How to draw a bridge bend under load?

Question

Mathematics

A beam is drawn, sticking out horizontally from the wall. A weight is placed on the free end. The larger the load, the stronger the beam bends smoothly.
What form does it take?

For one graphic project, I want to dynamically calculate such curved shapes, when one end is fixed, and some kind of force vector acts on the other. At the same time, it is not necessary to seriously delve into the strength of materials and take into account vibrations and fatigue of the material. You can roughly understand the mathematics of the form, with some simplification and approximation .
Hypotheses:
The shape of an arc of a circle with a center below the wall, the larger the load, the smaller the radius. Unlikely.
Upd. googled the catenary curve - maybe there will be half of the catenary curve here too?
Upd. 2Tried with chaincurve jsFiddle :
Upd. 3 is a similar question in English, where they write that “everything is complicated” and the curve will be of the 3rd or 4th order.

Answer 1

The basic foundations of the strength of materials - they just start with a spherical horse in a vacuum with an abstract ideal beam and its deformation and load diagrams -))
A non-abstract beam with a variable section is the next round

Answer 2

In the first approximation (the deformations are small, the beam is light) - half of the parabola.
To be exact, it can only be solved numerically.