Structural Engineering & Statics
where the load goes
Concept

Thermal restraint — where it appears

The holding of a heated or cooled member by the structure around it, so that part of its thermal strain is refused and becomes force. The force depends on the restraint's stiffness against the member's own, and if either yields some of it stays behind when the temperature returns.

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

The lining that carries less for being weaker. Bending moment and hoop thrust in a circular lining, against the lining's own bending stiffness, both as fractions of the free-ring values. The ground arrives already stressed — 500 kPa vertically and 300 horizontally at K₀ = 0.6 — and the difference between them tries to squash the hole into an ellipse. A lining stiff enough to refuse absolutely collects the whole distortion pressure, p₂R²/3 = 300 kNm/m; one flexible enough to go with the ground collects nothing, because there is no curvature change left to resist. The thrust is the flat line: it comes from the mean stress rather than the difference, so it does not move at all. Putting 8 joints in this ring drops the moment to 34% of the solid one and leaves the thrust exactly where it was, which is why a segmental lining is jointed and why the intuition carried over from a beam is inverted here.

The lining that is stronger for being weaker

A tunnel lining is not loaded. The ground arrives already stressed and the hole wants to squash into an ellipse; the lining's only job is to refuse, and how much moment it collects depends entirely on how hard it refuses. Make it stiffer and it takes more. Make it flexible — put joints in it, make it thin — and it takes almost none, while the hoop thrust it carries does not move at all.

structures · Tunnel ring The more it carries, the less it pulls when the fire is out. For a 9 m 457 × 191 × 67 steel beam held at its ends at 10 per cent of its own axial stiffness through the fire: the tension pulling on its ends once it has cooled (solid) and the largest tension while it is still hot (dashed), against the load it carries as a share of its cold plastic moment; dotted, the straight beam's 1,669 kN. At 0.10, 1,017 kN cold and 41 hot; at 0.39, 613 and 293; at 0.54, 477 and 397. From 0.56 the beam can find no sag that carries its load near the fire's peak and runs away.

The beam that pays the fire back in sag

A steel beam held at its ends and kept straight through a fire yields short on the way up and pulls on its connections with its plastic shortening on the way down. A beam carrying a floor does not stay straight. Squeezed by its own expansion, it forms a hinge at about 500 °C and bows out of the way, hangs as a catenary while still hot, and keeps a metre of sag when it cools — length it no longer has to be pulled for. It ends the fire pulling a third as hard as the straight beam, nearly whatever holds it, and the hard pull comes earlier instead: at 800 °C, on connections that are as hot as it is.

materials · Fire

Named alongside it

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

CatenaryConnection designConstruction sequenceEarth pressureFireFlexibility ratioHoop tensionImposed deformationJoint stiffnessLower-bound theoremPlastic hingePlastic strain

All concepts