146 KANSAS UNIVERSITY QUARTERLY. 



nates are equal, and either is the ordinate of the locus when z=x. 

 Hence if u' denote the ordinate of the locus we have 



, 8h 8hl2 

 2U =-— (2I — 2x)andu=z (n) 



We notice that this equation is not a function of P but is a function 

 of h and 1. The shape of the locus depends on the shape of the rib 

 and not at all on the loading. 



The curve D F E, Fig. i, shows this locus for the rib whose rise is 

 ID feet and span 50 feet. 



The moment at any point of the rib can be easily found when this 

 locus is constructed. For, if at the point where the load P, represen- 

 ted as in Fig. i, cuts the locus lines be drawn to the hinges they form 

 two segments of the special equilibrium polygon for this position of 

 the load. If through the extremities of P, drawn to scale, lines be 

 drawn parallel to these two segments and from their intersection a 

 perpendicular be dropped on P, this line represents H on the same 

 scale that P is drawn. Then the moment under the load is H multi- 

 plied by the distance between the locus and the rib measured on P, 

 and the moment at any other point of the rib equals H multiplied by 

 the vertical distance from the point to the equilibrium polygon. 



From M^=o=-j^ j 1— z *- -j 1— ^-k j- we have 



1 A 2^ 



z^l and z=~ , 



5k 

 The value z=:l is independent of x, that is, the moment at the right 

 hinge is zero for all values of x. The other value of z is a function 

 of x which approaches |1 as its limit as x approaches zero and equals 



x 

 t when— -=.67 approx. Hence this second point of zero moment 



on the right of P moves from |1 to 1, while P moves from x=o ta 

 x=.6j\. 



From Mi=o=z | P -^— _5_^k(l_z) i w^e have 

 z=o and z=l •< i — I 



xk 



From z=o we see that the moment is zero at the left hinge for all 



values of x. The other value of z is negative while x varies from o to 



1— X 

 .34I [found from i — | — — =0] and it moves from o to 1 while x 

 xk 



varies from .34I to 1. 



This value of z can be put in the form 



