Structural Engineering & Statics
where the load goes
Concept

Curved beam — where it appears

A member whose axis is curved in the plane of its bending, so that its fibres are of different lengths before any load is applied. Its strain varies as one over the radius rather than linearly, which puts the neutral axis inside the centroid and makes the stress a hyperbola.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Cut the throat, and the face carries a moment and a tension at once. A crane hook of trapezoid section, 50 to 120 mm radius and 40 to 15 mm wide, carrying 50 kN, drawn beside the section at the cut and the stress across it. The load hangs on a line through the centre of curvature, so cutting the throat and taking everything below the cut as the free body leaves a face carrying a direct tension of 50 kN and a moment of N·R = 3.985 kN·m about the section's own centroid, which sits a full R = 79.70 mm from the load line. The stress is a hyperbola, zero at r = 75.04 mm rather than at the centroid 4.65 mm outside it, reaching 248.7 N/mm² of tension at the inner fibre and 140.6 of compression at the outer. The straight-beam formula, drawn dashed, reports 161.7 N/mm² for the bending part against the true 222.8, and leaves the 26.0 N/mm² of direct tension out altogether — between them, 54% under the real peak, at the fibre where a hook actually breaks. This is why a hook is trapezoidal: both effects are worst inside, so the material goes there.

The bar that was bent before it was loaded

In a curved bar plane sections still stay plane, and the bending formula is wrong anyway. The fibres were different lengths before anything was applied, so an equal rotation of two plane faces produces unequal strain — the stress is a hyperbola, the neutral axis has moved inward, and a crane hook carries half as much again as My/I reports.

sections · Curved beam Cut the throat, and the face carries a moment and a tension at once. A crane hook of trapezoid section, 50 to 120 mm radius and 40 to 15 mm wide, carrying 100 kN, drawn beside the section at the cut and the stress across it. The load hangs on a line through the centre of curvature, so cutting the throat and taking everything below the cut as the free body leaves a face carrying a direct tension of 100 kN and a moment of N·R = 7.970 kN·m about the section's own centroid, which sits a full R = 79.70 mm from the load line. The stress is a hyperbola, zero at r = 75.04 mm rather than at the centroid 4.65 mm outside it, reaching 497.5 N/mm² of tension at the inner fibre and 281.3 of compression at the outer. The straight-beam formula, drawn dashed, reports 323.4 N/mm² for the bending part against the true 445.5, and leaves the 51.9 N/mm² of direct tension out altogether — between them, 54% under the real peak, at the fibre where a hook actually breaks. This is why a hook is trapezoidal: both effects are worst inside, so the material goes there.

The wide side goes inside

A crane hook's section is a trapezoid with its broad face towards the centre of curvature, and that is not a casting convenience. Turn the same section round — same area, same depth, same moment — and the stress at the fibre that breaks rises by thirty-nine per cent. The shape is doing two things at once, and only one of them is in a straight beam's arithmetic.

sections · Curved beam A curved I-beam's flanges curl. The cross-section of an I-beam bent to a radius of 3.0 m, its flanges 300 mm wide and 15 mm thick, with the flanges' radial movement exaggerated 47 times; dashed, where the flanges would be if they did not bend across their width. It is bent the way that opens the curve, so the outer flange is in tension and the inner in compression. A tensioned flange round a curve is pulled toward the centre of curvature and a compressed one pushed away from it, so here both are pressed toward the web, and each outstand bends like a cantilever from the web, its tip moving 0.80 mm. Bent the other way, both would curl away from it by the same amount. The bars are the stress along the beam across each flange: the full value at the web, falling toward the tips to 0.72 of it, so that the flange works as if it were 89 per cent as wide.

The flange that curls away from its stress

Bend an I-beam into a curve and each flange, carrying its stress round the bend, is pressed sideways by that stress. The outstands bend like cantilevers from the web, and in bending they move to a different radius and shed the very stress that pushed them. At a tight radius only a strip beside the web works. But the flange's loss of width arrives slowly, and a stress nobody computes for a straight beam arrives at once: the flange bending across its own width, harder than it is stressed along the beam.

sections · Curved beam The flange is bent hardest where it is thinnest, not where it curls most. The transverse bending stress at the root of the inner flange, where it meets the web, along a knee curved to 1,000 mm through a quarter turn, its flanges 600 mm apart on a 10 mm web, carrying 500 kN·m all round the curve; the dotted line is a yield stress of 355 N/mm². Uniform, 269 all round. Narrowing, 283 at the rafter end and 207 at the column end, largest where λ is; thinning, 191 at the rafter end and 397 at the column end, largest where λ is too. Tapered both ways, 205 at the rafter end and 325 at the column end: largest at the column end, where λ is 0.94 against 1.14 at the rafter end, because there the flange's own stress has nearly doubled.

The flange bent hardest where it curls least

A curved I-beam's flange curls towards its web and is bent across its own width at the root, by a stress that rises with the curling parameter λ. A portal knee's flange is cut from plate and usually tapers, and a taper changes λ along the curve. So the obvious guess is that the transverse stress peaks where λ is largest. For a flange that only narrows, or only thins, it does. Taper it both ways and the guess fails: λ is smallest at the narrow, thin end, and that is where the flange is bent hardest, because the same force is carried there on half the steel. The force the changing width hands to the web, meanwhile, is a few newtons a square millimetre.

sections · Curved beam

Named alongside it

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

Bending stressCentroidCurvatureEffective widthNeutral axisStress concentrationStress distributionElastic foundationElastic limitFatigueFlange curlingFree body

All concepts