175. In estimating the proper stiffness to be given to the haunches of curved ribs we know that the sum of the bending moments on them should not exceed those on straight beams equal in length to the chords of the ribs, and we know that the sum of the bending moments will vary as the length of the arc.

Let A be the length of the arc, L the length of the chord, v the ver. sine of the arc, and f a force acting in the dircction of the chord. Take 2/3 of v to represent the mean distance of the axis of the rib from the chord, and suppose the neutral axis to be coincident with the axis of the rib. Then we have for the sum of the moments on half the haunch f A/2 x 2/3v, and for the moments on the half of a beam equal in length to the chord,

W/2xL/2xL/4 ... WL2/16 = fAv/3 or W = 16f Av/3L2.

Substituting the above value of W in equation [8] (Art. 100)

D4= Wa L2, and putting L for the length A, we have for the strength at the haunches:

To calculate the strength at the crown. Let A be the length of the arc between the weight and the point P, and v its mean distance from the direction of the force; then equating the moments as before with those of a straight beam equal in length to the chord, we have f Av = W/2xL/2xL/4 or W = f Av/L2.

And substituting in equation [8], and again putting L for A, we have for the strength of the crown,