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I am analyzing a built-up C-channel for bending purposes (2 point loads of 1200 N each @ 30 cm symmetrically from the CL). The flanges are made of a different material than the web, they are also of a different thickness. The two materials have very different elastic and strength properties. Assume a perfect bond between the flanges and web.

enter image description here

So far i have used the "equivalent section" trick to convert the flanges' dimensions as if they were made of material 2 and to find the stresses in both the web and flanges, after this i ran an FEA on the beam and the stresses are very close to what the theory predicts.

The next and hardest step in the analysis is to figure out the deflection of the beam due to the bending load. While the "equivalent section" trick is helpful to find the maximum stresses, i'm not sure it can be used to find the actual deflection. Additionally, i need to figure out the shear center of the section and all formulas i have seen so far assume a uniform thickness in the web and flanges which is not my case. Furthermore, since the shear center will be located somewhere to the left of the section, there will be a resultant torque on the section which will induce a shear stress in both the flanges and web. While Roark's formulas for stress and strain provide guidance on how to calculate torsional shear stresses in C-channels, it generally assumes uniform material properties which is not the case of this channel. The materials are a ton weaker in shear than they are tension/compression so this is actually my main concern.

Can anybody recommend any resources, tools or tricks to solve this problem? I have access to my college's mechanics of materials books and even in the advanced edition i could't find anything that might help me. In theory i could simply use FEA software for this section but i will have nothing to verify the results against and I always like to at least have an analytical ballpark number to double check.

Any help would be greatly appreciated. Thank you fellas.

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    $\begingroup$ I'd need to think about the rest of the question, but I can tell you with certainty that the "equivalent section" method works just fine for deflections as well as stresses. After all, deflections are a matter of $EI$, and the equivalent section method simply modifies $I$ so you can adopt one $E$ and get a result equal to reality. $\endgroup$ – Wasabi May 5 at 1:52
  • $\begingroup$ But actually, is that red arrow simply showing how you changed the section for equivalent sections or is it the direction of the loads? I assume the former. If the latter, then you should've increased the height of the equivalent sections, not their width. $\endgroup$ – Wasabi May 5 at 1:55
  • $\begingroup$ @Wasabi .. thank you for your help. The arrow shows how i transformed the section. I believe this approach is OK since the rule is to "widen or narrow in a direction parallel to the neutral axis of the section so that each element remains at the same vertical distance from this NA". Let me know if you have any advice on the torsional analysis. Regards $\endgroup$ – JC ME May 5 at 4:22
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Assuming your sketch has been drawn to scale, it won't be easy to make a hand calculation of the location of the shear center. The challenge is not the use of two different materials but that the usual assumption of a thin-walled cross section won't be very accurate.

If you need the accurate location of the shear center, you will pretty much have to use a FEM with 3D elements. The shear flow in a thick-walled cross section is too complicated a topic to be worth the bother.

To calculate the approximate location using the approximation of a thin-walled cross section, you can use the following approach:

  • Calculate any cross section parameters you need (area, first moment of area and section moment of area) for the transformed cross section, which is very similar to using an equivalent section, but not quite. That is, you pick one material as the reference material (e. g. number 2) and multiply the contribution of the other material (1 in that case) with the ratio of E for the two materials. The difference is that you don't adjust the width or thickness of material 1. The transformed cross section does not have geometric representation as a cross section of a single material.

  • Assume the cross section is loaded in pure shear, i. e. a vertical shear load placed in the shear center.

  • Use Zhuravskii's shear stress stress formula to calculate the sum of horizontal shear in each flange in material 1. Clearly there will also be some horizontal shear in material 2, but based on the approximation of a thin-walled cross section, we're assuming it is a small contribution.

  • There will also be some vertical shear in the flanges in material 1, but we'll assume it's a small contribution. Then the centroid of vertical shear will be located in the centroid of the web.

  • The horizontal shear forces and the vertical will both contribute to a torsional equilibrium about the shear center but with opposite signs. The only unknown in that equation is the location of the shear section, so solve for that.

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  • $\begingroup$ Thank you for your help. I've actually learned a lot from this problem in the last couple of days. Cheers! $\endgroup$ – JC ME May 11 at 14:48

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