How the total potential energy is a unique function with respect to semi-tangential rotation?
Also how the tangent stiffness matrix of space elements based on semi-tangential moment is symmetric?
Dear Khaled,
refer the site below for further study.
The kinematic relationships traditionally employed for spatial Euler-Bernoulli beam elements do not necessarily result in a flawed energy equation for the flexural-torsional buckling analysis of straight beams subjected to non-uniform bending moments (or transverse shear forces). This paper has nicely written the concept of semi-tangential bending moments is nly viable due to the mistaken belief that a conservative moment must invariably be made up of forces fixed in direction, and to the assumption that bending moments are work conjugate with transverse displacement derivatives in the 3D realm
https://pdfs.semanticscholar.org/9ad0/b3d9453f7f2d56adf9f627c8f4e8eb487e60.pdf
refer another paper . Go through the hole of manuscript.
Chen, C. S. (1994) “Geometric nonlinear analysis of three-dimensional structures,” M.S thesis, Cornell University, Ithaca, NY.
This may be useful for you.
Ashish
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