Torsion usually happens since a load is not applied over the torsion axis (shear center). Torsion is likely to happen in combination with bending.

From the mechanics of material we recognize two types of torsion. Some
cross-sections produce only uniform St. Venant torsion. For these, only tangent
stress $\tau_t$ occur. Other, open cross-section might produce also
*normal* stress as a result of torsion load: **it depends whether there are constraints against
warping torsion.**

- Simple, uniform (St. Venant) torsion
- Non uniform torsion (with restricted warping)

We can meet simple torsion on two basic occasions:

- 1) full sections, 2) closed hollow profiles or 3) thin walled profiles whose members intersect at one cross-point: only tangent stress as a response, no significant warping present;
- an open cross-section: warping occurs but due to designed supports there are
**no constraints to prevent**it.

A beam subjected to torsion resists the load by its torsion resistance. Only tangent stresses $\tau_t$ occur.

In the response to the torsion and designed supports there are observed warping deformations along the member, for example open section is loaded by a torsion with one fixed support: the fixed support restricts warping and as a result there are not only tangent stresses (as in simple tension) but also normal stresses.

**Since the warping is being prevented, additional stresses affect section: warping torsion $\boldsymbol{\tau_{\omega}}$ and normal stress $\boldsymbol{\sigma_{\omega}}$.**

These cross-sections are not so common in steel structures. From mechanics of material

$$ \begin{equation} \tau = \frac{Tc}{I_t}, \end{equation} $$where $T$ is applied torque load, $c$ is radius, $I_t$ is torsion section constant.

For example rectangle has $I_t=\alpha bh^3$, $W_t = \beta bh^2$.

$b/h$ | $\alpha$ | $\beta$ |
---|---|---|

1 | 0.141 | 0.208 |

2 | 0.229 | 0.246 |

10 | 0.312 | 0.312 |

$\infty$ | 0.333 | 0.333 |

These sections are sources of large deformation from torsion. Then it depends on types of restraints:

- If warping is
**not**being prevented, then only simple torsion $\tau_t$ occur. - If warping
**is**being prevented, then the beam is stressed not only by $\tau_t$, but also by $\tau_\omega$ and $\sigma_\omega$.

The highest stress is at the member with maximum thickness:

$$ \begin{equation} \tau_{t,i} = \frac{T_{T,Ed}}{I_t}t_i \end{equation} $$$\tau_{t,i}$ | tangent stress from simple torsion |

$T_{T,Ed}$ | St. Venant torsion moment from external load |

$I_t$ | torsion section constant |

$t_i$ | thickness of considered element (highest stress at max $t$) |

$\alpha = $ | 1 | ... L sections |

1.2 | ... I, U rolled sections | |

1.3 | ... I high welded sections |

We have to find both components of non uniform torsion: $\tau_{\omega},\ \sigma_\omega$. The stress $\tau_{\omega}$ from restrained warping is evaluated as

$$ \begin{equation} \tau_{\omega} = \frac{T_{\omega,Ed}S_\omega}{I_\omega t} \end{equation} $$$\tau_{\omega}$ | tangent component of the warping stress (caused by restricted warping) |

$T_{\omega,Ed}$ | non uniform (warping) torsion moment from external load |

$S_{\omega}$ | sectorial static moment where the stress is being determined [m^{4}] |

$I_\omega$ | torsion warping constant (in tables) [m^{6}] |

$t$ | thickness of the wall where the stress is being determined |

where $\omega$ is sectorial area ([m^{2}]).

The other component of non uniform torsion $\sigma_\omega$, which acts as an axial stress is determined from bimoment $B_{Ed}$ from external load:

$$ \begin{equation} \sigma_\omega = \frac{B_{Ed}}{I_\omega}\omega \end{equation} $$Now remains the task to determine $B_{Ed},\ T_{t,Ed},\ T_{\omega,Ed}$. These components can be found from differential equations describing the beam: they are function of eccentricity and depend on load distribution on the beam and on type of supports. But since these DE are difficult to work with in practice, there are shorter ways.

Firstly torsion moment is transimtted by simple torsion $T_t$ and by bending torsion $T_{\omega}$: $T=T_{\omega} + T_t$ or in other words

$$ \begin{align} T_\omega &\cong T(1-\chi) \\ T_t & \cong T\cdot \chi \hskip2em \\ & \text{or} \nonumber \\ T_\omega &\cong Ve(1-\chi) \label{Torsion:eq2} \\ T_t &\cong V\cdot e\cdot \chi \label{Torsion:eq3} \\ B &\cong Me(1-\chi) \end{align} $$

The value $\chi$ is to be found by means of tables. An example follows, there exist also other, more descriptive tables:

$$ \begin{align} \chi &= 1 / [\beta + (\alpha / K_t)^2] \\ K_t &= L \sqrt{GI_t/{EI_\omega}} \end{align} $$Values $\alpha, \ \beta$ are specific to load distribution, supports of the beam and are taken from the table.

Restraints against torsion | Torsion distribution | $\alpha$ | $\beta$ | ||
---|---|---|---|---|---|

Supports at both ends | Simple support (warping allowed) | uniform full | 3.1 | 1.00 | |

general | 3.7 | 1.08 | |||

Fixed support (warping restricted) | uniform full | for internal forces at supports | 8.0 | 1.25 | |

for internal forces inside the span | 5.6 | 1.00 | |||

general | 6.9 | 1.14 | |||

Cantilever | Fixed support | general—for internal forces at supports | 2.7 | 1.11 |

For a simple supported beam

- it is possible to ignore effect of simple torsion $(T_{t,Ed})$ if $K_t \le 1$ and
- it is possible to ignore effect of constrained warping $(B_{Ed},\ T_{\omega,Ed})$ if $K_t \ge 15$.

According to the profile/section and loading we have a tangent and/or
normal stress $\sigma_\omega$ as an impact of torsion load. **The stress $\boldsymbol{\sigma_\omega}$ has to be considered
together with the stress from bending.**

The material property—value of yield tangent stress—is $f_y / \sqrt{3}$ (as in the case of shear). For the allowed yield stress $f_y / \sqrt{3}$ resistance can be determined using above formulas $(\ref{Torsion:eq2}),\ (\ref{Torsion:eq3})$:

$$ \begin{align} T_{Rd} &= T_{t,Rd} + T_{\omega,Rd}, \hskip2em \text{then} \\ T_{Ed} &\le T_{Rd} \label{Torsion:eqAs} \end{align} $$In $(\ref{Torsion:eqAs})$ $T_{Ed}$ is design torsional moment from the load, $T_{Rd}$ is resistance torsional moment.

Such sections are either available on the market or can be made in situ. They are much more effective in resisting torque than similarly configured open sections, often by orders of magnitude. Stresses $\sigma_\omega,\ \tau_\omega$ are negligible (commonly ignored)—compared to the simple torsion stress $\tau_t$ which is prevalent.

Contrary to open cross-section, the maximum shear $\tau_t$ is where thickness is the smallest:

$$ \begin{align} \tau_{t,i} &= \frac{T_t}{\Omega \cdot t_i}, \\ \Omega &= 2 A_s. \end{align} $$$T_t$ | (St. Venant) torsion moment from external load |

$\Omega$ | double of area onclosed by axes of elements |

$t_i$ | thickness of considered element |