Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/356

Rh 312 BRIDGES [ARCHES. boxes or troughs ; the block supporting the centre acts as a lid resting on the sand inside ; when the sand is allowed to escape the block sinks slowly down inside the box. &amp;lt;j 45. Comparison of Metal U ith Masonry Arches. Metal arched ribs may be used instead of rings of masonry to support a platform and roadway. These arched ribs con stitute true arches whenever, as is generally the case, all parts of the rib are compressed. The principles by which the stress on each part may be computed do not differ from those already explained for arches of masonry, but it is possible to calculate the stresses with much greater exactitude for continuous metal ribs than for voussoirs. With voussoirs we have seeu that the resultant thrust at the springing is indeterminate both in magnitude and position, but we shall see hereafter that the resultant thrust, which will be called t, at the springing of a metal arch is easily rendered deter minate. Supposing t and t v the thrusts due to a given load (fig. 56), to be known, then if the form of rib be made to correspond with any linear arch for the given distribution the compression at any section of the rib will be axial and uniformly distributed ; the arch will then be strained as a chain of the same length would be strained under the same distribution of loads, extension being substituted for com pression and dip for rise. Fig. 56 shows a rib of this kind with the approximate linear arch drawn as an equilibrated polygon by the method explained in 38. If the distribution of the load is altered the linear arch will also change, and the stress on each part of the rib will no longer be axial. The change in the form of the linear arch will generally be much greater for a mntal than for a masonry arch, because most metal arches have light open spandrils and a light roadway, so that the passing load is considerable in comparison with the permanent load. Not improbably the linear arch, when only one haunch of a metal rib is loaded, may pass quite outside the rib for a portion of its length if this rib is made, as is usually the case, of a form containing the linear arch for a sym metrically distributed load. On the other hand, it does not follow, as with masonry, that because the linear arch passes outside the rib the bridge will fail. The bending couple then produced can be resisted by the moment of the elastic forces of the cross section of the rib if the rib is made strong enough. In masonry the joints open so soon as the resultant pressure passes outside the middle third of the ring ; the couple required to produce equilibrium would then require a negative force or tension at the opposite edge, and masonry cannot supply this tension, but in a metal rib the couple or bending moment produced by the excentricity of the stress may be resisted by the stiffness of the rib acting as a beam subject to a bending moment. Thus the strength of an arch to resist flexure is a more important element in the metal rib than in the masonry structure. It would be false to say that the ring of voussoirs had no strength to resist flexure, for we have on the contrary seen that the moment of the elastic forces at any section of a stone ring does resist any distorting action produced by the load; but in masonry this moment should never exceed the comparatively small value consistent with the absence of tension on any part of any joint. The metal rib may with safety be sub jected to considerable tension in parts, and its strength to resist flexure can be easily increased and can be calculated with certainty. Moreover, by hinging the rib at one or both springings, as can be done with metal, the problem of determining the horizontal thrust (or total thrust) is sim plified, the position of the thrust being thereby rendered certainly axial at this point, and then by taking into account the actual deformation of each part of the rib a complete solution of the problem of its strength can be obtained. 46. Horizontal I krust of a Metal Arch or Rib hinged at the Abutments. By supporting a rib on pins or in cylindrical bearings (vide fig. 62) at the abutments we determine two points traversed by the thrust. The effect of allowing free rotation is necessarily to render the bending moment nil round the centre of rotation. Hence the resultant thrust must traverse the centre of the pin, or the centre of curvature of the bearing. Knowing the point of application of the thrust we have now to determine its magnitude. The vertical component v is the same as the load on the pier of a girder of the same span equally and similarly loaded, so that the problem reduces itself to the determination of li the horizontal component. Fig. 57. Let us -first consider a semicircular rib (fig. 57), bearing a load uniformly distributed along the horizontal platform of the bridge (neglecting the weight of the rib). The linear arch will pass through the centre of the bearings N and Q, and will be a parabola. Moreover, it will be that parabola which requires the rib to exert no internal forces due to its own elasticity, and tending either to push out or draw in the springings ; in other words, the rib, being supposed in equilibrium before the application of the weigh, ts, will not tend to act as a spring to increase or diminish the opening between N and Q. Nevertheless, as the semicircle cannot coincide with the parabola, most parts of the rib must be subject to bending moments, against which it will react as a bent spring. When the linear arch, as shown in fig. 57, passes above the axis of the rib at the crown, and below it at the haunches, the upper portion of the bent rib will act as a spring, tending by its reaction to diminish the distance between the ends N and Q, while the portions near the springing will be so bent as by their reaction to tend to increase that distance ; now, if, as is necessarily the case, the whole rib is not to act as a spring, tending either to close or open the ends N and Q, then the effect of the bending near the haunches must exactly neutralize the effect of the bending near the crown. We have now to find what direction of thrust at the springing will give a linear arch such that the above condition may be fulfilled. Let M be the bending moment acting at any given section, the centre or neutral axis of which is at a height ?/ (fig. 58) above the horizontal line joining the springings ; let this moment be con sidered constant for a short length AL of the rib measured axially along the rib ; let As be the short distance measured horizontally by which the moment M acting throughout the length AL would