Structural Engineering & Statics
where the load goes
Series

Rocking — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 0.90 m wide and 4.20 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 12.1° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done.

    The block that is safer for being bigger

    A block resting on the ground lifts off at an acceleration that depends only on its shape and not at all on its size, and then falls over at one that depends strongly on its size. Two objects of identical proportion begin rocking at the same instant and only the smaller one topples — which is why the slender water towers stood in Chile and the squat tanks beside them did not.

    part 1 · dynamics
  2. One pulse, two blocks of the same shape. Rotation as a fraction of the toppling angle, under a single 0.8 s sine pulse of 1.00 g, for two blocks of identical proportion whose sizes differ by a factor of 3. Both lift off at the same instant, because uplift depends on the shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46%. The kinks are impacts: there is no dashpot anywhere in this model, and the only energy the block loses is lost when it lands on its other corner, at a velocity ratio of 0.926 per landing — 14% of the energy each time, decided by the block's shape and by nothing else.

    The only damping is the landing

    A rocking block has no dashpot in it. The only energy it ever loses is lost at the instant it lands on its other corner, and how much is a property of the block's proportions — 14 per cent for a slender one and 38 for a stocky one. That single number decides whether it settles or goes over, and a real base does not deliver the value the theory computes.

    part 2 · dynamics
  3. The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.

    A wall that is allowed to lift

    A block that rocks inherits everything that decides whether it survives — a restoring moment set by its weight and shape that falls as it leans, and a loss of energy set by its proportions at each landing. Put a tendon through a wall and yielding bars across its base and both become design quantities, and the ratio between them decides whether the wall comes home.

    part 3 · dynamics
  4. After a yield, the prestress left does not remember the prestress put in. The prestress left in the tendon of a 2.0 × 8.0 m rocking wall of 400 kN once it has rocked to each rotation and come back upright, for a tendon yielding at 1,370 kN and stiffening by 20 kN a millimetre as the base opens, prestressed to 300 kN, 600 kN and 900 kN. Each keeps all its prestress until it yields — at 53.5 mrad, 38.5 mrad and 23.5 mrad — and then loses 20 kN for every further milliradian, so past 53.5 mrad the three lines are one: the yield force less the stiffness times the stretch. Every one of them is slack at upright past 68.5 mrad. With bars of 400 kN the wall re-centres only while 400 kN remains, dashed, and it keeps that only up to 48.5 mrad, whatever it was prestressed to.

    The tendon that forgets its prestress

    A rocking wall comes home because a tendon pulls it, and the tendon must not yield — yet the rotations that test the wall are exactly the ones that stretch it. Past its yield, the force a tendon keeps is its yield force less its stiffness times the stretch, whatever it was prestressed to, so the guarantee a design needs is a limit on rotation, and more prestress only reaches that limit sooner.

    part 4 · dynamics
  5. A fuse caps the force the strand ever sees. The force in the tendon line of a 2.0 × 8.0 m rocking wall of 400 kN, prestressed to 600 kN, with bars of 400 kN, against the opening of the base under the tendon, loaded to 79 mm and back: the strand alone, 20 kN/mm yielding at 1,370 kN, and the strand with a fuse yielding at 1,000 kN with a stiffness of 500 kN/mm in series, together 19.2 kN/mm and capped at 1,000 kN. The strand alone yields at 39 mm and is slack at upright after any opening past 69 mm; the fused line yields at 21 mm and is slack past 52 mm. In the fused line the strand carries the line's force and never more than 1,000 kN, so it stays elastic and all the plastic stretch, 58 mm at this opening, is in the fuse.

    The fuse that protects the strand and not the wall

    A rocking wall's tendon loses its prestress when it yields, so give it a fuse: a short link in series that yields first and is replaced afterwards. The strand then never passes 1,000 kN and stays elastic. But the fuse caps the restoring moment lower, so under the same 1.5 g pulse the wall rocks to 110 mrad instead of 89, and the line loses all its prestress just as the strand did. A new fuse gives the prestress back; the bars' stretch stays, and the next pulse takes the wall to 134 mrad.

    part 5 · dynamics

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