At full draw, the limb bends in a circular arc of certain radius. The bending causes a stretching, by tension, of the wood on the back of the bow, and a shortening, by compression, on the belly side. At any section of the limb, the neutral axis marks the boundary between the extended and compressed parts; along this line there is no stress. In the rectangular section it is midway between belly and back. The greatest stresses, or deforming forces, occur at the outermost surfaces, i.e., outside surface layer of the back and of the belly. If the tensile and compressive forces do not reach those at which the wood gives way, the limb bends without breaking. The amount of the force depends on two things: the thickness of the limb, and the radius of the arc in which it is bent. The force is proportional to the thickness and to the reciprocal of the radius. For a given radius, doubling the thickness doubles the force; and for a certain thickness, bending the limb to a sharper arc, say with half the radius, also doubles the force. There is a ratio of radius of the arc to thickness of limb that represents the average safe value for a particular kind of wood. For yew, this ratio is about 45, and may be slightly smaller; for osage, 35 seems to be safe. These figures represent rough guides only, since they vary greatly with individual staves; but they are useful to the bowyer as design data from which he can make a first approximation to the thickness of the limb. It is useful also to know that in a long bow which is straight in its zero or unbraced condition, the radius of curvature at full draw is approximately equal to the length of the limb. We may accordingly take the above figures as representing approximately the minimum permissible ratio of length to thickness. This rule of thumb does not apply to short or reflexed bows, or bows with large initial set, or string follow. When such bows are being designed, it is necessary to determine the total amount of bending from the zero position to full draw, and compute the corresponding value of the radius. This becomes complicated, and we need not consider it further, since our primary interest is in the straight longbow.
At this point, experience must be one's guide. It is necessary first to judge the stiffness of the wood, and its density. If it is very stiff, less thickness is required, for a certain length and width, to give a bow having the desired force at full draw. If the wood is comparatively dense, or "heavy", it will usually withstand greater compressive force than will "light" wood of the same species; and the number expressing ratio of radius to thickness may be taken somewhat smaller than the average figures given. As a general rule for limbs about 1¼ inch in width at the dip, it is necessary to employ a length of limb of 30 to 31 inches for a 50-pound yew bow. This, according to our rules of thumb, would call for a thickness of about 5/8". With a better-than-average piece of yew, one might go to 11/16", or perhaps even to ¾ ", thus securing a correspondingly stronger bow. The last figure would of course call for an exceptionally strong piece of yew, if compression failure is to be avoided.
Making a bow which will give the highest possible velocity to an arrow is a problem of balancing the variables against each other, and striking some compromises among them. In general, the shorter the limbs, the greater the cast; but the lower limit to length of limb is set by the length of arrow to be drawn, for this establishes the radius of curvature of the limb. We have seen that maximum permissible thickness of limb is set by this radius. If the thickness has been found, by the rule mentioned above, the question at once arises whether, with this thickness and length of limb, a bow of the desired weight can be made. The limb can be widened, to be sure, but a limiting width is soon reached. If, at the limiting width, the bow is not sufficiently strong, we must start over, and choose greater length of limb, so that greater thickness is permissible. To repeat in another way what has been said, the general rule is to design the limb with the greatest thickness permitted by the limiting strength of the wood, to give a bow of the desired weight.