7.1.1 The plastic moment

For pure bending $N = 0$, we use Eq. (7.2). The plastic moment $M_{y{\,}pl}$ about the neutral axis y in a plastic hinge can be computed from Eq. (7.3).

$$\int_{A_{t}}\sigma_{Y}dA - \int_{A_{s}}\sigma_{Y}dA = 0$$ (7.2)

$$\int_{A_{t}}z\sigma_{Y}dA - \int_{A_{s}}z\sigma_{Y}dA = M_{y{\,}pl}$$ (7.3)

where $\sigma_{Y}$ is the yield stress, ${A_{t}}$ - an area in tension, ${A_{s}}$ - an area in compression.
Equations (7.2) and (7.3) give

$$A_{t} - A_{s} = 0,$$ (7.4)

$$\left\vert S_{t}\right\vert + \left\vert S_{s}\right\vert = \frac{M_{y{\,}pl}}{\sigma_{Y}}$$ (7.5)

where ${S_{t}}$ and ${S_{s}}$ are accordingly the first moment of the area in tension and that of the area in compression about neutral axis.

Figure 7.2: Rectangular cross section

From Eq. (7.5) we get the expression for the plastic moment

$$M_{y{\,}pl} = W_{y{\,}pl}\sigma_{Y}$$ (7.6)

where the plastic section modulus is given by

$$W_{y{\,}pl} =\left\vert S_{t}\right\vert + \left\vert S_{s}\right\vert$$ (7.7)

For a rectangular cross section shown in Fig. 7.2, the plastic section modulus is

$$W_{y{\,}pl} = S_{t} + S_{s} = \frac{1}{2}bh\left( \frac{h}{4} + \frac{h}{4}\right) = \frac{bh^{2}}{4}$$ (7.8)

For a rectangular cross section, the elastic section modulus is

$$W_{y{\,}el} = \frac{bh^{2}}{6}$$ (7.9)

The ratio of the elastic section modulus to the plastic section modulus is

$$\alpha_{pl} = \frac{W_{y{\,}pl}}{W_{y{\,}el}} = \frac{\frac{bh^{2}}{4}}{\frac{bh^{2}}{6}} = 1.5$$ (7.10)

This ratio is termed the shape factor. The shape factor is a measure of the efficiency of a cross section in bending. Shape factors for some cross sections are given in Fig. 7.3.

Figure 7.3: Shape factor $\alpha_{pl}$ for different sections

Fig. 7.4 shows stresses at loading and unloading with a pure bending moment. At unloading, the stresses decrease (Fig. 7.4 g). The residual stresses on the cross section after removing a moment $M_{y}$ are shown in Fig. 7.4 h.

Figure 7.4: Cross section and stresses