Adrian Bejan | Tapered cantilever beam, from Design in Nature

Oct 13, 2025

Under the same loading, deflection, material, and maximum allowable stress, the video demonstrates the effect of changing the drawing from a solid round rod to a hollow tube. To a tapered solid beam, and finally to a hollow tapered tube makes the volume requirement smaller, because hollowing removes the inactive core and tapering reshapes the profile so that the outer fibers carry equal work, which leads to a round nose shape and to the simple rule that change in architecture corresponds to change in performance, with the combined use of hollowing and tapering giving the most efficient result in volume units.

Hollowing removes the inactive core and leaves an annulus with diameter, wall thickness, and length chosen so that stiffness and deflection match the starting case. This shifts work to the material that counts, lowering the expenditure to six volume units without altering the material.
Tapering keeps the beam solid and redistributes material along the span to avoid stress concentration at sharp features. With the same conditions, the tapered round beam reaches four volume units, demonstrating how one degree of freedom in shape multiplies the gain.
Using hollowing together with tapering sets the dorsal fiber and the ventral fiber at the maximum allowable stress from the clamp to the tip. The profile ends in a round nose and removes the inactive core while keeping the fibers equally loaded.
The round nose is selected to prevent breaking where sharp protrusions or indentations would concentrate stress. The bending moment, combined with the area moment of inertia, guides a smooth shape that can withstand the load with minimal material.

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