Clause 7.3.4 The development of formulae for the prediction of crack widths given in clause 7.3.4 of
EN 1992-1-1 follows from the description of the cracking phenomenon given above. If it is assumed that all the extension occurring when a crack forms is accommodated in that crack, then, when all the cracks have formed, the crack width will be given by the following relationship, which is simply a statement of compatibility:
w = Srmεm (D8.14)
where w is crack width, Srmis the average crack spacing andεmis the average strain.
The average strain can be more rigorously stated to be equal to the strain in the reinforcement,
taking account of tension stiffening,εsm, less the average strain in the concrete at the surface,
Fig. 8.7. (a) Load-deformation response for a member subjected to steadily increasing deformation. (b) Crack width-deformation response for a member subjected to steadily increasing deformation
εcm. Since, in design, it is a maximum width of crack which is required rather than the average,
the final formula given in EN 1992-1-1 is
wk= Sr, max(εsm-εcm) (D8.15)
Since no crack can form within S0of an existing crack, this defines the minimum spacing of
the cracks. The maximum spacing is 2S0, since if a spacing existed wider than this, a further
crack could form. It follows that the average crack spacing will lie between S0and 2S0. It is
frequently assumed to be 1.5S0.
It is in the calculation of Srm that the most significant differences arise between the
formulae in national codes. The distance S0, and hence Srm, depends on the rate at which
stress can be transferred from the reinforcement, which is carrying all the force at a crack, to the concrete. This transfer is effected by bond stresses on the bar surface. If the bond stress is
assumed to be constant along the length S0and that the stress will just reach the tensile
strength of the concrete at a distance S0from a crack, then
τπφS0= Acfct
whereτ is the bond stress, Acis the area of concrete, fctis the tensile strength of concrete and
φ is the bar diameter.
Takingρ = πφ2/4A
cand substituting for Acgives
S0= fctπφ/4ρ
From this,
Srm= 0.25kφ/ρ
where k is a constant depending on the bond characteristics of the reinforcement. This is the oldest form of relationship for the prediction of crack spacings. More recent studies have shown that the cover also has a significant influence, and that a better agreement with test results is obtained from an equation of the form
Srm= kc + 0.25k1φ/ρ
where c is the cover. This formula has been derived for members subject to pure tension. In
order to be able to apply it to bending, it is necessary to introduce a further coefficient, k2,
and to define an effective reinforcement ratio,ρeff. These modifications take account of the
different form of the stress distribution within the tension zone and the fact that only part of
the section is in tension. k2 andρeff can be derived empirically from tests. The resulting
formula is
Srm= 2c + 0.25k1k2φ/ρeff (D8.16)
Here, k1is a coefficient taking account of the bond properties of the reinforcement. A value
of 0.8 is taken for high bond bars and 1.6 for smooth bars. k2is a coefficient depending on the
form of the stress distribution. A value of 0.5 is taken for bending and 1 for pure tension. Intermediate values can be obtained from
k2= (ε1+ε2)/2ε1 (D8.17)
where ε1and ε2are, respectively, the greater and lesser tensile strains at the faces of the
member. ρeff is the effective reinforcement ratio, where As is the area of tension
reinforcement contained within the effective area of concrete in tension, Ac,eff. This is the
area of concrete in tension surrounding the reinforcement of a depth equal to 2.5 times the distance from the tension face of the member to the centroid of the tension reinforcement. A figure in the code gives definitions for other, less typical cases.
In design, it is not the average crack width which is required but a value which is unlikely to be exceeded. EN 1992-1-1 uses the characteristic crack width, which is defined as a width with a 5% probability of exceedance. It is found experimentally that a reasonable estimate of the characteristic width is obtained if the maximum crack spacing is assumed to be 1.7 times
the average value. In EN 1992-1-1, therefore, the maximum spacing is used, and this is assumed to be given by
Sr, max= 3.4c + 0.425k1k2φ/ρeff (D8.18)
The other parameter in the crack width equation is the average strain,εsm-εcm. This is
obtained from equation (7.9) in EN 1992-1-1, and is repeated below for convenience:
εsm-εcm=σs/Es- ktfct, eff(1 +αeρp, eff)/Esρp, eff≥ 0.6σs/Es (D8.19)
whereσsis the stress in the tension reinforcement calculated assuming a cracked section,αe
is the modular ratio (Es/Ec), and ktis a factor depending on the duration of the load (0.6 for
short-term loads and 0.4 for long-term loads).
There remains a situation where the formula described above can lead to a significant overestimate of the likely cracking. The reason for this may be understood by considering the element shown in Fig. 8.5. This is an unreinforced element subjected to an axial load applied at an eccentricity sufficiently large for part of the section to be in tension. If the load is sufficient, the section will crack. The formation of this crack will not result in failure but merely a redistribution of stresses locally to the crack. Some distance away from this crack the stresses will remain unaffected by the crack. It is found that the crack affects the stress distribution within a distance roughly equal to the height of the crack on either side of the crack. Thus, by the same arguments used earlier, the spacing of the cracks can eventually be expected to be between the crack height and twice the crack height. This leads to the following relationship:
Srm= h - x (D8.20)
Clause 4.4.2.4(8) where h is the overall depth of the section and x is the depth of the neutral axis. This formula
applies not only to members subject to axial compression but also to any situation where the cracks, when they form, do not pass right through the section. The effect of bonded reinforcement in the section is almost always to give a calculated crack spacing much smaller than given by the above equation. The equation does, however, give a maximum, or limiting, value for the spacing, and there are a number of practical situations where it can be used with advantage. One particular case is that of a prestressed beam without any bonded normal reinforcement. The bond of prestressing tendons or wires is often much inferior to that on normal high bond reinforcement. A safe estimate of the crack spacing, and hence the crack width, can be made by treating the prestress as an external load, calculating the depth of the tension zone under the loading considered and applying the above formula. Clause 4.4.2.4(8) permits this procedure.