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For example, metal tape measures can be extended and stay straight when in a "u" orientation, but collapse the other way round.

I presume the same phenomenon is why metal shelving has sheet metal on top, and flanges on the bottom rather than the top.

Why is this?

I'm guessing: it's because of the direction of compression/tension, and buckling. Since a long piece of material being compressed is more likely to buckle than a short piece. In one orientation, the buckling can only happen when the walls buckle across their height but in the other orientation, the compression acts across the entire length of the channel?

Both a common-sense approach, and a mathematical approach are welcome; I assume formulas are well established, using "moment of inertia" values for beam profiles or similar.

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You are correct.

A channel section in a "u" or "n" orientation is obviously asymmetric around its transversal x-axis. This means that its centroid is not located at the midpoint of its height. Instead, the centroid will be closer to its web than to the opening.

Since the stress at a given fiber at a distance $y$ from the centroid is equal to $$\sigma = \dfrac{My}{I}$$ in bending, this means that the stresses in the web will be lower than the stresses at the ends of the flanges.

Using your metal tape measure example, an open tape measure without other supports will behave like a cantilever, with its rotation fixed at the "mouth" of the tape measure. It will therefore be under negative bending moment (compression on bottom).

Since the tape measure is very slender, it is more sensitive to buckling due to compression than to simple collapse due to tension. Therefore, in order to avoid buckling, you want to have your tape measure in a "u" orientation, which will reduce the compressive stresses, while increasing the tensile stresses. In an "n" orientation, the stresses are inverted, with low tension and high compression (and therefore lower resistance to buckling).

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