FIGURES TO THE CALCULATION OF STRAINS ON FRAMEWORK. 445 



minate, and the frame may be self-strained. In these cases, therefore, the 

 reciprocal figure is useless to determine the stresses. 



Such a frame as this shown in Plate XVIII. fig. 5 ; in which, if the diagonals 

 and verticals were all adapted to resist tension and compression, the stresses could 

 not be determined by the use of reciprocal figures or any ordinary method of 

 computation. If, however, the verticals be alone suited to resist the compression, 

 the diagonals being fit to sustain tension only, the stresses become determinate, 

 half the diagonals being with any given load wholly inoperative. The reciprocal 

 figure can be used to discover which are the active members, as they may be 

 called, and what are the strains upon them. 



Fig. 5 a shows the reciprocal figure for a uniform load. 



The inactive members in Frame II. are not numbered. The component poly- 

 gons of fig. 5 a are shown in figs. 56, 58. These figures require no explanation 

 beyond that already given for frame ] . 



Fig. 6 shows the active and inactive members of Frame II. partially loaded. 

 The active members have arrows on them, showing the direction of the stresses. 



Fig. 6 a is the reciprocal of fig. 6, and figs. 6 a and 66 show the component 

 polygons as before. 



It must be remembered that in fig. 6 a, as in fig. 5 a, that the lines represent- 

 ing the stresses on B, C, D, E, F, G, H, I, all start from Z. 



In fig. 7 we have a third frame not unfrequently used in roof work. Fig. la is 

 the reciprocal of fig. 7, and is thus constructed. XY is the line of loads subdivided 

 at Z in the ratio of the loads borne by the two piers. It is further subdivided into 

 the parts 1,2,3,4,5,6, 7, the loads directly borne by each joint. The lines A, B, C, D 

 and F are in fig. la all drawn parallel to the top members of the same name in 

 fig. 7, and start from points in the line of loads determined by the subdivision 

 into partial loads. The lines a, b, c, d, 0,/all radiate from Z, and these two sets of 

 lines are joined by the zigzag line g, h, i,j, k, I, m, n, o, each of course parallel to the 

 corresponding member in fig. 7. The figure, although a little complex at first 

 sight, is extremely easily and rapidly constructed. In building it up out of 

 successive polygons we should as usual begin with the reaction of one pier; start- 

 ing at Z we draw the line ZX, return along load 1 directly borne by the pier, and 

 complete the first polygon by drawing Aa. The second polygon is A, 2, b, H, g, and 

 the remaining component polygons corresponding to each joint can easily be 

 traced in like manner. It will be observed that all the members except 

 A, B, C, D, E, F are in tension. 



Fig. 8 and fig. 8a (Plate XIX.) show a slightly different frame with the corre- 

 sponding reciprocal figure when uniformly loaded. h,j,l,n are in compression 

 instead of in tension, as in Frame III. 



VOL. XXV. PART II. 5 Y 



