Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/340

(fig. 18c). When the load covers a length x measured from the right hand pier, the shearing stress at all points beyond x towards the left is - -=- , when x is greater than ^L the above expression gives the maximum stress which can occur on that section with any distribution of the given uniform load. Thus the maximum shearing stress at the left end occurs when the whole bridge is loaded, and is fyvL ; the maximum stress at the centre occurs when the bridge is half loaded, and is equal to -gtt L. The maxi mum shearing stresses on the other half of the beam occur when the load comes on from the left side, and covers more than half of the beam; these stresses are equal in amount

to the stresses in the left half, but are positive in sign.

Fig. 18b.

The scales in figures 18, 18a, correspond to the shearing stresses in examples 1 and 2 for a span of 50 feet and loads of 10 tons and 1 ton per foot run respectively. The scale in fig. 186 corresponds to the shearing stresses in example 3 with a single passing load of 14 tons. The scale in fig. 18c, example 4, gives the maximum shearing stress which the advancing load of one ton per foot run can pro duce at each section. As a train leaves a bridge it produces the same shearing stresses as when it comes on to the bridge from the opposite end, the same portions being similarly loaded. The maximum shearing stress due to a passing load of this kind changes its sign at the centre of the span, as appears by diagram 18c.

Fig. 18c.

In IE girders with solid vertical webs the shearing stress is practically all borne by the web unassisted by the top or bottom members, it being clear that these would fold down at the sides under a small fraction of the total stress. Sufficient material is therefore employed in the web or upright plates to reduce the intensity of the stress to the desired amount. The shearing stress may, however, when ths web is a thin iron plate, cause failure by crumpling the web, or causing it to buckle, instead of by shearing it across. This tendency is prevented by stiffening the web with anefle or T irons rivetted to the sides. Mathematical analysis has not yet been very successfully applied to the determina tion of the amount of stiffening required ; experience has given a sufficient number of examples to guide the practical designer. In cast-iron girders the web is generally much in excess of the strength required to resist shearing.

§20. Factor of Safety.—In designing a girder the load which it will have to carry is multiplied by a number called the factor of safety, varying from 3 to C, and the girder is so designed that it shall not yield at any point with less than the load thus multiplied. If, for instance, the girder is practically to carry 1 ton per foot run, it is designed so that at no place shall it break or yield injuriously with less than say 5 tons per foot run. The multiplier is called the factor of safety. The factor of safety is required to allow for imperfections in the material as compared with picked specimens, for the wear and tear by which the strength of a structure is gradually reduced, for unforeseen loads, for jars and vibrations, for imperfection of theory, and for the sake of obtaining stiffness. This last property might be the subject of calculation, and in some cases must be separately examined. The particular factor employed depends on the judgment of the engineer. A larger factor of safety is required for a passing or moving load than for a permanent load, there being a greater uncertainty as to the stress which may be caused by vibrations or impulses due to what is sometimes called a live load. Moreover, the mere pre sence of a large permanent load tends by its inertia to diminish the dangerous effect of the impulses or stresses due to the passing load, so that the factor of safety should max. passing load load be chosen with reference to the ratio max. permanent being larger as this ratio increases. Eankine recommends that the factor of safety should for the moving load be double that employed for the permanent load. Sometimes the factor is more conveniently employed as a divisor to deduce the safe stress/! from the ultimate strength/ of the material, rather than as a multiplier for the load. Thus the same number of square inches will be obtained in the tension member of a wrought iron girder to bear 1 ton per foot run, whether we use in the calculations 25 tons as the value of /, the; ultimate strength of the material, and a load iv of 5 tons per foot run, or if we use 5 tons as/ p the safe stress on the material, and a load iv of 1 ton per foot run ; in short, if we call the factor of safety K, we may in equation 5, 14, 2/"I f use KM ? -, or making/! ? ~ , we may write

1. M d The same remark applies, of course, to equation 6, 14.

§21. Weight of Girders and Roadway Roadway.—When two

girders are employed for each line of way on a railway, the weight of the iron girders per foot run will, with the usual proportions, probably lie between O OOITL tons and 0-0025L tons, L being the span in feet. It follows from the theory given above, that for similar beams the quantity of material in the whole girder will be proportional to the square of the length, and, there

fore, the quantity per foot run will be proportional to the