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 $I_t=\alpha bh^3$, $W_t = \beta bh^2$.

$b/h$	$\alpha$	$\beta$
1	0.141	0.208
2	0.229	0.246
10	0.312	0.312
$\infty$	0.333	0.333

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.

Restraints against torsion		Torsion distribution		$\alpha$	$\beta$
Supports at both ends	Simple support (warping allowed)	uniform full		3.1	1.00
	Simple support (warping allowed)	general		3.7	1.08
	Fixed support (warping restricted)	uniform full	for internal forces at supports	8.0	1.25
		uniform full	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 $K_t \ge 15$.

Torsion on cantilevers

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.

III. Thin walled closed cross-sections

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

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