\section{Analytical analysis}

\begin{figure}[h]
    \centering
    \includegraphics[width=.7\textwidth]{Figures/box.png}
    \caption{Geometry used in analytical analysis. Black and red indicate the top/bottom surface and verticall walls respectively.}
    \label{fig:anal_box}
\end{figure}

An analytical analysis inspired by sandwich theory was implemented in python to acquire a reasonable starting point for the FEA.\cite{sandwich_theory, sandwich_theory_wiki} Optimal orientation of the layers in the vertical supports and in the top and bottom faces were determined with respect to minimizing the weight of the bridge while maintaining an exposure factor $f_E < \frac{1}{3}$, where $f_E = \max{(f_{FF}, f_{IFF})}$ is the Hashin-criterion exposure factor. One layup was chosen for the vertical support walls and another was chosen for the top and bottom surfaces.

The bending stiffness of the beam was computed as
\begin{equation}
    D = \int \int z^2 E_{11}(z,y)dzdy
\end{equation}

Where $E_{11}(z,y)$ is the Young's modulus as a function of position. The Young's modulus at any location was approximated as the homogenized Young's modulus of the layup at that position, computed as

\begin{equation}
    \Bar{E} = \frac{1}{d}\left(A_{xx} - \frac{A_{xy}^2}{A_yy}\right)
\end{equation}
Where d is the thickness of the layup.
This gives a beam stiffness of

\begin{equation}
    D = \frac{2}{3}\Bar{E}_c (B - b) \left(\frac{h}{2}\right)^3 + \frac{2}{3}B\Bar{E}_f\left[\left(\frac{H}{2}\right)^3 - \left(\frac{h}{2}\right)^3\right],
    \label{eq:stiffness_integral}
\end{equation}

where $\Bar{E}_c$ is the homogenized Young's modulus of the vertical walls, $\Bar{E}_f$ is the homogenized Young's modulus of the top and bottom surfaces, $B$ and $H$ are the outer width and height of the beam, and $b$ and $h$ are the inner width and height of the beam, as shown in Figure \ref{fig:anal_box}.

The stress in the top and bottom faces are assumed to be aligned with the $x$-axis, and can be computed as 
\begin{equation}
    \sigma_{x,f} = \frac{z\Bar{E}_f M_{max}}{D}
\end{equation}
where $M_{max}$ is the internal moment, as computed in section \ref{load_explanation}. The stress in the vertical walls is assumed to consist of a stress component $\sigma_x$ and a shear component $\tau_{xz}$, these can be computed as

\begin{equation}
    \sigma_{x,c} = \frac{z\Bar{E}_c M_{max}}{2D}, \hspace{1cm} \tau_{xz,c} = \frac{Q_x}{2D}\left[ \Bar{E}_c\left(\left(\frac{h}{2}\right)^2 - z^2\right) + \Bar{E}_f f(f + h)\right],
\end{equation}
where $Q_x = \frac{dM_{max}}{dx}$, and $f$ is the thickness of the top and bottom surfaces, as indicated in Figure \ref{fig:anal_box}. The exposure factor was computed at $z = \{\frac{H}{2}, \frac{h}{2}, 0\}$, as these are the regions with highest compressive strain in the upper face, highest compressive strain in the vertical walls and highest shear in the vertical walls respectively. The exposure factor of the beam was taken to be the maximum of the exposure factor in these regions. Finally, the required vertical wall thickness to achieve an exposure factor $f_E = \frac{1}{3}$ was computed for varying thickness of the upper and lower surfaces using different orientations for the layers in the vertical walls. A flowchart of the algorithm is presented in Figure \ref{fig:flowchart}, the results of the computation are displayed in Figure \ref{fig:wall_ratios}. 

\begin{figure}[h]
    \centering
    \includegraphics[width=.8\textwidth]{Figures/wall_ratios.png}
    \caption{Vertical wall thickness required to achieve an exposure factor of $f_E = \frac{1}{3}$ for varying thickness of top and bottom surfaces, using different layup orientations in the vertical walls. The top and bottom surface consist of only \SI{0}{\degree} oriented layers.}
    \label{fig:wall_ratios}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=.6\textwidth]{Figures/flowchart.png}
    \caption{Algorithm for computing wall thickness required to achieve an exposure factor $f_E = \leq \frac{1}{3}$, for varying thickness of the top and bottom surface. The top and bottom surfaces were set to consist of only \SI{0}{\degree} oriented layers.}
    \label{fig:flowchart}
\end{figure}

The relationship between the layup orientation in the vertical walls and the maximum deflection of the beam was also computed. Inspired by sandwich theory, the total deflection was taken to be a superposition of bending of the upper and lower surface, and shear deformation of the vertical walls. Treating the load from the vehicle as two point loads $P$ at a distance $a$ from each end of the beam allows the total deflection of the beam to be calculated as a superposition of the deflection resulting from each of these loads. The total, maximum deflection will then be given as

\begin{equation}
    \delta_{max} = 2 \left(\frac{P a}{48 D_b}  (3 L^2 - 4 a^2) + \frac{P a}{2 S}\right),
    \label{eq:max_deflec}
\end{equation}

Where $S$ is the shear stiffness of the vertical walls, and $D_b$ is the bending stiffness of the top and bottom surface. For a sandwich structure, $S$ is computed as $S = bhG_{c}$ where $G_c$ is the the shear modulus of the core material. For the box beam considered here, $G_c$ is taken to be the effective shear modulus of the vertical walls, computed as 
\begin{equation}
    G_c = \frac{A_{ss}}{B}
\end{equation}
where $A_{ss}$ is computed for the two vertical walls, separated by a "layer" of air with thickness $b$, with all elements in the 2D-stiffness matrix equal to zero. The maximum deflection as a function of layup orientation in the vertical walls and as a function of layup orientation in the top and bottom surfaces is displayed in Figure \ref{fig:ori_deflection}.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{Figures/ori_deflection.png}
    \caption{Maximum beam deflection as a function of layup orientation in the vertical walls (left) and in the top and bottom surfaces (right). Values as computed from equation \eqref{eq:max_deflec}.}
    \label{fig:ori_deflection}
\end{figure}

Due to the possibility of buckling occurring, a combination of layup orientations that minimizes the deformation was chosen as the starting point for the FEA analysis. As seen from Figure \ref{fig:ori_deflection}, this is achieved for layers oriented $\pm 9.5\si{\degree}$ in the vertical walls, and \SI{0}{\degree} in the top and bottom surfaces. 

The initial thickness of the vertical walls and the top and bottom surfaces was chosen by inspecting Figure \ref{fig:wall_ratios}. According to the figure, a top and bottom surface thickness of \SI{5}{\milli\meter} and vertical wall thicknesses of \SI{1.1}{\milli\meter} will give a maximum exposure factor of $\frac{1}{3}$. Figure \ref{fig:wall_ratios} also shows that this is not, per the analytical analysis, the combination that will give the lowest weight while maintaining a safety factor of 3. However, solutions with side wall thickness below \SI{1}{\milli\meter} are expected to produce buckling.

\subsection{Design of the profile's cross section}

It is noted that the analytical analysis that was implemented does not take into consideration the horizontal position of the vertical walls. In practice this means that the analytical approach does not differentiate between the geometry shown in Figure \ref{fig:internal_topo}, with five vertical supports, and the geometry shown in Figure \ref{fig:anal_box} with only two, given that the total thickness of the vertical supports is equal. To prevent the top surface from breaking due to bending in the transverse direction of the beam, a geometry with five vertical supports, as shown in Figure \ref{fig:internal_topo}, with a total thickness \SI{2.2}{\milli\meter} was used as the initial configuration.

\begin{figure}[h!]
    \centering
    \includegraphics[width=.5\textwidth]{fea/part.JPG}
    \caption{Internal topology of the beam. Outer dimensions are 500x250\si{\milli\meter}}
    \label{fig:internal_topo}
\end{figure}