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I am trying to learn how to mathematically prove that a steel tube (with wall thickness of ~ 1mm and external diameter of ~ 15 mm) will have ...

At any point along its length, the shaft must :

1) bend in such a way that the deflection is the same regardless of how the shaft is rotated about its longitudinal axis; and

2) twist the same amount in both directions

To make this even more difficult, I propose to bend channels into the shaft with 120 degree spacing, a depth of 2 mm and a width of 2mm (channels to be placed at 60 degrees, 180 degrees, and 300 degrees.) with a length, along the longitudinal axis, of 250 mm.

My issue is that I want to be able to mathematically prove that the combined resultant forces on the shaft will effectively remain constant, regardless of shaft rotation, due to the spacing of the channels. There is no requirement that I have found that states that all points along the tube must react in the same manner to other points along its shaft, in respect to flexing, bending or twisting

My hypothesis is that since the channels are spaced at 120 degree increments, the combined resultant forces when measured in any direction will be the same relative to the channel locations, but regardless of tube rotation, and can meet the requirements set forth above. I believe that if I can establish this theory mathematically, it will prove true in physical testing as well.

Any help that can be offered is greatly appreciated.

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# My dilema in proving a theory (bending a steel tube)

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