324 BRIDGES [SUBSTRUCTURE. spectively ; and w 2 and iv. 2 the proportions of the load on AB borne by B and A respectively. Then draw the line of loads DEFGH as sketched (fig. 89). From E and G draw lines parallel to AC and AB respectively, meeting in O. Join CD and OH. OD and Oil represent in magni tude and direction the thrust on C and on B. OF is the stress on the pin at A. These thrusts being known, the stress on each member of each frame can be easily com puted by a reciprocal figure or otherwise. This form is only inferior to the true framed arch or suspension bridge inasmuch as it is incapable of balancing the thrust due to the passing load on neighbouring spans. It is superior to the framed arch and suspension bridge inasmuch as it can not be strained by any change of temperature. VII SUBSTRUCTURE. 64. Preliminary. The substructure of a bridge com- prisas the piers, abutments, and foundations. Those por tions of the bridge usually consist of masonry in some form, including under that general head stone masonry, brickwork, and concrete. Occasionally metal work or woodwork is used for intermediate piers. When girders form the superstructure, the resultant pressure on the piers or abutments is vertical, and the dimensions of these are simply regulated by the sufficiency to bear this vertical load. When arches form the superstructure, the abutment must be so designed as to transmit the resultant thrust to the foundation in a safe direction, and so distributed that no part may be unduly compressed. The intermediate piers should also have considerable stability, so as to counterbalance the thrust arising when one arch is loaded while the other is free from load. For suspension bridges the abutment forming the anchorage must be so designed as to be thoroughly stable under the greatest pull which the chains can exert. The piers require to be carried above the platform, and their design must be modified according to the type of suspension bridge adopted. When the resultant pressure is not vertical on the piers these must be constructed to meet the inclined pressure. In any stiffened suspension bridge the action of the pier will be analogous to that of a pier between two arches. G5. Stability. When the magnitude and direction of the thrust borne by a pier, or abutment at the springing are known, the stability of any series of masonry blocks forming the pier or abutment may be studied by drawing lines showing the direction and magnitude of the resultant force on each joint. This may be done as for the voussoirs of an arch. The thrust on the upper block may be con- pounded with the weight of that block, the resultant com pounded with the weight of the next, and so forth, until the direction and magnitude of the thrust on the rock or earth foundation is determined. A better method of making the drawing is shown in fig. 91 ; find the centre of gravity C / of block 1, the centre of gravity C a of blocks 1 and 2 treated as a single mass, similarly C,,, for blocks 1, 2, and 3 Let AT be the direction of the thrust on the top bloek, and 0,3, a vertical line though C, cutting AT in B, ; let B,D, be the direction of the resultant of t, the thrust acting in the line AT, and w, the weight of the first block acting in the line C/B, ; and let D, be the point where the direction of this resultant cuts the first joint; similarly let l^D,, be the direction of the resultant of t and the weight w, + w tl of the first two blocks; 13,,/D,,, the direction of the resultant of t compounded with the weight w / + w fl + w llt of the three first blocks, &c., &c. This method of proceeding gives the direction and magnitude of each force and centre of pres sure D independently of the values obtained for the preceding joints. For stability the line BD must not make a greater angle with the normal to the joint than the angle of repose ; and the point D must nowhere fall beyond the edge of the joint ; for strength and safety the point D might be required to fall within the middle two-thirds of the joint, or within tho middle three- quarters. The theory by which the joints furthest from the centre of pressure would open when tho centre of pressure leaves the middle third cannot apply to such a structure as a masonry abut ment, all parts of which arc bonded together. Professor G. Fuller calculates the thickness of abutments by the following empirical rule, deduced from many practical examples: Let c = half the span in feet ; d = versed sine in feet ; t = thickness of abutment at springing ; then, for ilat arches, in which the span does not exceed 150 feet, and the ratio of the rise to the span is not less than 116, (c 1 - + d I . From the springing to the base the abutment may have a batter of 1 in 4. This gives an abutment the cubic capacity of which will be sufficient, but it may with ad vantage be divided into abutment proper and counterforts ; in semi circular arches t should be taken as the thickness of the abutment at a height above the springing equal to two-thirds of the radius. The maximum intensity of stress on the stone at the edge F might be approximately found by the theory explained in 8. This theory finds a useful application in calculating the maximum inten sity of stress which a given foundation might produce on the earth t -Jp - Q Fig- 92. or rock supporting it. Thus, let the section of the foundation under consideration be 1 foot in breadth, 8 feet in length from F to G, as shown in plaa fig. 92 ; let the centre of pressure D be 2 feet from FF, and let the total resultant thrust be about 16 5 tons, in clined so that the horizontal component is 4 tons and the vertical component 16 tons, then the mean vertical intensity of pressure per square foot is l s -" 2, and the maximum intensity along the line FF, is by equation 2, 8 2 Jl+(4x2x8)~ 2 j = 5 > this maximum intensity being 2^ times as great as the mean intensity. Obviously, although for the sake of appearance the courses of masonry in the abutment of an arch may be left horizontal in tho face, the stability is increased by inclining them at a proper angle, so that they lie normal to the thrust. The stability of abutments may be tested by taking moments round points in the joints selected as the points beyond which the thrust must not come. The same methods apply to the anchorages of suspension bridges, and to intermediate piers, which are intended to take a given horizontal thrust. When metal or wooden piers are adopted their weight will generally be insignificant, and such as may be neglected in calculating their stability. Metal-work piers or wooden piers usually consist of wrought or cast-iron frame work, and the stress on each part of the frame, as well as the resultant stress on the
foundations where each upright member reaches it, is