Structural Engineering & Statics
where the load goes
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Prestress limits — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four inequalities, and the wedge between them. The Magnel diagram: every limit on a prestressed section, plotted as a bound on 1/P against the eccentricity. Two of the four come from transfer, when the force is largest and the only moment is the beam's own weight, and two from service, when 20% of the force has been lost and the moment is 640 kNm. Each is linear in 1/P, which is the substitution that makes the problem a picture rather than a search. The shaded region is every force-and-eccentricity pair the section will accept: it is a wedge opening to the right, so the cheapest prestress is always at the largest eccentricity the cover allows — 1029 kN at e = 400 mm here. The section's kern is 241 mm, and every useful answer is outside it.

    Four inequalities and a wedge

    A prestressed section has to satisfy two stress limits when the force is largest and the load smallest, and two more when the force has relaxed and the load has arrived. Each is linear in one over the force — which turns a search for a prestress into a region on a page, and turns an impossible section into an empty one.

    part 1 · sections
  2. Each fibre has to earn its own prestress. For each section on a 12 m span carrying its own weight and 20 kN/m more: bars, the top and bottom moduli it has; ticks, the moduli its two pairs of limits need — the top fibre holding the service compression and the transfer tension, the bottom holding the service tension and the transfer compression, both from the same prestress. The symmetric I: top 28.9 against 17.4 needed, bottom 28.9 against 23.5 (× 10⁶ mm³) — a prestress exists; the tee: top 48.5 against 17.8 needed, bottom 22.7 against 24.0 (× 10⁶ mm³) — no prestress exists; the bulb-tee: top 46.0 against 17.7 needed, bottom 37.4 against 23.8 (× 10⁶ mm³) — a prestress exists.

    The flange the prestress cannot use

    The four stress limits on a prestressed section pair up by fibre: the bottom fibre has to hold the service tension and the transfer compression from the same force, and the top fibre the service compression and the transfer tension. Each pair is possible only if that fibre's own section modulus is large enough, whatever the eccentricity. On a symmetric section the two pairs are close to balanced. On a tee they are not: its wide flange multiplies the top fibre's modulus and hardly touches the bottom's, so a tee with half as much concrete again as a symmetric I cannot be prestressed for a load the I carries.

    part 2 · sections

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