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.

17 min read 7 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 neitherA 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, 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.free at one end10.8 mmno stressheld at both ends75.6 MPaheld by a spring of 100 kN/mm53.2 MPaE·α·ΔT = 210000 × 12×10⁻⁶ × 30 = 75.6 MPa, at every length
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 lengthStress 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.0102030405060020406080length of the member (m)stress (MPa)fully held: 75.6 MPaheld by a springfree movementto 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 alongA 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.free: 50.6 mm of camber, no stressheld: 33 kNm and ±18.9 MPa, no movementκ = α·ΔT/d = 0.450 × 10⁻⁶ per mm either way
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 curveFailure 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.5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither
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 neitherA 30 m steel member -25 °C warmer than it was built, in three conditions. Free, it grows -9.0 mm and carries nothing. Held, it moves nothing and carries -63.0 MPa — 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.free at one end-9.0 mmno stressheld at both endsno stressheld by a spring of 100 kN/mmno stressE·α·ΔT = 210000 × 12×10⁻⁶ × -25 = -63.0 MPa, at every length
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 mattersFour 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.K = 0.5both ends fixedK = 0.7one fixed, one pinnedK = 1both ends pinnedK = 2fixed at the base, free at the topsame column, same section, four ways of holding the endsthe load at which each buckles goes as 1 ÷ K² — a factor of sixteen across this row
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.

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.

One support too manyThe 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.simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment
Fig. 7 A beam with restraints added one at a time. Each additional restraint reduces the deflection and raises the moment — for applied load. For imposed strain the same restraints do something else entirely: they convert movement into force, and a structure with more restraints has more thermal stress and less thermal movement, with nothing to show for the trade.

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.

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

this support 0.01 lowmoment19.6 sagging24.5 hogging30.6 if the spans were simplereactions 14.0 38.5 38.5 14.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not 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. simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment 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. 0.60.811.21.41.61.8200.511.5span, relative to the firstthey cross heredeflection runs out at 1.40strength runs out at 1.54the limitstrengthdeflection 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. 1 d10 d100 d2.7 yr27 yr0246810time since the strain was imposedstress, N/mm²E × strain: 9.6tensile strength: 3.5what is left: 2.32the one-line shortcut at χ = 0.8: 3.31 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. 5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither 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.

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BucklingExpansion jointIndeterminacyRestraintSelf stressServiceabilityThermal gradientThermal movement