Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/358

Rh 314 BRIDGES [ARCHES. hut the ratio sc : sg is constant, and may be designated by the letter t ; so that we may write k (S,y. sg) 2y 2, from which equation we liud the value of k 6 If the cross section is not constant we have for k the more com plex expression ^y-w but sc = k.sg, so that the required ordinate sc is at once obtained in terms of k and the known ordinate sg. When the actual linear arch oc^c^ 3, &c. , has been thus obtained, it is easy to calculate the horizontal thrust ; for let s 5 c s be the maximum ordinate, the direction of the thrust will at this point of the curve be horizontal, and therefore calling W the weight on one side of this ordinate, and x the distance of its centre of gravity from the springing, we have 8 ........ W from which h can be found. h and v being known give the position and direction of the result ant thrust at the springing of the rib. The magnitude of the thrust at any other point is easily computed graphically or by moments Fig. 61. round the springing ; then the resultant thrust at a given point being known, the intensity of the stress on any part of a section at that point is to be computed by first resolving the thrust into two components, one normal to the section and one in the plane of the section ; the latter gives rise to a shearing stress (analogous to the force which causes one stone to slip on another in the masonry arch), while the component normal to the section will (if not axial) give rise to a uniformly varying stress, the magnitude of which at each distance from the axis can be computed by the formula given in 8. The value of h is determinate, if the direction of the rib be supposed fixed at the springing, but this cannot be ensured in large structures, and the_ theory need not therefore be developed. It simply requires 2 co = 0. When the rib (as is generally the case in existing bridges) abuts against a flat springing the exact value of h is indeterminate. When the rib is hinged the friction at the bearing renders the thrust indeterminate within limits depending on the possible bending moment at the springing due to the friction. 47. Process of Designing a Rib. In future designs of ribbed arches it is to be hoped that the practice will be adopted of allowing the rib freedom to turn at the springing. This can be done by ending the rib in a bearing, curved as in fig. 62 ; the resultant thrust will then be approxi mately axial, and the stress on every part of the rib can be determined with as much accuracy as on the several parts of a girder. When the span is large a cast-iron metal arch can be made lighter than a wrought iron girder for the same load, but the imperfection of the theory of the stresses on the ribs has hitherto led to great waste of metal in their construction. In what follows it is assumed that the resultant at the springing passes through the geometrical centre of the cross section of the rib. If the rib were to carry a load distributed only in one way we ought clearly to make the form of the axis of the rib coincide with a linear arch for that load. There would then be no bending moment on any part of the rib. As Fio;. 62. in practice we must provide for all the possible combinations of passing load, we need take little pains in designing the curvature of the rib a flat arc of a circle with a rise of say -^th will answer well. The semi circular or elliptical forms are not good, for no linear arch with any prac tical distribution of load can even approximately coincide with a form in which the rib springs vertically from the abut ments. The general character of the cross section should be similar to that for a girder, inasmuch as the rib will have to resist bending moments as well as direct compression. The depth need not, however, be nearly so great as the depth of a girder. Let a cross section be chosen in which the area is assumed as approxi mately say 5 per cent, more than that which would be sufficient to sustain the thrust resulting from a linear arch suitable for the maximum load and coinciding approximately with the axis of the rib. (If the load be nearly uniform per foot run of platform we may for this first approximation take h -, where d is the rise of the 8d linear arch above the springings.) With the rib thus de signed determine by the method given in 46 the actual linear arches resulting from the following arrangements of passing load (combined with the permanent load) : (1), Bridge half covered from one end; (2), three-quarters covered from one end ; (3), wholly covered ; (4), covered by the passing load over the middle half of the bridge, the haunches being unloaded. Draw these linear arches on the rib by the method given in 38, and choosing at each one of some eight or ten selected sections the two curves which most nearly approach the top and bottom flanges respectively, coir - pute the maximum intensity of stress on the top and bottom flanges at each section from the two thrusts corresponding to these two linear arches; where the stress is excessive add metal ; remove it where the maximum stress is less than the safe stress for the material. If no great change is made in the design this process will be sufficient, but if the cross section is seriously modified by the alteration we must make a second approximation by recalculating the linear arches for the new form of rib, and thus proceed by trial and error until the stresses corresponding to the actual linear arches are met by sufficient metal at all points. The rib need not be of uniform depth throughout, and may be increased in depth at the places where the stress due to bending moment has been found excessive. In large spans the effect of a change of temperature must be taken into account. This can be done by finding the linear arch given by the expression 1. -y- y = As , where As is the alteration in span which would result from the expansion or contraction of the span if free to expand or contract with the change of temperature. In a series of arches abutting against comparatively slender piers, account must be taken of the thrust trans mitted from the neighbouring nrch. This thrust will only be due to the passing load, and a part may be con sidered as taken by the pier ; the remainder which the