With any wood there is a maximum permissible combination of thickness and curvature that wilt avoid failure. With one batch of yew I have found it about as follows, using the length in place of the radius of curvature.
|Length of Bow||Maximum Thickness|
|6 feet||.90 inch|
|5½ feet||.75 inch|
|5 feet||.62 inch|
|4½ feet||.50 inch|
|4 feet||.38 inch|
|3½ feet||.27 inch|
|3 feet||.16 inch|
Having made any one bow from a given quality of wood, the thickness for any other length may be immediately determined from relationships like the above.
With such definite limits, the pull of any bow depends on its width of limb and that, in turn, is limited by the necessity of narrowing down at the handle without introducing too much weakness in the resulting shoulder. A 2-inch width with a 4½-foot bow gives about a 50-pound or 60-pound pull at 28-inch draw with ordinary material but that is about the limit of width. Any stronger simple bow of that material must be longer.
Keeping in mind the above limits of thickness, curvature and width, we deal with shape of the limbs and their cross-section. Sticking to our circular arc for simplicity and because it is one limit, the limb should be considerably bulged in width from the "straight taper" from handle to tip. (Apologies to Klopsteg and Hickman). The cantilever analysis leading to that unnatural shape assumed that distances from the tips fixed the increasing bending moments, a condition far from the truth because of the flexed shape at full draw and because the bending moment increases as the distance from the string and not from the tip.
Keeping the above limits in mind, a constant thickness bulged width of limb can easily be arrived at to give a circular arc, though special treatment is necessary at the tips to have them do work. Thinning them beyond the point where width equals thickness is the simplest way but fluting them is mathe-matically better if the material is sufficiently homogeneous.
Here enters another problem. If a bow limb varies in thickness, it cannot be bent to a circular arc without unequal fiber stress. Conversely, if of uniform thickness, it cannot be bent to varying curvature without unequal stress. Any inequality of stress means that some portion of the limb is not working all it should, which further means that the how would be more efficient, i.e., have better "cast" if it were redesigned.
I am not an advocate of the circular arc but there is no getting away from the fact that we must try to approach uniform stress over as much of the limb length as possible. On the basis that a bow weak in the middle is inefficient, a whip ended bow is in the right direction, the "circle" bow separating these two classes. A whip ended bow may be bent somewhat in the form of an ellipse having its minor axis about in line with the arrow. Such a bow will taper in thickness from the handle to tip to give uniform fiber stress in accordance with simple mathematical relationships, once the shape of ellipse is chosen. The width at all points is likewise fixed by the same laws if the desired pull is given and the special characteristics of the wood used.