444 PROF. F. JENKIN ON THE PRACTICAL APPLICATION OF RECIPROCAL 



to the various loads on the several joints, these unequal loads being, as before, 

 indicated by vertical lines in fig. 2. The two halves of the reciprocal figures are 

 now no longer symmetrical, but it is as easily drawn as the simpler case. Two of 

 the component polygons are shown, as in the previous case. The direction of the 

 stress on each member is found by going round each separate polygon, beginning 

 with some strain the direction of which is known. Thus at joint 7 the polygon 

 is c, 7, <#, P 2 , 0, and knowing that the direction of the weight 7 is down, we find at 

 once that P 2 and must both be pulled upwards. Care must be taken as before 

 to measure ABCDEFGH from the origin Z in each case. As soon as the values 

 of P and Pj have been determined, the most complicated arrangement of loads 

 presents no more difficulty than the very simplest, the typical form being identical 

 in all cases, and easily remembered. 



In figs. 3 and 3a the reciprocal figure for the same frame with a single load 

 at the centre is shown. The strains on a, b, c are represented by the lines xa, xb, xc ; 

 the strains on A, B, C by the lines zA, zB, zQ. This will be clear from an 

 inspection of the component polygons. 



In figs. 4 and 4 a the reciprocal figure for a single weight hung at any joint is 

 shown, and will readily be understood from the explanations already given. 



If the frame were inverted, and the loads applied at the top, the strains would 

 remain the same in amount, but be altered in direction in the diagonals ; the 

 reciprocal figures would be identical in form with those already given, but would 

 lie on the other side of the line of loads, as if simply turned over through 180° 

 on that line as a hinge. 



If the loads in Frame I. were applied at the 

 top joints instead of along the bottom, the 

 strains and the reciprocal figure would be modi- 

 fied, the component polygons for the bottom 

 joints being of the annexed type, fig. 3, and the 

 component polygons for the top joint of the annexed type, fig. 4. P 1 is also 

 placed above P, so that the upper and lower halves of the figure change places. 



It is not, however, necessary to recollect these 



Fig. 3. 



/ x\ changes of arrangement, since the known reaction 



f load /* at one pier and the first polygon of forces deter- 



mine at once the general arrangement of the 

 Flgi 4- figure. 



In the example just given the members of the frame are simply sufficient in 

 number to make the frame stiff; such a frame is incapable of being self-strained, 

 that is to say, any member might be lengthened or shortened without throwing 

 a strain on the other members. When this condition is fulfilled the stresses on a 

 frame under the action of known external pressure determinate ; but when more 

 members are used than suffice to render the frame stiff, the stresses are indeter- 



