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Reading here on Eurocode.

It says that when a cross section is subject to both normal force and moment their effect should be combined like this:

$$\frac{N_{Ed}}{N_{Rd}} + \frac{M_{y, Ed}}{M_{y,Rd}} + \frac{M_{z, Ed}}{M_{z, Rd}} \leq 1$$

Why this? Shouldn't we individually check that the ratio of neither design force to design resistance for force nor ratio of design moment to design resistance for moment are less than one?

Why are they summed together like this? I just don't get the mathematical reason for this.

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  • $\begingroup$ If any two are zero, then the other has to be less than 1... $\endgroup$
    – Solar Mike
    Jan 26 at 20:18
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    $\begingroup$ No, it doesn't say they "should be combined" like the equation. It says "As a conservative approximation they "may" by combined that way. If that approximation is too conservative for your design, then do it properly instead! $\endgroup$
    – alephzero
    Jan 26 at 20:37
  • $\begingroup$ @alephzero Fine, but I still don't see what that expression means... $\endgroup$
    – S. Rotos
    Jan 26 at 22:09
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If we look at each of those fractions individually, their meaning is obvious. Obviously, for a structure under axial load (or bending moment), the applied force must be lower than what the structure can hold.

But at a fundamental level, what matters isn't the force or moment that's applied, but the stress the structure is under. If you have a beam with a bending moment that's 99% of what the beam can support and then you apply tension equal to 50% of the resistance, you'll obviously have some areas where the combined tension from the bending moment and force is greater than what the beam can take and it'll collapse.

So the question is how to handle these combined cases.

For a simplistic calculation, we just need to think about how all of these loads are cumulative in at least one point of the beam: axial loads are uniform across the section, bending around the horizontal axis generates (for example) tension at the bottom fibers and moment around the vertical axis generates tension on the right-side fibers. So the bottom-right fibers get put into tension from all of these applied loads.

So we need to make sure that the combined stress from all of these loads is lower than the resistance (which I'll simplify to the yield strength $f_y$).

And to do so we just need to make the simple observation that each of those fractions describes what fraction of $f_y$ is taken by each of those solicitations. For example, for normal forces:

$$\begin{gather} \frac{\sigma_N}{f_y} = \frac{N_E/A}{N_R/A} = \frac{N_E}{N_R} \\ \therefore \sigma_N = \frac{N_E}{N_R}f_y \end{gather}$$

With bending moments, the function is different but the conclusion is the same.

So all we're saying with that sum of fractions is that the fraction of the resistance taken up by the normal force plus the fraction taken up by the horizontal and vertical bending moments must be less than one. If you want to make that explicit:

$$\frac{N_{Ed}}{N_{Rd}}f_y + \frac{M_{y, Ed}}{M_{y,Rd}}f_y + \frac{M_{z, Ed}}{M_{z, Rd}}f_y \leq f_y$$

There are some interactions we can take into consideration that allow us to go over this threshold to a certain extent, but this is a straightforward conservative solution.

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  • $\begingroup$ Ah, so that was very simple as I tought, I just failed to see it. Thank you! $\endgroup$
    – S. Rotos
    Jan 27 at 10:36

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