| δ = | 5 24 | wa4 EI | . |
The radius of curvature of the beam at D is given by the relation
R = EI/M.
37. Graphic Method of finding Deflection.—Divide the span L into any convenient number n of equal parts of length l, so that nl = L; compute the radii of curvature R1, R2, R3 for the several sections. Let measurements along the beam be represented according to any convenient scale, so that calling L1 and l1 the lengths to be drawn on paper, we have L = aL1; now let r1, r2, r3 be a series of radii such that r1 = R1/ab, r2 = R2/ab, &c., where b is any convenient constant chosen of such magnitude as will allow arcs with the radii, r1, r2, &c., to be drawn with the means at the draughtsman's disposal. Draw a curve
as shown in fig. 72 with arcs of the length l1, l2, l3, &c., and with the radii r1, r2, &c. (note, for a length ½l1 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 = av/b, 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 ft. in length would be drawn equal to 10 in.; then if the true radius at the centre were 10,000 ft., this radius, if the curve were undistorted, would be on paper 1000 in., but making b = 50 we can draw the curve with a radius of 20 in. The vertical distortion of the curve must not be so great that there is a very sensible difference between the length of the arc and its chord. This can be regulated by altering the value of b. In fig. 72 distortion is carried too far; this figure is merely used as an illustration.
38. Camber.—In order that a girder may become straight under its working load it should be constructed with a camber or upward convexity equal to the calculated deflection. Owing to the yielding of joints when a beam is first loaded a smaller modulus of elasticity should be taken than for a solid bar. For riveted girders E is about 17,500,000 lb per sq. in. for first loading. W.J.M. Rankine gives the approximate rule
Working deflection = δ = l²/10,000h,
where l is the span and h the depth of the beam, the stresses being those usual in bridgework, due to the total dead and live load.
(W. C. U.)