Art. XXIX.—On some points connected with the Construction of the Bridge over the Grey River at the Brunner Gorge.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 9, 1876, Page 310

Art. XXIX.—On some points connected with the Construction of the Bridge over the Grey River at the Brunner Gorge. By C. H. H. Cook, M.A.; Fell. St. John's Coll., Cam.; Prof. Math., Cant. Coll. [Read before the Philosophical Institute of Canterbury, 5th October, 1876.] It is not my intention to examine into the cause of the disaster which overtook the suspension bridge at the Brunner Gorge on the morning of the 28th

July last, but to call attention to the most noticeable peculiarity in its construction, and to investigate mathematically the tension on the wire ropes which supported the bridge. In order, however, to make the calculations and other remarks intelligible, it will be desirable to give a short description of the principal points connected with the structure. And here it will be well to state that this information has been derived partly from the Report, recently submitted to the General Assembly, of the Commissioners appointed to inquire into the cause of the accident; partly from direct communications from J. E. FitzGerald, Esq., who was a member of the Commission. The floor of the bridge was eight feet wide, and on each of its sides it was connected by means of suspending rods with a strong wire rope, or chain, which passed over two piers, each 25 feet high, one on each side of the river; each end of the chain was connected with an anchor-plate of cast iron attached to solid masonry, and intended to be built up with a mass of concrete. I may remark, in passing, that delay in executing this intention was, in the opinion of the Commission, the principal cause of the accident. The distance between the two piers over which the same chain passed was 300 feet; the distance between the two piers on the same side of the river was 30 feet. This width being much greater than the width of the floor of the bridge, 8 feet, it will be seen at once that the chains did not hang in a vertical plane. The lowest points of each chain may, it would seem, be taken to have touched the floor. None of the suspending rods on each side of that central point were vertical; the nearer the rods were to the banks of the river, the more and more did they slope outwards, till the the pair next the piers much have had their upper extremities nearly as far apart as the piers themselves, viz., 30 feet, whilst their lower ones were only eight feet apart. It is this peculiarity of construction which it is my purpose to examine mathematically, with a view of comparing the tension on the chain in this case with what it would have been had the usual method of construction been adopted, viz., that in which each chain as well as the suspending rods connected with it lie in a vertical plane, passing through a pair of piers. The object of this construction was, I believe, to stiffen the bridge. The bridge was intended to carry, at any one time, only a single truck loaded with coals, and never to have a locomotive upon it. It was estimated, therefore, that the load, in addition to that caused by the weight of the bridge itself, would not exceed ten tons. The weight of the bridge and suspending rods appears to have been 82 tons; the weight of each chain, seven tons; so that the total weight under which the bridge gave way was only 96 tons, no extra load being on it at the time.

The above description is all that is necessary for my purpose. As regards the investigation that ensues, I may remark that it is of the same degree of exactness as that given in treatises on civil engineering as applicable to the more ordinary method of structure. In the first place, I have been compelled to neglect the weight of the chain, because, though it is easy to form the equations of equlibrium when that weight is taken into account, yet they are unintegrable, or, at any rate, I believe so. This is to be the more regretted in this particular instance, because the weight of the chains formed a very appreciable part, rather more than one-seventh, of the total weight; in ordinary cases, of course the weight of the chains is insignificant, compared with that of the bridge, and no sensible error is made, therefore, in leaving their weight out of account. In the next place, I have treated the chains as forming a continuous curve, which is a departure, though a very slight one, from the case which actually occurs. I repeat that these suppositions are those usually made. For considering the equilibrium of either of the chains, take its lowest point as origin of co-ordinates, the vertical line through that as axis of z; the horizontal line through the same point, and parallel to the length of the bridge as axis of x; a line at right angles to both of them, that is to say, transverse to the bridge, as axis of y. Let w be the weight of a unit of the length of the bridge, T the tension of the rope at any point (x y z) in it; s the length of the rope measured from the lowest point up to the point (x y z); X Y Z, the resolved parts, parallel to the axes of co-ordinates, of the forces acting on the element ds in the neighborhood of (x y z); then the ordinary equations for equilibrium are:— X — ds d/ds (T dx/ds) = 0 (1) Y — ds d/ds (T dy/ds) = 0 (2) Z — ds d/ds (T dz/ds) = 0 (3) Since X = 0, there being no force in the direction of the axis of x, the first equation gives us at once T dx/ds = constant = c suppose (4) If T′ be the tension along suspending rod connected with the point (x y z) and θ the inclination of that rod to the vertical; then Z = T′ cos θ; Y = T′ sin θ. But, since there are two rods, one on each side of the bridge, which between them support a length dx of the bridge, therefore resolving vertically,— 2 T′ cos θ = wdx ∴ T′ = w/2 sec θ dx ∴ Z = wdx/2 Y =w/2 tan θ dx

Equations (2) and (3), therefore, become, d/ds (T dy/ds) - w/2 tan θ dx/ds = 0 (5) d/ds (T dz/ds) - w/2 dx/ds = 0 (5) (6) The latter of these equations integrates at once, and gives us,— T dz/ds - wx/2 = k but, when x = 0 we have dz/ds = 0, therefore k = 0 hence T dz/ds = wx/2 But from (4) T dx/ds = c Dividing one equation by the other,—- dz/dx = w/2c x Integrating, and observing that x and z vanish together, we get,— z = w/4c x2 (7) or, x2 = 4 c/w z (8) This is the equation to a parabola, and gives the parabola in which it is well known the chain would hang if everything were in a vertical plane. To integrate (5), we observe that tan θ = y/z, substituting this value we obtain,— d/ds (T dy/ds) - w/2 dx/ds · y/z = 0 But T dy/ds = T dy/dx · dx/ds = c dy/dx and from (7) y/z = 4cy/wx2 hence we obtain c d/ds (dy/dx) - 2c y/x2 dx/ds = 0 or x2 d2y/dx2 - 2y = 0 This can be reduced to a linear equation by the well-known substitution of putting x = e ø, and changing the independent variable to ø. In this way the complete integral will be if a and b are the constants of integration, y = ax2 + b/x And since x and y vanish together, b must be zero, hence the last equation reduces to y = ax2 (9) This equation together with (8) determines the form of the curve in which the chain hangs. The constants a and c, which enter into them, are easily determined from the condition that the chain passes through the top of the pier, whose co-ordinates are, adopting a foot as our unit of length, x = 150, y = 11, z = 25; and w = 82/300 = 41/150 tons

∴ a = 11/(150)2 . c = ¼ × 41/150 × (150)2/25 = 123/2 = tension at lowest point in tons. From equations (8) and (9) we deduce y/a - 4c/w z = 0 (10) the equation to a plane passing through the axis of x, that is, the straight line joining the feet of the suspending rods. Considering the curve in which the chain hangs, as determined by equation (10) combined with either (8) or (9), we see that the curve is that made by the intersection of the plane (10), with either of the parabolic cylinders (8) or (9), and hence is not only a plane curve, but is a common parabola. Next to calculate the tension at any point. We have already shown that T dx/dx = c ∴ T = c ds/dx ds/dx2 = 1 + dy/dx2 + dx/dx2 and dy/dx = 2 ax = 22x/(150)2 dx/dx = w/2c x = x/3 × 150 By making the necessary substitution it is then easy to find the tension at any point of the chain. At the point where the chain passes over the top of the pier x = 150, hence then dy/dx = 11/75 dz/dx = ⅓ Therefore the tension at that point is c √ 1 + 1/9 + 121/74 × 75 = 123/2 √ 752 + 252 + 121/75 = 41/50 √ 6371 = 65.45 tons nearly. The portion of the chain between the top of the pier and the anchor-plate will hang by its weight in a catenary, and the tension at the anchor-plate would be less than that at the top of the pier, by the weight of a piece of the chain equal in length to the vertical height between the two points. But, as we have neglected the weight of the chain all along, we must consider the tension on the anchor-plate to be equal to that at the top of the pier, viz., 65.45 tons. If the chain and suspensing rods had all lain in a vertical plane, equations (1) and (3) of our fundamental equations would have applied, and the chain would have then hung in the parabola whose equation is (8), and the c, which is a constant introduced by integration, would be the same as in the other case. It can be proved, as in the previous case, that the tension on the rope at the top of the pier would be c √ 1 + 1/9 = 123/2 × √ 10/3 = 41/2 √ 10 The ratio of the tension in the construction actually adapted to the tension which would have existed had the whole been in a vertical plane, is √6371/6250,

a fraction whose value will be found, on calculation, to be somewhat less than 1.01. It follows, therefore, that by adopting the construction explained in the foregoing part of this paper, the tension of the chain at the anchor-plate was not increased by more than one per cent. It might, perhaps, have been legitimate to assume that the chain would lie in a plane curvé; and then by reasoning similar to that used in the more usual case of a suspension bridge, it might easily be proved that the curve would be a parabola. But the method above given is perfectly general, and can readily be applied to the case in which the middle point of the chain is not attached to the floor of the bridge, but is a given height vertically above it, and in which the suspending rods are not in the same plane. Upon examination, the equations of equilibrium will be found to be integrable in this case also. I now wish to call attention to what appear to me to be serious defects in this bridge. 1. If there should have been any swaying of the bridge from side to side, inasmuch as the supporting chains did not hang in a vertical plane, there would have been a tendency to throw a great deal more than its due share of the burden on one chain. Mr. O'Conor, the District Engineer, says in the memorandum to the Commissioners, which forms Appendix A, attached to this Report:—“On the 24th of the month (three days before the accident) there was a heavy gale blowing down the gorge, which caused the bridge to sway to the extent of about six inches from side to side.” In such a case the windward of the two ropes would be unduly tightened, whilst the leeward one would have its tension suddenly diminished. The windward rope therefore would be in a very abnormal state of tension, and if the wind came in sudden and violent gusts, as I believe constantly happens in mountain gorges, the increase in tension of one rope and decrease in the other would be sudden, and might, I conceive, be disastrous to the bridge. I must mention, however, that it was contemplated to fix cross-braces, which it was expected would counteract the swaying completely, but these had not been fixed on the 24th when the swaying above alluded to was observed, nor do they appear to have been fixed at the time of the accident. 2. Each rope was made up of two sorts of material. The Commissioners' Report says:—“Each chain was to be composed of seven twisted wire ropes four and a half inches in circumference, laid side by side, and above these six other ropes, each made of thirty telegraph wires spliced together but not twisted, placed side by side, the whole united every ten feet by clips, forming a flat chain twelve inches in width, by two inches in depth.” Each rope therefore consisted of two portions entirely different from each

other in structure, and doubtless also different in stretching capacity. Change in temperature or the application of sudden strain, such as might be caused by any swaying in the bridge, would tend to throw the whole tension of either of these ropes on to only certain strands of it, and thus the effective strength of the rope might be most seriously reduced.