Structural Engineering & Statics
where the load goes
Concept

Contact pressure — where it appears

The distribution of normal stress across an interface where two bodies bear on each other, which is a result of their relative stiffness rather than an applied load. It is rarely uniform: a rigid foundation on soft ground concentrates it at the edges and a flexible one under the load.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The bearing that is drawn as a roller. The horizontal force a sliding bearing delivers, against the vertical load it is carrying, with its coefficient of friction on the same picture. The coefficient is not a constant: PTFE's falls as the contact pressure rises, and the standard fit is μ = 1.2/(10 + σ), so the bearing drawn is at 30.0 N/mm² and μ = 0.030 while the same bearing at a fifth of the load is at 0.075 — 2.5 times as much. The force curve is therefore strongly non-linear: a fifth of the load gives 50% of the force. Two readings follow and only one of them is usually taken. The largest force is at full load, 108 kN, and that is what the pier is designed for. The largest nuisance is at light load, where 54 kN of friction is 39% of the 140 kN of wind the bearing was put there to release the structure from. Cold makes it worse again: below about −5 °C the same bearing delivers 216 kN. A roller symbol on a drawing means this, and it is a pair of load cases rather than one, because friction opposes whichever way the deck happens to be going.

The roller that is not a roller

A sliding bearing is drawn as a roller and detailed as a sheet of PTFE, and it delivers a horizontal force of a few per cent of whatever it is carrying. The coefficient everybody quotes is the one at full design pressure, and PTFE's coefficient rises as the pressure falls — so the bearing is at its freest exactly where nobody checks it.

connections · Bearing friction The reaction lies inside the cone, so the block stands. A block of 48 on a plane at 22°, against a coefficient of friction of 0.6. Resolving across and along the plane gives a normal force of 44.5 and a friction demand of 18.0, against a capacity of μN = 26.7 — a ratio of 0.67. Added together the two make one contact reaction leaning 22.0° from the normal, and the admissible reactions fill a cone of half-angle arctan μ = 31.0°. Equilibrium is possible exactly when the demanded reaction lies inside that cone, which here it does. The weight enters neither the cone nor the lean: a block of any weight on this slope leans its reaction by the same 22.0°, which is why the angle of repose is a material property and the size of a heap of sand is not.

The area that is not in the equation

Friction is proportional to the force pressing two surfaces together and independent of how large they are, which sounds like an approximation and is not. The area is absent because the contact that carries the load is a tiny fraction of the contact that is drawn, and that fraction grows in exact proportion to the load.

equilibrium · Friction 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 Three grounds, three tipping pushes. The push against the lean it produces, in thousandths of a radian, for the 6 m square, 3,000 kN block on ground that bears 300 kPa at most, half of it mobilised at 7.5 mm of settlement, pushed 8 m up, its weight 5.0 m up, on the three grounds matched at their limit and at half of it; dashed, the rigid-plastic limit, 813 kN. Linear, capped: 776 kN at a lean of 13.2 thousandths; hyperbolic: 701 kN at a lean of 26.2 thousandths; S-shaped: 770 kN at a lean of 15.2 thousandths. The S-shaped ground behaves almost exactly like the capped one; the hyperbola, which never quite reaches its limit, tips 10 per cent sooner at twice the lean.

The ground that never quite gives way

On a bed of springs capped at the ground's bearing limit, a block pushed sideways tips at a definite push, below the limit the rigid-plastic parabola gives. Real ground is curved: it softens all the way to its limit and never quite reaches it. On such ground the block has no tipping push of its own. Its resistance creeps toward the limit, and the peak it does have is made by its own weight leaning with it — 14 per cent below the limit with the weight 5 m up, nearly at it with the weight at the base, where the lean simply runs away.

equilibrium · Overturning The linear answer has the ground holding the ends down. The contact pressure along a strip 16 m long, EI 5.40 × 10⁵ kN·m², on a bed of 50 × 10³ kN/m² per metre of its length, under a column of 1,000 kN. With springs that pull as readily as they push (dashed) the bed presses 197 kN/m under the column and pulls down on the ends at up to 34 kN/m — tension the ground cannot supply. With the pulling springs released (solid), the strip touches the ground only from 3.97 m to 12.03 m, 8.05 m in all, and presses 213 kN/m under the column, 8 per cent more; beyond that it carries nothing.

The ends the ground was holding down

Winkler's springs pull as readily as they push, and on a long footing strip the linear answer quietly uses that: it has the ground holding the strip's ends down. Take the pull away and the correction under the column is modest — eight or nine per cent on the pressure and the moment — but the shape of the answer changes completely. A weightless strip longer than about eight metres keeps exactly eight metres of itself on the ground, whatever length was poured, and the rest rises off as straight cantilevers carrying nothing. The middle-third rule turns out to be a property of rigid footings: an eight-metre strip lifts at a fiftieth of the eccentricity the kern allows.

internal-forces · Elastic foundation 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 foundationFactor of safetyOverturningSettlementShakedownSoil-structure interactionBearing capacityCoefficient of frictionFree bodyFriction

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