306 BRIDGES [ARCHES. masonry above the haunches of the ring; it is carried back between the spandrils to the pier or abutment. If the backing is not carried up to the roadway, as is seldom the case, the rough material employed between the backing and the roadway is called the fitting. The parapet rests on the outer spandrils. The abutments and piers have the same signification as in other bridges. The masonry arch differs from the superstructure of other bridges in the following respect : it clepsnds for its stability on the presence of a permanent load specially arranged, and so considerable in amount that the changes produced in the direction and magnitude of the stresses by the passing load are insignificant. The theories of the masonry arch often neglect the passing load entirely, and simply teach the student how to distribute the per manent load, so that the voussoirs may be in equilibrium. The permanent load consists of the ring, the backing, the filling, the spandrils, and the roadway. Inasmuch as the ring is that part of the structure which by its special strength and arrangement carries the superstructure in the same sense as a beam or chain carries it, the arch in this article will be treated simply as a ring of voussoirs springing from two abutments and loaded with weights, some permanent and some passing. Where the backing strengthens the arch, it becomes virtually part of the ring. 37. Equilibrium of a Single Voussoir. A block, such as a voussoir, ABCD, fig. 42, resting on one of its surfaces, such as the joint AB separating it from the next voussoir, is in equilibrium when the resultant of all the forces acting upon it (including its own weight) falls within the supporting surface, while the direc tion of this resultant makes an angle < with the nor mal to the surface less than the angle of repose ; (the tangent of the angle of repose is the coefficient of friction). If the resultant, as K,, falls without the surface, the block will heel over, pivoting on the edge A. If the resultant, as B /; , although falling within the surface of which AB is the trace, is yet much inclined to the normal, the block ABCD will slide up on the joint AB without heeling over. The block, if used as the voussoir of a bridge, must not only be in equilibrium under the forces applied to it, but must also be of sufficient strength to resist these forces. The intensity of crushing stress due to the external forces must nowhere exceed the safe crushing strength of the material. This latter condition would in most arches bo fulfilled by an extremely thin ring of stones or brick if the resultant passed through the geometrical centre of the joint AB in a direction normal to it. In that case the stress on the joint would be a uniformly distributed stress ; if, however, the resultant stress passes near one edge, the intensity of stress at that edge will be much greater than elsewhere, and would indeed be infinite if the resultant passed exactly through the edge at A or B ; while, therefore, the condition of equilibrium is satisfied if the resultant passes within either edge of the voussoir at no great inclina tion, the condition of strength requires that this resultant shall not cut the joint very near the edge, and the common practical rule is that it shall always fall within the middle third of the joint. This rule is based on the condition that the pressure on a joint shall nowhere be negative ; in other words, that no tension shall occur at any part of any joint. The principles explained in 8 show that the minimum Fig. 42. J. -f stress on any joint p p Q - ~ , or that the stress will be zero, when p = Px x
T . Let d be the depth of the rect angular joint, and b the breadth ; then d P hsnce Id 3 1= j^andfcj = 3, uuu/,0 = ^, 12P.r rj bd an equation expressing the condition that the centre of pressure lies at the edge of the middle third ; any greater value of x Q will give a negative value to p v We shall see that the actual resultant is, according to the theory practically in use, indeterminate within certain limits ; it is therefore useless to attempt to calculate the exact maximum stress on any one stone. In the rest of this article the ring is to be held to mean the middle third of the actual masonry, or brick ring, wherever the theory requires that the blocks are to resist practical loads. As bridges are subject to a sensibly equal load on all parts of their breadth between the parapets, it is usual to consider a portion of the ring one foot in width, each other strip being under precisely similar conditions. Similarly the joint may be spoken of for convenience as the line which is its trace, and the edge as the point which is its trace. The external forces which act on any voussoir are 1st, the vertical force, being the resultant of its own weight and the load which is directly over it ; 2d, the thrust from the voussoir above it ; and 3d, the reaction from the voussoir on which it rests (fig. 43). It is sometimes difficult to determine exactly what portion of the superincumbent load a voussoir may properly be said to carry, but a sufficient ap proximation is obtained for practical purposes by assuming that the mass vertically above any voussoir is carried by the voussoir when the back of the voussoir is not much inclined. If the materials had little co hesion, the direction of tho force produced by the load would not be vertical, but inclined at an angle depend ing on the coefficient of fric tion ; in practice, the direc tion of the force is uncertain and even variable with changes in the condition of the superincumbent filling. If, however, the stability of the arch is calculated with a reasonable margin or coefficient of safety, on the hypothesis that the force produced by the load is vertical, there is every probability that the arch will be stable under any actual stress which may arise in practice. 38. Equilibrium of any three Voussoirs; .Equilibrated Polygon. The simplest arch would be an arch of three voussoirs resting on two abutments, and any actual arch consisting of many voussoirs may be considered as com posed of successive triplets, the voussoirs on each side of which act as abutments. If, therefore, we can show the conditions of equilibrium for three voussoirs we shall have determined the conditions for the whole ring. Let three voussoirs be taken from any part of the ring (fig. 44), and let the lines 1, 2, and 3 represent the position of the resultants of the three known loads w lt w. 2 , and w 3 (including the weight of the voussoirs) borne by each voussoir. _ Let NA represent the position of the reaction t due to the abut ting voussoir on one side. Let A be the point where the prolonga
tion of the line NA cuts the line 1 ; then if the magnitude of the