# Analysis of Seismic Action on the Tie Rod System in Historic Buildings Using Finite Element Model Updating

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Analysis

#### 2.1. Experimental Investigation of the Structure

#### 2.2. Experimental Investigation of Tie Rods

## 3. Numerical Analysis

#### 3.1. Initial Numerical Model of the Cathedral

^{3}for material density. For the stone masonry, material density was 2490 kg/m

^{3}and the modulus of the elasticity was assumed as 20 GPa. The material behavior of stone was described by linear theory and homogeneous behavior was assumed. The sum of the material characteristics assigned to different cross-section properties of the global model of St. James Cathedral is shown in Table 2.

#### 3.2. Results of the Initial Numerical Model of the Cathedral

## 4. Finite Element Model Updating

#### 4.1. Manual FEMU

#### 4.2. Evaluation of the Updated Numerical Model Using Modal Assurance Factor Criteria (MAC)

#### 4.3. Results of the Evaluation of the Numerical Model

## 5. Analysis of Seismic Action of the Tie Rod Systems

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Cathedral of the Assumption of Mary, Zagreb: (

**a**) Fractured tie rod after its load bearing capacity was reached (

**b**) Damaged vaults after the March 22nd 2020 Zagreb earthquake.

**Figure 5.**Experimentally obtained modal shapes of the St. James Cathedral in Šibenik: (

**a**) first, (

**b**) second and (

**c**) third modal shape.

**Figure 6.**Characteristics record of frequency domain decomposition (FDD) for determination of natural frequencies of the St. James Cathedral in Šibenik (red and green lines indicate the dominant natural frequencies associated with mode shapes present on Figure 5).

**Figure 8.**Stress levels obtained in the (

**a**) aluminum tie rods (level R2) and (

**b**) cast steel tie rods (level R4) in the Cathedral of St. James in Šibenik (all indicated values are in MPa).

**Figure 10.**Experimentally obtained normalized mode shape vector for the selected measurement points p (1–7).

**Figure 11.**Deformation of the model updated finite element model of the cathedral due to seismic actions in (

**a**) x and (

**b**) y direction.

**Figure 12.**Graphical comparison of stress level in the aluminum tie rod in axis A–B and C–D at the level R2 for the initial state and seismic action in x and y-direction.

**Figure 13.**Graphical comparison of stress level in the aluminum tie rod in axis B and C at the level R2 for the initial state and seismic action in x and y-direction.

**Figure 14.**Graphical comparison of stress level in the cast steel tie rod in axis B-C, B and C at the level R4 for the initial state and seismic action in x and y-direction.

Mode, n | $\begin{array}{c}{\mathbf{f}}_{\mathbf{n}}^{\mathbf{e}\mathbf{x}\mathbf{p}}\\ \left(\mathbf{Hz}\right)\end{array}$ | Direction |
---|---|---|

1 | 3.75 | Bending around x axis, symmetrical |

2 | 5.55 | Bending around x axis, antimetrical |

3 | 5.85 | Second bending mode shape around the x axis |

**Table 2.**Material characteristics assigned to the different cross-section properties of the global numerical initial model of St. James Cathedral in Šibenik.

Material | Material Density, ρ (kg/m ^{3}) | Elasticity Modulus E (GPa) |
---|---|---|

Stone | 2491 | 20 |

Cast Steel | 7697 | 185 |

Aluminum | 2660 | 70 |

**Table 3.**Comparison of the numerically $\left({\mathrm{f}}_{\mathrm{n}}^{\mathrm{num}}\right)$ and experimentally $\left({\mathrm{f}}_{\mathrm{n}}^{\mathrm{exp}}\right)$ obtained natural frequency values.

Mode, n | Direction | ${\mathbf{f}}_{\mathbf{n}}^{\mathbf{num}}$ (Hz) | ${\mathbf{f}}_{\mathbf{n}}^{\mathbf{exp}}$ (Hz) | Natural Frequency Error (%) |
---|---|---|---|---|

1 | Bending around x axis, symmetrical | 3.13 | 3.75 | 16.5 |

2 | Bending around x axis, antimetrical | 5.14 | 5.55 | 6.5 |

3 | Second bending mode shape around x axis | 6.24 | 5.85 | 6.3 |

**Table 4.**Changing of the natural frequency values $\left({\mathrm{f}}_{\mathrm{n}}^{\mathrm{num}}\right)$ depending on the stone elasticity modulus $\left({\mathrm{E}}_{\mathrm{n}}^{\mathrm{num}}\right)$ for the different types of boundary conditions.

(a) Hinge | (b) Clamped | ||||||

Mode, n | 1 | 2 | 3 | Mode, n | 1 | 2 | 3 |

${E}_{\mathrm{n}}^{\mathrm{num}}$(GPa) | 27.5 | 26.0 | 25.5 | ${E}_{\mathrm{n}}^{\mathrm{num}}$(GPa) | 21.5 | 24.0 | 24.5 |

Average | 26.3 GPa | Average | 23.3 GPa | ||||

(c) Winkler springs | |||||||

Mode, n | 1 | 2 | 3 | ||||

${k}_{\mathrm{n}}^{\mathrm{num}}$(kN/m) | 15,000 | - | - | ||||

Average | 15,000 kN/m |

Model | Boundary Condition | ${\mathbf{E}}_{\mathbf{n}}^{\mathbf{n}\mathbf{u}\mathbf{m}}$ (GPa) | ${\mathbf{k}}_{\mathbf{n}}^{\mathbf{n}\mathbf{u}\mathbf{m}}$ (N/m) |
---|---|---|---|

1. HBC | ${\mathrm{u}}_{\mathrm{x}}{,\mathrm{u}}_{\mathrm{y}}{,\mathrm{u}}_{\mathrm{z}}=0$ | 26.5 | - |

2. CBC | ${\mathrm{u}}_{\mathrm{x}}{,\mathrm{u}}_{\mathrm{y}}{,\mathrm{u}}_{\mathrm{z}}=0$ ${\mathsf{\phi}}_{\mathrm{x}}{,\mathsf{\phi}}_{\mathrm{y}}{,\mathsf{\phi}}_{\mathrm{z}}=0$ | 23.2 | - |

3. ELS | ${\mathrm{u}}_{\mathrm{x}}{,\mathrm{u}}_{\mathrm{y}}=0$ | 25 | 15,000 |

Numerical Model | ||
---|---|---|

1. HBC | 2. CBC | 3. ELS |

(a) Numerically obtained normalized mode shape vector for the selected measurement points (1–7) | ||

${\mathrm{MAC}}_{\mathrm{M}1-\mathrm{E}}=\left[\begin{array}{ccc}0.988& 0.498& 0.626\\ 0.596& 0.943& 0.185\\ 0.805& 0.195& 0.816\end{array}\right]$ | ${\mathrm{MAC}}_{\mathrm{M}2-\mathrm{E}}=\left[\begin{array}{ccc}0.984& 0.313& 0.608\\ 0.780& 0.235& 0.971\\ 0.863& 0.245& 0.870\end{array}\right]$ | ${\mathrm{MAC}}_{\mathrm{M}3-\mathrm{E}}=\left[\begin{array}{ccc}0.996& 0.224& 0.661\\ 0.780& 0.166& 0.766\\ 0.891& 0.596& 0.722\end{array}\right]$ |

(b) MAC matrices for comparison of the numerically and experimentally obtained normalized mode shape vectors for the selected measurement points (1–7) | ||

(c) Graphical representation of MAC matrices for comparison of the numerically and experimentally obtained normalized mode shape vectors for the selected measurement points (1–7) |

**Table 7.**Values of forces and stress levels in the aluminum tie rods in the axis at the level R2 for the seismic action in the x and y-direction.

Tie Rod | 2 A–B | 3 A–B | 4 A–B | 5 A–B | 6 A–B | 1–2 B | 2–3 B | 3–4 B | 4–5 B | 5–6 B | 6–7 B |
---|---|---|---|---|---|---|---|---|---|---|---|

EarthquakeDirection | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) |

x | 15.68 | 16.26 | 16.84 | 16.84 | 14.10 | 18.92 | 14.16 | 15.36 | 15.32 | 16.72 | 17.79 |

y | 15.83 | 16.41 | 17.00 | 17.00 | 14.23 | 18.67 | 13.62 | 14.78 | 15.93 | 16.09 | 17.11 |

Tie rod | 2 C–D | 3 C–D | 4 C–D | 5 C–D | 6 C–D | 1–2 C | 2–3 C | 3–4 C | 4–5 C | 5–6 C | 6–7 |

EarthquakeDirection | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) |

x | 15.25 | 16.04 | 15.84 | 16.45 | 15.87 | 19.18 | 15.08 | 15.98 | 16.55 | 16.47 | 17.48 |

y | 15.39 | 16.18 | 15.98 | 16.60 | 16.01 | 18.45 | 14.51 | 15.38 | 15.92 | 15.85 | 16.81 |

**Table 8.**Values of forces and stress levels in the aluminum tie rod for the seismic action in x and y-direction.

Tie Rod | 2 B-C | 3 B-C | 4 B-C | 5 B-C | 6 B-C | 7 B-C | 7‒8 B | 7‒8 C |
---|---|---|---|---|---|---|---|---|

EarthquakeDirection | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) | σ_{n}(MPa) |

x | 50.26 | 43.85 | 47.52 | 46.83 | 40.51 | 66.83 | 71.35 | 72.18 |

y | 49.54 | 45.10 | 46.82 | 47.23 | 44.06 | 67.86 | 68.56 | 70.39 |

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**MDPI and ACS Style**

Ereiz, S.; Duvnjak, I.; Damjanović, D.; Bartolac, M. Analysis of Seismic Action on the Tie Rod System in Historic Buildings Using Finite Element Model Updating. *Buildings* **2021**, *11*, 453.
https://doi.org/10.3390/buildings11100453

**AMA Style**

Ereiz S, Duvnjak I, Damjanović D, Bartolac M. Analysis of Seismic Action on the Tie Rod System in Historic Buildings Using Finite Element Model Updating. *Buildings*. 2021; 11(10):453.
https://doi.org/10.3390/buildings11100453

**Chicago/Turabian Style**

Ereiz, Suzana, Ivan Duvnjak, Domagoj Damjanović, and Marko Bartolac. 2021. "Analysis of Seismic Action on the Tie Rod System in Historic Buildings Using Finite Element Model Updating" *Buildings* 11, no. 10: 453.
https://doi.org/10.3390/buildings11100453