116 SUPPLEMENT TO CHAP. VII. [CIIAP. HI. 



If the breadth is constant, h = hi./ ., or the elevation of the beam is a 



parabola with the weight at apex. If the cross-section is always timilar, 



that is, if -i -, we have b = -\-, and substituting in the equation above 



fii n hi 



h = hi / fl , which is a paraboloid of revolution. 



(a) Change of shape 

 From (8) we have 



d* y _ Pat _ Fa 



die* ~ El ~ 



where b and h are variable. If we suppose the height h constant and 



always equal to hi, then, as we have seen, b = bi T ; hence for rectangular 



I 



cross-section 



d*y_ 12 PI 

 das* E hi 3 bi* 



Integrating, since for * = I, 2- = 0, we have 



d x 



dy _ 12 P Ix 12 P Z* 



d x ~~ E hi* bi E hi* bT 

 Integrating again, since f or x = I, y = 0. we have 



_ 6 P I a' _ 12 PZ* x 6PZ 

 E hi* bi E /ti 3 6 1 E hi*bi 



For the maximum deflection x 0, and 



A = E hS bi' 

 The above value of y can be written 



r *p / * ft 



but _ . , , is 77-, the deflection of a beam of constant cross-section bi h,, 

 E bi hi z 2 



as already found. Calling this deflection A , we have 



q 



for the deflection at any point, or A = jr- Ao for the maximum deflec- 



tion. 

 In a similar manner, for constant breadth, we have 



