Theory of Stability

It is evident. that before any conclusions can be drawn concerning the strength or stability of a masonry arch, the position of the line of resistance must be known; or at least, limits must be found within which the true line of resistance must be proved to lie. But before the line of resistance can be found, the external forces and also the crown thrust must be determined.

It is clear that before we can find the stresses in a proposed arch and determine its dimensions, we must know the load to be supported by it. In other words, the strength and stability of a masonry arch depend upon the posi tion of the line of resistance; and before this can be determined, it is necessary that the external forces acting upon the arch shall be fully known, i.e., that (1) the point of application, (2) the direction, and (3) the intensity of the forces acting upon each voussoir shall be known. Unfortunately, the accurate determination of the external forces is, in general, an impossibility.

Pressure of Water.

If the arch supports water or other liquid, the pressure upon the several voussoirs is perpendicular to the extrados, and can easily be found; and combining this with the weight of each voussoir gives the several external forces. This case seldom occurs in practice.

Pressure of Masonry.

If the arch is surmounted by a masonry wall, as is frequently the case, it is impossible to determine, with any degree of accuracy, the effect of the spandrel walls upon the stability of the arch. It is usually assumed that the entire weight of the masonry above the soffit presses vertically upon the arch: but it is known certainly that this is not the case, for with even dry masonry a part of the wall will be self-supporting. The load sup ported by the arch can be computed roughly by the principle of § 631; but, as this gives no idea of the manner in which this pressure is distributed, it is of but little help. The error in the assumption that the entire weight of the masonry above the arch presses upon it, is certainly on the safe side; but if the data are so rudely approxi mate, it is useless to attempt to compute the stresses by mathe matical processes. The inability to determine this pressure consti tutes one of the limitations of the theory of the arch.

Usually it is virtually assumed that the extradosal end of each voussoir terminates in a horizontal and vertical surface (the latter may be zero) ; and therefore, since the masonry is assumed to press only vertically, there are no horizontal forces to be considered. But as the extrados is sometimes a regular curve, there would be active horizontal components of the vertical pressure on this surface; and this would be true even though the spandrel masonry were divided by vertical joints extending from the extrados to the upper limit of the masonry. Further, even though no active horizontal forces are developed, the passive resistance of the spandrel masonry—either spandrel walls or spandrel backing—materially affects the stability of an arch. Experience shows that most arches sink at the crown and rise at the haunches when the centers are removed (see Fig. 189, page 610), and hence the resistance of the spandrel masonry will materially assist in preventing the most common form of failure. The efficiency of this resistance will depend upon the execution of the spandrel masonry, and will increase as the deformation of the arch ring increases. It is impossible to compute, even roughly, the

horizontal forces due to the spandrel masonry.

Further, in computing the stresses in the arch, it is usually assumed that the arch ring alone supports the masonry above it; while, as a matter of fact, the entire masonry from the intrados to the top of the backing acts somewhat as an arch in supporting its own weight.

Pressure of Earth.

If the arch supports a mass of earth, we can know neither the amount nor the direction of the earth pressure with any considerable degree of accuracy (see Chap. XVIII—Re taining Walls,—particularly § 1008).

In the theory of the masonry arch, the pressure of the earth is usually assumed to be wholly vertical, even though it is well known that the pressure of earth, in general, gives active horizontal forces. An examination of Fig. 186 (page 609) will show how the horizontal forces add stability to an arch ring whose rise is equal to or less than half the span. It is clear that for a certain position and intensity of the thrust T, the line of resistance will approach the extrados nearer when the external forces are vertical than when they are inclined. We know certainly that the passive resistance of the earth adds materially to the stability of masonry arches; for the arch rings of many sewers which stand without any evidence of weakness are in a state of unstable equilibrium, if the vertical pressure of the earth immediately above the ring be considered as the only external force acting upon it.

The value and position of the horizontal components of the external forces are somewhat indeterminate. According to Ran kine's theory of earth pressure, the horizontal pressure of earth at any point can not be greater than 1 + sin times the vertical pres 1 — sin sure at the same point, nor less than 1 — sin times the vertical 1 + sin 0 pressure,-0 being the angle of repose.* If 0 = 30°, the above expression is equivalent to saying that the horizontal pressure can not be greater than three times the vertical pressure nor less than one third of it. Evidently the horizontal component will be greater the greater the cohesion and the harder the earth spandrel-filling is rammed into place. The condition in which the earth will be deposited behind the arch can not be foretold; but it is probable that at least the minimum value, as above, will always be realized. Hence we will assume that the horizontal intensity is at least on third of the vertical intensity; that is to say, h = e d 1, in which e is the weight of a cubic unit of earth—which may be assumed at 100 lb. per cu. ft.,—d the depth of the center of pressed surface below the top of the earth filling, and 1 the vertical dimension of the surface. On this assumption the values and the positions of the horizontal forces acting on the several voussoirs of any particular arch can readily be determined.

It would be more logical in determining the horizontal component of the earth pressure, to use the angle of internal friction (f 1000) instead of the angle of repose as above; but the laws of earth pressure are not known, and the above value of the horizontal component has been employed by the author in testing numerous voussoir arches, and seems to give results in accordance with experience; and hence it will be employed in this chapter.