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# 2.1.1 Bending and Axial Force

### Bending and Axial Force

The standards EN 1992-1-1, clause 6.1 or DIN 1045-1, clause 10.2 describe the calculation basis for the ultimate limit state design in detail. These regulations apply to bending with or without axial force, as well as to axial force only.

The mathematical state of failure occurs when the ultimate strains are reached. Depending on where these ultimate strains occur, the failure can be caused by the concrete or the reinforcing steel.

The following figure shows the allowable strain distributions for bending with and without axial force according to EN 1992-1-1, clause 6.1.

According to [3], the areas for strain distributions shown in the figure have the following meaning:

Area 1

This area appears in the case of a central tension force or a tension force with slight eccentricity. Only strains occur on the entire cross-section. The statically effective cross-section consists only of the two reinforcement layers As1 and As2. The reinforcement fails because the ultimate strain εud is reached.

Area 2

Area 2 appears in the case of pure bending and bending with axial force (compression and tension force). The neutral axis lies within the cross-section. The bending-tension reinforcement is fully used, meaning the steel fails when the ultimate strain is reached. Generally, the concrete cross-section is not fully used: The compressive strains do not reach the ultimate strain εc2u.

Area 3

This area appears only in case of pure bending and bending with axial force (compression). The steel's load-bearing capacity is higher than the load-bearing capacity of the concrete. The concrete fails because its ultimate strain εc2u is reached.

As in the areas 1 and 2, the concrete's failure is announced by cracks because the steel exceeds the yield point (failure with announcement).

Area 4

Area 4 appears in case of bending with a longitudinal compression force. It represents the transition of a cross-section mainly subjected to bending to a cross-section affected by compression. The concrete fails before the steel's yield point is reached because the possible strains are very small. This area results in a strongly reinforced cross-section. To avoid such a cross-section, a compression reinforcement is inserted.

Small steel strains in the tension zone result in failure without announcement (the bending-tension reinforcement does not start to yield).

Area 5

This area appears in case of a compression force with slight eccentricity (a column, for example) or a centric compression force. On the entire cross-section, only compressive strains occur.

The compressive strain on the edge that is less compressed is between 0 > εc1 > εc2. All compressive strain distributions intersect in point C.

Literatur
 [3] Avak, Ralf. Stahlbetonbau in Beispielen, DIN 1045 – Teil 1 : Grundlagen der Stahlbeton-Bemessung - Bemessung von Stabtragwerken. Werner Verlag, 5. Auflage, 2007