What are the forces on a cell in a tree vs on the surface of a tree

[bo reddude]

Homework Statement

Not a homework, just wondered all of sudden.

Relevant Equations

f = ma

Hey everyone.

Let's say you have a tall tree. say, 10 m tall. Take a cubic cell 1mm in dimensions. Suppose the cell is at 1 m high in the center of the trunk of the tree.

What are the forces acting on the cell? let's say tree cell's mass is t grams.

its weight is F= ma = t (9.8) = 9.8 t

it's exerting normal forces to the cell below, above and all around it? or no? Do cells around it exert normal forces? What happens to the sum of these forces? is there a diagram that shows this or similar scenario?

What about a same dimension cell on the surface of the tree at the same height? what are the forces acting on that?

What happens if I were to change the height of the cell to 2 m? it has less number of cells above it, so it should experience less weight from above?

I don't know where to begin to answer some of these questions. Any help is appreciated

 

[haruspex]

The net vertical forces on a horizontal slice through the tree obviously equal the weight above it, but these are not uniformly distributed across the slice. The newer growth at the perimeter will be somewhat plastic, so cannot bear much load.
Winds, and any tilt of the trunk, will also shift the load sideways. One side may even be under tension.

 

[jbriggs444]

What happens if I were to change the height of the cell to 2 m? it has less number of cells above it, so it should experience less weight from above?

If the cells are larger, there are fewer of them. But each one weighs more. The result is that both calculations will yield the approximately the same results.

A difficulty with analyzing large rigid structures by breaking them into small cells is that the result is statically indeterminate. There are many patterns of internal force that could support the structure. You cannot determine by simply looking at the large scale forces on the structure which internal pattern of stresses is present.

What you could do would be to set up a simple structure like a rectangular tower in an unstressed configuration, assume some figures for the bulk modulus (how it compresses under pure pressure) and the shear modulus (how it deforms under a shear load) and slowly relax the structure to see what shape it assumes and what pattern of stresses and forces results. This would require some programming skill.

I have no experience with finite element analysis, but this is how I would proceed.

If you got to the point of having a working simulation (learning a lot of programming in the process), you would probably discover that there are a number of ways in which a tower built from soft blocks can fall down.

 

[256bits]

Homework Statement: Not a homework, just wondered all of sudden.
Relevant Equations: f = ma

What happens if I were to change the height of the cell to 2 m? it has less number of cells above it, so it should experience less weight from above

Correct. The higher one goes up the tree, the less weight is bearing down. Stands to reason, doesn't it. The most highest top cell has zero weight acting from above acting upon it.

 

[256bits]

Homework Statement: Not a homework, just wondered all of sudden.
Relevant Equations: f = ma

it's exerting normal forces to the cell below, above and all around it? or no? Do cells around it exert normal forces? What happens to the sum of these forces? is there a diagram that shows this or similar scenario?

One can assume that the tree 'elements' are rigid, homogenous and not deformable, and that all elements at one height section experience only vertical forces.

If the assumption is that the elements are deformable, then the analysis and equations become somewhat more complicated,