THEORY OF STABILITY.
There are two classes of theories of the stability of the masonry arch—the line of thrust theories and the elastic deformation theories. The line of thrust theory considers the stability of the arch ring as depending upon the friction and the reactions between the sev eral arch stones; while the elastic theory regards the arch as a curved beam which depends for its stability upon the internal stresses devel oped in the material of the arch. Both theories can be applied to either a voussoir or a continuous arch, although usually the line of thrust theory is employed for the voussoir arch, and the elastic theory for the monolithic arch. There is no great difference between the two theories, although the elastic theory is a little more com plicated but a little more accurate. In this chapter the voussoir arch will be investigated by the line of thrust theory; and in the next chapter the monolithic arch will be considered by the elastic theory.
A clear comprehension of the nature of the line of resistance is fundamental in the theory of the voussoir arch.
If the action and reaction between each pair of adjacent arch stones be replaced by single forces so situated as to 'be in every way the equivalent of the distributed pressures, the line connecting the points of application of these several forces is the line of resistance of the arch. For example, assume that the half arch shown in Fig. 186 is held in equilibrium by the horizontal thrust T—the re action of the right-hand half of the arch—applied at some point a in the joint CH. Assume also that the several arch-stones fit mathematically, and that there is no adhesion of the mortar. The forces F„ F,, and F, represent the resultants of all the forces (including the weight of the stone itself) acting upon the several voussoirs. The arch stone CIGH is in equilibrium under the action of the three forces, T, F„ and the reaction of the voussoir IJEG. Hence these three forces must intersect in a point, and the direction of resultant pressure between the voussoirs CIGH and IJEG—can be found graphically as shown in Fig. 186. The point of application of R, is at b—the point where intersects the joint GI. The voussoir IJEG is in equilibrium under the action of R„ and resultant reaction between JEGI and JEDK, —and hence the direction, the amount, and the point of application (c) of R, can be deter mined as shown in the figure. R, and R, are de
termined in the same manner as and The points a, b, c, d, and e, called centers of pressure, are the points of application of the result ants of the pressure on the several joints; or they may be regarded as the centers of resistance for the several joints. In the for mer case the line abcde would be called the line of pressure, and in the lat ter the line of resistance. Strictly speaking, the line of resistance is a continuous curve circumscribing the polygon abcde. The greater the number of joints the nearer the polygon abode approaches this curve. Occasionally the polygon mnop is called the line of resistance. The greater the number of joints the nearer this line approaches the line of resistance as defined above.
If the four geometrical lines ab, bc, cd, and de were placed in the relative position shown in Fig. 186, and were acted upon by the forces T, F„ F„ F„ F,, and R, as shown, they would be in equilibrium; and hence the line abcde, or rather a curve passing through the points a, b, c, d, and e, is sometimes called a linear arch.