Plastic Analysis of Structures

Beyond elastic design

Elastic design treats a structure as failed when the extreme fibre first reaches yield. Ductile steel structures actually carry much more load: sections can yield fully, and indeterminate structures redistribute moments until enough plastic hinges form to turn the structure into a mechanism. Plastic analysis finds this collapse load. It is the basis of limit state design of steel (IS 800:2007 permits plastic analysis for suitable sections).

Assumptions

1. The material is elastic–perfectly plastic: linear up to $f_y$, then yields at constant stress (strain hardening ignored).
2. Plane sections remain plane.
3. Properties are the same in tension and compression.
4. Sections are compact enough to develop full plastic moments without local buckling; lateral and overall instability is prevented.
5. Effects of axial force and shear on the plastic moment are neglected (or treated separately).
6. Deformations are small until collapse; plastic hinges form at discrete points.

Yield moment and plastic moment

As moment increases on a section:

1. Elastic — stress linear, maximum $<f_y$.
2. First yield — extreme fibre reaches $f_y$: yield moment $M_y=f_y{Z}_e$.
3. Partly plastic — yielding spreads inward.
4. Fully plastic — the whole section at $\pm f_y$: plastic moment $M_p=f_y{Z}_p$.

In the fully plastic state the plastic neutral axis divides the section into two equal areas (not necessarily through the centroid).

Shape factor

It measures the reserve of strength beyond first yield — depends only on the section shape.

Plastic hinge

When $M=M_p$ at a section, it can rotate freely at constant moment — a plastic hinge. Unlike a real hinge, it resists the moment $M_p$ and it can "unload" elastically if the moment reduces.

• Hinges form at points of maximum moment: under concentrated loads, at fixed supports, at intermediate supports, at joints of frames, and at zero-shear points under distributed loads.
• The length of the plastic hinge (the zone over which moment exceeds $M_y$) depends on the shape factor and the loading: for a simply supported rectangular beam ($S$ = 1.5) it is $L(1-1/S)=L/3$ under a central load and $L\sqrt{1-1/S}=L/\sqrt{3}\approx 0.577L$ under a UDL.

Mechanism

A structure collapses when the number of plastic hinges is enough to form a mechanism:

for a complete collapse of the whole structure; a partial mechanism (e.g. one span of a continuous beam) may need fewer hinges. Common mechanisms: beam, sway (panel), combined, gable and joint mechanisms.

Theorems of plastic collapse

For a collapse load factor $\lambda$:

Virtual work for a mechanism

External work $=\sum W_i{\delta }_i$; internal work $=\sum M_p{\theta }_i$ (at every hinge, whatever the sign of rotation). Set them equal.