The stress nobody restrained

Every thermal argument in this collection so far has been about restraint. Heat a member and it wants to grow; stop it and it pushes; let it go and nothing happens. The whole subject seemed to be a question about bearings.

It is not, and the reason is the assumption that runs under every other page here. A section is required to stay plane. The temperature is under no such obligation.

Where the two disagree, the material is stressed — and on a concrete deck with the sun on it the disagreement is worth four newtons per square millimetre, in a member nobody is holding.

Which free body produced the number #

Take the cross-section as the free body, with no external force on it at all.

The temperature at height $y$ is $T(y)$, so the material there would like to strain by $\alpha T(y)$. It is only allowed to strain by $\varepsilon_0 + \kappa y$. The stress is $E$ times the difference:

$$\sigma(y) = E\left(\varepsilon_0 + \kappa y - \alpha T(y)\right)$$

and $\varepsilon_0$ and $\kappa$ are whatever they have to be for that stress to have no resultant. Two conditions, two unknowns:

$$\varepsilon_0 = \frac{1}{A}\int \alpha T\,dA \qquad \kappa = \frac{1}{I}\int \alpha T\, y\,dA$$

which is to say: the mean is the axial strain and the first moment is the curvature, and everything left over is stress.

For the deck drawn — 1.4 m deep, a 3 m slab on a 400 mm web on a bottom flange, with 18 °C at the top face falling away over the top 140 mm — that gives $\varepsilon_0 = 32$ microstrain and $\kappa = 6.3 \times 10^{-5}$ per metre. The residual stress field reaches 3.98 N/mm² of compression at the top face and 1.83 of tension 140 mm below it.

Its resultant force is $8 \times 10^{-14}$ kN and its resultant moment $3 \times 10^{-12}$ kNm. Those are not small numbers; they are zero, and the fact that they are zero is exactly why no equilibrium check anybody could make on the deck would ever find this stress.

The linear profile is the exception, and it is the one everybody uses #

Put a straight line into the integrals above. A linear profile is $\varepsilon_0 + \kappa y$ divided by $\alpha$ for some $\varepsilon_0$ and $\kappa$ — so the plane section can follow it exactly, the difference is zero at every fibre, and the stress is zero everywhere.

Measured rather than asserted: with a linear profile the peak self-stress computes to $10^{-14}$ N/mm², which is a double-precision zero.

That is why the subject has been about restraint until now. Uniform and linear are the two profiles under which a free member carries no stress, and they are the two profiles every simple treatment uses. Neither is what the sun does.

Two temperature fields no analysis can tell apart #

Here is the sharpest way to put it.

Take the real profile and compute its $\varepsilon_0$ and $\kappa$. Now construct the straight line with exactly those two: for this deck it runs from +6.28 °C at the top to −2.57 °C at the bottom.

The two profiles produce:

Every quantity a structural analysis computes is identical. The deck moves the same distance, bends by the same amount, pushes on its bearings with the same force and, if it is continuous, generates the same moment over its piers. And one of them cracks the deck soffit and the other does not.

The stress is a measure of how far the temperature profile is from being a straight line, and of nothing else.

Why the profile is never straight #

Concrete conducts heat badly and stores a great deal of it. The sun heats the top surface over a few hours; the heat diffuses downward at a rate set by the thermal diffusivity of concrete, which is about 0.05 m² per day; and over an afternoon it penetrates a few hundred millimetres.

So the profile through a 1.4 m deck at four o’clock is not a gradient. It is a hot skin over a body that has not yet noticed, and the thinner the skin the further the profile is from a straight line.

The night-time case is the mirror image and is often worse. The deck radiates to a clear sky, the surface cools below the body, the sign of everything reverses — tension at the top face, where the deck slab is already in tension over a pier, and the two add.

The section’s own shape decides how much #

The two integrals that fix $\varepsilon_0$ and $\kappa$ are weighted by width, so a section whose material is not evenly distributed through its depth gets a different answer from a rectangle of the same depth under the same profile.

That matters more than it sounds. The deck here has 3,000 mm of slab in its top 200 mm and 400 mm of web underneath, so the hot layer coincides almost exactly with the widest part of the section — the mean is dragged up and the curvature with it, and both are much larger than a rectangle of the same depth would have given. A box girder with a thin top slab and a heavy bottom flange behaves the other way round.

There is no shortcut here and no coefficient. The profile has to be integrated over the section that exists.

Where it actually matters #

Four newtons per square millimetre is not a strength problem. On a deck whose concrete has a cylinder strength of 40 it is a tenth of the compression capacity, and no member has ever failed because of it.

It is a cracking problem, and cracking problems are decided by tensile strength, which is about 3 N/mm² for the same concrete. A tension of 1.8 from the thermal field alone is more than half of it, and it is a bonus term added to whatever the load is already doing.

Three places it bites:

The soffit of a deck slab, at night. The cooling profile puts tension at the top and compression at the bottom in the slab, but the same profile through a box girder puts tension in the bottom flange of the box — which is where a prestressed deck’s designer has spent his whole budget keeping the tension out.

The web of a deep girder. The self-equilibrating field is largest where the profile is most curved, which for a heating case is a few hundred millimetres below the top face. On a box girder that is inside the web, where there is generally the least reinforcement.

Anywhere the section changes. The field depends on the section’s own $A$ and $I$, so a deck that widens or a web that thickens has a different self-stress field in each part, and the transition between them is a region this model says nothing about.

The history that made it a load case #

Self-equilibrating thermal stress was not discovered by anybody looking for it. It arrived with the first generation of long prestressed concrete box girders in the nineteen-sixties, as cracks in places where the calculations said there was compression to spare — soffits of spans that had never carried more than their own weight, webs of boxes at mid-height, the tops of decks over piers.

The measurements that settled it were made by instrumenting real bridges and reading the strain gauges through a day, which is how the standard profiles now in the codes came to be shaped the way they are: a hot skin over a body, with a kink in it, rather than a gradient. The shape is a summary of thermocouple readings and not a solution to anything.

What is worth carrying from that is the order in which it was understood. The stress came first, as cracks; the mechanism came second; and the reason it took so long is that every tool available for finding it — a moment diagram, a reaction, an equilibrium check on any free body at all — is blind to a field with no resultant.

Restraint is a separate question, and it adds #

Nothing above needed a bearing. If the deck is restrained — continuous over piers, or built into abutments — the curvature $\kappa$ is resisted, and a restraint moment $EI\kappa$ appears on top of everything else.

For this deck $EI$ is 8.9 × 10⁶ kNm², so full restraint of the curvature would give 566 kNm and continuity over one pier of a long viaduct gives about half of it — 283 kNm, and stresses of $\pm 0.75$ N/mm² at the extreme fibres.

The important part is that the two effects are additive and independent. The self-equilibrating field depends only on the shape of the profile; the restraint moment depends only on its first moment and on the boundary conditions. A designer who computes one and calls it the thermal effect has computed half the problem, and which half depends on which textbook was open.

Where the model stops #

The temperature profile is an input. Getting it is a heat-conduction problem with the sun, the sky, the wind and the surfacing in it, and codes supply standard profiles precisely because nobody wants to solve it. Every number on this page is downstream of a curve somebody else drew.

The material is linear and its modulus is one number. Concrete’s is not, and worse, the stress here is applied slowly and sustained for hours — so creep relieves a good part of it, and the peak that a linear elastic calculation gives is an upper bound on what is actually there after an afternoon.

And the profile varies only through the depth. A box girder is hotter on its south web than its north one, which puts a horizontal curvature and a twist into the same section and needs a two-dimensional version of every integral above.

What the pictures cannot show #

The stress block is drawn as a static shape. It is a moving one: the profile changes hour by hour, so the field peaks, reverses overnight and does it again tomorrow. It is a cyclic action, which puts it in the same family as fatigue rather than in the family of static load cases, and the number that governs is a range rather than a value.

Nor can they show that it is invisible. A stress field with no resultant does not appear in any bending-moment diagram, any shear diagram, or any reaction. Every figure on this site that draws a load path is drawing something this field is not on.

The assumption the figures rest on #

Plane sections, and only plane sections. Take that away — allow the section to warp out of its own plane — and there is no constraint left for the temperature to violate, and no self-stress.

That is not an idle observation. A very deep member, or one whose profile varies rapidly along the span, does not obey plane sections, and the real self-stress in it is smaller than these integrals say. So is the stress near a free end, where the field has to die out over a Saint-Venant length: the end face carries no stress at all, the full field exists a depth or so in, and the transition between them is a region none of this describes.

The practical form of that observation is worth having. The self-stress field is largest where the member is most beam-like, which is the middle of a long uniform span, and it fades wherever the assumptions this site is built on fade.

The ladder from here #

Later rungs on this anchor: the cooling profile and why it is the governing one for a prestressed deck. The heat-conduction problem that produces the profile in the first place, and how much of it a surfacing layer changes. Creep relief of a sustained self-stress, which is the same calculation as relaxation in a tendon. Differential temperature between two materials in a composite section, where the mismatch is in $\alpha$ rather than in $T$. Two-dimensional profiles in a box girder, with a transverse curvature and a warping stress. And the general statement this is one case of — an eigenstress, the stress left by any imposed strain field a plane section cannot follow, which covers shrinkage gradients, differential creep and welding residuals with the same three integrals.