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I would like to find the full (12 by 12) Timoshenko beam element stiffness matrix for a variable cross section with mass center axis and elastic center does not coincide. I have the full stiffness matrix for a constant cross section with mass center axis and shear center axis does not coincide as shown here. Now I need to find the stiffness matrix for a tapered beam such as the one below.

enter image description here

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  • $\begingroup$ Are they meant to coincide? $\endgroup$ – Solar Mike Jan 14 '18 at 9:02
  • $\begingroup$ I hope you mean mass center and shear center. No, Mass center and shear center does not coincide. $\endgroup$ – Paul Thomas Jan 14 '18 at 14:55
  • $\begingroup$ So why question that they don't coincide in the question / title ? $\endgroup$ – Solar Mike Jan 14 '18 at 16:39
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    $\begingroup$ If the mass center axis and shear center axis does not coincide then there would be bending and torsion coupling. So I would like to find the Timoshenko beam stiffness matrix with both bending and torsion coupling for a variable cross section. What I have provided in the .pdf in the link provided is the Timoshenko beam stiffness matrix for a constant cross section with bending and torsion coupling. $\endgroup$ – Paul Thomas Jan 14 '18 at 17:27
  • $\begingroup$ I put a close tag on this because variable cross section is entirely too broad. The cross section could be variable by width only, by height only, or by width and height. The section could vary parabolically or linearly. All of this makes an exact function of area and inertia needed, and each element's individual Gaussian integral resolved on an energy basis. Nominally in FEA, the cross section is broken up with each element having a different constant cross section, so the system can react appropriately. $\endgroup$ – Mark May 30 '18 at 20:04

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