2.3.4 Possible Load Situations

Possible Load Situations

The load is obtained by applying the principal axial forces n₁ and n₂, with the principal axial force n₁ always being greater than the principal axial force n₂ when taking the algebraic sign into account.

There are different load situations depending on the algebraic signs of the principal axial forces.

In a matrix of principal axial forces, you get the following designations of the individual load situations (n₁ is called n_I, n₂ is n_II):

Determining the design axial forces using Equation 2.5 through Equation 2.7 is described in the previous paragraphs for the load situations Elliptical tension and Hyperbolic state. For the load situation Parabolic tension, the design axial forces are obtained in the same way. The value k is to be applied with zero in Equation 2.5 through Equation 2.7.

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

Equations 2.5 to 2.7 are applied without changes, even if the two principal axial forces n₁ and n₂ 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 the 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 the direction of the individual reinforcement directions in order to be able to determine the minimum compression reinforcement that is required, for example, in EN 1992-1-1, clause 9.2.1.1. To this end, a statically required concrete cross-section is needed, which can only be determined by using the previously determined concrete compression forces in the direction of the provided reinforcement.

When determining the minimum compressive reinforcement, other standards manage without a statically required concrete cross-section that results from the principal axial force, transformed 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 as well. Studies have shown that the design with transformed compressive forces is the safe choice. The concrete compressions that occur in the direction of the individual reinforcement directions are designed.

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 has to be established.

Equations 2.5 through 2.7 are used without alterations. If the direction of the two main axial forces is identical with 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 sought again. For the direction of the compression strut, the two intermediate angles between the reinforcement directions are analyzed again. As with elliptical tension, the following applies: The assumption of a compression strut direction is deemed to be correct if a negative design force is actually 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, a compression reinforcement is determined.

In this load situation, the principal axial force n₁ is zero. Since the quotient k = n₂ / n₁ cannot be calculated anymore, you cannot use Equations 2.5 through 2.7 in the usual way. The following modifications are necessary.

$$\begin{gathered} n_{\alpha } = \frac{n_1 · \sin {\displaystyle \beta }{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle \gamma }{\displaystyle }{\displaystyle +}{\displaystyle n_2}{\displaystyle · }{\displaystyle \cos }{\displaystyle }{\displaystyle \beta }{\displaystyle }{\displaystyle · }{\displaystyle \cos }{\displaystyle }{\displaystyle \gamma }}{\sin {\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}{\displaystyle }{\displaystyle ·} \sin {\displaystyle }{\displaystyle (}{\displaystyle \gamma }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}} \\ {n}_{\beta } = \frac{n_1 · \sin {\displaystyle }{\displaystyle \alpha }{\displaystyle }{\displaystyle ·} \sin {\displaystyle }{\displaystyle \gamma }{\displaystyle }{\displaystyle +}{\displaystyle n_2} {\displaystyle · }\cos {\displaystyle }{\displaystyle \alpha }{\displaystyle }{\displaystyle · }\cos {\displaystyle }{\displaystyle \gamma }}{sin{\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle sin}{\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle -}{\displaystyle \gamma }{\displaystyle )}} \\ {n}_{\gamma } = \frac{-n_1 · \sin \alpha · \sin \beta + n_2 · \cos \alpha · \cos \beta }{\sin {\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \gamma }{\displaystyle )}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle (}{\displaystyle \gamma }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}} \end{gathered}$$

With these modified equations, the program searches for the design axial forces in the two reinforcement directions and for a design axial force for the concrete in the same 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 (as with parabolic tension). Again, the smallest value of all design axial forces decides which solution is chosen.