# 7.9: Summary

- Page ID
- 23528

You should now understand the basic principals of bending and torsion. You should be able to predict how a beam will respond elastically to a bending moment, from a knowledge of the Young's modulus, *E *, and the sectional geometry of the beam (from which the second moment of area, *I *, is derived). You should understand the relationship between the (local) bending moment, *M *, the beam stiffness (flexural rigidity), *EI *, and the resultant (local) curvature, *κ*.

The concept of torsion has been introduced, with the analogue of the bending moment being a torque, *T *, and the analogue of the curvature being the rate of twist of the beam, θ / *L *. The elastic constant controlling the behaviour is the shear modulus, *G *, and the sectional geometry analogue of the second moment of area, *I *, is the polar second moment of area, *I*_{P}.

You should also have an appreciation of the nature of the stress distribution within an elastically deformed beam and you should understand that, for a metallic beam, it's possible that these stresses could exceed the yield stress, σ_{Y} , so that plastic deformation could take place. In this case, there is a change in the relationship between the applied moment and the resultant curvature (so that a given increase in moment gives a larger increase in curvature). Furthermore, on removing the applied moment, the beam retains a residual curvature. Analogous phenomena can occur during torsion. These effects can be quantitiatively predicted.

Some implications of these analyses for the design of components and structures subject to bending moments and torques have been briefly outlined.

## Going further

### Books

- S.P. Timoshenko, J.N. Goodier,
*Theory of Elasticity,*McGraw-Hill International Editions, Third Edition, 1970. - J.M. Gere,
*Mechanics of Materials*, Nelson Thornes, Fifth SI Edition, 2001.

### Websites

- Simulation showing the stress distribution in 3-point bending: teaching.ust.hk/~mech101/3%20point%20bending.htm
- Video of the 3-point bending of a brass beam: https://www.doitpoms.ac.uk/vidlib/full_record.php?id=14
- Stress distribution In a polycarbonate bar undergoing 3-point bending: https://www.doitpoms.ac.uk/vidlib/full_record.php?id=36
- 3-point bending simulation for a sandwich beam: mms.ensmp.fr/Mms_scripts/Beam/index.php
- Simulation showing the stress distribution in 4-point bending: teaching.ust.hk/~mech101/4_point_bending.htm
- Stress distribution In a polycarbonate bar undergoing 4-point bending: https://www.doitpoms.ac.uk/vidlib/full_record.php?id=40
- Simulation of shear, moment and deflection on application of different forces to a beam: http://www.engr.umd.edu/HAMLET/BeamAnalysis/
- Sandwich panels: http://oneoceankayaks.com/Sandcore.htm
- The torsional pendulum: http://farside.ph.utexas.edu/teaching/301/lectures/node139.html
- Granta Design case studies by a former Cambridge student