This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.

When the center line of the rib is a parabola, we may lay off vm by drawing it at a height 2/3 of the rise above the line OB. Even when it is a circle, it is comparatively easy to compute with mathematical accuracy the height of OK, by computing the area of the segment OCB, and dividing it by the length of the line OB. As previously explained in Article 428, the two-thirds rule may be used even for circular arcs, when the arch is very flat. In this particular case, the two-thirds rule is far from applicable; and since for multi-centered curves no such rule is applicable, the general method will be here given. We divide the span (61.40) into 20 equal parts.

Practically this is most easily done by setting a pair of dividers by trial so that 10 equal spaces may be stepped off in the length of the half-span. From these division points on the line OB, erect perpendiculars to the center line of the arch rib OCB. The area of a curve bounded by a straight line at the bottom, and which has vertical and equally spaced ordinates, may be computed with very close accuracy by the adoption of Simpson's rule. If yo represents the ordinate at the beginning of the curve (and in this case yo = 0), while y1 up to yn represent the lengths of the several ordinates (ya being the last ordinate, and, in this case, also equal to 0; and n being equal to the even number of divisions, in this case 20), then the area may be expressed by the formula:

Area =OB / 3n {yo + 4 (y1 +y3 + .y(n-1) + 2(y2 + y4 + . . .y(n-2)+ yn }

......(59)

Applying this rule, we find that the area will be 640.50 square feet. Dividing this by the span, 61.40, we find that the height of vmi above the line OB will be 10.59 feet. The approximate two-thirds rule would give us 10.17. Making a rough interpolation in the tabular form of Article 428, we could say that for an angle 2 a equal to 106° 16'. the quantity to be added to the result by the two-thirds rule would be approximately 5 per cent. Adding 5 per cent to 10.17, we would have 10.68, which gives a rough check with the far more accurate value just found.

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