Structural Engineering & Statics
where the load goes
Generator

Strain goes as 1/r, so the stress is a hyperbola and its zero has moved

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
Strain goes as 1/r, so the stress is a hyperbola and its zero has moved. Bending stress across a trapezoid of 70 mm depth curved to R/h = 1.14, under 3.985 kN·m. The fibres are not the same length, so strain goes as 1/r rather than as r and the stress is a hyperbola: 222.8 N/mm² at the inner fibre against 161.7 from My/I, a factor of 1.378, and 166.6 of compression at the outer fibre against My/I's 219.4. The axis of zero stress is at r = 75.04 mm, 4.65 mm inside the centroid at 79.70 mm — 35.8% of the depth from the inner fibre rather than the 42.4% the centroid sits at. Everything divides by e, which is a difference of two nearly equal numbers, and the solver checks its closed form against a quadrature before anything is divided by it.

Strain goes as 1/r, so the stress is a hyperbola and its zero has moved. Bending stress across a trapezoid of 70 mm depth curved to R/h = 1.14, under 3.985 kN·m. The fibres are not the same length, so strain goes as 1/r rather than as r and the stress is a hyperbola: 222.8 N/mm² at the inner fibre against 161.7 from My/I, a factor of 1.378, and 166.6 of compression at the outer fibre against My/I's 219.4. The axis of zero stress is at r = 75.04 mm, 4.65 mm inside the centroid at 79.70 mm — 35.8% of the depth from the inner fibre rather than the 42.4% the centroid sits at. Everything divides by e, which is a difference of two nearly equal numbers, and the solver checks its closed form against a quadrature before anything is divided by it.

3 essays call curved-beam. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Sections and stress

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.

 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. Sections and stress

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.

 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. Sections and stress

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.

The library, page 1 of 7 — where curved-beam sits