2.6.4.12 Check of crack width

Check of crack width

The calculation value w_k of the crack width is determined according to Equation (7.8) of EN 1992-1-1, clause 7.3.4.

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

where

Table 2.2

sr,max	maximum crack spacing in final crack state (see Equation 2.74 or Equation 2.75)
εsm	mean strain of the reinforcement under governing action combination, including the effects of applied deformations and taking the concrete's effect of tension between the cracks into account (only the additional concrete tensile strain beyond the zero strain at the same level is considered)
εcm	mean strain of concrete between cracks

If the spacing of the rebars in the bonded reinforcement is not larger than 5 ⋅ (c + φ/2) in the tension zone, the maximum crack spacing for the final crack state may be determined according to EN 1992-1-1, Equation (7.11):

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

If the spacing of the rebars in the bonded reinforcement exceeds 5 ⋅ (c + φ/2) in the tension zone or if no bonded reinforcement is available within the tension zone, the limit for the crack width may be determined with the following maximum crack spacing:

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

The depth of the compression zone x in state II therefore has to be calculated for the check of the crack width. It is determined with the neutral axis depth ξ that is related to the depth of the structural element.

$x = \xi · h = \frac{0.5 + {\alpha }_e · {\displaystyle \frac{a_{s,\text{exist}}}{b · h}} · {\displaystyle \frac{d}{h}}}{1.0 + {\alpha }_e · {\displaystyle \frac{a_{s,\text{exist}}}{b · h}}}$

Furthermore, the maximum crack spacing is analyzed according to EN 1992-1-1, Equation (7.15):

$s_{\mathrm{r},\text{max}}=\frac{1}{\frac{\cos \theta }{s_{\mathrm{r},\text{max},x}}+\frac{\sin \theta }{s_{\mathrm{r},\text{max},y}}}$

where

Table 2.2

θ	angle between reinforcement in x-direction and direction of principal tension stress
sr,max,x  sr,max,y	maximum crack spacing in x- or y-direction

This equation is important if the first method, By assuming an identical deformation ratio of the longitudinal reinforcement for determining the design internal forces in the serviceability limit state, has been selected in the Settings for Analytical Method of Serviceability Limit State Design dialog box (see Figure 2.89).

In the third method (By taking into account the deformation ratio of the longitudinal reinforcement), on the other hand, the direction of the compression strut is determined according to Baumann. The limit angle of 15° is ignored because the crack width in this area is not governing.

For the calculation value of the crack width w_k according to Equation 2.73, we need to determine the factor (ε_sm - ε_cm) for each reinforcement direction and for the direction of the resulting strain.

The difference in the mean strain of concrete and reinforcing steel is determined according to , clause 7.3.4, Equation (7.9):

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

The maximum mean strain (ε_sm - ε_cm)_-z,res is obtained as the resulting mean strain of the individual reinforcement directions as 1.291 ‰.

To simplify the expression, we introduce symbols for the sought mean strain (ε_sm - ε_cm): s for the side length in the reinforcement direction, d for the partial length of the compression struts, l for the perpendicular to the compression strut, and ε.

The partial length d_γ-α is determined as follows for a selected compression strut inclination:

$d_{\gamma -\alpha } = \frac{1}{\tan \left(\gamma - \alpha \right)}$

The length is unitless (the perpendicular to the compression strut was included without unit).

Then the length s_γ-α is determined.

$s_{\gamma -\alpha } = \frac{1 + {\epsilon }_{\alpha }}{\tan \left(\gamma - \alpha \right)}$

If the reinforcement direction θ₁ forms the smallest differential angle with the principal moment m₁, we have to insert the previously determined difference in the mean strains (ε_sm - ε_cm)_θ1 of concrete and reinforcing steel for ε_α:

$s_{\gamma -\alpha } = \frac{1 + {\left({\epsilon }_{sm }- {\epsilon }_{cm}\right)}_{{\theta }_1}}{\tan \left(\gamma - \alpha \right)}$

If the reinforcement direction θ₂ forms the smallest differential angle with the principal moment m₁, we have to insert the previously determined difference in the mean strains (ε_sm - ε_cm)_θ2 of concrete and reinforcing steel for ε_α:

With the Pythagorean theorem, we can determine the value l_γ-α from the lengths d_γ-α and s_γ-α:

$I_{\gamma -\alpha } = \sqrt{s_{\gamma -\alpha }^2 - d_{\gamma -\alpha }^2}$

Since all formulas are based on an initial length of 1.0 units of length, the strain ε is determined as follows:

$\epsilon = I_{\gamma -\alpha } · -1.0$

This strain ε = (ε_sm - ε_cm) is checked again by means of the intermediate angle (β - γ).

For the determination of the SLS design internal forces according to the By assuming an identical deformation ratio of the longitudinal reinforcement method, the strain ratio of the reinforcements can significantly deviate from the assumed geometric strain ratio. To correctly determine the resulting strain ratio, the program therefore uses the strain of the reinforcement that is closer to the main action.

The calculated value of the crack width w_k is determined according to Equation 2.73.

In window 1.3 Surfaces, we have specified the maximum allowable crack width w_k = 0.3 mm. The following criterion of check for the governing resulting direction is thus obtained: