**ABC—XYZ**

310 the ring itself, keeping the masonry joints radial. The portion of the arch below the joint of rupture B is often considered as part of the abutment. If the load at the crown of an arch were very light, and the load at the haunches comparatively very heavy, the series of tentative curves drawn with various values of li would assume the character shown in fig. 51. The longest curve which can now be fitted into the ring (drawn with a thick black line in the figure) will probably approach the soffit at the crown and be tangent to the upper surface of the ring at the haunch. In fig. 51 the longest linear arch is shown as tangent to the soffit at C. This condition could seldom be secured ; with most loads the linear arch tangent to the back of the ring at B will cut the soffit at 0. Nevertheless, the value of h to be provided for will be that given by the linear arch tangent to the soffit. If this arch leaves the middle third at B, the ring must be thickened or efficient backing provided at this point. If the abutment yield an arch thus loaded would fail, as in fig. 5 la, but the case very seldom arises in practice. If the arch were not pointed at A, but curved so as to con tain the linear arch near the crown, the piece BAB would be lifted up as a whole without breaking at A. Fig. 51. The joints of rupture can be found for unsymmetrical loads as well as for symmetrical loads, but these joints will then not be at equal distances from the crown. Fig. 51. If the middle third of the ring be alone treated as effective, the designer, after finding the joint of rupture for a bridge of the usual form and with usual loads, need make no further calculation as to the arch above that joint. A linear arch which is tangent to the soffit at the joint of [ARCHES. rupture, and to the upper surface of the ring at the crown, will probably lie within the ring at intermediate joints, and will cut them at an angle not differing much from a right angle ; but the linear arch must be carried on below the joint of rupture, through the backing and the abutments, to see that it is nowhere too much inclined to the bedding joints, and never comes too near the edge of the effective masonry. The horizontal thrust determined by finding the joint of rupture on the hypothesis that the middle third of the ring is the only effective part will be a safe value; but the actual value may be considerably less, since the actual linear arch called into play may lie outside the middle third. Since we do not know the actual position of the resultant pressures on each voussoir, any refinement in calculating the maximum in tensity of stress due to these resultants would be useless. If the actual horizontal thrust were known, it would be easy to determine the couple acting on each joint and due to the distance between the resultant pressure and the centre of resistance of the joints ; then knowing this couple and the total thrust it would be equally easy by the principles in 8 to determine the maximum intensity of stress. Practi cally the thickness of the arch ring is determined by rules derived from experience, and the chief use of the above theory is to determine the dimensions of the abutments ; if, however, with a given load the joint of rupture were found much nearer the crown than the positions indicated above, it would be well to rearrange the permanent loads or to alter the form of the ring. 42. Professor George Fuller of Belfast has communi cated the following novel and very neat method of finding the linear arch of maximum rise (and therefore of minimum thrust) which can be drawn within the middle third of a given ring. In fig. 52 let the dotted curves GI and I1K bound the middle third of the ring. Let the span be divided into any convenient Fig. 52. number of parts at a, b, c . . . &c. Let the load on the half arch be subdivided into a corresponding number of parts, and each partial load referred to the vertical line passing through a,b,c. . . &c. Let the curve D 1 2 3 . . . A be a curve of bending moments for these loads, drawn to any convenient scale. This curve will also ( 30) be a linear arch for the given loads. Draw the straight line AB at any convenient inclination, cutting the horizontal line DB at B. Eaise the verticals al, 12, c3, . . . &c., from the points 1, 2, 3, &c. Where these cut the curve DA draw horizontal lines, cutting AB at 1 , 2 , 3 , ... &c. Since the ordinates of all possible linear arches are merely multiples or submultiples of the curve of bending moments, it follows that any other straight line from B to the vertical through A will have ordinates, which, if measured from DB along the verticals passing through 1 , 2 , 3 , &c., will be the ordinates of a linear arch, set off on the corresponding verticals passing through 1 and a, 2 and b, 3 and c, &c. AB might be called the development of the linear arch DA. Now let the curves CI and HK be developed in a similar way, so that, for instance, the ordinates measured from a to these curves are equal to the ordinates measured on the vertical passing through 1 from DB to the developments ] 1 Gf l and Hj Kj ; then it is clear that for the given loads any linear arch which lies within the middle third of the ring must, when developed, be represented by a straight line lying within the area Ij G : H] K 1} and consequently that the straight line BC, which starts from the lowest point B in this area, and is

tangent to the curve G l l lt will be the development of the curve of