Art. 88. 



RESILIENCE OF BEAMS. 



137 



For a ratio of span to depth of 10, the deflection due to 

 shear is but 3% of that due to bending; when this ratio is 5 the 

 percentage is 12, arid as the ratio decreases, the deflection due to 

 shear becomes more important than that due to bending. In very 

 short beams it is more rational, therefore, to neglect the bending 

 stress than the shearing stress ; this is also indicated by the fact 

 that the assumption, that plane cross sections remain plane after 

 bending, can not be true. The shearing stresses at the upper 

 and lower edges of the cross section are zero and therefore 8 8 is 

 zero the cross section remains perpendicular to the axis of the 

 beam ; toward the neutral axis the deformation increases so that 

 the cross section becomes curved. 



In beams with circular cross sections pins of pin-connected 

 trusses, for example the influence of the shear is somewhat 

 greater than in those with rectangular cross sections. 



The above considerations show the limits within which the 

 formulas of Arts. 81 to 86 and Art. 90 are applicable. 



88. Resilience of Beams. As explained in Art. 19, resil- 

 ience is equal to the work of deformation, and elastic resilience 

 is equal to the potential energy stored in a beam. 



If the deformation due to shear is neglected, only the bend- 

 ing stress, that is, tension and compression, need be considered 

 Equation (4) is applied to an element of volume, and the total 

 work of deforming the beam is then obtained by integration over 

 the cross section and length of the beam. 



Work per unit of volume = \ -J = \ -^^- by substitu- 

 tion from equation (12). 



Work per element of volume = 



Total work of deformation = -H-p- I _^_ 



Since 



/+ VI 

 v 2 dA = 

 V2 



I this becomes 



Elastic resilience of a beam 



/L 

 %** 



(41) 



In deducing the above equation it should be remembered 



