The problem of the stiffener and advice for the worm

We consider a rectilinear elastic stiffener of finite length welded longitudinally to the boundary of an elastic half-plane. One end of the stiffener is loaded by a longitudinal force, the other end is free. For a given total volume, the stiffener may be either short but thick or long but slender. We then show that there is a unique value of the length and of the cross-sectional area of the stiffener which minimizes the adhesive force transmitted by the welding under the load.