This is sometimes written in the form


3 JVjf where m = = the bending moment resulting from a unit load only in the place of W. This method of solution is then termed the unit load method.

Castigliano's theorem also applies to angular movements:

If the total strain energy expressed in terms of the external moments be partially differentiated with respect to one of the moments, the result is the angular deflection in radians of the point of application of that moment and in its direction

where M, is the actual or imaginary moment at the point where 6 is required. Deflections due to shear

Beam loading

Cantilever-concentrated end load W'


Simply supported beam-central concentrated load W

Simply supported beam - concentrated load dividing span into lengths a and b

Simply supported beam-u.d.l.

w L2 _ WL ÏÂG ~ 8ÏÏG


Energy is normally defined as the capacity to do work and it may exist in any of many forms, e.g. mechanical (potential or kinetic), thermal, nuclear, chemical, etc. The potential energy of a body is the form of energy which is stored by virtue of the work which has previously been done on that body, e.g. in lifting it to some height above a datum. Strain energy is a particular form of potential energy which is stored within materials which have been subjected to strain, i.e. to some change in dimension. The material is then capable of doing work, equivalent to the amount of strain energy stored, when it returns to its original unstrained dimension.

Strain energy is therefore defined as the energy which is stored within a material when work has been done on the material. Here it is assumed that the material remains elastic whilst work is done on it so that all the energy is recoverable and no permanent deformation occurs due to yielding of the material, i.e. strain energy U = work done

Thus for a gradually applied load the work done in straining the material will be given by the shaded area under the load-extension graph of Fig. 11.1.


*Area A

Fig. 11.1. Work done by a gradually applied load.


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