# Stretching of a Thin Plate with a Hole

## Model Data

**Type:**
quickstart
structural mechanics

**Physics Modes:**
plane stress

**Keywords:**
linear elasticity
validation

This example is a well known benchmark test case from structural mechanics, a thin plate with a circular hole in the center is subjected to a load and stretched along the horizontal axis.

The plate is thin enough to satisfy the two dimensional plane stress
approximation, and since both the problem and solution will be
symmetric around the hole it is enough to model a quarter of the
plate. The computational geometry here therefore consist of a 0.05 by
0.05 *m* square with a quarter of a circle with radius 0.005 *m*
removed from one corner. Due to the symmetry the displacement of the
left edge of the plate should be zero in the x-direction, and
similarly the y-displacement for the lower edge should also be
zero. Furthermore, a horizontal force of 1000 *N* is applied to the
right edge. With a plate thickness of 0.001 *m*, the resulting load
will be 1000/(2·0.05·0.001) *N/m ^{2}*.

Assuming that the plate is made of steel with a Poisson ratio of 0.3
and modulus of elasticity 210·10^{9} *Pa* then it is
expected that the maximum stress in the x-direction will be three
times the stress of a plate without a hole, that is
σ_{x} = 3·1000/(2·0.05·0.001) =
3·10^{7} *Pa* [1]. The computed stress along the
left side vertical boundary is compared against the theoretical
reference results in the figure, clearly showing very good agreement.

How to set up and solve the thin plate with hole example with the
FEATool graphical user interface (GUI) is described in the linked
step-by-step tutorial instructions. Alternatively, this tutorial
example can also be automatically run by selecting **Model Examples
and Tutorials** > **Quickstart** > **Thin Plate with a Hole** from the
**File** menu.

## Reference

[1] E.G. Kirsch, *Die Theorie der Elastizitaet und die Beduerfnisse
der Festigkeitslehre*, Zeitschrift des Vereines Deutscher Ingenieure,
Vol. 42, pp. 797-807, 1898.