394. The mechanics of the arch are almost invariably solved by a graphical method, or by a combination of the graphical method with numerical calculations. This is done, not only because it simplifies the work, but also because, although the accuracy of the graphical method is somewhat limited, yet, with careful work, it may easily be made even more accurate than is necessary, considering the uncertainty as to the true ultimate strength of the masonry used. The development of this graphical method must necessarily follow the same lines as in Statics. It is here assumed that the student has a knowledge of Statics, and that he already understands the graphical method of representing the magnitude, direction, and line of application of a force. Several of the theorems or general laws regarding the composition and resolution of forces will be briefly reviewed as a preliminary to the proof of those laws of graphical statics which are especially applied in computing the stresses in an arch.

395. Resultant Of Two Forces

The resultant of two forces, A and B, which are not parallel whose lines of action are as shown in Fig. 213a, and which are measured by the lengths of the lines A and B in diagram b, is readily found by producing the lines of action to their intersection at c. The two known forces are drawn in diagram b so that their direction is parallel to the known directions of the forces, and so that the point of one force is at the butt end of the other. Then the line R joining the points m and n in diagram b gives the direction of the resultant; and a line through c parallel to that direction, gives the actual line of that resultant. The line raw also measures the amount of the resultant. Note that diagram b is a closed figure. If an arrow is marked on R so that it points upward, the arrows on the forces would run continuously around the figure. If R were acting upward, it would represent the force which would just hold A and B in equilibrium; pointing downward, it is the resultant or combined effect of the two forces. We may thus define the resultant of two (or more) forces as the force which is the equal and opposite of that force which will just hold that combination of forces in equilibrium.

Fig. 213. Resultant of Two Forces.

Fig. 213. Resultant of Two Forces.

396. Resultant Of Three Or More Forces

This may be solved by an extension of the method previously given as shown in Fig. 213. The resultant of B and C (see Fig. 214) is R'; and this is readily combined with A, giving R" as the resultant of all three forces. The same principle may be extended to any number of non-parallel forces acting in a plane. The resultant of four non-parallel forces is best determined by finding, first, the resultant of each pair of the forces taken two and two. Then the resultant of the two resultants is found, just as if each resultant were a single force.

397. Resultant Of Two Or More Parallel Forces

When the forces are all parallel, the direction of the resultant is parallel to the component forces; the amount is equal to the sum of the component forces; but the line of action of the resultant is not determinable as in the above cases, since the forces do not intersect. It is a principle of Statics which is easily appreciated, that it does not alter the statics of any combination of forces to assume that two equal and opposite forces are applied along any line of action. From Fig. 215 b, we see that the forces F and G will hold A in equilibrium; that G and H will hold B in equilibrium; and that H and K will hold C in equilibrium. But the force G required to hold A in equilibrium is the equal and opposite of the force G required to hold B in equilibrium; and similarly the force H for B is the equal and opposite of the H for C. We thus find that the forces A, B, and C can be held in equilibrium by an unbalanced force F, two equal and opposite forces G, two equal and opposite forces H, and the unbalanced force K. The net result, therefore, is that A, B, and C are held in equilibrium by the two forces F and K. The resultant R is the sum of A, B, and C, and therefore the combined-load line represents the resultant R. The external lines of diagram b show that F, K, and R form a closed figure with the arrows running continuously around the figure; and that F and K are two forces which hold R, the resultant of A, B, and C, in equilibrium. By producing the lines representing the forces F and K in diagram a until they intersect at x, we may draw a vertical line through it which gives the desired line of action of R. This is in accordance with the principles given in the previous article.

Fig. 214. Resultant of Three Forces.

Fig. 214. Resultant of Three Forces.

Fig. 215. Equilibrium Polygon with Oblique Closing Line.

Fig. 215. Equilibrium Polygon with Oblique Closing Line.

Nothing was said as to how F, G, H, and K were drawn in a and b. These forces simply represent one of an infinite number of combinations of forces which would produce the same result. The point o is chosen at random, and lines (called rays) are drawn to the extremities of all the forces. The lines of force (A, B, and C) in diagram b (which is called the force diagram), are together called the load line. The line of forces (F, G, H, and K) in diagram a, together with the closing line yz, is called an equilibrium polygon.