The relationship between power and stress is found by bringing in the bending moment.
S and E should be the same for all limit sections. The bending moment M and the section dimensions will vary. The relationship is:
FIRST: M = Pd
P is the tension in the string (about 12% more than pull of bow)
d is the distance of any section from the string measured perpendicularly to the string. All at full draw, of course.
SECOND: M = SI/y
S is the fiber stress.
I is the moment of inertia of the section.
y is the distance from neutral axis to the extreme fiber.
For rectangular section limbs this simplifies to:
where W is width of section in inches and T is the thickness.
We can then say from the two equations for M,
Which gives us the relationship between tension in the string (or, very closely, the pull of the bow), the distance to any limit section, the stress in the outer fibers, and the width of the limb and its thickness a such point. If we know S from the wood and T from the thickness resulting from the curve desired, this formula gives the required widths of limb sections.
That much abused and mysterious term "cast" is undoubtedly no more than an attempt to combine into one term all the above relationships. If applied to wood, it covers only strength, weight and modulus of elasticity. If applied to a bow, it adds curvature, thickness, length and shape of section. As the combining of these qualities becomes more common knowledge "cast" sure will be stripped of its mysterious cloak.