Module 6 · Lesson 6.2

Crack width and deflection

The two checks that decide whether a floor is a success — and the empirical rule that usually replaces one of them.

Why this matters

Nobody complains that their office floor has an inadequate moment of resistance. They complain that the doors stick, the partition has cracked, and there is a visible line across the soffit. Serviceability is what the client actually experiences, and it governs the design of most concrete floors. It deserves better than being treated as a box-ticking exercise at the end of a calculation.

By the end of this lesson you should be able to

Explain crack width as spacing times strain difference
Say why bar spacing matters more than bar area
Apply the span-to-depth rule and state what it is really doing
Explain the two modifications and when they apply
Decide when the rule is not good enough and a calculation is needed

What you should already know

The cracked elastic section (previous lesson)
Combinations of actions, especially quasi-permanent (Module 2)

Crack width is a geometric idea

Strip away the empirical dressing and the crack-width formula says something obvious.

Between two cracks, the steel stretches more than the concrete does — the steel is carrying tension while the concrete beside it is largely relaxed. That difference in extension has to go somewhere, and where it goes is into opening the cracks at each end.

So the width of one crack is the difference in strain between steel and concrete, accumulated over the distance between cracks:

Crack width

w_{k} = s_{r, ma x} (ε_{s m} - ε_{c m})

What it calculates: The characteristic crack width at the surface.

$s_{r, ma x}$: Maximum crack spacing (mm)
$ε_{s m}$: Mean strain in the reinforcement (—)
$ε_{c m}$: Mean strain in the concrete between cracks (—)

This assumes

Quasi-permanent loading — this is a long-term effect
The section is cracked and elastic
The crack spacing expression is EMPIRICAL, a regression on measured crack patterns

In plain terms: The equation itself is exact — it is a definition. Everything difficult is in the two ingredients, and they are of different kinds. The strain difference is mechanics with an empirical tension-stiffening correction; the crack spacing is a pure curve fit. Knowing which is which tells you which parts you can reason about and which you must simply apply.

Why spacing matters more than area

The crack spacing expression has the form

s_r,max = 3.4c + 0.425 k₁ k₂ φ / ρ_p,eff

and the two terms say two different things.

The cover term, 3.4c. A crack cannot be controlled by a bar it is far from. Deep cover means the bar's influence has dissipated by the time the crack reaches the surface, so cracks form further apart and open wider. Increasing cover for durability makes cracking worse — a genuine conflict between two serviceability requirements, and one you cannot resolve by adding steel.

The bar term, φ/ρ_p,eff. For a given area of steel, using smaller bars more closely spaced reduces this term. Each bar controls the concrete near it, so more bars means more, finer, better-distributed cracks.

Two beams with identical steel areas — one with 2 H32, one with 4 H20 — have nearly the same strength and quite different crack widths. The one with more, smaller bars cracks better.

This is why crack control is fundamentally a detailing problem rather than a calculation problem, and why the code offers deemed-to-satisfy tables of maximum bar diameter and maximum bar spacing as an alternative to the calculation. In most ordinary buildings those tables are what is actually used.

Worked example

Crack width in a beam at service

Given

300 × 500 beam, d = 450 mm, C30/37
3 H25 tension bars, A_s = 1470 mm²
Cover to the link 35 mm, link diameter 8 mm, so cover to the main bar c = 43 mm
Quasi-permanent moment M_qp = 70 kNm
Cracked section from the previous lesson: x = 136.7 mm, z = 404.4 mm, α_e = 6.09

Find

The calculated crack width.

Assumptions

k_t = 0.4 for long-term loading
k₁ = 0.8 for ribbed bars, k₂ = 0.5 for bending
Short-term modular ratio used, which is conservative for the strain difference

Deflection, and the rule that usually replaces it

Calculating deflection properly is hard. It requires the cracked stiffness, which varies along the span; tension stiffening between cracks; creep, which depends on age at loading, humidity and member thickness; and shrinkage curvature, which deflects a singly reinforced beam even with no load on it at all.

So the codes offer a shortcut: satisfy a span-to-effective-depth ratio and the deflection is deemed acceptable without calculation.

Where this stops working

What the span/depth rule actually is

It is a curve fit, calibrated so that a member satisfying it will usually deflect less than span/250, with the increment after construction less than span/500.

It bundles together stiffness, creep, steel stress and reinforcement ratio into one number. It is not a deflection calculation, it does not tell you the deflection, and it carries no information about your particular member beyond the few variables in it.

It should be treated as a screening rule. Where the answer actually matters — long spans, brittle finishes, transfer structures, cantilevers, anything where a client will notice — calculate the deflection instead. The rule's reputation for conservatism is undeserved: it can be unconservative for lightly loaded long spans, and markedly conservative for heavily reinforced short ones.

Worked example

Span/depth check on a 7 m beam — and what to do when it fails

Given

Simply supported beam, span 7.0 m, d = 450 mm, b = 300 mm
C30/37 concrete
Tension steel required by the bending design: 1250 mm²
Tension steel provided: 1470 mm² (3 H25)
No compression steel; no brittle finishes

Find

Whether the beam passes the span/depth check.

Assumptions

K = 1.0 for a simply supported beam
ρ is based on the steel PROVIDED
The expression is empirical, and the constants are nationally determined

Practice

A simply supported beam has d = 500 mm and a basic span/depth ratio of 16.5. What is the maximum span it may have, in metres, with no modification factors?

Practice

A cantilever with d = 400 mm has a basic span/depth ratio of 18. What is its maximum length, in metres? Take K = 0.4.

Practice

A beam has σ_s = 200 N/mm² at quasi-permanent load, cover to the main bar 40 mm, H20 bars, and ρ_p,eff = 0.04. What is the maximum crack spacing s_r,max, in mm?

Check yourself

Two beams have identical steel areas. Beam A has 2 H32; beam B has 5 H20 (slightly more area). Which cracks better, and why?

Summary

w_k = s_r,max × strain difference — the equation is a definition, the spacing is a curve fit
Cover dominates crack spacing: more cover means wider cracks
For a given steel area, smaller bars closer together crack better
Crack control is a detailing problem more than a calculation problem
The span/depth rule is a screening device, not a deflection calculation
Calculate deflection where it matters: long spans, brittle finishes, cantilevers, transfer structures
A design that only passes after chasing modification factors should be calculated properly

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint