**ABC—XYZ**

FRAMES.] BRIDGES 317 same name in the frame (fig. 66). Tlun the triangle YXE is the reciprocal figure of the three lines YX, XK, EY in the frame, and represents the three forces in equilibrium at the point YXE of the frame. The direction of YX, being a thrust upwards, shows the direction in which we must go round the triangle YXE to find the direction of the two other forces ; doing this we find that the force XK must act down towards the point YXE, and the force EY away from the same point. Putting arrows on the frame diagram to indicate the direction of the forces, we see that the member E Y must pull and therefore act as a tie, and that the member XE must push and act as a strut. Passing to the point XEFAwe find two known forces, the load XA acting downwards, and a push from the strut XK, which, being in compression, must push at both ends, as indicated by the arrow, fig. 66. The directions and magni tudes of these two forces are already drawn (fig. G6a) in a fitting position to represent part of the polygon of forces at XEFA ; be ginning with the upward thrust EX, continuing down XA, and drawing AF parallel to A.F in the frame we complete the polygon by drawing EF parallel to EF in the frame. The point F is deter mined by the intersection of the two lines, one beginning at A, and the other at E. We then have the polygon of forces EXAF, the reciprocal figure of the lines meeting at that point in the frame, and representing the forces at the point EXAF; the direction of the forces on EX and XA being known determines the direction of the forces due to the elastic reaction of the members AF and EF, showing AF to push as a strut, while EF is a tie. We have been guided in the selection of the particular quadrilateral adopted by the rule of arranging the order of the sides so that the same letters indicate corresponding sides in the diagram of the frame and its reciprocal. Continuing the construction of the diagram in the same way, we arrive at fig. 66d as the complete reciprocal figure of the frame and forces upon it, and we see that each line in the reciprocal figure measures the stress on the corresponding member in the frame, and that the polygon of forces acting at any point, as IJKY, in the frame is represented by a polygon of the same name in the reciprocal figure. The direction of the force in each member is easily ascertained by proceeding in the manner above described. A single known force in a polygon determines the direc tion of all the others, as these must all correspond with arrows pointing the same way round the polygon. Let the arrows be placed on the frame round each joint, and so as to indicate the direction of each force on that joint ; then when two arrows point to one another on the same_ piece, that piece is a tie ; when they point from one another the piece is a strut. It is hardly neces sary to say that the forces exerted by the two ends of any one member must be equal and opposite. This method is universally applicable where there are no redundant members. The reciprocal figure for any loaded frame is a complete formula for the stress on every member of a frame of that particular class with loads on given joints. Some examples of these figures will be given, and the reader will easily construct others for himself. 54. Warren Girders Reciprocal Figure and Method of Computing Stresses by Method, of Sections . Case 1. The AVarren girder leaded at each top joint, figs. 67 and 67a. This diagram differs very slightly from that shown in fig. G6. The top and bottom members are in straight lines, and consequently the lines indicating the stresses on the bottom member are superim posed one on the other instead of radiating from Y; the loads XA and FZ are shown as directly borne by the piers. It is clear that if the road is supported by a platform reaching from the end joints to the piers, half of the load on these parts of the platform will be directly supported on the piers. These end loads are shown also in the subse quent diagrams. The truss is generally built of equilateral Fto. 67. triangles, and the inclination of the bracing to the horizon should never be less than 45. Case 2. AVarren girder loaded on top and bottom, fig. G8. 686 shows the polygon of external forces, and G8c shows half the reciprocal figure. These figures have been added to facilitate the comprehension of the complete reciprocal shown by G8a Case 3. AVarren girder with one load, not central, Gg. X A B / n A Wv*/ <r/^ A> / A -/ /^/ G / z x/^/o It J*i (j 6" .9

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-d, 69. The polygon RXZ, fig. G9a, represents the external

forces.