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\DOI{10.5802/crmeca.294}
\datereceived{2024-07-12}
\daterevised{2025-03-18}
\dateaccepted{2025-03-24}
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\dateposted{2025-04-10}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}


\title{A recurrence formula in reciprocal frame structures: the Da
Vinci bridge}

\alttitle{Une formule de r\'{e}currence pour les structures
r\'{e}ciproque : le pont de L\'{e}onard De Vinci}

\author{\firstname{Jean} \lastname{Aummonier}}
\address{Acad\'{e}mie militaire de Saint-Cyr Co\"{e}tquidan, CReC, 56380 Guer, France}

\author{\firstname{Jean-Eudes} \lastname{Windeck}}
\addressSameAs{1}{Acad\'{e}mie militaire de Saint-Cyr Co\"{e}tquidan, CReC, 56380 Guer, France}

\author{\firstname{Thomas} \lastname{Burkhard}}
\addressSameAs{1}{Acad\'{e}mie militaire de Saint-Cyr Co\"{e}tquidan, CReC, 56380 Guer, France}

\author{\firstname{Brewen}\nobreakauthor\lastname{Lev\^{e}que}}
\addressSameAs{1}{Acad\'{e}mie militaire de Saint-Cyr Co\"{e}tquidan, CReC, 56380 Guer, France}

\author{\firstname{Christelle Jeanne} \lastname{Combescure}\CDRorcid{0000-0002-4588-3682}\IsCorresp}
\addressSameAs{1}{Acad\'{e}mie militaire de Saint-Cyr Co\"{e}tquidan, CReC, 56380 Guer, France}
\address{University Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France}
\email[C. J. Combescure]{christelle.combescure@st-cyr.terre-net.defense.gouv.fr}

\begin{abstract} 
This article studies a reciprocal system referred to as the ``Da Vinci
bridge'' and first proposed by Leonardo Da Vinci in the Codex
Atlanticus written in 1495. The concept of reciprocal systems consists
in transferring forces using planar or elongated elements {in such a
way to form a closed circuit}. Reciprocal systems are usually built
from the repetition of a building block. In the case of the Da Vinci
bridge, the building block is an assembly of four beams. This article
uncovers a recurrence formula linking the number of building blocks to
the reaction forces on the most loaded beams of the bridge. It is
interesting to note that the recurrence formula differs if the system
consists in an even or odd number of building blocks. The {various
hypotheses of} the strength of material computations that led to the
recurrence formula are validated by comparison with an experimental
{realization}. This studies shows that, when designing reciprocal
systems, an optimization has to be carried out in order to chose
between using long beams or using a large number of building elements.
\end{abstract} 

\begin{altabstract} 
Cet article \'{e}tudie un syst\`{e}me r\'{e}ciproque appel\'{e}
\og pont de L\'{e}onard De Vinci \fg,
propos\'{e} pour la premi\`{e}re fois par L\'{e}onard de Vinci dans le
Codex Atlanticus r\'{e}dig\'{e} en 1495. Le concept des syst\`{e}mes
r\'{e}ciproques consiste \`{a} transf\'{e}rer les forces \`{a} l'aide
d'\'{e}l\'{e}ments plans ou allong\'{e}s de mani\`{e}re \`{a} former un
circuit ferm\'{e}. Les syst\`{e}mes r\'{e}ciproques sont
g\'{e}n\'{e}ralement construits \`{a} partir de la r\'{e}p\'{e}tition
d'un \'{e}l\'{e}ment de construction. Dans le cas du pont de
L\'{e}onard De Vinci, l'\'{e}l\'{e}ment de construction est un
assemblage de quatre poutres. Cet article d\'{e}montre une formule de
r\'{e}currence reliant le nombre d'\'{e}l\'{e}ments de construction aux
forces de r\'{e}action sur les poutres les plus sollicit\'{e}es du
pont. Il est int\'{e}ressant de noter que la formule de r\'{e}currence
diff\`{e}re si le syst\`{e}me est constitu\'{e} d'un nombre pair ou
impair d'\'{e}l\'{e}ments de construction. Les diff\'{e}rentes
hypoth\`{e}ses de calculs de r\'{e}sistance des mat\'{e}riaux ayant
conduit \`{a} la formule de r\'{e}currence sont valid\'{e}es par
comparaison avec une r\'{e}alisation exp\'{e}rimentale. Cette \'{e}tude
montre que, lors de la conception de systh\`{e}mes r\'{e}ciproques, une
optimisation doit \^{e}tre effectu\'{e}e afin de choisir entre
l'utilisation de poutres longues ou l'utilisation d'un grand nombre
d'\'{e}l\'{e}ments de construction.
\end{altabstract} 

\shortrunauthors

\keywords{\kwd{Reciprocal}
\kwd{Strength of materials}
\kwd{Da Vinci}
\kwd{Bridge}
\kwd{Recurrence}}


\altkeywords{\kwd{R\'{e}ciproque}
\kwd{R\'{e}sistance des mat\'{e}riaux} 
\kwd{De Vinci} 
\kwd{Pont}
\kwd{R\'{e}currence}}

\thanks{Military Academy of Saint-Cyr, Centre de Recherche de Saint-Cyr Co\"{e}tquidan}

\maketitle

\end{noXML}

\newcommand{\vx}{\vec{x}}
\newcommand{\vy}{\vec{y}}
\newcommand{\vz}{\vec{z}}

\section{Introduction}

{In 1495, Leonardo Da Vinci proposed an innovative bridge design
grounded in the concept of reciprocity, or self-support. This
particular design, illustrated in the Codex Atlanticus, as referenced
by a drawing~\cite{vinci} reproduced in
Figure~\ref{fig:leonardobridge}a, has been extensively analyzed in
engineering education settings and has been constructed in multiple
locations around the world. Instances can be found either as
decorations such as a sculpture in Freiburg, Germany, or as real
bridges in Mors{\o}, Denmark or in Vancouvert, Canada. Some companies
propose to sell elements to build reduced toy models of this bridge.
Example instructions to build this bridge are presented in
Figure~\ref{fig:leonardobridge}b. The so-constructed bridge presents
an out-of-plane curvature due to the thickness of the beams composing
it; were these beams fictively without thickness, the bridge would lay
flat.}

\begin{figure}[h!]
\includegraphics{fig01}
\caption{\label{fig:leonardobridge}(a) Bridge proposed by Leonardo Da
Vinci on p.~69 of the Codex Atlanticus.  Image
from~\cite{vinci}~\url{https://codex-atlanticus.ambrosiana.it/}.   (b)
Da Vinci bridge construction instructions
from~\url{https://xtremeimportskapiti.co.nz/products/da-vinci-bridge-3-pack}.}
\end{figure}


Reciprocal structures, also called self supporting structures, are
defined by Olga Popovic Larsen, as ``structures consisting of linear
flat or inclined elements which support each other and are arranged in
a way to form a closed circuit or unit''~\cite{PopovicLarsen2014}. The
closed circuit of the Da Vinci bridge consists of four beams that
support each other as presented in Figure~\ref{fig:recipunit}. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}\label{fig:recipunit}Reciprocal unit circled in
black on the simplest form of the Da Vinci bridge.} 
\end{figure}

{Reciprocal structures are usually built from the repetition of a
building block. In the case of the Da Vinci bridge, the building block
is an assembly of three beams composing a H shape (see subimage 
\circledN{2}~of Figure~\ref{fig:leonardobridge}b).} The reciprocal concept
allows to span long distances with short elements~\cite{Pugnale2014}
which is particularly convenient in bridge design, but most of all, it
allows to build robust structures without using any assembling
elements. An extensive review of reciprocal structures in the world is
presented in the book~\cite{PopovicLarsen2008} with a focus on their
architectural use. The kinematic behavior of such structures has been
studied by numerous researchers in the field of architecture or
mechanics~\cite{Brocato2011, Song2013}. {Software for reciprocal frame
structures has been developed, drawing upon these articles, to
ascertain the dimensions of the constituent elements necessary to
achieve a specified three-dimensional geometry using reciprocal
structures~\cite{Song2013}}. However, the mechanical behavior {of
reciprocal systems} in terms of forces has been studied {only} in a
very limited number of articles~\cite{Douthe2009, Greco2014,gelez2011behavior}. Among them, one could cite~\cite{Greco2014} which
established an equivalent continuous medium able to capture the
essential elastic properties of such structures.


To the {best of the} authors' knowledge, no {existing} study has been
conducted that allows {the mechanical design of the structural elements
constituting the bridge with respect to limit forces. Specifically, the
authors raise a query concerning the closed reciprocal unit such as the
one presented in Figure~\ref{fig:recipunit}. Considering} that the forces
transferred from one element of the unit to another constitute a closed
circuit, it {appears intuitive} that a recurrence formula could be
{derived, correlating} the size of a reciprocal structure with the
forces at the intersection of the elements. {The objective of this
article is to uncover this recurrence formula in the case of the Da
Vinci bridge.}\looseness=-1


In Section~\ref{sec2}, the closed form recurrence formula is
established and the hypothesis for this computation are listed. In
Section~\ref{sec3}, based on this recurrence formula, design rules are
set that predict the location of the maximum torque in the bridge thus
identifying the critical beam. This section also presents an
experimental validation of this prediction, thus validating the
hypothesis listed in the previous section. Finally, a conclusion on the
possibility to generalize these results is proposed.

\section{Closed-form recurrence formula for the Da Vinci bridge}\label{sec2}

\subsection{Model and hypothesis}\label{sec:model}

The Da Vinci bridge is modeled as an assembly of beams and studied
using strength of materials theory. Therefore, it is assumed that the
beams are elongated and that the material stays in its linear elastic
domain of behavior. The influence of the thickness of the beams on the
out-of-plane orientation of the beams is neglected, along with its
influence on the orientation of the reaction forces. As such, the
reaction forces are supposed to all be aligned with the direction
$\vz$. The influence of these hypotheses will be assessed in
Section~\ref{sec:experiments} by comparison with
experiments.

The Da Vinci bridge is studied for various bridge {orders} referred to
by their number $N$ of transversal beams. Transversal beams are the
beam oriented along $\vx$ {and colored in red}. Bridges with different
{orders} differ by the number of transversal beams that compose them
while keeping the dimension of the composing beams constant. This means
that different {orders} induce different bridge lengths in the
longitudinal direction $\vy$. Moreover, two important configurations
are differentiated based on whether the bridge contains an even or odd
number of transversal {red} beams. In odd cases, the load is applied as
a point load to the center of the central transversal beam. In the even
case, the load is evenly split in two and applied symmetrically
to the center of the two transversal central beams. In both cases the
load is considered to be applied downward in the out-of-plane
direction, i.e.\ along ${-}\vz$.


Top views of two bridges are presented in Figure~\ref{fig:exmppair} for
the even-order $N=2$ and odd-order $N=3$ cases, from left to right.
Thanks to the sy{mm}etry of both the structure and the loading, beams
of similar colors are considered to be subjected to the same loading
and thus have similar behavior. For the strength of material study, the
spacing between the longitudinal green and black beams is neglected.
Therefore, the structure can be considered to be sy{mm}etric with
respect to both the $(O\vx\vz)$ and the $(O\vy\vz)$ planes passing
through the center. Consequently, green and black beams in the even
case and blue and black beams in the odd case are considered to be
loaded similarly. The influence of this hypothesis will be assessed in
Section~\ref{sec:experiments} by co{mpa}rison with experiments.

\begin{figure}
{\vspace*{-1pt}}
\includegraphics{fig03}
\caption{\label{fig:exmppair}Top views of two studied bridges for the
(a) even-order $N=2$ and (b) odd-order $N=3$ cases. Contact points on
central transversal beams are indicated by circles and denoted $A^N$,
$B^N$, $C^N$ where $N$ is the beam order.}
{\vspace*{-1pt}}
\end{figure}


The reciprocal nature of the assembly is taken care of by considering
unilateral contact forces, meaning that the direction of the forces are
imposed based on the knowledge of the assembly: when a red beam is
above a green beam, the load from red to green goes downward in the
$\vz$ direction and vice-versa. {In Figure~\ref{fig:exmppair}, a beam
passing under another one is represented as cut.}

Finally, the transversal beams are supposed to be crossing the
longitudinal beams at their centers.{\vspace*{-3pt}}

 
\subsection{Recurrence study}

The present study is divided into 2 cases: 
\begin{enumerate}[(1)]
    \item Structures with an even number of transversal beams, or even-order.
    \item Structures with an odd number of transversal beams, or odd-order.
\end{enumerate}

As presented in the previous subsection, the study is carried out on
bridges of increasing sizes indexed by their number $N$ of transversal
beams {(or similarly their number of H-shaped building blocks) that will
be called order from now on.} Figure~\ref{fig:variousn} presents three
bridges structures with {orders} $N=2$ to $N=4$. {It can be observed
that the bridges are always constructed on the same manner, using the
repetition of a H-shaped building block composed of one transversal and
two longitudinal beams.} The building blocks are then assembled by
creating a top-bottom connection between the longitudinal and
transversal beams.


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:variousn}Example of three bridges configurations
corresponding to various {orders} $N$.}
\end{figure}

The recurrence relation is established on the reaction forces {$A^N$,
$B^N$ and eventually $C^N$ applied to the central transversal beam,
where the exponent $N$ refers to the bridge order.} {The locations of
these reaction forces are marked by circles in Figure~\ref{fig:exmppair}
for the even-order $N=2$ and odd-order $N=3$ cases. For clarity and in
anticipation of a recurrence proof, the reaction forces issued from the
order $N=2$ are indicated in the case $N=4$ on Figure~\ref{fig:variousn}
with a subtlety on the reaction force denoted $A'$. This force is a
modified $A^2$ reaction force which will be detailed in the coming
subsection. Note that in the even-order cases the reaction forces
applied to the other central transversal beam can be deduced by
sy{mm}etry.}


\subsection{Results and recurrence formula}

\subsubsection{{Even-order case}}

The {full} calculation is {now} detailed for the case $N = 2$. {This
case will serve as initialization of the recurrence proof.}

As presented in the ``model and hypothesis'' Section~\ref{sec:model}, the
structure has a sy{mm}etry plane $(O\vy\vz)$ passing through the
center. Moreover, the total load applied to the structure is a force of
total magnitude $F$ oriented along {the negative vector $\vz$. This
force is separated into two forces evenly divided} on the two central
transversal beams. By neglecting the spacing between the longitudinal
green and black beams, the fundamental principle of statics provides
the four reaction forces applied by the ground to the bridge. These
forces are all equal to $\vec{R}_{g}=({F}/{4})\vz$.

One of the black beam is now isolated and its free body diagram is
presented Figure~\ref{fig:blackbeameven}. The black beam is on two
supports, the ground and the $R_1$ red beam, while red beam $R_2$
applies a vertical force at the middle of the black beam {which is then
in a 3-point bending setting. Reaction} forces from beams $R_1$ and
$R_2$ are identified as forces $A^2$ and $B^2$, respectively,
corresponding to the forces at the contact points with similar names.



\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:blackbeameven}Free body diagram of one of the black
beam in the case $N=2$.}
\end{figure}


By application of the fundamental principle of statics in projection on
$\vz$ and the static moment theorem at the point of contact with the
ground, the reaction forces can be computed as{\unskip\break} follows:
{\begin{eqnarray*}
\left\{ 
\begin{array}{@{}l}
\displaystyle A^2=\frac{F}{4}\\[0.6pc]
\displaystyle B^2=\frac{F}{2}.
\end{array}
\right.
\end{eqnarray*}}\unskip


With this last computation, {both} reaction forces {at contact points
$A^2$ and $B^2$} are determined. These results are su{mm}ed up on
Figure~\ref{fig:resultsn=2}.


\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:resultsn=2}Reaction forces as function of applied
force $F$ on a bridge system constituted of $N=2$ transversal beams.}
\end{figure}



Repeating this process on structures of $N=4$ and $N=6$ transversal red
beams lead to the results gathered in Table~\ref{tab:n=2}. These results
concern the reciprocal unit presented in Figure~\ref{fig:resultsn=2} that
is located at the center of each bridge. 


\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:transverseven}Free body diagram of one of the
central transversal beams in any even-order case $N&gt;2$.}
\end{figure}

\begin{table}
\caption{Computation results of reaction forces on the central
reciprocal unit along with established recurrence formula\label{tab:n=2}}
\begin{tabular}{cccccc}
\thead
&amp; $N=2$ &amp; $N=4$ &amp; $N=6$ &amp; \ldots &amp; $N$ \\ 
\endthead
\raisebox{17pt}{}
$A^N$ &amp; $\displaystyle\frac{F}{4}$ &amp; $\displaystyle\frac{F}{2}$ &amp;
$\displaystyle\frac{3F}{4}$ &amp; \ldots &amp;
$\displaystyle\frac{FN}{8}${\vspace*{7pt}}\\
$B^N$ &amp; $\displaystyle\frac{F}{2}$ &amp; $F$ &amp; $\displaystyle\frac{3F}{2}$
&amp; \ldots &amp; $\displaystyle\frac{FN}{4}$ {\vspace*{4pt}}
\botline
\end{tabular}
\end{table}


Table~\ref{tab:n=2} presents a recurrence formula than is now proven
taking the case $N=2$ as initialization. When increasing the size of
the beam going from order $N$ to $N+2$, the reaction force at contact
point $B^N$ is unchanged while the reaction force at contact point
$A^N$ is modified by the fact that the applied load $\sfrac{F}{2}$ is
moved from this transversal beam to the new central transversal beam.
Consequently, the new reaction force, noted $A'$ is computed as
$A'=A^N+\sfrac{F}{4}$ with sy{mm}etry arguments. The longitudinal
beam where this reaction force $A'$ acts is in a in 3-point bending
setting with central load $B^{N+2}$. Consequently, the reaction force
$B^{N+2}$ is computed{\unskip\break}  as 
{\begin{eqnarray*}
&amp;\displaystyle B^{N+2}= 2 A' = 2 A^N + \frac{F}{2}.
\end{eqnarray*}}\unskip

Finally, the reaction force $A^{N+2}$ can be computed from a
fundamental principle of statics computation on the central transversal
beam. The free body diagram of this beam is presented on
Figure~\ref{fig:transverseven}. This computation leads to
{\begin{eqnarray*}
&amp;\displaystyle A^{N+2}=A^N+\frac{F}{4}.
\end{eqnarray*}}\unskip
Together with the initialization, these
results yield the following recurrence formulae, also presented in
Table~\ref{tab:n=2}: 
{\begin{eqnarray*}
&amp;\displaystyle A^N=\frac{FN}{8} \quad \text{and} \quad B^N=\frac{FN}{4}.
\end{eqnarray*}}\unskip

\subsubsection{{Odd-order case}}

{The study of odd-order number of transversal beams is now presented.}
In this case, the load applied to the structure is a force of intensity
$F$ applied at the mid-span of the central transversal beam.


The calculation is detailed for the case $N = 3$ {which serves as
initialization for the recurrence proof. This case is presented in
Figure~\ref{fig:exmppair}b.} In this case, longitudinal outmost black
and blue beams are first isolated and the fundamental principle of
statics gives the following results{, where $R_{R_i}$ is the reaction
force at contact point between the longitudinal black beam and the
transversal red beam $R_i$. Note that, since the transversal beam $R_2$
is the central beam, $R_{R_2}$ (resp.\ ${\color{blue}R_{R_2}}$)
corresponds to the reaction force at contact point $A^3$ (resp.\ 
$B^3$).}
{\begin{eqnarray*}
\left\{
\begin{array}{@{}l}
\displaystyle    R_{R_2}=A^3=\frac{F}{4} \\
\displaystyle    R_{R_3}=\frac{F}{2} \\[0.5pc]
\displaystyle   {\color{blue}R_{R_1}}=\frac{F}{2} \\
\displaystyle    {\color{blue}R_{R_2}}=B^3=\frac{F}{4}.
\end{array}
\right.
\end{eqnarray*}}\unskip

The red central transversal beam {$R_{2}$} can then be isolated. Its
free body diagram is presented in Figure~\ref{fig:redbeamn=3}. Using
reaction forces $A^3$ and $B^3$, the fundamental principle of statics
determines the last reaction force 
{\begin{eqnarray*}
C^3=F.
\end{eqnarray*}}\unskip

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:redbeamn=3}Free body diagram of one of the central
transversal beams in any odd-order case $N\geq 3$.}
\end{figure}


With this last computation, all three reaction forces of the system are
fully determined at the three contact points between the central
transversal beam {$R_2$} and the longitudinal beams. These results are
su{mm}ed up on Figure~\ref{fig:resultsn=3}.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:resultsn=3}Reaction forces as function of applied
force $F$ on a bridge system constituted of $N=3$ transversal beams.}
\end{figure}

Repeating this process on structures of $N=5$ and $N=7$ transversal red
beams lead to the results gathered in Table~\ref{tab:n=3}. These results
concern the reaction forces presented in Figure~\ref{fig:resultsn=3} that
are located on the central transversal beam of each bridge. 
The recurrence proof for the odd-order case stands on the same basis
as for the even-order case. The reaction force $B^N$ appearing on the
$N+2$ case is modified by the fact that the applied load $F$ is moved
to the new central transversal beam. The new reaction force $B'$ is
computed as $B'=B^N+\sfrac{F}{2}$. The longitudinal beam where this
reaction force $B'$ acts is again in a three-point bending setting with
central load $C^{N+2}$. The, the reaction force $C^{N+2}$ is computed as 
{\begin{eqnarray*}
 C^{N+2}=2B^{N+2}=2B^N+F.
\end{eqnarray*}}\unskip
The two other longitudinal beams acting
on the central transversal beam are in a three-point bending setting
both with central load $C^N$. From this fact, the remaining reaction
forces are computed as 
{\begin{eqnarray*}
&amp; \displaystyle A^{N+2}=B^{N+2}=\frac{C^N}{2}.
\end{eqnarray*}}\unskip
Altogether,
these results yield the following recurrence formulae also referenced
in Table~\ref{tab:n=3}: 
{\begin{eqnarray*}
&amp; \displaystyle A^N=B^N=F\frac{N-1}{8} \quad
C^N=F\frac{N+1}{4}.
\end{eqnarray*}}\unskip

\begin{table}
\caption{Computation results of reaction forces on the central
reciprocal unit along with established recurrence formulae\label{tab:n=3}}
\begin{tabular}{cccccc}
\thead
&amp; $N=3$ &amp; $N=5$ &amp; $N=7$ &amp; \ldots &amp; $N$ \\ 
\endthead
\raisebox{17pt}{}
(1) &amp; $\displaystyle\frac{F}{4}$ &amp; $\displaystyle\frac{F}{2}$ &amp;
$\displaystyle\frac{3F}{4}$ &amp; \ldots &amp;
$F\displaystyle\frac{N-1}{8}${\vspace*{7pt}}\\
(2) &amp; $\displaystyle\frac{F}{4}$ &amp; $\displaystyle\frac{F}{2}$ &amp;
$\displaystyle\frac{3F}{4}$ &amp; \ldots &amp;
$F\displaystyle\frac{N-1}{8}${\vspace*{7pt}}\\
(3) &amp; $F$ &amp; $\displaystyle\frac{3F}{2}$ &amp; $2F$ &amp; \ldots &amp; 
$F\displaystyle\frac{N+1}{4}$ {\vspace*{4pt}}
\botline
\end{tabular}
\end{table}


\subsubsection{Verification of the recurrence formulae}

All the recurrence formulae presented in Tables~\ref{tab:n=2} and
\ref{tab:n=3} have been verified by a full strength of material
computation on two case studies of bridges {of orders 15 and 16.} The
computations provided the following reaction forces {of $A^{16}=2F$ and
$B^{16}=4F$ in the case of the $N=16$ bridge and
$A^{15}=B^{15}=\sfrac{7F}{4}$ and $C^{15}=4F$ in the case of the $N=15$
bridge.} These results confirm the proposed recurrence formulae.

\section{Design rules and experimental validations}\label{sec3}
\subsection{Design rules}

From the previously established recurrence formulae, it is possible to
compute design rules for the Da Vinci bridge. Indeed, from the
establishment of recurrence formulae, it appears that the most loaded
beams in the Da Vinci bridge are the beams located around the center of
the bridge. The longitudinal bars are always loaded in three-point
bending while the transversal bars are in 7 point bending with varying
forces orientations depending on the considered beam. Using this
information, the recurrence formulae and basic strength of material, it
is possible to draw diagrams to determine the most loaded bar of the
whole bridge and thus design accordingly the Da Vinci Bridge.


Considering that the transversal and longitudinal beams can be of
different lengths, denoted $\ell$ and $L$, respectively, and that the
spacing of the longitudinal beams in the transverse $\vx$ direction is
denoted $t$, one can define the following two non-dimensional
quantities: the relative length ratio $\overline{\ell}=\sfrac{L}{\ell}$
and the spacing ratio $e=\sfrac{t}{\ell}$. From these quantities, phase
diagrams can be computed for each bridge size $N$, stating whether the
maximum torque is located in the longitudinal or the transversal beam,
and where it is located in the transversal beam (namely at the midspan
of the beam or on one of the side contact point). Phase diagrams differ
for even-order and odd-order cases. Examples of such phase diagrams are
displayed in Figures~\ref{fig::phaseodd} and~\ref{fig::phaseeven}.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig::phaseodd}Phase diagrams indicating location of
maximum torque for the odd cases $N=3$ and $N=11$.}
\end{figure}

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig::phaseeven}Phase diagrams indicating location of
maximum torque for the even cases $N=4$ and $N=12$.}
\end{figure}


These diagrams are built from the following results: 
\begin{itemize}
\item Maximum torque in the longitudinal beams for the odd case
$M_{\mathrm{max}}=\sfrac{F(N+1)L}{16}$ and the even case
$M_{\mathrm{max}}=\sfrac{FNL}{16}$;
\item Maximum torque on the side contact points of the transversal
beams for the odd case $M_{\mathrm{max}}=\sfrac{3F(N-1)e}{8}$ and for
the even case $M_{\mathrm{max}}=\sfrac{F(4-3N)e}{8}$ or
$M_{\mathrm{max}}=\sfrac{F(2-N)e}{8}$;
\item Maximum torque at the midspan of the transversal beams for the
odd case $M_{\mathrm{max}}=(\sfrac{F\ell}{4})-(\sfrac{F(5+3N)e}{8})$
and for the even case. $M_{\mathrm{max}}=\sfrac{F\ell}{8}$.
\end{itemize}
In the even case, the maximum torque differs on the side contact point
depending on the considered central bar, hence the two possible
expressions presented above.

An attentive reader would notice that the phase diagrams uncover the
existence of a limit spacing ratio {(ratio of spacing between
longitudinal bars to the length of transversal bars)} below which the
maximum torque is always located in the longitudinal beam; and more
precisely at the midspan of this beam. Moreover, it seems like the
maximum torque can be located on the side contact point of the
transversal bar only for bridges composed of enough reciprocal units.

\subsection{Experimental validation}\label{sec:experiments}


The presented computations have been validated by testing reduced size
Da Vinci bridges built from circular cross-section wooden beams. These
wooden beams have first been tested in three point-bending on an
Instron test machine to determine their limit stress. The wooden beams
have a cross-section diameter of 8~{mm}. The experiments have
been carried out on five specimens leading to a mean limit stress
$\sigma_l\approx140~\mathrm{MPa}$ with standard deviation of
25~{MPa}. It is believed that the high standard deviation is due
to the poor quality of the wood composing the beams.


The tested bridge was simply supported by two tables and a load was
applied by means of a carabiner at the center of the bridge. The bridge
structure was of {order $N=3$ (thus composed of $3$ transversal
beams).} An image of the test setup is presented in
Figure~\ref{fig:testbridge}. Longitudinal beams had a length of 30~{cm}
while transversal beam were 15~{cm} long, thus the
relative length ratio was chosen to be
$\overline{\ell}=\sfrac{L}{\ell}=2$. As for the spacing between the
longitudinal bars, it was minimal and thus corresponded to the diameter
of the longitudinal beams cross section. Consequently, the spacing
ratio was set to be $e=\sfrac{t}{\ell}\approx0.053$. According to the
phase diagram presented in Figure~\ref{fig::phaseodd}, these values
predict a failure of the longitudinal beam at its midspan with a
maximum force of 93~{N} corresponding to a total loading weight
of 9.5~{kg}. In the experiments, a medium weight of
11~{kg} was used to reach the failure of one of the central
longitudinal beam at its midspan, with a standard deviation of
1.5~{kg}. Both standard deviation of the structural and the
material experiment are coincident which lead us to validate the
predictions of the presented theoretical analysis.

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:testbridge}Image of the test setup in the case of a
bridge composed by square cross-section wooden beams.}
\end{figure}

\section{Conclusion}

In this paper, {recurrence formulae are established} that link the
number of transversal beams {of the classical Da Vinci bridge} to the
{reaction} forces on its central {transversal beam. These recurrence
formulae are} established for two bridge configurations depending on
the parity of the number of transversal beams composing the bridge.
Using the recurrence formulae, phase diagrams are established that
predict the location of the maximum torque in the beams composing the
bridge. These results are validated by experimental realization of a
reduced wooden {Da Vinci} bridge composed of three transversal beams.

The main result of this article being the recurrence formulae, one
could wonder if this recurring effect could be generalized to all
reciprocal systems. Indeed, these recurrence formulae state that, in
order to span a long distance and still hold a constant weight, an
optimum must be found between using a large number of short
longitudinal beams which would lead to an increase in the contact
forces between the beams and using long longitudinal beams submitted to
three point bending which could lead to large internal torque in these
beams. Could this result be generalized to the design of a reciprocal
floor? To which extent this result is limited by the use of strength of
material hypothesis? These questions still remains unanswered to the
authors and would merit further analysis probably including numerical
simulations.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that  could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Dedication}

The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Acknowledgments}

Authors wish to acknowledge the Military Academy of Saint-Cyr and the
Centre de Recherche de Saint-Cyr Co\"{e}tquidan for their material and
financial support.

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