# Torsion (6.2.7)

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

Distributed torsion moment loading on a channel to hollow-core slabs. The load acts besides the shear center

## Simple St. Venant and non uniform torsion with warping

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.

It depends what kind of supports restraint a beam. If beam is free against torsion (on the left), then no stresses associated to warping. Otherwise (on the right) constraint preventing warping can cause additional shear and normal stresses.
• Simple, uniform (St. Venant) torsion
• Non uniform torsion (with restricted warping)

### Simple St. Venant torsion

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.

### Non uniform torsion with restricted warping

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.  Warping stresses in flanges on thin walled open section; Bimoment is the distribution of axial stresses that is needed to reduce the warping of the section

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

## Three basic cases to deal with for assessment

### I. Solid cross-sections  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.

#### Rectangle

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

$b/h$$\alpha$$\beta$
10.1410.208
20.2290.246
100.3120.312
$\infty$0.3330.333
Tangential stress distribution for rectangle solid section from torsion

### II. Thin walled open cross-sections  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 $\sigma_\omega$.

#### Simple torsion case

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 $t$)
$$\begin{equation} I_t = \frac 1 3 \alpha \sum b_i t_i^3 \end{equation}$$
 $\alpha =$ 1 ... L sections 1.2 ... I, U rolled sections 1.3 ... I high welded sections

#### Constrained warping case

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 [m4] $I_\omega$ torsion warping constant (in tables) [m6] $t$ thickness of the wall where the stress is being determined
\begin{align} S_\omega &= \int_A \omega \ dA \\ \omega &= \int_{C_t} r\ ds, \end{align}

where $\omega$ is sectorial area ([m2]).

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.

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