Structural Engineering & Statics
where the load goes
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Arch buckling — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The arch does not squash; it leans. The first two buckling modes of the same rib, drawn against its undeformed shape at an exaggeration of a few hundred. The first, at wcr L³/EI = 49.7, is antisymmetric: one half rises while the other falls and the crown moves sideways. The second, at 117.1, is symmetric — the whole rib settling. The two differ by a factor of 2.36, which is why an arch is braced against sideways movement of its crown rather than against the load it is carrying, and why a tied arch with a single hanger at midspan is doing nothing for the mode that governs it.

    The arch that leans instead of squashing

    A masonry arch is asked whether a line of thrust fits inside it. A steel rib is asked a different question entirely: it is a column carrying an axial force along its whole length, and the mode it buckles in puts one half up and the other half down while the crown moves sideways.

    part 1 · stability
  2. The rib yields long before it would buckle. The largest combined stress, axial plus bending, anywhere in a 60 m two-pinned parabolic rib at a 12 m rise (EI 250,000 kN·m²) with a 12,000 mm² section 1,000 mm deep, against a factor on 15 kN/m of dead load over the span and 6 kN/m of live load over the left half together. Dashed: first order; solid: second order; dotted: the 355 N/mm² yield stress. First order, the rib reaches yield at a factor of 1.73; second order, at 1.18. The same load pattern, spread uniformly, would bifurcate the rib at a factor of 2.97, the right-hand edge — which is the number the eigenvalue reports and a number the rib never reaches.

    The weight that bends a rib it cannot bend

    A parabolic rib carries its own uniform dead load as pure thrust, with no bending in it at all. Put a live load on half the span and the rib bends in the shape of its own buckling mode from the first kilonewton, and the whole thrust, most of it from the dead load, multiplies that bending. The rib never reaches its buckling load. It yields well short of it, at a load its first-order check says it can carry.

    part 2 · stability

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