355. In designing bridges with curved ribs, it is important to ascertain the degree of curvature that may be given to the beams without impairing their elastic force, as the depth of each piece composing the rib will to a certain extent depend on the result.

The curvature to be given to a beam being sensibly uniform, the degree of uniform curvature to which the beam may be bent is inversely as its depth, and the radius of curvature will be directly as the depth.

Dr. Young has shown that the deflection of a beam uniformly curved is to that of one bent by a load placed in the middle of its length, as 3 is to 2.*

Barlow has determined by experiment the extent to which a beam will bend without the elasticity being de-

* ' Lectures on Nat. Philosophy.' stroyed.* From this data a rule for the depth of the beams which will admit of a given curvature, may be obtained.

Let x be the deflection found by experiment, upon a beam, of which half the length is y; then 3x/2 will be the deflection corresponding to a uniform curvature; and y2/3x = the radius of curvature, the deflection being small when compared with the length.

From Barlow's experiments we have the depth of the beam in inches: -

Wiebeking made some experiments on a large scale by bending beams, which tend to prove the accuracy of this method of finding the depth. He observed, that when several pieces were placed one upon another, they could be bent without fracture much more than a single piece.†

The radius of curvature of the ribs used in the bridge of Bamberg (Plate XL.) was 422 feet, which multiplied by .035 as above would give 14.77 inches for the depth, whereas the actual depths of the beams employed were from 13.5 to 15.5 inches. The rule giving nearly a mean between the two.