Structural Engineering & Statics
where the load goes
Concept

Plate girder — where it appears

A beam fabricated from plates rather than rolled, so its web can be far more slender than a rolled section's. That slenderness is what makes it efficient and what makes it liable to shear buckling, patch loading and a compression flange pressing into it with nothing applied.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 coefficient a web gets depends on where its neutral axis is. The plate buckling coefficient for an internal element against ψ = σ₂/σ₁, the ratio of the stresses at the two edges of the panel. Uniform compression is ψ = 1 and k = 4; a gradient running from compression to zero is ψ = 0 and k = 7.81; pure bending is ψ = −1 and k = 23.92, six times the value a column's flange gets. The 1800 × 12 mm web drawn here starts at ψ = −1.000 and k = 23.92 and ends at ψ = −0.910 and k = 21.63, because losing width from the compressed half drops the neutral axis and deepens the compression zone. The curve is steepest exactly where a bending web sits, so a small movement of the neutral axis costs 2.29 of coefficient.

Classified by a gradient it does not have

A web in bending is the one plate whose buckling coefficient cannot be looked up. It depends on the stress gradient, the gradient depends on where the neutral axis is, and the neutral axis depends on how much of the web the coefficient has just taken away — so the answer is a fixed point, and the calculation everyone does is its first term.

stability · Plate buckling 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 The hole in the web pulls the axis down after it. The effective section of a 1,500 × 8 mm web between 400 mm flanges 15.0 mm thick at the top and 25.0 mm at the bottom, in S355, bending with its top in compression, and the elastic stress on it. The top flange's outstands are slender and keep 91% of their width. The web is compressed over 959 mm of its depth and keeps 44% of that, as two strips next to the flange and the neutral axis, leaving a hole of 538 mm. The neutral axis sits 566 mm above the bottom, against 665 mm for the gross section, and the section keeps 78% of its gross elastic modulus: 9.91 × 10⁶ mm³ against 12.76.

The flange a slender web wants heavier

A girder's gross section is strongest with its flange steel shared equally, top and bottom. Its effective section is not, because a slender web loses the middle of whatever part of it is compressed, and moving steel into the compression flange raises the neutral axis and shortens that part. Forty millimetres of flange on a 1,500 by 8 mm web is best split 22 on top and 18 below, and split the other way round — 15 on top and 25 below, the shape of a composite girder's steel before its slab hardens — it keeps 70% of what the best split carries.

sections · Effective cross-section

Named alongside it

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

Plate bucklingStiffenerSlendernessWebLocal bucklingCompression flangeEffective widthFree bodyLoad pathNeutral axisSection classificationSection modulus

All concepts