Limitation of crack widths

Cracking shall be limited to an extent that will not impair the proper functioning or durability of the structure. The limit recommended values wmax for crack width are given, for relevant exposure classes and for quasi-permanent combinations, in the table 7.1 of EC2 :

0.4  mm  for classes X0, XC1

0.3  mm  for classes XC2,XC3,Xc4,XD1,XD2,XS1,XS2,XS3

The two value can be modified in Code and reinforcement options window.

Program controls cracking by means direct calculation of crack widths (§7.3.4 EC2).

The characteristic crack width wk may be obtained from the relation:

wk = sr,max (εsm - εcm)

where

sr,max        maximum crack spacing for final crack state according to eq. (7.11)

εsm        mean strain of reinforcement considering contribution of tension stiffening

εcm        mean strain in concrete between cracks

sr,max    = k3 ⋅ c + (k1⋅ k2 ⋅ k4 ⋅ Φ)/ ρeff

where

k3        factor (recommended value: 3.4)

c        concrete cover of longitudinal reinforcement

k1        factor to consider bond properties (0.8 for bars with high bond properties; 1.6 for bars with plain surface)

k2        factor to consider distribution of strain (0.5 for bending; 1.0 for tension only)

k4        factor (recommended value: 0.425)

Φ        is the diameter of tensile bars within Ac,eff

The setting of bond properties of bars k1 and of recommended values is possible in Code and reinforcement options window.

εsm - εcm = [σs - kt ⋅ fct,eff/ρeff ⋅ (1+αe ρeff)] /Es  ≥  0.6⋅ σs/Es

where

σs            stress in tension reinforcement under assumption of cracked section

kt        factor for creep of bond (kt = 0.6 for short-term loading (frequent comb.); kt = 0.4 for long-term loading (quasi-permanent comb.)

fct,eff        effective tensile strength of concrete at relevant point of concrete: in program is always equal to fctm

ρeff        = As/Ac,eff

As        total area of bars placed within Ac,eff

Ac,eff        effective tension area (surrounding the tension bars) of depth hc,ef, where hc,ef is the lesser of 2.5(h-d) , (h-x/3) or h/2 (see Figure 7.1 in EC2)

Program allows to calculate (see check box "Crack width (effective tension area)" in General data window) coefficient k2 with reference to the fibre of the area Ac,eff less tensioned and also with compression axial force, so to have:

k2 = (ε1+ε2)/2ε1                (7.14) EC2

where

ε1 is the greater and ε2 the lesser tensile strain at the boundary of Ac,eff in cracked section

The definition of Ac,eff is easy for predefined sections in uniaxial bending; but for biaxial bending or for general section the program implements the following criterion:

• the lengths 2.5(h-d) ,(h-x/3), h/2  are measured normally to neutral axis (so to have the effective inclined depth hc,eff )

• Ac,eff is the area obtained adding the concrete circular areas around all tensile bar within the depth hc,eff : the circle diameter is 7 times the diameter Φ of the correspondent bar.

The entire area of all section is discretized in little squares that are summed to give Ac,eff if their centre is within the depth hc,eff and within the above circle.

Minimum reinforcement areas check

If the section is in Low Ductility Class (only EC2 applies) program compares the areas As of bars included in the effective tension area  Ac,eff with the minimum tension area As,min specified by EC2:

As,min σs = kc k fctm Act                (7.1) EC2

where

Act        is the area of concrete in the tensile zone calculated just before formation of the first crack.

σs        is the absolute value of the max stress permitted in the reinforcement immediately after the formation of the crack

k        coefficient which allows for the effect of non uniform self-equilibrating stresses

= 1.0 for webs with h≤300 mm or flanges with widths less than 300 mm

= 0.65 for webs with h≥800 mm or flanges with widths greater than 800 mm

intermediate values may be interpolated

kc        coefficient which take account of the nature of stress distribution

= 1.0 for pure tension

= 0.4 [1-σc/(k1(h/h*)Fctm)] ≤ 1        for bending in webs of box section and T sections

= 0.9 Fcr / (Act fctm) ≥ 0.5                for flanges of box sections and T sections

σc        = NEd/bh        mean stress of the concrete acting in the part of the section under consideration

h*        = h        for h < 1.0 m

=1.0 m for h ≥ 1.0 m

k1        coefficient considering the effects of axial force on stresses

= 1.5 if NEd is  a compressive force

= 2h*/(3 h) if NEd is a tensile force

Fcr        absolute value of the tensile force within the flange immediately prior to cracking due to the cracking moment