Structural Engineering & Statics
where the load goes
Concept

End rotation — where it appears

The angle a member's axis makes with its original direction at a support. It is fixed by the deflection and the shape of the load rather than by the material or the span, and it is what a bearing, an expansion joint or a cladding gap has to accommodate.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

One coefficient, and nothing else in it. The deflected shapes of one beam under four load cases, each scaled so that its mid-span deflection is the same, with the tangent at the left-hand support drawn on each. The end rotation is that deflection times a coefficient that depends only on the shape of the load: 3.20 for a uniform load, 3.00 for a load at mid-span, 2.99 for a triangular load, 3.60 for a load on half the span. Every material property, every second moment and the span itself cancel out of the ratio θL/δ, so a beam at any deflection limit has an end rotation that is known before anything about it is: at L/360 it is 8.89 milliradians, or 0.51 of a degree.

The angle nobody limits

Every serviceability rule in this collection limits a displacement. What a bearing, a joint and a cladding gap actually have to accommodate is an angle — and the angle is locked to the displacement by a coefficient that contains no material, no section and no span.

deflection · End rotation The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 200 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 659 mm — 3.3 times the bearing, and 70% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to.

The support that is not a point

A reaction is drawn as a single arrow because the equilibrium equations only need its total. Underneath the arrow is a bearing of some width, delivering a pressure over that width, and almost everything a designer would like to know about the region near a support is a consequence of the width the arrow does not have.

internal-forces · Support width The rotation a bearing sees is made before it arrives. The end rotation of a 12 m pretensioned beam, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, from the prestress's release at three days to thirty years, with sagging positive. Release turns each end 4.8 milliradians upward; creep takes it to 7.9 by day 28, when the beam is set on its bearings. The slab brings it back by 0.9, the surfacing by 0.1, and thirty years of creep take it to 8.1 upward. The band after erection is the imposed load (0.9 down) and a night with the top cooler (0.6 down) above the line, and a sunny day with the top warmer (1.0 up) below it. Every value in the band is below zero: under every load it will carry, the beam's ends still point up, and a level bearing is turned the same way for its whole life.

The angle made in the casting yard

A bridge bearing is designed for the rotation of the beam it carries, and the rotation is listed as a sum of load, temperature, creep and a tolerance. Followed through the life of a pretensioned beam, the largest term is none of those. It is the upward turn the prestress gives the beam's ends in the casting yard, before any bearing exists, and under every load the beam will ever carry its ends still point up.

deflection · End rotation The joint's moment takes the sign of the day it was cast. For two 12 m pretensioned beams, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, made continuous over their pier by a joint cast with the slab: the moment the joint is asked to carry to hold their ends together, sagging positive, over thirty years, with the joint cast when the beams are 28, 90 and 365 days old; dashed, the joint's cracking moment either way, 140 kN·m. Cast at 28 days the joint is sagged to 211 kN·m; at 90 days it ends at 40; at a year it is hogged to −214.

The joint cast over the pier

Two pretensioned beams made continuous over a pier by a joint cast with the slab have ends that are still turning when the joint goes in, and the joint stops them. What it is then asked to carry depends on the day it is cast. At four weeks the beams' creep is still lifting their ends and the joint is sagged to half as much again as its cracking moment. At a year the beams have nearly stopped and the slab's shrinkage hogs the joint by as much the other way. Cast at about sixteen weeks, it is asked for almost nothing.

deflection · End rotation A joint poured twice ends near where the slab's day puts it. For two 12 m pretensioned beams, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, made continuous over their pier: the moment in the joint, sagging positive, over thirty years, with its first stage — the diaphragm and the beams' bottom connection — cast at 28 days and the slab at 42, 90 and 365 (solid), against the joint cast once, with each slab (dashed). With the slab at 42 days it ends at 139 kN·m against 160 poured once; with the slab at 90 days it ends at 41 kN·m against 40 poured once; with the slab at 365 days it ends at −165 kN·m against −214 poured once. The first stage starts each history early and the slab's weight kicks it into hog on the day it is cast; after that each solid line runs beside its dashed one.

The joint poured twice over the pier

A continuity joint over a pier can be cast in two stages — the diaphragm and the beams' bottom connection first, the slab weeks later — in the hope that the early pour catches the beams' creep and the late one leaves the slab's shrinkage little to do. It does neither. From the slab's day on the composite section holds exactly what a single pour on that day would have held, and the first stage adds only its own small account, of either sign. It trims a year-old slab's hog by a fifth, and it asks a joint with half the cracking moment to carry the beams through the months between.

deflection · End rotation

Named alongside it

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

BearingContinuityCreepPrestressShrinkageCompatibilityComposite sectionRestraint momentToleranceBending momentCamberCladding

All concepts