\section{Finite Element Analysis}

Using the results from the analytical analysis as an initial configuration, a finite element analysis was conducted with the goal of minimizing the weight of the beam while maintaining a Hashin-criterion exposure factor $f_E < \frac{1}{3}$. The major issue that can be handled by FEA which was not included in the analytical analysis is that of buckling in the vertical supports and in the top surface. Using FEA, the viability of a sandwich structure was also investigated. The final results of the finite element analysis beams B and D presented in Table \ref{tab:final_layups}. The bridge utilizing sandwitch structures has a far lower mass, but is extremely fragile if handled incorrectly, as discussed in detail in sections \ref{sec:sandwich} and \ref{sec:discuss}.

\begin{table}[h]
    \centering
    \caption{Notable beam structures investigated by finite element analysis. The letter ''f'' indicates a foam layer with properties as described in Table \ref{tab:mat_chara}. Beam A is the initial configuration chosen based on the analytical analysis, beam B is the result of optimization without a sandwich solution. Beam C is the initial configuration used for optimization with a sandwich solution, beam D is the final result of optimization with a sandwich solutions.}
    \begin{tabular}{c c c c c c c c}
    \toprule
    \multirow{2}{*}{Beam} & & Top & Bottom & Vertical &  Eigenvalue, 1st & \multirow{2}{*}{$f_E^{\max}$} & \multirow{2}{*}{Mass [kg]}\\
    & & Surface & Surface & Supports & buckling mode & &\\
    \midrule
    \multirow{2}{*}{A} & Orientation & \makecell{[0]$^{10}$} & [0]$^{10}$ & [9.5, -9.5]$_s$ & \multirow{2}{*}{0.1} & \multirow{2}{*}{0.0003} & \multirow{2}{*}{168} \\
     & Thickness & \makecell{[1]$^{10}$} & [1]$^{10}$ & [0.22, 0.22]$_s$ & & \\
     \midrule
    \multirow{2}{*}{B} & Orientation & \makecell{[0, 90, 0, 90]$_s$} & [0]$^6$ & [9.5, -9.5]$_s$ & \multirow{2}{*}{1.36} & \multirow{2}{*}{0.002} & \multirow{2}{*}{91.8} \\
     & Thickness & \makecell{[0.5, 0.2, 1.5, 0.1]$_s$} & [0.5]$^6$ & [0.4, 0.4]$_s$ & & \\
     \midrule
     \multirow{2}{*}{C} & Orientation & [0, 90, 0, 90, f]$_s$ & [0]$^6$ & [9.5, -9.5, f]$_s$ & \multirow{2}{*}{7.1} & \multirow{2}{*}{0.002} & \multirow{2}{*}{57.6}\\
     & Thickness & [0.5, 0.2, 1.5, 0.1, 5]$_s$ & [0.5]$^6$ & [0.4, 0.4, 5]$_s$ & & \\
     \midrule
     \multirow{2}{*}{D} & Orientation & [0, 90, f]$_s$ & [0] & [9.5, -9.5, f]$_s$ & \multirow{2}{*}{1.70} & \multirow{2}{*}{0.03} & \multirow{2}{*}{57.6}\\
     & Thickness & [0.4, 0.1, 5]$_s$ & [0.6] & [0.1, 0.1, 5]$_s$ & & \\
     \bottomrule
    \end{tabular}
    \label{tab:final_layups}
\end{table}

\subsection{Model characteristics}
ABAQUS Version 6.21 was used in order to compute the FEA analysis. The beam is simply supported at the extremities and distributed load due to gravity and two concentrated loads as consequence of the load of the car were applied. 
The total mass that has to handle one single beam is 1000kg (500kg per wheel), in order to simulate it, 6 masses of 167kg were added in 6 different point (3 for each wheel) as can be seen from figure \ref{fig:mass_wheels}  

\begin{figure}[h]
    \centering
    \includegraphics[width=.6\textwidth]{fea/mass.JPG}
    \caption{Mass distribution on the top of the beam in correspondence with the wheels.}
    \label{fig:mass_wheels}
\end{figure}

Different layup are applied for the different part of the beam as can be seen from \ref{fig:internal_material}. The supports will has the same layup but the two lateral walls has the offset set to $top$ while the central supports has the offset set to $middle$
For the element meshing it was used a mesh size of $50$ and a $Quad$ shape combined with a $Structured$ technique.
\begin{figure}[h!]
    \centering
    \includegraphics[width=.5\textwidth]{fea/property.JPG}
    \caption{Beam section with the different layup applied.}
    \label{fig:internal_material}
\end{figure}


\subsection{Plain laminate structure}
As shown in Figure \ref{fig:vert_buckle}, the vertical walls in the initial configuration buckle at very low loads, with a buckling eigenvalue of 0,78166. Additionally, Figure \ref{fig:initial_hashin} shows that the analytical analysis over-predicted the Hashin-criterion exposure factor.

\begin{figure}[h]
    \centering
    \includegraphics[width=.6\textwidth]{fea/buckling_support.JPG}
    \caption{Buckling occurs in the vertical walls if they are thinner than \SI{1.6}{\milli\meter}, with top and bottom surfaces of thickness \SI{10}{\milli\meter}.}
    \label{fig:vert_buckle}
\end{figure}

\begin{figure}[h]
    \centering
    \subfloat[Compressive]{\includegraphics[width=.45\textwidth]{fea/carbon_6_hashin_compressive.JPG}}
    \subfloat[Tensile]{\includegraphics[width=.45\textwidth]{fea/carbon_6_hashin}}
    \caption{Hashin-criterion exposure factor for beam A, shown in Table \ref{tab:final_layups}. Exposure factors are in the order of $10^{-3}$.}
    \label{fig:initial_hashin}
\end{figure}

Based on these results the thickness of the vertical walls was increased, while the thickness of the top and bottom surfaces was decreased. It is also noted that the bottom surface is only subjected to a tensile load therefore it will not buckle. Furthermore, it does not need to support the concentrated load of the wheels of the vehicle. Due to these considerations the bottom layer was chosen to be notably thinner than the top layer. The thickness of the vertical supports was increased until the first positive buckling mode was buckling in the top surface, with an eigenvalue of 1.36, as shown in Figure \ref{fig:first_top_buckle}. Further, a small number of \SI{90}{\degree} layers were introduced in the top surface to prevent deformation between the vertical supports. As shown in Figure \ref{fig:hashin_no_sand} the Hashin-criterion exposure factor was far below one.

\begin{figure}[h]
    \centering
    \includegraphics[width=.6\textwidth]{fea/carbon_21_buckle.JPG}
    \caption{First buckling mode for a beam with \SI{1.6}{\milli\meter} thick vertical supports oriented $\pm \SI{9.5}{\degree}$, \SI{4.6}{\milli\meter} thick top surface and \SI{3}{\milli\meter} thick bottom surface, with orientations as shown for beam B in Table \ref{tab:final_layups}.}
    \label{fig:first_top_buckle}
\end{figure}

\begin{figure}[h]
    \centering
    \subfloat[Compressive]{\includegraphics[width=.48\textwidth]{fea/carbon_21_Hashin_compr2.JPG}}
    \subfloat[Tensile]{\includegraphics[width=.48\textwidth]{fea/carbon_21_Hashin_tensile_section_3.JPG}}
    \caption{Hashin-criterion exposure factor for a beam with \SI{1.6}{\milli\meter} thick vertical supports oriented $\pm \SI{9.5}{\degree}$, \SI{4.6}{\milli\meter} thick top surface and \SI{3}{\milli\meter} thick bottom surface, with orientations as shown for beam B in Table \ref{tab:final_layups}. Maximum exposure factor is 0.002.}
    \label{fig:hashin_no_sand}
\end{figure}

Varying the orientation of the layers in the vertical walls to increase their resistance to buckling was attempted, but the reduction in stiffness lead to increased deformation and buckling in the top surface. Comparing the analytical value of the maximum deflection in left graph of figure \ref{fig:ori_deflection} with the value that results from the FEA (\ref{fig:beam deflection}), considering an angle of the fiber of the vertical walls of $\pm45$, we are able to verify that the results obtained with FEA regarding deformation are consistent with the analytical approach since the maximum value is approximately $31mm$ in both cases.

\begin{figure}[h!]
    \centering
    \subfloat{\includegraphics[width=.7\textwidth]{fea/carbon_21_U2_2.JPG}}
    \caption{Beam deflection for a angle of $\pm45\si{\degree}$ of the vertical wall, with all other properties equal to those of beam B, presented in Table \ref{tab:final_layups}. Maximum deflection is $\approx$\SI{30}{\milli\meter}.}
    \label{fig:beam deflection}
\end{figure}

\subsection{Sandwich structure}
\label{sec:sandwich}
To enable further reduction of the thickness of the laminates, the beam was modeled using sandwich structures. The sandwich structure prevents buckling, but does not increase the tensile strength of the walls. In fact, as can be seen from equation \eqref{eq:stiffness_integral}, the bending stiffness of the beam will be decreased if sandwich structures are used for the top and bottom surfaces while the total thickness of laminate is kept constant. Therefore, a sandwich structure was only used in the vertical supports and the top surface, as these are the only regions exposed to buckling.

Initially, a foam layer of \SI{10}{\milli\meter} was introduced in the top surface and vertical walls while keeping the thickness of the Carbon-Epoxy constant. This lead to a greatly reduced eigenvalue of the first buckling mode as shown in Figure \ref{fig:positive_buckling_mode}. The increased resistance to buckling allowed for reduction in the thickness of the laminates.

\begin{figure}
    \centering
    \includegraphics[width=.6\textwidth]{fea_sandwich/sand_buckling_beginnign.JPG}
    \caption{First positive buckling mode for beam C, presented in Table \ref{tab:final_layups}.}
    \label{fig:positive_buckling_mode}
\end{figure}

The thickness of the laminate faces in the sandwich structured walls was decreased until the eigenvalue of the first mode of buckling was 1.34. Further reduction of the thickness of the laminate layers in the vertical supports was not deemed viable due to production limitations. Reduction of the foam thickness in the top surface lead to buckling modes with eigenvalues below one. The final configuration using sandwich panels was thereby determined to be the structure shown in Figure \ref{fig:sandwich_beam_section}.

\begin{figure}[h]
    \centering
    \subfloat{\includegraphics[width=.7\textwidth]{fea/buckling_negative.JPG}}
    \caption{First mode with BLF negative on the bottom of the beam, for beam D, presented in Table \ref{tab:final_layups}}
    \label{fig:buckling_negative}
\end{figure}

It is noted that the extremely thin bottom layer is incapable of carrying any significant load. A buckling analysis shows that there is a buckling mode with eigenvalue -0.02, as shown in Figure \ref{fig:buckling_negative}. This indicates that the beam is likely incapable of supporting its own weight if carried or placed upside down.

\begin{figure}[h]
    \centering
    \subfloat{\includegraphics[width=.9\textwidth]{Figures/Beam section.png}}
    \caption{Cross-section of beam D, presented in Table \ref{tab:final_layups}.}
    \label{fig:sandwich_beam_section}
\end{figure}
