Adrian Bejan | Trunk, Canopy, Diversity, from Design in Nature
The video draws the trunk and the canopy in profile, sets the coordinate from the tip, and brings in a uniform wind that sees the big, so the drag per unit height acts on the canopy and builds a bending moment that the trunk must carry, with tensile in the dorsal fiber and compressive in the ventral fiber, and by asking for equal maximum allowable stress along the height the drawing leads to the lightest design, where the trunk shape follows from the same degree of freedom that sets the canopy. The result is that trunks are essentially conical. In contrast, canopies exhibit diversity without conflict with performance, with the Eiffel Tower shown as the tree with the sharpest top, all trunk and no canopy, carrying the pushing from the side through a truss-like structure.
The drawing sets the diameter of the trunk as a function of location, the canopy size as a function of its own level, and a uniform wind with frontal view and speed, so the drag per unit height is read as a slice-by-slice force that tries to turn the top clockwise while the trunk takes that load.
The bending moment at a given altitude is calculated by integrating the arrows of force with their arms, and the counter moment in the trunk comes from stresses that vary linearly across the section, with the area moment of inertia and the local diameter determining how the section resists.
By requiring the maximum allowable stress to be the same everywhere along the height, the drawing declares equal SMA and establishes a constant relationship between bending moment and diameter, so the trunk profile follows directly from the wind loading and the way the section works.
The canopy is treated as a power in its coordinate, which introduces a degree of freedom; the same degree of freedom appears in the trunk profile. Yet, the outcome is that for a cone, the trunk is a cone, for a round top, the trunk is very close to one, and for a Christmas tree with a sharp tip, the trunk is again very close to one, so the trunk shape is relatively insensitive to the crown.
The message is that diversity shows up at the largest scale, in the canopy that the wind sees. At the same time, the trunk remains essentially conical and transports the water flow rate upward per unit of volume. The example of the Eiffel Tower shows a tree with all trunk and no canopy, with a distribution of maximum allowable stresses through a truss-like structure that does not break under lateral forces.
I enjoy putting together Umit Gunes on the Constructal Law, and I’m glad to share it with you for free. I hope you find it engaging and worthwhile. Preparing each post takes considerable time and effort, so if you appreciate Umit Gunes on the Constructal Law‘s content and can support it, please consider choosing one of the paid subscription options.