simple length. The constants given above are derived from practice. The weight of girders for a common road, it placed from 7 to 8 feet apart, will be nearly the same as for railway girders of the same span. The weight of a cast- iron railway girder (two girders per way) will be about O OOoL tons per foot run. The weight of the roadway in a railway bridge will probably be from 14 to - 22 tons per girder, or double this for each line. For a turnpike road with metalling the weight will much exceed this, and
should in each case be computed.
§22. Design of a Girder.—(1.) From the span and load to be carried the engineer will determine the material and form to be employed. Cast-iron may in some districts be the cheapest material for girders under 30 feet span. Wrought iron I girders are very generally employed for spans of from 30 feet to 100 feet; beyond that span lattice or framed girders are more usually employed. For extreme spans exceeding, say, 300 feet, a hollow rectangle or tubular bridge may be used, carrying the road on its top or inside the tube. The depth of the cross section is limited by the consideration that the web must be suffi ciently stiff not to buckle ; but for this consideration the deeper a girder could be made the better. In practice the depth is made from ith to j^-th of the span. The engineer will also determine whether he will keep the depth of the girder constant throughout or diminish the depth at the ends. It is impossible to graduate the material so as to give absolutely uniform strength at all sections, but by diminishing the depth towards the ends, some material may be saved without attenuating the top and bottom members to such an extent as to be incon venient. When the general character of the design has thus been settled, the engineer will compute the probable weight of the girders and roadway or total permanent load ; he will next determine the passing load for which he intends to provide.
(2.) The value of M, the bending moment, must next be computed for a sufficient number of cross sections of the beam, and for various distributions of load. For a small cast- iron girder of uniform cross section a single value of M will be sufficient, computed for the section at the centre when the girder is wholly covered with the greatest uniform load and also supports the greatest single load at the centre of the span. When, as in larger girders, the design is intended to give a structure of approximately equal strength through out, the maximum value of M should be found for eight or ten sections ; this maximum value will be that obtained when the bridge is wholly loaded with its maximum uni form load and has the maximum single load resting just over the section in question.
(3.) The maximum shearing stress must next be calculated for each, of the above sections. The designer will bear in mind that the maximum stress occurs at the points of support, and that at the centre it is greatest when the bridge is half covered with the passing load.
(4.) The engineer can now compute the number of square inches S,. and S, required at each section in the upper and lower members consistently with the factor of safety he chooses to employ ; this he obtains from the expressions—
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1 S--^. S ~
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= -- f ci d It is here assumed that the best and strongest form of girder is employed, but if a mere square or circular beam is to be used, the cross section will be obtained by equating the values of M and p., using a safe modulus of rupture / v
(5.) The web will next be designed by giving it such a thickness as will, with the depth already fixed, supply the number of square inches required to reduce the stress per square inch to the safe or proof shearing stress, say 4 or 5 tons on wrought iron. When the web is a thin wrought iron plate it must be stiffened with _L or angle irons. In a cast-iron girder the web must have at least the number of square inches required by the shearing stress, but the exigencies of the foundry generally require a iesign resulting in a great excess of strength in this part of the beam, except in beams which are tapered towards the ends, as in fig. 19. With these beams care must be taken that the taper is not carried to excess so as to leave insufficient metal to resist the shearing stress at M and N.
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Fig. 19.
§23. Practical Details.—The designer must be practically acquainted with the forms in which his materials can be best procured. He must know the sizes in which iron or steel plates can be produced, and the forms best adapted for castings. Thus, in cast-iron beams the thickness of the web is at the bottom made equal to the thickness of the lower flange, and at the top to the thickness of the upper flange, in order to avoid permanent internal strains, which would result from unequal rates of cooling after being cast, if sudden changes of thickness in the metal were allowed. The engineer must also be familiar with the methods adopted of joining the several parts, as with the rivetting of wrought iron, the bolting together of large castings, the jointing of wood- work. He should also be acquainted with the various methods in which roadways are constructed and supported on existing bridges, and the manner in which the girders are braced one to another, so as to prevent vibration and lateral deflection due to the pressure of the wind. The examples of bridges described hereafter will give some information on these points. In long girders provision must be made by rollers, sliding plates, or suspension links for the expansion and contrac tion due to changes of temperature. The range in Great Britain may be taken as about 45 C. If the ends of the girder could be firmly secured at a constant distance apart this change of temperature would produce a stress of about 6 tons per square inch in wrought iron, and 3 tons per square inch in cast-iron. The result in practice would be that any attempted fastening of stone or iron work would be torn loose.
deflection at the centre under a known passing load is generally observed with the object of ascertaining whether the work has been properly done, for it is assumed that any defective material or bad jointing would increase the deflection beyond that calculated on the assumption of sound material and perfect workmanship. Sometimes the practical test applied is a rough one, a certain fraction of an inch being allowed per foot of span as a safe deflection. If an inspector of bridges, having authority, chooses to limit the deflection to a constant fraction of the span, the ratio of the depth to the span must be made sufficiently great to give the desired stiffness and maintained constant for all spans ; equation 5 below shows that when p l is kept constant and d is a given fraction of L, the deflection v will be proportional to the span. For the proof or maxi mum possible load, Rankine gives as the result of practice a value for the deflection of from -n^L to -g-fnj-L ; but on<3
foot deflection in a span of 200 feet would certainly be exces-