The structural efficiency of orthotropic stalks, stems and tubes

Abstract

An optimised structure is one which uses the smallest quantity of the best material to perform its function, with adequate safety factor or margin for error. Structural optimisation occurs not only in mechanical engineering, but also in nature: plants with hollow stems or stalks gain a height advantage, and are thus more efficient, by approaching the optimum shape. Here we consider the optimisation of orthotropic tubes, typifying, in a mechanical sense, stalk and stem. The stiffness and strength of orthotropic tubes of initially circular section are reviewed, and diagrams are proposed which allow the optimum section shape to be selected.

Appendices

Appendix 1 full solution for ϕ_A

The transition from fracture/yield to splitting is found by equating the respective moments of failure at A:

$$ \phi_A =\frac{A}{2}\pm \left(\frac{A^{2}}{4}-B\right)^{1/2} $$

(A1)

with

$$ A=\left[\frac{1+2.037\cdot \left(\frac{E_{||}}{E_\bot}\right) \left(\frac{\sigma_{t\bot}}{\sigma_{c||}}\right)^{2}}{1.259\cdot \left(\frac{\sigma_{t\bot}}{E_\bot}\right)^{2} \left(\frac{E_{||}\sigma_{t\bot}}{\sigma_{c||}^2}\right)}\right]\quad \hbox{and B}=\left(\frac{1}{1.236}\right)^{2} \left(\frac{E_\bot}{\sigma_{t\bot}}\right)^{2} $$

Appendix 2 failure through inadequate stiffness

The final criterion for the macroscopical mechanical performance is that of stiffness. Mosbrugger [42] classifies plants according to their structural behaviour: either the plant is a ’flexibility strategist’ and reduces external loads by bending or it is a ’stability strategist’ and has a structure which is stiff and strong enough to withstand the loads without much bending. As the principal load is that due to wind and the velocity of wind increases with height above ground, a flexible tree which bends in a strong wind reduces the moment arm of the net wind force, especially if elastic deformation of its crown reduces its down-wind profile. Tree trunks are frequently stability strategists, whereas their branches must be capable of bending to a quarter-circle. The curvature of the bamboo culm, C, can then be expressed as a function of the length, l, of the stem

$$ C=\frac{1}{R}\approx \frac{\pi}{21} $$

(A2)

Substituting this expression for C in Eq. 8 and inserting the result for c in Eq. 13b gives the bending moment, M, which bends the stem into a quarter-circle

$$ M=\frac{\sqrt{\pi}}{4\sqrt{2}} \left(\frac{r}{l}\right) A^{3/2} \phi^{1/2} E_{||} \left[1-\frac{3\pi^{2}}{8} \left(\frac{r}{l}\right)^{2} \phi^{2} \left(\frac{E_{||}}{E_\bot}\right)\right] $$

(A3)

The second expression in the square bracket is very small due to the high slenderness ratios of bamboo (ϕ = l/r = 550−1000, [43]) and may therefore be neglected. The moment which bends a stem to a quarter-circle may therefore be rewritten as

$$ M_4 =\frac{\sqrt{\pi}}{4\sqrt{2}} \left(\frac{r}{l}\right) A^{3/2} \phi^{1/2}E_{||} $$

(A4)

If the plant is to function as a flexibility strategist, it must be able to do this without failing by any of the other three mechanisms analysed above.