Structural Engineering & Statics
where the load goes
Concept

Catenary — where it appears

The hanging shape a cable takes under its own weight, and the mechanism a floor falls back on when it has lost its support. It carries load in pure tension, which is why it is the shape a failed floor falls back on and why the connections at its ends are what decide whether that helps.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The cable and the arch are the same curve. The shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is.

The shape that carries itself, and the arch that is its reflection

Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is.

structures · Funicular The same restraint, twice, with opposite signs. A 4 m strip of 200 mm slab whose ends cannot move apart, against deflection measured in its own thicknesses. The flat line is what a yield-line calculation gives, which is what the same strip would carry if its ends were free: 30.0 per unit width. The rising branch is compressive membrane action — the deflected strip is forced into an arch — and it peaks at 116.6, which is 3.89 times the yield-line load, at a deflection of 0.24 of the thickness. Past that the arch runs out of depth and the load falls back to the flexural one; past a deflection of one thickness there is no arch left and the reinforcement starts carrying the strip as a cable. It gets back to the arch's load at 2.17 thicknesses, which is one part in 9 of the span — a sag nobody would design for and exactly what a floor does instead of falling.

The force nobody put in the model

A slab strip whose ends cannot move apart is not the strip in the yield-line calculation. Deflecting shortens the chord between its ends, the ends do not come in, and the strip is forced into an arch — worth four times the load it was designed for, at a movement nobody would see.

internal-forces · Membrane action A cable alone goes to a kink, and a kink is not a road. A point load of 1000 at mid-span of a 900 m suspended deck. The upper shape is the cable with no girder at all: two straight lines meeting under the load, because a cable takes the funicular shape of whatever is on it and the funicular of a point load is a kink — 0.0083 radians of it here. The lower shape is the same cable with the girder present, peaking at 1.125 against the bare cable's 1.873. The girder is not carrying the load — it takes only 17% of it — it is spreading it, over a characteristic length of √(EI/H) = 183 m, and what reaches the cable is spread over that length rather than arriving at a point.

The deck is not there to carry the load

A cable takes the shape of whatever is on it, which is exactly the problem — under a point load its shape is a kink, and a kink is not a road. The stiffening girder exists to spread the load until what reaches the cable is something the cable's own shape is right for.

structures · Stiffening girder The cable pays the factor the plateau saved. The force a beam of two 8.0 m spans over the column, 600 mm deep, plastic moment 800 kN·m, its far ends held against rotation and against spreading (plastic collapse load 400 kN) reaches after a sudden loss divided by the load it carries — the dynamic factor — against that load as a share of the plastic collapse load. Below the plateau the beam is a linear spring and the factor is 2. On the bending plateau it falls to 1.18 at 0.90 times the collapse load, because a flat resistance does its work at full strength from the start. Past it the beam becomes a cable, whose resistance is again straight, and the factor climbs back: 1.73 at 1.5 times, 1.88 at 2, 1.97 at 3.

The cable that pays the factor back

A frame that loses a column carries the floor across the gap first by bending and then, as the beam sags, as a cable. The hope is that a path which stiffens as it deflects needs less than the factor of two a suddenly loaded spring does. On the bending plateau it does — 1.18 at nine tenths of the collapse load. But past one depth of sag the cable is a straight line again, and the factor climbs straight back toward two: 1.73 at one and a half times the collapse load. And the connections are asked for a tenth of a radian on the way.

structures · Robustness

Named alongside it

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

Horizontal thrustForm-findingFunicularMembrane actionProgressive collapseRobustnessSag ratioAlternative load pathCharacteristic lengthCollapse mechanismCompatibilityDeflection

All concepts