the ordinary suspension-bridge with vertical rods, if by any force the structure is caused to vibrate laterally, there is merely its own weight to resist such vibration, but in the section here shown any lateral movement causes the strain on one side rod to become greater than that on the other opposite it; hence a strong tendency to resume its normal. position. The angular disposition of the rods will bring upon them strains greater than the load in the ratio of their lengths to their vertical heights, and this will lead to a lateral stress on the chains, tending to overturn the piers inwards, therefore they must be held in position by struts connecting the tops of the opposite towers. ef represents our semi-chain, with its suspension-rods hi, jk, &c. The strains will be the same as on the half of a complete chain of the same form and fall, its span being double that of the semi-chain. In this bridge, as constructed, the road girder was not continuous, but consisted of a series of short longitudinal girders equal in length to the distance between two suspension-rods; and this to avoid the vibratory wave sent forwards in a continuous road girder by the rising of that girder in the bay next in front of that on which the load is entering; but the discontinuous arrangement has this disadvantage, that it does not distribute the load over several suspension-rods, as does the rigidly continuous road girder. It is obvious that the semi-chain will be much less liable to continued pendulous vibration than the complete chain, and from its position, lying as it does in an inclined plane, it will have on it an initial strain, which is of great service under lateral disturbance; for if it be supposed that the platform is thrown aside until the suspension-rods occupy the positions shown by the dotted lines s w', t z', there is evidently a great effort on the part of the rod 8 w' to swing back with its load into the normal position, while at the commencement of such return there is no resistance offered to it from the side %, although, as the platform approaches its proper place, the rising of the end z' of the rod t z checks. the onward vibration of the platform. In the ordinary bridge the suspension-rods appear to act like two isochronous pendulums connected by a link. Before leaving this subject I must mention the anchorage of the main chains. The chain cd, Fig. 52, may be carried over a saddle, and so brought into a vertical direction, and passing below the ground line a b, be anchored beneath a mass of masonry ef, the weight of which should be at least twice the maximum strain that can be put upon the chain. In another arrangement the chain is anchored behind a mass of masonry, as shown at i k, g h being the ground line. In this case the stability of the masonry is relied upon; in both cases the work must be so executed that the masonry behaves as if it were one solid mass: the method of executing this part of the work will find its place in the chapter on Foundations. CHAPTER IX. COLUMNS AND STRUTS. THE conditions under which materials yield and fail when subjected to compressive force are very various, and they have not been sufficiently ascertained to enable a rational theory of resistance to compression to be formed: hence empirical formulæ form the only resource. By empirical is meant a formula deduced directly from experiment, the laws of variation as well as the constants being obtained therefrom. As might be expected, the results so obtained are not so satisfying to the thoughtful mind as those derived from a combination of pure reasoning and experiment; but still in the absence of the latter we must, until more light shall be thrown upon the subject, be content to put up with the former. Under certain known conditions the stress may be calculated, as for instance when a force acts rectilineally on a curved member, and passing outside its section, gives rise to a bending moment. It may be interesting to examine the effects of loads on straight elements, assuming that they will bend under the superimposed load. Let there be a bar 3 feet 6 inches long and 1 inch square placed horizontally on supports 3 feet apart, and loaded at the centre with 0-2 ton; then, according to the formulæ in a former chapter, if the bar be of cast iron its deflection under this transverse strain will be, D= 4 W 13 = •2 X 33 = 0.385 inch; and the moment of 14.m 14 x 1 = 4 strain of the load producing this deflection will be M = W·l .2 x 3 15 foot tons. The central moment of strain due to a force acting on the ends will be equal to the intensity of the force multiplied by the central deflection, or W' D if W' the force on the end. To find then the end load to produce a central moment equal to the above, the two expressions must be equated, giving M = •15 W' D = W' x ;... W'=4.67 tons. This load, 0.385 though acting compressively in direction, produces both tension and compression on the bar, and the tensile resistance will be the measure of strength of the bar, taking half the moment as being upon half the section; that upon the half in tension is 0.075 ft. ton, or 0.9 inch ton. The maximum tensile strain will be found by inverting the expression M = 1 8 b d2 ; 2. 6 12 M = 10.8 tons, which would exceed the strength of ordinary cast iron, unless the resistance of flexure is included (as explained in the chapter on Bending Stress), including which the tensile resistance, as found from the ultimate moment 8,000 inch lbs., becomes 48,000 lbs. per square inch, or 214 tons per square inch. The deflection, however, will not be of the same character under compression by direct and transverse stress in respect to the moment of strain at any part; under the latter the strain commencing at each point of support increases simply as the distance from the point of support to its maximum at the centre; but under direct pressure the moment at any point is the load multiplied by the deflection at that point. Now it is easy to imagine a sample of iron having one place much weaker than the rest, or it may be, from some defect in the casting, that it is not actually straight; then the strain may pass axially from each end to such a point, the deflection all occurring at that a point. In saying the deflection all occurring at such a point, the additional deflection due to bending is intended, the form of the bar being somewhat of the form shown in Fig. 53; then wherever this point is situated will be the position of maximum deflection, and of maximum strain in consequence. b In the foregoing examination I have regarded the compressing strain as acting at the centre of the Fig. 53. bar, but practically that will only occur while the bar does not deflect, for as soon as deflection commences the pressure will act only on the sides a and b of the end sections of the bar, thus reducing the moment. But in the first place the load must act centrally, for an initial pressure at a and b would deflect the bar in a direction the opposite of that shown. The calculated (from experiment) resistance of such a column as that taken above is 8.1 tons, so if we consider the bar as breaking by deflection, it is clear that the tensile strength is not in this class of strain augmented by the resistance of flexure. One very striking difference in the behaviour of the material under the transverse and the compressive deflecting forces must be noted. Under transverse strain the ultimate strength of the bar is not affected by variations in the modulus of elasticity, which only affects the amount of deflection; but under the compressive force the less the modulus of elasticity the greater the deflection, and therefore the greater the moment of strain under a given compressing force, and therefore practically the less the ultimate strength. It is evident from this that the iron selected for elements |
