THE THEORY OF STRUTS 859 



Bending moment at A = u Wy 



Then El ^ = u - Wy 



ax 1 - 



3- - 



W 



W 



= n 2 (y a), where w 2 = ^r? 



JU 



, dz dy . d 2 z d*y 



If z = t/ - a then -7- = -j 2 and T~2 = T^ 



dx dx dx 2 dx 2 



d z z 



Hence -r- 9 = n 2 3 



(ir 2 



The solution of this is : 



z = A sin nx + B cos IMJ 

 and y a = Asinrw:+B cos nx 



dv 



then - = nA cos TW; wB sin nx 



dx 



but when # = 0, -^ = 0, then A = 



d# 



and T/ a = B cos nx 



Also, when # = 0, y = 0, then B = a 



Then y a == a cos n# 



t/ = a(l cos nx) 



when x = I, y = Q, then a(l cos n/) = 



That is 1 cos nl = 0, or cos nl = 1 



Hence nl must have the values 0, 2n, 4rc . . . 



Considering the least value, 2rc, for the zero value is clearly 

 inadmissible. 



nl= 27T 



= 



W 47c 

 El P 



and W = 



///. IF^^n one end is fixed and the other end free. 

 The free end must be so constrained that it will always be in 

 the same vertical line as the fixed end, while it should be free to 



