable to the ends of the deck instead of to the ground and the deck must carry the same pull as a compression — which acts on the same deflected shape with the opposite sign and cancels the stiffening exactly. A self-anchored bridge is designed by the theory the long suspension bridge was invented to escape, and the price grows as the square of the span.

structures · Stiffening girder A load on one span moves the other. The change in level along two 100 m spans of stressed ribbon at a sag of 2.00 per cent, carrying 35 kN/m, when 20 kN/m of live load is added to the left-hand span, with a rigid pier (dashed) and with a pier of 200,000 kN/m (solid). With a rigid pier the loaded span sags a further 195 mm and the other does not move. With a pier of 200,000 kN/m the pier top leans 25 mm towards the load, the loaded span sags 358 mm and the unloaded span rises 191 mm; the pier carries 4,953 kN of the 9,440 kN a rigid pier would.

The pier that carries only the difference

A stressed ribbon continued over a pier pulls on it from both sides, and under its own weight the two pulls cancel. Load one span and they do not: the pier is asked for the difference, nearly ten meganewtons for a crowd on one of two hundred-metre spans. Whether it gives it depends on its stiffness against a number the ribbon supplies itself, and whatever the pier does not carry, the ribbon turns into movement — a third of a metre of extra sag in the loaded span and a fifth of a metre of rise in the empty one.

structures · Stressed ribbon

Named alongside it

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

Geometric stiffnessCable stiffnessCompatibilityDeflection theorySecond-orderStiffening girderBending momentBucklingDeckFunicularLive loadLoad-sharing

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