0
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.