Remarkable historic timber roofs. Knowledge and conservation practice Part 2 - Investigation, analysis, and interventions

The structural solution called chevrons formant ferme is a typical invention of the French Gothic age. In this paper, the structural functioning of such kind of structures is considered and analyzed, referring to the two original different structures of the roof of Notre-Dame of Paris, destroyed by the fire of 2019. The results show the incredible skill of the builders of the Middle Ages in designing very effective timber structures and how these structures were conceived to respond to criteria of different nature.


INTRODUCTION
The unexpected birth and rapid explosion of what we call, from the Renaissance, Gothic architecture, is an unprecedented architectural phenomenon that appeared in the North of France during the 12th century, so huge and wide that, according to Gimpel [1], during less than three centuries, the French builders carried and used more stones than the Egyptians during the whole period of their civilization.The typical Gothic construction is the cathedral that, beyond the peculiarities of the Gothic structural elements, like flying buttresses, pointed arches, and rib vaults, is characterized by its dimensions: a Gothic cathedral is a high and large stone building.
What is often forgotten in the description and architectural studies of the Gothic cathedrals is that these buildings were covered by a timber structure.Though not as apparent and visible as the stone body of a cathedral, these timber constructions, in French, the charpentes, are impressive constructions, often having huge dimensions.This fact, along with other constraints of different natures, detailed below, forced the carpenters of the Middle Ages to invent innovative structural solutions, peculiar to this period and geographical area, that time proved to be very effective.
The true invention of the carpenters of the Gothic period is what in French is called the charpentes à chevrons formant ferme.This expression, whose translation could be "a carpentry with rafters forming common frames", indicates a particular structural solution that appeared in the French region around Paris during the first part of the 13th century.
Though this kind of structure has been studied extensively in the past by historians of architecture, rather curiously, no serious and complete structural studies of these constructions seem to exist in the literature.This fact has probably contributed to creating misunderstandings and false ideas about the structural functioning of these structures.
A detailed study of two roofing structures with chevrons formant ferme has been done recently on the two original charpentes of the choir and of the nave that covered Notre-Dame of Paris and were destroyed by the fire that occurred on April the 15th, 2019 [2].
Of course, the static scheme of the main and common frame, as well as of the bracing system, can change, each roofing structure being a peculiar case, but all of them share the same fundamental characteristics of having a main frame and some common frames that are placed at a short distance (say, of the order of 80÷100 cm) and by the absence of purlins.The rafters forming the common frames are not supported by purlins but are a part of the whole structure.The fundamental difference between the main and common frames is the absence, in these last, of the tie-beam: only collar ties are present in the common frames.This is a key point of the structural functioning of chevrons formant ferme charpentes.The reason for this choice made by the carpenters of the Middle Ages is analyzed below.Still, since now, it should be kept in mind that these choices (the absence of the tie-beam in the common frames and that of the purlins, with the consequence that the rafters are not carried but self-supporting) completely characterize this kind of structure: the rafters of the common frames actually form, with their collar ties, a (secondary) frame, whence the name of charpente à chevrons formant ferme.The whole system is then typically characterized by rafters (of the main and common frames) that are close together, which allows posing the wooden decking (the voligeage) directly on the structure, unlike in the structures composed of main frames, purlins, and rafters, see, e.g.[3,4], that as a consequence have a higher global thickness of the covering structure.
The objective of the present paper is to try to give a reconstruction of the structural ideas of the Gothic builders, using to this end the results of the cited paper, i.e., considering the destroyed roof of Notre-Dame as a representative paradigm of a chevrons formant ferme roofing structure.The indications given by the structural calculations show some facts describing clearly the statics of the timber structure and, rather likely, the structural thought of the builders.From these facts, it is also evident that some ideas carried on by architecture historians are inaccurate.All these points are discussed below.
Finally, this research aims to go beyond the mere descriptive analyses done so far and try to shed light on its real static behavior, how, presumably, it was thought by its ancient master-builders.In some way, it is an attempt to retrace the constructional thinking of the Gothics, their ideas in designing their charpentes, and to check whether or not some of the more common ideas on this matter are sound.
The paper is so structured: first, a description of a chevrons formant ferme structure is given, with some examples.Then, the structural analysis results on the destroyed charpentes of Notre-Dame are recalled and analyzed.On the basis of these results, we try, on one side, to refute or confirm some ideas of the past about this kind of structure and then to try to respond to the most delicate question: why the carpenters of the gothic age invented such a sophisticated and innovative structure?

DESCRIPTION OF A CHEVRONS FORMANT FERME ROOFING STRUCTURE
A covering structure with chevrons formant ferme is basically composed of four main structural elements, see Fig. 1: the main frame, in French ferme principale or chevron maître, the common frames (fermes secondaires or fermettes or chevrons), the bracing system (contreventement) and the wall plates (sablières).All these elements work together to carry the vertical (dead load) and horizontal (wind loads) actions.This modular unit, composed of the main frames and a few common frames, is then repeated to form the whole roofing structure.Usually, the charpente, so constituted, was covered by a wooden decking (the voligeage) and by tiles or lead plates.The correspondence between English and French technical terms is given in Tab. 1.
Fig. 1.The modular unit of a charpente with chevrons formant ferme; in blue, the main frame (chevron maître); in grey, the common frames (fermettes); in green, the bracing system and in brown, the wall plates (sablières).
e-ISSN 2421-4574 nal structure relying together the frames is schematically presented too.The parts of the charpente object of this analysis are those between frames FC4 and FC9 for the choir, and between FN1 and FN11, for the nave.They correspond to the regular part of the medieval charpente, the rest being the part of the structure constituting the apse of the choir that Lassus and Viollet-le-Duc reconstructed during the 19th century.Above the frames was a wooden decking, the voligeage, which was supporting the lead plates, nailed on it.This structural organization of the charpente gives it a three-dimensional structural functioning, which is not the case for the charpentes with main frames, purlins, and rafters.This is a fundamental aspect of the charpentes à chevrons formant ferme, which is detailed below.Before, a brief description of the charpente of Notre-Dame , used as a paradigm for the structural analysis, is given in the next section.

THE ANCIENT CHARPENTES OF NOTRE-DAME
The roof of Notre-Dame of Paris that burnt in the fire of April the 15th, 2019, was composed of three distinct charpentes, built at different periods ( [5,6] and Fig. 2): the choir charpente, made after 1220, probably from 1225 to 1230; the nave charpente, slightly subsequent, presumably built from 1230 to 1240; the transept charpente, entirely rebuilt during the restoration work of Lassus and Viollet-le-Duc after 1843, along with the spire and the first frames of the nave and choir nearby the spire.The charpente built in the 19th century was of the type with main frames, purlins, and rafters and will not be considered here.
The whole timber structure (also named the combles in French) had an overall length of 115.6 m, was 13 m wide, and was 9.75 m high.The two charpentes of the nave and choir were both à chevrons formant ferme but with different schemes.This fact, considered in detail below, suggests an evolution in the constructive thinking of the ancient carpenters [6,8].The longitudinal scheme of the combles is presented in Fig. 3a.The three parts of the charpente are clearly indicated as well as the notation of the main frames, as usually adopted.The longitudi-Fig. 2. Views of the combles: (a) in the choir (image source: [7]); (b) in the nave (image source: [8]); (c) in the transept (photo by the Author).the nave's charpente presents minor differences in some frames.This is probably due to maintenance operations, done or not done, during the centuries.Here, being interested in analyzing the static functioning of the structure, we will consider just the modular structural unit around FN7, considered the most representative frame for the nave's charpente.
Though Dubu [10] said that the charpente was made with chestnut wood, it is certain that it was realized with the wood of oak trees, while the voligeage was made of fir wood, cf.[9], [8].The number of trees employed for the structure was vast, so the charpente was called la forêt, the forest.According to F. Épaud, who In the choir of Notre-Dame , the modular unit was composed of one main frame and four common frames, spaced ~82 cm, for a whole length of ~4.1 m.In the nave, this distance has been reduced to ~3.5 m, with frames spaced ~75 cm.The main frames of the choir and nave of Notre-Dame , as well as the model of a fermette, which is sensibly the same in the two cases, are shown in Fig. 3b-d.These schemes have been reconstructed using mainly [9], who made the first dendrochronological analysis of the charpente in her MSc thesis, and [7].In the same figure, the French names for the different pieces of the structure are also indicated.Unlike the choir's charpente, relatively homogeneous from FC4 to FC9,  uniquely, and are still a matter of historical debate today, cf.[12][13][14][15].What is important for the purposes of the present study is that these transformations necessarily forced the carpenters to adopt a structural scheme different from the previous one.This is a crucial point to be analyzed below.

THE MECHANICAL MODELS
The charpentes of the choir and the nave are studied through a Finite element (FE) analysis.Each of the two structures is modeled as a truss, i.e., as an assembly of elastic rods pinned at the ends and each intermediate intersection with another beam.This choice is motivated by, on the one hand, the great uncertainties that exist in the ancient charpentes, where the gaps between the timber struts can be rather crucial because of the long drying of the wood pieces (as shown by the research of F. Épaud [16], the charpentes of the 12th and 13th centuries were realized with green wood), which implies a practically null couple at the joints, see also [17].On the other hand, this assumption does not substantially affect the general structural functioning of the charpente, which is the study's objective.
The scheme of the main frames of the two charpentes and the common frames are shown in Fig. 4. The support points are denoted by the labels S1 to S10; all of them are modeled as frictionless unilateral supports, i.e., able has deeply studied the timber structures of the Middle Ages, it was composed of the wood of about 1000 oaks, almost all of them with a diameter of ~25 to 30 cm and 12 m high, a small part with a diameter of ~50 cm and 15 m high, corresponding to about 3 hectares of forest.

BRIEF HISTORICAL ACCOUNT OF THE NOTRE-DAME CHARPENTES
The first choir's charpente was probably finished before 1182 ( [11], page 16; [12]) when the choir was consecrated.Subsequently, a new charpente was erected: on the choir between 1225 and 1230 and in the nave between 1230 and 1240 [5].These new charpentes reused some timber beams of the original roof; in fact, several pieces of them showed unused mortices or mi-bois notches, a clear sign of reuse [11].
The reconstruction of the charpente was the consequence of a set of changes made to the cathedral during its construction.In particular, the guttering wall, i.e., the upper part of the clerestory, was raised about 2.70 m above its original height, a fact that had several consequences on the structure of the new charpente, as discussed below.
The reasons for this reconstruction are not well known, and historians still debate on this point: according to [13], the changes were done just for a matter of style, while [14] suggest that they were made to improve the structural response.Whatever the reasons for these modifications on the still unfinished cathedral, they are significant, do not concern the charpente  The dimensions of the beams composing the charpente are reported in Tab.2; the minimum diameter d min of a trunk to obtain the corresponding cross section is also indicated.As observed in [8], most parts of the charpentes can be obtained by trunks with a moderate diameter, less than ~30÷35 cm.A thickness of 2 cm for the voligeage, made of fir wood (density ~500 kg/m 3 ), has been considered.For the catwalk, the mass has been evaluated to ~240 kg for each structural unit of both the charpentes.The global volumes and masses of wood for each structural unit are summarized in Tab. 3.
The data above show that the quantity of wood is practically the same for the two charpentes, though the total mass per unit length is ~19% greater for the nave's charpente.In consideration of these data, what can be said is that the change of the structural scheme was not dictated by economical issues.The results of the structural analysis, shown below, suggest another possible reason: the carpenters of the 13th century probably searched for a better structural response.In fact, the main frame of the nave has an improved mechanical behavior than that of the choir; in short, it needs less wood to obtain the same stiffness: the nave's main frame is lighter than the choir's one.However, the nave's charpente unit is heavier than that of the choir to exert only a purely contact reaction.The question of frictionless contact is examined below in section 3.5.In particular, the support points S3 to S10 can exert only upward vertical forces, while in S1 and S2, two unilateral reactions are exerted, an inward horizontal one and a vertical upward one.It is important to notice that as a consequence of the deformations produced by the loads, some of the support points can detach from their footing.The two structural units, the choir and the nave, are modeled through three-dimensional structural schemes shown in Fig. 5.The sablières are wall plates running from one main frame to another; they are just posed onto the top of the guttering walls and transmit the horizontal forces from the common to the main frames.They are hence bent in the horizontal plane.Each beam of the FE model is modeled as an Euler-Bernoulli rod.The boundary conditions imposed on the points at the ends of the liernes and sablières specify the continuity of displacements and rotations with the corresponding elements of the adjacent structural units.In this way, the simulation done on a singular structural unit represents the global structural response of the charpente for each one of its parts, in the assumption of uniform loading all over the charpente, which is the case for the own weight and, at least to a first approximation, for the wind action.ficient to completely characterize a cylindrically orthotropic material like oak wood [20].In particular, the data taken from EN338 correspond to the wood class D50.Considering this insufficient data, the wood has been modeled as an isotropic material with Young's modulus E = 12500 MPa and a Poisson's ratio ν equal to 0.25.Because the structure is almost exclusively solicited by axial forces and bending moments, this choice does not substantially affect the way the structure works.For what concerns the density ρ, the value of 710 kg/m 3 , generally accepted for dry oak wood, has been taken.because of the greater weight of the bracing system and of the common frames of the nave.

MATERIAL CHARACTERISTICS
The mechanical characteristics of oak wood are not constant; they change with the state of the wood, green or dry, and with the plants: as a matter of fact, a certain degree of uncertainty is unavoidable.In order to fix the necessary material parameters, the scientific publication of CIRAD [18] and the European norm EN338 [19] have been used.However, these data are not suf- Tab. 2. Dimensions of the wood beams, as deduced from [7]; for each section, b is the width, h its height, d min is the minimum diameter of the trunk, A the area, J 1 and J 2 the moments of inertia, J 0 the polar moment of inertia and µ the linear density of mass.
and distributed on the windward (overpressure) and leeward (suction) sides of the roof.The actions applied to the charpente are sketched in Fig. 6, and the values of the loads are detailed in Tab. 4.

LOADING CONDITIONS
Two loading conditions have been considered: own weight and own weight plus the wind.Following the European norm EUROCODE 1 [21], the wind action has been modeled as a static load, orthogonal to the surface,   the choir, nodes S3 on the North side are more charged, while nodes S4, S5, S8, and S10 are inactive; nodes S6, S7 and S9 are also active, but with a reaction force far below that of S3 and node S1 is charged only horizontally, to entirely absorb the wind thrust, while S2 is charged in the vertical direction, as an effect of the roof's slope; • in the nave, nodes S2, S4, S6, S8 and S10 are inactive, while S1, S3, S5, S7 and S9 active; to remark the high reaction at node S5, consequence of the structural scheme; • the slope of the roof, ~55°, ensures a stabilizing moment of the wind force distribution w 1 on the windward side, Fig. 6, which explains the positive reaction in nodes S2, S6, and S9 of the choir and the high reaction at node S5 of the nave's charpente; • for both the charpentes and load conditions, the distribution of the reactions on the top of the guttering walls is far from being uniform: the largest part of the load is transmitted by the main frames, which play a fundamental role in the structural functioning of the charpente.
In Tab. 5, the resultant of the reactions are also indicated.This allows an estimation of the global loads and weights, which are summarized in Tab. 6.

DISPLACEMENT FIELD
The deformation of the modular structural units of the two charpentes is represented in Fig. 8; the values of the displacements of nodes A and B in Fig. 6 are shown in Tab. 7. In all the cases, the magnitude of the displacements is minimal.To remark that for the nave's charpente submitted to only vertical loads, the vertical displacement of point B, in correspondence with the middle of the tie-beam, is only 56% of the same displacement for the choir's structure.Moreover, the horizontal displacement of point A, the frame's top, of the nave is 44% of that of the choir.These data allow assessing the difference in stiffness of the nave and choir's charpentes; see section 3.6.

RESULTS OF THE NUMERICAL SIMULATIONS
The results of the FE simulations on the two charpentes are presented below.For understanding the structural functioning of the structure, especially in view of a critical analysis of the existing literature on the subject, introduced further, the following outputs of the numerical simulations have been considered: the distribution of the reaction forces, the displacements, the stresses, the support conditions, and in particular the role played by friction contact, and a modal analysis used to have an appraisal of the global stiffness of the structure.All these aspects are presented hereafter.

REACTION FORCES
The reaction forces at all the support points, S1 to S6 for the main frame F1, S7 to S10 for the secondary frames F2, F3 and F4, F5 (cf.Figs. 4 and 5), are detailed in Tab. 5 for the two charpentes and loading conditions.These reactions are also represented in Fig. 7. Observing these data and figures, the following remarks can be made: • the distribution of the reactions for the loading condition OW (own weight) is almost, but not exactly, symmetric between the South and North sides; this is the consequence of small asymmetries in the main frames structure; • still, for the OW loading condition, some of the contact forces are null, e.g., for the supports S7 and S10 of the choir's fermettes, or S7 for the F4 and F4 fermettes and S9 for the F2 and F3 fermettes of the nave's charpente; • the vertical reactions in S1 and S2, i.e., at the level of the corbels supporting the jambes, are far less than that absorbed by nodes S3 to S6, on the guttering wall's top, for the choir's charpente, while it is higher in the nave, due to the different arrangement of the structure; • for the loading condition OW+W (Own Weight + Wind), the jambes play a major role and strongly affect the distribution of the reactions; in particular, the set of supporting nodes changes: in whether or not the material is still in the elastic range [20].Because wood has a brittle behavior and a different strength in tension and compression, the Hoffman criterion has been used for checking the strength of the structure [22].By this criterion, for a plane state of stress, which in the case of bent beams, the matter is still in the elastic range if the failure index F H , defined as

STRESSES
In Tabs.8 and 9, the worst combination of internal actions, i.e., the one causing the highest stress value, is given for each different type of beam section, cf.Tab. 2. N is the axial force, positive when tension, M 1 and M 2 are the bending moments, T the shear force, σ max a,d σ min the highest and lowest values of the normal stress in the section, τmax the maximum of the shear stress.The wood being transversely isotropic, a strength criterion for anisotropic materials must be used to check The characteristic values of X t , X c , and S have been chosen once more according to [18] and the European norm EN338, [19]: X t = 30 MPa, X c = 29 MPa, S = 4 MPa.The results for F H sup are shown in Tabs.8, 9: F H sup << 1 in all the cases.In small words, the structure of both the charpentes is very feebly stressed: the matter is everywhere far below the elastic limit state.The most stressed pieces are the jambes on the leeward side under the action of the wind: F H sup = 0.381 for the choir's charpente and 0.496 for nave's one.The entraits are less stressed: the system of the intermediary supports is effective in reducing its bending.
is not greater than one.In the case of beams, σ yy = 0; moreover, in order to simplify the calculation, for each beam, we calculate F H for σ xx = {σ max , σ min } and σ xy = τ max as reported in Tabs.8 and 9, though, these stress values generally do not occur at the same point of the same beam.In this way, we obtain, for each beam, an upper bound F H sup for F H , and the beam is in the elastic range if  the nodes S1 to S10, a fixed support has been considered to simulate a perfect contact, i.e., a support with sufficient friction to stop any sliding.This is the condition implicitly understood when the horizontal thrust of the charpente on the top of the guttering walls is supposed to exist.The new simulation results are detailed in Tab. 10, where H is the total horizontal thrust of the charpente and V is the total vertical reaction for each structural unit applied to the top of the guttering walls.

ABOUT THE TRANSMISSION BY THE FRICTION OF THE HORIZONTAL FORCES
The results presented in the previous Sections have been found in the assumption that the horizontal forces cannot be taken up by friction.We ponder now on whether this assumption is or is not correct.To this end, a new simulation, considering the own weight uniquely, has been done, with changed boundary conditions: for each one of greater than 50 cm for all the cases, are extremely high and should cause the overturning of the wall.Also, if the wall had a greater thickness, the eccentricity should be too large to ensure a safe equilibrium of the system charpente-guttering walls.We can hence check the physical possibility for the system to develop effective friction forces.The values of the horizontal, R x , and vertical R y , contact forces on the top of the guttering walls for the model with fixed bilateral supports are shown in Tab.11.If we consider a friction coefficient ν = 0.7 for the contact between wood and stone, value usually admitted in such a case for a dry contact, then it appears clearly that the ratio R x /R y exceeds ν in several cases.At the same time, it is close to it in other cases.In addition, the contact between the wood of the charpente and the stone of the guttering walls is probably far from being perfect: infiltrations of dust and rainwater cannot be excluded, especially if one considers that the contact is actually unilateral and that, as shown in the numerical simulations presented in the section 3.4.1,some nodes of the charpente can slightly lift up under the action of the loads (cf.Fig. 8).It should also be mentioned that, as the same word sablières indicates (sable is the French word for sand), these wall plates were put in place over a layer of sand, to better ensure the contact between the charpente and the wall's top.Of course, this greatly decreases the friction forces, and the builders of the Middle Ages were likely aware of that.We check the global equilibrium of the guttering wall 2.70 m below its top, i.e., at the same level as the corbels supporting the timber consoles of the charpente.This was the level of the top of the clerestory walls before the modifications started around 1220, and it is, to a good approximation, the free-standing height of the guttering walls.If we consider the thickness of the guttering walls of 60 cm and the density of the limestone of 2,400 kg/m 3 , the weight V w of this part of the guttering wall is ~160,000 N for the choir (length of 4.1 m) and of ~136,100 N for the nave (length of 3.5 m).The total vertical load V t at the level -2.70 m with respect to the top of the wall can hence be calculated, as well as the overturning moment M of the horizontal thrust, and finally, the eccentricity e of V t with respect to the centroid of the wall's section, cf.Tab. 10.The values of the eccentricity e so calculated,

ANALYSIS OF THE RESULTS AND THEIR INTERPRETATION
We now ponder the structural functioning of a charpente à chevrons formant ferme.We start with an analysis of the results of the structural study presented above.Then, on this basis, we try to draw some conclusions on the structural thought of the carpenters of the Gothic Age in France and, at the same time, compare these conclusions with the ideas and hypotheses existing in the literature to confirm or refute them.

THE MAIN RESULTS OF THE STRUCTURAL ANALYSES
The main points arising from the structural analyses presented above are: • an intense concentration of the vertical reactions in correspondence with the main frames, while the vertical forces in correspondence with the supports of the common frames are much lower; • the impossibility of transferring the horizontal forces from the charpente to the stone structure Finally, the transmission, by friction, of horizontal forces between the charpente and the stone structure should be not only dangerous for the structure's safety but also rather uncertain or even impossible, physically speaking.

MODAL ANALYSIS OF THE CHARPENTE
In order to assess the difference in the structural behavior, namely in terms of stiffness, between the charpentes of the nave and of the choir, a modal analysis of the two charpentes has been performed.In fact, because the weight of the structural units of the two charpentes is practically the same (cf.Tab.6), the higher the frequency, the higher the stiffness.For this analysis, the FE model is the same one used in the previous section for checking the friction mechanism.The first five modes of the choir's and nave's structures are presented in Fig. 9, where the corresponding frequencies are also indicated.What is apparent is that the nave's charpente has a greater stiffness than the choir's.In fact, if we compare the fundamental frequencies, mode 1, we can observe that the frequency of the nave's charpente is ~2.5 times that of the choir's one.This clearly indicates a better structural conception of the nave's charpente with respect to that of the choir.In [23], some similar considerations are done, but with some differences.Viollet-le-Duc implicitly admits the role played by the system blochets-sablières, but he imputes its adoption for another reason.According to him, the invention of the charpentes with chevrons formant ferme with such important slopes served to adapt the roofing structure to the reduction of the thickness of the walls of the Gothic cathedrals, with respect to the Romanesque architecture, which rendered challenging to pose a charpente with main frames, purlins, and rafters.The only structural consideration made by Viollet-le-Duc concerns the high value of the slopes that decreases the bending of the rafters, so allowing the use of wood beams of relatively small cross sections (an important point, considered further): he never gives an explanation of the global functioning of the charpente, though it seems probable that he understood the danger of the transmission of the horizontal thrust from the common frames to the top of the guttering walls by friction.
The point of view of [24] is different: the increase of the roofs' slope in the Gothic period is essentially used to decrease the horizontal thrust.This interpretation cannot be considered correct, cf.Sect.3.5.Also, the point of view of Pol Abraham [25], as reported in [3], is mechanically wrong: according to him, to explain the deformations of the Gothic vaults, he implicitly admits that there is a transfer, by friction, of the horizontal thrust of the charpente to the top of the guttering walls.The results presented above clearly indicate that this is not possible.The point of view of [3] is ambiguous: on the one hand, he strongly supports the idea of Viollet-le-Duc of the increase of the roofs' slope to use trunks of small diameters; on the other hand, he does not make any real structural consideration and tacitly, in the end, he seems to accept the point of view of Pol Abraham.
• Distribution of the loads: a common idea about the charpentes with chevrons formant ferme is that this structural scheme allowed an almost uniform distribution of the loads on the top of the guttering walls.As demonstrated in section 3.4.1,this is far from reality: the loads transferred by the charpente to the guttering walls are unevenly distributed, much higher in correspondence with the main frames than in the secondary ones.The below by friction and the fact that such a mechanism of transfer could cause the overturning of the guttering wall; • the low levels of stress everywhere in the charpente; • the slight deformation of the structure; • the better design of the main frame of the nave with respect to that of the choir.

CRITICAL ANALYSIS OF THE RESULTS
Based on these results, the following conclusions can be drawn about the structural functioning of a charpente à chevrons formant ferme.
• A three-dimensional structural functioning: the structural behavior of a charpente with chevrons formant fermes is rather complicated and cannot be reduced to a simple planar scheme.The fundamental difference between the main and common frames is the tie-beam, used only for the main frames.Consequently, the horizontal thrust at the base of each secondary frame needs to be equilibrated differently; otherwise, the bending of the rafters would severely deform the frame.This is done by the system composed of the sablières and the blochets, see Fig. 3d.At the base of any rafter of a secondary frame, a blochet transfers the forces from the rafter to the sablières, restrained by the tie-beams of two successive main frames.The sablières are hence bent and sheared, in the horizontal plane, by the thrust at the base of the rafters of the secondary frames, and in this way, they transfer these horizontal thrusts, caused by the vertical loads, to the tie-beam of the main frames.Any modular structural unit of the charpente is hence self-equilibrated in the horizontal plane and does not apply any horizontal thrust to the stone structure below: a charpente with chevrons formant ferme is designed to transmit only vertical forces to the top of the guttering wall, while the horizontal thrusts engendered by the vertical loads are self-equilibrated by the system composed by the rafters, blochets, sablières, and tie-beams of the main frames.In particular, friction is not the mechanism of transfer of the horizontal thrust of the secondary frames to the top of the guttering walls.
• The wind thrust: placed above such high constructions, the Gothic builders certainly did not ignore that the wind forces on the roof of a cathedral acted upon the stone structure below with a considerable thrust; also, as seen above, they knew that this thrust was impossible to be counterbalanced by friction, see section 3.5.There are few studies on how Gothic cathedrals, and their roofing structures, withstand the wind forces; namely, for what concerns Notre-Dame [14], with a historical perspective or [27] with nonlinear analysis.The carpenters of Notre-Dame invented an effective system to transfer the wind action to the underlying stone structure: it is composed of the brackets (jambes) of the main frames with their windbraces (aisseliers) and chapeau, Fig. 10; in [9] this set of struts is indicated as the console.It is usually said that the console is used as a vertical support of the main frame or also for relieving the bending of the tie-beam.These ideas are not likely: the support given by the sablières is largely sufficient, and it should be easier and less expensive to use more posts for decreasing the bending of the tie-beam.In addition, observing the main frames of the nave, we can see that the console should support the entrait just where there is already a vertical tie, which is redundant.For the choir's frames, the span of the entrait is divided into three parts, which is largely sufficient to relieve it in bending, as confirmed by the numerical simulations presented in section 3.4.3.Actually, the true reason that led the carpenters of the 13th century to use the system of the console was to transfer the wind force to the lower part of the guttering wall: the horizontal thrust flows as an inclined force to the bottom of the guttering wall through the inclined aisseliers and the stone corbels close to the vault level, so improving the strength of the clerestory structure to the action of the wind considerably.This is also corroborated by the presence of a strong shear key connecting the tie-beam and the console, Fig. 10, whose role is to transfer the wind thrust from the tie-beam to the console, while it is completely useless for transmitting vertical forces.It is likely that the only reason that pushed the carpenters of the cathedral to introduce the consoles was to dispose of an excellent device to safely transmit the wind action on the roof to the stone structure below.reason for that is exactly the global structural organization that the carpenters gave to the charpente, i.e., its three-dimensional functioning.
Probably inspired by a rough two-dimensional analysis of the structure or also, perhaps, suggested by ideological positions, both of them far from the physical reality, the idea that the chevrons formant fermes were used to distribute the vertical load almost uniformly on the top of the clerestory walls does not correspond to reality.Moreover, we can affirm that the Middle Ages carpenters did not consider the correspondence of load-points for the charpente and the stone structure beneath, simply because they never coincide.In [26], also cited by [3], this discrepancy is severely criticized, based, on the one hand, upon a rather ideological point of view and, on the other hand, on a static idea that cannot be considered as valid, that of a structure that functions as an array of planar independent frames.In addition, the idea that the carpenters on one side and the masons of the stone structure on the other one worked separately, without interacting, is likely to be false: the stone corbels put at the base of the guttering walls exactly in correspondence with the main frames indicate that actually the wall construction of the cathedral was well planned and also conceived for the roofing structure.ture's equilibrium without the bracing system.There are three bracing systems in the choir of Notre-Dame's combles; in the nave, they are five (cf.Fig. 5).
• The structural differences between the two charpentes: if the differences between the two charpentes are attentively considered, it appears that all the structural action of the nave's designer is oriented to increase the stiffness, and so the stability of the charpente: all the structural changes made with respect to the choir, i.e., the evolution of static scheme for the main frame, the reinforcement of the bracing system and the greater sections used for the fermettes, go in the direction of a stiffer structure.Actually, it is not the strength that is substantially improved because the level of stresses in the two charpentes is similar, Sect.3.4.3,but the structure's stiffness, as confirmed by a comparative modal analysis, section 3.6.While a stress analysis was, without doubt, out of the means of the builders of the Middles Ages (the concept of stress was introduced by Cauchy in the 19th century [28]), an embryonic perception of the stability and hence of the stiffness, of a structure can have been in the abilities of the Gothic carpenters.It can be acquired through experience, especially during the construction phases.Thanks to this experience and ability, some particularly wise carpenters can have improved the technique, like in the case of Notre-Dame.Moreover, nowhere in the combles did the stresses reach important values: all the structure was scarcely solicited.Hence, all seem to indicate that the Gothics were mainly guided by increasing the structure's stiffness.Anyway, the change of the static scheme from the choir's charpente to the nave's one was certainly not dictated by economical reasons because the mass of wood for unit length along the longitudinal axis is greater for the nave.
• The global structural functioning: from what was said above, the global structural functioning of a charpente with chevrons formant ferme emerges clearly: the main frames collect and absorb, through the sablières, all the horizontal thrust caused by the vertical loads so that no horizontal action is conferred to the top of the guttering walls.Through the bracing system, the main frames also take on the part of the vertical load of the The system of the consoles is also useful for another reason: the charpente of a Gothic cathedral was, typically, built before the construction of the high vault, see, e.g.[3].During this phase, the charpente was essential to counterbalance the flying-buttresses thrust, ensuring the connection between the two sides of the clerestory before the construction of the vault.The consoles well assumed this structural role: the tie-beams equipped with the two consoles could balance the inward thrust applied to the two opposite clerestory walls by the flying-buttresses.In the case of Notre-Dame, after the rise of the guttering wall, the top of the flying-buttresses was too far below the charpente to assume such a kind of thrust balance without a device, the consoles, acting down below the charpente.
• The bracing system: it is generally thought that a system composed of longitudinal beams (the girts) and braces placed at different levels was exclusively used by the carpenters as a bracing system.Indeed, the bracing system was important, especially during the constructive phases, to ensure the global stability of the charpente and, once built, to withstand longitudinal horizontal forces (though this is also provided, and more effectively, by the intersection of the nave/choir and transept charpentes; also the voligeage, acting as a sort of a plate connected to the frames, certainly contributed to the longitudinal stiffness of the structure).However, its true role was another one: the bracing system was used by the Gothic carpenters to relieve the bending of the collar ties (faux entraits) of the common frames and to transfer an important part of the vertical load from the secondary frames to the main ones.The numerical simulations (cf.section 3.4.1)confirm the effectiveness of the bracing system in transferring the vertical loads to the main frames, and, most importantly, it also helps in decreasing the bending in the rafters of the fermettes and the sablières because also the horizontal outward thrust at the base of the rafters is diminished by such a structural organization.Also, thanks to the bracing system, the carpenters did not need to use vertical ties in the fermettes to sustain the faux entraits.In short, the overall behavior of the charpente, conferred by the set of main and common frames and sablières, was insufficient to ensure the struc-TEMA: Technologies Engineering Materials Architecture 28 e-ISSN 2421-4574 ic scheme different from the scissor truss, but they were also faced to the problem of scarcity of trunk of sufficient dimensions (it is famous the adventurous pilgrimage of Abbot Suger to find trees of adequate size for the roof of the Royal Abbey of Saint-Denis), see [3].
This was the true problem for the carpenters of the period: to dispose of wooden pieces of sufficiently large dimensions.In particular, the use of long entraits (in Notre-Dame, they have a length of 13 m) must dispose of beams of a significant section to withstand the rod's bending.This was not so easy in the France of the period, and the use of a charpente with few entraits was hence almost compulsory.So, to solve a structural problem (efficient roofing structures of large dimensions) with the scarcity of sufficiently large timber beams, the carpenters of the beginning of the 13th century invented the charpente with chevrons formant ferme.Other great cathedrals were covered with this type of structure, among the still-existing ones Amiens [29] and Bourges [30].With this solution, they brilliantly solved not only the problem of scarcity of sufficiently large wooden beams but transformed the statical problem, adopting a solution that allows eliminating the horizontal thrust at the top of the guttering walls, which are too high to withstand such forces.This solution witnessed a radical change in the structural thought of the carpenters of the period: they were able to pass from a bi-to a three-dimensional scheme, where all the parts of the structure interact together.In the end, we can appreciate how deep and subtle was the structural knowledge of the master builders of the Middle Ages.secondary frames that otherwise should be too much solicited in bending, like the sablières.The wind thrust is not transferred to the stone structure by friction on the wall's top, which could cause the tipping of these ones at their base, but by the system of the consoles and uniquely in correspondence of the main frames, the wind thrust is passed to the stone structure at the base of the wall on the leeward side as an inclined force.Along with the weight of the upper part of the wall, this allows the stone structure to withstand the action of the wind safely.Globally, the functioning of such a structure is three-dimensional; it is based upon strong collaboration and interaction of all the parts of the structure, and contrarily to what is often said, just like the underlying stone structure of the cathedral, it is a system that mainly transfers the loads at some points, in correspondence of the main frames, and not continuously.On the whole, we can today appreciate the ability of the carpenters of the Middle Ages, that were capable of inventing a very effective structural system well before the discoveries of structural mechanics; such timber structures have spanned the centuries, which witnessed their effectiveness, their only and true danger seems to be the neglect of humans.
A question remains: for what reasons did the Gothics invent the charpentes with chevrons formant ferme?The following section is devoted to answering this question.

CONCLUSION: WHY THE CHARPENTES À CHEVRONS FORMANT FERME?
The invention of the charpentes à chevrons formant ferme actually responds to an evolution of Gothic architecture, evident just in the case of Notre-Dame [23]: the rise of the guttering walls above the top of the vault.Before such a modification, it was not possible to use frames with tie-beams, because the vault's top was higher than the footing of the frames.The solution adopted by the carpenters was probably that of scissor trusses that do not have a tie-beam.We do not know what the reason for the raising of the guttering walls was, a solution later adopted also in other cathedrals; what is certain is that the carpenters of the 13th century could then adopt a stat-

Fig. 3 .
Fig. 3. Schemes of the charpentes of Notre-Dame: (a) longitudinal scheme; the parts of the structure studied in this paper are in red; thick lines indicate the main frame and the longitudinal bracing system, while thin lines represent the common frames; (b) scheme of a main frame of the choir; (c) scheme of a main frame of the nave; (d) scheme of a common frame.

Fig. 4 .
Fig. 4. From left to right: schemes of the main frames of the choir, the nave, and the fermettes.The supports are indicated by letters S1 to S10.

Fig. 5 .
Fig. 5. Schemes of modular structural units of the choir, left, and nave, right.

Fig. 6 .
Fig. 6.Scheme of the actions on the charpente; p: lead tiles and voligeage load; w 1 , w 2 , w 3 : wind load; f 1 , f 2 : the load of the catwalk. w

Fig. 7 .
Fig. 7. Distribution of the reaction forces in the original state; top, the choir's charpente; bottom, the nave's one.

Fig. 8 .
Fig. 8. Deformation of the structural units of the charpentes, in the original state, for their own weight (OW) and for their own weight plus the wind (OW+W) (displacements magnified).

Fig. 10 .
Fig. 10.The system of the console for the transfer of the wind thrust to the guttering walls; the arrow indicates the shear key between the tiebeam and the chapeau of the console (by the Author).