Crossbow Limb strength calculation

We gonna use this example as a basis for the current calculation. There we are evaluating the Limb stiffness and calculating Limb's tip displacement during deformation under applied force (string tension force). So, we have the Limb with a rectangle cross-section of varying size.

Here we gonna calculate Limb strength by given sizes and material specs. Sometimes we happen to know the crossbow parameters before we design the Limb, so we know how much the Limb will deform when the weapon is loaded. Therefore the distance Δ of the Limb's tip between the undeformed state (with NO string installed) and fully loaded state is given.