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I'm trying to model a larger system and working through a conceptual issue. I'm familiar with the concept of expressing the stiffness of a beam as an equivalent spring stiffness like in the image below:

spring stiffness

I'm curious if there's a way of doing to rotational analog to this, expressing the resistance of a beam to a rotation as an angular stiffness (akin to replacing it with a torsional spring). I drew this mock up to illustrate the conditions

undeflected

And here is an illustration of the system upon deflection. Here, there is a 30 degree deflection. The white link is fixed horizontally (i.e. grounded), while the grey link is free to rotate about the blue joint. Consider the joint to be frictionless and the links to be massless/infinitely stiff. And let's assert that the beam is supported at the ends of the links through roller joints (can translate and rotate).

enter image description here

First off, I'd like to know if there if there is even a closed form way to express rotational stiffness of beams. If so, I'd like to be able to express this as a stiffness about the original axis of rotation (the joint axis) in a closed form like the linear example above. But this might be an issue since the axis through the centroid translates as the beam is rotated. Is this an issue? Would the stiffnesses about the 2 axes be different? If so, could the parallel axis theorem be used to simply express the one stiffness at the other axis?

Also, here's an image of the beam alone with some quantities. But these are fake values, I just care for the symbolic representation since this is just a mock up of the concept I'm trying to understand.

enter image description here

Please let me know if there is information I left out and I appreciate any help or resources you could point me to.

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You seem to have two conceptual misunderstandings. I'll try going over them. First, the replacing of the beam for a spring is tied to a specific load pattern. If you placed the load $m$ shown in the first figure at any point other than $L/2$ the equivalent stiffness would no longer be what's shown on the picture. This is not a big issue, but still important to keep in mind.

The second, more important issue is that the set-up you show in the second and third picture is simply a beam with an imposed displacement and twist on one end, so its not a purely rotational phenomenon. If you wanted to make a purely rotational analogy you'd have to neglect the twist of the beam at the end, which would contradict your set up.

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