What is meant by the rotation capacity of a cross section?

It is said:

The ratio of the ultimate rotation to the yield rotation is called the rotation capacity of the section. The yield and the plastic moments together with the rotation capacity of the cross-section are used to classify the sections.

I understand the concept of yield moment, it is the moment at which the extreme fiber of the cross section reaches the yield. But the page also talks about yield rotation. Rotation of what? I understand that we normally measure the rotation of the ends of the beam, but here it is supported rigidly at the ends, so what rotation are we measuring here?

If we thought about a simply supported beam so we could measure the rotation of the end, isn't the yield rotation a property of the beam, rather than the cross section? If the beam is longer, wouldn't the required rotation (to make the critical cross section reach yield) probably be smaller, making the yield rotation dependent on the whole beam rather than just cross section?

Rotation of the section over the length with the particular moment.

Skip forward to the end: in a theoretical section made of a material with perfect plasticity (no hardening, infinite strain capacity) at the plastic moment, there is a plastic hinge. Once you've reached the fully plastic moment, any increment of moment will result in infinite rotation of that hinge.

Obviously real beams made of real material don't exhibit infinite rotation in a hinge.

In most real sections formed of real plastic material (like steel) there are other mechanisms at play, and often you can't actually get the whole section plastic. With respect to classification of sections, the most significant other effect is local buckling. Some part of the section reaches a limiting compressive force and buckles, so limiting the compressive force that element can carry, thus limiting the moment the section can carry. So you get a sort of hinge type behaviour in the section (no greater moment can be carried) without plasticity across every part of the section.

Furthermore, you can get a behaviour where the full plastic moment is generated, but as rotation occurs then an element buckles (rather than squashes). As the hinge rotates, the moment the section carries drops dramatically (just like a classic strut buckling - once it has buckled the axial load drops compared to the load that caused it to buckle). In this case you don't get a long flat plateau on your moment/rotation curve, you get a buckling type behaviour and the moment plummets past the peak.

Thus, you get the classes of section described, illustrated in your reference at figure 4.

A slender section doesn't even get to first yield - you get a local buckling (typically of the flanges) before the section even gets to yield. When the flanges buckle they can't take any more compressive force, so you get a hinge-ish behaviour, there's a localised rotation.

A semi-compact section can carry enough moment that some part of it has started to yield before the flanges buckle, but then they do buckle before you get to the whole cross-section at yield stress.

Compact and plastic sections both achieve the full plastic moment, and differ in the post-yield behaviour. If you had a square or circular solid cross section with nothing to buckle you can generate the plastic hinge and then just keep bending it, getting vey high concentrated rotations (i.e. hinge-like) without any reduction in the moment carried. However, quite a few sections that can generate the full plastic moment can't actually sustain that moment with significant rotation - when they start rotating the outstands actually buckle. Then the moment carried drops dramatically. So you can achieve the plastic moment, but you don't achieve an actual plastic hinge.

Note that the current UK (and Europe) codes - EC3 (aka EN1993) has classes 1 to 4. Class 1 can form a plastic hinge, class 2 can generate the plastic moment but not a hinge - local buckling prevents the hinge, class 3 gets to yield moment but cannot get to plastic moment, and class 4 doesn't even get to yield stress at any point in the section before some part of it buckles - i.e. the same classification, just different labels.

Possibly (arguably) figure 4 could be thought of as rotation per unit length (i.e. some function of curvature), but these behaviours occur over a finite length of beam (a buckle, for example, needs some length). To be rigorous, the graph is probably the rotation (i.e. change in angle) of an elemental but finite length of beam with a constant moment along the whole length of that element, thus giving rise to a finite rotation across the element. It's not curvature, because that breaks down at a plastic hinge.

Mathematically, the graph (and discussion) is probably conflating (and muddling) delta-theta/delta-position with d-theta/d-position (something that engineers are somewhat prone to do, causing mathematicians to shake their heads and tut).

• Thank you for the thorough answer. So my question basically boils down to this: the diagram 4 shows moment against rotation for a cross section. What does rotation mean here? Does it mean the angle it turns to after being subjected to moment? Or are we really talking about curvature, the second derivative of deflection? Dec 7 '20 at 21:25
• Rotation of an elemental but finite length of beam with a constant moment along the whole length of that element. It cannot be curvature because that makes no sense at a hinge (or part hinge). I'll add that to teh answer. Dec 8 '20 at 8:11
• Rotation, as in an angle? But doesn't the angle an element makes depend on the beam as a whole? I mean, we can twist a beam at some other end causing it to bend, so now other parts of the beam have some rotation but no moment across their cross section.. Dec 8 '20 at 9:17
• I don't see how the rotation of a single cross section or finite element is determined by the moment on that element alone. Dec 8 '20 at 9:19
• I'm just repeating myself. Rotation of an elemental but finite length of beam with a constant moment along the whole length of that element. Rotation across that element. Difference in angle between each end of that element. You have a non-zero-length element of beam subject to a moment, so some rotation occurs across that element. Dec 8 '20 at 16:37