# Flexible vegetation

## Main Question or Discussion Point

I want to know how to determine the shear stress a plant undergoes when it is pulled perpendicular to the plant length and then bends in the direction you are pulling. Would I use a cantilever beam equation? What if the force pulling the plant is at different locations along the length of the plant (i.e. if I pulled at the base vs. pulling at the top)? Or at many places at once?

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Danger
Gold Member
Welcome to PF, Czarnm.
Going by the title of the thread, I was expecting a conversation about rubber trees. Since that's not the case, I'm afraid that I have nothing to contribute.

Andy Resnick
Sure- you could use the cantilever equation (with caveats):

EI$$\frac{\partial^{4}y}{\partial s^4} = 0$$

Where E is the Young's modulus, I the moment of inertia, 's' the coordinate that deforms with the beam, etc. etc.

You probably want to start with a free end and a built-in end for the boundary conditions- the fixed end position and slope are zero, the bending moment at the free end vanishes, and the force 'F' is applied at the free end as well:

$$y(0) =\frac{\partial y}{\partial s}\right)_{s=0} = 0$$

$$\frac{\partial^{2} y}{\partial s^{2}}\right)_{s=L} = 0$$

$$\frac{\partial^{3} y}{\partial s^{3}}\right)_{s=L} = F$$

If you are applying a force at different locations, your boundary conditions will change as well.

In order to extract out the shear stress, I think you need to be careful- one could calculate the bending energy by calculating the curvature along the length, for example.

In any case, you need to know Young's modulus which is experimentally determined.

Basically, this is why engineers have moved to finite element analysis software platforms to calculate all this stuff for them.