points represents the resultant in magnitude and direction. As in the case of the force triangle, it is immaterial in what order the forces are applied as long as they all act in the same direction around the polygon. A force polygon is analogous to a traverse of a field in which the bearings and the distances are measured progressively around the field in either direction. The conditions for closure in the two cases are also identical. It will be seen that any side in the force polygon is the equilibrant of all the other sides, and that any side reversed in direction is the resultant of all the other sides. Equilibrium of Concurrent Forces.-The necessary condition for equilibrium of concurrent coplaner forces therefore is that the force polygon close. This is equivalent to the algebraic condition that Σ horizontal components of forces o, and Σ vertical components of forces = 0. If the system of concurrent forces is not in equilibrium, the resultant can be found in magnitude and direction by completing the force polygon. The resultant of a system of concurrent forces is always a single force acting through their point of intersection. Algebraic Resolution. In calculating the stresses in a truss by algebraic resolution, the fundamental equations for equilibrium, (1) and (2), for translation are applied (a) to each joint, or (b) to the members and forces on one side of a section cut through the truss. (a) Forces at a Joint.-The reactions having been found, the stresses in the members of the truss shown in Fig. 4 are calculated as follows: Beginning at the left reaction, R1, we have by applying equations (1) and (2) The stresses in members 1-x and 1-y may be obtained by solving equations (4) and (5). The direction of the forces which represent the stresses in amount will be determined by the signs of the results; if compressive stresses are assumed as positive, tensile stresses will be negative. Arrows pointing toward the joint indicate that the member is in compression; arrows pointing away from the joint indicate that the member is in tension. The stresses in the members of the truss at the remaining joints in the truss are calculated in the same way. The direction of the forces and the kind of stress can always be determined by sketching in the force polygon, for the forces meeting at the joint as in (c) Fig. 4. It will be seen from the foregoing that the method of algebraic resolution consists in applying the principle of the force polygon to the external forces and internal stresses at each joint. Since we have only two fundamental equations for translation (resolution) we can not solve a joint if there are more than two forces or stresses unknown. Where the lower chord of the truss is horizontal as in Fig. 5, we have by applying fundamental equations (1) and (2) to the joint at the left reaction the plus sign indicating compression and the minus sign tension. Equations (6) and (7) may be obtained directly from force triangle (c). (b) Forces on One Side of a Section.—The principle of resolution of forces may be applied to the structure as a whole or to a portion of the structure. If the truss shown in Fig. 6 is cut by the plane A-A, the internal stresses and external forces acting on either segment, as in (b) will be in equilibrium. The external forces acting on the cut members as shown in (b) are equal to the internal stresses in the cut members and are opposite in direction. Applying equations (1) and (2) to the cut section Now, if all but two of the external forces are known, the unknowns may be found by solving equations (8) and (9). If more than two external forces are unknown the problem is indeterminate as far as equations (8) and (9) are concerned. In the Warren truss in Fig. 7 the stresses at a joint may be calculated by completing the force polygon as at the left reaction in (b) Fig. 5. Applying equations (1) and (2) to a section as in (c) = shear in the panel. Therefore the stress in 2-3 (R1P) sec = shear This analysis leads directly to the method of coefficients as explained in detail Graphic Resolution.—In Fig. 8 the reactions R1 and R, are found by means of the force and equilibrium polygons as shown in (b) and (a). The principle of the force polygon is then applied to each joint of the structure in turn. Beginning at the joint Lo the forces are shown in (c), and the force triangle in (d). The reaction R1 is known and acts upward, the upper chord stress 1-x acts downward to the left, and lower chord stress I-yacts to the right closing the polygon. Stress 1-x is compression and stress 1-y is tension, as can be seen by applying the arrows to the members in (c). The force polygon at joint U1 is then constructed as in (f). Stress 1-x acting toward joint U1 and load P1 acting downward are known, and stresses 1-2 and 2-x are found by completing the polygon. Stresses 2-x and 1-2 are compression. The force polygons at joints L1 and U2 are constructed, in the order given, in the same manner. The known forces at any joint are indicated in direction in the force polygon by double arrows, and the unknown forces are indicated in direction by single arrows. The stresses in the members of the right segment of the truss are the same as in the left, and the force polygons are, therefore, not constructed for the right segment. The force polygons for all the joints of the truss are grouped into the stress diagram shown in (k). Compression in the stress diagram and truss is indicated by arrows acting toward the ends of the stress lines and toward the joints, respectively, and tension is indicated by arrows acting away from the ends of the stress lines and away from the joints, respectively. The first time a stress is used a single arrow, and the second time the stress is used a double arrow is used to indicate direction. It will be seen that the upper chords are in compression, while the lower chord is in tension. The stress diagram in (k) Fig. 8 is called a "Maxwell diagram" or a "reciprocal polygon diagram." The notation used is known as Bow's notation, in which points in the truss diagram become areas in the stress diagram, and areas in the truss diagram become points in the stress diagram. The method of graphic resolution is the method most commonly used for calculating stresses in roof trusses and simple framed structures with inclined chords. For the analysis of the stresses in roof trusses, see the author's book, "The Design of Steel Mill Buildings." Warren Bridge Truss.-In Fig. 9 the dead load stresses in a Warren bridge truss loaded on the lower chord, are calculated by the method of graphic resolution. In the stress diagram the loads are laid off from the bottom upwards. The details of the solution can easily be followed by reference to Fig. 9 and Fig. 8. It will be seen that the upper chord of the truss is in compression, while the lower chord is in tension. (3) MOMENTS.-In calculating the stresses in a truss by moments, the fundamental equation for equilibrium for rotation Σ moments of forces about any point = 0 is applied to parts of the structure. Equation (3) may be solved either by algebra or by graphics. Before applying equation (3) to the parts of a structure it will be necessary to discuss a few fundamental principles. Equilibrium of Non-concurrent Forces.-If the forces are non-concurrent (do not all meet in a common point), the condition that the force polygon close is a necessary, but not a sufficient condition for equilibrium. For example, take the three equal forces P1, P2 and P3, making an angle of 120° with each other as in (a) Fig. 10. The force polygon (b) closes, but the system is not in equilibrium. The resultant, R, of P. and P, acts through their intersection and is parallel to P1, but is opposite in direction. The system of forces is in equilibrium for translation, but is not in equilibrium for rotation. The resultant of this system is a couple with a moment considered negative and counter-clockwise positive, (c) Fig. 10. in (a) Fig. 10 is a couple with a moment = + Ph. P1h, moments clockwise being |