(Botanical) branch bending under gravity

Question

I'm a PhD student in maths, and attended my last physics class some 15 years ago, so you can imagine my competences in the field. My supervisor (also not a mechanist) cant tell me how to proceed either, and after having spent already way too much time on wikipedia to try to understand the elementary concepts, I am turning to you, here is my problem:

Given is a discrete curve in $\mathbb{R}^3$, i.e. an ordered set of points $x_1,...,x_n$ all in $\mathbb{R}^3$, representing the branch of a (botanical) tree. $x_1$ denotes the point where the branch in question branches off from the trunk, $x_n$ is the branches ending point. This discrete curve may be curved and twisted arbitrarily. At each point $x_i$, there is a mass $M_i$ concentrated. Moreover all the branch radii, $r_i$, at point $x_i$ are known (which - I think I at least got this right - is relevant to calculate the "second moment of area"). (If it helps, I would also have a continuous version of the problem, i.e. a continuous curve $s\to g(s)$ instead of the points etc...)

The discrete curve $[x_1,...,x_n]$ describes the branch without gravity being taken into account and the task is to find the new curve $[x_1'(=x_1),x_2'...,x_n']$ resulting from the original curve when taking gravity into account. (By the way the trunk itself is considered to be unaffected by the branches growth and weight, thus remains vertical).

Apparently the problem is trivial if the curve is not twisted, in the way that all $x_i$ lie in one and the same plane perpendicular to the ground (discussed on the second half of page 2 of "Bending of apricot tree branches under the weight of axillary growth" by Alméras et al., in: Trees. Structure and Function, 2002.). However, I have unsuccessfully searched for a generalization for twisted branches.

Could you point me in the right direction? Or, if applicable, tell me that the problem is much more complicated than I think. Thanks a lot in advance.

PS: If, as it seems to me in the article mentioned, there exists a very easy way to approximate the real solution to some extend, making possibly questionable assumptions (the "small deflection assumption" seems to be of that sort), that's fine by me. I do not require an exact solution, any (halfway justified) rough approximation works perfectly for me.

Answer 1

The standard engineering approximation, which makes a lot of simplifications, you should be able to find in any book by the name of Mechanics of Materials or Strength of Materials. The general ideas would be as follows...

First, within these simplifications elasticity is linear, so you can consider each force independently, and calculate the total stress, deformation, deflection... as the sum of the individual ones.

Each force will produce, at each cross section of your branch, a reaction force and moment. The force will have a component perpendicular to the cross-section (tensile force) and another in its plane (shear force). The moment will also have a perpendicular component (torsion moment) and another in the same plane (bending moment). You will need to figure this out for every cross section of your branch.

For long, straight, constant circular cross-section beams, each of these produce a specific set of strains and stresses, which translate into a definite set of displacements. These are the kind of things you will find in the books I mentioned above, or in this document. If your beam is twisted, and does not have a constant, or circular cross section, the approximation will be less accurate.