When the whole slab warms or cools together, it wants to grow or shrink as one piece. The ground beneath will not let it slide freely, and the resulting drag produces frictional stress.
The engineering answer is to cut the slab into shorter pieces with joints. This post covers the frictional stress formula and the design of expansion and contraction joints.
Frictional Stress
Frictional stresses are developed due to seasonal variation of temperature. In this case there is no temperature gradient across the thickness.
This is the essential contrast with warping stress. Daily variation heats the top more than the bottom, producing a gradient and hence bending. Seasonal variation changes the entire slab uniformly — no gradient, no bending. Instead the slab simply wants to get longer or shorter.
If the slab tries to contract due to a temperature fall, as in winter, it tries to move inward, and friction with the ground resists that movement.
The Derivation
Consider half the slab length. The frictional force resisting movement equals the weight of that half multiplied by the coefficient of friction. Equating that to the force carried by the slab cross-section:
B × (L/2) × h × γconc × f = B × h × σ
Cancelling B and h from both sides:
σ = f γconc L / 2
Why L/2 and Not L
The slab tries to contract from both sides, hence half the length of the slab is taken.
This is the point most often missed. A slab contracting towards its own centre has two halves, each moving inward from its own end. The maximum accumulated friction is felt at the middle, and it is generated by only half the slab length pulling in from each side.
Note also the consequence: during summer, compression develops in the slab instead, since expansion is resisted the other way.
Types of Joints in Cement Concrete Pavement
- Expansion joint
- Contraction joint
- Construction joint
Expansion Joint
The purpose is to allow expansion of the pavement due to rise in temperature with respect to the construction temperature.
Design Data
| Item | Value |
|---|---|
| Maximum joint thickness specified by IRC | 2.5 cm |
| Load transfer | Dowel bars, developing bending, bearing and shearing stresses |
| Filler compression assumed | 50 % of its thickness during expansion |
The Half-Compression Rule
The fillers provided at the expansion joint are assumed to be compressed by 50 percent of their thickness during expansion. Hence the gap of the joint should be twice the expansion in concrete.
If the filler has original thickness δ, expansion of the slab compresses it to a maximum of δ/2. So:
L α ΔT = δ / 2 → L = δ / (2 α ΔT)
| Symbol | Meaning |
|---|---|
| L | Maximum spacing between expansion joints |
| δ | Gap of the expansion joint |
| α | Coefficient of thermal expansion |
| ΔT | Rise in temperature |
The left side, LαΔT, is simply the free expansion of a slab of length L. Setting it equal to δ/2 rather than δ is what builds in the 50 percent filler compression allowance.
Contraction Joint
Provided to control cracks due to shrinkage and moisture variation.
The Idea Behind It
To regulate the crack — that is, to ensure that the crack forms at a predetermined location — the slab is weakened at certain intervals. Those locations are called contraction joints.
This is an unusual and rather elegant piece of engineering. The designer accepts that the slab will crack, and rather than trying to prevent it, deliberately weakens the slab at chosen points so the crack appears where it can be sealed and maintained instead of wandering randomly across the pavement.
Why Cracking Occurs
- During the initial curing period, shrinkage occurs in the concrete, and if resisted, tensile stress develops in the slab.
- Fall of temperature will also develop tensile stress in the slab.
Case A: No Reinforcement Provided
Equating the frictional force over half the slab length to the tensile capacity of the concrete section:
σ = f × γconc × A × (L/2) ÷ A = f L γconc / 2
L = 2σ / (f γconc)
| Symbol | Meaning | Typical Value |
|---|---|---|
| σ | Permissible tensile stress in concrete | 0.8 kg/cm2 if not given |
| f | Coefficient of friction between concrete and base | 1.5 |
| L | Spacing between contraction joints | — |
| γconc | Unit weight of concrete | 24 kN/m3 |
When reinforcement is not provided, the maximum spacing between contraction joints is taken as 4.5 m.
Case B: Reinforcement Provided
When reinforcement is provided in the slab, it is assumed that all tension is taken by the reinforcing steel.
σst Ast = B × (L/2) × h × γconc × f
L = 2 σst Ast / (B h γconc f)
| Symbol | Meaning |
|---|---|
| σst | Permissible tensile stress in steel |
| Ast | Area of steel in the complete width of slab |
| f | Coefficient of friction |
| B | Width of slab |
| h | Thickness of slab |
Comparing the Two Cases
Both formulas have the same shape — tensile capacity divided by frictional demand per unit length. What changes is where the capacity comes from:
| Without Reinforcement | With Reinforcement | |
|---|---|---|
| Tension carried by | Concrete | Steel alone |
| Formula | L = 2σ/(fγconc) | L = 2σstAst/(Bhγconcf) |
| Typical result | Short spacing, max 4.5 m | Longer spacing possible |
Steel is far stronger in tension than concrete, so reinforcing the slab allows a much greater joint spacing — which is exactly why reinforced concrete pavements have fewer joints.
Construction Joint
The third type, provided where construction work stops and later resumes — for example at the end of a day’s paving.
All Three Joints Compared
| Joint | Purpose | Governed By |
|---|---|---|
| Expansion | Allow expansion due to temperature rise above construction temperature | δ, α and ΔT |
| Contraction | Control cracks due to shrinkage and moisture variation | Tensile capacity versus friction |
| Construction | Provided where construction work is interrupted | Construction sequence |
Formula Summary
| Quantity | Expression or Value |
|---|---|
| Frictional stress | σ = f γconc L / 2 |
| Expansion joint spacing | L = δ / (2αΔT) |
| Maximum expansion joint gap | 2.5 cm (IRC) |
| Filler compression assumed | 50 % of thickness |
| Contraction joint, no reinforcement | L = 2σ/(fγconc) |
| Contraction joint, with reinforcement | L = 2σstAst/(Bhγconcf) |
| Max contraction joint spacing, unreinforced | 4.5 m |
| Coefficient of friction f | 1.5 |
| Unit weight of concrete | 24 kN/m3 |
| Permissible tensile stress in concrete | 0.8 kg/cm2 if not given |
Quick Revision Notes
- Frictional stress arises from seasonal variation, with no temperature gradient across the thickness.
- σ = fγconcL/2, where L/2 is used because the slab contracts from both sides.
- Summer produces compression; winter contraction produces tension.
- Three joint types: expansion, contraction, construction.
- Expansion joint maximum thickness = 2.5 cm as specified by IRC.
- Dowel bars at expansion joints develop bending, bearing and shearing stresses.
- Filler is assumed to compress by 50 %, so the joint gap is twice the expansion.
- L = δ/(2αΔT) for expansion joint spacing.
- Contraction joints weaken the slab deliberately so cracks form at predetermined locations.
- Without reinforcement, L = 2σ/(fγconc), with maximum spacing 4.5 m.
- With reinforcement, all tension is taken by steel: L = 2σstAst/(Bhγconcf).
- Standard values: f = 1.5, γconc = 24 kN/m3, σ = 0.8 kg/cm2 if not given.
Mistakes Students Commonly Make
- Using the full length L instead of L/2 in the frictional stress formula. The slab contracts from both sides.
- Setting LαΔT equal to δ instead of δ/2, which ignores the 50 percent filler compression allowance.
- Confusing the two contraction joint formulas. Without steel the capacity comes from concrete; with steel it comes from σstAst.
- Forgetting the 4.5 m maximum for unreinforced contraction joints.
- Thinking contraction joints prevent cracking. They regulate it, ensuring cracks form at chosen locations.
- Confusing frictional stress with warping stress. Seasonal variation gives friction; daily variation gives warping.
- Assuming frictional stress is uniform along the slab. It is zero at the ends and maximum at the centre.
- Mixing up Ast with area per metre. It is the area of steel in the complete width of the slab.
Conclusion
Seasonal temperature change makes the whole slab want to grow or shrink, and friction with the ground resists it — producing a stress that grows with slab length. Joints are the remedy. Expansion joints leave a gap for the slab to grow into, sized on the assumption that the filler compresses by half. Contraction joints deliberately weaken the slab so that shrinkage cracks appear where the designer chooses. And reinforcement, by carrying all the tension in steel, allows those joints to be spaced much further apart.
Frequently Asked Questions
What causes frictional stress in a rigid pavement?
Seasonal variation of temperature, which changes the overall slab temperature without producing any gradient across the thickness, combined with friction between the slab and the ground resisting the resulting expansion or contraction.
What is the frictional stress formula?
σ = fγconcL/2, where f is the coefficient of friction, γconc is the unit weight of concrete and L is the slab length.
Why is half the slab length used?
Because the slab tries to contract from both sides towards its centre, so only half the length contributes to the friction accumulated at the critical section.
What are the three types of joint in cement concrete pavement?
Expansion joints, contraction joints and construction joints.
What is the purpose of an expansion joint?
To allow the expansion of the pavement caused by a rise in temperature relative to the construction temperature.
What is the maximum expansion joint thickness?
2.5 cm, as specified by IRC.
Why is the joint gap twice the expansion?
Because the filler at the joint is assumed to compress by only 50 percent of its thickness during expansion, so the gap must be twice the expected movement.
What is the expansion joint spacing formula?
L = δ/(2αΔT), where δ is the joint gap, α the coefficient of thermal expansion and ΔT the rise in temperature.
What is the purpose of a contraction joint?
To control cracks due to shrinkage and moisture variation, by weakening the slab at chosen intervals so that cracks form at predetermined locations.
What is the maximum contraction joint spacing without reinforcement?
4.5 m.
How does reinforcement change contraction joint spacing?
When reinforcement is provided, all tension is assumed to be taken by the steel, giving L = 2σstAst/(Bhγconcf). Since steel is much stronger in tension than concrete, a considerably larger spacing becomes possible.