84 STRENGTH OF MATERIALS 



stress forms a moment equal to M. This moment is zero wlu-ii fir* 

 applied, and gradually increases to its full value, its average < 

 being \M. Therefore the work done by the normal 

 cross section is -. M~dx 



Hence the total work of deformation for the entire beam is 



Problem 91. As an application of the above, find the dcfU-< ti..n at the renter of 

 a simple beam of length I, bearing a single content rat.-d l..ad /' at tl..- . 



Solution. Let D denote the deflection at the center. Then the external ' 

 of deformation is W=\ PD 



At a point distant x from the left support the bending moment to If = , and 

 consequently the internal work of deformation is 



Therefore 



whence 



Problem 92. Find the internal work of deformation for a recta i 

 beam 10ft. long, 10 in. deep, and 8 in. wide, which bears a uniform loal 



per foot of length. 



73. Impact and resilience. In the preceding article an expression 

 was deduced for the work done by the stress in producing deforma- 

 tion of a beam. If the stress lies within the elastic limit ! tin- 

 material, the body returns to its original shape upon removal of tin* 

 external forces, and the internal work of deformation is given nut 

 again in the form of mechanical energy. The internal work of 1 

 mation is thus a form of potential energy, and from this point of 

 view is called resilience. 



The work done in straining a unit volume of a material to the 

 elastic limit is called the modulus of resilience of the material 



