Structural Engineering & Statics
where the load goes
Concept

Stiffening girder — where it appears

The deck beam of a suspension bridge, whose job is to spread a concentrated load along the cable rather than to carry it across the span. Its required stiffness depends on how much the cable's tension stiffens it, which is why deflection theory and self-anchoring change its size so much.

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 Double the span and the deck is nearly three times nearer buckling. The factor on its own dead load at which a stayed deck buckles, against the main span, on logarithmic axes, for a deck of 0.50 × 10¹² N·m² under 150 kN/m with stays every 15 m from towers 0.40 of the half-span tall. At 300 m it is 40.7; at 600 m, 14.7; at 1,000 m, 6.9; at 2,000 m, 2.3. Beyond 600 m the line falls with a slope of 1.53; it crosses ten, below which a design has to follow the deck's deflections into its forces, at 805 m.

The deck the stays squeeze

Every stay of a cable-stayed bridge pulls the deck towards its tower, and the deck carries all those pulls as a compression that is largest at the tower — 95 MN in a 1,000 m span under 150 kN/m. The stays hold it up and hold it straight, as springs under a strut. The compression grows with the square of the span and the springs soften with it, so the factor on its own weight at which the deck would buckle falls as the span to the power one and a half: 41 at 300 m, 6.9 at 1,000 m, 2.3 at 2,000 m. A deck four times stiffer only doubles it, and the long stays' sag, which softens them for traffic, hardly enters.

structures · Stiffening girder

Named alongside it

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

Second-orderBucklingCable stiffnessCompatibilityDeflection theoryGeometric stiffnessSuspension bridgeBending momentCritical loadDeckElastic foundationFunicular

All concepts