96 



STEENGTH OF MATERIALS 



Since P l =.2^ - P 2 , ^ = - 1, and hence 



2 or<L 



_ P js I 



Putting = 0, and substituting for P, its value in terms of P,, 



whence 



and consequently 



PI = i wl 



80. Principle of least work. Differentiating partially with respect 



dW 



to P, both members of the equation - - = />,-, we have 



As the load increases the deflection increases, and vice versa. There- 



O -r\ 



fore, since dZ> t . and 



ffW 



- is also positive. But, from the differential calculus, 



t . have the same sign, - - is positive and hence 

 * 



W, 



are the conditions that W shall be a minimum. Consequently, the 

 reactions of a continuous beam, calculated from the condition - = 0, 



are such that they make t he 

 work of deformation a mini- 

 mum. 



In Article 73 it was pointed 

 out that the internal work <f 

 deformation is a form of po- 

 tential energy. The above is 

 thus a special case of what 

 FIG. 77 is known as the principle of 



least work, the general > 



ment of this principle being: For stable equilibrium the potc, 



energy of any system is a minimum. 



