One diaphragm is nearly none

A box girder will not keep its shape, and half of an eccentric load’s torsion is spent distorting the section rather than twisting it.

The hero is what that costs when nothing is done about it: a 4.0 by 2.5 m box, 60 m long, carrying 45 N/mm at 2.0 m off the axis, with 73 N/mm² of longitudinal stress at its corners from distortional warping — 83 per cent of the bending stress the girder was designed for.

The cure is a diaphragm. This rung is about how many, and where, and the answer is not the one the rules of thumb give.

The decay length is the only number that matters #

The distortion is a beam-on-elastic-foundation problem along the girder’s length: the section wants to become a rhombus, the plates’ own bending resists it, and a disturbance introduced at one point dies away along the span with a characteristic length.

For this box that length is 23.6 m. It comes out of the plate thicknesses, the box’s proportions and its material — and not out of the span, the load or the eccentricity.

Which means a diaphragm has a radius of action. It holds the section square at its own cross-section, and its effect fades over 23.6 m in each direction. Beyond that the girder does not know it is there.

The relationship is not linear and it is not close to linear. Zero, one, two and three interior diaphragms give 73, 49, 18 and 10 N/mm². The first one buys 24; the second buys 31; the third buys 8.

That shape — little, then a lot, then little again — is what a decay length looks like from the outside. While the spacing is longer than the decay length each diaphragm acts alone and the girder between them is untouched. Once the spacing comes inside it, the diaphragms start helping each other and the stress collapses. Past that, there is nothing left to remove.

Where the decay length comes from #

The 23.6 m is the only quantity in this essay that is not read off a figure, so it is worth assembling.

The girder is a beam on an elastic foundation, laid along its own length. The “beam” is the section’s warping stiffness — as the box distorts, its corners displace longitudinally, and resisting that longitudinal displacement is a bending action along the span. The “foundation” is the section’s transverse frame stiffness — the four plates form a closed rectangular frame, and squashing it into a rhombus bends all four at their corners.

The characteristic length of any beam on an elastic foundation is $\ell = \sqrt[4]{4EI/k}$, with $EI$ the beam’s stiffness and $k$ the foundation’s per unit length. Here $EI$ is the distortional warping constant and $k$ is the transverse frame stiffness, and the fourth root is what makes the answer so insensitive: a factor of sixteen in either quantity is a factor of two in the length.

That fourth root explains the whole of the previous section’s trap. Thinning the plates from 20 mm to 12 reduces the warping constant by roughly the ratio of thicknesses and the frame stiffness by roughly its cube, so the ratio $EI/k$ grows by about the square — and the length grows by its square root, which is why it went up rather than down.

It is exactly the arithmetic a beam on ground has, applied along a girder instead of across a foundation, and the two share the same practical moral: the characteristic length is the thing to compute, and everything else is read off it.

The sweep, and where the knee is #

The flat right-hand end is the part worth staring at. A designer who doubles the diaphragm spacing from 40 m to 80 m has changed nothing, and one who reduces it from 40 m to 30 m has changed almost nothing either. All of the available benefit lives inside one decay length, and a rule expressed as a fraction of the span cannot know where that is.

Which is the practical instruction: compute the decay length first, then space the diaphragms at a fraction of it. Half a decay length is a good target and gets the distortional stress to about seven per cent of the bending stress; a quarter buys little more.

The trap in the thin-plated box #

That figure is the reason this essay is a rung rather than a rule.

The decay length goes as the fourth root of a ratio between the longitudinal warping stiffness and the transverse frame stiffness of the section. Thinning the plates reduces both — and it reduces the transverse frame stiffness faster, because that stiffness goes as the cube of the plate thickness while the warping stiffness goes as the first power.

So a flimsier box distorts more and distorts over a longer length, and the two effects say opposite things to a designer reading only the decay length. A rule of “space them at half a decay length” applied to the thin box gives diaphragms 14.3 m apart and a distortional stress of 11 per cent; applied to the thick one it gives 11.8 m and 7 per cent. The flimsier girder gets fewer diaphragms and a worse answer.

The correct reading takes both numbers off the curve: the spacing and the stress it delivers. The decay length says where the knee is; it does not say how high the curve is above it.

Across the three boxes on this page the decay length runs from 16.7 m to 28.6 — a factor of 1.7 — and the stress ratio at one decay length from 9 per cent to 55, a factor of six. No fraction of the span could track either.

What the diaphragm is holding #

The diaphragm’s job is to resist the second set, and reading it off that figure says what the diaphragm actually is.

It is a plate in shear. The distortional load set is four edge forces going round the section in a pattern that racks it, and the diaphragm resists them by shearing in its own plane. Its thickness is a shear check, not a bending one.

And it needs to be connected on all four sides. A diaphragm welded to the webs and not to the flanges resists nothing, because the racking pattern needs both pairs of edges held.

Which is why the access hole in a diaphragm matters more than it looks. A large central hole removes shear area from the middle of a plate carrying shear, and the standard detail — a hole with a stiffened rim, offset from the centre — is a shear-panel detail rather than an architectural one.

The other half of the torque is not the diaphragm’s problem at all. The pure-torsion set is carried by Bredt’s shear flow round the closed cell, which requires the section to be closed and requires nothing to be held square. Slitting the box destroys that half and leaves the distortional half untouched, which is the cleanest demonstration available that the two are independent.

What the stress is doing to the section #

The 73 N/mm² is a longitudinal stress at the corners, and where it sits in the section decides how much trouble it is.

It adds to the bending stress at two corners and subtracts at the other two. The distortional warping pattern is antisymmetric about both axes, so one diagonally opposite pair gets the sum and the other gets the difference — and which pair depends on which side the load is on.

That is a real difficulty for a bridge, because the load’s eccentricity reverses. A vehicle in the left lane loads one diagonal pair and one in the right lane loads the other, so every corner of the box sees the full range, and the section has no favourable corner to hide the detail in.

And the stress is at a welded corner. The web-to-flange weld is a longitudinal fillet running the whole length of the girder, at a fatigue category well below the parent plate, and the distortional stress is largest exactly there. That is the mechanism behind most of the cracking found in older steel box girders: not a bending overload, but a distortional range at a longitudinal weld that no bending calculation contained.

There is a second consequence at serviceability that gets forgotten. The distortion is a change of shape, so it moves the deck surface. A box that racks by a few millimetres carries its deck plate with it, and on a bridge with a stiff surfacing that movement is a crack in the wearing course rather than a stress in the steel. A movement with no limit against it is the general shape of that problem, and distortion is one of its clearest instances.

Where the diaphragms have to be regardless #

Three positions are decided by something other than the decay length, and they are worth listing because they are usually most of the diaphragms in a real girder.

At the supports. A bearing applies its reaction over a small area of the bottom flange, and without a diaphragm the section distorts locally under it. This one is a strength requirement rather than a distortion-control one.

Under a concentrated load. A crossbeam, a hanger, or a patch load that would crush the web on its own — anything that puts a large force into one plate needs the section held square where it lands.

And at a change of section. Where the plate thickness steps, the decay length steps with it, and the discontinuity itself generates distortion.

So the sequence in practice is: place the diaphragms the other requirements demand, compute the decay length, and add intermediate ones until the spacing is comfortably inside it. On a short span that often adds none, because a 30 m box with diaphragms at both supports already has them 30 m apart against a 23.6 m decay length, and one at midspan brings the spacing to 15 m — which is the answer this whole essay arrives at from the other direction.

The cost side, which decides the answer #

Nothing above has priced a diaphragm, and the pricing is what makes this a design decision rather than an optimisation.

A diaphragm is expensive out of proportion to its weight. It is a plate cut to fit the inside of the box, with a stiffened access hole, welded on four edges from inside a confined space — and the welding is overhead on one edge and in a corner on all four. A single diaphragm can cost several times a tonne of plate elsewhere in the girder.

It also has to be got in. On a fabricated box the diaphragms go in before the last plate is closed, which fixes the assembly sequence; on a site-spliced girder there is a diaphragm at every splice whether the distortion wanted one or not.

And each one is an inspection point for the life of the structure. Access holes are how a box girder is inspected, so their number, size and alignment are a maintenance decision as much as a structural one.

So the sensible design is the smallest number that gets the spacing inside the decay length, placed where other requirements wanted diaphragms anyway. The curve’s flat right-hand end is what makes that safe: there is no penalty for being a little generous with the spacing, provided it is inside the knee, and the whole of the benefit is captured by the time the spacing reaches half a decay length.

The opposite error is the one worth guarding against, and it is cheap to make on a drawing. A regular spacing chosen as a fraction of the span, on a girder whose plates thin towards midspan, puts the widest effective spacing exactly where the decay length is longest and the section flimsiest — which is an average taken over a member that changes, and it fails in the same way.

There is one arrangement that removes the decision entirely and is worth naming for contrast. A box with cross-bracing at close centres instead of plate diaphragms solves the same racking with a triangulated frame rather than a shear panel — lighter, cheaper to weld, and easy to put in at spacings a plate diaphragm could never justify. It is the standard steel detail for exactly that reason, and it changes none of the arithmetic on this page: the bracing’s stiffness enters as the foundation modulus, the decay length is computed the same way, and the spacing rule is the same rule. What changes is the price per restraint, which moves the economic answer from three diaphragms to a dozen frames.

That substitution is the ordinary way this subject is handled in practice, and it is why plate diaphragms in a real girder appear almost only at the three positions the previous section listed — supports, concentrated loads and section changes — with the distortion control done by something lighter in between. The plate diaphragm is a strength member that happens to control distortion; the cross frame is a distortion member that carries nothing else.

What to carry away #

The decay length is the design quantity. 23.6 m on this box, from the plate thicknesses and the proportions, and not from the span.

The first diaphragm is the least useful. Zero, one, two and three give 73, 49, 18 and 10 N/mm², because a single diaphragm at midspan is more than a decay length from each end.

A thinner box is worse and its decay length says otherwise. 20 mm to 12 mm lengthens the decay from 23.6 m to 28.6 and raises the stress at one decay length from 20 per cent to 55.

And the diaphragm is a shear panel. Its load set is four edge forces racking the section, which is why it must be connected on all four sides and why its access hole is a shear detail.

Where the model stops #

The diaphragms are rigid. A real diaphragm is a plate with a hole in it and a finite shear stiffness, so it does not fully restore the section and the curve between two of them does not return to the ideal.

The distortion is the first harmonic only. A section can distort in higher patterns, and a load applied near a diaphragm excites them; the beam-on-elastic-foundation model has one shape in it.

Nothing here is a fatigue calculation, which is where distortion does most of its damage. The corner stresses computed here cycle with the traffic, at a welded detail, and a distortional stress that is 83 per cent of the bending stress is a stress range of the same order at a poor category.

And the box has two cells nowhere on this page. A multi-cell box is a different distortion problem with more shapes available and an interior web that may be doing nothing about any of them.

The ladder from here #

Later rungs on this anchor: 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. Distortional fatigue at the web-to-flange weld, which is where box girders actually crack. The multi-cell box, where the number of distortional shapes grows with the number of cells. Distortion under a moving load, where the worst position is not the worst position for bending. And the concrete box, where the section’s own transverse frame is stiff enough that the whole problem changes character.

Vlasov set out the general theory of thin-walled beams with deformable cross-sections in the 1940s, and the distortional part of it sat unused in bridge design for twenty years — because the boxes being built were small enough, and stiffened enough, for the problem not to appear. It appeared when spans grew and plates thinned, and the two box girder collapses of 1970 at Milford Haven and West Gate, both during erection, are the reason the diaphragm is now a designed member with its own calculation rather than a piece of the fabrication.