Plastic moment and shape factor of a beam

Definitions

• Yield moment $M_y$: the moment at which the extreme fibre first reaches $\sigma_y$.
• Plastic moment $M_p$: the moment when the whole section has yielded (a plastic hinge forms).
• Shape factor $S = M_p/M_y = Z_p/Z$: depends only on the section shape.

Assumptions: elastic-perfectly plastic stress-strain curve, same behaviour in tension and compression, plane sections remain plane, and no axial force.

Derivation (rectangular section)

Yield moment

$$M_y = \sigma_y Z = \sigma_y\,\frac{bh^2}{6}$$

Elastic-plastic stage. Let the elastic core have half-depth $y_0$ ($y_0 \le h/2$). The stress is $\sigma_y y/y_0$ in the core and $\sigma_y$ in the outer zones.

$$M = \underbrace{\sigma_y\,\frac{b\,(2y_0)^2}{6}}_{\text{core}} + \underbrace{2\sigma_y\,b\,\left(\frac h2 - y_0\right)\left(\frac{h/2 + y_0}{2}\right)}_{\text{outer zones}} = \sigma_y\,b\left(\frac{h^2}{4} - \frac{y_0^2}{3}\right)$$

Fully plastic ($y_0 \to 0$):

$$M_p = \sigma_y\,\frac{bh^2}{4} = \sigma_y Z_p,\qquad Z_p = \frac{bh^2}{4}$$

Shape factor

$$S = \frac{M_p}{M_y} = \frac{bh^2/4}{bh^2/6} = 1.5$$

Numerical values ($b = 60$, $h = 100$, $\sigma_y = 250$ MPa)

$$\begin{aligned} M_y &= 250\times\frac{60\times100^2}{6} = 250\times10^5 = 25.0\ \text{kN m} \\ M_p &= 250\times\frac{60\times100^2}{4} = 250\times1.5\times10^5 = 37.5\ \text{kN m} \\ \end{aligned}$$

Elastic core 60 mm deep, so $y_0 = 30$ mm:

$$M = 250\times60\left(\frac{100^2}{4} - \frac{30^2}{3}\right) = 15000\times(2500 - 300) = 33\times10^6\ \text{N mm} = 33.0\ \text{kN m}$$

(A value between $M_y$ and $M_p$, as expected.)

Collapse load (simply supported, central load $W$, span $L = 3$ m)

At collapse a plastic hinge forms under the load, with $M_{max} = WL/4 = M_p$:

$$W_p = \frac{4M_p}{L} = \frac{4\times37.5}{3} = 50.0\ \text{kN},\qquad W_y = \frac{4M_y}{L} = 33.3\ \text{kN}$$

Answer: $M_y = 25$ kN m, $M_p = 37.5$ kN m, shape factor $= 1.5$, $M = 33$ kN m for a 60 mm elastic core, and the collapse load is $50$ kN (first yield at $33.3$ kN).