Structural Engineering & Statics
where the load goes
Generator

The curved-beam generator

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 movedBending 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.5060708090100110120-200-1000100200300radius (mm)stress (N/mm²)neutral axis r = 75.04centroid 79.70222.8 N/mm²curved barMy/Ithe straight-beam answer

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.

1 essay calls 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.

the cutcentre of curvatureN = 50 kNR = 79.70 mmr_i = 50r_o = 120My/I248.7 N/mm²r = 75.04centroid 79.70free body: everything below the cut, so the face carries N = 50 kN and M = N·R = 3.985 kN·m248.7 N/mm² of tension inside against 140.6 of compression outside, on a neutral axis 4.65 mm off the centroid 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.

The library, page 1 of 5 — where curved-beam sits