Structural Engineering & Statics
where the load goes
Concept

Box girder — where it appears

A beam whose section is a closed cell, usually rectangular or trapezoidal, with the deck as one flange. The closed form gives it a torsional stiffness hundreds of times an open section's, and the price is that the cell must be held square by diaphragms or it distorts along its length.

Named by 13 essays across one field — each of them below, with the objects they name alongside it.

An eccentric load is three load cases, and only two of them are checked. A line load of 40 N/mm at 1.5 m from the axis of a 3.0 by 2.0 m box, replaced by the three cases it is equivalent to. Bending is the load on the axis. The torque 60 kNm per metre then splits into a set of edge forces that drives Bredt's shear flow and distorts nothing, and a set with the flange forces reversed — 10.0 kN/m up one web and down the other, 15.0 kN/m across the flanges — which carries no torque at all and squashes the rectangle into the rhombus drawn behind it. Its generalised load is exactly half the torque, so a box girder spends half of an eccentric load's torsion on changing its own shape, and no torsion calculation contains that half.

The section that will not keep its shape

A box girder is closed, so torsion costs it almost nothing. What an eccentric load actually does to it is something a torsion calculation contains no term for — the rectangle becomes a parallelogram, in its own plane, along the whole length of the span.

sections · Box distortion The lag follows the shear, so it is worst at the supports. Effective width along the span of a simply supported beam under a uniform load, summed over 25 odd harmonics. Each harmonic has its own half-wavelength L/n and its own, smaller, effective width — 0.815 for the first, 0.400 for the third, 0.269 for the fifth — and near a support the short harmonics carry a bigger share of what little moment there is. So the working fraction is 0.603 at the support against 0.839 at mid-span, a difference of 23.6 percentage points on the same flange. The single sinusoid's answer, 0.815, is drawn as the flat line, and it is only right at mid-span. This is the behaviour a code reproduces by shortening L_e near a support, and the reason it does is that shear lag is driven by w‴, which is the shear force.

The flange works least where the shear is largest

Shear lag is driven by the shear force rather than by the moment, so the effective width of a wide flange is not a property of the beam. It is a function of position along it, worst at the supports, and a single number quoted for a whole span is right at mid-span and nowhere else.

sections · Effective width The distortion runs the length of the span, and a diaphragm stops it. Longitudinal stress at a corner of the box from distortional warping, along a 60 m span carrying 45 N/mm at 2.0 m off the axis. With no interior diaphragm it peaks at 73 N/mm², which is 83 per cent of the bending stress the girder was designed for. Two interior diaphragms take it to 18. The governing length is Winkler's: the distortion decays over 23.6 m, so a diaphragm helps its neighbours only if it is closer than that, and past it the spacing stops mattering.

One diaphragm is nearly none

A box girder's distortion decays over a length the section decides, and on a sixty-metre span that length is twenty-four metres. So a single diaphragm at midspan sits further from each end than the distortion can reach and removes a third of the problem; three diaphragms remove nine tenths. The spacing rule is not span over five — it is a property of the plates.

sections · Box distortion Two webs of one box, carrying different amounts of the same load. The equivalent stress on each web of a 3.0 × 2.0 m box along the half span, under 40 N/mm at an eccentricity of 1500 mm, with the bending stress alone drawn for comparison. The two webs carry the same bending and opposite torsion, so one gets the sum and the other the difference: at the quarter point the loaded web is at 116 N/mm² and the far web at 11, a ratio of 10.9. A check made on the section rather than on the wall reports the average of two numbers that differ by that much.

Three actions on one web

A load applied off the centreline of a box girder is three actions at once, and every textbook decomposes it into them. What the decomposition does not say is that all three land on the same piece of plate — added on one web and subtracted on the other, so the two walls of one section differ by a factor of ten.

sections · Multicell torsion The interior web pulls the shear centre toward itself. The shear centre of a two-cell box 600 mm wide and 300 mm deep on its centrelines, 10 mm walls, its interior web 200 mm from the left, its distance from the left against the interior web's thickness as a multiple of the other walls', on a logarithmic scale. The single box's middle is 300 mm (dashed) and the web is at 200 (dotted). A web a hundredth as thick as the walls moves it 0.6 per cent of the way; a tenth, 5; as thick as the walls, 27; half way at 5.4 times; a hundred times, 92.

The web that moves the centre and not the twist

A thin plate closing a channel into a box changes its torsion at once and its shear centre hardly at all. An interior web dividing a box into two cells does the reverse. Off the middle of a 600 mm box, a web as thick as the walls takes 38 per cent of the vertical shear — more than either outer web — and pulls the shear centre 27 mm toward itself, while the torsion constant changes by less than one per cent. The two walls are the same kind of wall; what differs is which flow they are part of.

sections · Shear centre The flange over the support works over a third of its width. The effective width of the flange as a share of its overhang along two continuous 20 m spans with a flange overhang of 3.0 m each side of the web, the support bearing over 1.0 m, under a uniform load, left out where the moment is under three tenths of its peak (solid), with EN 1993-1-5's factors over the regions they apply to (dashed) and the simply supported span's mid-span value (dotted). At the sagging peak, 7.6 m from the end, the flange works over 0.81 of its width; over the support, over 0.33. EN 1993-1-5 gives 0.83 in the span and 0.34 over the support, from effective lengths of 17.0 and 10.0 m; a single span of 20 m has 0.88 at mid-span.

The reaction that is made of short waves

Run a wide-flanged girder continuously over a support and its worst moment moves to the support. That moment is made by a reaction, a reaction is made of short waves, and a short wave puts its stress next to the web. Over the support of two 20 m spans with a 3 m overhang the flange works over a third of its width, against four fifths at the sagging peak, so the stress at the web there is 4.2 times the sagging peak for a moment only 1.7 times as large — and how much flange works depends on what the support bears on.

sections · Effective width An inclined web takes the shear centre off its line. Where the shear centre of a trapezoidal box 1,500 mm deep, 3,000 mm across the top and 2,000 mm across the bottom (15 mm top plate, 12 mm elsewhere) lies as an interior web from the top flange's middle is leaned so that its foot moves along the bottom flange (solid, dots at each position), against where it would lie with a vertical web at the inclined web's mid-depth position (dashed); the cross marks the box with a vertical web at the middle, 791 mm up. A vertical web moves it sideways only, within 2 mm of that height. The inclined web lifts it as it leans: with its foot 500 mm left of the middle it is −66 mm across and 800 mm up; 500 mm right, 66 across and 800 up.

The web that leans and lifts the centre

In a rectangular box an interior web pulls the shear centre sideways toward itself and leaves its height alone, because a vertical web cannot carry horizontal shear. Most steel box girders are trapezoids, and their webs lean. Leaning the outer webs puts the shear centre below the centroid instead of above it; leaning an interior web lifts the shear centre as well as pulling it, by up to 35 mm in a 1.5 m box; and a vee of interior webs meeting at the bottom flange drops it by as much as 200. A web that leans carries horizontal shear, and whatever carries horizontal shear decides where the shear centre is in height.

sections · Shear centre The tub girder's shear centre at three stages of its building. A steel trough 1,500 mm deep, 3,000 mm between its web tops and 2,000 mm across its 12 mm bottom flange, with 12 mm webs and a 400 × 20 mm flange on each web, drawn to scale. Open, as it is lifted, its shear centre lies 706 mm below the bottom flange, outside the steel entirely. Closed across the top by a bracing truss equivalent to a plate 0.50 mm thick, which carries shear and no bending stress, it is still 538 mm below. With a 250 mm concrete deck cast and hardened across the web tops it is 1,127 mm above the bottom flange, inside the box. The steel's centroid, 673 mm up, does not move until the deck is cast.

The box that is a trough until its deck is cast

A composite tub girder is a box only once its concrete deck has hardened. Before that it is an open steel trough — two leaning webs and a bottom flange — whose shear centre lies seven hundred millimetres below its bottom flange, in the air, and whose resistance to twisting is the thickness of its plates cubed. A light truss across the top is what holds it together while the deck is poured, and it is a plate for shear and nothing else: an angle of ordinary size is worth half a millimetre of steel. That half-millimetre multiplies the trough's torsion constant two thousandfold, and leaves its shear centre almost exactly where it was. The bracing closes the box for twisting; it does not close it for the shear centre.

sections · Shear centre A girder softer to short waves sheds a little of its support moment. For two continuous 20 m spans of a girder whose flanges are 6.0 m wide and 30 mm thick, 1.20 m apart on 24 mm of web — the flanges 97% of its stiffness — under a uniform load, each interior support 1.0 m long: the bending moment along the girder per unit load, sagging up — for a prismatic girder (dashed), which is what a global analysis gives with one effective width per span, and for the girder whose every wave of moment bends it with that wave's own effective flange (solid). The interior reaction falls by 0.88%, and because the support moment is the difference of two much larger moments, that becomes 4.6% of the support moment, 47.0 to 44.8; the sagging peak rises by 2.9%, 28.1 to 28.9.

The reaction moves one per cent and the moment five

Over the support of a wide-flanged continuous girder the flange works over a third of its width, so the girder there should be softer and shed moment into its spans. It does — but it is the reaction that the softness changes, by less than one per cent, and the support moment, a small difference of two large ones, moves by five. The stress spike over the support follows the reaction rather than the moment and eases by two. A prismatic analysis is right about the moments to a few per cent, and the spike stays a problem of the section.

sections · Effective width Rigid enough long before it is a plate. A box 4.0 m wide and 2.5 m deep on a 60 m span, 45 kN/m at 2.0 m off its axis, with three diaphragms, each diaphragm of racking stiffness K, against K over the girder's own stiffness against a point distortional load, 15.0 × 10⁹ N·mm/rad: the largest warping stress (solid) and the largest distortion, which drives the walls' transverse bending (dashed), each over its value with rigid diaphragms. At ten times the girder's stiffness the warping stress is 1.05 of the rigid value and the distortion 8.37 times it; at a hundred, 0.89 and 1.50; at a thousand, 0.99 and 1.05. Marked: a cross-frame of two 1,000 mm² diagonals (26 times) and a 10 mm plate (524 times).

The diaphragm that is a few angles

A box girder's diaphragm holds its section square against the distortion an eccentric load drives. It does not have to be rigid; it has to be stiff against the girder's own resistance to a point racking load, which is small — twice the section's frame stiffness times its decay length. A cross-frame of two 1,000 mm² diagonals is twenty-six times that and holds the warping stress at the rigid value, while a 10 mm plate is five hundred times it. The cross-frame lets the section rack three times as far as a plate, which leaves the walls bending at 5 N/mm² instead of 1.5, against 275 with no diaphragm at all. And its diagonals carry nearly a rigid diaphragm's force whatever their stiffness, so it is sized for strength.

sections · Box distortion Ring frames close together are a continuous foundation. The corner-angle change along a 60.0 m box 4.0 m wide and 2.5 m deep, its top flange an orthotropic deck of 14 mm plate, its webs 14 mm and its bottom flange 20 mm, under 45 kN/m at 2.0 m from its centre line, in milliradians, with ring frames whose crossbeam has a second moment of 1.7 × 10⁹ mm⁴ every 4.0 m and every 12.0 m (solid), and each row smeared into a continuous foundation of the same stiffness per metre (dashed). At 4.0 m the two are indistinguishable, 1.12 and 1.12 at the worst point; at 12.0 m the box distorts between its frames, to 3.45 against 3.33. Without any frames the same deck distorts 145.

The deck whose frame is its crossbeams

A steel box girder resists distortion as a beam on an elastic foundation, the foundation being its walls' transverse bending. When the top flange is an orthotropic deck, that foundation is almost absent between the crossbeams and concentrated at them, so the box sits on a row of springs, not a bed. Smeared into a bed, the springs give the right answer to within a few per cent as long as they are closer together than the bed's own decay length — and the rule holds for every crossbeam stiffness on one curve. Past that spacing the frames act as separate diaphragms and the warping stress between them doubles.

sections · Box distortion Whether the cross-frames' spacing matters depends on the crossbeams between them. For the 60.0 m orthotropic box 4.0 m wide and 2.5 m deep under 45 kN/m at 2.0 m from its centre line: the largest distortional warping stress against the spacing of cross-frames of two 1,900 mm² diagonals, with nothing between them (dashed), with open tee crossbeams every 4.0 m (light) and with ring frames every 4.0 m (solid); dotted, the ring frames alone, 4.1 N/mm². At 8 m, 3.8, 3.8 and 3.1 N/mm²; at 20 m, 21.8, 21.7 and 7.1; at 30 m, 60.8, 59.2 and 7.3.

The stiff frame among soft ones

A steel box girder with an orthotropic deck is held in shape by a frame at every crossbeam, and its designers usually add a stiffer frame — a pair of diagonals across the box — at every few crossbeams as well. Whether the soft frames between can be left to do the work depends on one length: how far a distortion reaches along a box held by the soft frames alone. Ring frames welded into the webs and bottom flange hold it to six and a half metres, and then cross-frames twenty metres apart barely change the distortion and add a peak of warping stress at each one, where a stiff frame grips a box that was distorting gently. Crossbeams that are only tees under the deck reach twenty-one metres, and then the box between the cross-frames distorts as if the crossbeams were not there.

sections · Box distortion The diagonals over the support are the girder's shortest-lived members. The years a diagonal's connection of the top bracing of a 1.5 m deep composite tub girder under a 250 mm deck, continuous over two 40 m spans, 480 kN lorries passing 2.5 m off its centre line lasts at two million such lorries a year, along a span, for fatigue categories 50, 71 and 90, logarithmic and drawn at 1,000 years where the range is below the category's cut-off and the life is without limit; dotted, 100 years. Where the deck is whole every category lasts without limit. At the worst diagonal, 34.5 m along, category 50 lasts 2 years, category 71 lasts 8 years, category 90 lasts 25 years.

The bracing nobody took out

The light truss that closes a steel tub girder while its deck is poured is usually left in once the deck has hardened, because taking it out from under a finished slab is more trouble than it is worth. Over most of a span that is harmless: the deck lifts the neutral axis to the top flanges and takes ninety-nine per cent of the shear round the top, and a passing lorry puts five newtons per square millimetre in a diagonal. Over an interior support, where the deck is cracked to its bars, the same diagonals sit six hundred millimetres from the axis and are handed a share of the torque, and each heavy lorry puts forty in them — the shortest-lived connections in the girder, on members nobody designed for anything after the day the concrete went off.

sections · Shear centre

Named alongside it

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

Shear flowWarpingDiaphragmDistortionTorsionShear centreCharacteristic lengthContinuityEccentricityEffective widthElastic foundationFatigue

All concepts