XVl] 



OF RECIPROCAL DIAGRAMS 



697 



W 



Fis. 342. 



Professor Fidler's words, "Every diagram of moments represents 

 the outline of a framed structure which will carry the given load 

 with a uniform horizontal stress in the principal members." 



In the following diagrams (Fig. 342, a, b) (which are taken 

 from the original ones of Cul- 

 mann), we see at once that the 

 loaded beam or bracket (a) has 

 a '"danger-point"' close to its 

 fixed base, that is to say at the 

 point remotest from its load. 

 But in the parabolic bracket 

 (b) there is no danger-point at 

 all, for the dimensions of the 

 structure are made to increase pari passu w^ith the bending- 

 moments : stress and resistance vary together. Again in Fig. 340, 

 we have a simple span (A), with its stress diagram (B) ; and in 

 Fig. 341 we have the corresponding parabolic girder, whose 

 stresses are now uniform throughout. In fact we see that, by a 

 process of conversion, the stress diagram in each case becomes 

 the structural diagram in the other*. Now all this is but the 

 modern rendering of one of Gahleo's most famous propositions. 

 In the Dialogue which we have already quoted more than oncef, 

 Sagredo says "It would be a fine thing if one could discover the 

 proper shape to give a soUd in order to make it equally resistant 

 at every point, in which case a load placed at the middle would 

 not produce fracture more easily than if placed at any other 

 point." And Gahleo (in the person of Salviati) first puts the 

 problem into its more general form ; and then shews us how, by 

 giving a parabolic outline to our beam, we have its simple and 

 comprehensive solution. 



In the case of our cantilever bridge, we shew the primitive girder 



* The method of constructing recijnocal diagrams, in which one should represent 

 the outlines of a frame, and the other the system of forces necessary to keep it 

 in equilibrium, was first indicated in Culmann's Graphische Statik; it was greatly 

 developed soon afterwards by Macquom Rankine {Phil. Mag. Feb. 1864, and 

 Applied Mechanics, passim), to whom is mainly due the general application of the 

 principle to engineering practice. 



j Dialogues concerning Two New Sciences (1638) : Crew and Salvio's transljition, 

 p. 140 seq. 



