# Calculation of crack widths `w`_{k}

_{k}

## Eurocode 2 part 1-1: Design of concrete structures 7.3 Crack control

The crack width, `w _{k}`, may be calculated as follows:

w = _{k}s⋅(_{r,max}ε - _{sm}ε) _{cm} | (7.8) |

where:

`s`_{r,max}- is the maximum crack spacing
`ε`_{sm}- is the mean strain in the reinforcement under the relevant combination of loads, including the effect of imposed deformations and taking into account the effects of tension stiffening
`ε`_{cm}- is the mean strain in the concrete between cracks.

*The difference of the mean strains ε_{sm} - ε_{cm}*

(7.9) |

where:

`σ`_{s}- is the stress in the tension reinforcement assuming a cracked section,

see application for a rectangular section or application for a T-section `E`_{s}- is the design value of the modulus of elasticity of the reinforcing steel, see § 3.2.7 (4)
`α`_{e}- is the ratio
`E`/_{s}`E`_{cm}with

`f`_{ct,eff}- is the mean value of the tensile strength of the concrete effective at the time when the cracks may first be expected to occur:

`f`=_{ct,eff}`f`or lower, (_{ctm}`f`(_{ctm}`t`)), if cracking is expected earlier than 28 days `ρ`_{p,eff}-
= ( `A`+_{s}`ξ`⋅_{1}`A'`)/_{p}`A`_{c,eff}(7.10) with

`A`_{s}- the cross sectional area of reinforcement
`A'`_{p}- the area of pre or post-tensioned tendons within
`A`_{c,eff} `A`_{c,eff}- the effective area of concrete in tension surrounding the reinforcement or prestressing tendons of depth,
`h`, where_{c,ef}`h`is the lesser of 2,5(_{c,ef}`h`-`d`), (`h`-`x`)/3 or`h`/2 (see Figure 7.1) `ξ`_{1}- the adjusted ratio of bond strength taking into account the different diameters of prestressing and reinforcing steel:
`ξ`=_{1}(7.5) with

`ξ`- the ratio of bond strength of prestressing and reinforcing steel, according to Table 6.2
`Φ`_{S}- the largest bar diameter of the reinforcing steel
`Φ`_{P}- the diameter or equivalent diameter of prestressing steel:

`Φ`= 1,6⋅√_{p}`A`_{P}for bundles, where`A`_{P}is the area of a prestressing steel,

`Φ`= 1,75⋅_{p}`Φ`for single 7 wire strands,_{wire}

`Φ`= 1,20⋅_{p}`Φ`for single 3 wire strands, where_{wire}`Φ`is the wire diameter._{wire}

`k`_{t}- is a factor dependent on the duration of the load:

`k`= 0,6 for short term loading,_{t}

`k`= 0,4 for long term loading._{t}

*The maximum crack spacing s_{r,max}*

• Where the bonded reinforcenlent is fixed at reasonably close centres within the tension zone (spacing ≤ 5(`c` + `Φ`/2), cf. Figure 7.2), the maximum crack spacing `s _{r,max}` may be calculated as follows:

= s_{r,max}k_{3}c + k_{1}k_{2}k_{4}Φ / ρ _{p,eff} | (7.11) |

where:

`Φ`- is the bar diameter. Where a mixture of bar diameters is used in a section, an equivalent diameter,
`Φ`, should be used._{eq} `c`- is the cover to the longitudinal reinforcement
`ρ`_{p,eff}- see
*the difference of the mean strains*above `k`_{1}- is a coefficient which takes account of the bond properties of the bonded reinforcement:

`k`= 0,8 for high bond bars,_{1}

`k`= 1,6 for bars with an effectively plain surface (e.g. prestressing tendons)._{1} `k`_{2}- is a coefficient which takes account of the distribution of strain:

`k`= 0,5 for bending,_{2}

`k`= 1,0 for pure tension._{2}

Intermediate values of`k`should be used for cases of eccentric tension or for local areas:_{2}`k`= (_{2}`ε`+_{1}`ε`)/(2_{2}`ε`)_{1}(7.13) where

`ε`is the greater and_{1}`ε`is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section._{2} `k`_{3}- is a Nationally Determined Parameter, see § 7.3.4 (3)
`k`_{4}- is a Nationally Determined Parameter, see § 7.3.4 (3).

• Where the spacing of the bonded reinforcement exceeds 5(`c` + `Φ`/2) (cf. Figure 7.2), or where there is no bonded reinforcement within the tension zone, the maximum crack spacing `s _{r,max}` may be calculated as follows:

s = 1,3(_{r,max}h - x) | (7.14) |

where:

`h`- is the overall depth of the section (see Figure 7.1)
`x`- is the neutral axis depth of the section (see Figure 7.1).

This application calculates
the crack width `w _{k}`
from your inputs.
Intermediate results will also be given.

First, change the following option if necessary: