Structural Engineering & Statics
where the load goes
Equilibrium

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.

Assumes The load that depends on what carries it, Weight is the only thing resisting it and The free body is a choice, and choosing it well is the whole skill.

18 min read 8 figures The load must go somewhereThe statics of things that do not moveWhich failure arrives first

Every load so far in this collection presses down or sideways, arrives from something built or from the weather, and is resisted by a strength. Hydrostatic uplift is none of those things. It presses up, it arrives from a water table nobody controls, it is resisted by weight, and the check that governs it contains no material property of any kind.

A substructure below the water table displaces water, and the water pushes back. That is Archimedes rather than statics, and the arithmetic is the same either way: the pressure on the underside of the base slab is the depth of water above that slab times the unit weight of water, applied over the whole plan area.

A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 0 m down. The head on the underside of the base slab is 6.0 m, so the pressure there is 58.9 kN/m² over the whole plan — 35.3 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 0.53. The structure floats if the water reaches 2.82 m below the ground, and a base slab alone would have to be 2.45 m thick to hold it down.
Fig. 1 A 20 × 30 m substructure dug 6 m into ground whose water table stands at the surface. The head on the underside of the base slab is the full 6 m, so the pressure there is 58.9 kN/m² over 600 m² — 35.3 MN pushing upward, against 18.7 MN of concrete. The factor against flotation is 0.53, and the structure floats.

The check with no strength in it

Write it out:

factor=weight+friction on the walls+tie-downγw h A\text{factor} = \frac{\text{weight} + \text{friction on the walls} + \text{tie-down}}{\gamma_w\,h\,A}

There is no fyf_y, no fckf_{ck}, no EE, no section modulus and no member anywhere in that expression. It is a weight divided by a weight of water, and the only design variables available are:

  • add weight — thicken the slab, add ballast, count the building above;
  • add friction — mobilise shear on the outside of the retaining walls;
  • tie it down — anchors or tension piles into the ground beneath;
  • or stop the water arriving — drainage, which changes hh rather than resisting it.

That is the whole list. Nothing about making the structure stronger appears on it.

A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 0 m down. The head on the underside of the base slab is 6.0 m, so the pressure there is 58.9 kN/m² over the whole plan — 35.3 MN of it, pushing upward. Nothing about the structure changes that number. What resists it is weight: 41.0 MN of concrete and whatever is built above, giving a factor of 1.16. The structure floats if the water reaches -0.97 m below the ground, and a base slab alone would have to be 2.45 m thick to hold it down.
Fig. 2 The first item on the list, taken to the point where it works on its own. The same substructure with its base slab at 2.45 m instead of 0.9 — the thickness at which 600 m² of concrete weighs 35.3 MN and matches the uplift exactly. With the walls counted as well the weight is 41.0 MN and the factor 1.16, and the basement has lost more than a metre and a half of the depth it was excavated to obtain. The thickness bought that as weight and not as strength: no term in the check knows what grade the concrete is.

Every remedy on the list is bought at that kind of price, and the fourth is the only one that changes the numerator instead of the denominator. It is worth seeing what the denominator does on its own before pricing anything.

The only load case that gets better as the water goes away. Factor against flotation for a 20 by 30 m substructure 6 m deep, against how far down the water table stands. With the water at the surface the head under the slab is the full 6 m and the factor is 0.53; it reaches one at about 2.82 m and rises steeply after that, because the head falls linearly while the weight resisting it does not move at all. The curve has no material property anywhere in it: it is a weight divided by a weight of water, and the only design variables are ballast, tie-down and drainage.
Fig. 3 Factor against flotation against the depth of the water table below ground. With the water at the surface it is 0.53; it reaches one at 2.82 m and rises steeply after that, because the head falls linearly while the weight resisting it does not move at all.

The curve has a shape worth noting: it is a reciprocal, not a straight line, because the resistance is fixed and the action is falling. So a small change in the assumed water table near the critical depth changes the factor a great deal, and a large change well above it changes almost nothing.

Which free body produced the number

The free body is the whole substructure, cut on a plane just beneath its base slab and on planes just outside its walls.

Across the bottom face acts the water pressure, γwh\gamma_w h upward, uniform over the whole plan. It is uniform because the base is horizontal and hydrostatic pressure depends only on depth.

Across the outside of the walls acts water pressure horizontally, which balances itself, and shear on the soil interface if the wall is rough and the soil is willing to supply it.

And downward acts the weight of everything inside the cut: the slab, the walls, the floors, the building above.

That uniform pressure over a flat base has its resultant at the centroid of the plan, which is why flotation has no lever arm in it and why the check is a force balance rather than a moment one. It is the only stability check on this site that never takes a moment about anything.

There is nothing else, and in particular there is no reaction from the ground. The structure is not resting on anything — that is precisely the condition being investigated. A free body with no bearing reaction is unusual in this subject and it is the reason the check feels unfamiliar.

Choosing the free body is the whole of the method, and here the choice is unusually stark. The cut has to go under the base slab, because a cut anywhere higher leaves the uplift on the wrong side of it and produces a structure standing comfortably on ground that is trying to lift it — a free body that passes a check the real substructure fails, drawn correctly and taken in the wrong place.

The load that does not scale with the building

A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 3 m down. The head on the underside of the base slab is 3.0 m, so the pressure there is 29.4 kN/m² over the whole plan — 17.7 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 1.06. The structure floats if the water reaches 2.82 m below the ground, and a base slab alone would have to be 1.23 m thick to hold it down.
Fig. 4 The same substructure with the water table at 3 m. The head has halved, the pressure with it, and the factor is 1.06 — the structure is stable by a margin that depends entirely on a number measured in a borehole some years earlier.

Uplift is the one action in this collection with the following property: it does not grow when the building does.

A taller building has more wind, more seismic mass, more gravity load, more foundation, more of everything. Its uplift is unchanged, because uplift depends on the plan area of the substructure and the depth of the water — neither of which the storeys above alter. So the resistance grows and the action does not.

storeys above weight factor
0 18.7 MN 0.53
1 23.5 0.67
2 28.3 0.80
3 33.1 0.94
4 37.9 1.07
A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 0 m down. The head on the underside of the base slab is 6.0 m, so the pressure there is 58.9 kN/m² over the whole plan — 35.3 MN of it, pushing upward. Nothing about the structure changes that number. What resists it is weight: 37.9 MN of concrete and whatever is built above, giving a factor of 1.07. The structure floats if the water reaches -0.44 m below the ground, and a base slab alone would have to be 2.45 m thick to hold it down.
Fig. 5 The last row of that table, drawn. The same substructure with four storeys standing on it: the uplift is the 35.3 MN it always was, because nothing built above ground alters the plan area or the head, while the weight has gone from 18.7 MN to 37.9 and the factor with it from 0.53 to 1.07. The water table that would float it has moved from 2.82 m down to −0.44 m — above the ground surface, which is to say nowhere.

Three and a half storeys of ordinary building are enough to hold this substructure down. Which means the governing case is not the finished building at all — it is the substructure standing empty, which is a real condition that exists for months during construction, and it is the case the design has to satisfy.

That is the same structure of argument as the most dangerous day and it arrives from the opposite direction: there, the capacity was missing; here, the load that provides the capacity is.

Ballast, friction, and tying it down

Ballast is the direct answer and it is expensive in the currency that matters underground: to take this substructure from 0.53 to a factor of 1.10, an extra 1.40 m of concrete is needed over the whole plan. That is 840 m³ of concrete occupying 1.4 m of a basement that was excavated in order to have space in it.

Wall friction is the cheap answer and the one hardest to justify. The outside of a 6 m deep retaining wall has 600 m² of contact with the soil, and mobilising even 20 kN/m² of shear on it is worth 12.0 MN — which takes the factor from 0.53 to 0.87 on its own.

Wall friction is a force that is whatever it needs to be, up to a limit, and every property of that class applies to it here. It is not a force that exists until the structure tries to rise. It develops only as the structure moves. And the movement it needs in order to reach the value quoted for it is more than a basement can tolerate.

That last point is why wall friction is usually taken conservatively or ignored. It needs relative movement to develop, the movement it needs is millimetres, and a basement that has risen by millimetres has already broken its own waterproofing.

Tension piles or ground anchors are the reliable answer. They convert a weight problem into a strength problem, which is the one thing everything else on this page cannot do — and they reintroduce a material property to a check that had none.

Five loads behind one wall, and the water is the biggest. The horizontal pressure on a 6 m wall retaining soil at 18 kN/m³ with a friction angle of 30°, a surcharge of 10 kPa and the water table 0 m down, drawn once as the profile the wall feels and then once per term. The terms are surcharge 20.0 kN/m at 3.00 m, soil at the water table 0.0 kN/m at 3.00 m, submerged soil 61.1 kN/m at 2.00 m, water 176.6 kN/m at 2.00 m, and they sum to 257.7 kN/m — matched to 6e-13 by integrating the drawn profile numerically. The largest single term is the water, at 176.6 kN/m: water has no shear strength, so its coefficient is exactly one where the soil's is 0.333, and it acts on top of the soil's effective stress rather than instead of it. The combined resultant sits at 2.08 m above the base, 0.346 of the height rather than the third point at 2.00 m that a pure triangle would give.
Fig. 6 The load that depends on what carries it: the same wall, from the side, with the water separated from the soil. The water pressure on the wall is the largest single component and it is the same water that is pushing up under the slab — one water table producing a horizontal load, a vertical load and a buoyant reduction of the soil’s own weight, all at once.

Drainage is a structural decision

The fourth option changes hh rather than resisting γwhA\gamma_w h A, and it is the only one that removes the load rather than balancing it.

An underdrain beneath the slab, connected to a pumped sump, holds the water table at the level of the drain. That reduces the head to nearly zero and the uplift with it, and it turns a structural problem into a mechanical one — with the whole of the structure’s stability now depending on a pump.

Which is why it is usually refused. The design life of a basement is a century; the design life of a pump is fifteen years; and the failure mode of the arrangement is that the pump stops and the structure floats. It is the load that will not hold still in a form no analysis can see: a load whose magnitude depends on a maintenance regime. Where drainage is used, it is used with a relief valve — a one-way valve in the slab that admits water and floods the basement rather than letting it lift, which is a decision to accept a flooded basement in preference to a broken one — an alternative load path whose alternative is not structural at all.

The water behind a wall pushes harder than the soil does. Retained soil drawn as the fluid it is equivalent to — the density a real liquid would need in order to push on the wall as hard, which is Ka·γ — against the soil's friction angle, for unit weights of 18 and 20 kN/m³. Water is the horizontal line at 9.81, and the curves cross it at 17.1° and 20.0°: above those angles the soil is the lighter load and below them it is the heavier one. At the 30° the other views are drawn at, a soil of 18 kN/m³ pushes like a fluid of 6.00 kN/m³ — 0.61 of what the water in it pushes with. So a wall holding saturated ground is carrying more water than soil, the water term does not care about the friction angle at all, and a blocked drain is the commonest way a retaining wall is lost.
Fig. 7 The same decision seen on the wall rather than under the slab. Retained soil is drawn as the fluid that would push on the wall as hard — KaγK_a\gamma — against the soil’s friction angle, with water as the horizontal line at 9.81 kN/m³. At the 30° used everywhere on this page, soil of 18 kN/m³ pushes like a fluid of 6.00, which is 0.61 of what the water standing in it pushes with; the curves cross the water line only at 17.1° and 20.0°, so for any soil worth retaining the water is the larger of the two loads. The drain that removes the head under the slab is holding the wall as well.

The general form is worth keeping. Uplift is one of very few structural actions that can be designed away rather than designed for, and every scheme that does so trades a structural certainty for a mechanical one.

Ballast placed below the water counts at its submerged weight

The first item on the list of remedies — thicken the slab — hides an arithmetic trap that reverses a good deal of its value, and it is the same halving the last section of this essay warns about for soil, arriving where nobody expects it.

Thickening a base slab can be done in two directions, and they are not worth the same. Add the concrete upward, into the basement, and the formation level does not move: the head stays at 6 m, the uplift stays at 35.3 MN, and every cubic metre of concrete is worth its full 24 kN/m³. Add it downward, by digging deeper, and the underside of the slab goes with it — so the head grows by exactly the thickness added, and the uplift grows with it.

On the 600 m² plan here, one extra metre of slab is 14.4 MN of concrete either way. Taken downward it also adds 9.81×1×600=5.899.81 \times 1 \times 600 = 5.89 MN of uplift, so the net gain is 8.51 MN rather than 14.4. Ballast placed by deepening the excavation is worth 59% of the same ballast placed by raising the floor, and the ratio is nothing but (24−9.81)/24(24 - 9.81)/24 — the submerged unit weight of concrete divided by its bulk weight.

Read the other way, that is the general rule and it applies to every remedy on the list. Any material added below the water table resists uplift at its submerged weight, because it displaces water in doing so. Concrete at 24 kN/m³ counts at 14.2; a mass-fill of sand at 19 counts at 9.2, which is less than half; and steel kentledge at 77 counts at 67, which is why the only efficient ballast is a dense one. It is also why the 1.40 m of extra concrete quoted above for a factor of 1.10 is the figure for concrete placed inside the box: dug down instead, the requirement is 2.37 m, and the basement has lost most of a storey.

Weight against weight, twice

There is one other check in this collection built the same way. Overturning is also a ratio of one weight to an action, it also contains no strength, and it is also passed or failed by the structure as a rigid body rather than as an assembly of members — weight is the only thing resisting it there too. What that check has and this one does not is a second failure mode standing beside it.

Both failures are decided by the same two numbers. Factors of safety against overturning and against sliding, for a body 4 m wide weighing 900 kN under a wind pressure of 1 kN/m², as its height grows. Overturning falls as the square of the height and sliding as the first power, so they cross: below 34.6 m the body overturns at a factor of one, and uplift at one edge has already begun at 20.0 m — a ratio of exactly √3, whatever the numbers are.
Fig. 8 The other weight-against-weight check, and the shape flotation does not share. Factors against overturning and against sliding for a 4 m wide body weighing 900 kN under 1 kN/m² of wind, swept against its height: overturning falls as the square of the height and sliding as the first power, so the two cross. This body overturns at a factor of one at 34.6 m and has lifted one edge already at 20.0 m — a ratio of exactly √3 whatever the numbers are. Flotation has one mode, no lever arm and no friction coefficient, so it has no curve to cross.

The comparison is worth making because it reveals what is unusual about flotation. Overturning has a lever arm, so the geometry of the body matters. Sliding has a friction coefficient, so a material property sneaks back in. Flotation has neither: it is the only check in this collection whose answer is unchanged by every dimension of the structure except the two that set its plan area and its depth.

The slab itself

A second calculation follows the first, and it is the one the thickness on the drawing is actually for. Once the substructure is held down — by piles, by anchors, or by the columns it carries — the base slab spans between those hold-downs with the uplift pressure applied to its underside. A 58.9 kN/m² upward load on a slab spanning 8 by 6 m is a substantial slab, and it is designed upside down.

This is where the thickness argument returns, correctly. The slab is not thick because thickness resists pressure; it is thick because it has to span between the piles or columns that hold it down, under a uniform pressure applied from below, with the reinforcement in the top face where a designer’s hand is not used to putting it.

Every check that belongs to a slab belongs to this one, with the sign of the loading reversed. That includes a punching check made on a perimeter, which here has the column pushing down through the slab while the water pushes up around it — the same geometry as a flat slab and the same control perimeter, with the load inside the perimeter subtracted for a different reason than usual, because it is genuinely acting on the column rather than crossing the perimeter at all.

The two ways of counting the same weight

There is a bookkeeping trap in this subject that is worth spelling out, because both ways of doing the arithmetic are correct and mixing them is not.

The buoyancy view. The structure displaces a volume of water; the upthrust is the weight of that volume; the structure floats if it weighs less. This is Archimedes and it needs the whole submerged volume, walls and voids included.

The pressure view. The water pushes up on the underside of the base slab at γwh\gamma_w h and horizontally on the walls, and the horizontal components cancel. This is the free-body calculation made above, and it needs the plan area and the head, and nothing else.

The two give the same answer for a closed box, and they do not look as though they should. The reconciliation is that the pressure view’s uplift, γwhA\gamma_w h A, is exactly the weight of a prism of water of plan area AA and depth hh — which is the volume the structure displaces below the water table, provided its walls are vertical and its base flat.

Where they diverge is instructive. A structure with a stepped base, a projecting toe, or a sloping wall has a displaced volume that is not hAhA, and the buoyancy view is then the reliable one; a structure with an open top below the water table has no buoyancy argument at all and the pressure view still works. Pick one and stay in it, and never add a buoyant upthrust to a hydrostatic pressure on the same surface.

A factor of one on the whole does not mean a factor of one on the parts

Every number on this page is a single ratio for a single substructure, and a single ratio quietly assumes the thing it describes is rigid enough to be weighed as one object. Real substructures are not one object, and this is where the check most often passes while the structure lifts.

Consider the ordinary case: a basement whose plan carries a tower over part of it and a two-storey podium over the rest. Take the tower half at four storeys and the podium half empty. The whole structure weighs 18.7 MN of concrete plus, say, 19.2 MN over the tower half — 37.9 MN against 35.3 MN of uplift, a global factor of 1.07, and the check passes.

Now cut the free body down the middle. The tower half carries 9.35 + 19.2 = 28.6 MN against 17.7 MN of uplift, a factor of 1.62. The podium half carries 9.35 against the same 17.7, a factor of 0.53 — unchanged from the empty case, because nothing was added to it. One half of the structure is trying to rise by 8.3 MN while the other half is holding down with 10.9 MN to spare, and the two halves are connected by the base slab.

So the global check has not been wrong; it has been answering a different question. What it establishes is that the structure will not float away as a body. What decides whether it is damaged is the distribution, and the 8.3 MN that the podium cannot carry has to travel through the slab to the tower — as bending in the base slab over a span nobody drew, as tension in the walls between them, and as a hogging moment concentrated at the line where the loading changes.

Two rules follow, and both are habits rather than calculations. Take the free body at every plausible line, not only around the whole: a cut anywhere on plan is a legitimate free body, and the worst one is wherever the weight above changes most abruptly. And apply the same test in time as well as in space — the tower half reaches its four storeys months before the podium does, so a structure that is safe in every completed configuration can pass through one that is not, which is the construction case this essay has already identified as governing and which the plan-cut version simply repeats a level down.

Where the model stops

The water table is a design assumption, not a measurement. It is measured in boreholes at one time of year, in one season of one decade, and the value used for design is a conservative envelope of what it might become. The whole calculation is exact and its principal input is not.

The soil’s own weight has been left out. Soil above any part of the structure — a heel projecting beyond the wall, a backfill over a lower basement — counts, and it counts at its submerged weight below the water table, which is roughly half its bulk weight. Getting that halving wrong is the commonest arithmetic error in the subject.

Nothing here is a seepage calculation. In a low-permeability soil the full hydrostatic head may take a long time to develop, and in a fissured rock it may develop instantly. The uplift used in design is the long-term steady state, which is correct and is not what the structure sees on the first day.

And the factor of safety is applied differently from every other one on this site. Uplift is a permanent action favourable to nothing, so it is factored up while the weight resisting it is factored down — and the two partial factors are applied to quantities that are both simply weights. There is no characteristic strength to take a fraction of, which puts the check in the same family as overturning and outside every other one on this site.

What the pictures cannot show

The uplift arrows in the hero figure are drawn as a row of discrete arrows with a pressure band beneath them. The real action is a continuous pressure on a surface, and the drawing has to make it look like a set of forces in order to show that it is there at all.

Nor can any figure show that the water is on the other side of the concrete. Every pressure drawn here acts on a surface that is also a waterproofing problem, and the two disciplines meet at exactly the plane this calculation is made on.

And the factor of 0.53 is a number about one substructure with one water table. It is drawn near enough to failure to make the arithmetic legible, which is a choice about a figure rather than a statement about basements.

The ladder from here

Later rungs on this anchor: tension piles and ground anchors, and the pull-out mechanisms that decide their capacity. The submerged unit weight of soil, and why the buoyancy of the ground is the term most often dropped. Uplift during construction, where the dewatering is switched off in stages and the governing case is a partly built structure with the pumps already stopped. Seepage and the flow net, which is where the pressure distribution comes from when the ground is not uniform. Relief valves and the decision to flood. And the historical case: several large basements and dry docks have floated, and in every recorded instance the structure was intact, the calculation was right about the structure, and the water table was not where the calculation said.

Where this goes next

Five loads behind one wall, and the water is the biggest. The horizontal pressure on a 6 m wall retaining soil at 18 kN/m³ with a friction angle of 30°, a surcharge of 10 kPa and the water table 2 m down, drawn once as the profile the wall feels and then once per term. The terms are surcharge 20.0 kN/m at 3.00 m, soil above water 12.0 kN/m at 4.67 m, soil at the water table 48.0 kN/m at 2.00 m, submerged soil 27.2 kN/m at 1.33 m, water 78.5 kN/m at 1.33 m, and they sum to 185.7 kN/m — matched to 7e-8 by integrating the drawn profile numerically. The largest single term is the water, at 78.5 kN/m: water has no shear strength, so its coefficient is exactly one where the soil's is 0.333, and it acts on top of the soil's effective stress rather than instead of it. The combined resultant sits at 1.90 m above the base, 0.317 of the height rather than the third point at 2.00 m that a pure triangle would give. The load that depends on what carries it Every other load in this collection is a number the structure is given. Retained soil is not — it pushes with a fraction of its own weight, and the fraction is decided by how far the wall moves. Six millimetres of retreat on a six-metre wall takes a third off the load, and being held still puts it back. 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. The same beam, cut at x = 5. A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier. The free body is a choice, and choosing it well is the whole skill Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty. The reaction lies inside the cone, so the block stands. A block of 100 on a plane at 15°, against a coefficient of friction of 0.35. Resolving across and along the plane gives a normal force of 96.6 and a friction demand of 25.9, against a capacity of μN = 33.8 — a ratio of 0.77. Added together the two make one contact reaction leaning 15.0° from the normal, and the admissible reactions fill a cone of half-angle arctan μ = 19.3°. 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 15.0°, which is why the angle of repose is a material property and the size of a heap of sand is not. The force that is whatever it needs to be Every other force in statics has a value the equations produce. Friction has an inequality instead, so it takes whatever value equilibrium demands and the bound only ever says no — which means a problem with friction in it has a range of answers rather than one.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

BallastBuoyancyControl perimeterDrainageEffective depthEquilibriumFactor of safetyFlotationFree bodyHydrostatic pressureOverturningPunching shearTension pileUpliftWater table