Theory of Stability

There are three criteria, corresponding to the three modes of failure, by which the stability of an arch may be judged. (1) To prevent overturning, it is necessary that the line of resistance shall everywhere lie between the intrados and the extrados. (2) To prevent crushing, the line of resistance should intersect each joint far enough from the edge so that the maximum pressure will be less than the crushing strength of the masonry. (3) To prevent sliding, the angle between the line of resistance and 'the normal to any joint should be less than the angle of repose ("angle of friction ") for those surfaces; that is to say, the tangent of the angle between the line of resistance and the normal to any joint should be less than the coefficient of friction (1 931).

Stability against Rotation.

An arch composed of incom pressible voussoirs can not fail by rotation as shown in Fig. 189, unless the line of resistance touches the intrados at two points and the extrados at one higher intermediate point (see Fig. 193, page 618); and an arch can not fail by rotation as shown in Fig. 1.90, unless the line of resistance touches the extrados at two points and the intrados at one higher intermediate point (see Fig. 193). The approximate factor of safety against rotation (1 939) at any joint is equal to half the length of the joint divided by the distance between the center of pressure and the center of the joint; that is to say, in which 1 is the length of the joint and d the distance between the center of pressure and the center of the joint. For example, if the center of pressure is at one extremity of the middle third of the joint, d = i l; and, by equation 1, the factor of safety is three. If the center of pressure is 1 from the middle of the joint, the factor of safety is two.

It is customary to require that the line of resistance shall lie within the middle third of the arch ring, which is equivalent to specifying that the approximate factor of safety for rotation shall not be less than three.

Stability against Orushing.

The method of determining the pressure on any part of a joint has already been discussed in the chapter on masonry dams (see pages 470-76). When the total pressure and its center are known, the maximum pressure at any part of the joint is given by formula 19, page 472. It is in which P is the maximum pressure on the joint per unit of area; W is the total normal pressure on the joint per unit of length of the 'arch; t is the depth of the joint, i.e., the distance from intrados to extrados; and d is the distance from the center of pressure to the middle of the joint. This formula is general, provided the masonry is capable of resisting tension; and if the masonry is assumed to be incapable of resisting tension, it is still general, provided d does not exceed 1.

For the case in which the masonry is incapable of resisting tension and d exceeds 1, the maximum pressure is given by formula 23, page 475. It is

If the Lie of resistance for any arch can be drawn, the maximum pressure can be found by (1) resolving the resultant reaction per pendicular to the given joint, and (2) measuring the distance d from a diagram of the arch similar to Fig. 186 (page 609), and (3) sub stituting these data in the proper one of the above formulas (the one to be employed depends upon the value of d), and computing P. This pressure should not exceed the safe compressive strength of the masonry.

Unit Pressure. In the present state of our knowledge it is not possible to determine the value of a safe and not extravagant unit working pressure. The customary unit appears less extrava gant when it is remembered (1) that the crushing strength of masonry is considerably less than that of the stone or brick of which it is composed (see § 581; and § 622-23 respectively), and that we have no definite knowledge concerning either the ultimate or the safe crushing strength of stone masonry (§ 582-84) and but little con cerning that of brickwork in large masses (§ 622-29); and (2) that all the data we have on crushing strength are for a load perpendicular to the pressed surface, while we have no experimental knowledge of the effect of the component of the pressure parallel to the surface of the joint, although it is probable that this component would have somewhat the same effect upon the strength of the voussoirs as a sheet of lead has when placed next to a block of stone subjected to compression (§ 14).

On the other hand, there are some considerations which still further increase the degree of safety of the usual working pressure. (1) When the ultimate crushing strength of stone is referred to, the crushing strength of cubes is intended, although the blocks of stone employed in actual masonry have less thickness than width, and hence are much stronger than cubes (see § 17, § 78, and § 657). To prevent the arch stones from flaking off at the edges, the mortar is sometimes dug out of the outer edge of the joint. This procedure diminishes the area under pressure, and hence increases the pressure; but, on the other hand, the edge of the stone which is not under pressure gives lateral support to the interior portions, and hence increases the resistance of that portion (see § 657). It is impossible to compute the relative effect of these elements, and hence we can not theoretically determine the efficiency of thus relieving the extreme edges of the joint. (2) The preceding formulas (2 and 3) for the maximum pressure neglect the effect of the elasticity of the stone; and hence the actual pressure must be less, by some unknown amount, than that given by either of the formulas.