Added a place for the Documentation

Documentation/Lid_Correction.tex

+\documentclass{article}
+
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{subcaption}
+\usepackage[a4paper,left=1.1in,right=1.1in,top=1.1in,bottom=1.1in]{geometry}
+
+\renewcommand{\familydefault}{\sfdefault}
+
+\begin{document}
+    \section*{Correction of Normal Load by Lid
+Displacement}\label{correction-of-normal-load-by-lid-displacement}
+
+Because the tie rods which measure the shear force produce an angle that
+is dependent on the current lid position, the normal force $F_N$
+measured by the machine is not equal to the effective normal stress
+$F_{Neff}$. It has to be corrected with respect to the currently
+measured shear force $F_{S1}$ and the current angle $\beta$, that is
+dependent on the current lid position $d_{lid}$.
+
+\begin{figure}[h!]
+\centering
+  \includegraphics[width = .7\columnwidth]{RST_Angle}
+  \caption{Formation of an angle between the lid and the sensor.}
+\end{figure}
+
+For this we use a simple decomposition of the resulting force
+$F_R=F_{s1}$ into the effective shear force $F_S$ and the normal
+force $F_N$ excerted by pulling on the lid at an oblique angle
+$\beta$:
+
+\begin{equation}
+  F_R = \sqrt{F_S^2 + F_N^2}
+\end{equation}
+\begin{equation}
+  F_S = F_R \, cos \, \beta
+\end{equation}
+\begin{equation}
+  F_N = F_R \, sin\, \beta
+\end{equation}
+
+The angle is defined by the length of the tie rods $l_{rod} = 412
+\,mm $ and the lid displacement $d_{lid}$ with respect to the
+reference point at zero lid displacement $d_0 = +\,9\,mm$. The usual
+range of lid displacement is a few millimeters upwards (negative) and
+downwards (positive).
+
+\begin{equation}
+  \beta = sin^{-1} \, \frac{(d_0-d_{lid})}{l_{rod}}
+\end{equation}
+
+\begin{figure}[h!]
+
+\centering
+\begin{subfigure}[t]{0.45\textwidth}
+\includegraphics[width = \textwidth]{angle_vs_lid}
+\caption{Angle as a function of the lid position.}
+\end{subfigure}
+~
+\begin{subfigure}[t]{0.45\textwidth}
+\includegraphics[width = \textwidth]{correction_factors}
+\caption{Factors needed to compensate the difference between measured and effective forces.}
+\end{subfigure}
+
+\end{figure}
+
+    The angle only shows a very small variation between 2$^\circ$ and
+-1.5$^\circ$, where negative shows a downward slope and positive an
+upward slope, in contrast to the definition of $d_{lid}$. As a result,
+the normal force component of the resulting force is changing from a
+very small positive value into a small negative value. This shows in
+correspondence to the position of the lid, that if the lid is higher
+than the sensor ($d_{lid} > d_0$) then it pushes down onto the sample
+resulting in a higher normal stress and vice versa. The shear stress
+component has a maximum when $d_{lid}=d_0$ because here the tie rods
+are parallel to the direction of shear.
+
+
+
+    The above graph shows the factors by which the measured forces (or
+stresses) have to be corrected. The machine does this for the internal
+determination of the normal load that is applied onto the lid, but the
+measurement is done using an external controller which only measures the
+pure signal. This leads to the following corrections with $F_R$ being
+the measured shear force and $F_N$ the measured normal force:
+
+\begin{itemize}
+\item Effective normal force:
+  \begin{itemize}
+    \item $F_{Neff} = F_N - F_R \, c_{FN}$
+  \end{itemize}
+\item  Effective shear force:
+  \begin{itemize}
+    \item $F_{Seff} = F_R \, c_{FS} $
+  \end{itemize}
+\end{itemize}
+
+\end{document}