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

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.

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).