The flange works least where the shear is largest

A wide flange is not all there, because longitudinal stress reaches it only through shear along its junction with the web, and the parts furthest from the web arrive late. That essay computes an effective width for a section. This one is about the fact that there is no such thing.

The lag is driven by the shear #

The governing equation for shear lag contains the third derivative of the deflection, which is the shear force — not the second, which is the moment.

That single fact organises everything else. A quantity driven by shear is largest where the shear is largest, and on a simply supported beam under a uniform load the shear is largest at the supports and zero at mid-span. So the lag is worst exactly where the moment is smallest, and mildest where the moment is largest.

The consolation is real but partial. The flange is worked hardest at mid-span, where the lag is mild, so the peak stress in the whole member is less affected than the numbers suggest. What is affected is the flange force near a support — which is where the shear connection, the splices and the curtailment all live.

There is a second consequence of the same derivative that is easy to miss. A quantity driven by shear is zero wherever the shear is zero, so at mid-span of a uniformly loaded simply supported beam there is no lag being generated at all — the flange there is fully effective in the sense that nothing is currently spreading. What the figure reports at mid-span is not zero lag but the accumulated consequence of everything that happened either side of it, which is why the mid-span value is 0.839 rather than 1.0.

That distinction matters for a member whose shear diagram is unusual. A beam under a central point load has constant shear either side of the load and a step at it; its lag is uniform along each half and discontinuous at the load. A cantilever under a tip load has constant shear throughout. Neither behaves like the curve above, and neither is what an effective-width table was fitted to.

Every harmonic has its own effective width #

The mechanism behind the variation is worth setting out because it explains the shape of the curve.

Expand the moment diagram as a Fourier series. A simply supported parabola is a sum of odd sinusoids: the first with half-wavelength $L$, the third with $L/3$, the fifth with $L/5$.

Each harmonic produces its own shear-lag problem, and the effective width depends on the ratio of the flange width to the half-wavelength, not to the span. So the first harmonic on this beam achieves 0.815 of the flange, the third only 0.400, and the fifth 0.269. A short wave lags far worse than a long one, because the shear that feeds the flange reverses before the stress has finished spreading.

Near mid-span the first harmonic carries almost all of the moment and the effective width is close to its value. Near a support the first harmonic contributes almost nothing — a sinusoid is zero at its ends — so what little moment there is comes disproportionately from the short harmonics, which lag badly. The curve falls.

That is also why the effect is worse on a short span.

The effective width is set by the span and not by the flange, which is the least intuitive result in the subject and the one that decides how a wide-flanged member is proportioned.

Which free body produced the number #

The free body is a strip of the flange, cut parallel to the web and running along the member.

Crossing the two ends of that strip is the longitudinal force in it. Crossing its inner face — the cut parallel to the web — is a shear flow. The strip has nothing else on it: no load, no support, nothing at its free edge.

So the only way the strip’s longitudinal force can change is through the shear on its inner face, and the only way stress can get into the outer parts of the flange is by being handed outward, strip by strip. The flange is loaded through its side, and the lag is the delay that transmission takes.

That is the same statement as the shear nobody draws, read along the flange instead of through the depth, and it is a relative of how far a wrong load reaches: both are about a disturbance spreading through a solid over a distance set by the geometry rather than by the material.

The distance in question is what the next section is about, and it has an exact value.

The ceiling nobody quotes #

There is a width past which no amount of flange helps, and it is not the code’s.

An infinitely wide flange loaded by a sinusoidal shear flow of half-wavelength $L$ achieves an effective width of $L/2\pi$ per side. That is a closed-form elasticity result and it puts a hard ceiling on the whole subject: a flange more than about a third of the span wide is mostly decorative, whatever it is made of and however thick it is.

The code’s $L/8$ sits 21.5 per cent below that, with a straight line where the elasticity has a curve. The 8 is a round number and the 2π is the answer, which is a fair description of a great many design rules and is worth knowing when one is being applied outside the range it was fitted over.

How big the error is, and where it lands #

It is worth converting the working fractions into the quantities a design actually uses, because the effect is not uniform across them.

Peak stress. At mid-span on the 24 m span, the flange is 81.5 per cent effective, so the stress at the web is 1.23 times what plane sections give. On the 10 m span it is 2.07 times. That is a strength number and it is the one usually quoted.

Deflection. The section’s second moment is reduced by roughly the same fraction as the flange area that works, so the member is softer than the gross section suggests — about 10 per cent on the 24 m span, and considerably more on the short one. That is a serviceability number and it is usually forgotten, because effective width is presented as a stress correction.

Flange force at a splice. The force to be transferred across a splice is the total in the flange, which is unchanged — the lag redistributes stress without changing the resultant. So a splice is designed for the same force whether or not the lag is considered, and the bolts nearest the web are the ones actually carrying it.

And the shear connection. The flow into the flange is what causes the lag, so it is unchanged in total and concentrated near the web. On a wide composite flange the studs nearest the web work hardest, which is a two-dimensional version of the connection being busiest where the beam is not.

Four quantities, one phenomenon, and only the first of them appears in the design rule.

What the codes do about the variation #

A design rule cannot easily print a curve, so it reproduces one by adjusting an input.

The standard device is to define an effective span $L_e$ that is not the physical span: about 0.85 of it in an end span, 0.70 in an internal span, and — the important one — about 0.25 of the sum of the two adjacent spans in the region either side of an interior support.

Since the effective width is proportional to $L_e$, shortening $L_e$ near a support narrows the effective width there, which is exactly the behaviour in the first figure. The code is not modelling the harmonics; it is fitting the answer they produce, and the fit is good enough over the range of proportions real bridges have.

That has a consequence for anybody using the rule outside that range. A member with a very wide flange, a very short span, or a moment diagram that is not roughly parabolic — a cantilever, a beam under a point load, a member in a frame — has harmonic content the fit was not made against, and the tabulated $L_e$ factors are then standing in for something they were not calibrated on.

Over a support, where two effects arrive together #

The region beside an interior support of a continuous member is where all of this concentrates, and there are three reasons rather than one.

The shear is largest there, so the lag is worst — the first figure’s mechanism.

The moment reverses, so the moment diagram has a zero nearby and its local harmonic content is short-wavelength by construction. That is the same statement in a different language and is why the effective span is defined from the distance between points of contraflexure rather than from the span.

And in a composite member the slab is in tension there, so it is cracked and its contribution is the reinforcement alone. The effective width is then a width of reinforcement, and the two reductions multiply.

The result is that a continuous composite deck’s hogging region has an effective flange perhaps half what a naive calculation gives, at the section carrying the largest moment in the member. It is the governing section for exactly the reason it is the worst-modelled one, which is why bridge codes give it more clauses than the whole of the sagging region.

The sign that catches people out #

There is a case where the effective width is larger than the flange, and it is worth naming because it looks like an error in the arithmetic.

Under some loadings — a cantilever near its root, a continuous member near a point of contraflexure, the region beside a concentrated load — the stress across the flange does not fall monotonically away from the web. It can rise, so that the free edge is more highly stressed than the junction, and the “effective width” computed as the equivalent rectangle then exceeds the real width.

That is negative shear lag, it is a genuine elastic result rather than a numerical artefact, and it happens where the shear flow feeding the flange changes sign within a distance comparable to the flange width. The stress that arrived earlier is still there while the new flow is running the other way, and the two superpose.

Its practical consequence is small in a strength check and large in a measurement. A strain gauge on the free edge of a flange near a support can read higher than one at the web, which is the opposite of what the design model predicts, and the usual reaction to that reading is to doubt the gauge.

The same effect in a connection #

The mechanism is not peculiar to bridge decks, and its smallest instance is one every steelwork detailer meets.

An angle bolted through one leg uses part of itself, and the reduction factor $U = 1 - \bar{x}/L_c$ is shear lag with the span replaced by the connection length. The outstanding leg is fed only through the connected one, over a length of a few hundred millimetres, and the further its centroid is from the connected face the less of it arrives.

Same physics, four orders of magnitude apart in size, and the two design rules look nothing like each other: one is a fraction of a span and the other is a fraction of a connection length. What they share is that both are a length over which the stress has to spread divided into a distance it has to spread across.

What to carry away #

Shear lag follows the shear, so it varies along the member. Worst at the supports, mildest at mid-span, and a single quoted effective width is the mid-span value.

The span sets the effective width and the flange does not. A wider flange lowers the working fraction almost as fast as it raises the width, and past $L/2\pi$ per side it buys nothing at all.

And the worst-modelled region is the governing one. Over an interior support the shear peaks, the harmonic content shortens, and in a composite member the slab has cracked — three reductions arriving together at the section with the largest moment.

Why it is a modern problem #

Shear lag is old physics and a new design problem, and the reason is a change of shape rather than a change of knowledge.

A riveted plate girder of 1900 has a flange perhaps 400 mm wide on a span of 20 m — $b/L$ of 0.01 per side, at which the working fraction is above 0.99 and the effect is smaller than the rivet holes. Nobody computed it because nobody needed to.

A welded box girder of 1965 has a flange 6 m wide on a span of 40 m — $b/L$ of 0.075, at which the fraction is around 0.93, and a cantilever deck slab wider still. A concrete box for a modern viaduct is wider again relative to its span, and a slab spanning between widely spaced beams is effectively a flange of enormous width.

The trend has been toward wider flanges on shorter spans in every material, driven by the fact that a wide flange is cheap to make once it is welded rather than riveted and once concrete is cast rather than assembled. Every step in that direction moves a member up the sweep curve toward the ceiling, and the ceiling is $L/2\pi$ regardless of anything a designer does.

That is the reason a subject with a nineteenth-century solution acquired its design rules in the 1970s: the physics did not change, and the sections did.

Where the model stops #

The flange is elastic and uncracked. Past first yield the stress distribution flattens, because a yielded strip near the web carries no more and the outer flange catches up. The effective width at the ultimate limit state is therefore wider than the elastic one, and codes say so.

One term is used for the profile. The parabolic shape assumed across the flange is a one-term approximation to Airy’s solution, which is why the sweep is drawn no further than b/L ≈ 0.22 — beyond that the one-term answer rises above the exact ceiling and is optimistic.

The shear flow is assumed sinusoidal. The harmonic decomposition handles that for a simply supported beam under a uniform load and not for a cantilever, where the series is different and the boundary conditions are the interesting part.

Nothing here is a stability calculation. A wide compression flange that lags is also a wide plate that can buckle, and the two reductions interact — the effective width from lag and the effective width from buckling are different quantities that a code multiplies together with no theory behind the product.

The flange is assumed to be fed only by its own web. A multi-web deck has flange strips fed from both sides, and the lag between webs is a different problem from the lag on an overhang — closer to a plate spanning between lines of shear than to a cantilever off one.

And the web is assumed rigid in its own plane. In a box girder the two webs move relative to each other, the flange distorts, and shear lag arrives combined with distortion in a way a single-web model cannot describe.

Two other reductions of a plate’s usable width sit beside this one and are frequently confused with it. An angle connected through one leg uses half of itself for the same reason — the force has to spread from where it enters — while what is left after a plate ripples is a stability reduction with no shear lag in it at all. They are separate effects on the same dimension, and a plate that ripples can do both at once.

The ladder from here #

Later rungs on this anchor: the energy derivation in full, with the second freedom written out. The exact Airy solution and its sign reversal near a support. Negative shear lag, and the loadings that produce it. Effective width after the strip next to the web has yielded. Shear lag interacting with plate buckling in a wide compression flange, where two effective widths are multiplied without justification. Shear lag in the torsion and distortion of a box section, which is a different flow with the same delay. And the finite-element answer, which needs none of this and reports a number nobody can check by inspection.

The subject arrived with the box girder. A plate girder’s flange is narrow enough that the lag is a few per cent, and nobody needed to compute it; a box girder’s is a third of its span, and the four box-girder collapses of 1969 to 1971 put shear lag, plate buckling and distortion into the same set of design rules at the same time. The elasticity had been available since Airy, and what made it a design problem was a shape.