**ABC—XYZ**

ARCHES.] BRIDGES 309 joint. Greater or less elastic resistance in the stone corresponds to greater or less curvature in the surface of the joint. A small dis tortion of the arch will restore equilibrium when the curvature is great, or when the stone has a high modulus of elasticity. The ring with plane bed-joints is in stable equilibrium, and adapts itself to new distributions of load for precisely the same reasons as the model with curved joints, but in the one case the couple called into play to move the voussoir is actually cancelled by the new posi tion which the points of contact assume ; in the other case it is balanced by the equal and opposite couple resulting from the resistance to motion due to the hardness of the stone. The preceding paragraph showed how to determine whether an arch was in equilibrium when a known reaction was applied at one abutment ; the experiment and reasoning now given show that the incipient yielding of an arch under loads will produce a reaction at the abutments suited to keep the whole ring in equilibrium, provided only an equi librated polygon can be drawn, cutting the joints within the ring at suitable angles. 41. Practical Investigation of the Stability of a given Arch under a given Load Joint of Rupture. This investi gation resolves itself into finding that equilibrated polygon or linear arch which can be drawn within the (middle third of the) ring from the crown to the lowest jiossible joint of the ring (or to the springing if this be possible). This lowest possible joint must in any case be treated as the springing of the arch, and if the linear arch goes out of the (middle third of the) ring above the actual springing, as will be the case in all semicircular or elliptical rings, masonry must be provided in the backing capable of taking the actual thrust into the abutment and constituting the real arch, which often differs widely from the form indi cated by the ring of stones in the face. The linear arch in a circular or segmental bridge loaded simply by its own weight generally has a smaller radius of curvature than the ring at the crown, and a much larger radius towards the haunches. Consequently, the longest linear arch which can be drawn within the ring will approach the upper surface of the rin at the crown and the soffit towards the haunches. Fig. 49 shows a series of linear arches, all drawn for the same A)ad, and all tangent to the upper surface of the crown of the arch, differing only in being the result of different horizontal thrusts. The curve drawn with a thick black line, tangent to the soffit, is clearly the longest linear arch which can be drawn within the ring. Any smaller value of the horizontal thrust h would give a linear arch like curve 3, and any larger value of k would give a linear arch like curve 1, and both these values of hare incompatible with equili brium for the whole arch down to joint C ; if, therefore, the arch fails by the yielding of the abutment, or of the lower portion of the ring, the failure will first be apparent at the joints A and 15, where this black line is tangent to the ring, and at joint C, where the linear arch cuts the back of the ring. Smaller values of h will keep the stones in equilibrium above and below joint 1!, but unless the arcli below the joint 13, as well as the abutment, can resist the tendency of the arch to spread, or, in other words, supply at least the horizontal reaction h required for this linear arch, the joint B will open at the top, the centre joint A will open at the bottom, the joint C will open at the back, and the crown fall in as shown in fig. 49. The joint B, where the longest linear arch is tangent to the soffit, is called tih.e joint of rupture. The value of h required to make a linear arch tangent to the back of the ring at the crown pass through the edge of the joint of rupture at the soffit, is larger than the value of h required to give a linear arch passing through the edge of any other joint at the soilit ; at the same time, it is the smallest value of h consistently with which the arch can remain in Fig. 49a. equilibrium down to B and from B to C. In circular arches the joint of rupture generally makes an angle of about 30 with the hori zontal plane ; in elliptical arches the angle is usually about 45^. Its position is easily found as follows : Let y lt ?/ 2 , 7/ 3 , &c. (fig. 50), be the heights of the upper surface of the crown A above any points Fig. 50. Bj, B.,, B 3 at the lower edges of the soffit ; let W lt W.,, W 3 be the weights of the portions of the arch with its load carried by the ring from Bj to A, from B 2 to A, from B 3 to A, &c. (The load is in the fig. assumed to be symmetrically disposed relatively to the centre of the span.) Let x v x. 2 , x 3 be the horizontal distances of the centres of gravity of w v w. 2 , 3 from the points B a , B 2 , B 3 , &c. ; then taking moments round B 1( B 2 , B 3 in succession, we have, if the linear arch be assumed to pass through any point B W* - hy taking the successive values of h for a series of joints B, we shall find that one joint gives a maximum value. This value corresponds with that of the linear arch tangent to the soflit (of the middle third) at the joint of rupture ; for this arch has the maximum thrust of any passing through the points Bj, B 2 , &c., as appears by simple inspection of fig. 49. The joint of rupture can thus 1 e tentatively found, and the value of h, or the thrust which the abutment must resist, is obtained at the same time. If the backing is carried well up above C, a larger value of h than that obtained by this method would be consistent with the stability of the arch, and might actually occur ; but we need not provide for this larger value, since the yielding of the abutment under it would diminish the thrust till it fell to the value as above detennincd. If the abutments could resist this thrust, the bridge would then remain in equili brium. If the arch is flat there may be no joint of rupture, and in that case the value of h is to be taken as that given by a linear arch passing through the bottom of the (middle third of the) spring ing and tangent to the crown of the arch, i.e., to the summit of the middle third of the ring. When the apparent springing lies much below the joint of rupture, we find that the linear arch leaves the ring on the upper surface at a joint (C) lower down, where failure must result by the opening of the joint at the lower sur face, xmless the pressure is taken by masonry outside the ring. It is for this purpose that the lacking is required.

Obviously the best mode of supplying backing is to thicken