Structural Engineering & Statics
where the load goes
Deflection

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.

Assumes Stiffness is not strength, and usually it is the one that governs, One support too many, and what it costs to know and Strong enough and still falls over.

18 min read 8 figures Scale changes everythingOne support too many

Every load in this collection so far has been a force. Something pushes, the structure resists, and the resistance is proportional to the push. A temperature change is not like that at all: nothing pushes. What happens is that the material’s unstressed length changes, and whether that produces a movement or a force is decided entirely by what is holding the ends.

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.
Fig. 1 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 — which is E·α·ΔT, and contains neither the length nor the area of the member. 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 middle case is the one worth staring at. Seventy-six megapascals is a fifth of the yield stress of ordinary steel, produced by a summer’s day, in a member nobody has loaded — and the same number arises in a two-metre strut and a two-hundred-metre girder, because

σ=E α ΔT\sigma = E\,\alpha\,\Delta T

has no dimension in it anywhere.

Why the length cancels

The reasoning is two lines and it is worth doing slowly, because the result is so counter-intuitive that people re-derive it every time they meet it.

A free member of length LL grows by αΔTL\alpha \Delta T L. To push it back to its original length takes a force NN producing a shortening of NL/EANL/EA. Setting the two equal:

αΔTL=NLEA⟹N=EA αΔT⟹σ=E αΔT\alpha \Delta T L = \frac{NL}{EA} \quad\Longrightarrow\quad N = EA\,\alpha\Delta T \quad\Longrightarrow\quad \sigma = E\,\alpha\Delta T

The length appears on both sides and cancels. A longer member expands more and is more flexible, in exactly the same proportion, and the two effects annihilate. The area cancels for the same reason: a bigger member produces a bigger force and has more area to spread it over.

What is left is a strain comparison. Thermal expansion is a strain of αΔT\alpha \Delta T — 360 microstrain for steel at 30 °C — and holding it produces the stress that strain would produce mechanically. A restrained temperature change is an imposed strain, which is the same category of thing as a support that has settled or a shrinking concrete slab — and, like both of them, it is a case where stiffness and strength part company entirely — and it behaves the way imposed strains always do: proportional to stiffness, unrelated to load, and invisible to statics.

A longer member is more restrained, not less

Perfect restraint is a fiction. Real ends are held by something with a stiffness — an abutment, a bearing, the rest of the structure — and the split between movement and force is the ratio of two flexibilities in series: the member’s own L/EAL/EA and the support’s 1/k1/k.

The stress has no length in it and the movement is nothing but length. Stress and movement against member length, for a 30 °C change. Held rigidly, the stress is 75.6 MPa at every length there is — E·α·ΔT, with no L, no A and no I anywhere in it. Held by a spring of 100 kN/mm the answer climbs with length rather than falling, because a longer member hands the same spring more movement to absorb: 45% of full restraint at 10 m and 83% at 60 m. The free movement, plotted to its own scale, reaches 22 mm.
Fig. 2 Stress and movement against member length for a 30 °C change. Held rigidly the stress is 75.6 MPa at every length there is. Held by a 100 kN/mm spring the stress climbs with length rather than falling — 28% of full restraint at 5 m and 83% at 60 m — because a longer member hands the same spring more movement to absorb. The free movement, plotted to its own scale, reaches 21.6 mm at 60 m.

That curve reverses the intuition that long members are the ones with thermal problems and short ones are safe. Long members have movement problems, which are visible, require joints, and are dealt with. Short restrained members have stress problems, which are invisible, and the shorter the member the closer its own stiffness is to the thing holding it — right down to a stub bracket welded between two heavy flanges, where the restraint is essentially perfect and the full 75.6 MPa arrives.

That is the thermal version of a rule this collection keeps meeting: the forces in a redundant structure follow stiffness, and the stiffest path takes the most. Here the “load” is an imposed strain and the stiff element is the one that ends up carrying it.

The gradient is a curvature, and it has the same two limits

Uniform heating is the simple case. A structure warmer on one face than the other has a gradient, and the same argument runs a field further along: free to move it curves, held down it carries moment.

Warmer on top: the same two limits, one field along. A 30 m deck 15 °C warmer on top than underneath, over a depth of 400 mm. Free to move, it takes a curvature of 0.450 per km and lifts 50.6 mm at midspan, carrying no stress at all. Held down, it carries 33 kNm and ±18.9 MPa at the extreme fibres and does not move. Every real deck is somewhere between, and where it sits is decided by the bearings rather than by the deck.
Fig. 3 A 30 m deck 15 °C warmer on top than underneath, over a depth of 400 mm. Free to move it takes a curvature of 0.45 per km and lifts 50.6 mm at midspan, carrying no stress at all. Held down it carries 33.1 kNm and ±18.9 MPa at the extreme fibres and does not move. Every real deck is between the two, and where it sits is decided by the bearings rather than by the deck.

The curvature is αΔTgrad/d\alpha \Delta T_{\text{grad}} / d, which does contain a dimension — the depth — so a shallow deck curves more than a deep one for the same temperature difference across it. The restrained moment EIκEI\kappa then contains the second moment as well, so the gradient case has none of the dimension-free tidiness of the uniform one.

A 50 mm lift on a 30 m span is a serviceability event by itself. On a bridge it is the reason bearings are designed to accept rotation, and on a building it is why the top storey of a tall structure with an exposed core is where the cracking appears — the core is warmed by the sun on one side and tries to bow, and the floors are attached to it.

The temperature that buckles something nobody loaded

A restrained member that is heated goes into compression, and a member in compression is a column.

The column curve. Failure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.
Fig. 4 The column curve, which is the calculation a restrained heated member has walked into without anybody applying a load. The critical stress is π²E/λ², the thermal stress is EαΔT, and setting them equal removes E from both sides.

Equating EαΔTE\alpha\Delta T with π2Ei2/L2\pi^2 E i^2/L^2 gives

ΔTcrit=π2i2αL2\Delta T_{\text{crit}} = \frac{\pi^2 i^2}{\alpha L^2}

and the modulus has gone. The temperature rise that buckles a fully restrained member is a fact about its geometry and its coefficient of expansion, and nothing else. For the member above — radius of gyration 241 mm — it is 1,919 °C at 5 m, 480 °C at 10 m, 53.3 °C at 30 m and 13.3 °C at 60 m.

The last of those numbers is the point. A 60 m member held at both ends and warmed by thirteen degrees has buckled, and thirteen degrees is the difference between a cool morning and a warm afternoon. This is not hypothetical: it is the mechanism behind rail buckling in hot weather, behind the bowing of long unjointed pipe runs, and behind the classic construction failure in which a long brace is welded in on a cold morning and is found bowed at noon.

The defence is not strength. It is either a joint that lets the movement happen, or a member stocky enough that ΔTcrit\Delta T_{\text{crit}} is comfortably above anything the weather can do — and continuously welded rail takes the second route, resisting buckling by being held down along its whole length by ballast rather than by being strong.

Cooling is the dangerous direction for concrete

Everything above was written for a temperature rise, which puts a restrained member into compression. The other sign is worse for most structures, and it is the one that produces visible damage.

It moves, or it pushes. Never both, and never neither. A 30 m steel member 25 °C colder than it was built, in three conditions. Free, it shortens 9.0 mm and carries nothing. Held, it moves nothing and carries 63.0 MPa in tension — 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: 2.7 mm of movement and 44.4 MPa, and the split is decided by the spring rather than by the member.
Fig. 5 The same member 25 °C colder than it was built. Free, it shortens 9.0 mm. Held, it carries 63.0 MPa — in tension now, because the restraint is preventing a contraction rather than an expansion. The arithmetic is identical and only the sign has changed; what has changed with it is which materials can survive it.

Steel does not mind: 63 MPa of tension is a fifth of yield, in a material equally good either way. Concrete does mind, because its tensile strength is around 3 MPa, and the restrained thermal contraction of a concrete member exceeds that at a temperature drop of about ten degrees.

The consequence is that a long restrained concrete element cracks, and the design question is not whether but where and how wide. That is why a ground slab is cast in bays with joints between them, why a retaining wall has vertical crack-inducing grooves at regular spacing, and why the reinforcement in a wall that carries no load at all is sized by a calculation about crack widths rather than about strength.

It also explains a detail that looks superstitious from outside: early-age thermal cracking, in which a thick concrete pour warms by the heat of its own hydration, is restrained by the older concrete it was cast against, and cracks as it cools back to ambient. Nothing external has happened at all. The load case is the concrete’s own chemistry, restrained by the concrete’s own earlier self, and it is an imposed strain like every other one in this essay.

Where the restraint comes from decides the buckling length too

The buckling calculation two sections above used a pinned member of length LL, which is the conservative reading and rarely the real one. A rail is not pinned at two points; a pipe run is held by hangers along its length; a long brace is bolted through a gusset that offers some rotational restraint. Each of those changes the effective length, and the critical temperature goes as its inverse square.

The ends decide the length that matters. Four columns of identical height and section, buckling under four sets of end conditions. The effective length factor is the fraction of the column that behaves like a pin-ended one, and the buckling load goes as its inverse square.
Fig. 6 The four end conditions and the effective length each implies, from the same member. The critical temperature of a restrained heated member is π²i²/αL_eff², so moving from pinned ends to fixed ones multiplies it by four, and moving from pinned to a free top divides it by four. A sixteen-fold range in the temperature a member can take, from a detail nobody thought of as thermal.

Continuously welded rail takes the extreme version of this route. It is not free to expand — the whole point is to have no joints — and it does not buckle at thirteen degrees because it is held sideways along its entire length by ballast, so the wavelength it would have to buckle in is very short and the effective length correspondingly tiny. The rail is stressed to something like the full EαΔTE\alpha\Delta T and is prevented from going anywhere by a lateral restraint that has to be maintained: ballast that has been disturbed by track work is ballast whose restraint has gone, which is why speed restrictions follow maintenance in hot weather.

That is the same argument as a brace that need not be strong, arriving from an unexpected direction: what holds a hot rail down is a stiffness rather than a strength, and it is supplied by loose stones.

Thirty degrees from what?

Every ΔT\Delta T on this page has been quoted without a datum, and the datum is a construction event rather than a climate statistic.

A member is stress-free at the temperature it was at when its restraint was completed — when the weld cooled, when the concrete set, when the bolts were tightened, when the last brace went in. That temperature is the zero of everything above, and nothing about it is the mean of anything. It is whatever the weather was on the morning of one particular operation.

The consequence is that the effective temperature range is not the annual range but the annual range measured from the closing temperature, and closing a structure at one end of the range doubles the excursion at the other. A frame braced on a cold winter morning at 0 °C in a climate that reaches 35 °C sees a rise of 35 and a fall of 10; the same frame closed at 20 °C sees 15 and 30. Same building, same weather, same restraint, and the compressive stress that buckles things differs by a factor of more than two — decided by nothing but a date.

The industry that takes this most seriously is the one the previous section ended on. Continuously welded rail has a specified stress-free temperature — a “rail neutral temperature”, commonly around 27 °C in a British climate — and it is not left to the weather. The rail is deliberately tensioned or heated at installation until it is stress-free at the chosen value, so that the summer compression the whole system fears is measured from a datum near the top of the range rather than near the middle. Setting it low is what makes rail buckle; the neutral temperature can also drift downward over years as the rail creeps and the ballast moves, which is why it is measured and periodically restored.

Almost nothing else in construction is that careful, and the reason is not that it matters less but that it is nobody’s item. The stress-free temperature of a steel frame is not on the drawings, not in the specification, and not recorded anywhere after the fact — so a thermal calculation done to three figures is being applied to a range whose origin nobody wrote down.

Two practical residues. Where a structure has a genuine thermal sensitivity, specifying the closing temperature is available and cheap — “the bracing shall be finally bolted at a steel temperature between 10 and 20 °C” is a line of specification that halves a stress. And where it has not been specified, the honest design range runs from the coldest plausible closing to the hottest plausible service and back, which is wider than the climate.

And the structure is not at air temperature

The second correction runs the same way and is usually larger.

A dark roof in full sun reaches 30 °C or more above the air around it. An exposed steel member on a south elevation does something similar; a shaded one on the north face does not. A concrete core inside a building sits near the internal temperature all year while the columns outside it follow the weather, which is the differential that makes the top storey crack.

So the design ΔT\Delta T has three parts, and only one of them is meteorological: the air temperature range, the solar gain of the particular surface, and the difference between elements of the same structure that are in different environments. The third is the one that produces stress rather than movement, because it is a differential inside a restrained system rather than a uniform change the whole structure shares.

Every joint is an admission

The engineering response to all of this is to decide, deliberately, which of the two states the structure is in — and the movement joint is how the first is chosen.

A 60 m building frame in a 40 °C annual range moves αΔTL\alpha \Delta T L = 28.8 mm end to end, or ±14.4 mm about its middle. That is the number a joint has to accept, and it is why joint spacings in codes are quoted in metres rather than as a function of anything: the movement is proportional to length and the tolerable movement is a property of the joint hardware.

The alternative is to accept the stress. Both choices are legitimate, and the reason to be explicit about which was made is that a structure detailed as though it could move, and built so that it cannot, gets the stresses without anybody having computed them. That is the commonest way for a thermal problem to become a defect: a movement joint filled with mortar, a bearing seized with corrosion, a slab cast tight against a column it was meant to slide past.

And the gap itself is not a thermal calculation, which is the last thing this essay has to say about joints. The 28.8 mm above is one term in a sum, and the other terms are of the same size or larger: the shortening of a floor as it deflects, the shrinkage of the concrete, the sway of the frame under wind, and the tolerance on where the two faces of the joint were actually built. Only the first two are computed by anybody, and the largest of the five is the one that is not a structural quantity at all.

The gap is a sum of five things and only one of them is computed. What a 25 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.
Fig. 7 What a 25 mm movement joint is asked to accommodate, under three ways of combining the five terms. Adding every term at its extreme asks for 33.2 mm and assumes the hottest day, the fullest floor, the whole of the shrinkage and the worst-placed wall all arrive together, which has about a 1.5% chance of happening; treating them as independent asks for 16.0 mm. The ordering across the top bar is the finding: the largest term is tolerance at 10.0 mm and the smallest is deflection at 3.2 mm, which is a tenth of the total and the only one anybody computes carefully.

The thermal term is the second largest of the five and it is the only one on the list that reverses. Shrinkage runs one way and stops, tolerance is fixed on the day, deflection accumulates downwards — but the temperature term is ±\pm about its datum, so a joint sized for the summer closure has to open by the same amount in the winter, and a joint that has been packed solid with debris in July is a restrained member in January.

The number that makes it all small enough to live with

Steel’s coefficient of expansion is 12×10−612 \times 10^{-6} per °C and concrete’s is about 10×10−610 \times 10^{-6}, and the near-coincidence is the reason reinforced concrete works at all. A 20% difference in expansion coefficients would tear the two apart in the first summer; two parts in twelve is a stress of a few megapascals that the bond absorbs.

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.
Fig. 8 The mismatch strain a 40 °C change puts into seven pairs of materials that construction bonds together, which is the difference of their two coefficients times the temperature. Steel against concrete is 80 microstrain, 17 per cent of the larger of the two coefficients and by far the smallest pairing on the list; it puts 0.223 N/mm² of tension into the concrete, which is 7.7 per cent of the concrete’s tensile strength and 1.9 per cent of the 12 N/mm² a fully restrained member would have carried. Every other pair drawn is worse, and the same bar in aluminium is worse by a factor of two and a third.

Read down that list and the point is not that steel and concrete match well. It is that they are the only pair on it that does.

That coincidence is not a coincidence in any deep sense — it is a property of two particular materials that happens to hold — but it is worth naming because it is load-bearing for a whole construction industry, and because it fails for other pairings. Aluminium at 23×10−623 \times 10^{-6} against steel at 12 is a factor of two, so an aluminium facade on a steel frame must be detailed for differential movement; glass at 9 against an aluminium frame at 23 is the same problem, which is why a glazing gasket is a structural component.

The general form is the one to carry: wherever two materials are joined and restrained, the imposed strain is the difference of their expansions, and it produces stresses in proportion to how firmly they are attached — which is the component-stiffness reading of a joint applied to a material interface rather than to a bolt.

What the picture cannot show

The temperature is uniform through the member and constant along it. A real structure has a temperature field, varying by the hour, with the sun on one side; the uniform-plus-gradient decomposition used above is the first two terms of it, and it misses the local effects that crack things.

Nothing here creeps. A concrete member restrained against a slow temperature change relaxes a large part of the stress, so the elastic answer is a substantial over-estimate for anything sustained. Steel does not, at ordinary temperatures, which is why the same calculation is honest for one material and conservative for the other.

The restraint is drawn as a spring with a number on it. Estimating that number is the whole difficulty in practice: the restraint to a floor slab is the columns, the cores, the cladding and the friction of everything sitting on it, and it is known to perhaps a factor of three. Since the stress is proportional to the restrained fraction, so is the answer.

Where the ladder goes

The first rung is the composite case: two materials with different coefficients bonded together, which curves like a bimetallic strip and carries a self-equilibrating stress field with no external load at all.

The second is the one this essay skirted: time. Concrete’s shrinkage is an imposed strain of 300×10−6300 \times 10^{-6} — the same size as a 25 °C temperature change and permanent — and it arrives over years while creep is relaxing the stress it causes. The two effects are of opposite sign and comparable size, and getting the answer needs both.

The third is the design question the whole field turns on: whether to allow movement or resist it. It is one of the few decisions in structural engineering with no technical answer at all — both are correct, they lead to entirely different buildings, and the choice is made on grounds of maintenance, waterproofing and appearance rather than on anything computable.

Where this goes next

3 continuous spans against 3 simple ones. The bending moment in a continuous beam whose support 1 has settled by 0.01. Three curves: the moment the load causes, the moment the settlement causes on its own — dashed, peaking at 73.5, and in equilibrium with no applied load at all — and their sum, which is what the beam carries, peaking at 73.5 against 24.5 without the settlement. The settlement field is proportional to EI: a stiffer beam is punished harder for the same movement, which is the opposite of every intuition load-carrying gives. The support that moved A redundant structure knows things statics cannot see. Settle one support by ten millimetres and a complete set of bending moments appears — in equilibrium with no load at all, and larger for a stiffer beam. Where the peak moment goes when a restraint is added. The same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics. One support too many, and what it costs to know Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question. Which limit arrives first. Utilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry. Stiffness is not strength, and usually it is the one that governs A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all. 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. The column curve. Failure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it. Strong enough and still falls over A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.

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

The 8 essays that link to this one and share the most of its objects, of 33 that link here.

The objects this essay names

Each one links to every other essay that touches it.

BucklingExpansion jointIndeterminacyRestraintSelf-stressServiceabilityThermal gradientThermal movement