Structural Engineering & Statics
where the load goes
Concept

Stiffener — where it appears

A plate welded to a girder web to carry a local force or to shorten the panel that buckles. Bearing stiffeners take a reaction directly into the flanges, and intermediate ones divide the web so that its buckling is checked against a panel rather than against the whole depth.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

stability · Patch loading Pull it along the girder and it just unfolds. One period of a 30° trapezoidal corrugation, 300 mm of flat and 260 mm of incline, and the same period pulled along the girder's axis. The fold opens by bending the inclined panels out of the web's own plane, so the axial flexibility contains the plate's t³ where a flat web's would contain t — and the effective modulus that comes back from solving the cell as a frame is 222 N/mm², which is 10.6 parts in ten thousand of the steel's 210 GPa. A web with a thousandth of the stiffness carries a thousandth of the stress, which is why the flanges of a corrugated girder carry the whole moment and why the section has 9 per cent less second moment than the flat-webbed girder it replaces. The fold buys freedom from stiffeners and pays for it here.

The web that carries no bending

A corrugated web needs no stiffeners, because the folds give it in one direction a depth it does not have in its thickness. In the other direction the same folds make it an accordion — and a web that cannot be stretched cannot carry a bending stress at all.

sections · Corrugated web The width nobody drew. A gusset plate with a brace bolted to it over 240 mm, and the width the profession has agreed to pretend is carrying the force. Everything else on this site arrives with a cross-section; a gusset does not, because it is a piece of steel with something attached somewhere in the middle of it and there is no geometry that says how much of it is working. The answer is the Whitmore section: assume the force spreads at 30° from the first fastener and take the width it has reached at the last, b_eff = w + 2L·tan30° = 367 mm. That is 4.08 times the width anything is actually attached to, and the rule comes from a 1952 master's thesis. It has since been checked against finite element work and holds to about ten per cent, which is fortunate, because moving the assumed angle by ten degrees moves the answer by 34%. On this plate the check that governs is not the stress the rule was written for: it is the Whitmore section buckles, at 721 kN against 1564.

The width nobody drew

Every other member in this collection arrives with a cross-section. A gusset plate does not — it is a piece of steel with a brace bolted to it somewhere in the middle, and no geometry says how much of it is working. The profession's answer is a thirty-degree spread from a 1952 master's thesis, it invents three quarters of the area being checked, and the check it was written for is not the one that governs.

connections · Gusset 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.

stability · Flange induced One plate, three systems, one stress. The stress at the rib-to-deck weld of an orthotropic deck, split by which system produced it. The main girder contributes 120 N/mm², the crossbeam 34 and the trough 13, and they add to 167 because the deck plate is the top flange of all three at once. A calculation that treats the crossbeam as a support rather than a structure understates the total by 26 per cent. The stress RANGE at that weld is 47 N/mm², which is the number the whole deck is designed by, because the weld is a low fatigue category running the entire length of the bridge.

One plate and three structures

An orthotropic deck is a single steel plate stiffened by troughs, sitting on crossbeams, sitting on main girders. Nothing about that is unusual until it is noticed that the plate is the top flange of all three, so a wheel standing on it loads every one of them at once.

sections · Orthotropic deck A 10 mm plate, and the width it can be. The elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 462 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 900 mm only 46 per cent of it is still working.

The coefficient that is not four

A plate's buckling stress carries a coefficient that looks like a constant and is not. It is 4 for an internal element, 0.43 for an outstand and 23.9 for a panel in shear — and the width a 10 mm plate may be runs from 152 mm to 1,130 across that range.

stability · Plate buckling A buckled panel is a truss that nobody drew. A 1500 × 2000 panel of 8 mm web, at d/t = 188. It buckles in shear at 41.0 N/mm², which is 492 kN — and it then carries 1187 kN, 2.41 times as much, because the tension diagonal takes over from the compression one that has gone. The band runs at 18.5° with a membrane stress of 348 N/mm² over a width of 788 mm, and it pulls on the flange at 280.2 N per millimetre of its length. A web that never buckled at all would have reached 2460 kN, so the panel ends at 48% of a stocky web's capacity on a fraction of its steel.

The tension has to pull on something

A buckled web carries its shear on a diagonal band of membrane tension, and the band pulls sideways on the flanges and stiffeners that bound it. That pull is the design output nobody plots — it runs from 72 to 603 newtons per millimetre across ordinary panel proportions, it is largest exactly where the panel is most efficient, and at the end of the girder there is nothing beyond to take it.

stability · Tension field What a torsional brace buys, and the stiffness it takes. The critical moment of an 8 m beam against the stiffness of a single rotational restraint at midspan — a cross-frame or a stiffened connection to a secondary beam, resisting the twist rather than the sideways movement. It climbs from an unbraced 143 kNm to the same two-half-wave plateau of 447 a lateral brace reaches, and gets to 99% of it at 253 kNm/rad. Nothing in the calculation refers to a height, which is the difference that matters: a torsional brace cannot be put on the wrong flange because it is not attached to a flange in the sense the lateral one is.

Twice the moment, four times the brace

A lateral brace has to be on the right flange and its demand is very nearly linear in the load. A torsional brace has no flange to be wrong about, and its demand is exactly quadratic — so the restraint that is indifferent to where it is attached is the one that gets expensive fastest.

stability · Beam bracing 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.

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.

stability · Patch loading The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything.

The plate that is only as big as it is thick

A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all.

connections · Base plate Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 200 mm, carrying a reaction of 300 kN through three bolts, as utilisations, for the end as coped and with three repairs: an 8 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.46, shear 0.41, tear-out 0.57 and local buckling 0.46. The doubler thickens the web, which is most of what a coped tee is, and lowers all four; the edge plate gives the tee back a flange and lowers only flexure and buckling, leaving shear and tear-out exactly where they were. What governs: as coped, tear-out at 0.57; doubler, tear-out at 0.30; edge stiffener, tear-out at 0.57; both, tear-out at 0.30.

The repair that fixes the wrong check

A coped beam end that fails its checks is usually repaired by welding a plate along the edge the cope left, because the cope removed a flange and the plate puts one back. On ordinary proportions that repair restores the check that was not failing. The check that governs is carried by the web, and only a plate on the web reaches it.

connections · Coped beam Two models of one buckled web. The shear resistance of a web 1500 mm deep and 8.0 mm thick in steel of 355 N/mm², as a share of its plastic shear 2,460 kN, against the spacing of its intermediate stiffeners as a multiple of the depth, by Basler's tension field, by the rotated stress field with a rigid end post and with a non-rigid one, and by buckling alone; the dots at the right-hand edge are the same web with no intermediate stiffeners. With stiffeners at the depth Basler gives 0.71 and the rotated field 0.50; at three depths 0.38 and 0.42; with none, 0.14 — buckling alone — and 0.41. The two cross at a spacing of about 2.6 depths. Basler's reserve is a band the stiffeners anchor and it narrows to nothing as they move apart; the rotated field's is spread over the whole web and anchored at its ends, and the spacing reaches it only through the buckling stress.

Two reserves for one buckled web

A slender web keeps carrying shear after it buckles, and design uses two models of how: Basler's band of tension anchored on the stiffeners, and the rotated stress field spread over the whole web and anchored at its ends. Given the same 1,500 × 8 mm panel they agree about the buckling and disagree about everything after it — and in particular about what an intermediate stiffener is for. One says stiffeners save 37 per cent of the steel; the other, 1 per cent.

stability · Tension field

Named alongside it

The objects these essays reach for when they reach for this one.

Plate bucklingLoad pathSlendernessPlate girderWebLocal bucklingShear bucklingPost-bucklingAnchorageBearingBlock shearBrace

All concepts