Crack width and deflection of an RC beam
Strength checks show a member will not fail; the serviceability checks show it will look and work right: cracks narrow enough and sag small enough. Once concrete cracks, only the compression zone and the steel resist bending, so the member gets softer. Creep and shrinkage make it softer still over time. This calculator works out the real stiffness and the real crack width, instead of using a span/depth rule. Every number below comes from the free STATIX crack width and deflection calculator with the inputs in step 1.
1. The problem
| Input | Value |
|---|---|
| Design code | SANS 10100 (see note) |
| Support / load shape | Simply supported, UDL |
| Span L | 6 m |
| Dead Gk | 15 kN/m |
| Live Qk | 12 kN/m |
| Dead point Gp | 40 kN |
| Live point Qp | 30 kN |
| Occupancy ψ2 | Residential / office (0.3) |
| ψ2 (custom) | 0.3 |
| Shape | Rectangle |
| Width b | 300 mm |
| Depth h | 600 mm |
| Effective d | 550 mm |
| Cover c | 40 mm |
| Bar φ | 20 mm |
| Number n | 3 |
| Grade fck | 30 MPa |
| wmax (custom) | 0.3 mm |
| SLS combination | Characteristic (Gk+Qk) |
| Strength basis | Cylinder fck |
| Creep φ(∞,t0) | 2.0 |
2. Materials over time
Concrete creeps under a sustained load, so its long-term modulus is lower: E_c,eff = E_c / (1 + φ). The steel is that many times stiffer than the concrete (the modular ratio), more so in the long term.
3. Service moment and cracking
Deflection and cracking are checked under the service load: on the SANS and BS routes the full service load (G_k + Q_k), as SANS 10100-1 Annex A and BS 8110-2 take it; on the Eurocode route the quasi-permanent load (the part that is on most of the time). The section cracks when the moment passes the cracking moment M_cr.
4. The cracked section
In a cracked section the concrete below the neutral axis is ignored. The neutral axis depth x and the cracked second moment of area I_cr come from the transformed section (the steel counted as concrete × the modular ratio). Between cracks the concrete still helps (tension stiffening): the real stiffness lies between cracked and uncracked, by the factor ζ.
5. Deflection
The curvature (from the moment and the stiffness) is integrated along the span; shrinkage adds its own curvature because the steel restrains it. The total must stay below span/250, and the part after the partitions go in below span/500.
6. Crack width, BS 8110-2 method
The strain at the level of the crack, less the help of the concrete between cracks, times a distance that depends on how far the point is from the nearest bar (a_cr), gives the crack width.
7. Crack width, EN 1992-1-1 7.3.4
The mean strain difference between steel and concrete times the maximum crack spacing gives w_k. The crack spacing depends on the cover, the bar size and how much steel sits in the effective tension area around the bars.
The calc sheet shows each formula with its clause; the support conditions change the moment and the deflection coefficient.
Open the crack width and deflection calculator
A worked example for learning, not a design. Check every value against the current code and your own project, and have the design checked and signed by a registered professional engineer.
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