# What is the difference between plastic and compact cross sections?

I'm studying beam cross section classifications here.

It is explained that the difference between plastic and compact cross sections is that while both do develop their plastic moments, only plastic cross sections have sufficient rotation capacity to form a collapse mechanism.

I don't quite understand what happens when a compact cross section undergoes local buckling. In the text there is an example of a rigidly supported beam. When enough moment is applied, the ends develop plastic hinges and from there on the beam acts as a simply supported beam with moments on the ends. With more rotation, the middle also develops a hinge and the structure collapses.

So let's consider this scenario with both a plastic and a compact cross section. Plastic cross section develops the plastic hinge, and has enough rotation capacity to form the collapse mechanism. But what happens with the compact cross section? By definition it also forms the plastic hinge, but this one doesn't have the required rotation capability to form the collapse mechanism, so what happens with this one?

So how does the compact cross section form a plastic hinge and not a complete collapse mechanism?

Plastic cross-sections can develop full plastic moment, $$\theta= \theta_{M_p}$$, and maintain it even under additional torque causing additional rotation.

They can undergo additional rotation without loss of strength till $$\ \theta_u$$.

Consider a plastic fixed, fixed, section loaded with a uniform torque $$\tau= t/m,$$ with the full plastic moment at the supports. If you keep increasing the load it will support it until you reach $$\theta=\theta_u>\theta_p$$.

Compact sections however do not have reserve ductility to maintain their strength beyond the rotation $$\theta= \theta_{M_p}.$$ They fail due to the local buckling of the flanges. They do not have a wide plateau of ductility.

• Compact section don't have a large plateau of ductility and small rotation capacity, but what is the decisive amount of rotation a section has to be able handle in order to be a plastic section? If a section is plastic instead of compact, what exactly does this mean in terms of forming a collapse mechanism? Dec 13, 2020 at 17:00
• it means the ratio t to b, is such that the beam under excessive torque would warp and suffer saint Venants torsion to a higher level. of course, the beam will form the 3rd hinge eventually after $\theta u$ rotation. Dec 13, 2020 at 17:10
• Great answer! Very clear explanation of material performance and hence why we have different steel grades for construction of safer structures Apr 11 at 15:05

Simply put, a compact section is one that can develop a plastic hinge prior to local buckling of the flange or the web, whereas a non-compact section will buckle at a lower stress level ($$\sigma$$ < fy) before the entire section attains full plasticity.

A plastic hinge is formed when the entire cross-section has stressed to yield as shown below.

So the ideal design is to have a compact section, or with adequate bracing, to ensure that plastic hinge can form at the end supports but not to occur anywhere in the span that would trigger the collapse mechanism.