## Calculating stiffness of a box section

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I started Googling how to calculate this, then thought F**k it, just ask UKC, someone will know

I have a frame made from box section aluminium. The box section is 35mm high, 6mm wide and 1mm wall thickness. The frame is a rectangle of about 1300 x 700, if that matters. I want to make it a lower profile, so the box secton is only 15mm high. Obviously going from 35mm to 15mm will make it a lot less stiff, so to make it this high with the same stiffness I will need to increase width and wall thickness. I can make it as wide as I like, and weight is not really an issue, I could even use solid bar if necessary.

Looking at available aluminium stock I will probably use 2mm wall thickness extrusion. What is the best shape to use? A 15mm square? Are two 15mm square sections bonded together side by side stronger than a 15x30 box? Will either be as stiff as the current 35x6x1?

Depends which axis you measure the stiffness in - major axis (i.e. in the 'strong' direction) or minor axis (bending it in the weaker/thin direction). Assuming major axis:

The second moment of area of the old box (35 x 6, 1mm thick) is 0.946cm^4. If you use 2mm wall thickness and 15mm height, you would need 53mm width for the equivalent stiffness. If you use solid bar it would need to be 34mm wide to be the same stiffness.

You would need three 15x15x2 boxes side by side to be the same stiffness as the 35x6x1.

Post edited at 16:42

That's perfect, thanks. It's interesting that i need three 15x15s or one 53x15, so it appears that a single box works out stronger (for a given weight) than multiple boxes.

It's the major axis of the 35x6 that counts. Since I'm making it wider my new frame is going to end up much stronger than the old one in the other direction.

Does it have to be made from aluminium? Steel is a potentially more elegant option (i.e. a thinner frame) - you could have a nice stainless steel frame welded up if you don't have the kit yourself.

That did vaguely cross my mind, but getting SS fabricated is going to be way more expensive than sticking some ally box together.

Read up on 2nd moment of area; it's a relatively simple concept that's just a wee bit too much for me to (be bothered to) explain.

for your purposes; same material & similar I value = similarly stiff. you can get the Ixx, Iyy values form section data from suppliers.

generaly:

hollow section to solid = negligible increase in stiffness,

thicker section = small difference in stiffness

change on shape = significant difference in stiffness (see 2nd M of A)

I presume this isn't a safety critical thing, you're house isn't going to collapse onto your head. if stiffness really is your critical factor; there's no substitute (at this scale of project) for build it and see.

15 x 15 x 2mm and 30 x 15 x 2mm stainless tube are standard sizes available anywhere as they are used for balustrades, about a tenner per m for 30 x 15.

Edit, want it stiffer then slip some 10mm inside.

Post edited at 18:33

Stiffness of a beam is proportional to the cube of its depth so if you double the depth of a beam its stiffness in increases by a factor of 8 etc..

Stiffness is also directly proportional to the modulus of elasticity of the material you make it from.

Modulus of aluminium is 68.9 GPa, modulus of steel is 200 GPa so if you just changed material from aluminium to steel stiffness would increase by 200/68.9.

> Stiffness of a beam is proportional to the cube of its depth so if you double the depth of a beam its stiffness in increases by a factor of 8 etc..

For a solid rectangular section, not necessarily for hollow sections.

What are you planning to do with it e.g what forces, which direction and how long do you want it to last?

You're right, I read up on it and it's straightforward. Since I'm sticking to the same material the only relevant factor is the 2nd moment of area. There are even online calculators for all the standard extrusions, so I can plug in all the available options and see how they perform.

Thanks all

> For a solid rectangular section, not necessarily for hollow sections.

I was simplifying...

To improve the effectiveness of structural elements in bending, there are two basic things you have to do.

1. Choose a shape where sufficient material is significantly displaced from the axis of bending.

2. Ensure that the shape also has sufficient stability in the axis of banding.

So at (1), the section modulus increases in proportion to the square of that distance displaced and in proportion to its width. This means reducing the height is a bad thing to do and you very quickly lose strength in bending by doing that. You do not lose or gain as much by changing width.

At (2), one would typically find that by choosing a shape will too much of the material displaced from the bending axis, there is no longer sufficient material in the webs or sides of the shape to hold its shape when under stress. In such a case, the shape is unstable and collapses.

These are the reasons that an I-shaped beam or column is so popular in important structures. A hollow section, such as the rectangle you describe, is also a structurally effective shape but can be more difficult to join to other elements.

https://en.wikipedia.org/wiki/Section_modulus

Post edited at 16:27

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