Structural Engineering & Statics
where the load goes
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Secondary prestress — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A reaction with no load, and the moment it bends the beam with. The prestress moments in a 2-span beam. The primary moment is −P·e, the tendon acting on its own section, and it reaches 540 kNm over the middle support. The secondary moment is what is left when the primary is taken off the total, and it is 306 kNm — 57% of the primary, with the same sign, so it does not cancel anything. It comes from the middle support refusing to let the beam lift: 51.0 kN pressing down there and 25.5 kN lifting at each end, a reaction set that sums to -2e-13 because nothing external was applied. Its diagram is straight between supports to 2.0e-13% of its own peak, which it has to be: reactions are point forces and a point force puts no curvature in a span.

    The prestress that pushes back

    On a simply supported beam a tendon is an internal matter and changes no reaction. Put the same beam on three supports and the tendon lifts it off the middle one, the support refuses, and the force it takes to hold the beam down is a reaction produced with no load applied at all.

    part 1 · internal-forces
  2. A reaction with no load, and the moment it bends the beam with. The prestress moments in a 2-span beam. The primary moment is −P·e, the tendon acting on its own section, and it reaches 1440 kNm over the middle support. The secondary moment is what is left when the primary is taken off the total, and it is 720 kNm — 50% of the primary, with the same sign, so it does not cancel anything. It comes from the middle support refusing to let the beam lift: 102.9 kN pressing down there and 51.4 kN lifting at each end, a reaction set that sums to 0e+0 because nothing external was applied. Its diagram is straight between supports to 3.6e-13% of its own peak, which it has to be: reactions are point forces and a point force puts no curvature in a span.

    The tendon that can be moved

    Lift a continuous beam's tendon at its interior support without changing its drape and nothing about the beam's total moment changes. The primary falls, the secondary rises by exactly as much, and the pressure line stays where it was — which turns a parasitic effect into a quantity a designer can place.

    part 2 · internal-forces
  3. The shear is read from the pressure line, not the tendon. The first 9.6 m of a two-span beam of 16 + 16 m, 400 × 900 mm, prestressed with 2,400 kN on a parabolic tendon and carrying 45 kN/m, with depths drawn from the top face (−450 mm) to the bottom (+450 mm): the tendon's centre (dashed) and the pressure line, where the resultant of the prestress actually acts once the supports have pushed back (solid). 0.8 m from the support the tendon falls 87.00 mm per metre, which is the slope the rule P·sin θ reads, worth 209 kN; the pressure line falls 76.39 mm per metre, worth 183 kN. The difference is the secondary shear, 26 kN, which the supports put back.

    The relief that belongs to the pressure line

    An inclined tendon carries part of a prestressed beam's shear, and the rule is to subtract its vertical component, P times the slope of the tendon. In a continuous beam the tendon also provokes reactions, and the reactions make a shear of their own — a constant in each span, too small to look at. It is not small where it matters. Beside the end support of a two-span beam it doubles the shear left for the links, and moving the tendon without changing the beam at all can make the tendon's slope report that there is none.

    part 3 · internal-forces

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