# How to describe the rotation of opposite ends of a body due to an applied moment

### Summary

A short way to phrase this question is:
Shear describes the translation of one part of a body relative to another given an applied force.
What is the equivalent rotational term? That is,

What describes the rotation of one part of a body relative to another given an applied moment of force?

### In more detail:

I'm trying to find the right word for a particular type of distortion. I have an assembly of items connected to be approximately solid. I mount one flat to a wall, and put some force or moment onto the opposite flat side so that the assembly distorts. In particular, I want to correctly describe the relative rotation of the two flats (in any direction), and I want this term to describe only the rotation, and exclude any translation (so that if it only shears without twisting I'd like the quantity to be zero).

I thought of torsion, but definitions of torsion always seem to refer to a bar or an elongated axis of some type. Is a bar required for "torsion" to be meaningful? I thought of twist, but that doesn't necessarily imply a dynamic quantity, so, for example, one could freeze in a twist, and something can be twisted without forces applied. I don't think bend is correct because that could just refer to translation. Is there a better word that I'm not thinking of, or even a concise phrase?

If there is a word that matches my needs, a reference that I could review would be very helpful.

### Why "bending" is not the word I need

Below is a sketch of two objects that are bending along the same curve (both by intuition and the equations), yet the rotation of the ends is different in each. I'm interested in a word or phrase for describing the rotation of the ends (or a tendency for the ends to rotate or not, etc).

Also, according to wikipedia, bending "characterizes the behavior of a slender structural element" under load, but I'm not interested in slender elements (and only drew the above to illustrate bending). The first is the common way a beam would bend, but not everything bends like a beam, and I am hoping to find a word that characterized the particular difference illustrated here.

### Things that don't bend like normal beams

A reasonable model is a highly anisotropic material, rigid in one axis but less so in the others. For example, a bundle of parallel fiber optic cables.

I think another would be a tube filled with water at high pressure, like a fire hose. (I don't know how this bends, but I doubt it bends like a beam.)

Here's a sketch of a structure that doesn't bend like a beam, in that the side parallel to the wall stays parallel, even though the rest of the structure bends in a normal way. The black dots are meant to represent rotating joints. Even if this item were long and slender I think the ends would stay parallel. The point of the third picture is that you don't necessarily get to see what's on the inside (but it could still be important to describe that the two walls don't rotate with respect to each other in response to an applied force).

• The drawing you've shown here has a concentrated force. You will always have shearing deformations in such a situation. That shearing might be irrelevant (an assumption in Euler-Bernoulli theory as described in HDE's answer). The only way to have zero shearing deformations is with a couple: two equal forces in opposite directions separated by a certain lever arm. In your drawing, considering X as the horizontal and Z as vertical, two loads of 1kN in $\pm$Z separated by 1m in X will cause 1kNm in bending and no shear (if the loads are not directly on the object).
– Wasabi
Jun 3 '15 at 15:55
• You really need to stop using the word torque in this question. It's not torque. Torque is a moment about the longitudinal axis of a member. Whenever you refer to torque in your question, you really should use the term moment instead. Jun 4 '15 at 9:14
• I think the relationship you're trying to describe would properly be called the dihedral angle. Jun 4 '15 at 14:25
• @AndyT: thanks, changed it. (As an explanation, I'm far more familiar with the physics terminology where the terms are used interchangeably to refer the general case.) Jun 4 '15 at 14:31
• @AndyT I strongly disagree that "it's not torque." That there is a convention in certain fields to use "torque" with a more specific meaning does not invalidate the general physical meaning, with which every engineer who wants to communicate outside of their niche should be comfortable. The diagrams make quite clear what force is being applied and where, so there's no ambiguity.
– Air
Jun 4 '15 at 17:27

What you're describing is a subset of the general concept of rigidity (or stiffness). I think the measurement you're looking for is the inverse of rotational stiffness (as defined by Wikipedia):

A body may also have a rotational stiffness, k, given by $$k=\frac {M} {\theta}$$ where

M is the applied moment
θ is the rotation

You could say something like, "The rotational stiffness of the assembly is very high, such that the rotation of the free end with an applied bending moment is nearly zero."

If you need to come up with a term for the inverse of rotational stiffness, you might get some mileage out of "skew," as in, "The end of the assembly opposite and parallel to the wall will not tend to skew with respect to the wall when any force is applied."

Informally, this use of "skew" may just make sense to the reader; formally, it's a bit tricky to justify, and an audience of engineers may second-guess it. You could argue that there are more parallel lines between parallel planes than there are between intersecting planes, because all parallel lines between intersecting planes must be parallel with the line of intersection, therefore intersecting planes are more "skewed" because there are fewer ways to pair lines between them that are not skew. You'd be on shaky ground with the mathematicians, though.

That's my direct answer to your vocabulary/documentation question. The rest of this will be a bit more free-form, starting with a response to your comment on AndyT's answer:

my assembly doesn't really have clear surfaces (think of two spheres stuck together). So instead of, say, "torsional rigidity", I'd need something like "given any two virtual surfaces, one close to the binding surface and one close to the free end, when a force is applied to the free end, the two virtual surfaces will experience only a small relative rotation".

You'll need to at least define these virtual surfaces initially. One of them should be easy, since it's a plane defined by a wall. If the wall is curved, you might use the plane having a normal vector that averages the axes of the fasteners used to mount the assembly, which passes through some "centroid of mounting" (call this the "virtual wall" if you like). A more complex geometry will demand a more complex explanation; efforts to reduce the complexity of that explanation are unreasonable if and when they impair the effectiveness of the explanation.

Beams and shafts aside, the difference between rotational stiffness and torsional rigidity has to do with the nature of the forces applied. Rotational rigidity corresponds to bending moments, while torsional rigidity corresponds to twisting moments; in a torsional context, your virtual plane(s) would be planes of rotation. The same would not be true in a bending context.

For an analogy, consider the motion of a Rubik's Cube. When you twist one of the layers of the cube, the angle through which the layer moves is within the plane of the cube. Now, imagine twisting a softball-sized lump of rubber in the same way that you would twist a Rubik's Cube. Because the rubber is solid, the half in one hand is constrained relative to the half in the other hand, and the moment you apply deforms it. How much it deforms depends on its torsional stiffness; the internal stresses are shear stresses.

Likewise, consider the motion of an accordion. The two ends of the accordion are initially parallel but, when you hold the accordion in your two hands, you can manipulate it so that the ends are not parallel (because of its flexible diaphragm). Now recall that lump of rubber. Holding it in the same grip as before, you can apply a bending moment to deform the rubber in a different way, corresponding to bending rather than twisting. How much it deforms now depends on its rotational stiffness; the internal stresses are normal stresses.

If we have a slender beam or shaft, torsion is only practically relevant with respect to a single axis, and bending is only practically relevant with respect to the other two (i.e., the major and minor axes of a beam). If we have a lump of rubber, bending and torsion could occur and be significant in any direction. We don't have the advantage of simple equations developed for classical beams and shafts (or even less-simple methods used for, e.g., short beams), but "torsion" and "bending" are still meaningful.

If you define virtual planes, the angle between the two planes quantifies the relative rotation you're after. Ethan48's suggestion of using the term "dihedral angle" is technically correct, but probably more abstract/general than you need—identify the planes and say the angle between them is $\theta$. If you know that angle to be much smaller than $M$, don't even go through the trouble of naming the inverse of rotational stiffness; just claim that $\theta \approx 0°$, justify it with "high rotational stiffness" and be done with it.

Remember, if you go through the trouble to formally define virtual planes, that points fixed to the plane when the assembly is unloaded must remain collinear so that the virtual plane doesn't warp into a curved surface under loading. (Do you want to measure the angle between curved surfaces? I don't!)

• Excellent! This helps a lot, and the evolution from the standard objects to lumps of rubber was particularly helpful. I have a few questions. 1) I understand what you mean by torsion rigidity vs rotational stiffness, but I wonder whether torsion might be a subset of rotation, since it would make sense to me that a rotational stiffness would refer to any deformation that can be described by motion of the surface about a fixed axis (ie, at least one point on the surface that doesn't translate, like what's normally meant by rotation)... Jun 5 '15 at 4:33
• Also, 2) Can you recommend a book that describes these things very well. I'm ok to pay some cash for a good book, but I don't have access to a good library and I don't want to make risky purchases of expensive texts. I do own the Oxford Dictionary of M.E. and it says "rotational stiffness See torsional rigidity", and generally hasn't been up to the task of these issues. Jun 5 '15 at 4:45
• Torsion is not a subset of rotation in this context. For any arbitrary motion or deformation, you can define a frame of reference such that at least one point on the surface doesn't translate. I'm not aware of any way to change bending rotation into torsional rotation by redefining the frame of reference (in Euclidean space). Nor can you turn a normal stress into a shear stress, or vice versa. The lump of rubber example is supposed to help you understand intuitively that these are different bulk states; you hold it in the same manner, with arbitrary orientation, but apply force differently.
– Air
Jun 5 '15 at 16:30
• Unfortunately, I can't recommend any particular book. Some of this answer is based on my education but much of it is extrapolated and a priori. Approach it as you would any other answer—with a healthy dose of skepticism.
– Air
Jun 5 '15 at 16:44
• OK, skepticism noted. I'm also quite sure that I understand your distinction on bending rotation vs torsion rotation, and the internal stresses make this very clear as distinct consequences. It's only that the words "bending rotation" and "torsion rotation" make it seem that "rotation" might be the correct term for the superset since it's in both, but I'm fine to leave the question alone. Thanks again. Your input and essay have been very helpful! Jun 5 '15 at 16:55

In Euler-Bernoulli beam theory, this is known as bending, or, if the "beam" continues to oscillate, dynamic bending.

In this specific case - assuming that "up" is "up" in real life - this is called the first vertical bending. It is dynamically shown here:

Alternate terms include deflection (given here) and bending action (given here), though I'm less certain of the applicability of the latter.

• Thanks for you answer. I don't, though, think the terms you mention are what I'm looking for, as none of them specifically apply to or describe rotation. In the picture you show, for example, it's true that the end of the bar is rotating, but the general definition and equations of "bending" are about displacement not rotation, so it could still bend but not rotate. I specifically need the descriptor to be zero in the case of distortions that don't lead to rotation. Jun 3 '15 at 2:02
• @tom10 Ignore the picture and look at the Euler beam equations. They seem to specifically describe what you are asking about. HDE226868, You might want to add the equations to your post.
– hazzey
Jun 3 '15 at 2:21
• @hazzey: the equations are specifically describing curvature (a displacement related concept), whereas I am looking for a description of rotation. In a beam, or any body with a particular set of properties, these are all interrelated and can all seem like the same thing. But in a beam the rotation of the end results from how compression and stretching work for a beam, but something with very different properties would have a different rotation at the end, and therefore one need a different word for it. Jun 3 '15 at 14:51
• @tom10 Is your situation non-elastic? Are you trying to define a specific situation involving plastic deformation? If your objection to the word "rotation" is based on considering the piece as a whole, then maybe you need to add a virtual joint where the rotation takes place.
– hazzey
Jun 3 '15 at 15:11
• In beam theory, bending (divided by the beam's stiffness) is the derivative of the (tangent of the) angle of rotation, which is itself the derivative of the deflection. So, in effect, bending causes a rotation which causes a deflection.
– Wasabi
Jun 3 '15 at 15:40

First picture

If I understand the picture and explanation correctly, you are looking for a word to describe the change from "A" to "A+dA". In which case the word you need, in a generic sense, is rotation.

If you want something more specific to your particular situation, you are looking for the rotation of one plane relative to another.

Laughably, after writing this and re-reading the original question, relative rotation is actually used in the description! Is there any reason why you didn't think was the correct term?

Second picture

The top and bottom beams in the middle part are bending. For the overall structure plane sections have not remained plane. This would take the generalised description of distortion.

Summary

You have asked: What describes the rotation of one part of a body relative to another given an applied torque? The simple, generic, answer is distortion. But you are complicating things by looking at a body rather than a member. For a member, the rotation caused by a moment is, quite simply, bending. But you have rejected this because you are looking at a larger structure which has complicated relative stiffnesses etc. This means you lose the nice precise words and have to rely on generic ones instead. Hence why distortion is probably the most accurate, but isn't specific to rotation.

• Thanks! Relative rotation is certainly an option, and I might need to use it. The problem is that in my document: 1) I can't use pictures, and 2) my assembly doesn't really have clear surfaces (think of two spheres stuck together). So instead of, say, "torsional rigidity", I'd need something like "given any two virtual surfaces, one close to the binding surface and one close to the free end, when a force is applied to the free end, the two virtual surfaces will experience only a small relative rotation". Could do it though. (Also, upvote, esp for thinking of meanings rather than models.) Jun 3 '15 at 16:37
• @tom10 If you want to use brief language, you should start by rigorously defining the virtual surfaces in your document. Just like you would spell out an organization's full name the first time you mention it, with the abbreviation, and from then on use only the abbreviation. The more pithy you want to be throughout, the more verbose you have to be in your definitions at the beginning.
– Air
Jun 3 '15 at 18:09
• @Air: true, unless there's a standard term with a known meaning for the concept of "rotational shear". Finding out whether such a term exists is the purpose of this question. If possible, I don't want to start a document by describing what a horse is, if I can just say "horse". (Given the responses here though, apparently if there is a term for "rotational shear", it's not super common, but I'm fine with an uncommon term so long as it's applicable and clearly defined in, say, a technical dictionary.) Jun 3 '15 at 18:46
• The main problem from what I can tell is that we (or at least I) are having a hard time understanding exactly what sort of object we're talking about. I at least have no clue what you mean when you say "think of two spheres stuck together" and how that relates to the figure in the original post, which seems to me very uniform and beam-like. That being said, I'm a civil engineer, and you know what they say: to a hammer, everything looks like a nail.
– Wasabi
Jun 3 '15 at 19:43
• @Wasabi: I did get your comment immediately, btw, I just don't know what to say so I'm trying to come up with an answer. I never mentioned beams, and my picture specifically doesn't show any bending (ie, no compression) only rotation of the relevant surface. From wikipedia: "bending characterizes the behavior of a slender structural element", but my picture was specifically drawn to be pudgy. My rephrased summary at the top of my question is quite independent of specific structure.. did you see that? Jun 3 '15 at 20:54