Structural Engineering & Statics
where the load goes
Concept

Contraflexure — where it appears

A point along a member where the bending moment passes through zero and the curvature changes sign. Its position is a property of a load case rather than of the structure, so a splice placed there carries no moment for one arrangement of load and some for every other.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The moment does not stop at the end of the beam. A portal frame of 8 m by 4 m with fixed bases, carrying 20 kN/m on the beam. The bending moment is drawn on the tension side of every member, and it runs round the corner without a break: 65.2 kNm arrives at the end of the beam and 65.2 kNm leaves down the column, which is the same number, since joint rotational equilibrium is one of the equations the frame solve satisfied. Midspan carries 94.8 kNm, and the two add to 160.0 — the 160.0 kNm of a simply supported span, to 0.0e+0 kNm. The corner takes 61% of the wL²/12 a fully built-in beam would have carried, because the columns are springs rather than walls: the beam-to-column stiffness ratio is 1.27. The beam's moment crosses zero 0.92 m from the corner and the column's 1.33 m above its base.

The moment that goes round the corner

At a rigid knee the bending moment does not stop at the end of the beam. It turns and runs down the column, and in the same instant the beam's shear becomes the column's axial force — while the block of steel that has to carry the turn appears on no member diagram anywhere.

internal-forces · Corner moment The end bolts do the work and the middle ones very nearly nothing. A lap of 10 bolts at 75 mm pitch transferring 800 kN between two plates, with the force each bolt actually carries drawn above it and the flat line a division by the bolt count would have given drawn behind. The end bolts carry 1.16 of their nominal share and the middle ones 0.89. The reason is not in the bolts: at the leading end the first plate is carrying everything and the second nothing, so the two strain at different rates and the slip between them is largest there. In the middle they strain alike, there is almost no slip, and a bolt with no slip across it transfers almost no force. The mean over the worst is 0.861, and the end bolt has to slip 1.16 mm before the rest catch up.

The joint that has to be as good as the member

A splice exists because members come in lengths and structures do not. It has to deliver the same force, at the same stiffness, in the same distribution across the section, through a discontinuity — and each of those three requirements is met by a different feature of the detail, with the third one usually left to look after itself.

connections · Splice The shear goes round the corner instead of across it. A 6-panel Vierendeel girder, 21 m by 6000 mm, under 300 kN at mid-span. There is no diagonal in it, so each panel's 150 kN of shear is carried as bending in the chords: the curves drawn along them are the chord moments, and every one passes through zero at the middle of its own panel. The local moment is the panel shear times the panel length over four, 131.3 kNm, and it adds to an axial force of 263 kN from the global moment at the same point. The girder deflects 109.09 mm against 6.53 mm for the same members triangulated — 16.70 times — and 98% of that movement is chord bending that a diagonal would have removed entirely.

The frame is a girder stood on end

Every unbraced building frame is a Vierendeel girder turned through ninety degrees, and the identification is not an analogy — it is the same equations with the axes swapped. Which means the frame inherits results that read as absurd for a building — more bays is stiffer, a wider building is not, and doubling one section property halves the sway.

structures · Vierendeel However heavy the chord, ordinary posts cap its share. The heavy chord's share of the shear in a panel of uniform shear in a Vierendeel girder of 2 m panels, 1.5 m deep, against how many times stiffer it is than the light chord, from the one-panel model (lines) and from a whole girder (dots). Splitting by second moment (dashed) runs from 0.5 towards 1. With posts as stiff as the light chord the share is 0.54 at a ratio of 4 and still only 0.55 at 30; posts 4, 16 and 100 times stiffer reach 0.65, 0.80 and 0.93 at 30. The posts, not the chord, set the ceiling. The dots are a twenty-panel girder at its sixth panel, far from both support and load: with ordinary posts 0.50, 0.53, 0.56, 0.60, 0.65, 0.69 at ratios of 1, 2, 4, 10, 20, 30. They follow the model until the heavy chord's own second moment becomes a noticeable part of the girder's — 31 per cent at a ratio of 30 — and it starts to carry the global moment by bending as a beam.

The heavy chord the posts will not let work

A Vierendeel girder carries its shear as bending in its two chords, and when one chord is four times as stiff as the other the obvious rule gives it four fifths of the shear. With posts as stiff as the light chord it takes 0.56. The heavy chord's end moments have to be carried round each joint by a post, the post bends, and the heavy chord turns with its joints instead of bending between them. No amount of extra steel in that chord takes its share past 0.55 in the middle of a long girder; only the posts can, and they have to be about 150 times as stiff as the light chord to make the rule come true.

structures · Vierendeel The hogging region cannot give its studs away. For a 457 × 191 × 67 steel beam under a 130 mm slab, continuous over two 12.0 m spans under 30 kN/m, 40 studs a span: the largest slip (solid, left scale) and the force in the slab's bars over the middle support (dashed, its value in the legend), against the density of studs beyond the point of contraflexure as a share of the density elsewhere, the span's total held at 40. At the same density everywhere the largest slip is 0.68 mm and the bars carry 344 kN; at half the density beyond contraflexure, 1.04 mm and 251 kN; at a quarter, 1.69 mm and 118 kN. The support's hogging moment falls from 448 to 398 kN·m as the bars stop working.

The slip that peaks where the moment turns

A simply supported composite beam slips most at its supports, and its studs are crowded there. Make the same beam continuous over a middle support and the slip peaks twice: at the end support, and again, just as hard and the other way, at the point of contraflexure — not over the middle support, where the shear is largest but the slip falls to nothing. Studs graded toward the supports, as a simple span's would be, miss the second peak and buy a tenth; graded to the slip they buy more than a third. And the hogging region cannot give its studs away to the span, because its edge is where the second peak is.

internal-forces · Composite action

Named alongside it

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

ContinuityLoad pathLoad sharingSlipStiffnessVierendeelBending stiffnessBolt groupChordComposite actionConnectionContinuous beam

All concepts