Structural Engineering & Statics
where the load goes
Concept

Imposed deformation — where it appears

A change of length asked of a structure by temperature, shrinkage, creep or settlement rather than by a load, which produces force only where it is restrained. The force it produces is proportional to the stiffness of whatever restrains it, so for this kind of action flexibility helps and strength does not.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 30% of the plastic moment with a 20°C profile it never yields at all; At 60% of the plastic moment with a 120°C profile it shakes down; At 85% of the plastic moment with a 200°C profile it ratchets, at 1.9% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains.

The structure that settles down, and the one that walks

A load that is safe applied once may not be safe applied ten thousand times. Nothing about that is fatigue — the structure never breaks, it simply arrives somewhere slightly further round every cycle, until it has arrived somewhere unusable.

materials · Shakedown The stress that leaks away. A restrained shrinkage strain of 300 microstrain in concrete of modulus 32000 N/mm². Ignoring creep it produces 9.60 N/mm², which is above the tensile strength of 3.5 and predicts that every restrained concrete member ever cast has cracked. Counting creep by the superposition integral leaves 2.32 N/mm² after 27 years, and the one-line age-adjusted shortcut at the usually quoted ageing coefficient of 0.8 leaves 3.31. The two disagree — this creep function implies an ageing coefficient of 1.32, not 0.8 — and both are below the tensile strength, so the conclusion turns on counting creep at all rather than on how it is counted.

The strain that was imposed, and the stress that leaked away

Multiply a restrained shrinkage strain by the modulus and the answer is three times the tensile strength — which predicts that every restrained concrete member ever cast has cracked. Most have not, and the reason is that the material creeps while it is being stressed.

materials · Relaxation Four camber rules, and what each leaves on the finished beam. The same 12 m composite beam, cambered against four different things, followed through its own load history. Positive is a sag and negative a hog, and the point at the left of each line is the shape it was fabricated to. Cambering against the wet concrete leaves 12.7 mm of sag at the end and a flat beam on the day the slab is poured; cambering against the total load leaves the beam dead flat when fully loaded and hogged 37.9 mm — one part in 316 of the span — before anything is on it at all.

Built to the wrong shape on purpose

A cambered beam is fabricated curved upward so that load bends it down to something like straight. Nothing in the analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads count.

deflection · Camber Two differences up the same building, peaking in different places. Differential shortening between a perimeter column and the core of a 40-storey building, plotted up the height. The part driven by load peaks at level 20 — exactly half way up, because a floor near the top has almost nothing built above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it to shorten. The part driven by shrinkage does not care what is above it at all and accumulates all the way to the roof. Their sum is worst at level 40, at 43 mm, which across a 9 m bay is a floor out of level by one in 208.

The columns are shorter than the core

Every column in a tall building gets shorter as the building is built on top of it, and the core beside it gets shorter by a different amount. The floors between them tilt by the difference — and the difference is largest exactly half way up, because a floor near the top has almost nothing built above it and a floor near the bottom has almost nothing beneath it.

deflection · Differential shortening One coefficient, and nothing else in it. The deflected shapes of one beam under four load cases, each scaled so that its mid-span deflection is the same, with the tangent at the left-hand support drawn on each. The end rotation is that deflection times a coefficient that depends only on the shape of the load: 3.20 for a uniform load, 3.00 for a load at mid-span, 2.99 for a triangular load, 3.60 for a load on half the span. Every material property, every second moment and the span itself cancel out of the ratio θL/δ, so a beam at any deflection limit has an end rotation that is known before anything about it is: at L/360 it is 8.89 milliradians, or 0.51 of a degree.

The angle nobody limits

Every serviceability rule in this collection limits a displacement. What a bearing, a joint and a cladding gap actually have to accommodate is an angle — and the angle is locked to the displacement by a coefficient that contains no material, no section and no span.

deflection · End rotation Two drawings of one deck, and they are not the same structure. A 112 m viaduct on five supports, articulated two ways. Above, the fixed point is at the left abutment: the far end has to be given 45 mm of movement, and the friction of every sliding bearing runs one way, so the fixed support takes 660 kN before any wind or braking is applied. Below, the fixed point is at the middle pier: the largest joint halves to 22 mm and the friction now cancels across the fixed point, leaving 0 kN. The movement arrows are drawn at 900 times the scale of the deck, because a 45 mm movement on a 112 m span is thinner than the line the deck is drawn with. Nothing about the deck, the loads or the ground has changed between the two.

Where the structure is allowed to move

One drawing decides how big every movement joint on a bridge is and where every horizontal force goes, it takes an afternoon, and it appears on no calculation sheet. Move the fixed point from an abutment to the middle pier and the largest joint halves and the horizontal force on that support drops from the whole of the friction to none of it.

structures · Articulation The gap is a sum of five things and only one of them is computed. What a 30 mm movement joint is asked to accommodate, by three combination rules. The top bar is every term at its extreme, added: 33.2 mm, which assumes the hottest day, the fullest floor, the whole of the shrinkage and the worst-placed wall arrive together. The chance of that is about 1.5%. The bottom bar treats them as independent and asks for 16.0 mm. The middle bar is the rule used for actions and almost never for movements — one term at its full value and the rest at their coincidence factors — and gives 25.5 mm. The segments across the top bar are the terms themselves, and the ordering is the finding: the largest is tolerance at 10.0 mm, which is not a structural quantity at all, and the smallest is deflection at 3.2 mm — the only one anybody computes carefully, and 10% of the total.

The gap nobody computed

A movement joint is sized by adding up everything the structure will do to it, and the deflection calculation — the only term anybody computes carefully — is usually the smallest one in the list. The largest is a construction tolerance, which is not a structural quantity at all, and the sum of the extremes is nearly twice what treating them as independent would ask for.

deflection · Movement budget 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 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.

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.

structures · Propped excavation The lining that carries less for being weaker. Bending moment and hoop thrust in a circular lining, against the lining's own bending stiffness, both as fractions of the free-ring values. The ground arrives already stressed — 500 kPa vertically and 300 horizontally at K₀ = 0.6 — and the difference between them tries to squash the hole into an ellipse. A lining stiff enough to refuse absolutely collects the whole distortion pressure, p₂R²/3 = 300 kNm/m; one flexible enough to go with the ground collects nothing, because there is no curvature change left to resist. The thrust is the flat line: it comes from the mean stress rather than the difference, so it does not move at all. Putting 8 joints in this ring drops the moment to 34% of the solid one and leaves the thrust exactly where it was, which is why a segmental lining is jointed and why the intuition carried over from a beam is inverted here.

The lining that is stronger for being weaker

A tunnel lining is not loaded. The ground arrives already stressed and the hole wants to squash into an ellipse; the lining's only job is to refuse, and how much moment it collects depends entirely on how hard it refuses. Make it stiffer and it takes more. Make it flexible — put joints in it, make it thin — and it takes almost none, while the hoop thrust it carries does not move at all.

structures · Tunnel ring Two identical pipes, and one carries three times the other. Load per metre on a buried conduit against the depth of cover, in trench widths, with the weight of the prism of soil directly above it drawn between them. A conduit laid in a narrow trench is stiffer than nothing and softer than the sides: the backfill settles relative to the undisturbed ground, the friction on the trench walls acts upward, and the conduit gets 64% of the prism. Lay the same conduit on the ground and build an embankment over it and it is now stiffer than the fill beside it, the interior prism settles less, the friction acts downward, and it gets 172% — a factor of 2.71 between two pipes with nothing different but which way the ground moved. The equation is Janssen's, the same one a silo wall obeys, with a trench for a silo; both curves start on the prism line, because with no depth there is no shear to redistribute anything. This is why a flexible pipe is buried rather than a rigid one: making the conduit weaker moves it down the page.

The pipe decides what the soil weighs

A buried conduit is not loaded by the soil above it. It is loaded by whatever share of that soil the relative movement leaves it — and which way the shear on the sides of the prism acts depends on whether the conduit settles more or less than the ground beside it. Two identical pipes under identical fill, one carrying two thirds of the prism and one carrying nearly twice it.

equilibrium · Soil arching Five millimetres short, and a hundred kilonewtons in every member. An X-braced bay 6 m by 4 m in which one diagonal was fabricated 5 mm short, with the force that leaves in every member. Nothing is applied to this frame. The forces are the self-stress state the frame's one redundancy supports, scaled so that the diagonal is pulled back to the length it should have been: tension in both diagonals at 100 kN, compression in the four members round the outside, and the whole set in equilibrium with nothing. That is 25% of the force the diagonal was sized to carry, and it is there for the life of the structure. Take one diagonal out and the frame becomes determinate: the short member then simply puts the joint somewhere else, and the structure is in the wrong place instead of under stress. Redundancy is bought, and this is the price.

Built to the wrong length

A redundant structure's members do not have independent lengths. Choose all but one and geometry decides the last, so a member made a different length has to be pulled or pushed into place — and the force required stays in the structure for as long as the structure does. Nothing has been applied to it, there is no load case and no factor, and the members are carrying real force.

connections · Fit-up Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 20% of the plastic moment with a 40°C profile it never yields at all; At 50% of the plastic moment with a 150°C profile it shakes down; At 90% of the plastic moment with a 260°C profile it ratchets, at 51.0% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains.

The map with three regions

A structure carrying a constant load and a cycling temperature has three possible fates and only one of them is a collapse. It can stay elastic, it can yield once and then stop, or it can gain a little more deformation every cycle for ever — and the third has no failure load at all.

materials · Shakedown The stress that leaks away. A restrained shrinkage strain of 320 microstrain in concrete of modulus 34000 N/mm². Ignoring creep it produces 10.88 N/mm², which is above the tensile strength of 3.8 and predicts that every restrained concrete member ever cast has cracked. Counting creep by the superposition integral leaves 1.72 N/mm² after 55 years, and the one-line age-adjusted shortcut at the usually quoted ageing coefficient of 0.8 leaves 3.05. The two disagree — this creep function implies an ageing coefficient of 1.67, not 0.8 — and both are below the tensile strength, so the conclusion turns on counting creep at all rather than on how it is counted.

The stress that leaks away

Creep makes a load's deflection grow and an imposed strain's stress shrink, and the second is why restrained concrete does not crack as often as an elastic calculation says. The same material property runs both ways, and which way it runs depends on whether the structure was given a force or a movement.

materials · Creep Half of the creep is the member drying out. The creep coefficient of a concrete of mean strength 38 N/mm² loaded at 28 days, in a member of notional size 150 mm, against time since loading: exposed to air at 50 per cent relative humidity, and the same member sealed so that it cannot dry. After a year the exposed member's coefficient is 1.93 and the sealed member's 0.81; after 50 years they are 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member's final creep, is what drying adds. The sealed member's coefficient has no size in it; the drying part is where the member comes in.

The creep that belongs to the member

A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether.

materials · Creep The humidity through a slab's depth, and the strain it asks for. A slab 200 mm thick drying through its top face only into air at 50 per cent, with its underside sealed by a steel deck. Left, the pore humidity against depth at 28 days — 50 per cent at the surface and 100 at the base; 90 days — 50 per cent at the surface and 100 at the base; 1 year — 50 per cent at the surface and 99 at the base; 5 years — 50 per cent at the surface and 74 at the base; 20 years — 50 per cent at the surface and 51 at the base. Right, the free strain each of those profiles asks for, taken as shrinkage at the local humidity. At 28 days the top wants to be 461 microstrain shorter than it was and the bottom 0; at 20 years the two are 461 and 456, and the gradient that produced the curl has gone. Not one of the profiles is straight, and a section that stays plane cannot deliver any of them.

The slab that dries from one face

A creep coefficient is one number for a member, and a member drying through one face does not have one. Give every depth its own humidity and the section a strain profile it cannot deliver, and two things follow that no single coefficient contains — a six-metre slab lifts eleven millimetres at its edges with nothing on it, and its top surface is past cracking before it has been loaded.

materials · Creep Three moment diagrams for one pair of beams. Two 15 m spans carrying 12.0 kN/m, drawn sagging downward. As built they are simple spans: 337.5 kN·m at each midspan and nothing over the middle support. Built monolithic they would carry 337.5 kN·m of hogging over the support and 168.8 at midspan. Loaded at 28 days and made continuous at 60, creep takes them 46 per cent of the way from the first to the second: 155.2 kN·m over the support and 259.9 at midspan, after twenty years in which nothing about the loading changed. The support had no moment on the day it was cast and no drawing of the finished structure shows why it has one now.

The support that had no moment when it was cast

Two beams are set on their bearings, carry their own weight for a month, and are then stitched together over the middle support. Nothing about the loading changes afterwards. Twenty years later the stitch is carrying 155 kilonewton-metres, because the concrete went on creeping and the joint would not let it — and how much arrives is decided by a crane schedule.

materials · Creep Where a tendon's force goes, over fifty years. The loss of stress in a tendon stressed to 1300 N/mm² and released at 7 days into a member of notional size 300 mm at 70 per cent humidity, with the three causes stacked. At 28 days the total is 61.9 N/mm²; at a year 131.5; at fifty years 190.8, which is 14.7 per cent of what the tendon started with. Creep supplies 107.7 of that, drying and autogenous shrinkage 50.6, and the steel's own relaxation 32.5. Half the loss has happened by 119 d and nine-tenths by 8 y.

The prestress the member takes back

A tendon is stretched, locked off against the concrete, and then has to hold that extension while the concrete underneath it shortens by itself. Fifteen per cent of the force goes, most of it to creep, and how much goes is decided by the shape of the member and the air it stands in rather than by anything about the steel.

materials · Creep

Named alongside it

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

ServiceabilityCreepShrinkageFree bodyAgeingStiffnessConstruction sequenceDeflectionEffective modulusRelaxationRestraintSuperposition

All concepts