Flexure under Line Load

The graph below allows you to test the influence of different line loads/displacements and effective elastic thickness on the flexure of a plate. Just drag the sliders to modify the values. You find a short description of the governing equation below.

Young's Modulus [GPa]: 10 25 75 100 Poisson's Ratio: .25 .5

Plate Flexure due to Line Load

Flexure, \(w\), of an elastic plate due to a line load / end displacement, \(w_0\), is

\[w = w_0 e^{-x/\alpha}\left( \cos{x \over \alpha} + \sin{x \over \alpha} \right)\]

If the plate is broken then the sine term is dropped.

\(x\) is the distance from the line load and \(\alpha\) is

\[\alpha = \left( {4D \over \left( \rho_m - \rho_w \right) g} \right)^{1/4}\]

where \(D\) is the flexural rigidity, which is related to Young's modulus \(E\), Poisson's ratio \(\nu\), and the effective elastic thickness \(h\) through

\[D = {E h^3 \over 12\left( 1 - \nu^2 \right)}\]

The other variables are the density of the mantle, \(\rho_m\), the density of water, \(\rho_w\), and the gravity acceleration, \(g\).

A more detailed derivation can be found in Geodynamics by Turcotte and Schubert.