Three actions on one web

Two cells, one equation, and a web with nothing in it is about how a closed section shares a torque between its walls. This is about what else is on those walls at the same time.

An eccentric load on a box girder is decomposed, in every treatment of the subject, into a symmetric part that bends the girder, an antisymmetric part that twists it, and a distortional part that changes the shape of the cross-section. The decomposition is exact and it is the only way to make the problem tractable.

It is also where most treatments stop, and stopping there hides the fact that all three arrive on the same plate.

Two walls of one cross-section, at one station, under one load case, differing by a factor of ten.

Which free body produced the number #

The free body is a length of the box, cut normal to the axis, and what crosses the cut is a shear flow round the closed circuit and a normal stress distribution.

Take them one at a time, because their signs are the whole argument.

Bending contributes a shear flow $VQ/2I$ in each web, both in the same direction — downward on the section, toward the neutral axis from both flanges. It also contributes a normal stress that varies linearly down the depth and is the same in both webs.

Torsion, as a closed circuit, contributes Bredt’s $T/2A_m$ round the cell: up one web and down the other. It adds to one web’s bending flow and subtracts from the other’s, and it contributes no normal stress at all in a section free to warp.

Distortion contributes a longitudinal warping stress that is largest at the corners and reverses sign across each wall, and a transverse plate bending in the walls that has nothing to do with the girder’s span at all.

So the loaded web carries $q_b + q_t$ and the far web carries $q_b - q_t$, and both carry the bending normal stress plus or minus the warping one. On the box here that is 107 + 50 against 107 − 50 in shear flow, and the combination with the normal stresses gives the factor of ten in the figure.

The mean of the two is what a check on the section returns, and the mean is very nearly what the bending-only calculation gives. That is the whole reason this is easy to miss: every summary number is right and no wall is described by any of them.

The one that is genuinely two problems #

Distortion is worth separating from the other two because it is not a section calculation at all.

Bending and pure torsion are both statements about a rigid cross-section: the shape stays what it was and the stresses follow from equilibrium on that shape. Distortion is what happens because the shape does not stay: the box goes from rectangle to parallelogram, and the resistance to that comes from the walls bending transversely as a four-bar frame, not from anything longitudinal.

That gives distortion two consequences on the same wall, and one of them surprises everybody the first time.

A longitudinal warping stress, which adds to the bending stress at the corners in the way any warping stress does.

And a transverse bending stress in the plate, which on this box is 151 N/mm² — larger than either of the longitudinal terms. It is a stress across the wall rather than along it, so it does not add to them directly; what it does is use up the wall’s capacity in the second principal direction, and it is the reason a slender box wall can be at yield in a check nobody performed.

What the diaphragms are actually for #

The size of the distortional term is not a property of the box. It is a property of how far apart its diaphragms are, which is what the rung on that anchor is about and is the reason the numbers here are as large as they are: this box has no interior diaphragms at all.

Which reorders the three actions by how much can be done about them. Bending is fixed by the span and the load. Torsion is fixed by the eccentricity, and a closed box carries it so cheaply that nobody minds. Distortion is a design variable, bought in diaphragms, and it is the only one of the three whose size is chosen.

That is the practical hierarchy, and it explains a fabrication decision that otherwise looks arbitrary: diaphragms in a box girder are spaced by a stress calculation and not by anything to do with handling or stability, and the spacing that results is usually two to four times the box’s depth.

It also settles an argument about where to put the load. Two lanes of traffic at the kerbs of a wide box produce almost no torque between them and a great deal of distortion each; one lane at one kerb produces both. The governing case for the webs is therefore a single eccentric lane, and the governing case for the diaphragms is a pair of them — two different load arrangements deciding two parts of the same section, which is the shape of finding an envelope is not a structure exists to warn about.

The eccentricity that costs nothing on average #

Sweeping the load across the box shows the arithmetic in its clearest form.

The mean is flat because the torsional flow is added to one web and subtracted from the other in equal measure, and the distortional warping does very nearly the same thing. Eccentricity is free on average and expensive on the wall, which is the single most useful sentence about box girders under traffic.

It also explains why the governing load case for a bridge box is a lane load at the kerb rather than a heavier load in the middle. A lane of traffic on the centre line loads both webs at 56 N/mm²; the same lane at the edge loads one at 116. A load case that adds nothing to the total can double a wall, and no check that reduces the section to a moment and a torque can see it.

The same box made wide and shallow #

Everything above is a proportion rather than a number, and the proportions move a long way across the box girders that get built.

Three effects, in three directions, from one change of shape. It is worth reading them off because they say which box shape is sensitive to what.

Torsion gets cheaper as the box gets bigger, in both directions: Bredt’s flow is the torque over twice the enclosed area, so a wide box carries the same torque at a lower flow. That is the property everybody knows about closed sections and it is the reason a box is chosen at all.

Distortion gets much worse as the box gets wider, because the transverse frame’s stiffness depends on the walls’ plate bending over their own lengths — a longer flange is a much more flexible member of that frame. The decay length grows with it, so the distortion from a load spreads further as well as growing.

And the two do not trade off against each other. A designer who widens a box to reduce its torsional flow has increased its distortional stress and lengthened the distance over which it acts, and the diaphragm spacing that was adequate before is no longer.

Where it decides something on a real deck #

A concrete or steel box carrying a road is the case this argument was developed for, and the sequence in a real design runs the other way round from the sequence here.

The section is chosen for bending, because bending is what a span costs. The box form is chosen for torsion, because a deck carrying eccentric traffic on a curved alignment needs torsional stiffness and an open section has essentially none — the internal force with no diagram is the one that decides the form. Then the diaphragms are spaced for distortion, which is the only variable left.

What that leaves as the check nobody has done is the wall. The bending calculation gave a stress on a section; the torsion calculation gave a flow round a circuit; the distortion calculation gave a spacing. The web is where all three are, and it appears in none of the three calculations as an object.

The consequence in practice is a web thickness chosen from the larger of two shear checks, and a fatigue detail — the web-to-flange weld — assessed on a shear range that is the sum of a bending range and a torsional range with the sign taken from whichever lane the traffic is in. That is a real calculation on a real bridge, and it is the direct descendant of the figure at the top of this page.

A surprising place the same addition turns up #

The pattern generalises past boxes, and the general statement is worth having: when a load is decomposed into components for the convenience of the analysis, the components have to be recombined on the object that carries them, and the object is usually smaller than the one the decomposition was written about.

A bolt group’s elastic analysis does exactly this: the moment and the direct force are resolved separately and then added vectorially at one bolt, which is a different bolt from the one either component alone would find. The two-way slab splits its load into two spanning directions and then has to reinforce the corner where both are largest. And a section under biaxial bending has a neutral axis that obeys neither of the two moments that produced it.

In every one of them the decomposition is exact, the recombination is elementary, and the failure is to stop after the first step because each component’s answer looks like a complete one.

The history, which is a bridge that was built before the theory #

Box girders in steel are a post-war form, and the distortional half of this argument arrived after they were being built rather than before.

The torsional half is old. Bredt published the closed-tube shear flow in 1896 and it was in use immediately, because it is one line and it answers the question a closed section was chosen for. The bending half is older still.

Distortion is different. It needs the transverse frame’s stiffness, a warping function for a section that changes shape, and a differential equation of the beam-on-elastic-foundation kind — and it was worked out in the 1960s and 1970s, in parallel in Japan, Germany and Britain, largely because box girders had started to fail. The 1970 collapses at Milford Haven, West Gate and Koblenz were all in steel boxes and all during erection; none of them was distortion, and all of them produced the review that made box-girder behaviour a subject with numbers in it rather than a form with a reputation.

What that history leaves is a design method assembled in the order the pieces became available: an exact torsional theory from 1896, a bending theory older than that, and a distortional theory from the 1970s that most texts still present as a correction to be checked rather than as one of three co-equal actions. The order in which a subject was understood is a poor guide to which of its parts governs, and a designer meeting a wide shallow box for the first time will find that the newest of the three theories is the one deciding the plate thickness.

What to carry away #

Three sentences, and the third is the one that generalises.

An eccentric load on a box is three actions, and they are exactly separable. The two webs of the section receive them with opposite signs, so one carries the sum and the other the difference, and no summary quantity — a moment, a torque, a mean stress — describes either of them.

And the check has to be made on the wall. That is a statement about where a limit state lives, and it is the same statement as the free body being a storey rather than a column or the mechanism needing all four corners: the object the calculation is about has to be the object that fails, and choosing it is the part of the work that no formula supplies.

Where the model stops #

One box, one span, simply supported. In a continuous box the torque distributes between spans through the torsional stiffness rather than statically, and the distortional decay from an internal support interferes with the one from the next diaphragm.

The shear flows are elastic and the walls are stocky. A slender web at 157 N/mm of flow may buckle rather than yield, and the buckling check for a plate under shear plus a longitudinal stress plus a transverse one is a different family of calculation with an interaction of its own.

Distortional shear is neglected. The distortional set carries a small shear flow of its own which is left out here, as it is in most treatments; on a very flexible box it is not negligible and the honest version is a full beam-on-elastic-foundation analogy with all four stress resultants.

And the section is single-cell. A multi-cell box shares its torque between cells by the equation the rung below solves, and the internal web carries the difference between two cell flows — so it gets the bending flow plus a difference that may be very small, which is precisely the web with nothing in it.

What the pictures cannot show #

They cannot show what the wall does with a two-dimensional stress state.

Every number on this page is either a longitudinal stress or a shear, and the equivalent stress combines them by von Mises. The transverse plate bending is a third component in the other direction, and combining all three properly needs the full stress tensor at a point on the plate rather than three quantities computed by three separate one-dimensional theories.

That is the honest limitation of a superposition of beam theories: each of the three actions is exact within its own model, the models use different assumptions about the section, and there is no theory here at all for what their sum does to the material. A shell analysis is what answers it, and the value of the decomposition is that it says which of the three to look at first.

The assumption the figure rests on #

That superposition holds.

It does, exactly, while everything is elastic and the deformations are small — which is what makes the decomposition legitimate in the first place. The two places it stops holding are worth naming because both occur in real boxes.

Once a wall buckles, its stiffness in one direction changes and the shear flow redistributes to the walls that have not, so the three actions stop dividing the way the elastic solution says.

And once anything yields, the distortional term redistributes first, because it is the one held by transverse plate bending rather than by anything longitudinal. A box that has yielded locally at a corner has shed most of its distortional stress and kept all of its bending, which is a far more benign state than the elastic numbers suggest — and is the reason the large warping stresses on this page have not caused the failures they look as though they should.

The ladder from here #

Later rungs on this anchor: the multi-cell box with the shear-flow superposition done fully, where each cell’s rate of twist has to match and the internal webs carry differences. Shear lag on the flanges of the same box, which is a fourth action with its own longitudinal stress and its own effective width. Cells that are not rectangles — a trapezoidal box, a ship’s hull — where the areas and perimeters have to be computed before the system can be assembled. The plastic torque of a multi-cell section, from the sand-heap analogy with islands in it. And the box under combined bending, shear and torsion at the ultimate limit state, where the three actions are no longer superposed at all and the answer is a yield surface.