Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Bending and Torsion of Beams Springs
Bending and Torsion of Beams AimsBefore you startIntroductionPole-vaultingBending moments and beam curvaturesMaximising the beam stiffnessBeam deflections from applied bending momentsTwisting moments (torques) and torsional stiffnessSpringsPlastic deformation during beam bendingSummaryQuestionsGoing further
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Springs

A collection of assorted springs

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An interesting example of torsion is provided by the deformation that takes place during the loading of springs (torsional coils). Of course, these have a wide range of engineering applications. They are normally made of (high yield stress) metals. (Ceramics are too brittle, while polymers are insufficiently stiff: fibre composites are also unsuitable - see below.) When a spring is loaded (compressed or extended), the deformation experienced by the wire is one of pure torsion. This is illustrated in the diagram below.

Illustration of how the application of an axial load, F, to a spring generates torsional deformation of the wire and hence axial extension of the spring.

The torque acting on the wire is given by

\[T = F\left( {\frac{D}{2}} \right)\]

in which F is the axial force and D is the coil diameter. It can be shown ( click here ) that the shear stress within the wire (at a distance r from the core) is given by

\[\tau = \frac{{Tr}}{{2I}}\]

in which I is the bending second moment of area (NOT the polar moment), and the shear strain in the wire is related to the change in axial extension of one turn of the coil, s , by the expression

\[\gamma = \frac{{2sr}}{{\pi {D^2}}}\]

Measurement of the extension (per turn) of a spring, as a function of the applied force (first carried out systematically by Robert Hooke , in his pioneering work on the nature of elasticity) is a very convenient method of obtaining elastic constants. The ratio of τ to γ , obtained from the above equations, gives the shear modulus, G . The loading geometry is such that a large axial extension (per turn) is generated, while the strains within the material remain low, particularly for springs with a large ratio of D to w . Of course, this is exactly why springs are of practical use - they accommodate large deflections or displacements without the material being strained beyond its elastic limit (which is small for all materials except rubbers).

It's interesting to note why springs are not normally made of fibre composites. The natural orientation for the fibres would be along the length of the rod (wire) to be formed into a coil. However, these fibres would have very little effect on the shear modulus in a transverse section of the rod, which is the property that controls the elastic extensions of the spring. It might as well have been made solely of the polymeric matrix (although such springs have a very low stiffness). It's only by winding fibres into the hoop direction of the rod that the shear stiffness of transverse sections would be boosted. However, this is impractical, at least for anything but very large scale springs, since it would require the fibres to adopt higher curvatures than would normally be possible.

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