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In "Theory of Elastic Stability" by Timoshenko and Gere, the authors say for a beam in pure torsion,

For a beam of thin-walled open section it can be assumed with reasonable accuracy that the shearing stress at any point is parallel to the corresponding tangent to the middle line of the cross section and is proportional to the distance from that line.

What is meant by the "tangent to the middle line"? A cross section would be a two dimensional shape. For an I-beam, it would be the shape of an I. But I can see two reasonable definitions of what a middle line could be, since an I has two axes of symmetry. And even then, I'm not sure what a tangent to the line is, other than the line itself.

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  • $\begingroup$ I beam is not a thin walled open section. By thin walled open section, it may mean a pipe or a hollow rectangular stock, or similar entity with a hollow interior surrounded by a relatively thin wall. Middle line is the middle of the thin wall- halfway between inside and outside. $\endgroup$
    – Abel
    Sep 7 at 4:16
  • $\begingroup$ What is your understanding of "tangent"? $\endgroup$
    – Solar Mike
    Sep 7 at 5:08
  • $\begingroup$ @Abel I found this Wikipedia article, which includes I beams among thin walled open sections en.wikipedia.org/wiki/…. I now think it means a beam whose cross section can be approximated as the welding of one dimensional curves, but is not topologically equivalent to a circle. $\endgroup$
    – Mark
    Sep 7 at 16:52
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    $\begingroup$ Maybe an I beam is, although I thought open meant hollow and open to outside at both ends. Either way middle of the wall is what you are after. $\endgroup$
    – Abel
    Sep 8 at 12:09

2 Answers 2

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The figure indicates a thin-walled open section and the thickness $t, \ $ and the tangent to the midline.

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figure

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I found a manuscript https://s28490.pcdn.co/wp-content/uploads/2019/06/asm-B840.pdf that clarifies the situation.

enter image description here

The "middle line" mentioned in the quote is not strictly a mathematical line, but may be a curve, or in the case of an I beam or T beam, a join of more than one line segments, which describes the shape of the cross section if the width of the walls were shrunk down to zero.

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