Structural Engineering & Statics
where the load goes
Concept

Strain rate — where it appears

How quickly a strain arrives, which no strength quoted from a testing machine records. Steel is about twice as strong at blast rates as at test rates and only eight per cent stronger at earthquake rates, so the correction is either decisive or negligible with very little in between.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

How fast the strain arrived, which a quoted strength does not record. The dynamic increase factor on strength against strain rate, over eight decades. Steel follows Cowper and Symonds' fit, whose constant D = 40.4 s⁻¹ is not an arbitrary parameter — it is the rate at which the material is exactly twice as strong. Concrete in tension follows the model code's two-branch curve and is steeper. The four marked regimes are the argument: a testing machine works at about 10⁻⁴ per second, an earthquake at 5 × 10⁻³, a vehicle impact at a half, a blast at a hundred, and the enhancement across them runs 1.00, 1.08, 1.32, 2.04. So this is a correction that is either negligible or decisive with very little in between, which is why no seismic code carries it and every blast code does. What does not rise is the modulus, which is a lattice property, and the ultimate strength rises only a third as much — so the ultimate-to-yield ratio closes from 1.56 to 1.23 and the material has less warning left in it than it started with.

The steel that is stronger in a millisecond

Every strength quoted anywhere in this collection was measured at about a ten-thousandth of a strain per second, because that is what a testing machine does, and nothing on a drawing says so. Load the same steel a million times faster and its yield stress rises by a third.

materials · Strain rate What the shape of a load in time is worth, for two load shapes. The peak displacement as a multiple of the static deflection, against the load's duration divided by the structure's natural period, for two load shapes: a load that rises linearly, then stays; a rectangular pulse, then nothing. The lines are closed forms and eight dots are the peak of a complete time integration of an oscillator of 0.300 s period under that load, agreeing with the line to within 0.19% everywhere.

The load that is over before it has moved

A blast delivers an enormous pressure for a few milliseconds. Everything else in this field asks what force a structure can carry; a load that has come and gone before the structure has travelled any distance is not asking that question, and the answer turns out to depend on the mass and the ductility with the strength barely in it.

dynamics · Blast The same steel, brittle in January. Fracture toughness against temperature for a 25 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = -60 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 200 N/mm², which goes as the square of it. At +20 °C this plate carries a 764 mm flaw and at -20 °C it carries 199 mm — a factor of 3.8 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 27% of the flaw a 10 mm plate does. Loading it in a millisecond shifts the whole curve another forty degrees.

The same steel, brittle in January

Every other material property in this collection is a number. Toughness is a curve, and the axis it runs along is temperature. Between twenty degrees and minus twenty a structural steel does not get gradually weaker — it changes the mechanism by which it fails, and the flaw it will tolerate falls by a factor of four.

materials · Transition temperature A filler that knows the speed meets the collision hardest at its start. The contact force against the depth crushed in the hardest collision, 500 t and 300 t closing at 2.85 m/s with 761 kJ, on a 40 mm filler between 500 t and 300 t, crushable over 24 mm: uniform (dashed), with 50 per cent of its strength at that speed depending on the speed, and with all of it. Each is as strong as it must be to stop the collision in 24 mm. Uniform: 30.5 MN at first contact; 50 per cent on speed: 38.4 MN at first contact; all on speed: 63.4 MN at first contact. The crushing speed is highest at the first instant and falls to nothing, so a filler whose strength follows it spends its force at the start; the same area under a falling line needs a higher peak than under a level one, and a purely viscous filler's is twice the uniform's.

The filler that knows how fast it is hit

A filler graded through its depth meets the small collisions softly and the hard one with a strong layer that has to be very strong, because every kilojoule the soft layer misses must be absorbed in less depth. A filler whose strength rises with the speed it is crushed at ranks collisions by speed instead, with no layers at all, and treats the middling collisions better than grading does. But it meets every collision hardest at its first instant, when the crushing speed is greatest, and a force that falls as the collision proceeds needs a higher peak to absorb the same energy in the same depth. A purely viscous filler that stops the hardest collision in 24 mm hits it with exactly twice the force of a uniform one.

dynamics · Pounding Where each filler's depth goes. The crushable depth left in each filler between buildings of 500 t and 300 t, periods 0.8 and 1.2 s, 50 mm apart, through a record of six one-second sine cycles peaking at 0.35 g, for 40 mm fillers crushable over 24 mm, each just strong enough to stop a 2.85 m/s collision from new. Uniform: 5 mm when its hardest contact arrives at 2.76 m/s, 0 mm at the end. Graded: 12 mm when its hardest contact arrives at 2.76 m/s, 6 mm at the end. 50% on speed: 0 mm when its hardest contact arrives at 2.88 m/s, 0 mm at the end. The graded filler's soft layer goes in the first middling contacts and its strong layer is barely reached; the other two lose depth to every middling contact and have little or none left when the largest arrives.

The filler that is used up before the worst

A crushable filler between two buildings is sized for the hardest collision it might meet, meeting it new. An earthquake does not oblige: the hardest contact arrives after a dozen middling ones, and each has taken some of the depth. Through a record of six cycles, a filler half of whose strength follows its crushing speed is dense before the worst contact arrives, and a uniform one has five millimetres left. A graded filler still has twelve. Yet the graded filler meets the worst contact at 56 MN and the spent rate-sensitive one at 39. Keeping depth is not the same as being gentle, and the order of the contacts barely matters.

dynamics · Pounding

Named alongside it

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

ImpactImpulseBlastDuctilityEnergy dissipationFatiguePlastic hingePoundingSeismic gapToughnessBound theoremsBrittle failure

All concepts