Art. 88. 



BESILIENCE OF A BEAM. 



139 



Resilience == = A 4 bhL = 



It is to be noted that this equation shows, that for the same 

 form of cross section, the resilience of a beam does not depend 

 upon the size or dimensions of the cross section, or the length of 

 the beam, but only upon its volume. 



Since the resilience is a measure of the shock-resisting qual- 

 ities of a beam, the question arises, What is the resilience of a 

 beam for a load not gradually applied! This case is similar, of 

 course, to that explained in Art. 19. If it were possible to apply 

 a load instantaneously, its velocity being zero when it begins to 

 act on the beam, the total work done would be P$. If a gradu- 

 ally applied load produces the same deflection, the total work 

 done must be the same, so that 



1/2 p y = p. y and P = 2 Pi. 



An instantaneously applied load will, therefore, produce 

 twice the deflection, and consequently twice the stress, that the 

 same load gradually applied produces. 



When a load is gradually applied, no sensible part of the 

 work done is spent in giving a velocity to the beam, but all the 

 energy becomes potential the beam is capable of doing work 

 when it springs back. When the same load is applied instan- 

 taneously, only half of the energy will have been converted into 

 potential energy, when the deflection is equal to the static de- 

 flection ; the other half will have given an increasing velocity to 

 the beam and the load, and is converted into potential energy 

 during the last half of the deflection, when the velocity decreases. 

 The beam cannot remain in this extreme position because the 

 deflection is twice that for the load at rest ; the potential energy 

 is converted into kinetic energy again, the velocity increasing to 

 the point of the static deflection, and then decreasing to zero 

 deflection, after which, it deflects again, that is, it vibrates past 

 the point of static deflection. 



Experience teaches that these vibrations diminish in ampli- 

 tude and finally cease. This is due to various frictional resis- 

 tances. 



As explained in Art. 5, twice the static deflection never 

 results from suddenly applied loads, but it may be much larger 

 even, if the load falls from a height h. In this case the applica- 



