Estimating the strength of Notre-Dame's vaults after the fire

A heritage disaster, the Notre-Dame fire of 2019 is also an opportunity to increase our knowledge: the debris of the famous cathedral is a precious witness to the past! This series follows the scientific construction site at Notre-Dame, where charred wood and metal parts reveal their secrets. After initial episodes on the structure and origin of the wood, we now turn our attention to the cathedral's structure and, in this^4th installment, its masonry.

Stéphane Morel, University of BordeauxFrédéric Dubois, University of MontpellierJean-Christophe Mindeguia, University of BordeauxPaul Nougayrede, École Nationale Supérieure d'Architecture (ENSA) Paris-Malaquais - PSL; Pierre Morenon, INSA Toulouse and Thomas Parent, University of Bordeaux

In April 2019, in the wake of the fire that struck Notre-Dame Cathedral, the CNRS and the French Ministry of Culture set up the "Notre-Dame Scientific Workshop" to federate and organize initiatives from the French scientific community.

Our "Structures" working group is concerned with the mechanical assessment of the cathedral's load-bearing structures - particularly masonry and framework. We were very quickly called upon by the restoration team to assess the current stability of the cathedral's high vaults, which had been affected by the fire.

While some parts of the vaults collapsed during the fire, notably due to impacts with elements of the spire or roof structure, the vast majority remained in place. This is hardly surprising, given that the vaults were originally designed as a fire protection system, preventing burning elements from falling down! They have fulfilled their role, but their stability is crucial to the safety of the site.

What's more, the vaults are a true architectural masterpiece in the Gothic style - and are, of course, being conserved to the highest possible standards.

Unfortunately, no modern calculation method is currently available in engineering design offices to accurately model the mechanical behavior of such structures, in order to assess site safety and the effectiveness of the support measures implemented by the project manager.

This is why we need to call on scientific expertise... and, faced with highly complex mechanisms, our working group has had to develop new approaches.

A jewel of masonry, a challenge for modellers

In the ^twentieth century, masonry was gradually replaced by steel and reinforced concrete in the construction of large buildings, and these modern materials became the focus of calculation/modeling efforts.

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Moreover, the mechanical behavior of a masonry structure is extremely complex to grasp: dressed ashlar masonry such as that of Notre-Dame Cathedral is a composite, anisotropic (material whose mechanical properties vary according to the direction considered in the material) and heterogeneous material, made up of ashlar blocks joined by thin joints of lime mortar, with the stone-mortar interface constituting a zone of mechanical weakness.

Damage to dressed ashlar masonry is most likely to occur at the stone-mortar interface, and as a result, masonry cracking will occur according to perfectly identified patterns.

Stable and flexible construction

Thanks to this construction method, Notre-Dame has a high degree of mechanical stability, characterized by great flexibility: relative displacements between blocks are permitted by cracking at the stone-mortar interfaces, which induces a significant capacity to dissipate mechanical energy via friction at these cracking planes.

Cracking at the stone-mortar interface is what gives masonry its rich mechanical behavior, while at the same time making precise mechanical modeling a complex task.

Although the mechanical behavior of masonry is currently the subject of numerous developments, none of the methods developed to date can claim to provide an exhaustive description of the behavior of this heterogeneous material.

The studies carried out by our scientific consortium were therefore based on the parallel application of various complementary mechanical modeling methods, in order to obtain a reliable estimate of the post-fire mechanical behavior of Notre-Dame's high vaults.

These methods are based on :

• or a "continuous" approach: masonry is modelled as a single continuous material with elastic and fracture properties equivalent to those of the composite masonry material;
• or a block-by-block or "discrete" approach: the interactions between blocks describe the mechanical behavior imparted by mortar joints and their interfaces.

The divided character of masonry behavior, conferred by the assembly of blocks and the morphological influence of the fitting, will be more naturally and rigorously apprehended by the discrete approach compared to the continuous approach, but at the cost of longer mesh generation and model calculation times. On the other hand, the discrete approach will fail to describe block failure, whereas the continuous approach will describe it accurately.

These two examples illustrate the complementary nature of the discrete and continuous approaches, which can be used in parallel to define more precisely the simulated mechanical responses of the structures being modeled.

What are the differences before and after the fire?

A pre-fire assessment was first carried out to quantify the evolution of the vaults' stability after the fire.

This initial work shed light on the construction stages of the vaults and buttresses. Pre-fire modelling also revealed that the morphological differences between the thin vaults of the choir (12 to 15 centimetres thick) and the thicker vaults of the nave (19 to 25 centimetres thick) resulted in a lower thrust of the vaults than of the buttresses in the choir, and vice versa in the nave.

Modelling of the fire identified the physical phenomenon responsible for most of the post-fire damage to the cathedral: thermal expansion.

In fact, the "swelling" of materials due to the increase in their temperature during the fire seems to be a more important factor than the reduction in the mechanical properties of the materials themselves (this reduction is linked to the rise in temperature and the saturation of the materials by the fire extinguishing water).

Simulate reinforcement techniques

On this basis, we were able to simulate the solution for reinforcing the burnt-out vaults chosen by the project management team, in order to assess the risk-benefit ratio of this solution and the possible adaptations that would increase its effectiveness (for example, with regard to the modulus of elasticity and thickness of the screed, or the behavior of the vault-screed complex under mechanical stress).

The work of the "Structures" working group is scheduled to continue until 2024, when restoration work on the cathedral should be completed. At the same time, members of the group are developing a hybrid masonry modeling tool, involving the simultaneous use of discrete and continuous approaches.

Frédéric Dubois (LMGC-Montpellier), Paul Taforel (MiMeTICS engineering, LMGC spin-off), Pierre Morenon (LMDC-Toulouse TTT platform), Maurizio Brocato and Paul Nougayrede (GSA-Paris), Jean-Christophe Mindeguia, Thomas Parent and Stéphane Morel (I2M-Bordeaux, study coordination) are co-authors of this article..

Stéphane Morel, University Professor, University of BordeauxFrédéric Dubois, CNRS Research Engineer, University of MontpellierJean-Christophe Mindeguia, Senior Lecturer in Civil Engineering, University of BordeauxPaul Nougayrede, Doctoral student in architecture, École Nationale Supérieure d'Architecture (ENSA) Paris-Malaquais - PSLPierre Morenon, Civil Engineer and researcher in LMDC's technology transfer division. A specialist in numerical structural calculation methods.., INSA Toulouse and Thomas Parent, Senior Lecturer in Civil Engineering, University of Bordeaux

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