12 F-h${}^2$ analysis

The window consists of several blocks:

• •

  Parameters allows the user to set the width of the moving average window. The value 1 corresponds to no smoothing. This value is saved in settings.

• •

  Run perform calculation and display curve, see section 12.2.

• •

  Save save parameters and results to given file.

• •

  Graph

  • Top: display the loading curve and the smoothed curve.

  • Bottom: display the $F/h^2$ and the $\mathrm{d}F/\mathrm{d}{h}^2$ curves.

  Stepwise zooming/unzooming can be performed by selecting a range with the mouse and pressing the Zoom/ Unzoom buttons. The graph is restored to its original size by the Restore button. Zooming in the two graphs is independent.

1. 1.

   We use a moving average with a fixed width and constant weight. This means we substitute a value with its average with $s$ values to the left and to the right, $w=2s+1$

   $${\widehat{x}}_i=\frac{1}{w}\sum _{j=-s}^s{x}_{i+j}.$$

   The value $w=1$ corresponds to the original data. There is only one moving average type defined for both depth and load. Increasing the value of $w$ noise becomes less influential but important small effects can get lost as well. Therefore, the value should not be too large, below 11 is recommended.

2. 2.

   The ratio $F/h^2$ is calculated for for each data pair $(\widehat{h},\widehat{F})$ of the (smoothed) loading curve and plotted as a function of the (smoothed) depth $\widehat{h}$.

3. 3.

   The derivative $\mathrm{d}F/\mathrm{d}{h}^2$ is calculated for for each data pair $(\widehat{h},\widehat{F})$ of the (smoothed) loading curve and plotted as a function of the (smoothed) depth $\widehat{h}$. The derivative is done numerically as the ratio of the derivatives of $F$ and $h$ with respect to the (time)step or index $i$

   $$\frac{\mathrm{d}F}{\mathrm{d}{h}^2}=\frac{\mathrm{d}F}{\mathrm{d}i}{\left(\frac{\mathrm{d}{h}^2}{\mathrm{d}i}\right)}^{-1}.$$

   The numerical derivatives are calculated using the three-point formula for equally spaced data.