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

DoITPoMS Teaching & Learning Packages Bending and Torsion of Beams Derivation of equations
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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Derivation of equations

Consider the small elemental length of the wire shown in the figure, subtending an angle dβ at the axis of the coil. The torsional shear stress within the wire, τ, can be found by noting that it varies linearly with distance from the centre of the wire ( τ = κ r , where κ is an unknown constant and r is the distance of the element from the centre of the wire) . The torque can now be expressed in terms of internal forces in the wire. The force in an individual element of the wire is given by the torsional shear stress, τ, multiplied by the area of the element, 2 π r dr . Therefore, the elemental torque is simply this force multiplied by the distance of the element from the neutral axis of the wire, r . Summation of these elemental torques gives the total torque:

\[T = \int\limits_0^{w/2} {\tau (2\pi r\;{\rm{d}}r)r = 2\pi K\int\limits_0^{w/2} {{r^3}} } {\rm{d}}r = \frac{\pi }{{32}}K{w^4}\]

\[\tau = Kr = \frac{{32Tr}}{{\pi {w^4}}} = \frac{{Tr}}{{2I}}\]

where I is the second moment of area for a solid cylinder (the wire), given by:

\[I = \frac{{\pi {w^4}}}{{64}}\]

The local shear strain in the material is given by

\[\gamma = r\frac{{{\rm{d}}\theta }}{{{\rm{d}}L}} = \frac{{r\;{\rm{d}}\theta }}{{(D/2){\rm{d}}\beta }}\]

in which dθ /dL is the rate of twist along the length of the wire. The incremental angle of twist of the wire is therefore related to the corresponding incremental rotation angle of the coil by

\[{\rm{d}}\theta = \gamma \frac{{(D/2)}}{r}{\rm{d}}\beta \]

The axial displacement of the coil, due to the twisting of the elemental section, can therefore be written

\[{\rm{d}}s = \left( {\frac{D}{2}} \right){\rm{d}}\theta \; = \left( {\frac{D}{2}} \right)\gamma \left( {\frac{D}{{2r}}} \right){\rm{d}}\beta \]

The axial displacement associated with the twist in one complete turn of the coil is thus

\[s = \left( {\frac{{{D^2}}}{{4r}}} \right)\gamma \;\int\limits_0^{2\pi } {{\rm{d}}\beta } = \frac{{\pi {D^2}}}{{2r}}\gamma \]

The local shear strain in the wire is therefore related to the increase in spacing between adjacent turns in the coil by

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

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