§11.13 Strain Energy 269 i.e.the partial differential of the B.M.term containing W is identical to the result achieved if w is replaced by unity in the B.M.expression.Using this information the Castigliano expression can be simplified to remove the partial differentiation procedure,thus 6= [Mm ds (11.17) EI where m is the B.M.resulting from a unit load only applied at the point of application of W and in the direction in which the deflection is required.The value of M remains the same as in the standard Castigliano procedure and is therefore the B.M.due to the applied load system, including W. This so-called"unit load"method is particularly powerful for cases where deflections are required at points where no external load is applied or in directions different from those of the applied loads.The method mentioned previously of introducing imaginary loads P and then subsequently assuming P is zero often gives rise to confusion.It is much easier to simply apply a unit load at the point,and in the direction,in which deflection is required regardless of whether external loads are applied there or not (see Example 11.6). 11.13.Application of Castigliano's theorem to angular movements Castigliano's theorem can also be applied to angular rotations under the action of bending moments or torques.For the bending application the theorem becomes: 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, Le. M OM ds EI aMi (11.18) where M is the imaginary or applied moment at the point where 6 is required. Alternatively the"unit-load"procedure can again be used,this time replacing the applied or imaginary moment at the point where 6 is required by a"unit moment".Castigliano's expression for slope or angular rotation then becomes 0= (Mm.ds EI where M is the bending moment at a general point due to the applied loads or moments and m is the bending moment at the same point due to the unit moment at the point where 6 is required and in the required direction.See Example 11.8 for a simple application of this procedure. 11.14.Shear deflection (a)Cantilever carrying a concentrated end load In the majority of beam-loading applications the deflections due to bending are all that need be considered.For very short,deep beams,however,a secondary deflection,that due to$1 1.13 Strain Energy 269 i.e. the partial differential of the B.M. term containing W is identical to the result achieved if W is replaced by unity in the B.M. expression. Using this information the Castigliano expression can be simplified to remove the partial differentiation procedure, thus a=ps EZ (11.17) where m is the B.M. resulting from a unit load only applied at the point of application of W and in the direction in which the deflection is required. The value of M remains the same as in the standard Castigliano procedure and is tkrefore the B.M. due to the applied load system, including W. This so-called “unit load method is particularly powerful for cases where deflections are required at points where no external load is applied or in directions different from those of the applied loads. The method mentioned previously of introducing imaginary loads P and then subsequently assuming Pis zero often gives rise to confusion. It is much easier to simply apply a unit load at the point, and in the direction, in which deflection is required regardless of whether external loads are applied there or not (see Example 11.6). 11.13. Application of Castigliano’s theorem to angular movements Castigliano’s theorem can also be applied to angular rotations under the action of bending If the total strain energy, expressed in terms of the external moments, be partially diferentiated 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, moments or torques. For the bending application the theorem becomes: i.e. (11.18) where Mi is the imaginary or applied moment at the point where 8 is required. Alternatively the “unit-load procedure can again be used, this time replacing the applied or imaginary moment at the point where 8 is required by a “unit moment”. Castigliano’s expression for slope or angular rotation then becomes where M is the bending moment at a general point due to the applied loads or moments and m is the bending moment at the same point due to the unit moment at the point where 8 is required and in the required direction. See Example 11.8 for a simple application of this procedure. 11.14. Shear deflection (a) Cantilever carrying a concentrated end load In the majority of beam-loading applications the deflections due to bending are all that need be considered. For very short, deep beams, however, a secondary deflection, that due to