Structural Engineering & Statics
where the load goes
Concept

Cantilever — where it appears

A member supported at one end and free at the other, whose root moment is the whole of the load times its distance from the support. It is the arrangement with the largest moment for a given span and load, which is why it appears as a construction condition far more often than as a finished one.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

The corner columns take what the middle ones did not. Axial stress in the columns across one flange face of a 30 by 40 m framed tube, at the base. Plane sections says the flat line: every column on the face at the same distance from the neutral axis, therefore at the same stress. The solved distribution is the curve — 69.4 N/mm² at the corner against 18.2 in the middle, a ratio of 3.81. The middle columns lag because the only route the axial force has into them is the in-plane shear of the spandrel frame, bay by bay from the corner. The face is carrying its resultant on an effective width of 51 per cent, and the tube deflects as though its second moment were 72 per cent of the gross.

The corner columns take more than their share

A framed tube is a hollow cantilever, and a hollow cantilever's flange ought to be uniformly stressed. It is not, and the reason is that the only route the axial force has into a column in the middle of a face is the in-plane shear of the frame — one bay at a time, from the corner inwards.

structures · Framed tube Every section is hogged and sagged before the bridge exists. The bending moment envelope of a launched deck, section by section along its own length, taken over every position of the launch. In service each section has one sign; during the launch 100 per cent of them see both, because each passes over every pier and through every span on its way out. The worst launch moment is 42568 kNm against 40500 in service, and with no launching nose at all it would be 185977. That is why a launched bridge is a constant-depth box with symmetric flanges: the design case is not a load, it is a history.

Every section was somewhere else

A bridge pushed out over its piers subjects each of its cross-sections to a history rather than to a load case. Every one passes over every support and through every span, so the design envelope is the envelope of envelopes — and no in-service condition produces it.

structures · Launched bridge Two diagrams for one load, and the second one has no straight-beam ancestor. Bending moment and torsion round a 90° arc of radius 6 m under a uniform load, built in at one end. The bending peaks at 576 and the torsion at 329, 57% of it. Both are zero at the free end and largest at the support, which is where a curved cantilever's bearing has to hold a torque it was probably not asked for.

The torque that has nowhere to go

A curved beam on two supports splits its torsion between them, and the two halves cancel at mid-span. A curved cantilever has one end, so every increment of torque accumulates toward it — and the largest action at the root of a curved balcony is one that a straight beam does not have at all.

internal-forces · Curved in plan The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything.

The plate that is only as big as it is thick

A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all.

connections · Base plate At the tip it is the tension flange that swings. The buckled shape of a 3.00 m cantilever of 457 mm universal beam carrying a load at its tip at the shear centre, root built in: slope and warping held, seen from above: the sideways movement of the top flange and of the bottom flange along the length, scaled so the larger is one. The cantilever hogs, so its bottom flange is in compression, and that flange bows out furthest 0.58 of the way from the root, where it moves 0.15 — and has come back to 0.00 by the tip. The top flange, in tension, swings out to the full 1.00 at the tip. The section twists as it goes, so the flange that is doing the buckling is not the flange that moves most at the end. Critical moment at the root: 1266 kN·m.

The flange that swings is in tension

A cantilever hogs, so its bottom flange is the one in compression and the one that buckles. At its tip the bottom flange barely moves: the section turns about it, and the flange that swings out is the top one, in tension. So the restraint that matters at a cantilever's tip is on the flange every rule for spans calls the wrong one — and the root, which a drawing shows as a single line, moves the critical moment by a factor of thirty-four.

stability · Moment gradient In a V-shaped canyon the rings carry the water, and the top rings more than theirs. The water pressure down the crown of a dam 100 m high in a V-shaped canyon 200 m wide at the crest, 5 m thick at the crest and 25 m at the base, with its rings at 133.6° (dashed), divided between the horizontal ring at each depth (solid) and the vertical cantilever at the crown (thin) so that the two deflect alike. At 4 m below the crest the ring carries 195% of the water there — the cantilever, loaded from below and free at the top, leans on it; at 49 m the cantilever carries 11%; at 89 m, 6%. Over the whole reservoir the cantilever carries 10% of the water and the rings the rest.

The water the valley floor takes

An arch dam is not a stack of rings. At its crown it is also a vertical cantilever fixed into the valley floor, and the water at each depth is shared between the ring there and the cantilever in whatever proportion makes them bend alike. In a V-shaped canyon the short lower rings are stiff and carry nearly everything, and the top rings carry up to twice their own water because the cantilever leans on them. In a U-shaped canyon the cantilever takes half the reservoir and bends the heel into tension that the ring formula never sees.

equilibrium · Curved surface pressure Out of the bed, the beam loses its own weight's help. For the 12 m pretensioned beam, 300 × 700 mm, with eight strands at 220 mm below its centroid: its self-weight moment along its length in kN·m, sagging positive — in the casting bed, which holds it at its ends (dashed), and on bearers 1.0 m in from each end (solid). Over the cantilevers the stored beam hogs, 2.6 kN·m at the bearers; between them its moment is the bed's less a constant 31.5 kN·m, the bearers' reactions acting a metre closer in. Mid-span drops from 94.5 to 63.0. That constant is the relief the top fibre loses everywhere between the bearers: 1.29 N/mm² for each metre of inset.

The beam that leaves the bed

A pretensioned beam's debonding is designed for the moment its strands are cut, when the casting bed holds it at its ends and its own weight is the largest relief its top fibre will get. Then it is lifted by loops a little in from its ends and stored on bearers a little further in. Setting the supports a metre in takes 31.5 kN·m off the self-weight moment along the whole length between them, not just at the ends, and the schedule that passed transfer with 0.07 N/mm² to spare has room for 51 mm of it. In the yard's conditions the bearers may come in 0.57 m; a metre in needs a third strand sleeved, and past 2.2 m nothing does.

sections · Prestress limits The yard's schedule has no room for the beams above. For the 12 m pretensioned beam, 300 × 700 mm, with eight strands, stored on bearers 1.0 m in from each end, debonded for the bed and the yard, with two beams stacked on it: the top fibre's stress along half the beam, N/mm², compression positive, with the stack lined up and landing 100 and 250 mm outside its bearers; dotted, the yard's limit of −3.0. Lined up, the worst is −2.98; 100 mm out, −3.24; 250 mm out, −3.63. Every beam above costs the top fibre 0.13 N/mm² for every 100 mm, at every section between the bearers.

The beam at the bottom of the stack

A precast yard stacks pretensioned beams to save space, each on bearers resting on the one below. If the bearers line up, the weight above goes straight down through them and the bottom beam never knows it is there. If they land outside its own bearers, the stack hogs the bottom beam at every section between them and takes from its top fibre relief that its debonding schedule had already spent. A schedule built for the yard leaves the bearers of a stack of three seven millimetres to be out of line. A bottom beam with a schedule of its own can be given half a metre.

sections · Prestress limits On the road the beam's top fibre has nothing left. The top fibre along the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, the worst of the two bounces at each section: on its yard bearers standing still and bouncing ±30 per cent, and on a trailer carrying it 1.5 m from its front end and 2.5 m from its rear standing still and bouncing; dotted, the yard's limit of −3.0 N/mm². Yard bearers, still, −2.98 at 2.0 m; yard bearers, ±30 per cent, −3.26 at 9.8 m; trailer, still, −4.68 at 9.5 m; trailer, ±30 per cent, −4.88 at 9.5 m.

The beam the haulier holds

A pretensioned beam leaves the yard on a trailer, carried on a turntable and a bogie the haulier places, and bounced on them at road speed. The bounce works both ways: a wheel rising adds to the beam's weight over its overhangs, and a wheel falling away takes weight off — and with it the sag that was relieving the beam's top fibre between its supports. On the yard's own bearers a bounce of 30 per cent takes as much from that fibre as a stack placed 100 mm out of line. A rear overhang of 2.5 m takes five times as much. With the yard's schedule, a trailer may hold this beam only within half a metre and three quarters of a metre of its ends.

sections · Prestress limits The road is the harshest stretch of the beam's journey. The worst top fibre, at any corner of any section, of the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, in N/mm² with tension negative: on the yard's bearers; on the haulier's trailer, 1.5 m and 2.5 m from its ends with a bounce of ±30 per cent; lifted by loops 1.0 m in from each end on slings at 60° with a snatch of ±20 per cent; slewed at 0.10 g; and set down on bearings sloping 1 per cent across it; dashed, the stored concrete's limit of −3.0. Yard bearers, −2.98; trailer, −4.88; lifted, −2.97; slewed, −3.13; set down, −1.79.

The beam hung from its loops

A pretensioned beam is cast in a bed, stored on bearers, hauled on a trailer and then lifted by a crane onto its bearings — and at every stage its top fibre, which the strands put in tension, has to stay inside the limit its debonding schedule was drawn for. The trailer, with a bouncing wheel and supports where the haulier put them, was the harshest stage the yard essays found. The last few metres, from the trailer to the bearings, are gentler: hung from loops at the yard's bearer positions the beam is on its bearers again, set down it is on the support its middle was designed for, and only a slew of the crane pushes it a little past its limit. Where the loops are cast decides how much margin the lift has, and the best place for them is not where the yard put its bearers but where the bed held the beam: at its ends.

sections · Prestress limits Where real roots sit on the warping curve. The elastic critical moment of a 3.0 m cantilever of 457 mm universal beam with its tip load at the shear centre, with its root holding the lateral slope and resisting warping through a spring, against the spring as a multiple of EIw/L on a logarithmic scale — from 521 kN·m free to warp to 1,266 built in — with four real roots placed on it. Top storey, column ends: 2.5 times EIw/L, 791 kN·m, 36% of the way; 15 mm plate, column runs on: 5.9 times EIw/L, 945 kN·m, 57% of the way; 40 mm plate, column runs on: 6.8 times EIw/L, 970 kN·m, 60% of the way; 15 mm plate, column boxed: 19.4 times EIw/L, 1,124 kN·m, 81% of the way.

The root that belongs to the column

A cantilever's root must hold its flanges against warping at about fifty times the cantilever's own warping stiffness to count as built in, and doubles its buckling moment if it does. A flush end plate is the usual root, so it ought to be the plate's thickness that decides. It is not. The flanges warp by turning in plan in opposite directions, the plate passes that to the column as two opposite torques, and an open column twists between them more easily than the plate bends: three quarters of the root's flexibility is the column. An ordinary root is worth about six times the cantilever's stiffness, halfway up the curve, and a thicker plate barely moves it. Boxing the column does.

stability · Moment gradient

Named alongside it

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

DebondingPrestressPretensioningTransferWarpingContinuityEffective lengthImpact factorLateral-torsional bucklingLiftingSelf-weightStiffness

All concepts