Structural Engineering & Statics
where the load goes
Concept

Thermal movement — where it appears

The strain a temperature change asks a member for, which costs nothing if the member is free and produces stress in proportion to how much it is not. Providing for it is a joint, resisting it is a force, and the choice between the two is the whole of how a long structure is detailed.

Named by 14 essays across 7 fields — each of them below, with the objects they name alongside it.

It moves, or it pushes. Never both, and never neither. A 30 m steel member 30 °C warmer than it was built, in three conditions. Free, it grows 10.8 mm and carries nothing. Held, it moves nothing and carries 75.6 MPa in compression — which is E·α·ΔT and contains neither the length nor the area of the member, so the identical stress arises in a two-metre strut. Held by a spring it does some of each: 3.2 mm of movement and 53.2 MPa, and the split is decided by the spring rather than by the member.

The movement nobody applied

A temperature change is the only load in this collection that a structure can decline. Let it move and it produces a movement with no stress; hold it and it produces a stress with no movement — and that stress contains no length, no area and no second moment, so a bracket and a bridge girder carry exactly the same one.

deflection · Thermal movement The strain it wants, the strain it is allowed, and the difference. A bridge deck 1.40 m deep with 18 °C at the top face falling away over 10% of the depth. The left curve is the free thermal strain αT(y); the straight line beside it is what a plane section will actually take, ε₀ + κy with ε₀ = 32.0 microstrain and κ = 0.063 per km. The right-hand block is E times the difference, and it reaches -3.98 N/mm² of compression at the surface and 1.83 of tension 140 mm below it. Its resultant force is 8.3e-14 kN and its resultant moment 3.0e-12 kNm, which is what self-equilibrating means: the field is invisible to every equilibrium check that could be made on the member.

The stress nobody restrained

A bridge deck lying loose on its bearings, with nothing holding it anywhere, develops four newtons per square millimetre when the sun comes out. The stress is not caused by restraint. It is caused by plane sections, and it is invisible to every equilibrium check that could be made on the member.

sections · Thermal gradient 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 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 Restraint is a fraction, and the length decides how far up it reaches. A 20 m wall 3.0 m high cast against a base that has already hardened — a length-to-height ratio of 6.67. The base holds the bottom of the wall at R = 0.50 and the top of it at 0.304, decaying as 0.609 to the power of the height in wall heights. The free contraction is 380 microstrain, of which 84 per cent is the wall cooling from its own hydration peak and the rest is drying; the concrete's own strain capacity is 50. Everything to the right of the dashed line cracks, which here is the bottom 3.00 m of it. Nothing has been loaded.

The steel decides how many, not how much

A wall cast on a base that has already set cools, tries to contract, and is not allowed to. What follows is not a stress problem with a strength on the other side of it. The movement is going to happen; the only question the reinforcement gets to answer is how many pieces it is divided into.

internal-forces · Restraint cracking Steel and concrete happen to match, and nothing else on the list does. The mismatch strain a 40 degree change produces in seven pairs of materials that engineering bonds together, which is the difference of their coefficients of expansion times the temperature. Steel against concrete is 80 microstrain — 17 per cent of the larger coefficient, and by far the smallest on the list. It puts 0.223 N/mm² of tension into the concrete, 7.7 per cent of its tensile strength and 1.9 per cent of the 12 N/mm² a fully restrained member would have carried. Reinforced concrete works because of a coincidence in the third significant figure of two numbers nobody chose, and the same bar in aluminium would put in two and a third times as much.

The coincidence reinforced concrete stands on

Steel expands at twelve microstrain per degree and concrete at ten. Nobody chose either number, they are not equal, and the seventeen per cent between them is the smallest mismatch of any pair of materials engineering bonds together — which is the reason the most-used structural material on earth does not tear itself apart every summer.

materials · Thermal compatibility The summer that is worse than the one before it. The earth pressure behind an integral abutment, summer by summer, as a multiple of the at-rest value it started at. A 60 m deck expands by 10.8 mm at each end and pushes the abutment into the backfill. Granular soil under cyclic strain densifies, so the same movement next year needs a higher pressure to achieve, and K climbs from 0.38 toward 0.96 — a factor of 2.49 on the force, reached after about a century. The design load on an integral abutment describes the bridge's whole life rather than a load case, and it is the only load in this collection that gets larger because time has passed rather than because something was added.

The summer that is worse than the last

An expansion joint is a hole in a deck that leaks salt water onto the bearings underneath it. Remove it and the thermal movement does not go away — it goes into the soil behind the abutment, twice a day for a hundred and twenty years, and granular soil under cyclic strain gets denser.

structures · Integral bridge The pier grips, gives, and grips again. A sliding bearing carrying 3000 kN on a pier head of 20 kN/mm, dragged by a deck expanding at 1.7 mm an hour, with a static coefficient of 0.05 and a kinetic one of 0.03. The force in the pier climbs while the bearing grips, reaches 150 kN, and falls in a fraction of a second to 30 kN: the pier springs back under only the kinetic friction, overshoots the 90 kN that friction would hold it at, and grips again. The swing is 120 kN — 2.00 times the 60 kN between the two coefficients — and the pier head jumps 6.00 mm each time, three times in 12 hours.

The pier that moves in jumps

A sliding bearing whose static friction is larger than its kinetic friction does not release a slow thermal movement as a drift. It grips, gives and grips again, and each time the force in the pier swings by twice the difference between the two coefficients — whatever the pier is made of.

equilibrium · Friction Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither.

The order the loads arrived in

Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

equilibrium · Friction A load on one span moves the other. The change in level along two 100 m spans of stressed ribbon at a sag of 2.00 per cent, carrying 35 kN/m, when 20 kN/m of live load is added to the left-hand span, with a rigid pier (dashed) and with a pier of 200,000 kN/m (solid). With a rigid pier the loaded span sags a further 195 mm and the other does not move. With a pier of 200,000 kN/m the pier top leans 25 mm towards the load, the loaded span sags 358 mm and the unloaded span rises 191 mm; the pier carries 4,953 kN of the 9,440 kN a rigid pier would.

The pier that carries only the difference

A stressed ribbon continued over a pier pulls on it from both sides, and under its own weight the two pulls cancel. Load one span and they do not: the pier is asked for the difference, nearly ten meganewtons for a crowd on one of two hundred-metre spans. Whether it gives it depends on its stiffness against a number the ribbon supplies itself, and whatever the pier does not carry, the ribbon turns into movement — a third of a metre of extra sag in the loaded span and a fifth of a metre of rise in the empty one.

structures · Stressed ribbon The same rib, fixed, bends at both ends as well as the crown. The bending moment that axial shortening leaves in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m², EA 20,400,000 kN) under 60 kN/m, sagging upward, from a frame solve of the rib. Dashed: two-hinged, sagging throughout, 29.4 kN·m at the crown. Solid: fixed at the springings, 57.9 kN·m sagging at the crown and 112.4 kN·m hogging at each springing. The funicular load itself leaves no moment in either rib; all of this is the 0.11 and 0.63 per cent of the thrust the two ribs lose by getting shorter.

The springings that make shortening worse

Fixing an arch at its springings is the stiffer, cheaper and usual way to build one in concrete, and it makes the arch six times as sensitive to its own shortening. The thrust it loses acts at the elastic centre, two thirds of the way up, so the moment lands at the springings as well as the crown, twice as large and the other way round — at the section the fixed arch is designed at, not away from it.

deflection · Rib shortening The pier reads the difference between the two spans' temperatures. Contours of the pier force between two 100 m spans at a sag of 2.00 per cent with the deck hung on one, the tendons at 1,100 N/mm² when finished, on a pier of 200,000 kN/m, every half meganewton, over the warming of the hung span's tendons (across) and of the bare tendon's (up) since they were cut. With neither warmer the pier carries 3,080 kN; warming the bare tendon 30 °C alone gives 4,126 kN, the hung span alone 2,342 kN, and both together 3,392 kN. The contours are straight and parallel: the pier gains 35 kN for each degree the bare tendon warms and loses 25 for each degree the hung span does, so a warming both share moves it only 30 per cent as far as the same warming on the bare tendon alone, and what the pier feels is mostly how much warmer one span's strand is than the other's.

The strand in the sun beside the strand in the shade

Halfway through hanging a stressed ribbon, one span is a sagging cable full of segments and the next is bare tendon stretched nearly straight, and the pier between them carries the difference of their pulls. A warm day takes tension off both — and at the flat sag a ribbon is laid at, the hung span is so nearly all stretch that it loses three quarters of what the bare one does. What moves the pier is not the weather but the shade: strand in the sun beside strand under concrete, 35 kN on the pier for every degree between them. No sun slackens the bare tendon. The hour does change something else, which is the level the joints are cast at.

structures · Stressed ribbon

Named alongside it

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

RestraintServiceabilityArticulationFrictionImposed deformationSelf-stressBearingIndeterminacyMovement jointSelf-equilibratingSliding bearingBond

All concepts