Structural Engineering & Statics
where the load goes
Concept

Series stiffness — where it appears

The stiffness of two or more flexibilities acting one after the other, obtained by adding the flexibilities rather than the stiffnesses. The softest element governs, so improving the stiffest buys almost nothing.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A joint is springs in series. The five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 25227.71 kN·m per radian.

A joint made of springs in series

A connection's stiffness is not a property of the connection. It is the series combination of the flexibilities of everything the force passes through — so the softest component decides, and stiffening any of the others changes almost nothing.

connections · Component method How much of a deflection belongs to the beam. The share of the total deflection that is the beam's own bending, against the stiffness of what it sits on. A 8 m beam on two supports under a uniform load: on rigid supports every millimetre is the beam's, and the share falls away as the supports soften until almost none of it is. The beam drawn beside this figure sits at 51% — so 49% of what it does is happening somewhere a beam calculation never looks. The two flexibilities are in series, which means the softer one governs and stiffening the other buys nothing.

The deflection that belongs to the support

A beam calculation answers a question about a beam sitting on things that do not move. Real ones sit on bearings, on other beams and on columns that shorten, and every one of those is a spring in series with the member — so a deflection is the sum of two things and only one of them is a property of the beam.

deflection · Support flexibility The bolt force is the larger of two lines. What an M20 bolt in a 25 mm tee flange actually carries, against the tension applied to the flange. Preloaded to 171 kN it starts there and climbs at Φ = 0.185 — the bolt's own stiffness over the bolt's plus the clamped plates', 857 against 3781 kN/mm — so 18 per cent of every kilonewton applied reaches it and the rest is unloading the contact. At 138 kN the contact runs out and the line joins the one an ordinary bolt has followed from the start, climbing at 2.12. The two lines meet, so the strength is the same either way; what differs is the slope, by a factor of 11.5. The flange's own mechanism is at 172 kN, comfortably past the crossing.

The bolt that was already stretched

Prying is a lever that needs the flange to lift before it can act, and a preloaded bolt does not let it. The bolt carries a fifth of every kilonewton applied until the plates part, and everything after that is the ordinary calculation — which is why preload changes the fatigue answer by a factor of a thousand and the strength answer by nothing at all.

connections · Prying 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.

dynamics · Rocking

Named alongside it

The objects these essays reach for when they reach for this one.

Joint stiffnessLever armPryingAxial shorteningBearingBolt tensionCapacity designClamping forceColumn webComponent methodConnectionDeflection

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