Boundary conditions of a bending plate

2017-12-09 23:01:14

I'm trying to find the boundary conditions for the following problem:

A plate with length 2L is placed on supports at x = L/2 and x = - L/2. The plate is deforming elastically under its own weight (maximum displacement bowing up at x = $0$). Both ends of the plate are free boundaries.

The goal is to eventually solve the equation DW'''' = q(x) for the right half of the plate (x > $0$). Where D is the flexural rigidity $$\frac{Eh^3}{12(1-\nu^2)}$$

$E$ is Young's Modulus, $\nu$ is Poisson's ratio, h is the thickness of the plate, and q = -${\rho}$gh.

I think I have two of the four boundary conditions (DW'''= $0$ at x = L and DW'' = $0$ at x = L), but I'm having trouble finding the boundary conditions at x = $0$.