2.2.4 Crack Width Control

Crack Width Control

The limit diameter of reinforcing bars with max Ø_s is checked in accordance with EN 1992-1-1, clause 7.3.3 (2) as follows.

${\varnothing }_s =\hspace{0.17em}{\varnothing }_s^{*} · \frac{f_{ct,eff}}{2.9} · \frac{k_c · h_{cr}}{2 \left(h - d\right)} \text{for bending}$

${\varnothing }_s =\hspace{0.17em}{\varnothing }_s^{*} · \frac{f_{ct,eff}}{2.9} · \frac{h_{cr}}{8 \left(h - d\right)} \text{for uniformly distributed}\text{ tensile }\text{normal stresses}$

where

• Ø_s^* :  limit diameter according to Figure 2.3
• f_{ct, eff} : effective tensile strength of concrete at relevant point of time, in this case f_ctm
• k_c : factor for considering stress distribution in tension zone, see Chapter 2.2.3
• h_cr :  depth of tension zone immediately before cracking occurs
• h : overall depth of cross-section
• d : effective depth up to the centroid of outside reinforcement

The maximum rebar spacing max s_l is specified according to EN 1992-1-1, Table 7.3 (see Figure 2.4).

The characteristic crack width w_k is determined according to EN 1992-1-1, clause 7.3.4, Eq. (7.8).

$w_k = s_{r,max} · \left({\epsilon }_{sm} - {\epsilon }_{cm}\right)$

Equation 2.12 EN 1992-1-1, Eq. (7.8)

where

If the rebar spacing in the tension zone is not greater than 5 ⋅ (c + Ø/2), the maximum crack spacing for the final crack state may be determined as follows according to EN 1992-1-1, clause 7.3.4 (3):

$s_{r,\text{max}} = k_3 · c + \frac{k_1 · k_2 ·k_4 · \varnothing }{{\rho }_{p,\text{eff}}}$

Equation 2.13 EN 1992-1-1, Eq. (7.11)

where

If the spacing of rebars within the bond exceeds 5 ⋅ (c + Ø/2) or if there is no reinforcement within the bond in the tension zone, the following limit value of the crack width may be assumed:

$s_{r,\text{max}} = 1.3 · \left(h - x\right)$

Equation 2.14 EN 1992-1-1, Eq. (7.14)

Applying equations (7.11) and (7.14) are "optional" rules within the meaning of the Eurocode. Internal study of these two crack spacing equations has shown that the explicit differentiation when applying equation (7.14) to rebars with a larger spacing than 5 ⋅ (c + Ø/2) does not always lead to the desired crack width. We analyzed cross-sections with slightly different rebar spacings in the range of 5 ⋅ (c + Ø/2). For T-beam-like cross-sections and a bar spacing of 1.01 ⋅ [5 ⋅ (c + Ø/2)] using Eq. (7.14), the result was a smaller crack spacing than with Eq. (7.11) and a bar spacing of 0.99 ⋅ [5 ⋅ (c + Ø/2)]. This would mean that when you increase the reinforcement content, the crack width increases as soon as you fall below the limit value of the rebar spacing 5 ⋅ (c + Ø/2). To put it clearly: The calculated crack width in a zone without reinforcement is smaller than in a reinforced zone!

In the program, the crack spacing is calculated using equation (7.11) by default. Optionally, it is possible to activate s_r,max as the upper limit according to equation (7.14). As a result of the circumstance described above, the upper limit value is always taken into account, regardless of the available rebar spacing in the tension reinforcement.

The difference of the mean strain of concrete and reinforcing steel is determined as follows according to [1] 7.3.4 (2), Eq. (7.9).

${\epsilon }_{sm} - {\epsilon }_{cm} = \frac{{\sigma }_s - k_t · {\displaystyle \frac{f_{ct,\text{eff}}}{{\rho }_{p,\text{eff}}}} · \left(1 + {\alpha }_e · {\rho }_{p,\text{eff}}\right)}{E_s} \ge 0.6 · \frac{{\sigma }_s}{E_s}$

Equation 2.15 EN 1992-1-1, Eq. (7.9)

where

• σ_s : stress in tension reinforcement assuming a cracked cross-section
• k_t : factor for creep of bond
  • k_t = 0.6 for short-term loading
  • k_t = 0.4 for long-term loading
• f_ct,eff : effective tensile strength of concrete at relevant point of time (in this case f_ctm)
• α_e : ratio of moduli of elasticity E_s / E_cm
• ρ_eff : effective reinforcement ratio