Among the optimum conditions to be met in a bow so that it may have minimum virtual mass are uniform stress distribution and uniform energy density in the limbs. Consider any given section in a bent limb. Let the distance from the neutral axis of the section to the outermost fiber on the compression (belly) side be *dc *and the distance to the outermost fiber of the tension (back) side be *dt *(Fig. 7). The stresses in these outermost layers are directly proportional to the aforementioned distances and inversely proportional to the radius of curvature ρ* *at the point in question. Hence to achieve uniform compressive stress in the belly, the ratio *dc/ρ *must be uniform for the entire length of limb; and to achieve uniform tensile stress in the back, the ratio *dt/ρ *must likewise be uniform throughout the length of limb.

Click for a larger image Fig. 7. The maximum fiber stresses in a bow are directly proportional to the distances of the outermost fibers from the neutral axis and to the curvature 1/ρ |

In wood samples tested at the U. S. Forest Products Laboratory and elsewhere, the strength in tension is between two and three times as great as that in compression. It follows that, for any given radius of curvature, the distance *dt *should be less than the distance *dc.*

In the traditional English longbow the requirements mentioned are far from being met. The cross-sectional shape of the English longbow is something approaching semicircular, with the flat part of the section constituting the back, or tension, side of the bow, and the rounded portion constituting the belly, or compression, side. There are variations from the circular shape on the compression side, and there has been much discussion among old-time bowyers and archers whether the correct shape is a semicircle or a parabola or some other curve. None of these, obviously, is correct. If we sketch the cross section and remember that the neutral axis passes through the center of mass of the section, we see that in all these sections the neutral axis is farther from the rounded, or compression, side than it is from the flat, or tension, side. It puts excessive stresses in section areas least able to withstand them. Furthermore, the area of section available to take the compressive stresses at the outermost fiber is greatly reduced by the rounding. Thus the section of the English longbow (Fig. 8) is about as wrong as can be. The consequence of this design

Click for a larger image Fig. 8. Section of limbs of bows showing relative distances of outermost fibers from neutral axes, (a) Section of the traditional English bow. (b) Bow of rectangular section. (c) Bow of trapezoidal section, in which the stresses more nearly approximate the ultimate strength characteristic of wood. |

is that the wood is usually overworked in compression and underworked in tension. Compression failures are usual and are avoided only by making the limbs so long that the radius of curvature is kept large. Another defect is that, because of the failure to adhere to the principle of uniformly stressing the sections, the radius of curvature of the bent limb at full draw is, in most cases, a variable one with shorter radii and consequent excessive bending in certain portions of the limb, with very little in others. For the reasons mentioned, much of the wood is stressed far below safe limits of strength. This, together with the large length of limb, makes for large virtual mass and a sluggish-acting bow.