Torsion

Torsion refers to the twisting of a specimen when it is loaded by couples (or moments) that produce rotation about the longitudinal axis. Applications include aircraft engines, car transmissions, and bicycles, etc.

Units

Force X distance [lb-in or N-m]

Notation and Convention

Fig: Convention

Taken from TAM251 Lecture Notes - L5S11

$ \phi > 0 $ : counter clockwise
$ \phi < 0 $ : clockwise

Fig: RightHandRule

Taken from TAM251 Lecture Notes - L5S11

Torque and angle of twist follow the right hand rule sign convention. When positive, using the right hand, the thumb points outward from the shaft and the fingers will curl in the direction of the positive twist/torque.

Equilibrium

The stress distribution in the shaft is not known.

Statically indeterminate: must consider shaft deformations.
Multi-planar: equilibrium requires the existence of shear stresses on the faces formed by the two planes containing the axis of the shaft

Fig: Equilibrium

Taken from TAM251 Lecture Notes - L5S3

Assumptions

For circular shafts (hollow and solid): cross-sections remain plane and undistorted due to axisymmetric geometry
For non-circular shafts,: cross-sections are distorted when subject to torsion
Linear and elastic deformation

Fig: TorsionLines

Taken from TAM251 Lecture Notes - L5S5

Shear Stress and Strain

Shear Strain: Geometry of Deformation

Fig: GeometryDeformation

Taken from TAM251 Lecture Notes - L5S6

The angle of twist increases as x increases. The twist rate is given by:

$$ \frac{d\phi}{dx} = \frac{\gamma}{\rho} = \frac{\gamma_{max}}{c}\ $$

Moving terms we get:

$$ \gamma = \rho \frac{d\phi}{dx}\ $$

Hence, the shear strain ($ \gamma $):

is proportional to the angle of twist
varies linearly with the distance from the axis of the shaft
is maximum at the surface

Shear Stress: Torsion Formula

Geometry:

$$ \gamma = \rho \frac{d\phi}{dx}\ $$

Hooke's Law:

$$ \tau = G\gamma\ $$

From equilibrium:

$$ \frac{d\phi}{dx} = \frac{T}{GJ}\ $$

Elastic torsion formula:

$$ \tau = \frac{T\rho}{J}\ $$

The shear stress in the elastic range varies linearly with the radial position in the section

Fig: StressDistribution

Taken from TAM251 Lecture Notes - L5S8

Note: shaft under torque T rotating at angular speed w transmits power

$$ P=T\omega\ $$

Shear Stress: Polar Moment of Inertia

Solid Shaft (radius and diameter):

$$ J = \int_0^R{\rho^2dA} = \int_0^R{\rho^2(2\pi\rho d\rho)dA} =2\pi \int_0^R{\rho^3d\rho} = \frac{\pi R^4}{2} = \frac{\pi D^4}{32}\ $$

Hollow Shaft (inner and outer radius):

$$ J = \frac{\pi}{2}(R_o^4-R_i^4) = \frac{\pi}{32}(D_o^4-D_i^4)\ $$

Shear Stress: Symmetry in Axial Planes-?? /!\BSM: not sure what this topic is; can probably omit

**Reference pages have a broken link image X2 here and no text**

Shear Stress: Angle of Twist

From observation:

The angle of twist of the shaft is proportional to the applied torque
The angle of twist of the shaft is proportional to the length
The angle of twist of the shaft decreases when the diameter of the shaft increases

Angle of Twist:

$$ \phi = \frac{TL}{GJ}\ $$

Torsional stiffness:

$$ k_T = \frac{GJ}{L}\ $$

Torsional flexibility:

$$ f_T = \frac{L}{GJ}\ $$

Miscellaneous

Types of Material Failure

Ductile materials generally fail in shear

Fig: ductile

From reference pages

Axial: maximum shear stress at

Torsion: maximum shear stress at

Brittle materials are weaker in tension than shear

Fig: brittle

From reference pages

Axial: maximum normal stress at

Torsion: maximum normal stress at

Thin-walled hollow shafts /!\ BSM: we do not cover this topic in class/hw/exams. Not sure if/when this topic is covered in ME curriculum.

In general, the maximum shear stress is given by

$$ \phi = \frac{TL}{GJ}\ $$

For thin-walled shafts:

$$ \tau_{max} = \frac{T}{2tA_m}\ $$

where

$$ \begin{align} A_m &= \pi R_{ave}^2 \\ R_{ave} &= \frac{R_o + R_i}{2} \end{align} $$

Note that is NOT the cross sectional area of the hollow shaft!

Gear Systems - do we want to add?

BSM: I think it's a nice idea to add some basic info/equations. The biggest is the constraint of the statics version of the ``no slip'' condition between gears (mated gears twist through the same arc length).

Fig: GearSystem

Taken from TAM251 Lecture Notes - L5S15

TAM 2xx References