**BRIDGES**

**297**

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—

1 S--^. S ~

= -- 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*.—When a bridge has been erected its

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-