fundamental difference to be observed will be that the curve assumed by the neutral .surface will not be that of a circular arc, so that equation 1 is no longer true. Where, however, the law according to which R varies can be stated
algebraically, integration will give the value of v.Combining equations 2 and 3 for class 1, we have 1 M L- WL 3 i ,. AT ., LW^ we have -yoc -^-. Now this and as proportional equation can be shown to hold good for beams of uniform cross section and uniform depth, which may be called beams of class 2, also for beams of uniform cross section and uniform breadth, which may be called beams of class 3, hence for the three classes of beams we may write
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WL 3 7 ....... v = n ~m> where n will have different values in. the three classes and for each distribution of load.
Similarly it can be proved, that where beams are so designed that the loads produce the same maximum value p 1 of the stress on the outer elements, we have the deflec tion proportional to ^, equation 4 being one case of the general law ; we may therefore write
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where n l is a constant differing for each class of beam and each distribution of load.
Table X. gives the values of n and t for the three classes of beam, and for two distributions of load. The value of Pj_ in the case of a beam of uniform cross section applies to the stress at the centre ; the stresses elsewhere in that beam will be less.
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Description of Beam. W at Centre. w uniformly distributed. re i n "l Uniform strength and depth . . Uniform strength and breadth Uniform cross section i aH A A i. i A i 0-0178 B TsT "8 01427 TS
In any actual bridge girder the deflection should lie between the value calculated for the beam of uniform strength and that calculated for the beam of uniform cross section.
§25. Graphic Method of finding Deflection.—Divide the span into any convenient number n of equal parts of length I, so that nl ? L ; compute the radii of curvature R l7 E,.,, R 3 for the several sections. Let measurements along the beam be represented according to any convenient scale, so that calling L x and Z x the lengths to be drawn on paper, we have L ? aL x ; now let r v ? , r 3 be a series of radii such that r v ? -r , r. 2 ? ~ , &c., where 6 is any convenient constant chosen of such magnitude as will allow arcs with the radii rj, r 2 , &c., to be drawn with the means at the draughts man s disposal. Draw a curve as shown in fig. 21 vith arcs of the length l v I.,, l. A , &c., and with the radii r v t 2 , &c. (note, for a length ^1 L at each end the radius will be infinite, and the curve must end with a straight line tangent to the last arc), then let v be the measured deflection of this curve from the straight line, and V the actual deflection of the bridge; we have V ? v-, approximately. This method distorts the curve, so that vertical ordinates of the curve are drawn to a scale b times greater than that of the horizontal ordinates. Thus if the horizontal scale be one-tenth of an inch to the foot, a ? 120, and a beam 100 feet in length would be drawn equal to 10 inches ; then if the true radius at the centre were 10,000 feet, this radius, if the curve were undistorted, would be on paper 1000 inches, but making b ? 50 we can draw the curve with a radius of 20 inches. If we now measure the versed sine of an arc drawn with a length 10 inches and a radius 20 inches, we shall approximately find it equal to 64 inches, hence V/- / 1 54 inches. The vertical distortion of 50 . the curve must not be so great that there is any very sensible difference between the length of the arc and its chord. This can be regulated by altering the value of b. In fig. 21 dis tortion is carried much too far ; this figure is merely used as an illustration, and is not to be taken as an example.
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Fig. 21.
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Fig. 22.
called a continuous girder. The distribution of the stresses iu a continuous girder differs very materially from that in a simple girder, as will be at once apparent by the inspec tion of fig. 22, which shows the way in which a continuous girder of two spans and two simple girders bend when employed to carry equal weights across equal openings. The continuous girder when both spans are loaded is bent upwards at C over the centre pier ; in other words, the bending moment at this and neighbouring points is negative. The direction of the flexure changes at certain sections, as at A. and A 15 i.e., the bending moment is positive on one side of these sections, negative on the other side ; and at the section where the direction of flexure changes the bending moment is nil. Again, when only one of two simple girders is loaded, the girder over the second span is not bent in either direction, but with the continuous girder there may be a negative bending moment produced through
out the whole unloaded span as shown in fig. 23. Con-