138 



RESILIENCE OF A BEAM. 



Art, 88. 



that M and / may vary along the length of the beam, that is, 

 with x, and v varies in the cross section. 



The same result may be gotten by considering the stress sdA 

 on an elementary fiber and the distance vd$ through which it 

 acts (Fig. 89). The force increases from zero directly as the 

 deformation increases (within the elastic limit), so that the 

 mean force is y 2 sdA. Substituting the values s and cZ< from 

 equations (12) and (6) Art. 81, and integrating as before, equa- 

 tion (41) results. 



An expression for the work done by the shear will be quit 

 similar to equation (4). 



Equation (41) is an expression representing the potential 

 energy stored in a beam, and this must be equal to the extern a 1 

 work done by the load deforming the beam, if the elastic limit 

 is not exceeded. Thus, in a cantilever beam with a load P at 

 the end, 



Work=iP yiUM = 



(L x) 2 dx 



/ being constant. (See Fig. 94). 

 Upon integration there results 



iJfynox = 2^ 



which is the same as was found in Art. 82. The deflection at a 

 single load may be thus conveniently gotten. In the case of Art. 

 84, the integration would have to be between the limits x = Q 

 and l ly and x^ = and L, the total work being the sum of that 

 performed upon the two parts of the beam. 



If the deflection at each load is known, the resilience may, 

 of course, be calculated from the external work performed. Thus, 

 in a beam supported at its ends and carrying a load at its middle, 



PIP 



the elastic resilience will be equal to i/ 2 P 4g ^ (See Fig. 95 for 



the deflection). This may be put in terms of the maximum 

 moment or the maximum fiber stress. 

 therefore P = 



8l1 



Resilience = 



For a particular form of cross section , a rectangular one 

 for example, this will become 



