A concrete slab is stressed even when no vehicle is on it. The sun heats its top surface while the ground keeps its underside cool, and the slab tries to bend — but its own weight will not let it. That restrained bending produces warping stresses, and they can rival the stresses from wheel loads.
This post covers where temperature stresses come from, how the slab behaves by day and by night, the warping stress formulas, and the critical stress combinations used in design.
Where Temperature Stresses Come From
Temperature stresses develop in cement concrete pavement due to variation in slab temperature and the resistance against deformation provided by the weight of the slab and friction between slab and ground.
Two ingredients are needed, and both are essential:
- A temperature change that makes the slab want to move
- A restraint that stops it moving freely
Remove either one and there is no stress. A free slab could expand, contract or curl as it pleased without any stress at all. It is the restraint — the slab’s own weight and the friction beneath it — that converts thermal movement into stress.
Two Types of Temperature Variation
| Variation | Effect | Stress Produced |
|---|---|---|
| (a) Daily variation | Results in a temperature gradient across the thickness of the slab | Warping stress |
| (b) Seasonal variation | Results in an overall change in slab temperature | Frictional stress |
The distinction is fundamental and worth fixing firmly.
Daily variation heats the top faster than the bottom, so the slab is warmer on one face than the other. There is a gradient through the thickness, and that makes the slab want to bend.
Seasonal variation changes the whole slab together — top and bottom alike. There is no gradient, so the slab does not want to bend. It wants to expand or contract as a whole, which is resisted by friction with the ground.
This post deals with the first. Frictional stress from seasonal variation is covered separately.
Warping: What the Slab Does by Day and by Night
Daily variation leads to warping of the slab. The temperature differential between top and bottom causes curling, that is, warping stress in the pavement.
Day Time
The slab tries to expand at the top, but its weight tries to resist it — hence compression develops at the top.
During the day the sun heats the surface. The top is hotter than the bottom, so the top fibre wants to grow longer. If it could, the slab would curl downwards at the edges like a dome. The slab’s weight holds the edges down, and that restraint puts the top into compression.
Night Time
The slab tries to contract at the top but is restrained by its weight — hence tension develops at the top.
At night the surface loses heat to the sky faster than the ground beneath cools. The top becomes colder than the bottom and wants to shrink, curling the edges upwards. Again the weight resists, and this time the top goes into tension.
Why Night Is More Dangerous
| Day Time | Night Time | |
|---|---|---|
| Top of slab is | Hotter | Colder |
| Top tries to | Expand | Contract |
| Restrained by | Weight of slab | Weight of slab |
| Stress at top | Compression | Tension |
Concrete is strong in compression and weak in tension. So although warping occurs in both cases, the night-time condition is the one that threatens cracking — and, as we shall see, it combines dangerously with corner loading.
Warping Stress Formulas
Interior
σinterior = (E α t / 2) × [ (Cx + μ Cy) / (1 − μ2) ]
Edge
σedge = maximum of ( Cx E α t / 2 , Cy E α t / 2 )
Corner
σcorner = [ E α t / (3 (1 − μ)) ] × √(a / l)
Notation
| Symbol | Meaning | Value |
|---|---|---|
| E | Modulus of elasticity of concrete | 3 × 105 kg/cm2 |
| α | Thermal expansion coefficient of concrete | — |
| t | Temperature differential between top and bottom of the slab | — |
| Cx, Cy | Coefficients depending on Lx/l and Ly/l respectively | — |
| l | Radius of relative stiffness | — |
| μ | Poisson’s ratio | 0.15 |
| a | Radius of contact | — |
These stresses are tensile stresses.
Reading the Formulas
All three share the group Eαt, and that combination has a clear meaning. If a piece of concrete of thermal coefficient α undergoes a temperature change t, its free strain would be αt. Prevent that strain entirely and the stress becomes Eαt. So Eαt is the stress that full restraint would produce, and each formula simply scales it by geometry.
Note also that Cx and Cy depend on the slab dimensions divided by l — so once again the radius of relative stiffness governs behaviour, exactly as it did for load stresses.
Critical Combination of Stresses
Load Stresses Ranked
- Corner stress is maximum, as there is discontinuity in both directions.
- Interior stress is minimum.
- Edge stress is intermediate.
Temperature Stresses Ranked
Temperature stress is critical at the edge and interior, and minimum at the corner.
Why is warping stress smallest at the corner? Because warping stress depends on restraint, and restraint comes from the weight of the slab holding it flat. At a corner, there is very little slab beyond the point in question, so the restraining weight is minimal — the corner is nearly free to curl. Little restraint means little stress.
Note the opposition: load stress is maximum at the corner while warping stress is minimum there. The two effects rank in opposite orders.
The Design Conclusion
In combination of wheel load and temperature, the edge region is most critical. Hence design is done using edge region stress, and checking is done for the corner region.
This makes sense once you see the ranking. The corner has the biggest load stress but almost no warping stress. The interior has big warping stress but the smallest load stress. The edge is the only position where both are substantial, so their sum is greatest there.
The Three Critical Combinations
1. Summer, Mid-day
σload, edge + σwarping, edge − σfriction
2. Winter, Mid-day
σload, edge + σwarping, edge + σfriction
3. Mid Night
The critical combination here is for the corner region:
σload, corner + σwarping, corner
Understanding the Signs
The frictional term changes sign between summer and winter, and the reason is physical.
- In summer, the slab expands and friction resists that expansion, producing compression. Compression relieves the tensile stresses, so it is subtracted.
- In winter, the slab contracts and friction resists that contraction, producing tension. Tension adds to the other tensile stresses, so it is added.
Consequently the second combination becomes less severe than the first in terms of net outcome, but both must be examined.
Why Mid Night Omits Friction
Frictional stresses are generally assumed constant along the length, but in reality they are zero at the ends and maximum at the centre of the slab.
Since the mid-night combination concerns the corner — which is at the end of the slab — the frictional stress there is effectively zero. That is why the seasonal variation term does not appear in the third combination.
Formula Summary
| Quantity | Expression |
|---|---|
| Interior warping stress | (Eαt/2)[(Cx + μCy)/(1 − μ2)] |
| Edge warping stress | max(CxEαt/2, CyEαt/2) |
| Corner warping stress | [Eαt/(3(1 − μ))]√(a/l) |
| Summer mid-day combination | σload,edge + σwarp,edge − σfriction |
| Winter mid-day combination | σload,edge + σwarp,edge + σfriction |
| Mid night combination | σload,corner + σwarp,corner |
Quick Revision Notes
- Temperature stress needs both a temperature change and a restraint.
- Daily variation gives a gradient across the thickness and causes warping. Seasonal variation changes the whole slab and causes frictional stress.
- Day time: top expands, weight resists, compression at top.
- Night time: top contracts, weight resists, tension at top.
- Warping stresses are tensile stresses, all containing the group Eαt.
- Cx and Cy depend on Lx/l and Ly/l.
- Load stress: corner maximum, interior minimum, edge intermediate.
- Temperature stress: critical at edge and interior, minimum at corner because restraining weight there is least.
- Design at the edge; check at the corner.
- Friction is subtracted in summer and added in winter.
- Frictional stress is zero at the ends and maximum at the centre, which is why the mid-night corner combination omits it.
Mistakes Students Commonly Make
- Saying tension develops at the top during the day. It is compression by day and tension at night.
- Assuming warping stress is greatest at the corner. It is minimum there, because restraint from slab weight is least.
- Designing for the corner. Design uses the edge; the corner is only checked.
- Adding friction in summer. It is subtracted, because expansion produces compression.
- Including friction in the mid-night corner combination. Frictional stress is zero at the ends.
- Confusing the two variations. Daily gives warping; seasonal gives frictional stress.
- Using μ2 in the corner formula. The corner denominator is 3(1 − μ), not (1 − μ2).
- Forgetting the square root over a/l in the corner warping formula.
Conclusion
A concrete slab is stressed by weather alone. Daily heating and cooling puts a temperature gradient through its thickness, so it tries to curl — and its own weight will not let it, which is what creates warping stress. By day the top is compressed, by night it is in tension, and the night condition is the dangerous one for concrete. Because load stress peaks at the corner while warping stress peaks at the interior and edge, the edge turns out to be where both are large together — which is why rigid pavements are designed at the edge and merely checked at the corner.
Frequently Asked Questions
Why do temperature stresses develop in concrete pavement?
Because of variation in slab temperature combined with resistance against deformation provided by the weight of the slab and friction between the slab and the ground. Without restraint, thermal movement would produce no stress.
What are the two types of temperature variation?
Daily variation, which produces a temperature gradient across the slab thickness and causes warping, and seasonal variation, which changes the overall slab temperature and causes frictional stress.
What happens to the slab during the day?
The top of the slab tries to expand but is resisted by the weight of the slab, so compression develops at the top.
What happens at night?
The top tries to contract but is restrained by the weight of the slab, so tension develops at the top.
What is warping or curling?
The bending of the slab caused by the temperature differential between its top and bottom surfaces, which produces warping stress in the pavement.
Why is warping stress minimum at the corner?
Because warping stress arises from restraint, and at the corner the restraining weight of the slab is least, so the corner is nearly free to curl.
Which region governs the design of a rigid pavement?
The edge region, because in combination of wheel load and temperature stress the edge is most critical. Checking is then carried out for the corner region.
Why is frictional stress subtracted in summer and added in winter?
Because in summer the slab expands and friction produces compression, which relieves tensile stress, while in winter the slab contracts and friction produces tension, which adds to it.
Why is friction omitted from the mid-night combination?
Because frictional stress is zero at the ends of the slab and maximum at the centre, and the mid-night critical combination applies at the corner, which is at the end.
Are warping stresses tensile or compressive?
The warping stresses given by the standard formulas are tensile stresses.