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Overview

1 Introduction

2 Theoretical Background

2.6.4.9 Minimum reinforcement for crack control

Minimum reinforcement for crack control

The minimum reinforcement area for crack control is determined according to EN 1992-1-1, clause 7.3.2, Equation (7.1).

$a_{s, min} = \frac{k_{c} \cdot k \cdot f_{c t, eff} \cdot A_{c t}}{σ_{s}}$

where

Table 2.2
k_c	coefficient for considering the influence of the stress distribution in the cross-section prior to cracking as well as the change of the internal lever arm
k	coefficient for considering non-uniform self-equilibrating stresses, which lead to the reduction of restraint forces
f_ct,eff	mean value of the effective tensile strength of the concrete, to be expected when the cracks occur
A_ct	area of the concrete tension zone (part of the cross-section or partial cross-section that is calculated to be in tension in the uncracked state under the action combination that leads to the formation of the first crack at the gross cross-section)
σ_s	absolute value of the maximum allowable stress in the reinforcement immediately after crack formation

The maximum bar diameter d*_s is determined according to EN 1992-1-1, clause 7.3.3 (2) depending on the actually provided diameter d_s from the rearranged Equation (7.6N).

$d_{s} = d_{s}^{*} \cdot \frac{f_{c t, eff}}{2.9} \cdot \frac{k_{c} \cdot h_{c r}}{2 \cdot (h - d)}$

where

Table 2.2
d_s	adjusted maximum bar diameter
d*_s	maximum bar diameter according to EN 1992-1-1, Table 7.2 (see Figure 2.96)
h	overall depth of cross-section
h_cr	depth of tensile zone immediately prior to cracking while taking the characteristic values of prestress and axial forces under the quasi-permanent action combination into account
d	effective depth up to centroid of outside reinforcement

Image 2.96 Maximum bar diameter of reinforcing bars according to EN 1992-1-1, clause 7.3.3

In our example, determining the minimum reinforcement at the plate's bottom surface is excluded by opening the following dialog box in the Limit of Crack Width tab of window 1.3 Surfaces. In it, the check boxes for the Bottom (+z) reinforcement have to be cleared.

Image 2.97 *Further Settings for Min. Reinforcement Due to Restraint* dialog box in window 1.3 *Surfaces*

The limit diameter d*_s,-z,φ1 for the reinforcement direction φ₁ at the top side of the plate is determined according to Equation 2.72.

$d_{s, - z, ϕ_{1}}^{*} = 12 \cdot \frac{2.9}{2.9} \cdot \frac{2 \cdot (200 - 170)}{0.4 \cdot 100} = 18 mm$

Image 2.98 Limit diameter for reinforcement direction φ₁

Analogously, the limit bar diameter d*_s,-z,φ2 is obtained for the reinforcement direction φ₂:

$d_{s, - z, ϕ_{2}}^{*} = 12 \cdot \frac{2.9}{2.9} \cdot \frac{2 \cdot (200 - 158)}{0.4 \cdot 100} = 25.20 mm$

Image 2.99 Limit diameter for reinforcement direction φ₂

In window 1.3 Surfaces, the allowable crack width w_k,max is given as 0.3 mm (see Figure 2.97). With the maximum bar diameters d*_s,-z,φ1 = 18.00 mm and d*_s,-z,φ2 = 25.20 mm, we can interpolate the allowable stress σ_s from EN 1992-1-1, Table 7.2N (see Figure 2.96).

$σ_{s, - z, ϕ_{1}} = 240 + \frac{280 - 240}{16 - 25} \cdot (18.00 - 16) = 231.11 {N/mm}^{2}$

$σ_{s, - z, ϕ_{2}} = 200 + \frac{200 - 160}{25 - 32} \cdot (25.20 - 25) = 198.86 {N/mm}^{2}$

These allowable steel stresses are also shown in Figure 2.98 and Figure 2.99.

The steel stress in the direction φ₂ is governing.

The area of the concrete tension zone in the cross-section is determined as follows:

$A_{c t} = b \cdot \frac{h}{2} = 100 \cdot \frac{20}{2} = 1 000 {cm}^{2}$

Thus, according to Equation 2.71, the following minimum reinforcement for reinforcement direction 2 is obtained:

$a_{s, m i n, ϕ_{2}} = \frac{0.4 \cdot 1.0 \cdot 2.9 \cdot 1 000}{198.86} = 5.83 {cm}^{2} / m$

For this reinforcement direction, the applied reinforcement is greater than the minimum reinforcement. The following check criterion is thus obtained:

$\frac{a_{s, m i n, - z, 2}}{a_{s, e x i s t, - z, 2}} = \frac{5.83}{11.31} = 0.516$

Image 2.100 Check criterion for minimum reinforcement

Parent section

2.6.4 Serviceability Limit State Designs