# Elastoplastic behavior during torsion

## Plastic moment

Consider a bar with a thick-walled circular cross-section. The bar is modeled by an elastic-ideally-plastic material behavior, in shear, according to the figure below.

Let us determine the moment corresponding to initial yielding. This is obtained by using the relations in the previous section and considering the case when $\tau_{\rm y}$ for $r=b$ (where the stress is largest).

$$\tau_\mathrm{y} = \frac{ \Mv_\mathrm{y} b}{\Kv} \quad \Rightarrow \quad \Mv_\mathrm{y}= \frac{\tau_\mathrm{y} \, \Kv}{b}$$

We can also determine the plastic moment $\Mv_\mathrm{u}$ where the entire cross-section is in a plastic state. In this case the stress is $\tau=\tau_{\rm y}$ for all $r$ which gives

$$\Mv_\mathrm{u}=\int_A \tau_{\mathrm y} \, r \, {\mathrm d}A= \tau_{\mathrm y} \int_a^b r \,2 \pi r \, \dr= \tau_{\mathrm y} \, \frac{2 \pi}{3} (b^3-a^3)$$

An illustration of the corresponding stress state is shown in the figure below.

Note
The plastic moment $\Mv_\mathrm{u}$ is the limit load the cross-section can withstand before failing.

### Plastic strengthening

The plastic strengthening is defined as:

$$\beta = \frac{\Mv_\mathrm{u} – \Mv_\mathrm{y}}{ \Mv_\mathrm{y} } = \ldots =\frac{4b}{3} \cdot\frac{ b^3-a^3}{b^4-a^4}-1$$

and is a factor indicating how much the torque can be increased if one allows the cross-section to reach a fully plastic state, compared to initial yielding. For thin-walled cross-sections $\beta=0$, since the cross-section reaches a fully plastic state as soon as one point starts to yield.