Structural Engineering & Statics
where the load goes
Structural form

Every prop has its own worst day

A braced excavation has no finished state worth analysing. It is dug in stages, a level of props goes in at each stage, and a prop's force is largely fixed the moment it is installed — so the force to design it for is the largest it sees during a sequence that appears on no calculation sheet, and which for the middle prop here is nearly two and a half times what the finished arrangement gives.

Assumes The load that depends on what carries it, The structure that was never complete and The free body is a choice, and choosing it well is the whole skill.

12 min read 7 figures The load must go somewhereOne support too manyWhich failure arrives first

Every structure in this collection so far has been analysed in the state it will spend its life in. A braced excavation cannot be, because it does not have one: it is a different structure every week, and each of those structures is the one that decides some part of it.

The sequence is fixed by the method. Dig a metre or two; install the top level of props; dig down to the next level; install those; dig on. At every stage the wall is a beam on the props installed so far, cantilevering below the lowest of them into whatever passive resistance the remaining soil can offer. And — this is the whole of the argument — a prop’s force is largely locked in when it goes in. The wall has already moved by whatever it moved before the prop arrived; what happens afterwards is that the wall below the prop deflects and the prop holds the position it was given.

The prop load is decided by the digging, not by the hole. Prop forces in a 12 m excavation propped at 3 levels, with the force each prop reaches at any stage of the sequence drawn thick and the force the finished arrangement gives it drawn thin. The pale dots are the individual stages. The middle prop reaches 214 kN while the dig is at 9.5 m and finishes at 88 — a factor of 2.44 between the two, and the larger one is not in the final analysis anywhere. Terzaghi and Peck's apparent pressure diagram, a rectangle of 49.4 kN/m², reproduces the total of the staged maxima to 2% — which is what it is: an envelope of measured prop loads, back-figured into a pressure, and a shape nothing on a wall is ever loaded with.
Fig. 1 Prop forces in a 12 m excavation propped at three levels. The thick bars are the largest force each prop reaches at any stage of the dig; the thin ones are what the finished arrangement gives it. Only the bottom prop’s two are the same.

Which free body produced the number

The wall, at one instant, from its top down to the centroid of the passive resistance beneath the current formation level — with the props installed so far as supports and the active pressure as the load.

That is a beam on n+1n+1 supports, where nn is the number of props in place, and the reactions are the prop forces. The choice worth defending is where the bottom support goes. The embedded length below formation is not a support at a point; it is a distributed passive resistance, and representing it as a support at the centroid of that block — the “fixed earth” assumption — is what makes the first stage analysable at all. Before the first prop there is only one support, and a beam on one support with a free rotation is a mechanism.

The active pressure is Ka(γz+q)K_a(\gamma z + q) over the retained height, and the assumption in that is the weakest thing here. The load that depends on what carries it is why: the active state requires a movement of about a five-hundredth of the height to mobilise, and a wall propped near its top has not been allowed that movement near the top. Which is exactly the reason the profession stopped calculating with triangles and started measuring.

Which coefficient applies is decided by how far the wall moves. The pressure coefficient on a 6 m wall against the wall's own movement, negative into the soil and positive away from it. A wall that has not moved carries K₀ = 0.500. Letting it retreat 6.0 mm — one thousandth of the height — lets the soil carry its own weight on shear and brings the coefficient down to 0.342, close to the active limit of 0.333. Pushing it the other way reaches 2.87 against a passive limit of 3.00, but only after 150 mm — 25 times as far. Both halves of the axis are at the same scale, which is the argument: the curve has a kink at the origin, with a stiffness of 26962 on the active side and 16177 on the passive, and mobilising passive resistance fully would move the wall further than anything standing on it can tolerate. A wall designed for the active state and then not allowed to move carries 1.50 times what it was checked for.
Fig. 2 Why a triangle is the wrong shape near a prop. The active pressure is not a property of the soil; it is what the soil delivers once it has been allowed to move, and a wall held near its top has not been. The at-rest pressure is nearly twice the active one and is what a restrained wall receives.

The stage nobody draws

The top prop is the case that makes the argument, and it is decided by a structure that exists for about two days.

Before the first prop goes in the wall is a cantilever. It has a small retained height, so the pressure is small; but it has no support at all near the top, so the whole of that small pressure is carried by a moment at the toe and a rotation of the whole wall. When the prop goes in, it goes into a wall that has already leaned, and it goes in tight.

By the end of the dig that wall is a multi-span beam with three supports, the top span is short, and an elastic analysis of the finished arrangement gives the top prop very little — 43 kN per metre here. The cantilever stage gave it 62, which is 43% more, and it is the number that matters.

The middle prop is worse. It reaches 214 kN per metre while the dig is at 9.5 m — deep enough for the pressure above it to be large, and before the bottom prop exists to take any share of it — and by the last stage it is down to 88. A factor of 2.44 between a prop’s worst day and the day the drawings describe.

Only the bottom prop’s worst day is the last one, and only because it has had no other.

The prop load is decided by the digging, not by the hole. Prop forces in a 18 m excavation propped at 4 levels, with the force each prop reaches at any stage of the sequence drawn thick and the force the finished arrangement gives it drawn thin. The pale dots are the individual stages. The middle prop reaches 204 kN while the dig is at 9.5 m and finishes at 154 — a factor of 4.00 between the two, and the larger one is not in the final analysis anywhere. Terzaghi and Peck's apparent pressure diagram, a rectangle of 74.1 kN/m², reproduces the total of the staged maxima to 3% — which is what it is: an envelope of measured prop loads, back-figured into a pressure, and a shape nothing on a wall is ever loaded with.
Fig. 3 The same effect on a deeper dig with four levels. Adding props does not remove it, because every level has its own first day — and the deepest prop is always the one whose two numbers agree.

The diagram that is not a pressure

Terzaghi and Peck’s response to all of this, published from the Chicago and Berlin subway measurements of the 1940s, was to stop calculating the pressure and to publish an envelope.

Their apparent pressure diagram is a rectangle — 0.65KaγH0.65K_a\gamma H for a sand, 0.20.2 to 0.4γH0.4\gamma H for a clay — and it is not a distribution of pressure on a wall. It is a shape which, when carved up between the props by tributary lengths, reproduces the largest force each prop was measured to reach. Nothing is ever loaded like it: the real pressure is neither rectangular, nor constant with time, nor the same at any two stages.

On the excavation drawn, the rectangle is 49.4 kN/m², its total is 593 kN per metre, and the sum of the staged maxima is 581. Two per cent, from a diagram fitted to measurements eighty years ago against a stiffness analysis of a sequence.

That is the same species of object as the envelope is not a structure, and it deserves the same warning. An envelope satisfies no equilibrium: the apparent diagram used to compute a wall bending moment gives an answer that is wrong, usually too large near the top and too small near the bottom, because it was never a moment envelope. It is a set of prop loads and should be used for the props.

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.
Fig. 4 What is actually behind the wall, and why a single rectangle is doing so much work. Soil, surcharge and water, each with its own distribution and its own coefficient — and the water is usually the largest of the three and the one with no K in front of it.

Why the load is an imposed deformation

There is a way of seeing the whole problem that makes several of its oddities obvious at once: a prop is not loaded, it is displaced.

The soil behind the wall wants to move into the hole. The wall moves with it until something stops it, and the prop is what stops it. So the force in the prop is set by how much movement it refuses, times its own stiffness — which is the arithmetic of a lack of fit, or a settled support, or a restrained thermal movement, and not the arithmetic of a load.

Three consequences follow, and each of them inverts an intuition brought from load-carrying structures.

A stiffer prop takes more load. Doubling its axial stiffness roughly doubles its force, for exactly the reason a member built to the wrong length develops more force in a stiffer frame. There is no reward for oversizing.

Preloading a prop is a design decision with a cost. Jacking a prop against the wall before the dig continues reduces the wall’s movement, which is often the point — a movement limit protects the buildings next door. It also increases the prop’s force by the preload, directly, and increases the pressure the soil delivers by pushing it back toward the at-rest state.

Letting the wall move is the cheap way to reduce everything. More movement means more of the soil’s own strength mobilised, less pressure, smaller prop loads and a smaller wall bending moment. What it costs is settlement behind the wall, and that is usually what the whole scheme is really being designed against.

One soil, one wall, and a factor of nine. The pressure on a 6 m wall retaining dry soil at 19 kN/m³ with a friction angle of 30°, in the three states Rankine's theory allows, drawn to one scale and with no surcharge so that the three thrusts stand in the ratio of the three coefficients exactly. Active is 0.333 and 114.0 kN/m; at rest is 0.500 and 171.0 kN/m; passive is 3.000 and 1026.0 kN/m. That is a spread of 9.00 from end to end, decided entirely by which way the wall moved and by how far — and the active and passive pair are exact reciprocals, Ka·Kp = 1.000. All three resultants sit at the same third point, 2.00 m above the base, because all three profiles are the same triangle scaled.
Fig. 5 The three states one soil can be in, and the movement that separates them. Active needs a fraction of a per cent of the height and gives the smallest pressure; passive needs ten times as much movement and gives the largest. Everything about a propped wall is a negotiation on this axis.

Measured before it was calculated

It is worth knowing that this is one of the few parts of the subject where the measurements came first and the theory has still not entirely caught up.

Terzaghi’s Chicago subway work in 1936–38 and the Berlin measurements of the same decade were made because the calculations of the day did not work. Struts were being designed to a Rankine triangle and were failing, or were coming out absurdly overdesigned at the bottom and marginal at the top, and nobody could say why. Instrumenting the struts and plotting what they actually carried produced a shape that was roughly uniform with depth, which no pressure theory predicted, and the profession adopted the measured shape rather than the theory.

That order of events explains two features of the practice that otherwise look careless. The apparent diagram has no derivation — it is a fit, and it is presented as one. And it is soil-type-specific in a way a mechanics-based result would not be: separate rectangles for sand, stiff clay and soft clay, with the clay ones expressed as a fraction of γH\gamma H rather than through a KK at all, because the measurements said so.

The mechanism was worked out afterwards and is the one this essay opens with: the shape is uniform because it is a superposition of stages, and the top of the wall was propped when the hole was shallow. A rectangle is what a maximum over a family of triangles of growing height looks like. The empirical diagram turned out to be an envelope, and the envelope turned out to have a reason.

Built as two beams, used as one. Bending moments in a two-span beam erected as simple spans under 12 kN/m and made continuous before the remaining 18 kN/m arrived, against the same beam built continuous from the start. The support moment is 324 kNm rather than 540 — 60% of it — and the midspan moment is 378 rather than 270, which is 140%. Both diagrams are in equilibrium with the same total load; they differ only in when the joint was made, which appears nowhere on the drawing.
Fig. 6 The same reasoning on a building frame rather than a hole. A structure assembled in stages carries each stage’s load on the structure that existed at the time, so the final arrangement’s analysis is a description of the last increment and not of the total — and the difference has to be accumulated rather than computed once.

Reading a wall’s instruments

Because the loads are sequence-dependent and the model is weak, deep excavations are the most heavily instrumented temporary structures anybody builds — and the instruments are worth understanding as a structural argument rather than as monitoring.

Strain gauges on the props measure the thing being designed, and they measure it against the temperature correction, which is the difficulty: a prop’s strain includes its own thermal strain, and separating the two needs a reference. A gauge reading rising through an afternoon is not necessarily a wall moving.

Inclinometers in the wall measure the deflected shape, which is the integral of the curvature and therefore of the bending moment — so a wall’s moment can be recovered from its shape by differentiating twice, and this is one of very few structures where the deflection is the primary measurement and the force is inferred. It is the area of a diagram run backwards.

Ground settlement behind the wall measures the criterion the scheme usually exists for. The relationship between wall movement and ground settlement is empirical too: the settlement trough extends about twice the excavation depth behind the wall, and its volume is roughly the volume the wall moved.

What the instruments enforce is a discipline the calculation cannot: the observational method, in which the design is a set of predictions with trigger values, and the response to exceeding one is a pre-agreed action rather than a new analysis. That is the honest engineering answer to a problem whose loads are decided by a sequence and whose soil is known to a factor of two.

The resultant has left the middle third and the heel has lifted. A retaining wall 6 m high on a 3.5 m base, with a 1 m toe and a 2.10 m heel, holding soil at 18 kN/m³ under a surcharge of 10 kPa with the water table 2 m down. The thrust is 185.7 kN/m acting at 1.90 m above the base, so the overturning moment about the toe is 353 kNm per metre run. Against it the wall musters 338 kN of weight — stem, base and the column of soil standing on the heel — and that weight acts at 61.9% of the base rather than at its middle, which is what a heel is for. The restoring moment is 734 kNm, a factor of 2.08 against overturning and 0.91 against sliding. The base resultant lands 0.625 m off centre against a middle third of ±0.583 m, so the pressure under the base runs from 200 to 0 kPa and 0.12 m of it has lifted off. A rigid block carrying the same weight under the uniform pressure that matches this moment returns 1.68 instead of 2.08, because a block centres its weight and a wall does not.
Fig. 7 The permanent cousin, for contrast. A retaining wall stands on its own and is checked for sliding, overturning and bearing in its finished state — one structure, one set of checks, and no sequence. Everything that makes a propped excavation difficult is absent from it.

The temperature nobody put on the drawing

A steel prop across an excavation is a long compression member in the open air with both ends fixed against a very stiff structure, and it heats up.

Twenty degrees on a 30 m prop is 7 mm of expansion that has nowhere to go, and the force that produces is EAαΔTEA\alpha\Delta T — which for a 500 mm circular hollow section is several hundred kilonewtons, comparable with the soil load itself. It is the reason props are shaded, sprayed, insulated or instrumented, and the reason a prop failure is more likely on a hot afternoon than under a full dig.

The same reasoning as the propping load: it is an imposed deformation, so the force is proportional to the stiffness of the prop, and a stiffer prop is worse. The movement nobody applied is the general statement, and this is its most dangerous instance, because the structure it happens to is temporary and lightly designed.

Where the model stops

The props were rigid supports. A real prop shortens under load, and the wall moves with it. Modelling the prop as a spring redistributes force away from it — usefully — and is why a soil-structure analysis of an excavation gives smaller prop loads than a beam-on-rigid-supports one. The rigid model is the conservative one for the prop and the unconservative one for the wall’s movement.

The soil was elastic and the pressure active. Neither is true through a sequence: the soil unloads as the dig proceeds, which is a stiffer path than loading; and the pressure at any level depends on what that level has done, so the “active” assumption is a claim about movement that has to be checked rather than assumed. A stiff clay near the top of a deep wall may still be at rest.

Water was ignored. It should not be. In a permeable soil the water pressure is the largest single term behind the wall, has no KaK_a in front of it, and does not care how much the wall moves. Most excavation failures are water failures.

And base stability is a different check entirely. A deep excavation in soft clay can fail by the base heaving upward — the soil outside flowing under the wall and into the hole — which is a bearing capacity problem turned upside down and is not sensitive to the props at all. The ground is a mechanism is the mechanism, run the other way up.

The generalisation

The idea to take away is that for a structure built in stages, the design case is a stage and not the structure, and that the analysis has to be run over the sequence rather than at the end of it.

That is not confined to excavations. The structure that was never complete is the same statement for a building frame, where a column’s worst axial load can occur before the floors that will brace it exist. A cantilevered bridge built out from its piers is at its most heavily stressed the day before the two halves meet. A precast frame is a mechanism until the joints are grouted. And the most dangerous day is the general observation that the erection condition is often the governing one.

What makes the excavation the cleanest case is that its sequence is not optional. A building can sometimes be propped, or its floors cast in a different order, or a temporary brace added; a hole has to be dug from the top, and every prop has to go in before the soil beneath it is removed. The sequence is a property of the problem, so the envelope over it is a property of the problem too — and a designer who computes only the final state has not made a conservative simplification. They have analysed a structure that never carried the load.

Where this goes next

The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 12 kN/m of wet concrete and 18 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 292 MPa; propped, the finished composite section takes everything and reaches 186 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two. The structure that was never complete Every analysis in this collection is of a finished structure loaded once. Real ones are built in pieces, and each piece carries whatever was present at the moment it became structural — so the stress in a member depends on when it arrived, which appears nowhere on any drawing. 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. The envelope is not a state of the structure. Every arrangement of the imposed load on three spans — 8 of them, since each span is loaded or not — drawn faintly, with the greatest sagging and greatest hogging at each station drawn over them. Each faint curve is a real state of equilibrium and satisfies the free-moment identity exactly: mid-span ordinate minus the mean of the end moments is wL²/8, to 3e-16 of it. The envelope satisfies it nowhere, missing by up to 23% — because it is assembled from different load cases at different stations and no arrangement of load produces it. seven of the 8 arrangements are needed to build it; the rest never govern anywhere. The envelope is not a structure A continuous beam whose imposed load may sit on any span has eight load cases, and every one of them is a genuine state of equilibrium. The curve the design is made against is not one of them — it is assembled from different cases at different stations, and it fails the identity all eight satisfy exactly. How stiff a brace has to be before the frame stops swaying. The effective length factor of a swaying portal against the stiffness of a horizontal spring at its head. The curve starts at k = 1.317, the unbraced value, and falls to 0.774 — the factor for the same frame with its head held — at a brace stiffness of 23.2 EI/L³. Past that point the frame buckles in the non-sway mode, which the brace does not restrain, and further stiffness buys nothing at all. The threshold is worth stating as 1.41 N꜀ᵣ/L, which is the form the number is memorable in: for a storey carrying a thousand kilonewtons over four metres it is about 0.35 kN per millimetre of sway. Against the frame's own lateral stiffness of 12.0 EI/L³ it is a factor of 1.93. The most dangerous day is before it is finished A structure is analysed once, complete, with every restraint present. It spends weeks in states nobody drew — a beam landed with no deck on it holds 17% of the moment its section is worth, a frame not yet braced buckles at a third of the load it will, and a bolt not yet tightened is a pin where the analysis assumed a fixity.

Named alongside this one

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

The objects this essay names

Each one links to every other essay that touches it.

Active pressureApparent pressureConstruction sequenceEarth pressureEnvelopeFree bodyImposed deformationLoad arrangementLower bound theoremPassive pressurePropped excavationRetaining wallStiffnessTemporary worksWall movement