Strain Energy 255 The volumetric or "dilatational"strain energy per unit volume is then -2[o,+2+P门 6E and the shear,or"distortional",strain energy per unit volume is 12G[o1-2P+(o2-032+(a3-1}] 1 The maximum instantaneous stress in a uniform bar caused by a weight Wfalling through a distance h on to the bar is given by (+] The instantaneous extension is then given by 6-% If this is small compared to the height h,then 2WEh AL For any shock-loaded system the instantaneous deflection is given by -±(+】 where 6,is the deflection under an equal static load. Castigliano's first theorem for deflection states that: If the total strain energy expressed in terms of the external loads is partially differentiated with respect to one of the loads the result is the deflection of the point of application of that load and in the direction of that load (see Examples 11.5 and 11.6): aU i.e. Deflection in direction of In applications where bending provides practically all of the strain energy, CM2ds [MOM =aw]2EI ds EI aW This is sometimes written in the form 6= Mm E where m OM aw 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.Strain Energy 255 The volumetric or “dilatational” strain energy per unit volume is then and the shear, or “distortional”, strain energy per unit volume is 1 - [(ai- 02)2 + (02 - 4 + (03 - (71)21 12G The maximum instantaneous stress in a uniform bar caused by a weight W falling through a distance h on to the bar is given by 2 WEh A- The instantaneous extension is then given by dL 6=- E If this is small compared to the height h, then //2 WEh\ For any shock-loaded system the instantaneous deflection is given by 6 = 6, [ 1 * J( 1 +;)I where 6, is the deflection under an equal static load. Castigliano’sfirst theorem for tiefiction states that: If the total strain energy expressed in terms of the external loads is partially diyerentiated with respect to one of the loads the result is the defection of the point of application of that load and in the direction of that load (see Examples 11.5 and 11.6): au aw i.e. Deflection in direction of W = - = 6 In applications where bending provides practically all of the strain energy, This is sometimes written in the form 8M 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. aw