### A vault’s deformational analysis in the Cathedral

**ABSTRACT:**

This study is a method to calculate the deformations in historical masonry’s constructions. It can be realized using: a 3-D model, the applied loads of force and the degrees of freedom for each component and material characteristics. The calculation is done using finite elements (FEM-Finite Elements Method). This study lends you a 3D model of deformations numerically referenced with a legend colour.

The study has been applied to the vaults of the Cathedral of Tortosa. The method allows comparisons between different hypotheses of construction.

**INTRODUCTION:**

Tortosa’s Cathedral was constructed until 1347. It is a Gothic cathedral, who is placed in the site of the previous Romanic cathedral. The building is formed by a basilica with three parts and side chapels. The cover is made with stone vaults.

We have done a study on the structural behaviour of *Girola*’s vault, in Tortosa’s cathedral.

We have made this study by using a three-dimensional model of *Girola’s* vault, knowing the construction materials and the weight it receives. We have obtained a model which allows us to see the parts that are more likely to be deformed.

We repeated this procedure with two different hypotheses, the first is made without ceiling and, the second one is made with. These two models allowed us to compare their distribution of loads and the vault’s deformations.

**PREVIOUS CONSIDERATIONS:**

MATERIAL:

Santa Maria’s cathedral is done with a single material. It uses a type of rock: sandstone. Their resistance values are: E = 5e10 N/m2; δ = 280N/m3; ѵ = 0.25.

This rock has a very high resistance to compression, but zero traction.

STATE CHARGES:

The analyzed vault receives two different charges: the first charge is applied in the outer edge and the second charge is applied in the inner edge. The outer edge, receives the charge of the top wall. This wall has dimensions of 2.7 · 0.9 · 0.5 m3, multiplied by the density of the stone, giving a total of 340N. The inside edge, gets a wall load of 2.7 · 0.45 · 3m3, giving a total of 1020N.

DEGREES OF FREEDOM:

We considered that the base has no degree of freedom so it can’t be moved in any direction. However, the vault limits placed on his left and right has got two degrees of freedom. The limits can move in X and Z directions.

**PROCESS:**

1.Collecting data on the Cathedral.

2.Materials characteristics and construction elements.

3.Making a 3D model of the vaults (AutoCAD).

4.Meshing the model (SalomeMeca).

5.Enter the characteristics of the model (materials).

6.Calculation.

7.Analysis of results.

**HYPOTESIS:**

*
HYPOTHESIS 1: *Modeling the vaults without ceiling.

*HYPOTHESIS 2: *Modeling the vaults and ceilings without stone key associated with them.

*HYPOTHESIS 3: *Modeling the vaults and ceilings with a stone key associated with them.

**METHODOLOGY:**

We have made a three-dimensional modeling with AutoCad program. The complete vault is modeled with its four pillars (although it’s only needed a half). There has been one model for each of the hypotheses.

Each model we imported into Rhinoceros program has been transformed into . Step format. Then we introduced the model into SalomeMeca by Linux.

After entering geometry of the vaults, we meshed it. We have chosen the parts to be able to be charged. Finally, we have determinate the freedom of movement of each piece.

After apply the weights, we obtain a deformation model. We have adjusted the parameters to differentiate areas that have higher or lower tensions. It has been applied in the first hypothesis.

We increased the contrast for greater precision and to be able to differentiate the vaults that are more deformed.

**CONCLUSIONS:**

In the first hypothesis model and its graphic, we see that most part of the vault is subjected to a very high compression effort. In contrast, when we get to the base, tensions are gradually increased (as below, more tension).

Due to the geometry and mass of the elements there is a higher deformation in the vaults than in the walls. We also see the stone key who receive high deformations. Furthermore, the lateral limits have a different behaviour. While the outer limit is a continuous mass (wall), the inner limit is an arch, that concentrates efforts on the vertical elements: the props.

**Researchers team: Maria Franch, Gerard Olivé and Cristina Olucha **