Overturning — the series · Structural Engineering & Statics

Weight is the only thing holding it down. A body 2.5 m wide and 6 m tall weighing 120 kN, under a wind pressure of 1 kN/m². The wind delivers 48 kN and an overturning moment of 144 kNm about the leeward toe; the weight restores 150 kNm, a factor of 1.04. The resultant lands 1.20 m from the centre against a middle third of ±0.42 m, so the base is lifting over 2.35 m of its width.

Weight is the only thing resisting it

A structure that is strong enough everywhere can still be blown over, and nothing in its material properties has any part in whether it is. The whole answer is a weight and a width — and the failure begins long before anything tips, at the moment one edge stops pressing down.

part 1 · equilibrium

Weight is the only thing holding it down. A body 1.6 m wide and 4.5 m tall weighing 22 kN, under a wind pressure of 1 kN/m². The wind delivers 2 kN and an overturning moment of 4 kNm about the leeward toe; the weight restores 18 kNm, a factor of 4.35. The resultant lands 0.18 m from the centre against a middle third of ±0.27 m, so the base is still wholly in bearing.

Whether it tips or slides

A free body pushed sideways has two ways of leaving, and which one it takes is decided before any load is known. The condition is a width divided by a height set against a coefficient of friction, and the weight, the wind pressure and the depth of the body all cancel out of it.

part 2 · equilibrium

The resultant has left the base, and it tips. A body on three supports weighing 60 kN, pushed sideways by 16 kN at a height of 3 m in the plan direction 270°. The push moves the resultant of weight and push 0.80 m from under the weight, and the base — the convex hull of the supports, shaded — lets it go 0.75 m that way before the edge drawn heavy becomes a tipping line: a factor of 0.94, found both along the ray and by moments about that edge. The dashed rosette is the same reach in every direction, from 0.75 m toward the middle of the nearest edge to 1.50 m toward the furthest support. To hold it the support opposite the tipping edge would have to pull 1.3 kN, which a support standing on the ground cannot do, so it lifts and the body turns about that edge.

Half as far between the legs

A body standing on feet, legs or pads has for its base the polygon its supports enclose, and how far its weight can be pushed before it tips depends on which way it is pushed. A three-legged stand pushed toward the gap between two legs has exactly half the reach it has pushed toward one of them.

part 3 · equilibrium

The push that tips it, in every direction. The tipping push in each plan direction, drawn as a distance from the centre, for a 6000 kN body with its weight 15 m up and the push at 18 m. On rigid ground the curve is the hull's: weakest along the axes at 1333 kN. On four equal pads it shrinks, and more toward the corners. With the pads as built — stiffnesses 20000, 20000, 8000, 20000 kN/m — it is no longer symmetric: the weakest direction is 15°, on the soft pad's side, at 1039 kN.

It tips inside its own hull

On rigid ground a body tips when its resultant reaches the edge of its base, and how stiff its supports are has nothing to do with it. On pads that settle, the body leans as it is pushed, the lean moves its weight, and the push that tips it falls by one number a site engineer already has: settlement times the height of the weight, over the square of the half-width. Toward a corner the loss doubles, and one soft pad makes the weakest direction one nobody checks.

part 4 · equilibrium

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.

part 5 · equilibrium

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.

part 6 · equilibrium

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.

part 7 · equilibrium