Structural Engineering & Statics
where the load goes
Concept

Soil stiffness — where it appears

The force per unit displacement that the ground under or around a foundation offers, for each direction of movement and rotation. With the mass it carries it sets a machine foundation's natural frequencies, and the same ground radiates waves away, so stiffness and radiation damping come from one body of soil.

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

Damping that is computed rather than measured. Radiation damping of a 2.5 m block on soil with a shear wave speed of 178 m/s, against how heavy the block is made — the horizontal axis is a multiple of the 150 tonne block drawn. At that mass the three modes are at 47, 29 and 13 per cent of critical. None of this is dissipation: the energy leaves as waves travelling away into the half-space, so the quantity is a geometrical coupling and it can be computed from the size of the footing, the density of the soil and its shear modulus. Every curve falls as the block gets heavier, because a heavier block moves less for the same wave field — which is the one counter-intuitive thing here: mass buys frequency and costs damping.

The damping that is radiated

Every response in this field is quoted with a damping assumption attached, because damping is measured rather than designed and the measurement has a factor of two in it. A machine block on the ground is the exception: its damping is not dissipation at all, and it can be computed from three numbers none of which is a material property of anything that dissipates.

dynamics · Vibration isolation Set into the ground, the block is quieter only while its sides radiate. Displacement at the top per kilonewton of machine force for a 150 tonne block 3.2 m across and 3.2 m deep, on a logarithmic scale: on the surface, dashed, and with 1.60 m of it set into the ground, first with the side soil radiating as the side-layer model gives it and then with no radiation from the sides at all, dashed. On the surface it peaks at 191.8 µm per kN at 5.5 Hz. Embedded and radiating, it peaks at 11.0 µm per kN at 8.2 Hz. With the same embedment and silent sides the resonance moves up to 8.4 Hz and peaks at 168.7 µm per kN, because the lower mode's damping falls from 3.2 per cent to 1.6 per cent. At 5.5 Hz the three give 191.1, 8.0 and 9.5 µm per kN.

The damping that comes through the sides

A machine block set into the ground is held at its sides as well as its base, and the side soil does two things at once. It lifts the height at which the ground's resistance acts toward the block's centre of mass, which weakens the coupling between sway and rocking without ever removing it. And it radiates, which is what actually flattens the resonance — so the quiet an embedded block promises rests on the stiffness of backfill nobody measured.

dynamics · Vibration isolation Ground already loaded rocks stiffer than springs that are not. The moment on the column against its rotation through a storm that takes the moment to ±1,000 kN·m and back, for an 8 m strip of EI 540 MN·m² on a bed of 50 MN/m² under a 1,000 kN column: on elastic springs that cannot pull, and on ground that unloads and reloads 4 times as stiffly as it first loads, in the first storm and the third. On elastic springs the column turns 1.418 milliradians at the peak, a rocking stiffness of 0.70 GN·m/rad, the same every storm. On the remembering ground it turns 1.116 in the first storm and 0.868 in the third — 0.90 and 1.15 GN·m/rad — because rocking unloads half the strip, and ground that has carried the column unloads stiffly.

The footing the storms press down

A footing strip under a column that a storm rocks one way and then the other was expected to dig itself a bowl: its ends loaded half the time, compacted, and left hanging over the hollow. On ground that unloads and reloads four times as stiffly as it first loads, the bowl appears and the strip fills it. The column settles 26% more over three storms and then stops; the footing rocks 22% less than it did in its first storm and is 64% stiffer than on springs that forget, because rocking unloads half the strip and ground that has carried the column unloads stiffly. The storms change the settlement and the stiffness, and leave the moments almost where they were.

internal-forces · Elastic foundation A prevailing wind leans the footing, and the lean stops. The 8 m strip of EI 540 MN·m² on a bed of 50 MN/m² under a 1,000 kN column on ground that unloads and reloads 4 times as stiffly as it first loads, rocked by storms that push to 1,000 kN·m one way and swing back all, three quarters, half, a quarter or none of that the other: the column's tilt at rest after each of eight storms, in milliradians, positive the way the storms push. After eight: 0.011, 0.064, 0.117, 0.171, 0.218. Each settles within three storms; the lean is set by how unequal the storms are, not by how many there have been.

The footing a prevailing wind leans

A footing rocked by storms that reverse evenly settles into a symmetric bowl and stops. A footing under a frame in a prevailing wind is pushed harder one way than the other, and its ground keeps the difference. It leans — and then it stops leaning, within three storms, at a tilt set by how unequal the storms are and how large the largest has been. Fifty years of storms lean it further only in the years when a storm beats every one before it, and between those years nothing that happens to the footing moves it at all.

internal-forces · Elastic foundation

Named alongside it

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

RockingSoil-structure interactionContact pressureDampingElastic foundationMachine foundationNatural frequencyRadiation dampingResonanceSettlementShakedownSubgrade modulus

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