446 FROF. F. JENKIX ON THE PRACTICAL APPLICATION OF RECIPROCAL 



Figs. 9 and 9 a show Frame IV. and its reciprocal figure when not uniformly 

 loaded. 



The reciprocal figure now begins to appear very complicated, but it is drawn 

 on precisely the same plan as fig. 8 a; but the lines of loads being no longer equally 

 subdivided, the reciprocal figure no longer presents two symmetrical parts. 



Figs. 10 and 10a with 11 and 11a show frames commonly used as roofs, with 

 reciprocal figures. They are only simplified cases of the roof already described. 



Fig. 11 a may be compared with tig. 76 in Rankine's " Applied Mechanics." 

 The series of figures 75 in the same work are true elementary reciprocal figures. 



Figs. 12 and 12 a show a simple roof uniformly loaded, which is drawn in 

 order to render more intelligible the comparatively complex case in figs. 13 and 

 13 a. Fig. 13 shows the roof under a series of external forces which are no longer 

 parallel, but represented by the inclined lines 1, 2, 3, which have been some- 

 what arbitrarily chosen as corresponding to a possible distribution of stresses 

 produced by the lateral and vertical pressure of wind. These external forces are 

 met by the two reactions P and P 1 at the piers, calculated on the hypothesis 

 that each pier or wall takes half of the horizontal strain. 



Fig. 13 a is the curiously distorted reciprocal figure which results from these 

 assumptions. It is drawn by precisely the same rule as the comparatively simple 

 figs. 12 a and 7 a. In each the lines «, b, c, d radiate from a centre Z, which 

 divides the lines P and P x representing the reactions on the piers. In each the 

 members A, B, C, D diverge from points separating the successive loads on the 

 joints 1, 2, 3 ; but in fig. 13a the line of loads 1, 2, 3 with the lines P and P : 

 representing the reactions at the piers, build up a polygon enclosing a space, 

 whereas in figs. 7 a and 12 a this polygon was represented by two straight lines 

 superimposed. 



Again, if the zig-zag line corresponding to the diagonals be traced, it will be 

 found to run in an essentially similar manner in figs. 12 a and 13 a; thus I joins d 

 and c in both, the end of the line d having been determined by its intersection 

 with D, both starting from Z and X. Looked at by the light of fig. 12 a, fig. 

 13 a becomes readily intelligible, and serves to show how the theory of reciprocal 

 figures can be applied to the most complex conditions of stress which are con- 

 ceivable, without any greater essential complication than occurs in the simplest 

 examples. 



As a final example, the reciprocal figure is given of a braced suspension bridge 

 or arch uniformly loaded, figs. 14 and 14 a. The strains are drawn on the hypo- 

 thesis that the direction and magnitude of the resultant thrust are known. This 

 thrust can be determined by Professor J. Clerk Maxwell's method for calcu- 

 lating the equilibrium of Frames, published at the same time as his account of 

 reciprocal figures. 



In conclusion, a few words may be said of the advantage of the diagrams of 



