Frame element stiffness with non-axis-aligned anisotropic area moment of inertia?

Question

I've found equations for the $12\times12$ symmetric stiffness matrix for an element of a frame (a truss with welded joints): http://what-when-how.com/the-finite-element-method/fem-for-frames-finite-element-method-part-1/ second-to-last equation, $\textbf{k}_e=[\cdots]$.

However, that assumes the area-moment-of-inertia tensor is diagonal (i.e., that the beam can have an anisotropic section, but the anisotropy must be axis-aligned). In the general case, there's an additional $I_{yz}$ term, but it's not clear where it goes. I'm guessing it goes in the $(v, \theta_y)$ and $(w, \theta_z)$ entries scaled by $3E/2a^2$ and in the $(v,w)$s and $(w,v)$s scaled by $3E/2a^3$, but for all of those, It's not clear to me what the signs would be.

Answer 1

You formulate the matrices in a local coordinate system where local $x$ axis is along the length of the beam, and the local $y$ and $z$ axes are the principal axes of the beam section. In that coordinate system, $I_{yz} = 0$ by definition. ($I_{xy}$ and $I_{xz}$ are also zero).

Finally, you transform the equations in the the global coordinate system using equations 6.13, 6.14 and 6.15 in your reference.