Parabolic shape in Bow (not arrow!)
This is not an easy question to answer since the system is quite complex. Besides, the way the stick will bend depends on the type of stress you submit it to, in other words, what the boundary conditions of the system are.
A simple model for a beam of length $\ell$ is given by the following differential equation
$$\frac{d^4 Y}{dx^4} = \frac{W(x)}{EI} \; \text{ for } 0 < x < \ell \; ,$$
in which $Y$ is the vertical displacement of the beam if it is aligned horizontally. $W(x)$ is the vertical load per unit length, that is essentially the force to which the beam is submitted. Note that since this model is 1D, it doesn't account for torsion or shears along other directions than the vertical one. $E$ is Young's modulus of elasticity and $I$ is the moment of inertia of a cross section of the beam about the axis. Depending on boundary conditions, one can describe different types of problems:
- $Y=Y'=0$ for a clamped beam.
- $Y=Y''=0$ for a hinged or simply-supported ends.
- $Y''=Y'''=0$ for free ends.
This equation is known as the Euler-Bernoulli equation for beams.
So, depending on what you want to model exactly, you'll have to solve this equation for an appropriate choice of boundary condition and load $W(x)$. For instance, you could take one clamped end and another free end on which a load is concentrated to simulate a hand holding a beam still while the other is trying to push an end. In this case, the solution will not be given by a parabola but by a third degree polynomial.
This does nothing yet to solve the problem for more complicated stresses, for instance along other directions than the vertical or for torsions. From there on, this really becomes an engineering question. If you want to delve more deeply into the subject, you should read up on materials science. One thing is clear though, the more complex the system, the less likely you'll obtain a conic section while bending.
The shape you are seeking is the "elastica". Its differential equation is simply obtained by moment/ force F equilibrium as ... bending moment or curvature proportional to y, EI is the proportionality constant by Euler- Bernoulli Law.
$$ y^{' '}( x)= -( F/EI) . y(x) (1 + y^{'2}(x))^{3/2} $$
In mechanics of materials when shallow bow arch is considered,
$$ (1 + y^{'2})^{3/2} $$
is taken as unity in approximation. It is a sine wave. When shallow i.e., slope small compared to unity, sine waves,circles, parabolas,..all have same second degree approximation... as a parabola.
Large Deformation Here you see deflections of a beam when large forces are applied.
EDIT1
It can be simplified to two terms, in terms of elliptic integrals of the first and second kinds.
@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