Structural Engineering & Statics
where the load goes
Series

Rib shortening — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A tenth of a per cent of the thrust is all of the moment. The thrust a two-hinged arch loses to its own axial shortening, against rise-to-span. The flexibility equation's denominator has two terms — ∫y²ds/EI for bending and ∫cos²θ ds/EA for shortening — and their ratio is about (15/8)(i/f)², the square of the radius of gyration over the RISE. The loss is a little more than that ratio, because the released rib also shortens under its own shear, and at the 10 per cent rise drawn it is 0.11 per cent of the thrust, which sounds like a rounding error and is not: a parabolic arch under a uniform load is funicular, so the rigid solution has NO crown moment at all, and the 0.11 per cent that the rib shortening removes from the thrust leaves 29 kNm behind. The correction that is a tenth of a per cent of the thrust is a hundred per cent of the bending. At a two per cent rise the loss is 2.6 per cent, because a shallow arch's thrust is enormous and its lever arm is not.

    The arch that gets shorter

    A parabolic arch under a uniform load is funicular, so the perfect solution gives it no bending at all. Then the rib shortens under its own thrust by a tenth of a per cent, and every kilonewton-metre of moment the arch will ever carry comes from that.

    part 1 · deflection
  2. The same rib, fixed, bends at both ends as well as the crown. The bending moment that axial shortening leaves in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m², EA 20,400,000 kN) under 60 kN/m, sagging upward, from a frame solve of the rib. Dashed: two-hinged, sagging throughout, 29.4 kN·m at the crown. Solid: fixed at the springings, 57.9 kN·m sagging at the crown and 112.4 kN·m hogging at each springing. The funicular load itself leaves no moment in either rib; all of this is the 0.11 and 0.63 per cent of the thrust the two ribs lose by getting shorter.

    The springings that make shortening worse

    Fixing an arch at its springings is the stiffer, cheaper and usual way to build one in concrete, and it makes the arch six times as sensitive to its own shortening. The thrust it loses acts at the elastic centre, two thirds of the way up, so the moment lands at the springings as well as the crown, twice as large and the other way round — at the section the fixed arch is designed at, not away from it.

    part 2 · deflection
  3. How stiff an abutment has to be to count as fixed. The shortening moments in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m as shares of the fixed rib's, against the rotational stiffness of each springing as a multiple of EI/L on a logarithmic scale: at the springings (solid, −112 kN·m when fixed) and the crown (dashed, 58 kN·m fixed, 29 two-hinged). The springing moment is half the fixed value at kθ = 11.3 EI/L and nine tenths at 102 EI/L; at EI/L itself it is 8 per cent. The crown moment never falls below the two-hinged rib's 51 per cent of the fixed value.

    The abutment that spreads before it turns

    A fixed arch is fixed only if its abutments hold still, and they have two ways not to. Turning releases the springing moment, but slowly: an abutment has to be a hundred times the rib's EI/L to hold nine tenths of it, and a footing on stiff clay under a slender concrete rib already is. Spreading does the opposite. The rib's own shortening is worth 13.6 mm of spread, so an abutment that gives 7 mm a side under the thrust doubles every secondary moment in the arch — and a footing on dense sand gives 8.

    part 3 · deflection
  4. The gaps that open after the arch is closed. For the 60 m fixed concrete rib rising 6 m under 60 kN/m, closed at 28 days with its abutments jacked apart to take up the day's gap: the gap between the rib and its springings that each movement opens after closure, in millimetres, against the years since, on a log scale — the rib's creep under its thrust, its shrinkage, and the ground consolidating under the thrust by 1.0 times its 8 mm immediate spread with 90% done at 10 years. After a hundred years they are 24.8, 16.2 and 8.0 mm, 48.9 in all, against the 13.6 mm the thrust shortened the rib on the day. The rib resists them while it creeps itself, so the gap it acts as if fixed against is 20.9 mm (fine dashes).

    The jacks the arch forgets

    Jack a fixed concrete arch's abutments apart before the crown is closed, by exactly the gap its thrust and its footings open on the day, and the springings start with no moment at all. A century later they carry 173 kN·m — half as much again as the 112 the jacks were set to remove. The thrust's own moment never relaxes; the jacks' correction does, as the rib creeps; and shrinkage and the ground's consolidation arrive after the jacks have gone. No single setting cancels movements that run on different clocks.

    part 4 · deflection
  5. Closing late beats jacking again on the same date. For the 60 m concrete rib rising 6 m under 60 kN/m, struck at 28 days, on ground that spreads 8 mm at once and as much again as it consolidates, 90 per cent in 10 years: the springing moment after a century, against a date in the arch's first years — left three-hinged and closed on that date (solid), or fixed at striking and jacked back to no moment on that date (dashed); dotted, fixed and jacked at striking only, −173. At three months, −124 against −140; at a year, −84 against −117; at ten years, −15 against −76. The difference is the memory of the years before: the jacked arch's early restraint goes on relaxing after the second jacking, and the open arch had none.

    The arch that waits with its hinges open

    A concrete arch fixed on the day it is struck has to restrain every millimetre its rib creeps and shrinks and its ground settles for the rest of its life, and jacking it again a year later cannot make it forget the year before. Struck as a three-hinged arch and left that way, it takes the fastest part of every movement for nothing, because a determinate arch has no moment to spend on a change of length. Closed at a year it ends the century with half the springing moment of the arch fixed at striking, and a third less than the arch jacked again on the same date. The price is a crown that has dropped 78 mm while it waited, and a rib that is no longer the shape of its own thrust.

    part 5 · deflection

All series