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With the FAGUS My-N diagram, we can see graphically and numerically all the possible limit states, for one or several amounts of reinforcement that our section may have.

Being more precise, we observe the points that fulfill certain analysis parameters, which **will generally **correspond to a limit state, but may have other conditions, such as a given maximum steel stress, so that the pairs of M-N values make the stresses to be as given.

In general, in this diagram, those My-N points that collapse the section (ULS) are presented, but there is also a third variable that comes into play in determining the bending resistance of the sections.This third variable is the moment about Z axis (Mz) and implies that, in space, there is a surface formed by the My-Mz-N combinations that collapse the section.

The question that leads us to write this article is: what plane "cut" of that surface of limit states is the FAGUS My-N diagram showing us?

Logically, the surface of the limit states of our section will be formed by infinite My-Mz-N combinations.In order to represent in a flat diagram the My-N pairs that collapse the section, it will be necessary to set any parameter.This parameter for FAGUS is the curvature of the section about Z axis (Χz), thus being our deformation planes: free Y curvature (Χy≠0) and fixed Z curvature (Χz=0).

Forcing that there is no curvature in Z will lead to moments about Z that must appear to compensate the rotation of the main axes of the section. Let's see it.

We take a non-symmetrical section like the one in the following figure:

If we ask FAGUS to obtain the My-N diagram for this section and for the amount of the existing reinforcement:

In principle, if we take any point on this diagram and perform an efficiency analysis, we should get an efficiency equal to 1.00. Taking one of the points of the diagram at random, for example the one indicated in the following table, and assuming that Mz=0 (because no values of this variable appear in the table):

We see, first, that the section is rotating about an axis that is not horizontal. This rotation of the axis is necessary for the section to be in equilibrium when the forces N and My act.

Furthermore, looking at the numerical results, we can verify that, in the ultimate limit state, the moment Mz would be zero and that the curvature in Z would be different from zero:

We also see something surprising: the efficiency is 1.03>1.00, so the point given by the introduced values of N=115.9 kN, My=57.8 kNm and Mz=0 kNm is not part of the diagram (which would imply an efficiency of 1.00) but is external to it. Furthermore, in the table above, we see that the ultimate forces are not the ones introduced, taken from the tabular data of the diagram. If the values of N and My entered were the forces at the ultimate limit state, they would be the values shown in the "Ultimate state !ELU" table above.

This is because, in reality, the diagram is hiding something from us and that is that the ultimate moment Mz is not zero but has a value.

This value, as we have seen, is not given to us in the data of the diagram, but we can obtain it if we force the program to perform an efficiency analysis with uniaxial bending, that is, forcing the curvature in Z to be zero.

Entering the same N and My values as before, but checking the "Uniaxial bending" option:

Now we obtain an efficiency of 1.00 and we see that the axis of rotation of the section is horizontal.

If we see the numerical results:

We see that an ultimate moment Mz of -8.4 kNm appears. This moment is what, together with the introduced My, makes the section rotate with respect to a horizontal axis or, what is the same, that the curvature in Z is zero, as we also see in the previous table.

This would be equivalent to doing an efficiency analysis without forcing it to be with uniaxial bending, but introducing Mz=-8.4 kNm:

As we can see, the same results are obtained.

As a conclusion, we can say that the My-N interaction diagram is showing us the projection on the Mz=0 plane of the set of points that collapse the section. These points can have an Mz coordinate, which will not be shown to us, **but which will only exist if we have a section with non-symmetrical geometry, whose axis of rotation is not coincident with the horizontal **axis (Y axis). If we have a symmetrical section (which will be usual), the cutting plane of the surface in the space My-Mz-N will coincide with the plane Mz=0 and, carrying out an efficiency analysis with the efforts corresponding to any point from the diagram, we will obtain an efficiency of 1.00.

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