The diaphragm that is a few angles

One diaphragm is nearly none took a box girder 4.0 m wide and 2.5 m deep over a 60 m span, loaded 2.0 m off its axis, and found that the distortion an eccentric load drives — the section refusing to keep its shape — decays along the girder over a length the section decides, 23.6 m for this box. A diaphragm holds the section square where it stands, and its reach is that decay length. Three at the quarter points brought the warping stress at the corners from 73 N/mm² to 10.

Every diaphragm there was rigid: a full plate across the box, welded on all four sides, holding the corner angle exactly. The essay closed on what a diaphragm actually is — “diaphragm stiffness properly modelled, and the residual distortion a real plate leaves; cross-bracing instead of a plate diaphragm, which is the standard steel detail and is a truss solving the same racking.” This essay gives the diaphragms a stiffness, and asks how stiff they need to be.

The yardstick is the girder’s own stiffness #

A diaphragm resists racking: the change of the box’s corner angle, $\gamma$, from a right angle. Its stiffness is the distortional moment it supplies per radian of $\gamma$, in N·mm per radian. To say whether a diaphragm is stiff, there has to be something to compare it with, and the natural thing is the girder itself.

The girder resists distortion as a beam on an elastic foundation: the section’s frame bending is the foundation, with a stiffness $K_d$ per unit length, and its distortional warping is the beam. A beam on an elastic foundation pushed at one point far from its ends deflects there by $P\lambda/2K_d$, where $1/\lambda$ is the decay length, so its stiffness against a point load is

$$K_g = \frac{2K_d}{\lambda} = 2K_d \times \text{decay length}.$$

For this box $K_d$ is $3.19 \times 10^5$ N·mm per mm per radian and the decay length 23.6 m, so $K_g = 15.0 \times 10^9$ N·mm per radian. A diaphragm whose racking stiffness is large compared with $K_g$ holds the section nearly square where it stands; one comparable with $K_g$ shares the job with the girder and holds it only partly.

The two curves say that “rigid enough” is not one threshold. The warping stress at the corners — the longitudinal stress the original essay was about — is at its rigid value once the diaphragms are about ten times the girder’s point stiffness, and wanders within 10 per cent of it beyond. The distortion itself, the corner angle the section is left with, needs a thousand times. A diaphragm can be stiff enough for the stress it was put there to control and still let the section rack several times further than a rigid one would.

The reason is in what each quantity measures. The warping stress is made of the curvature of $\gamma$ along the span, and a diaphragm ten times stiffer than the girder’s own resistance already forces the distortion curve into nearly the rigid shape between diaphragms. The distortion itself is the height of that curve, and at the diaphragm it is whatever the diaphragm’s own flexibility lets through: the reaction divided by the stiffness. Halving that needs twice the stiffness, every time.

A few angles against a plate #

What sort of diaphragm is ten, or a hundred, or a thousand times the girder’s point stiffness?

A plate diaphragm welded round the inside of the box is sheared by the racking: the box’s corner angle changes by $\gamma$, and the plate inside it is strained in shear by exactly $\gamma$. Its stiffness is its shear stiffness over its whole area, $K = G\,t\,b\,h$. A cross-frame — two diagonals across the box, one in tension and one in compression as it racks — is only stretched by the racking’s effect on its length: a diagonal of length $d$ lengthens by $\gamma\,bh/d$, so the pair has $K = 2E\,A\,b^2h^2/d^3$.

The plate is two orders of magnitude stiffer than the frame. A 10 mm plate is 524 times the girder’s point stiffness; a cross-frame of two 1,000 mm² diagonals — a pair of 80 by 80 by 7 angles, say — is 26 times. Both are past the ten at which the warping stress is held; the plate is past the thousand at which the distortion is too.

The comparison is the geometry of the two load paths. A plate resists racking directly, by shear across its whole area; a cross-frame resists it at second hand, by the change in length racking makes in its diagonals, which is a fraction $bh/d^2$ of what a plate’s shear would be at the same corner angle, multiplied by the frame’s far smaller area. It is the difference between a truss and a web once more: a truss’s stiffness against shear lives in a few members’ axial strains, a web’s in all of its area.

What the bracing lets through #

With plates the distortion is a row of small humps between diaphragms that fall nearly to nothing at each, the rigid picture. With cross-frames the section racks at the cross-frames too — 1.8 milliradians at each, against the 2.0 of the humps between — so the curve is nearly flat at about two milliradians the whole way. With a single light diagonal it is a broad hump of 7.5 milliradians that the diagonals barely dent. All three are tiny next to the 112.5 milliradians of the box with no diaphragm.

That flat curve is why the warping stress is held while the distortion is not. The warping stress lives in the curvature of this curve, and a flat curve at two milliradians has as little curvature as a row of humps at nought; the plate bending in the walls lives in its height.

The plate bending the cross-frames leave #

The walls of a distorting box bend transversely, each plate curving between its corners as the rectangle becomes a rhombus. That bending is largest at the corners, which are where the webs are welded to the flanges, and its stress range under traffic is what makes box girders crack at their corners rather than anywhere the bending stresses would suggest.

The plate bending is proportional to the distortion, so it follows the dashed curve of the first figure. Cross-frames of two 1,000 mm² diagonals leave 5.0 N/mm² of transverse bending at the corners; doubling their area, 2.9; a 10 mm plate, 1.6; a rigid diaphragm, 1.5.

In proportion, the cross-frame leaves three times as much as a plate. In absolute terms it leaves 5 N/mm², against the 275 the box would carry with nothing, and a stress range of 5 N/mm² at a fillet-welded corner is below the cut-off of any detail category a box girder’s corner is given. For this box, a cross-frame of two angles does the job a plate does, and the plate’s extra stiffness buys a reduction from 5 to 1.6 N/mm² that the corners did not need.

That is a statement about this box’s proportions and this load. A thinner-walled box, with a longer decay length and a softer section frame, has a smaller point stiffness and makes every cross-frame relatively stiffer; a box loaded by a heavy wheel line much nearer one web puts more into distortion and makes 3.4 times a small number less small. The yardstick is what carries over: a diaphragm’s stiffness is measured against $2K_d/\lambda$, and the figure of merit for the corners is the transverse stress it leaves, not its stiffness alone.

The force follows the load #

A diaphragm’s other job is to carry the distortional reaction: the share of the eccentric load’s racking that it takes out of the girder at its station. In a cross-frame that reaction is carried as axial force in the diagonals, $N = R\,d/2bh$.

The force barely depends on the frame’s stiffness. A frame of two 1,000 mm² diagonals, 26 times the girder’s point stiffness, carries 166 kN in each diagonal; a frame a tenth as stiff, 161; a rigid diaphragm would carry 182. Only when the frame becomes softer than the girder itself, at the left of the figure, does its force fall away.

That is how a support behaves, and the diaphragm is a support for the distortion. Once it is stiff compared with what it supports, it takes the share of the load its position gives it, the way the stiffer path takes the load and keeps taking it however much stiffer it is made. So a lighter cross-frame does not attract a lighter force. It has to be sized for the 166 kN whatever its area, and the stiffness it then has, for an ordinary steel at an ordinary stress, is already ample: two 1,000 mm² diagonals at 166 kN are at 166 N/mm², under half their yield, and 26 times the girder’s point stiffness. A cross-frame is designed for strength, and its stiffness comes free. A plate diaphragm, sized by its stiffness or by its handling, carries the same reaction as a shear stress of a few newtons per square millimetre.

More frames, or stiffer ones #

The original essay’s lesson about spacing survives. A single diaphragm of any kind leaves 19 to 25 N/mm² of transverse bending; two leave 5 to 8; three, the quarter points, bring every kind under 5. Past that the rigid diaphragms and the plates go to nothing and the cross-frames to a floor of a few newtons per square millimetre, which is the racking their own flexibility lets through at every station: seven light frames leave 2.3, eleven leave 1.5.

The choice between more frames and stiffer ones is therefore a choice about which floor is acceptable. Seven light frames do what three heavy ones do; three plates do better than either. For a corner whose detail category puts its cut-off at a few tens of newtons per square millimetre, all of them are enough, and the cheaper detail — fewer, lighter frames, bolted rather than welded all round, with access through them for inspection — is the right one.

Where a plate is still the answer #

The diaphragms above are intermediate ones, between the supports, and their job is the distortional half of an eccentric load: the half of the torque that goes into changing the section’s shape rather than twisting it. At a support the job is different. The bearings are under the webs, or under one web each, and the whole of the girder’s torque — the pure-torsion half as well, which in the span travels round the closed cell as Bredt’s shear flow, the closed section’s whole advantage, without asking anything of a diaphragm — has to be turned there into a pair of vertical reactions. The support diaphragm is where the cell’s shear flow is collected and handed to the bearings, and it carries that flow as shear across its own area.

That is a plate’s job. A cross-frame at a support would have to carry the cell’s whole shear flow round its corners as diagonal forces, and at a bridge’s end the torque is the eccentric load’s whole lever arm times half the span: for this box, $45 imes 2.0 imes 30 = 2{,}700$ kN·m, a shear flow of $2{,}700 imes 10^6/(2 imes 4{,}000 imes 2{,}500) = 135$ N/mm round the cell and several hundred kilonewtons in each diagonal. The support diaphragm is a plate, usually stiffened, with a manhole in it, and the original essay’s rigid diaphragm is a good description of it.

The distinction matters because the two are often detailed alike. A girder designed with plate diaphragms everywhere, because the support ones had to be plates, carries three or four plates in its span whose job a pair of angles would do, and every one of them is a welded detail all round the inside of the box — a ring of fatigue details, each with its own stress range from the wheels passing over it, where a bolted frame would have had none.

The beam-on-elastic-foundation picture says the same in its own terms. A support is the end of the beam, where a beam on ground is stiffest to rotate and most sensitive to how it is held; an interior station is far from both ends, where the beam’s own stiffness against a point load is $2K_d/\lambda$ and the diaphragm need only beat that. The end diaphragm is holding the whole beam; the interior ones are each holding a decay length of it.

The two stiffnesses by hand #

The plate: $G = 205{,}000/2.6 = 78{,}800$ N/mm², and a 10 mm plate across a 4,000 by 2,500 mm box has $K = 78{,}800 \times 10 \times 4{,}000 \times 2{,}500 = 7.9 \times 10^{12}$ N·mm/rad, which is 524 times $K_g$.

The cross-frame: the diagonal is $\sqrt{4{,}000^2 + 2{,}500^2} = 4{,}717$ mm long. A racking of $\gamma$ moves one corner of the box sideways by $\gamma h$ relative to the other end of its diagonal, and lengthens the diagonal by $\gamma h \cdot b/d = \gamma \times 2{,}120$ mm. Each 1,000 mm² diagonal has an axial stiffness of $205{,}000 \times 1{,}000/4{,}717 = 43{,}500$ N/mm, so the pair stores $2 \times \tfrac12 \times 43{,}500 \times (2{,}120\gamma)^2$ and $K = 2 \times 43{,}500 \times 2{,}120^2 = 3.9 \times 10^{11}$ N·mm/rad: 26 times $K_g$, and a twentieth of the plate.

The diagonal force from the reaction: the most loaded frame’s distortional reaction is about $7.0 \times 10^8$ N·mm, and $N = R\,d/2bh = 7.0 \times 10^8 \times 4{,}717/(2 \times 10^7) = 166$ kN.

What the model leaves out #

A cross-frame is drawn as two pin-ended diagonals. Real cross-frames have chords along the flanges and gussets at the corners, and are bolted. The chords stiffen the frame a little; bolt slip in clearance holes softens it, and at the strokes of a few tenths of a millimetre that racking of a milliradian or two produces across a 4.7 m diagonal, a clearance of a millimetre or two is a frame that does nothing until it has taken up its slack. Pretensioned bolts, or a frame welded to its gussets, avoid that; the stiffness above is the frame’s once its bolts bear.

The compression diagonal does not buckle. A 1,000 mm² angle 4.7 m long buckles at well under the 166 kN it is asked for, and a cross-frame whose compression diagonal has buckled is a single tension diagonal with half the stiffness — 13 times $K_g$ rather than 26, still past the ten at which the warping stress is held. Designers either size both diagonals as struts or accept the tension-only frame; the figures say the second is usually enough.

The diaphragm is a point. A plate diaphragm has a thickness and its own out-of-plane stiffness, and a cross-frame has a depth along the girder; both are treated as concentrated at one section. And the load is a uniform line load at one eccentricity, where a real bridge carries wheel lines that move. A moving load’s distortion is largest wherever it stands, and the worst station for a cross-frame’s force is not the worst station for the corner’s bending, which is between frames.

A diaphragm is a support, not a wall #

The rigid diaphragm was the right idealisation for the question of how many diaphragms and where: it gave a decay length and a radius of action, and those survive. It was the wrong one for the question of what a diaphragm must be. A diaphragm supports the girder against distortion, and like any support it has to be stiff compared with what it supports and strong enough for the share of the load its position gives it. For this box the first is satisfied by a pair of angles, two orders of magnitude short of a plate, and the second decides their size.

Still open: the box whose walls are not plates #

Every wall here is a plain plate whose transverse bending is what the section’s frame stiffness is made of. The deck of a steel box bridge is not: it is an orthotropic plate, with longitudinal troughs welded under a thin deck plate and crossbeams below them at the cross-frame stations, and its transverse bending stiffness comes from the crossbeams rather than from the plate. A deck like that has a section frame whose stiffness is concentrated at the crossbeams and nearly absent between them, so the foundation of the beam-on-elastic-foundation picture is no longer continuous. Whether a box with an orthotropic top flange has a decay length at all, or distorts between its crossbeams as a row of panels each held at its ends, and whether its cross-frames then have to be at every crossbeam rather than at the spacing a decay length would choose, is the question the deck puts to everything this essay assumed about the walls.