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I'm a computational person, and I don't quite understand how rotational DoFs are manifested in the linear elasticity equations. I understand the translational DoFs are simply the displacement components. When specifying boundary conditions for linear elasticity simulations, we typically specify the displacement and/or traction boundary conditions at the boundaries. Where do the rotational boundary conditions come in? Are they manifested in the traction boundary conditions?

This is not very intuitive to me and I was hoping someone can explain.

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In the general theory of 3D elasticity, there is no need for rotation variables.

However many engineering structures such as shells and beams have one or two dimensions that are small compared with the others. Representing the behaviour through the "thin" dimension as the difference between the translational displacements at the "top" and "bottom" surfaces is very badly conditioned numerically. To overcome that problem, the behaviour can represented approximately as the translation of the mid-plane (or more generally, the "neutral axis" of the object), and the slope (or rotation) of the cross section.

A simple example of this type of approximation is Euler-Bernouilli beam theory.

The formulation for shells and plates are similar in principle, but more complicated because the shell itself may be curved in one or both directions (e.g. part of a cylinder or a sphere).

The reaction "force" corresponding to a constrained rotation variable is a moment - i.e. the finite value of the product $Fd$ when two equal and opposite forces $F$ are separated by a distance $d$, in the limit as $F$ increases to infinity and $d$ decreases to zero.

These approximations are useful in most engineering situations because of St. Venant's principle, which says that the overall response of a structure does not depend on the exact details of how the loads are applied, so long as the different applied loads are statically equivalent.

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  • $\begingroup$ Ah I see. I am working specifically with beams. But I'm using a 3-D linear elasticity code to perform analysis of the beam subject to a 4 point bend, and I've been quite confused about the concept of rotational DoFs and how they apply to boundary conditions. Typically with a 4 point bend, the 2 bottom supports are modeled as a fixed and roller condition. With a fixed boundary condition, articles online stated that both the translational and rotational DoFs are fixed, whereas with the roller condition, the rotational DoFs are not constrained. $\endgroup$ – anonuser01 Mar 19 '19 at 16:28
  • $\begingroup$ Be careful of the difference between pinned (translations fixed but rotations free) and fixed (translations and rotations both fixed). To model a bending test, you need the beam pinned to constrain its axial displacement (even though there is no axial load applied) but the test does not constrain the rotations anywhere along the beam. $\endgroup$ – alephzero Mar 19 '19 at 17:45
  • $\begingroup$ For a straight, thin beam, the rotation degrees of freedom just represent the slope of the beam at a point. For thick beams that isn't quite true, and for curved shells it gets more complicated defining what you really mean by "the slope," but it's enough to understand a beam model of a 4-point bend test. $\endgroup$ – alephzero Mar 19 '19 at 17:50
  • $\begingroup$ Yes, I think another confusion for me was how fixed vs. pinned can be specified in a 3-D linear elasticity solver. When you look at a commercial solver, you typically can specify a location as "fixed," but I don't know if they mean fixed as in just the displacements are fixed, or if displacements and rotations are both fixed. Any insight? $\endgroup$ – anonuser01 Mar 19 '19 at 17:58
  • $\begingroup$ How you specify it depends what program you are using, but in any decent structural solver you should be able to constrain each of the 6 degrees of freedom at a node (or grid point) independently, and leave the others DOFs at that grid free. In general you need to model things like sliding joints, hinges, etc, between parts of an engineering structure. That requires control over how the individual degrees of freedom at two coincident nodes are connected and/or constrained. $\endgroup$ – alephzero Mar 19 '19 at 23:51
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Just answering the traction part of your question. A traction on the surface of a shell or beam that has a component tangent to the surface is equivalent to forces on the translational DOFs of the nodes plus moments on their rotational DOFs. enter image description here

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