## Optimization of a computational model for the restoration of the intrados of the Charles Bridge (Prague)

The Charles Bridge is part of the world heritage: such piece of Gothic architecture must be maintained as a remain of history. Urgent attention should be paid to the intrados of the arches, where the masonry exhibits advanced state of weathering. Maintaining historical masonry is a delicate work. Usually, the technique of patching is used: the damaged surface of stone units is removed, and replaced with new elements of compatible stone, bonded to the old part with traditional mortar. Nonetheless, in the case of the actual structure, the weathering jeopardizes the mechanical properties of the stone. Therefore, some units should be entirely changed.

When important intervention is done on historical building, the motivations for it must be justified, as well as the low impact of the intervention – both when work is in progress and in the final state. Structural analysis of the structure based on a computational model is often carried out. Combined with site investigation, the computational model is used to understand the current state of the structure according to its loading history. It also enables to prepare the intervention and to predict its consequences. Finally, it is a valuable tool for the maintenance after the intervention.

To prepare the restoration of the intrados of the Charles Bridge, a computational model is required to minimize the impact of the replacement of weathered stones on the global deformation of the arch and, locally, on the increase of stress or the development of tensile stress around the stone. The methodology would be used by engineers to design and validate the stages of replacement.

**In this thesis, the author Maïa Laffineur** **focuses on the development and discussion of a methodology to model the replacement stage by stage of some damaged stones of the intrados of the Charles Bridge in Prague.**

Modeling strategy should be established to study the arches of the bridge, based on linear finite element analysis. Two points should be treated with particular attention. First, the modeling of different stages, with, for each stage, changes in the static scheme and in the properties of the elements that constitutes the replaced stone. Then, the modeling of the reactivation of the newly inserted stone, to describe the transfer of loads from the structure to the new block.

This computational model would then be a tool to determine the best replacement procedure to minimize global deformation of the arch and prevent local increase of stress and development of tensile stress.

## 1. The Charles Bridge – History

The Charles Bridge (Karlu ̊v Most, Figure 1.1) is one of the most important monuments of the city of Prague. It is classified in the National Historic Monuments list and by UNESCO Commitee. It was built between 1357 and 1406, when Charles IV was king of Bohemia, Ger- many, Italy, and the Holy Roman Empire. The legend reports that the king laid the first stone 5:31am on 9 July 1357, which would give the construction great strength according to numerology beliefs (1357 9, 7 5:31). He entrusted Petr Parlér the charge to design and build the bridge.

The bridge links the two historical areas of Malá Strana and Nové Mesto over the Vltava

River, the main river crossing the city of Prague (see figure 1.2). It replaced the Judith Bridge that collapsed in 1273 due to several floods (see figure 1.3a). As it was on a strategic trade road through Eastern Europe, a wooden bridge was built quickly, that was later replaced by this monumental stone bridge.

The construction of the Charles Bridge represented a challenge in its time (see figure 1.3b). The bridge is 515.76 m long, the spans between two pillars are between 16.62 m and 23.38 m. The width of the deck is 10 m. It is composed of 16 arches. On the bridge parapet are several baroque sculptures, all representing saints or scenes from the Bible. On the river banks stand two watchtowers, another emblems of the city. The construction was first supposed to support the merchant caravans crossing the city. Later, the bridge was adapted for trams and buses. Nowadays, it is only pedestrian so the loads have reduced a lot through times.

## 1.1. Intervention

Since the Charles Bridge was the main and only way to cross the Vltava river in a perimeter of 100 km around Prague during a long time, it was maintained in a good state. Nowadays, its cultural patrimony is the reason for maintenance since other bridges were built in the city.

The main causes for the deterioration of the state of the bridge are:

• self weight

• weathering (wind, erosion of the water, aggressive salts…)

• floods

• thaw and frost cycles (there can be an annual variation of temperature of 40 ◦C)

• movement of foundations

The flood episodes, some very damaging, are compiled in the Table 1.1. Some parts of the bridge collapsed due to floods, and pillars were as well severely damaged. Floods can cause angular rotation, subsidence, and shift on a bridge pillar. They cause tensile stress in the arches, which leads to local damage of the masonry.

From 1966 to 1975, an important restoration work was started. However, this intervention was afterwards highly criticized because cement mortar was used instead of traditional materials. The deck was strengthen with a reinforced concrete slab and hydro-insulation was installed. It is also the time when the bridge was closed to vehicles in order to preserve it. Concerning the stones, the technique of patching was used: only a superficial part of the brick was replaced with new stone and bonded to the remaining part with cement. Though this method is advised, rather than replacing the whole block when it is possible, it requires compatible properties of stone and mortar, which was not the case for the Charles Bridge. Later, the patches turned into odd colors (beige to yellow), and some parts of the arch did not behave as massive masonry: the bricks were not interdependent anymore. The development of biochemical decay in the layer between the old stone and the new patches fostered the global damage of the bridge. This intervention is one of the explanation for the advanced degradation of certain stones of the arches.

Another important intervention took place in 1990, to assure the overall stability of the bridge and its bearing capacity. This time, a multiscale approach was considered, in order to treat both local problems and their consequence on the global behavior.

The final intervention occurred from 2007 to 2010. It aimed at the rehabilitation of the foundations as well as the hydro-insulation of the pavement on the deck. Some smaller works were done at that time (repair of the parapet, renewal of gas lighting). This very recent intervention was the opportunity to install a monitoring system to control the temperature and the moisture content, and an to create an information database* (https://iskarluvmost.fsv.cvut.cz/)*.

From then on, no major operation was done on the bridge. However, the development of more and more powerful numerical tools enabled to have a better knowledge of the behavior of the bridge through several research programs and modeling campaigns at the Czech Technical University. However, some parts of the Charles Bridge present a concerning state.

## 2. **Modeling of the intrados**

In this thesis, the author tries to find a suitable numerical approach in order to optimize the replacement of some specific highly damaged stones of the arches of the Charles Bridge. The 14th arch is taken as an example since it is the most damaged one. There are two main tracks to develop and to achieve it. First, a numerical model that can capture the global behavior of the arch of the bridge and the local consequences around the replaced stones needs to be developed. Then, based on the results, and with the aim of reaching a reasonable time of the reconstruction and minimizing additional damage to the bridge or to adjacent stones, general recommendations could be drawn. This thesis focuses on the first part of the problematic that consists in the building of a computational model.

The two main criteria of the study are therefore:

• The global vertical deformation of the arch due to the axial shortening of the arch when a stone is removed.

• The increase in local stress concentration that may deteriorate the integrity and good mechanical properties of the surrounding stones.

The present study does not concern the ultimate state or the stability of the arch and will not consider damage mechanics of the stones or of the structure. The restoration of the bridge must not affect the structure beyond serviceability state.

One of the main difficulties of this work is that there is no similar question academically treated before. All that can be found concerns the chemical and mechanical compatibility of the new pieces of stone. For this reason, the work is done incrementally, to improve step by step the modeling of the physical phenomena, and results in two different models.

## 2.1. Results

Many information can be taken out the FE analysis and are of interest to describe exhaustively one specific case of replacement. In the following chapters, only specific values are studied to fulfill the objective of building a suitable computational model.

**Deformation**

The chosen procedure of replacement of the stones should minimize the vertical deformation. To calculate it, the vertical displacement of the top left node of the model is recorded. Deformations are then calculated as follows:

where Ly = 3.68 m is the height of the section of arch studied and H = 0.7 m is the height of the stone.

Since the model is simply supported, it is also important to examine the horizontal displacement u.

**Increase in stress concentration**

In the following chapters, the stress distribution more often refers to the compressive stress in y direction: σyy (negative values for compression). The stress in the other direction, σxx, and the shear stress τxy are other relevant quantities.

## 3. Homogeneous model

This model only considers ashlar units of sandstone, with no mortar: it is homogeneous and isotropic. The stones are arranged side-by-side with no interface between two units, the mechanical properties only depend on the state of damage of each stone. The linear constitutive law of the material and the homogeneity of the model make it a very simplified approach. That way, it is possible to roughly model the behavior of the section of arch when a stone is replaced.

Since the material is linear elastic, the solution is obtained without iterations in a very reasonable computational time. One can notice that compared to the size of the stone blocks, the layers of mortar are very thin (0.02 m), therefore it makes sense to assume that the response of a model composed of stones only can be treated as representative of the real behavior.

**3.1 Creating the computational model**

First, a script in Matlab is used to generate a regular finite element mesh with 60 × 60 elements, then the geometry of the section is drawn to match the mesh (see Figure 3.2). After that, the construction stages, and the stones and the corresponding finite elements to be removed and inserted are defined. Finally, the loading and boundary conditions are listed in the input file.

The building process of the final file is summarized in Figure 3.1.

The elements that are used are isoparametric quadrilateral, 4 nodes, plane-stress elements.

Each node has 2 degree of freedom (x and y), and there are 4 Gauss integration points for the element (see PlaneStress2d in OOFEM, Figure 3.3).

**3.2 Discussion on the model**

It is from the global point of view that this model can be helpful. Though the reactivation procedure cannot be captured, the global behavior is still relevant without it. Indeed, because of the homogeneity of the material and the simplicity of the geometry, it can run quite fast. Using this approach, many different combinations of replacement procedures could be analyzed without bothering for the reactivation, and trying only to minimize the global deformation.

Further studies could be carried out with this model to understand the influence of certain parameters on the global deformation:

• the maximum width of a column of stones that can be removed during one stage, compared to the width of the restored area

• the maximum area of stone that can be replaced during one stage

• the minimum number of stages that are needed to replace all the stones

This homogeneous isotropic model cannot be used to describe the reactivation of a new stone unit during the process of replacement of elements of historical masonry. Since it cannot capture correctly the response during activation it does not make sense to analyze the local stresses around the replaced stone, though it is important to check that there is no local damage due to tensile stress for instance.

From a computational point of view, preparing a model is complex. The geometry of the stones must match the mesh, though it is irregular – most of the time, the geometry is first drawn, then the mesh is applied to it. Moreover, the geometry is built up by creating the stones row by row, which is a limit for the cases when the stones do not have the same height on one row. This was not a problem for the Charles Bridge, since the masonry is quite regular. The topology of the replaced stone must also be specified manually, which starts to be a problem when the number of stones increases. In Figure 3.8, it can be noticed that the top right stone that stays in place during the replacement process does not have proper boundary conditions for the combination chosen. For each new combination, specific complexities appear either linked to the boundary conditions of the surrounding stones. This study shows another limit of this semi-automatic model. Therefore general recommendations should be found from quite simple combinations before building one complex and exhaustive model.

In conclusion, this model is not suitable for big, irregular structures where the inserted components need to be activated.

## 4. Heterogeneous model

This model is heterogeneous and takes into account the presence of 0.02 m mortar layers. In order to describe the activation procedure, the former homogeneous model used eigenstrain conditions on the newly inserted stone. In this model, the loading of the new stone unit is modeled with a stress applied in the vertical direction on the newly inserted stone.

**4.1 Creating the computational model**

Several softwares are used to build this model. First the geometry is written in Matlab, then the mesh is generated automatically with T3D. From this mesh, an input file for OOFEM is built. The different steps are summarized in the Figure 4.1.

The heterogeneous model is made of perfectly regular coursed masonry in order to minimize the number of parameters of the model and to focus on the modeling of the reactivation phenomena (see Figure 4.2a). The stones will be refereed as they are numbered in this scheme. For this model, the stones coordinates are generated automatically with Matlab and are stored in an array, so that the sets of replaced stone can be changed easily.

For the meshing, the stones and the mortar layers are defined in patches, each patch is then meshed automatically according to the size of element required. This means that each unit of stone is a single patch, surrounded with at least 8 rectangular mortar patches (3 at the top and bottom, and 2 on the sides, as in Figure 4.3b).

In the Figures 4.2b and 4.3a, the mesh can be observed. The stone is meshed with rough elements, meanwhile the mortar patches are refined. The top patches, especially, present numerous elements. It is required for the boundary conditions of the stone during the reactivation. In total, the model counts 15491 nodes and 30569 elements (this number varies with the number and size of stones to be restored. Here it corresponds to the replacement of one regular unit). The elements that are used are triangular, 3 nodes, with a constant strain plane-stress elements. Each node has 2 degrees of freedom, and there is 1 Gauss point for the element (see TrPlaneStress2d in OOFEM, Figure 4.4).

**4.2. Application to the reduced area of arch**

Finally, the current modeling strategy is applied to the investigated area of arch. The geometry of the model is closed to the geometry presented in the previous chapter.

The same stones are replaced as in the previous model, here numbered: 38, 37, 22, 6, 36, 21, 4, 5, 32, 17, 18, 2, 23, 8. The section is divided into 4 columns, and the complete restoration is done in 4 stages. Because of the geometry of the stones, it was hard to define the combinations for all stages. A further study would consist in optimizing the procedure.

The Figures 4.26 and 4.27 presents the states of displacement and stress. In the graph 4.25 is the deformation of the model.

**4.3. Limits of the model**

This heterogeneous model can capture the procedure of reactivation of the new stones. Therefore, it can be used to study local increase of stress. It appears that under the assumption of linear elastic analysis, and for the case when the activation process is respected (fulfilled pressure on the newly inserted stone), simplification in the methodology can be used. Instead of creating one model of N stages, N models of one stage each describe quite well the final results. Its strongest drawback currently is the fact that it is very complex to prepare and that the possibilities of geometry and of number of stages is very limited.

The investment of time to prepare the model might not be worthy; therefore it would be interesting to develop an automatic tool to print the geometry and the stages. Another limit of the current model is the linearity of the material. Further studies should be carried out to treat the case of non-linear material.

## 6. Conclusion

The current work led to the development of two methodologies to model the intrados of the Charles Bridge in order to optimize the replacement procedure of the most weathered stones.

In order to have a low impact on the existing structure, the constraints are the minimization of the axial shortening as well as the prevention of local increase of stress around the replaced stone, or development of tensile stress. The specific points that the models can capture are the intervention in stages – and consequently changes in the static scheme – as well as the process of reactivation to transfer the loads from the structure to the new block.

Only a representative area of the most damaged masonry arch is examined: 5 rows of 7 to 9 blocks of sandstone (3.5 m × 4.5 m). It is modeled as an unfolded 2D element loaded in the vertical plane, as a plane stress problem. The reduced area is simply supported on its bottom edge, and free of constraint on lateral edges; no additional friction due to the connection with the bridge is added. The material constituting the masonry behaves linearly. The geometry is implemented with semi-automatic scripts using Matlab and Python. The linear elastic finite elements calculation is performed with the software OOFEM.

One model is homogeneous and isotropic. It is really simplified due to this homogeneity. Because of the fact that each weathered stone must be implemented by hand, the geometry of the blocks is rough. It is not satisfactory to capture the reactivation phenomenon and to describe the behavior around the stone: the local increase of stress and the development of tensile stress is not accurate. The solution proposed to model the local scale uses eigenstrain activation of the stone and needs to be applied manually for each block dimension and stiffness, which is not suitable for more than four or five damaged stones.

Nonetheless, the homogeneous model is efficient to capture the global behavior of the arch. Indeed, it gives consistent results concerning the axial shortening, even without the reactivation of the units. Since the computational time is very low, it can be used to get a rough idea of the response of the arch when replacing some stones.

Another model takes into account the mesostructure of masonry by modeling the layers of 0.02 m of mortar. It can describe the replacement procedure step by step, including the reactivation of a stone after the removal of the damaged one. Nonetheless, it is complex to create and only a limited number of stages can be currently implemented because the input file is semi-automatic. Since the analysis is only linear elastic, optimization must be performed before using it more widely.

The heterogeneous model can be used to study the local response of the arch during the replacement process, to minimize the local increase of stress around the stone and to check that no tensile stress develops. Though the modeling can become complex, it was shown that under the assumption of linear elastic analysis, very similar results are obtained when replacing the stones in one stage, or creating as many models as the number of stones and summing the results of the four models. This remark only applies when the reactivation is complete and the final stress in the new stone is the same as in the structure.

The methodology that was developed with these models is satisfactory to model the phases of the intervention stage by stage, with corresponding changes in properties of the stone, in boundary conditions and in static scheme. The reactivation procedure is also captured in the second model by means of the application of a uniform distributed load on the newly inserted stone.

Some recommendations can already be drawn out of the present work to optimize the procedure of restoration.

- The stones should be replaced column by column since the loads transfer through lateral elements of masonry when a stone is removed.
- On the contrary, too many stones in the same row shall not be replaced during the same stage.
- The procedure of reactivation must be followed strictly: if the reactivation pressure is not high enough, the impact on the structure is not minimized.
- A combination of both models can be used. With the homogeneous model, a replacement procedure that minimizes the global deformation can be established. The heterogeneous model is then used to validate the procedure from the point of view of local increase of stress around the removed stones.
- Under the condition that the activation procedure is respected (full activation pressure applied), the heterogeneous model gives similar results when creating one model with four different stages or four different models with one stage. This solution, that is easier and low time demanding, can be applied to optimize the replacement procedure.

**6.1 Perspectives**

Further studies could be carried out with these models to understand the influence of certain parameters on the global deformation:

- the maximum width of a column of stones that can be removed during one stage, com- pared to the width of the restored area,
- the maximum area of stone that can be replaced during one stage, compared to the area that requires restoration,
- the minimum number of stages that are needed to replace all the stones.

The results coming from these studies might enable to establish criteria on the maximal deformation that the arch can suffer to guide engineers in the preparation of site intervention.

For a wider application of the present models, the computational codes must be optimized. Both the complexity of the geometry and high number of stages cannot be treated for now. Other studies should be carried out to treat the case of non linear material.

The specificity of these numerical solutions is that they can model a site procedure which has a temporary though strong impact on the structures. In the case of historical heritage, where the material in place shall be conserved as remains of the past, such modeling is very interesting to prepare an optimized site work. Because the arch was modeled unfolded with plane stress hypothesis, the present results can be applied to any wall loaded in its plane.

Another specific aspect of this methodology is the modeling of stages to reflect the different phases of construction on site. A set of time functions rules the presence or absence of finite elements and their properties. Such modeling strategy, in stages, can be applied to other engineering fields: the construction in general, but also mechanics.

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