# RF-CONCRETE Surfaces Version 5

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## RF-CONCRETE Surfaces Version 5

# 2.3.4 Possible Load Situations

### Possible Load Situations

The load is obtained by applying the principal axial forces n_{1} and n_{2}, with the principal axial force n_{1} always being greater than the principal axial force n_{2} considering the algebraic sign.

Different load situations are distinguished based on the algebraic sign of the principal axial forces.

In a matrix of principal axial forces, we obtain the following designations of the individual load situations (n_{1} is called n_{I}, n_{2} is called n_{II}):

The determination of design axial forces using Equation 2.5 through Equation 2.7 is described in the previous chapters for the load situations *Elliptical tension* and *Hyperbolic state*.
For the load situation *Parabolic tension*, the design axial forces are obtained using the same formulas.
The value k is to be assumed as zero in Equation 2.5 through Equation 2.7.

Now we will explain the design axial forces for the following design situations.

Equation 2.5 through Equation 2.7 are applied without changes, even if the two principal axial forces n_{1} and n_{2} are negative.
If a negative design axial force results for each of the three reinforcement directions, none of the three provided reinforcement directions is activated.
The concrete is able to transfer the principal axial forces by itself, that is, without the use of a reinforcement mesh in tension stiffened by a concrete compression strut.

The assumption that concrete compression forces in direction of the provided reinforcement are introduced to resist the principal axial forces is purely hypothetical. It is based on the wish to obtain a distribution of the principal compression forces in direction of the individual reinforcement directions in order to be able to determine the minimum compression reinforcement that is required, for example, by EN 1992-1-1, clause 9.2.1.1. To this end, a statically required concrete cross-section is necessary. It can only be determined using the previously determined concrete compression forces in the direction of the provided reinforcement.

When determining the minimum compressive reinforcement, other standards forgo a statically required concrete cross-section resulting from the transformed principal axial force into a design axial force. However, for a unified transformation method across different standards, the principal compressive forces are transformed in the defined reinforcement directions for these standards, too. Studies have shown that the design with transformed compressive forces is on the safe side. The concrete pressures occurring in the direction of the individual reinforcement directions are verified.

However, if at least one of the design axial forces is positive after the transformation, the reinforcement mesh is activated for this load situation. Then, as described in chapter 2.3.2 and chapter 2.3.3, an internal equilibrium of forces in the form of two reinforcement directions and one selected concrete compression strut is to be established.

Equation 2.5 through Equation 2.7 are used unaltered. If the direction of the two main axial forces is identical to the direction of both reinforcement directions, the design axial forces are equal to the principal axial forces.

If the principal axial forces deviate from the reinforcement directions, the equilibrium between a compression strut in the concrete and the design axial forces in the reinforcement directions is searched for again. For the direction of the compression strut, the two intermediate angles between the reinforcement directions are analyzed again. The same is applied for the elliptical tension: The assumption of a compression strut direction is deemed to be correct if a negative design force is indeed assigned to the compression strut. If allowable solutions are found for both compression strut directions, the smallest value of all design axial forces determines which solution is chosen.

If the design axial force for a reinforcement direction is a compressive force, the program first checks whether the concrete can resist this design axial force. If this is not the case, the program determines a compression reinforcement.

In this load situation, the principal axial force n_{1} is zero.
Since the quotient k = n_{2} / n_{1} cannot be calculated anymore, we cannot use Equation 2.5 through Equation 2.7 as usual.
The following modifications are necessary.

${n}_{\alpha}=\frac{{n}_{1}\xb7\mathrm{sin}{\displaystyle \beta}{\displaystyle}{\displaystyle \xb7}{\displaystyle}{\displaystyle \mathrm{sin}}{\displaystyle}{\displaystyle \gamma}{\displaystyle}{\displaystyle +}{\displaystyle {n}_{2}}{\displaystyle \xb7}{\displaystyle \mathrm{cos}}{\displaystyle}{\displaystyle \beta}{\displaystyle}{\displaystyle \xb7}{\displaystyle \mathrm{cos}}{\displaystyle}{\displaystyle \gamma}}{\mathrm{sin}{\displaystyle}{\displaystyle (}{\displaystyle \beta}{\displaystyle}{\displaystyle -}{\displaystyle}{\displaystyle \alpha}{\displaystyle )}{\displaystyle}{\displaystyle \xb7}\mathrm{sin}{\displaystyle}{\displaystyle (}{\displaystyle \gamma}{\displaystyle}{\displaystyle -}{\displaystyle}{\displaystyle \alpha}{\displaystyle )}}\phantom{\rule{0ex}{0ex}}{n}_{\beta}=\frac{{n}_{1}\xb7\mathrm{sin}{\displaystyle}{\displaystyle \alpha}{\displaystyle}{\displaystyle \xb7}\mathrm{sin}{\displaystyle}{\displaystyle \gamma}{\displaystyle}{\displaystyle +}{\displaystyle {n}_{2}}{\displaystyle \xb7}\mathrm{cos}{\displaystyle}{\displaystyle \alpha}{\displaystyle}{\displaystyle \xb7}\mathrm{cos}{\displaystyle}{\displaystyle \gamma}}{sin{\displaystyle}{\displaystyle (}{\displaystyle \beta}{\displaystyle}{\displaystyle -}{\displaystyle}{\displaystyle \alpha}{\displaystyle )}{\displaystyle}{\displaystyle \xb7}{\displaystyle}{\displaystyle sin}{\displaystyle}{\displaystyle (}{\displaystyle \beta}{\displaystyle -}{\displaystyle \gamma}{\displaystyle )}}\phantom{\rule{0ex}{0ex}}{n}_{\gamma}=\frac{-{n}_{1}\xb7\mathrm{sin}\alpha \xb7\mathrm{sin}\beta +{n}_{2}\xb7\mathrm{cos}\alpha \xb7\mathrm{cos}\beta}{\mathrm{sin}{\displaystyle}{\displaystyle (}{\displaystyle \beta}{\displaystyle}{\displaystyle -}{\displaystyle}{\displaystyle \gamma}{\displaystyle )}{\displaystyle}{\displaystyle \xb7}{\displaystyle}{\displaystyle \mathrm{sin}}{\displaystyle}{\displaystyle (}{\displaystyle \gamma}{\displaystyle}{\displaystyle -}{\displaystyle}{\displaystyle \alpha}{\displaystyle )}}$

With these modified equations, the program searches for the design axial forces in the two reinforcement directions and for one design axial force for the concrete in a similar way. If a reinforcement direction is identical to the acting principal axial force, its design axial force is the principal axial force. Otherwise, solutions with a compression strut between the two reinforcement directions are found again.

The formulas presented above are used according to Equation 2.13.

If the principal axial force runs in a reinforcement direction, solutions for a compression strut direction between the first and the second reinforcement direction or the first and third reinforcement direction are analyzed (like for the parabolic tension). Again, the smallest value of all design axial forces decides which solution is chosen.