# Thermo-Mechanical Analysis of a Stone Masonry Wall
## Subjected to Fire Using FEM and DEM Approaches

Guenser Colin, Morenon Pierre, Domede Nathalie, Corn Stéphane, Salgues Marie

Marie

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Guenser Colin, Morenon Pierre, Domede Nathalie, Corn Stéphane, Salgues Marie. Thermo-Mechanical Analysis of a Stone Masonry Wall Subjected to Fire Using FEM and DEM Approaches. IB2MaC 2024 - 18th International Brick and Block Masonry Conference, Jul 2024, Birmingham, United Kingdom. pp.556-567, 10.1007/978-3-031-73310-9_39. hal-04968144

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Thermo-Mechanical Analysis of a Stone Masonry Wall
Subjected to Fire Using FEM and DEM Approaches

Guenser Colin¹[0000-0003-0464-0083], Morenon Pierre²[0000-0001-5118-5584], Domede Nathalie ¹[0000-0002-3791-0820], Corn Stéphane ³[0000-0002-9819-0425], Salgues Marie ³[0000-0002-8801-6730]

¹ LMDC, INSA/UPS Génie Civil, 135 Avenue de Rangueil, 31077 Toulouse cedex 04 France

² Toulouse Tech Transfer, 18 route de Narbonne, CS 24246, 31432 Toulouse Cedex 04, France

³ LMGC, IMT Mines Ales, Univ Montpellier, CNRS, Ales, France
guenser@insa-toulouse.fr

**Abstract.** The DEMMEFI (Discrete Element Method, Finite Element Method, FLe) research project focuses on developing a numerical tool for post-fire diagnosis of stone masonry structures, with a particular interest in assessing the restoration feasibility for historic buildings like the Notre-Dame de Paris cathedral. This paper presents the validation process of micro-scale thermomechanical behaviour laws integrated into Finite Element Method (FEM) and hybrid FEM/Discrete Element Method (DEM) models, using experimental data from masonry walls subjected to thermal and mechanical loading. Through a methodology involving experimental data analysis, numerical simulations, and model refinement, the study evaluates the effectiveness of the ENDO3D (for 3D anisotropic damage) model in accurately predicting cracking schemes and deflection in masonry structures under fire conditions. Additionally, insights into the influence of key parameters on temperature propagation and out-of-plane deflection are discussed. The findings underscore the importance of these factors in numerical modelling, enabling further work to refine models and ultimately macro-scale simulations of stone masonry structures. This research contributes to advancing engineering practices for the preservation and restoration of cultural heritage sites constructed from stone masonry.

**Keywords:** Masonry, Fire, Thermo-Mechanical behaviour, Finite Element Method, Discrete Element Method

# 1 Introduction

The DEMMEFI research project aims to develop a numerical tool for the post-fire diagnosis of stone masonry structures. This tool is intended to assist in modelling masonry structures and determining whether a building, such as the Notre-Dame de Paris cathedral, can be safely restored after minimal repairs or if structural reinforcement is necessary. Two primary challenges must be addressed: first, accurately accounting for the thermal damage on constituent materials, such as stone blocks and mortar joints, and second, simulating the masonry material as a domain divided by weak zones
introduced by the joints. A 3-D multiscale study is being conducted to account for the microscale behaviour of materials within a macroscale simulation of a masonry structure. Such multiscale studies were carried out by other authors concerning fired-clay brick masonry [1,2] and many concerning reinforced concrete structures [3].

In the beginning stages of the project, a hybrid tool combining the Finite Element Method (FEM) and the Discrete Element Method (DEM) was developed to model 3-D masonry structures on a macroscale. This hybrid tool has already been validated on masonry structures like vaults under seismic loading [4] and walls under shear tests [5]. Additionally, as part of the same research project, specific thermomechanical behaviour laws have been created and validated on a microscale for soft limestone (see Fig. 1) and aerial lime mortar, based on materials tested after being heated at high temperatures, such as those reached during a fire, and cooled afterwards [6].

<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Thermomechanical laws based on [6], established for Saint-Maximin limestone: stress-strain behaviour in tension and compression at 20°C (green), 300°C (yellow) and 600°C (red).</figcaption></figure>

The present work aims to validate these micro-scale thermomechanical behaviour laws by incorporating them into a FEM model and a hybrid FEM/DEM model of a stone masonry wall subjected to fire and subsequent cooling. This validation process is essential prior to calculating the entire Notre-Dame de Paris cathedral, using either the hybrid FEM/DEM method or the FEM method.

## 2 Method

The thermomechanical behaviour of masonry walls subjected to fire and cooled down was validated at the meso-scale level through a series of experimental tests that were numerically simulated using a FEM model and a hybrid FEM/DEM approach. The experimental program, which was reproduced from scientific literature, involved heating one side of two masonry walls to fire-like temperatures. The validation process comprised three main steps. First, all available experimental data regarding the original
experiments was collected and used as input in the simulation. Second, the thermome-
chanical laws were adapted to match the materials used in the experiment, including
their strength, rigidity, and bulk density. Third, the results from the numerical simula-
tions were compared to the available experimental results, and any discrepancies in the
wall model were identified and corrected. This analysis enabled the formulation of hy-
potheses and their testing, ultimately improving the wall model and enhancing our un-
derstanding of its behaviour.

3 The Experimental Program

The present study features an experimental program involving a masonry structural element subjected to thermal and mechanical loading. The experimental data were obtained from tests published in the literature [7]. In this program, two masonry walls measuring 2.95 meters by 2.92 meters each were subjected to a standard fire test (EN ISO 834-1) on one side for two hours. The walls were constructed with Lutetian limestone blocks (Saint VAAST, with a mean compressive strength of 7.4 MPa) bonded with 10 mm thick mortar joints (NHL 3.5). The walls were allowed to cure for 25 days and 29 days for the first and second walls, respectively. The bottom and top faces of the walls were secured to a reinforced concrete frame with mortar layers of 20 mm and 67 mm, respectively. The first wall was loaded vertically to 1.25 MPa prior to the fire test, while the second wall was unloaded and placed inside a closed RC frame. The lateral sides of both walls were padded with rockwool to prevent any lateral mechanical load and confine the heat to the fire-exposed face only.

The results available show the evolution of temperature in the walls over time, measured by various thermocouples that were placed at different depths in the wall. In addition, the evolution of the wall's out-of-plane deflection with time and the cracking scheme were monitored using Digital Image Correlation (DIC). A displacement transducer was placed in the centre of the wall to measure the out-of-plane displacement at a specific point of interest. The unloaded wall was heated during two hours and then cooled naturally. The loaded wall was heated until it reached a failure criterion on the 27th minute: wall contraction rate was higher than 9 mm/min (measurement made in A, see Fig. 2 left). It then completely collapsed on the 34th minute. It is worth noting that a subsequent study conducted under the same conditions and with very similar materials [8] did not record any premature failures, which suggests that a local default or loading asymmetry may have been responsible for the observed failure.

4 The Numerical Models

The calculation of temperature fields is separate from the mechanical models, allowing
for the thermal simulation to be performed prior to the mechanical simulation. The
computed temperature fields are then provided to the mechanical models.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Experimental setup of the tested loaded wall, figure inspired by [7]. Front and side view.</figcaption></figure>

**4.1 Thermal FEM model**

The thermal properties of limestone and mortar were obtained from an experimental campaign carried out as part of the DEMMEFI research project [6]. The limestone used in the campaign, Saint-Maximin limestone, was similar in appearance, strength, and bulk density to the Saint-Vaast limestone, and the mortar used was made from NHL 3.5 lime. Thermal conductivity and heat capacity were measured as a function of temperature from 20°C to 600°C. Additional conductivity values up to 1000°C as well as thermal expansion as a function of temperature were obtained from another study conducted on the same limestone [9]. The 3D thermal model of the wall, shown in **Fig. 3(a)**, was simulated using CAST3M software. Preliminary studies had shown the great temperature gradient close to the heated surface, the mesh used was therefore based on cubic elements of varying thickness along the Y axis (see **Fig. 3(b)**).

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** 3D thermal model of the wall, with (a) the temperature boundary condition and (b) the mesh used.</figcaption></figure>
The thermal model demonstrated that the water content in the wall significantly influences heat propagation due to water vaporization. The water content can fluctuate depending on the construction method and curing time of the wall, and was not assessed in the experiment detailed here. Since the heat capacity was measured in [6] with dry samples, it was modified to account for the presence of liquid water in the wall. The water vaporization was modelled as a peak around 100°C in the heat capacity (see Fig. 4), and the height of the peak was adjusted to align with the experimental temperatures measured inside the wall. The thermal expansion was measured on stone of the same type and provenance in [10].

<figure><img src="image_5.png" /><figcaption>Fig. 4. Thermal expansion $\alpha$ [10] and heat capacity $C_p$ used in the thermal model of the wall.</figcaption></figure>

## 4.2 Mechanical FEM Model

The FEM is a widely utilized technique for analysing complex structures under quasi-static mechanical loading. This study employs the ENDO3D mechanical model within the CAST3M software [11], an anisotropic elasto-plastic model that considers tensile and compressive damage. This model is governed by three Rankine criteria for tension (one in each direction), one Drucker-Prager criterion for shear/compression, and one CamClay criterion for porosity collapse under tri-axial compression. The latter criterion is particularly suitable for modelling the crushing of relatively soft materials with high porosity. The ENDO3D model also allows for reproducing the thermal damage of the material by reducing three mechanical characteristics independently (compressive and tensile strengths and Young modulus), making it particularly well-suited for this application. The reduction factors follow the function described in Eq. (1), different parameters being used for each of the mechanical characteristics.

$$r = p + (1 - p)e^{-(\frac{\theta - \theta_{th}}{\theta_k - \theta_{th}})^m} \quad (1)$$

Where:

$r$ = reduction factor

$p$ = ultimate reduction factor

$\theta$ = temperature

$\theta_{th}$ = temperature at start of reduction

$\theta_k$ = characteristic temperature value
$m = \text{curvature of the reduction law}$

In the FEM mechanical model of the walls (see Fig. 5(a)), blocks, and mortar joints are modelled in a detailed approach. The mesh used is based on cubic elements of varying thickness along the Y axis. Due to the size of the wall and the complexity of the ENDO3D model, the wall was divided into six elements only along the Y axis. For the unloaded wall, boundary conditions are blocked in all directions on the upper and lower faces of the wall. For the loaded wall, a compressive load of 1.25 MPa is first imposed on the bottom face, and the vertical displacement of the bottom face is then locked in position before starting the fire.

The main mechanical parameters for limestone and mortar, such as strength and rigidity, were obtained from the available experimental data. The remaining parameters required for the ENDO3 model, such as Poisson's ratio, strain at peak stress, or fracture energy in tension, were determined through the experimental campaign presented in [6].

<figure><img src="image_6.png" /><figcaption>Fig. 5. Boundary conditions and meshes used (a) for the FEM model and (b) for the hybrid FEM/DEM model.</figcaption></figure>

## 4.3 Mechanical Hybrid FEM/DEM Model

DEM is typically less utilized in the civil engineering field than FEM for structural modelling. However, it does account for interfaces prone to failure and their associated failure mechanisms, such as joint tension cracking or joint slip that were already described in [12]. In this model, blocks are considered non-linear in accordance with the FEM model previously described (ENDO3D), and they interact with each other through a Frictional-Cohesive Zone Model contact law at their interfaces (FCZM) (see Fig. 6). In this model, the blocks keep their own dimensions and mechanical properties, and the FCZM models the mortar/block interface plus the mortar joint, hence the initial gap [5]. This model allows for interface slippage and/or block separation, which is essential for reproducing common masonry failure mechanisms. The FCZM model is more complex
than FEM and requires a significant amount of computational power. The parameters for the FCZM model were obtained through an experimental campaign conducted within the DEMMEFI project.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Hybrid modelling approach: (a) masonry components and (b) assembly modelling strategy [5].</figcaption></figure>

# 5 Results and Discussion

## 5.1 Temperature propagation

The temperature field is determined using the thermal model at all points of the mesh. In order to compare with experimental data, the progression of temperature over time is plotted at selected points of interest (Fig. 7(a)). This plot displays the temperature changes in the wall at the specified points over time. The computed temperature field closely aligns with the experimental data in the regions near the heated surface where the temperature gradient is high. However, it loses accuracy in the unheated area where temperature increases slowly. This discrepancy may be due to the way water content was modelled in the wall, which assumed a homogenous distribution throughout the wall. In reality, water content is likely higher in the core and lower near the surface. This error mainly affects the low temperature zones (less than 100°C) where the mechanical properties of the limestone are only slightly impacted. Therefore, the computed temperatures can be considered suitable for mechanical simulations.
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** (a) Evolution of experimental and numerical temperature with time and (b) graphical visualisation of the temperature gradient through the wall after 7200s of heating.</figcaption></figure>

## 5.2 Out-of-plane deflection

The unloaded wall that did not experience premature failure and provided data for the entire duration of the heating and cooling process was chosen as the reference for studying the mechanisms at work. As shown in **Fig. 8**, which depicts the out-of-plane deflection of the wall over time, two distinct phases can be observed in the experimental data. The first phase involves rapid deflection, reaching 3 cm within 20 minutes, followed by a slower deflection phase that stabilizes at 4.5 cm after 2 hours of heating.

After conducting several tests with the aid of the FEM model, it was determined that the boundary conditions play a critical role in the stabilization phenomenon, and more specifically, the crushing of the mortar at the boundary. A possible explanation for this may be a defect in the boundary mortar caused by inadequate filling of the 60mm gap between the wall and the top RC frame. This hypothesis was tested using the CamClay criterion, with different values of porosity attributed to the mortar at the boundary, as shown in **Fig. 8** for porosity values of 0.1, 0.3, and 0.7. It can be observed that higher porosity leads to greater total deflection of the wall, and that a porosity of 0.3 results in out-of-plane deflection that is very close to the experimental observation. It should be noted that another hypothesis, which was not tested in this study, proposes that porosity is equally distributed in every mortar joint rather than just the boundary joints.

The loaded wall was also simulated with a porosity of 0.3 in the boundary mortar, and demonstrated a lack of rapid deflection phase, as shown in **Fig. 8**. The deflection only reached 3.8 mm after 25 minutes of heating. This does not contradict the previous explanation, as the immediate application of the vertical load on the wall causes the porosity in the boundary mortar to collapse, leaving little porosity for thermal dilatation to crush. Consequently, the wall behaves as if there were almost no porosity in the boundary mortar.
<figure><img src="image_9.png" /><figcaption>**Fig. 8:** Evolution of out-of-plane deflection with time, measured at the centre of the wall.</figcaption></figure>

**5.3 Cracking scheme**

The experimental tests were conducted using the DIC instrumentation, which was capable of capturing the initiation and localization of cracks on the unheated face of the wall. In **Fig. 9**, the DIC cracking monitoring results of the unloaded wall are compared with the crack opening values computed by the FEM and hybrid models in colour maps.

After 30 minutes of heating, both numerical models showed that all mortar joints and FCZM contact points were cracked open by at least 0.3 mm. While the DIC map-colours do not provide an explicit validation of this value due to the absence of a colour scale, the cracking scheme reported after 30 minutes appears to be consistent with the position of mortar joints.
<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9:** Evolution of the cracking scheme of the unloaded wall with time, for (a) the experimental wall (based on [7]), colour scale was not specified), (b) the FEM model, and (c) the hybrid model.</figcaption></figure>

After 120 minutes of heating, the DIC data demonstrated two significant straight cracks at each lateral end of the wall, accompanied by secondary vertical cracks in the centre of the wall, which was also accurately reproduced by the FEM model. At the
time of writing, the hybrid model simulation had not yet reached 120 minutes of simulated time, but the initiation of the vertical cracks at each side of the wall was still discernible (Fig. 9(c)).

The cracking scheme described were consistent across all the porosity scenarios tested in section 5.2, with the only variations being the magnitude of the crack openings.

## 5.4 Stress distribution inside the wall

The examination of stress distribution across the thickness of the wall highlights the underlying mechanisms that lead to deflection and cracking observed previously. In Fig. 10, several stress profiles extracted from the FEM model at different time steps are presented, starting from 900 seconds of heating to 7200 seconds of heating. The top graph shows data from the centre of the wall and demonstrates a maximum compressive stress located between 1 cm and 2 cm from the heated face at all time steps. On the other hand, compressive stress tends to be very low when looking beyond 4 cm from the heated surface, and even turns into tensile stress at some point. This stress state, directly caused by the restricted thermal expansion of the hot face, induces a bending moment responsible for the deflection observed in Section 5.2.

When examining the stress distribution at the top of the wall (Fig. 10(b)), it is observed that the bending moment is not yet formed before 900 seconds, and only becomes noticeable after 1800 seconds of heating. This data indicates that the boundary mortar acts as a hinge at the beginning of the test, allowing for fast deflection, and gradually turns into a fixed support as its porosity collapses.

<figure><img src="image_13.png" /><figcaption>**Fig. 10:** Vertical normal stress distribution along the depth of the unloaded wall at several time steps, (a) at the centre of the wall and (b) at the top-centre of the wall. (the fineness of the mesh was limited due to constrained computation time)</figcaption></figure>
# 6 Conclusion

Based on our findings, the ENDO3D model has proven effective in both accurately modelling the cracking scheme and predicting deflection in masonry structures under fire conditions, making it a valuable tool for engineers and researchers in the field. Furthermore, the Hybrid FEM/DEM model that uses the ENDO3D model and FCZM contact law appears to be very promising, but requires longer computation times. Additional calculations are still ongoing.

Our research has revealed that the water content of the wall is a key parameter when simulating temperature propagation within the wall. Furthermore, deflection is influenced by the internal bending moment caused by restricted thermal expansion and can vary significantly based on the boundary conditions. Therefore, careful attention must be paid to the boundary conditions when designing experiments or numerical simulations. Porosity at the boundary condition has been found to be a key factor. Concerning the cracking scheme, both the FEM model and hybrid model have successfully reproduced it, and the effect of the boundary conditions is less significant on the result in this case.

Looking forward, additional tests will be conducted with a new hypothesis that considers porosity to be evenly distributed throughout all mortar joints rather than just at the boundary. Ultimately, the hybrid model will be utilized in a process to homogenize the masonry material, enabling the macro-scale simulation of the Notre-Dame de Paris cathedral using a FEM model.

# Acknowledgements

The authors wish to extend their gratitude to the Agence Nationale de la Recherche for their support of their research works through the research project ANR-20-CE22-0004. This work was also facilitated by access to the CHP resources of the CALMIP computing centre, which was made possible through allocation P22062.

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