# Secondary Reinforcement According to DIN EN 1992-1-1 9.2.1 to Ensure Ductile Structural Component Behavior

### Technical Article

By using the minimum reinforcement, the tensile stress, which has been absorbed by the concrete before in the event of initial crack formation, should be covered by a reinforcement. This tensile stress can be descirbed by the so-called crack moment. The crack moment is the loading which generates a stress distribution in the considered cross-section resulting in an initial crack formation. By arranging the minimum reinforcement, it should be ensured that the initial crack formation does not lead to failure of the component.

#### Determination of the Minimum Reinforcement

The crack moment M_{cr }is determined as follows for a rectangular cross-section without axial force:

${\mathrm M}_\mathrm{cr}\;=\;{\mathrm f}_\mathrm{ctm}\;\cdot\;{\mathrm W}_\mathrm c\;=\;{\mathrm f}_\mathrm{ctm}\;\cdot\;\frac{\mathrm b\;\cdot\;\mathrm h²}6$

The minimum reinforcement for this loading results in:

${\mathrm A}_{\mathrm s,\min}\;=\;\frac{{\mathrm f}_\mathrm{ctm}\;\cdot\;{\mathrm W}_\mathrm c}{\mathrm z\;\cdot\;{\mathrm f}_\mathrm{yk}}$

Assuming that d ≈ 0.9 h and z ≈ 0.8 d, it follows:

${\mathrm A}_{\mathrm s,\min}\;=\;\frac{\;{\mathrm f}_\mathrm{ctm}\;\cdot\;\mathrm b\;\cdot\;\left({\displaystyle\frac{\mathrm d}{0,9}}\right)^2}{6\;\cdot\;0.8\;\mathrm d\;\cdot\;{\mathrm f}_\mathrm{yk}}\;\approx\;0.26\;\cdot\;\mathrm b\;\cdot\;\mathrm d\;\cdot\;\frac{\displaystyle{\mathrm f}_\mathrm{ctm}}{{\mathrm f}_\mathrm{yk}}$

The minimum reinforcement can be determined in general according to [2] as follows:

${\mathrm A}_{\mathrm s,\min}\;=\;\frac{{\mathrm M}_\mathrm{cr}\;+\;\mathrm N\;\cdot\;(\mathrm z\;-\;{\mathrm z}_{\mathrm s1})}{\mathrm z\;\cdot\;{\mathrm f}_\mathrm{yk}}\;=\;\frac{{\mathrm f}_\mathrm{ctm}\;\cdot\;{\mathrm W}_\mathrm c\;+\;\mathrm N\;\cdot\;\left(\mathrm z\;-\;{\mathrm z}_{\mathrm s1}\;–\;{\displaystyle\frac{{\mathrm W}_\mathrm c}{{\mathrm A}_\mathrm c}}\right)}{\mathrm z\;\cdot\;{\mathrm f}_\mathrm{yk}}$

with

M_{cr} = crack moment, for bending and axial force ${\mathrm M}_\mathrm{cr}\;=\;\left({\mathrm f}_\mathrm{ctm}\;-\;\frac{\mathrm N}{{\mathrm A}_\mathrm c}\right)\;\cdot\;{\mathrm W}_\mathrm c$

N = with N < 0 as compression force

f_{ctm} = mean tensile strength of concrete

f_{yk} = characterizes the value of the yield strength of reinforcement steel

S_{c} = section modulus of concrete cross-section in state I

A_{c} = surface of the concrete cross-section in state I

z = lever arm of internal forces in state II

z_{s1} = distance between centroidal axis minimum reinforcement and centroidal axis concrete cross-section

#### Particularities When Designing Surfaces: RF-CONCRETE Surfaces

For the calculation in surfaces, the following particularity appears according to NCI on 9.3.1.1 (1): "For two-way spanning slabs, minimum reinforcement as in 9.2.1.1 (1) only needs to be provided along the major axis." Since the main direction has not to run in one of the reinforcement directions, the minimum reinforcement will be arranged in the reinforcement direction which is the nearest to the main direction. In this context it may happen that the minimum reinforcement will be arranged partly in reinforcement direction 1 and partly in reinforcement direction 2.

For the option "Reinforcement direction with the main tension force in the considered element" it is clearly shown in Figure 02 that the minimum reinforcement is only designed once per side and direction. This corresponds to the standard, it may, however, generate a reinforcement which seems to be unusual.

Figure 01 - Setting: Reinforcement Direction With Main Tension Force in the Considered Element

Figure 02 - Result: Reinforcement Direction With Main Tension Force in the Considered Element

For a minimum reinforcement with a direction defined by the user, a more uniform image of the reinforcement distribution emerges. This can be clearly seen in Figure 04.

Figure 03 - Setting: Reinforcement Direction User-Defined

Figure 04 - Result: Reinforcement Direction User-Defined

The intermediate values used in the calculation can be clearly seen in RF-CONCRETE Surfaces by using the [Info] button in the calculation details.

Figure 05 - Extract From Calculation Details in RF-CONRETE Surfaces

#### Particularities When Designing Members: RF-CONCRETE Members

In principle, the calculation of the secondary minimum reinforcement in RF-CONCRETE Members corresponds to the calculation in Surfaces. One particularity is that for members with segmented cross-sections, such as box sections or T-beams, the effective width for the determination of the crack moment has to be considered.

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