Structural Engineering & Statics
where the load goes
Concept

Soil-structure interaction — where it appears

The two-way dependence between a structure's response and the ground beneath it, in which the foundation's motion is not the free-field motion. It lengthens the period and adds radiation damping, and it can move a structure into a more damaging part of the spectrum rather than out of one.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

The building does not care how far it went down; it cares how much it tilted. Five footings on soil that is 35% as stiff under one of them, carrying 60 kN/m. They settle between 12 and 54 mm, and the number that matters is neither of those: it is the angular distortion between neighbours, 4.32 per thousand, or one in 231 — against a limit of one in 500 for cracking in finishes, which this does not. A building that went down half a metre uniformly would be undamaged and would need a new front step; this one has moved a twentieth as far and has cracked.

The settlement that matters is the difference

A building that goes down half a metre uniformly is undamaged and needs a new front step. One that goes down a twentieth as far, unevenly, has cracked. The superstructure can even the difference out — and the only way it can do so is by carrying the difference itself, as a force.

deflection · Differential settlement Most of a long pile is doing nothing. Head deflection against embedded length, both measured in the pile's own characteristic length T = 14.67 m. A pile shorter than about two T is a lever with nothing holding its foot and deflects several times as much; past four T the curve is flat to within a per cent, because the ground below that depth is never asked for anything. A lateral check is a check on the top four T of a pile, however deep it goes for its axial load — and adding length to fix a lateral deflection is the one remedy that does not work.

The pile that is too short to bend

A laterally loaded pile is long or short in its own characteristic length, and the two are different structures. The slender pile bends and the ground below four characteristic lengths never hears about it; the monopile is two lengths deep, rotates about a point, and every tabulated coefficient written for the first case is wrong for the second.

internal-forces · Lateral pile Two separate checks, and the block that is neither. Amplification of the steady displacement at a machine 2.0 m above the base of a 150 tonne block 5.0 m across and 2.0 m deep, against forcing frequency. Checked separately, the block sways at 11.2 Hz with 29 per cent of critical damping and rocks at 15.0 Hz with 13 per cent, and the rocking check's peak is 3.99. Because its mass sits above its base the two are one system, with modes at 9.9 Hz and 20.9 Hz — one below both checks and one above — and the block's own peak is 2.49 at 9.7 Hz.

The frequency below both checks

A machine block is checked for sway and for rocking as if they were two oscillators, each with its own spring, its own radiation damping and its own natural frequency. A block whose mass sits above its base is one oscillator with two modes, and neither of them is a sway or a rocking — one sits below both checks, one above both, and the lower one is where the machine resonates.

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 The vertical response on layers of different depth. The vertical amplitude of a 150 tonne block 5.0 m across on soil with a shear wave speed of 178 m/s, under a harmonic force of fixed size, as a multiple of its static deflection on a half-space, against frequency. On a half-space the mode is at 12.5 Hz and the peak is 1.10. On 10.0 m the mode is at 14.3 Hz against a cut-off of 7.4 and the peak is 1.03. On 6.0 m the mode is at 15.5 Hz against a cut-off of 12.3 and the peak is 1.74. On 4.5 m the mode is at 16.3 Hz against a cut-off of 16.4 and the peak is 5.84. The layer stiffens the footing and lifts the frequency a little; what changes the picture is the cut-off, which rises as the layer thins and, once it passes the mode, takes the radiation damping away and leaves only the soil's own 5 per cent. Radiation below the cut-off is taken as nothing and its rise above it as linear, which is the usual idealisation of the exact layered solution.

The rock that sends the waves back

A machine block on the ground is damped by the waves it launches, and a half-space lets them all escape. Put rock four and a half metres down and none of them can: below the soil layer's own lowest frequency there is no wave that travels, and a vertical mode damped at 47 per cent of critical keeps the soil's own 5. Its resonance grows fivefold, and the heavy block the textbook rule recommends is the one that loses its damping on the deepest ground.

dynamics · Vibration isolation How stiff an abutment has to be to count as fixed. The shortening moments in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m as shares of the fixed rib's, against the rotational stiffness of each springing as a multiple of EI/L on a logarithmic scale: at the springings (solid, −112 kN·m when fixed) and the crown (dashed, 58 kN·m fixed, 29 two-hinged). The springing moment is half the fixed value at kθ = 11.3 EI/L and nine tenths at 102 EI/L; at EI/L itself it is 8 per cent. The crown moment never falls below the two-hinged rib's 51 per cent of the fixed value.

The abutment that spreads before it turns

A fixed arch is fixed only if its abutments hold still, and they have two ways not to. Turning releases the springing moment, but slowly: an abutment has to be a hundred times the rib's EI/L to hold nine tenths of it, and a footing on stiff clay under a slender concrete rib already is. Spreading does the opposite. The rib's own shortening is worth 13.6 mm of spread, so an abutment that gives 7 mm a side under the thrust doubles every secondary moment in the arch — and a footing on dense sand gives 8.

deflection · Rib shortening 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 Pressed and pulled, the footing leans once and stops. The column's tilt at rest after each of eight storms reversing fully to 300 kN·m each way, on the 8 m strip of EI 540 MN·m², weighing 320 kN, on a bed of 50 MN/m² under a column carrying 1,000 kN, on ground that unloads and reloads 4 times as stiffly as it first loads, for loads swinging by 0, 50, 100, 150, 200 per cent. 0 per cent: −0.007 milliradians after the first storm and 0.003 after the eighth; 50 per cent: 0.112 milliradians after the first storm and 0.103 after the eighth; 100 per cent: 0.109 milliradians after the first storm and 0.091 after the eighth; 150 per cent: 0.113 milliradians after the first storm and 0.095 after the eighth; 200 per cent: 0.114 milliradians after the first storm and 0.102 after the eighth. The lean is towards the end the heavier half of each storm presses; it stops within three storms whether or not the strip ever hangs clear.

The footing that hangs from its column

A footing under a bracing bay's column is pressed hard when the wind pushes one way and pulled when it swings back, and if the pull is larger than the footing's weight it hangs from the column, clear of the ground, for part of every storm. It looks like a footing free to come down somewhere new each time, and walk. It does not walk. The ground under it keeps the memory of every press, and a footing that is not touching the ground cannot change that memory; it comes back down into its own bowl, leans once towards its harder press, and stops within three storms.

internal-forces · Elastic foundation

Named alongside it

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

RockingSubgrade modulusElastic foundationContact pressureNatural frequencyRadiation dampingResonanceSettlementShakedownSoil stiffnessDampingMachine foundation

All concepts