In the first part of this section, commencing at Art. 71, the method was developed of ascertaining the strains in the various parts of a frame by the parallelogram or triangle of forces. The method, so far as there explained, is adequate to solve simple cases; but when more than three pieces of a frame converge in one point, the task by that method becomes difficult. This difficulty, however, disappears when recourse is had to the method known as that of "Reciprocal Figures, Frames, and Diagrams of Forces," proposed by Professor I. Clerk Maxwell in 1867. This is an extension of the method by the triangle of forces, and may be illustrated as follows:
Let the lines in Fig. 50 represent, in direction and amount, four converging forces in equilibrium in any frame, as, for example, the truss of a roof; let the lines in Fig. 51 be drawn parallel to those in Fig. 50, in the manner following, namely: Let the line A B be drawn parallel with the line of Fig. 50 which is between the corresponding letters A and B, and let it be of corresponding length; from B draw the line B C parallel with the line of Fig. 50 which is between the letters B and C, and of corresponding length; then from C draw CD, and from A draw A D, respectively parallel with the lines of Fig. 50 designated by the corresponding letters, and extend them till they intersect at D. The lengths of these two lines, the last two drawn, are determined by the point D where they intersect; their lengths, therefore, need not be previously known, The lengths of the lines in Fig. 51 are respectively in proportion to the several strains in Fig. 50, provided these strains are in equilibrium. Fig. 51 is termed a closed polygon of forces. A system of such polygons, one for each point, in the frame where forces converge, so constructed that no line representing a force shall be repeated, is termed a diagram of forces. This diagram of forces is a reciprocal of the frame from which it is drawn, its lines and angles being the same. The facility of tracing the forces in the diagram of forces depends materially upon the system of lettering here shown, and which was proposed by Mr. Bow, in his excellent work on the Economics of Construction. In this system each line of the frame is designated by the two letters which it separates; thus the line between A and B is called line A B; that between C and D is called line CD; and so of others; and in the diagram the corresponding lines are called by the same letters, but here the letters designating the line are, as usual, at the ends of the line. Any point in a frame where forces converge is designated by the several letters which cluster around it; as, for example, in Fig. 50, the point of convergence there shown is designated as point A B C D.
This invaluable method of defining graphically the strains in the various pieces composing a frame, such as a girder or roof-truss, is remarkably simple, and is of general application. Its utility will now be exemplified in its application to framed girders, and afterwards to roof-trusses.