The Anatomy of Bridgework - William Henry Thorpe

E × Df × 2 = R, radius of curvature in feet,

and the radius of curvature for the diagram by

12 × R × F² = r, radius for plotting, in inches (5)

E being the modulus of elasticity, D the girder’s depth in feet, f the mean of the extreme flange stresses per square inch of gross area, and F the fraction indicating scale as ¹⁄₄₈, where ¹⁄₄ inch = 1 foot. The curve, being plotted, shows by direct scaling the movement of any point relative to its original position. Near the ends of the curve where the radii may be of considerable length, the arcs may be drawn with the help of template curves, or even set out as pieces of “straight.”

When the curve is laid down so that its chord equals the span to scale, the method involves an error of excess in the resulting deflection or droop which is as much as 7 per cent. when the mean radius for plotting equals the span as drawn, or when the droop of curve approaches one-eighth of the span. As the exaggeration of curvature is made less pronounced, this error rapidly diminishes, till for a droop of about one-sixteenth the percentage is one-fourth part of that above given. This excess in the droop of curve may be amended by the following expression:—

droop - (droop³chord² × 3·73) = corrected droop, or deflection.

For some purposes it may be preferable to amend the radii for plotting, so that the curve, as laid down, shall be correct, which may be effected by the formula here given, to be applied to each value of r, as first ascertained:—

r + (chord² r × ·0625) = corrected plotting radius.

If, however, the length of curve is made equal to the span (the chord then being less), and the radii for plotting as given by [(5)] are used, the result will for most purposes be sufficiently precise, though there will now be an error of a contrary kind, which, for a curve having a droop of one-eighth, will be about 2 per cent. too little. A somewhat similar method of setting out deflection curves is described by Professor Fleeming Jenkin in the article “Bridges” of the “Encyclopædia Britannica,” but without corrections.