# Local strain energy density for a plate subjected to in-plane linear load

1. Dec 5, 2011

### mp87

Dear all,

I would like to know from you the solution about this problem (which is not a homework, but a topic of my Master thesis!): I need the strain energy density related to a circle of radius r0 centered in an arbitrary point of a square plate, under the boundary conditions described in the attached picture (on the right edge a linear stress is applied, which ranges from 0 to S). The value I obtained, under the plain strain hypothesis, is:

$SED=\frac{1-\nu^2}{2\,E}\,\left(\frac{S}{h}\right)^2\,\left[\frac{r_0^2}{4}+(h+y_c)^2\right]$

where h is the edge of the square plate and yc the y coordinate of the center. The fact is that it doesn't match with the FEM solution (which is surely right, since it was obtained by my Supervisor .

Can you please derive the equation and compare it with mine?

Mattia

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2. Dec 5, 2011

### mp87

Oops, I found the mistake! It was hidden inside the choose of the reference system :P

Here's the correct formula:

$SED=\frac{1-\nu^2}{2\,E}\,\left(\frac{S}{h}\right)^2\,\left[\frac{r_0^2}{4}+(h-y_c)^2\right]$

Best regards,

Mattia