Structural Engineering & Statics
where the load goes
Concept

Ballast — where it appears

The bed of broken stone a railway track sits in, holding the sleepers in place by the interlock between its own particles. That interlock is lost if the deck beneath it accelerates hard enough to fluidise the stone, which is why a high-speed bridge is governed by an acceleration limit rather than by a stress.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 2 m down. The head on the underside of the base slab is 4.0 m, so the pressure there is 39.2 kN/m² over the whole plan — 23.5 MN of it, pushing upward. Nothing about the structure changes that number. What resists it is weight: 18.7 MN of concrete and whatever is built above, giving a factor of 0.80. The structure floats if the water reaches 2.82 m below the ground, and a base slab alone would have to be 1.64 m thick to hold it down.

A basement is a boat

Every load in this collection presses down and is resisted by strength. Hydrostatic uplift presses up, is resisted by weight, and does not care what is built on it — so the check contains no material property at all. It is a ratio of two weights, and one of them is water.

equilibrium · Uplift The worst speed is not the fastest one. Peak deck acceleration against train speed, for a 20 m span at 6.25 Hz under 10 axles 18 m apart. The spikes are not a numerical artefact and they are not about how heavy the axles are: a regularly spaced train is a forcing function with a frequency v/d, and where a multiple of it lands on the bridge's own frequency each coach arrives in step with the motion the last one left. The arithmetic is v = d·f₁/k, which puts peaks at 405, 203, 135, 101 km/h — all of them operating speeds. What fails first is the acceleration rather than any stress: ballast loses its interlock at about 3.5 m/s², and strength does not appear in the equation at all. Here the limit is first passed at 376 km/h.

The train that arrives in time with itself

A single load crossing a span is a mild problem. A train is not one load — its axles are regularly spaced, so the forcing has a frequency of its own, and where a multiple of it lands on the bridge's frequency each coach arrives exactly in step with the motion the last one left behind.

dynamics · Moving load resonance The toe that runs out of ground. A block 6.0 m square on ground that bears 300 kPa at most — 10800 kN over the whole base — pushed 8.0 m up, weighing 3000 kN, drawn at three stages of the push with the lean exaggerated eight times and the contact pressure under the base; the dashed line is the ground's bearing limit. At a push of 629 kN it touches the ground over 3.9 m of its 6.0 and has not yet brought the ground to its limit anywhere. At a push of 770 kN it touches the ground over 2.5 m of its 6.0 and has brought the ground to its limit under 0.8 m. At a push of 776 kN it touches the ground over 2.3 m of its 6.0 and has brought the ground to its limit under 1.1 m. The last is the tipping push: the heel has lifted, the toe is pressing the ground at its limit over a widening block, and the lever arm of the weight about that block is shrinking as the block grows.

The ballast that helps it over

On rigid ground a heavier block is harder to tip, in exact proportion to its weight — that is the whole of the overturning check. On ground with a bearing capacity it is not. The toe presses the ground to its limit, the weight's lever arm shrinks as the yielded block under the toe grows, and the tipping push peaks when the block weighs half of what the ground under it can bear. Past that, every tonne of ballast added to steady it brings it closer to going over.

equilibrium · Overturning Weakest the day it is finished. The factor of safety against tipping of a 15,000 kN tower on a 10 m square base over soft clay (undrained strength 35 kPa, gaining 0.22 of the effective stress the tower puts into it, consolidating with a 90 per cent time of 7.1 years), through twelve years, for construction times of three months, one year and four years; its wind push grows as it rises. Built in three months, it bottoms out at 1.14 on the day it is finished; built in one year, it bottoms out at 1.33 on the day it is finished; built in four years, it bottoms out at 1.60 on the day it is finished. After that every curve climbs toward 1.93, the factor once the clay has consolidated. Had the finished weight arrived in an instant the factor would be 0.92; on ground that could not fail it would be 3.00.

The ground that arrives after the weight

A block's resistance to tipping on ground that can yield is a parabola in its weight, with its peak at half of what the ground can bear. On soft clay the ground's capacity is not fixed: it grows as the clay drains under the weight, and it grows years after the weight has arrived. A tower on such ground is at its weakest on the day it is finished — 1.33 against the 1.93 it will have once the clay has drained — and a silo filled in a week walks over the top of the parabola and down the far side, which is the other way a structure can be built.

equilibrium · Overturning

Named alongside it

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

Factor of safetyOverturningBearing capacityEquilibriumFree bodyAcceleration limitBuoyancyConsolidationConstruction sequenceContact pressureControl perimeterDamping

All concepts