The deck whose frame is its crossbeams

The diaphragm that is a few angles gave a steel box girder’s diaphragms a finite stiffness and found that a light cross-frame of four angles does nearly everything a full plate does, because the yardstick for a diaphragm is the girder’s own stiffness against distortion, and a few angles pass it easily. Behind that yardstick was the picture of a box girder used ever since one diaphragm is nearly none: the box resists distortion as a beam on an elastic foundation, the foundation being the transverse bending of its four walls, continuous along the span, and the beam its distortional warping.

That essay closed on the box where the picture is least obviously true. The deck of a steel box bridge is rarely a plain plate. It is an orthotropic deck: a thin plate stiffened underneath by longitudinal troughs and carried every few metres by crossbeams, and its stiffness across the box comes from the crossbeams, not from the plate. A box like that has its frame concentrated at the crossbeams and almost nothing between them. This essay asks whether it still has a decay length, and whether the beam-on-a-bed picture survives.

A box with no frame between its frames #

The box is the earlier essays’ — 60 m long, 4 m wide, 2.5 m deep, loaded with 45 kN/m at 2 m off its centre line — with its top flange replaced by an orthotropic deck: a 14 mm plate, with the troughs under it running along the span. The webs are 14 mm and the bottom flange 20 mm. Its walls’ own resistance to distortion, the plate bending of a 14 mm deck, 14 mm webs and a 20 mm bottom flange round a 4 by 2.5 m frame, is almost nothing: $K_d = 2.5 \times 10^5$ N·mm per radian per millimetre of span. Left to that, the box distorts 145 milliradians under its eccentric load — eight degrees, which is not a girder but a mechanism. The troughs add warping area but no transverse stiffness, because they run the wrong way.

What holds the box square are ring frames. At each crossbeam the deck’s crossbeam — an inverted tee 600 mm deep, acting with a metre of deck plate, with a second moment of $1.7 \times 10^9$ mm⁴ — is welded to a vertical stiffener on each web and a transverse stiffener on the bottom flange, each with $1.5 \times 10^8$ mm⁴, making a closed frame round the box. Solved as a four-member frame under the distortional forces, as the box’s walls are, one ring’s stiffness against the corner-angle change is $1.70 \times 10^{11}$ N·mm per radian. Every 4 m along the span, that is $4.3 \times 10^7$ per millimetre if spread out — 170 times what the walls alone provide.

The question is whether it can be spread out. The box is now a beam on discrete springs, each holding the corner angle locally, with the walls’ warping rigidity carrying the distortion between them. Solved that way, with each ring a finite-stiffness diaphragm exactly as in the previous essay, and again with the rings smeared into a continuous foundation of their stiffness divided by their spacing, the two answers can be compared directly.

Four metres apart, there is no difference #

The hero figure shows them along the span. With rings every 4 m, the discrete and smeared distortions lie on top of each other: 1.12 milliradians at the worst point either way, the box’s corner angle rising from nothing at the end diaphragms over a few metres and staying almost level along the middle forty. An orthotropic box with ring frames every 4 m is a beam on an elastic foundation, to within the thickness of a line. The foundation is the rings, smeared.

With rings every 12 m the smeared answer is three times larger, because the same rings are spread over three times the length; and now the discrete answer departs from it. The box is held near the smeared value at each ring and bulges between them, to 3.45 milliradians against the smeared 3.33.

At 4 m the ripple between rings is 0.07 milliradians, six per cent of the distortion; the warping rigidity of the walls carries the distortion smoothly across a 4 m gap. At 12 m it is 0.48, fourteen per cent, and the curve has a shape of its own between each pair of rings: it dips where a ring holds it and bulges where nothing does. The box is beginning to behave not as a beam on a bed but as a beam on separate supports.

The decay length decides #

The previous essays’ central length is the decay length, the distance over which a disturbance to the distortion dies away. For the smeared box it is set by the rings’ stiffness per metre against the walls’ warping rigidity. With rings every 4 m it is 6.5 m. The next figure moves the rings apart.

The warping stress tells the same story as the distortion, more sharply. Up to about 8 m the discrete and smeared stresses agree within a few per cent: 4.3 and 4.1 N/mm² at 4 m. Beyond that they part. At 12 m the discrete rings give 9.7 against the smeared 7.0; at 20 m, 20.3 against 8.9, more than twice. The smeared foundation’s own decay length grows slowly as the rings spread, because the stiffness per metre falls, and the spacing overtakes it at about 10 m. Past that, a smeared bed is a fiction: the box distorts freely over most of each gap and is held only at the rings, which are then diaphragms in the previous essay’s sense, a decay length or more apart.

One rule for every crossbeam #

That is one box and one ring. The next figure asks whether the rule is general.

Divide each spacing by the smeared box’s decay length at that spacing, and the ratio of discrete to smeared stress collapses onto a single curve for crossbeams a third as stiff, as stiff, and three times as stiff as the one drawn. Below half a decay length the ratio stays within ten per cent of one; near one decay length it is 1.04 to 1.06; at twice the decay length it is 2.3. Whether ring frames can be smeared depends on one dimensionless number, their spacing over the decay length their own smeared stiffness gives the box.

This is the result the beam that sits on the ground would predict for any beam on springs. A Winkler foundation is a limit of discrete springs, and the limit is reached when the springs are closer than the distance over which the beam’s deflection can change shape. That distance is the decay length; a spring spacing well inside it lets the beam bridge between springs as if they were continuous. The ring frames are springs, the walls’ warping is the beam, and the rule is Winkler’s.

It also answers the previous essay’s question in the form a designer can use. The decay length of a smeared orthotropic box is

$$\ell = \left(\frac{4 E I_{dw}\, s}{K_{ring}}\right)^{1/4},$$

and since the spacing appears inside the fourth root, it grows only slowly as the rings are spread. Crossbeams at the 3 to 4.5 m spacing an orthotropic deck needs for its own sake, to carry wheel loads across its troughs, sit at about half to two-thirds of the decay length for the rings drawn, where the smeared answer is within six per cent. A stiffer ring shortens the decay length and pushes that ratio up, but only as the fourth root of its stiffness. For the spacings decks are actually built at, the beam-on-a-bed calculation is not merely an approximation for an orthotropic box; it is accurate.

What each crossbeam carries #

The rings are springs, and a spring holding a distortion carries a force. The next figure reads it out as each ring frame’s moment.

The rings along the middle of the span each carry about 53 kN·m of transverse bending at their corners, and the rings near the quarter points 57, where the distortion’s shallow peaks in the hero figure sit. Near the supports the rings carry less, because the end diaphragms hold the box square and leave the rings little distortion to resist. This moment arrives in each crossbeam on top of the wheel loads it was designed to carry across the deck, and it is largest where the box is loaded eccentrically for the whole span, which is exactly the load case a crossbeam designed for local wheels will not have been checked for.

The moment is set by the distortion each ring holds and its stiffness, and both are smeared quantities: a ring’s moment is its share, $s$, of the distortional load the smeared foundation carries, less the small part the walls’ warping carries along the span. Doubling the spacing doubles each ring’s moment and leaves their sum unchanged, until the spacing passes the decay length and the rings near the middle of each gap start doing more than their share.

A stiffer crossbeam buys little #

The last figure varies the ring’s stiffness at the 4 m spacing.

A crossbeam a tenth as stiff gives a peak warping stress of 5.2 N/mm²; the one drawn, 4.3; ten times as stiff, 4.2. Two things flatten the curve. The ring’s stiffness is limited by its weakest members, the web and bottom stiffeners, so ten times the crossbeam’s second moment is much less than ten times the ring’s stiffness. And the peak warping stress is not along the middle of the span but about five metres in from each end, where the distortion has to rise from nothing at the end diaphragm to its level along the middle. A stiffer foundation lowers that level, but it also shortens the distance over which the rise happens, and the warping stress is set by the curvature of that rise. Under a uniform load on a beam on an elastic foundation, the level of the distortion falls as the foundation’s stiffness and the distance over which it rises as its fourth root, so the end stress falls only as the square root.

Rigid diaphragms at the same 4 m spacing would give 0.9 N/mm², a fifth of the rings’ value. The difference is that a rigid diaphragm holds the corner angle at zero, so the distortion never builds up along the middle and there is no rise at the ends to warp the walls. A ring frame of finite stiffness holds the distortion at a level, not at zero, and the box’s end diaphragm, which is rigid, imposes the rise that the warping stress comes from. Warping stresses of a few N/mm² are small beside the bending stress in such a girder, which is why orthotropic boxes are designed with ring frames rather than diaphragms at every crossbeam; but the location of the peak, near the end diaphragm and away from midspan, is where a fatigue check of the corner welds should look.

A ring is a frame with no diagonal #

A ring frame resists distortion the way the truss with no diagonals resists shear: a closed rectangle with rigid corners, carrying by bending in its members what a diagonal would carry by axial force. That is why it is so much softer than a diagonal of the same steel, and why the previous essay’s four angles — a triangulated cross-frame — could match a plate diaphragm while the ring frame here, with members far heavier than four angles, falls well short of rigid, as the last figure showed. A ring frame is chosen not for stiffness but because it leaves the box open: there is no bracing across the inside of the girder, so inspection, services and fabrication pass through it.

Inside the ring, the distortion is shared by stiffness, as it always is when members share a displacement. The stiffest path takes the load is the rule, and it explains the two results that seem odd at first. The crossbeam, by far the heaviest member, carries the largest moment but contributes the least flexibility; the stiffeners, light by comparison, decide how much the ring resists. And the walls’ own plate bending, which in a box with thick plates is the whole of the foundation, here carries less than one part in a hundred of the distortion, because the rings are so much stiffer than the plates between them.

Three actions at the same corner #

Distortion is one of three actions an eccentric load puts into a box, and they meet at the same corners. Three actions on one web found that the web under the load carries the sum of the bending shear, Bredt’s torsional shear flow and the distortional warping stress, and the far web their difference. On the orthotropic box, the warping stress at the corners is small — 4.3 N/mm² at its peak with rings every 4 m, against a bending stress at midspan of more than 100 — and it peaks near the ends, where the bending stress is low and the shear is high. In a static check it hardly matters.

It matters where the section that will not keep its shape first said it would: at the welds. The ring frames’ corner moments, 53 to 57 kN·m in each, pass through the welds that join the crossbeam to the web stiffener and the stiffener to the web, and they cycle with every eccentric vehicle. The deck’s own wheel loads cycle the same welds. A ring frame designed for the deck’s local loads, and checked separately for the box’s global bending, can have its largest stress range in a load case neither check contains — a lorry in the outer lane, distorting the box, while its wheel stands over the crossbeam.

The ring, by hand #

The ring frame’s stiffness can be checked from the frame’s members. Under a distortional load the crossbeam and the bottom stiffener bend in double curvature across the box’s width, and the web stiffeners across its depth. For a closed rectangular frame of members $I_t$, $I_w$, $I_b$, the corner-angle stiffness is of the order $E/(b/I_t + b/I_b + 2h/I_w)$ times a geometric factor. With $b/I_t = 4{,}000/1.7 \times 10^9 = 2.4 \times 10^{-6}$, $b/I_b = 2.7 \times 10^{-5}$ and $2h/I_w = 3.3 \times 10^{-5}$, the stiffeners contribute 96 per cent of the flexibility: the crossbeam, by far the largest member, is barely bending. That is why ten times its stiffness changes the ring’s by only a few per cent, and why a designer who wants stiffer rings should stiffen the web and bottom stiffeners, not the crossbeam.

The smeared decay length then follows. The distortional warping rigidity is $I_{dw} = b^2h^2(bt_f + ht_w)/96$, which with the deck plate’s 14 mm and the webs’ 14 mm is $10^{14} \times 91{,}000/96 = 9.5 \times 10^{16}$ mm⁶; with $K_{ring}/s = 4.3 \times 10^7$ N·mm per radian per mm and the walls’ small share added, $\ell = (4 \times 2.05 \times 10^5 \times 9.5 \times 10^{16}/4.3 \times 10^7)^{1/4} = 6{,}540$ mm — the calculation’s 6.5 m.

Troughs, deck plate and rigid ends #

The troughs’ own warping. The longitudinal troughs add to the deck’s area and so to the box’s distortional warping rigidity, which lengthens the decay length and makes the smeared picture better still; the walls here are plain plates, so the decay length is, if anything, underestimated.

The deck plate between crossbeams. Between rings, the deck plate bends across the troughs under wheel loads, a local action this calculation leaves to the deck’s own design. It carries almost none of the distortion, but its bending stress adds to the warping stress at the same welds.

Rigid ends and a uniform load. A concentrated eccentric load — a heavy vehicle in one lane — drives a local distortion that decays over the same decay length; with rings closer than half of it, the local peak is smeared as well as the uniform one, but the ring nearest the load carries a larger share than its neighbours.

Still open: the box whose rings are cross-frames #

Every ring here is a closed frame of stiffened plates, bending at its corners. The commonest alternative in steel boxes is a lighter cross-frame — the four angles of the previous essay, triangulated across the box — at every second or third crossbeam, with the crossbeams between them left as open tees that bend only with the deck. Then the springs are of two kinds: stiff cross-frames at a wide spacing and soft crossbeams at a close one. Whether the soft rings in between carry enough of the distortion that the cross-frames can be spaced as diaphragms are, or whether the soft rings are so soft that the box between cross-frames distorts as if they were not there, is the question that the two kinds of spring put to a single decay length.