316 BRIDGES [FRAMES. be neglected except where a member of the frame is em ployed to carry the roadway. Fig. 64 shows a common form of bridge truss known as a Warren girder, with lines indicating external forces ap plied to the joints ; half the load carried between the two lower joints next the piers on either side is directly carried by the abutments. The sum of the two upward vertical reactions must clearly be equal to the sum of the loads. The lines in the diagram represent the directions of a series of forces which must all be in equilibrium ; these lines may, for an object to be explained in the next paragraph, be conveniently named by the letters in the spaces which they separate instead of by the method usually employed in geometry. Thus we shall call the first inclined line on the left hand the line AG, the line representing the first force on the top left hand joint AB, the first horizontal member at the top left hand the line BH, &c.; similarly each point requires at least three letters to denote it ; the top first left hand joint may be called ABHG, being the point where these four spaces meet. In this method of lettering, every enclosed space must be designated by a letter ; all external forces .must be represented by lines outside the frame, and each space between any two forces must receive a distinctive letter ; this method of lettering was first pro posed by Mr K. H. Bow (Economics of Construction}, and is convenient in applying the theory of reciprocal figures to the computation of stresses on frames. When the weight of the truss is small it is usual to refer the weights of the parts of the truss itself to the same joints as carry the roadway, and to treat all other joints as unloaded. The reactions at the points of supports of a framed arch or suspension bridge are inclined, as in fig. G5 ; the manner Fig. 65. of computing the stresses on these frames when the direction of these reactions is known will be first explained, and subsequently the manner of finding this direction will be given. 53. Reciprocal Figures. Prof. Clerk Maxwell has given (Phil. Mag. 1864), the following definition of reciprocal figures : " Two plane figures are reciprocal when they consist of an equal number of lines so that corresponding lines in the two figures are parallel, and corresponding lines which converge to a point in one figure form a closed polygon in the other." Let a frame (without redundant members), and the external forces which keep it in equilibrium, be represented by a diagram constituting one of these two plane figures, then the lines in the other plane figure or the reciprocal will represent in direction and magnitude the forces between the joints of the frame, and, consequently, the stress on each member as will now be explained. Reciprocal figures are easily drawn by following definite rules, and afford therefore a simple method of computing the stresses on members of a frame. The external forces on a frame or bridge in equilibrium under those forces may, by a well-known proposition in statics, be represented by a closed polygon, each side of which is parallel to one force, and represents the force in magnitude as well as in direction. The sides of the polygon may be arranged in any order, provided care is taken so to draw them that in passing round the polygon in one direction this direction may for each side corres pond to the direction of the force which it represents. This polygon of forces may, by a slight extension of the above definition, be called the reciprocal figure of the external forces, if the sides are arranged in the same order as that of the joints on which they act, so that if the joints and forces be numbered 1, 2, 3, 4, &c., passing round the outside of the frame in one direction, and returning at last to joint 1, then in the polygon the side represent ing the force 2 will be next the side representing the force 1, and will be followed by the side representing the force 3, and so forth. This polygon falls under the definition of a reciprocal figure given by Clerk Maxwell, if we consider the frame as a point in equilibrium under the external forces. 1Z -LZ Fig. 66. Fig. 666. Fig. 66 shows a frame supported at the two end joints, and loaded at each top joint. The loads and the supporting forces are indicated by arrows. Fig. 66a shows the reciprocal figure or polygon for the external forces on the assumption that the reactions are slightly inclined. The lines in fig. 66a, lettered in the usual manner, correspond to the forces indicated by arrows in fig. 66, and lettered according to Mr Bow s method. When all the forces are vertical, as will be the case in girders, the polygon of external forces will be reduced to two straight lines, fig. 66b, superimposed and divided so that the length AX represents the load AX, the length AB the load AB, the length YX the reaction YX, and so forth. The line XZ consists of a scries of lengths, as XA, AB DZ, representing the loads taken in their order. In sub sequent diagrams the two reaction lines will, for the sake of clearness, be drawn as if slightly inclined to the vertical (as practised by Mr Bow). If there are no redundant members in the frame, there will be only two members abutting at the point of support, for these two mem bers will be sufficient to balance the reaction, whatever its direction may be ; we can therefore draw two triangles, each having as one side the reaction YX, and having the two other sides parallel to these two members ; each of these triangles will represent a poly gon of forces in equilibrium at the point of support. Of these two triangles, shown in fig. 66c, select that in which the letters X and Y are so placed that (naming the apex of the triangle E) the
lines XE and YE are the lines parallel to the tAvo members of the