SUSPENSION BRIDGES.] BRIDGES 303 hand pier, and H the horizontal component. We know that the vertical component is equal to the whole reaction at the pier when the same load is carried by a girder ; then, taking moments about the joint in question, we have but the second member of the equation is the bending moment m 3 for the section at a distance 7 from the pier in a girder of similar span and similarly loaded, therefore, whatever may be the value of H, the values of y v y v y 3 , &e., are proportional to the bending mo ments. If, then, a curve of bending moments with the ordinates m lt m. 2 , ??1 3 , &c., be drawn for a given distribution of load, we can, with a pair of proportional compasses, construct any number of equilibrated curves, by making the values of?/ in these curves simply proportional to the values of m in the curve of bending moments, and by selection among these a curve of any required length may be found. If H be unity, the ordinates y lt y%, y 3 , &c., are equal to the bending moments. 31. Chain Loaded uniformly along a Horizontal Line. If the lengths of the links be assumed indefinitely short, the chain under given simple distributions of load will take the form of compara tively simple mathematical curves known as catenaries. The true catenary is that assumed by a chain of uniform weight per unit of length, but the form generally adopted for suspension bridges is that assumed by a chain under a weight uniformly distributed relatively to a horizontal line. This curve is a parabola. From equation 1, 30, remembering that S.wl in this case will wl? be equal to -g , we see that the horizontal tension II at the vertex for a span L (the points of support being at equal heights) is given by the expression II - JTX n or, calling x the distance from the vertex to the point of support, The value of H is equal to the maximum tension on the bottom flange, or compression on the top ilange, of a girder of equal span, equally and similarly loaded, and having a depth equal to the dip of the suspension bridge. Consider any other point F of the curve, fig. 34, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered ; the vertical component V will be simply the sum of the loads between and F, or wx. In the triangle FDO, let FD be tangent to the curve, FC vertical, and DC horizontal ; these three sides will necessarily be proportional respectively to the resultant tension along the chain at F, the vertical force V passing through the point D, and the horizontal tension at O ; hence = DC ~ hence DC is the half of OC, proving the curve to be a parabola. The value of K, the tension at any point at a distance x from the vertex, is obtained from the equation R 2 = IP Let i be the angle between the tangent at any point having tlu co-ordinates x and y measured from the vertex, then Let the length of half the parabolic chain be called s, then The following is the approximate expression for the relation be tween a change As in the length of the half chain and the corre sponding change Ay in the dip : (Ay)"- = 4wA?/ -" or, neglecting the last term, and 6. A S = 3 x 3 x = 5 -A*. From these equations the deflection produced by any given stress on the chains or by a change of temperature can be calculated. If the points of support are not at equal height (fig. 35) call the heights above the vertex y and y l5 and the horizontal distances of Fig. 35. the vertex from the points of support x and x l ; let y and y l be given, and x, x : unknown. The horizontal stress at the vertex will be the same as if the bridge were composed of two symmetrical halves, each having a span 2x and a dip y, or of two symmetrical halves, with a span 2x 1 and a dip y 1 ; in other words _ we
- ~oT7
hence 7. />, z zL 2/1 7TV*; thus, to find the horizontal position of the vertex, we have only to subdivide the span in the ratio Vy ; /y l we may then calculate the strains on one side of the vertex as for half a bridge with the span 2x and the dip y, and on the other side of the vertex as for half a bridge with the span 1x l and the dip y v . The device of piers of different heights may be used with advantage when it is desired to throw a larger portion of the weight of the bridge on one pier than the other, because of a difference in the soundness of the foundations, or for other reasons. Th^stresses on the loaded and un loaded portion of the chains between the piers and the anchorage are easily determined by methods similar to those which have been given for the stresses on each part of the main span. The same methods also give the direction of each successive link, and of the final links leading to the anchorage. 32. Practical Details. The chains of suspension bridges are either long wire ropes or true chains made of links pinned together. Wire ropes allow the strongest
known material to be adopted, namely, steel wire, which