Plastic hinge and its length

Plastic hinge

A plastic hinge is a section of a beam where the plastic moment $M_p$ has been reached: the whole depth has yielded, so the section rotates freely at a constant moment $M_p$, like a hinge that still carries $M_p$. In reality yielding spreads outwards from the section of maximum moment over a finite length, called the plastic hinge length $L_p$.

How the length is determined

The hinge length is the length of the member over which the bending moment exceeds the yield moment $M_y$, so part of the depth is yielded. It follows from the BMD at collapse by finding the points where $M = M_y = M_p/S$ ($S$ = shape factor).

Simply supported beam, central point load $P$: $M = \dfrac{P}{2}x$ for $x \le L/2$ and $M_{max} = \dfrac{PL}{4} = M_p$. The yield moment is reached at $x_y = \dfrac{L}{2S}$ from each support, so

$$L_p = L - 2x_y = L\left(1 - \frac1S\right)$$

For a rectangular section ($S = 1.5$): $L_p = L/3$.

Simply supported beam, UDL $w$: $M = \dfrac{wLx}{2} - \dfrac{wx^2}{2}$ and $M_{max} = \dfrac{wL^2}{8} = M_p$. Setting $M = M_p/S$ gives $x_y = \dfrac L2\left(1 - \sqrt{1 - 1/S}\right)$, so

$$L_p = L\sqrt{1 - \frac1S}$$

For a rectangular section: $L_p = 0.577L$.

So the hinge length depends on the shape factor and on the loading and span. (The larger $S$, the longer the hinge; in the idealised analysis, $L_p \to 0$.)