Structural Engineering & Statics
where the load goes
Stability

A column nine hundred millimetres long

The patch-load check asks how much of a web a flange can spread a wheel over, and answers in a plate-buckling reduction that throws seven tenths of it away. A pair of stiffeners does not improve that answer. It replaces the question with a different one, from a different family, with a different failure in it.

Assumes The load that chooses its own length, Strong enough and still falls over and The section that cannot reach its own strength.

15 min read 7 figures Which failure arrives firstThe load must go somewhereGeometry beats material

A load that chooses its own length ends with a sentence that has to be taken on trust: a bearing stiffener changes the problem completely rather than improving it. This is what the changed problem is.

The unstiffened check is a plate one. The flange bends under the bearing, yield lines spread the load along the web over an effective length the flange’s stiffness and the web’s strength decide between them, and the web yields over that length — and then a slenderness reduction takes most of it away, because a long thin panel under an edge load buckles before it yields. On the girder that essay used, a 1,200 × 8 mm web with 300 × 20 flanges, the yield resistance is 1,586 kN, the reduction factor is 0.283, and what is left is 450 kN.

The stiffened check is not a plate one at all.

The stub column a bearing stiffener makes, in plan. A plan through the girder at the bearing. The 8 mm web runs across; the pair of 100 × 12 stiffeners stands off it; and the shaded strip of web either side — 15ε t_w, or 98 mm each way — is the width that buckles with the stiffeners rather than independently of them. Together they are an area of 3962 mm² with a second moment of 9.01·10⁶ mm⁴ about the web's centreline, a radius of gyration of 48 mm over a buckling length of 900 mm — 0.75 of the depth, because the flanges hold the ends. That is a slenderness of 0.25, at which the column curve returns 0.98: the stub column reaches 98 per cent of its squash load, and the section's own strength is very nearly the whole answer.
Fig. 1 A plan through the girder at the bearing. The web runs across, the pair of 100 × 12 stiffeners stands off it, and the shaded strip either side — 15ε t_w, 98 mm each way — is the width of web that buckles with the stiffeners rather than independently of them. Area 3,962 mm², radius of gyration 48 mm, buckling length 900 mm: a slenderness of 0.25, and a resistance of 1,373 kN.

Three times the web’s own answer, from two plates weighing eighteen kilograms.

The check changes family, and that is the whole of it

It is worth being exact about what has been swapped for what, because “the stiffener helps” is not a description of anything that happens.

The web’s patch resistance is unaltered by the presence of a stiffener next to it. It is still 450 kN, computed from the same flange and the same web with the same yield lines. Nothing about the plate is different.

What the stiffener supplies is a second path, and the load takes it. The bearing plate sits on the flange, the flange sits on the stiffeners, and the stiffeners take the reaction down to the other flange as a compression member. The web is no longer being asked to carry a patch; it is being asked to be part of a column, which it is much better at.

So the two numbers on this page are not two estimates of one quantity. They are the capacities of two different structures, and the design decision is which one to build.

Which free body produced the number

The free body is a horizontal slice through the girder at the bearing, cut just below the top flange and just above the bottom one.

Crossing the top cut is the reaction, spread over the stiff bearing length. Crossing the bottom cut is the same force, delivered to the support. The material inside the cut is the two stiffener plates and the web between and beside them — and the question is at what load that piece of material stops standing up straight.

That is a column, and the only three things a column needs are an area, a second moment about the axis it will bend about, and a length. All three are arguable and all three are in the figure.

The area includes web. A width of web adjacent to a stiffener is held straight by it and buckles with it rather than away from it, so it belongs to the column. The conventional width is 15ε t_w each side, which on this 8 mm web is 98 mm — a strip that contributes 195 mm² × 8 = 1,562 mm² of the total 3,962, which is 39 per cent of the area and almost none of the second moment.

The axis is the web’s own centreline. The pair of plates is a cruciform, and its strong axis is the one containing the web, which is enormously stiff and is not the axis anything buckles about. The weak axis is the one across the web, about which the two outstands are lever arms — which is why an outstand buys second moment as the cube and area only linearly.

And the length is not the depth. The flanges restrain the ends of the stiffener against rotation, so the buckling length is conventionally 0.75 h_w: 900 mm on a 1,200 mm web.

The factor of 1.8 that does not matter

That last convention deserves a moment, because it is the one that looks most like a fudge and is the one that matters least.

A buckling length of 0.75 rather than 1.0 is a factor of 1.78 on the critical load, which sounds decisive. Compute it both ways and the resistance moves from 1,373 kN to 1,314 — four per cent.

What the stub column carries, against the size of its outstand. The resistance of a pair of bearing stiffeners on a 1200 × 8 web, against the outstand of the plates, with the unstiffened web's patch resistance of 450 kN drawn across it. The pair overtakes the web it is standing on at 40 mm of outstand, which is smaller than any stiffener anybody details. At the 100 mm drawn the answer is 1373 kN — 3.1 times the web's own — and the curve is very nearly a straight line, because the area grows with the outstand while the slenderness that would reduce it is already small enough not to matter. The dashed line is the squash load: the gap between the two is the whole of the buckling check.
Fig. 2 The stub column’s resistance against the outstand of its plates, with the unstiffened web’s 450 kN drawn across. The dashed curve is the squash load — the area times the yield stress, with no buckling in it at all — and the gap between the two is the entire buckling check. The stub column overtakes the web it stands on at 40 mm of outstand, which is smaller than anybody details.

The reason is that a stub column is not slender. A radius of gyration of 48 mm over a length of 900 gives λ = 19, and the reference slenderness for this steel is 76, so λ̄ = 0.25 — inside the plateau of the column curve where χ is 0.98 and buckling has almost nothing to say. Halve the length or double it and the answer is still on the flat part.

This is worth carrying beyond bearing stiffeners. A member’s buckling length is decisive only where the member is slender, and a great deal of argument goes into effective-length factors for members that are not. The place the same argument bites is the opposite one: a stiffener designed as though buckling did not exist at all, at an outstand past its own class limit, where the failure is local and the length never enters.

The stub column a bearing stiffener makes, in plan. A plan through the girder at the bearing. The 8 mm web runs across; the pair of 150 × 10 stiffeners stands off it; and the shaded strip of web either side — 15ε t_w, or 98 mm each way — is the width that buckles with the stiffeners rather than independently of them. Together they are an area of 4562 mm² with a second moment of 2.44·10⁷ mm⁴ about the web's centreline, a radius of gyration of 73 mm over a buckling length of 900 mm — 0.75 of the depth, because the flanges hold the ends. That is a slenderness of 0.16, at which the column curve returns 1.00: the stub column reaches 100 per cent of its squash load, and the section's own strength is very nearly the whole answer.
Fig. 3 The same web with a wider, thinner plate: 150 × 10 rather than 100 × 12. The area is larger, the radius of gyration is larger, the slenderness falls to 0.16 and the column curve returns 1.00 — but the outstand is 150 mm against the 114 that 14ε t_s permits, so the plate is class 4 and buckles locally before the section reaches the squash load this arithmetic has just promised.

A stiffener has two checks and they pull in opposite directions: the section classification wants a stocky plate, and the stub column wants a wide one. The efficient stiffener is at the corner where the outstand is exactly 14ε t_s and both are satisfied at once, which is a design rule that falls out of two calculations rather than being handed down.

Against a thicker web

The alternative to two plates is more web, and the comparison is not close.

Two ways to carry a bearing reaction, against the thickness of the web. The unstiffened web's patch resistance and the stiffened stub column's, both against web thickness, for a 1200 mm girder with a 100 mm bearing. The web's own curve rises as roughly the square of the thickness — 1.95 over this range — and the stiffened one rises only through the strip of web that acts with the plates, so the two converge slowly. They do not cross anywhere in this range: no web thick enough to overtake a pair of plates is drawn here. At the 8 mm web this girder has, the stiffened path is 3.1 times the unstiffened one, and the 900 kN reaction needs it. The choice is between steel everywhere along the girder and two plates at two points, which is why the answer is nearly always the plates.
Fig. 4 The two load paths against web thickness. The web’s own resistance rises as the square of the thickness — 1.95 fitted over this range — and the stiffened path rises only through the strip that acts with the plates, so they converge slowly and do not meet inside the range drawn. A 900 kN reaction needs the stiffener at any web thickness this girder would sensibly have.

The web’s curve is steep, so the arithmetic is genuinely arguable: a 25 mm web carries 4,181 kN as a patch and beats the stiffened 8 mm web four times over. What settles it is that the web is everywhere and the stiffener is at two points.

A 30 m girder with a 1,200 mm web carries 283 kg of web per millimetre of thickness. Going from 8 mm to 25 to carry a bearing reaction costs four and three-quarter tonnes; the two stiffeners cost eighteen kilograms. The patch check governs at supports and under any wheel that stops there, and the rest of the web is sized by shear, by flange-induced buckling, or by nothing at all.

That ratio is why the bearing stiffener is not a refinement but the normal answer, and why the unstiffened patch check earns its keep precisely where a stiffener is unavailable: under a travelling wheel, under launching rollers, at a support during erection before the plates are welded on.

The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 300 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 759 mm — 2.5 times the bearing, and 60% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to.
Fig. 5 What the unstiffened web does with a bearing three times as wide. Tripling the stiff bearing from 100 to 300 mm lengthens the spread from 559 to 759 mm and raises the resistance from 450 to 524 kN — 16 per cent for three times the plate, because the spread the flange supplies is most of the length either way.

The same argument at twice the depth

Everything above scales, and it scales in the direction that makes the stiffener more attractive rather than less.

Two ways to carry a bearing reaction, against the thickness of the web. The unstiffened web's patch resistance and the stiffened stub column's, both against web thickness, for a 2000 mm girder with a 150 mm bearing. The web's own curve rises as roughly the square of the thickness — 1.93 over this range — and the stiffened one rises only through the strip of web that acts with the plates, so the two converge slowly. They do not cross anywhere in this range: no web thick enough to overtake a pair of plates is drawn here. At the 12 mm web this girder has, the stiffened path is 3.7 times the unstiffened one, and the 2000 kN reaction needs it. The choice is between steel everywhere along the girder and two plates at two points, which is why the answer is nearly always the plates.
Fig. 6 A 2,000 × 12 web with 500 × 30 flanges, 180 × 20 stiffeners and a 2,000 kN reaction. The unstiffened patch resistance is 1,006 kN and the stub column is 3,766 — a factor of 3.7 rather than 3.1, and both curves have moved up and to the right without their relationship changing.

The stub column’s slenderness on that girder is 0.22, lower than on the shallower one, which is not the obvious result: the web got deeper, so the buckling length grew by 500 mm, and the stiffener got wider, so the radius of gyration grew by more. Deeper girders have stockier bearing stiffeners, because the outstand is proportioned to the flange width and the flange width to the span rather than to the depth.

What does change with depth is the unstiffened side of the comparison. The patch resistance contains tw3/hwt_w^3/h_w in its buckling term, so a deep thin web is worse at patch loads than a shallow one of the same thickness — the deep girder’s 12 mm web carries 1,006 kN where the shallow girder’s 12 mm would carry 989 with two thirds of the depth.

Where the number 15ε comes from, and what it is standing in for

The strip is the one term in the area that nobody measures, and it is worth knowing what kind of quantity it is before trusting a resistance that is 39 per cent made of it.

A plate held straight along one edge and free along the other buckles at a stress that falls with the square of its width. Set that stress equal to the yield stress and rearrange, and the width at which a plate can just reach its own yield is a multiple of its thickness times ε — the same rearrangement that produces every class limit in the section tables and every effective width in a slender one. For an outstand restrained at one edge the multiple is around 14; for the web strip beside a stiffener, restrained at one edge by the stiffener and continuing into more web at the other, the convention takes 15.

So the number is not a measurement of how far the stiffener’s influence reaches. It is the width of web that could carry yield stress if it had to, which is a different question with a similar answer, and the reason it is used is that the stub-column calculation wants an area at full stress rather than a stress distribution.

That substitution is everywhere in this subject and it is worth naming once. An effective width is always a fiction of the same shape: a real distribution replaced by a uniform one over a smaller area, chosen so the totals agree. The buckled plate’s effective width does it for a compression flange, shear lag does it for a wide flange at a support, and the strip here does it for a web beside a plate. All three are the same device, they are all calibrated rather than derived, and none of them describes anything the material is doing.

The reaction that arrives at a point, and the one that does not

There is a class of member for which none of this works, and it is worth putting beside the girder because it is where the patch check and the stub column meet.

The column curve. Failure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.
Fig. 7 The column curve the stub column is being read off, drawn out to a slenderness of 120. A bearing stiffener at λ = 19 sits at the far left, where the real curve and the squash line are the same to within a per cent or two. Everything the effective-length argument is about happens past λ = 60, which is where a member has to be before its length is worth arguing over.

A rolled beam at a support has no stiffener and usually needs none: a 400 mm web at 8 mm is a much stockier plate than a 1,200 mm one, its patch reduction factor is 0.71 rather than 0.283, and it carries 521 kN as a plain web against the 1,123 a pair of small plates would give it. The check passes, so nothing is drawn, and the same load path is being used with none of this arithmetic done.

What that means is that the two calculations on this page are not alternatives a designer chooses between so much as two regions of one problem separated by web slenderness. Below about 60 the web carries the patch and the stiffener is not needed; past about 120 the web’s reduction factor has fallen far enough that only the stub column is worth having. The girder here, at 150, is well inside the second region, and a support that is not a point is where the transition is usually met.

Where the model stops

One reaction, applied vertically, at a stiffener that is there. A skewed support, a bearing that is off the centreline of the stiffener, or a reaction with a horizontal component puts a moment into the stub column, and a member at λ̄ = 0.25 is much less tolerant of an eccentricity than of a length.

The stiffener is fitted to both flanges. A stiffener cut short of the bottom flange — which is common, because welding into the tension flange is a fatigue detail nobody wants — has no bottom end restraint and no bearing there, and the load path it was supposed to complete stops in mid-air. That detail is decided by the detail category and it changes the structural model, which is a combination worth watching for.

And the weld is not in any of this. The stiffener has to be connected to the web for the strip of web to act with it and for the load to arrive at all, and the connection is a shear flow along the length of the plate. Nothing in the stub-column calculation checks it, which is the usual arrangement: a connection is not a point and the member calculation assumes it away.

What the pictures cannot show

The section drawing is a plan, and it cannot show the one thing a bearing stiffener is most often criticised for: fit.

A stiffener carries load by bearing against the underside of the flange, and it does that only if it touches. A plate cut 2 mm short is a plate that carries nothing until the girder has deflected enough to close the gap, and the web it was supposed to relieve is carrying the patch load meanwhile. That is why bearing stiffeners are specified as fitted — machined or ground to the flange — and why the specification note is a structural requirement rather than a workmanship one. It is the same argument as a member built to the wrong length, at a scale of millimetres and with no redundancy to absorb it.

The other absence is time. Every number here is a resistance, and the reaction that will be applied to it during launching or jacking arrives at whatever point the girder happens to be over, which is what the unstiffened check is for.

The assumption the figure rests on

That the strip of web is either fully effective or absent.

15ε t_w is a step function standing in for something continuous. The web adjacent to a stiffener is restrained by it and the restraint decays with distance, so the true participation is a smooth curve and the effective width is an integral under it. The convention picks a width at which the integral comes out about right, and it is the same device as the effective width of a buckled plate and the shear-lag width of a flange — an area chosen so that a uniform stress over it gives the right total.

On this section it is 39 per cent of the area, so an error of a quarter in the convention is an error of a tenth in the resistance. On a thicker web it is more, and on a girder whose stiffeners are close enough together for their strips to overlap it is double-counted — which is the one place the convention fails outright rather than approximately.

The history, which runs the other way round

Bearing stiffeners are older than the check that justifies them by about a century. Riveted plate girders from the 1850s onward carry a pair of angles at every support and under every heavy load as a matter of course, and the reason given in the manuals of the period is that the web would otherwise crumple — a word that is a description of the failure and not a model of it.

The stub-column treatment arrived with the elastic column theory it borrows, and the strip of web with the effective-width ideas of the 1930s and 1940s. The unstiffened patch check is younger still: the yield-line spread and the plate-buckling reduction that make up the 450 kN on this page belong to the 1970s and 1980s, when welded girders with webs far thinner than any riveted one made it a question anybody had to answer.

So the practice came first, the calculation for the stiffened case second, and the calculation for the case with no stiffener last — which is the usual order when a detail is cheap. Nobody needed to know what an unstiffened web could carry while every girder had stiffeners at every load point, and the check became necessary at exactly the moment when webs got thin enough and loads mobile enough for the stiffener to be unavailable.

The ladder from here

Later rungs on this anchor: the interaction between a patch load and bending, which is where the check bites on a crane girder and where neither of the two calculations here is the governing one. Intermediate transverse stiffeners, which are a different member with a different job — they divide a panel and hold a tension field’s anchorage rather than carrying a reaction, and their requirement is a rigidity rather than a strength. Web crippling in cold-formed sections, where the corner radius means no yield-line model fits and the treatment is wholly empirical. Longitudinal stiffeners under a patch load, where the buckling coefficient changes and the panel that buckles is not the panel that was drawn. And the launching case in full, where the patch load travels the whole length of the girder and every section is checked at the position that is worst for it.

Where this goes next

The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 100 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 559 mm — 5.6 times the bearing, and 82% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to. The load that chooses its own length Every other load in this collection arrives over a length somebody decided. A wheel on a crane girder does not — the flange bends under it and spreads it along the web, and how far it spreads is an output of the flange's own stiffness against the web's own strength. The effective length is 5.6 times the bearing that produced it. A stiffener is a boundary condition, and it is bought at a threshold. The buckling stress of a 2400 × 12 mm plate with one longitudinal stiffener, against how rigid that stiffener is. Below γ the stiffener rides on the buckle and the plate takes the whole-width mode; at γ the stiffener stays straight and the plate buckles between stiffeners at 74 N/mm², 4.0 times the bare plate's 18.5. Above γ nothing further happens at all, because the sub-panel mode does not know the stiffener is there. The curve is a ramp and then a horizontal line, so a stiffener at twice γ is exactly as good as one at γ. Here γ = 31.5, which asks for an outstand of 144 mm; the 150 mm one drawn gives γ = 35.5, a margin of 1.13. The rib that is a boundary condition A rib on a plate is not a member carrying load. It is a line the buckle is not allowed to cross — and it becomes one at a threshold. Below the required rigidity it rides on the buckle and buys a fraction; at the threshold it stays straight and the plate buckles between stiffeners; above it, nothing further happens at all. A transverse load with nothing applied. Web slenderness against web thickness, with the limit the flange's own curvature sets. A flange carrying 6213 kN and curved to a radius of 592 m needs 10.5 N per millimetre of radial force to stay on its curve, and the only thing available to supply it is the web. Nothing has been applied to the girder: the load comes from the deflected shape, which is why a straight beam has none of it and a beam at a plastic hinge has a great deal. Setting the radial force against the web's own plate-buckling resistance gives, in four lines, h_w/t_w ≤ k·(E/f_yf)·√(A_w/A_fc) — the form the codes use, arrived at without them. The constants differ: an elastic flange strain gives k = 1.34 and the rule uses 0.3, a factor of 4.5, and the gap is the curvature assumed. k goes as the inverse square root of the flange strain, so 0.3 is a flange strained to 3.4% — which is what a plastic hinge does to it. The rule is not conservative; it is written about a different beam. The web that is crushed from inside A plate girder's compression flange is curved by the beam's own deflection, and a curved force needs a transverse load to stay on its curve. The only thing available to supply it is the web. So a deep girder can buckle its web vertically with nothing applied to it at all, and the rule that prevents it is the only clause in the codes about a load no load case contains. The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 200 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 659 mm — 3.3 times the bearing, and 70% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to. The support that is not a point A reaction is drawn as a single arrow because the equilibrium equations only need its total. Underneath the arrow is a bearing of some width, delivering a pressure over that width, and almost everything a designer would like to know about the region near a support is a consequence of the width the arrow does not have.

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Bearing stiffenerColumn curveEffective lengthLoad pathPatch loadingPlate bucklingPlate girderRadius of gyrationSection classificationSlendernessSquash loadStiffenerStub columnWeb