# Analysis of Cylindrical Masonry Shell in St. Jacob`s Church in Dolenja Trebuša, Slovenia—Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Stresses and Typical Damage Patterns of Masonry Shells

_{x}, N

_{y}

_{,}N

_{xy}and bending moments M

_{x}, M

_{y}

_{,}M

_{xy}the stresses which occur in the shell can be written as:

_{x}, σ

_{y}and τ

_{xy}within the shell thickness is linear. Perpendicular shear stresses τ

_{xz}in τ

_{yz}as a consequence of shear forces have a parabolic distribution, but their values are generally small in comparison with other stress components and can usually be neglected.

## 3. Analysis of Masonry Shell in St. Jacobs Church

#### 3.1. Characteristics of the 2004 Earthquake

#### 3.2. Structural Characteristics and Damage of Analyzed Masonry Shell

#### 3.3. Static Analysis of Masonry Shell

- Load case »1«: Self weight and rubble layer:$$\sum 1.35\ast G+\sum 1.35\ast {G}_{N}$$
- Load case »2«: Self weight, rubble layer, vertical displacement of outer supports on one side by Δ
_{z}= −0.5 cm:$$\sum 1.35\ast G+\sum 1.35\ast {G}_{N}+\sum 1.5\ast {\Delta}_{z}$$ - Load case »3«: Self weight, rubble layer, maintenance team:$$\sum 1.35\ast G+\sum 1.35\ast {G}_{N}+\sum 1.5\ast {Q}_{V}$$
- Load case »4«: Self weight, rubble layer, horizontal displacement of all supports on one side by Δ
_{y}= 0.5 cm:$$\sum 1.35\ast G+\sum 1.35\ast {G}_{N}+\sum 1.5\ast {\Delta}_{y}$$ - Load case »5«: Self weight, rubble layer, temperature difference:$$\sum 1.35\ast G+\sum 1.35\ast {G}_{N}+\sum 1.5\ast {\Delta}_{T}$$

_{c}) and tensile stress (σ

_{t}) should be smaller than the design compressive (f

_{cd}) and/or design tensile strength (f

_{td}) of the material, respectively. Values of the characteristic compressive (f

_{c}) and tensile strength (f

_{t}) were taken from the Technical Report, reference No. 6 and were obtained from investigations of similarly built stone masonry walls and vaults studied both in situ and in the laboratory of ZRMK. Design values are obtained by dividing the characteristic compressive (f

_{c}) and tensile strength (f

_{t}) of the material with the material safety factor γ

_{M}, as given by the code:

_{M}is assessed as γ

_{M}= 2.0. Calculated stresses should be smaller than permissible values. In the case of tension we obtain:

## 4. Discussion of the Results

_{z}= −0.5 cm) of outer supports on one side was considered which simulates differential settlement of foundations. In this case, high tensile stresses occur in the top area of the shell (Figure 11), which may cause the recorded longitudinal crack. Differential settlement of foundations can thus be one of possible causes for the registered crack pattern. However, the results of this load case otherwise show high stress levels also in other parts where no damage of the actual shell has been recorded.

_{T}= 15 °C) show that such loading combination is critical for collateral vaults but cannot be responsible for the registered longitudinal crack.

## 5. Possible Repair and Strengthening Measures

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Critical parts of the shell where the bending stress state can be formed (adapted after [3]).

**Figure 2.**Some typical crack configurations for barrel vault [4].

**Figure 4.**Registered crack pattern of analysed barrel vault before rehabilitation in 2013 [6].

**Figure 6.**3D FE model (

**a**) and cross section (

**b**) showing changing thickness of analysed barrel vault.

**Figure 7.**Load case simulating vertical displacement of outer supports on one side (

**a**) and horizontal displacement of all supports on one side (

**b**).

**Figure 9.**Selected points corresponding to stress values given in Table 1.

Point | 157 | 200 | 269 | 955 | 1174 | 1293 | 1301 | |
---|---|---|---|---|---|---|---|---|

LC1 | σ_{11top} [kN/m^{2}] | −41.28 | −36.73 | 52.70 | −33.07 | −33.53 | −71.39 | −68.44 |

σ_{11bot} [kN/m^{2}] | 44.92 | 7.10 | −97.54 | 1.48 | −2.49 | −26.30 | −30.91 | |

σ_{22top} [kN/m^{2}] | −119.26 | 124.74 | −12.98 | −147.14 | −156.93 | −177.86 | −179.27 | |

σ_{22bot} [kN/m^{2}] | −181.40 | -28.88 | -259.61 | −64.44 | −86.14 | −61.21 | −61.43 | |

LC2 | σ_{11top} [kN/m^{2}] | −120.11 | −265.17 | 567.59 | 120.82 | 13.75 | −180.22 | −177.48 |

σ_{11bot} [kN/m^{2}] | 50.19 | 513.14 | −447.47 | 29.13 | 86.94 | 85.03 | 83.61 | |

σ_{22top} [kN/m^{2}] | −240.61 | 344.94 | 321.98 | −158.93 | −334.60 | −531.38 | −560.17 | |

σ_{22bot} [kN/m^{2}] | −282.48 | 113.76 | −886.58 | −257.32 | −47.17 | 164.13 | 199.86 | |

LC3 | σ_{11top} [kN/m^{2}] | −75.67 | −66.08 | 91.63 | −44.53 | −136.14 | −354.65 | −158.88 |

σ_{11bot} [kN/m^{2}] | 82.37 | 12.28 | −161.91 | −17.07 | 50.18 | 148.84 | −42.61 | |

σ_{22top} [kN/m^{2}] | −204.97 | 196.11 | −32.57 | −209.95 | −363.76 | −608.23 | −436.52 | |

σ_{22bot} [kN/m^{2}] | −305.60 | −48.38 | −431.64 | −167.37 | −86.61 | 174.44 | 17.66 | |

LC4 | σ_{11top} [kN/m^{2}] | −63.50 | −66.64 | 82.45 | −60.25 | −57.92 | −112.90 | −107.81 |

σ_{11bot} [kN/m^{2}] | 71.79 | 5.53 | −153.84 | 10.56 | 1.21 | -38.65 | −46.36 | |

σ_{22top} [kN/m^{2}] | −212.72 | 186.28 | −39.18 | −269.19 | −277.32 | −301.05 | −302.35 | |

σ_{22bot} [kN/m^{2}] | −272.82 | −49.41 | −421.87 | −88.60 | −131.64 | −103.18 | −104.44 | |

LC5 | σ_{11top} [kN/m^{2}] | −84.97 | −103.62 | 42.07 | −72.93 | −67.18 | −123.88 | −118.42 |

σ_{11bot} [kN/m^{2}] | 94.06 | 54.80 | −120.62 | 20.64 | 8.32 | −30.73 | −38.42 | |

σ_{22top} [kN/m^{2}] | −228.95 | 167.47 | −67.97 | −265.28 | −269.97 | −296.05 | −297.56 | |

σ_{22bot} [kN/m^{2}] | −268.14 | −3.97 | −404.63 | −91.91 | −138.79 | −107.81 | −109.23 |

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**MDPI and ACS Style**

Uranjek, M.; Lorenci, T.; Skrinar, M.
Analysis of Cylindrical Masonry Shell in St. Jacob`s Church in Dolenja Trebuša, Slovenia—Case Study. *Buildings* **2019**, *9*, 127.
https://doi.org/10.3390/buildings9050127

**AMA Style**

Uranjek M, Lorenci T, Skrinar M.
Analysis of Cylindrical Masonry Shell in St. Jacob`s Church in Dolenja Trebuša, Slovenia—Case Study. *Buildings*. 2019; 9(5):127.
https://doi.org/10.3390/buildings9050127

**Chicago/Turabian Style**

Uranjek, Mojmir, Tadej Lorenci, and Matjaž Skrinar.
2019. "Analysis of Cylindrical Masonry Shell in St. Jacob`s Church in Dolenja Trebuša, Slovenia—Case Study" *Buildings* 9, no. 5: 127.
https://doi.org/10.3390/buildings9050127