TITLE: Parabolic shape in Bow (not arrow!)
QUESTION [11 upvotes]: This is what I am thinking for some days. And I think here are some experts who can answer this question.
If I bend any stick made with material that uniform density and its shape is cylindrical (more like stick than real cylinder), will the shape of it converge to parabola? If so, how long it'll be? I mean from the time I start to bend it and to the time It breaks. Between these two times when this will form a real parabola?

REPLY [2 votes]: @Raskolnikov:
The question from OP relates to Mechanics of Materials/Strength of Materials/ Large deformation Non-linear theory.
In Beam structures loads are applied laterally to the Beam axis. Engineer’s theory of Bending (ETB) you mentioned neglects term $  (1 + y^{'2})^{3/2} $.  Inclusion of this term and associated large deflections and beam rotations ( (comparable to stick thickness and $\pi/2$ respectively) lead to more accurate results by such non-linear analyses.A sample from internet:

In Engineering mechanics of materials a Column is distinct from a Beam with load F acting along its axis and classified as such separately. The column even exhibits no deformation until a critical force F as eigenvalue $ \pi^2 EI/l^2$ is applied.. this is the Euler Column buckling load. By ETB with lateral load we may end up trying to reconcile a circle with a shallow parabola.
Euler’s law has M/EI equaling curvature for both linear ( like ETB) and non-linear situations. Laid threadbare, ETB has been developed to suit the needs of civil engineering( and Applied Mechanics) with small beam deformations and tangential rotations of Beam. It is not suited to deal with deformations e.g., in thin fiber-optic wires, high elongation fiberglass pultruded rods,big bending in fiberglass in sports of pole vault made from S-glass or bamboo Bows mentioned here, or even a sheet of paper bent to form a loop, thin sheets rolled into cylinder forms and such.
The fourth order ODE given is not suitable for large deformations which she is inquiring about.
Loops of Elastica