DIAGRAMS. G55 



According to this method each force is drawn in its proper position in 

 the diagram of configuration of the frame. Such a diagram might be useful 

 as a record of the result of calculation of the magnitude of the forces, but 

 it would be of no use in enabling us to test the correctness of the calculation. 



But we have a graphical method of testing the equilibrium of any set of 

 forces acting at a point. We draw in series a set of lines parallel and pro- 

 portional to these forces. If these lines form a closed polygon the forces are in 

 equilibrium. We might in this way form a series of polygons of forces, one for 

 each joint of the frame. But in so doing we give up the principle of drawing 

 the line representing a force from the point of application of the force, for all 

 the sides of the polygon cannot pass through the same point, as the forces do. 



We also represent every stress twice over, for it appears as a side of both 

 the polygons corresponding to the two joints between which it acts. 



But if we can arrange the polygons in such a way that the sides of any 

 two polygons which represent the same stress coincide with each other, we 

 may form a diagram in which every stress is represented in direction and 

 magnitude, though not in position, by a single line which is the common 

 boundary of the two polygons which represent the joints at the extremities 

 of the corresponding piece of the frame. 



We have thus obtained a pure diagram of stress in which no attempt is 

 made to represent the configuration of the material system, and in which every 

 force is not only represented in direction and magnitude by a straight line, 

 but the equilibrium of the forces at any joint is manifest by inspection, for 

 we have only to examine whether the corresponding polygon is closed or not. 



The relations between the diagram of the frame and the diagram of stress 

 are as follows : 



To every link in the frame corresponds a straight line in the diagram of 

 stress which represents in magnitude and direction the stress acting in that link. 



To every joint in the frame corresponds a closed polygon in the diagram, 

 and the forces acting at that joint are represented by the sides of the polygon 

 taken in a certain cyclical order. The cyclical order of the sides of the two 

 adjacent polygons is such that their common side is traced in opposite directions 

 in going round the two polygons. 



The direction in which any side of a polygon is traced is the direction 

 of the force acting on that joint of the frame which corresponds to the polygon, 

 and due to that link of the frame which corresponds to the side. 



