# Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique

^{*}

## Abstract

**:**

## 1. Introduction

- The tie rods are generally located at elevated positions;
- High-accuracy displacement sensors should be used due to small vertical displacements. These should be placed on a previously-determined referent location;
- The strain gauge installation can be complicated at elevated positions.

_{n}), flexural stiffness (EI), mass (m), and boundary conditions. The method was verified on a historical building case study. One characteristic tie rod in this building was analyzed in detail. Based on these analyses of the characteristics of a single tie rod and the experimentally-determined values of the natural frequencies on the other tie rods in the building, the force in all the tie rods was determined reliably.

## 2. Analytical Solution for Lateral Tie Rod Vibration

_{z}) are acting in a vertical direction of an element and are responsible for balancing a weight of segment that varies in time along with element ($\mathsf{\rho}\mathrm{d}\mathrm{x}{\partial}^{2}\mathrm{w}/{\partial}^{2}\mathrm{t}$). The sum of vertical forces is equal to the product of the mass of the element and acceleration (Equation (1)):

_{z}and P

_{z}) and axial bending moment (M

_{y}). Substituting the bending moment with flexural stiffness (EI) and taking the second derivative of the deflection of beam (${\mathrm{M}}_{\mathrm{y}}=-\mathrm{E}\mathrm{I}{\partial}^{2}\mathrm{w}/\partial {\mathrm{x}}^{2}$) and a component of axial force (P

_{z}= Ptgα = P∂

_{w}/∂

_{x}, Equation (2b)) gives Equation (3)

_{n}is the nth natural frequency, l is a span of tie rod, m’ is mass per unit length ($\mathrm{m}\prime =\mathsf{\rho}\mathrm{b}\mathrm{h}$, b—width, and h—height of cross section) and κ is a boundary condition parameter, as presented in Table 1. By rearranging the previous equation, the boundary conditions (κ) can be determined:

_{n}), properties (EI, m′), and boundary conditions (κ), we can determine the axial force of a tie rod:

## 3. Methodology for Boundary Conditions and Axial Load Identification

## 4. Case Study Using the Proposed Methodology

#### 4.1. Description of Structure

#### 4.2. Experimental Identification of Dynamic Properties of Tie Rods

#### 4.3. Numerical Simulation

^{3}. Once the modal parameters were experimentally obtained, a numerical model was updated while changing the boundary conditions from fixed to hinged. To construct a real-life model, boundary conditions were additionally tuned with spring coefficients (stage 2 of the proposed methodology).

#### 4.3.1. Initial Model

#### 4.3.2. Updated Model

_{1}, P

_{2}) in the FE model, the results of natural frequencies were in good agreement with those measured for both mode shapes (Figure 13). The tuning of dynamic properties of the FE model enabled us to determine the actual tension in the tie rod (Table 4).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Cathedral of St. James in Šibenik (Croatia). Tie rods are supporting the walls and arches.

**Figure 3.**(

**a**) Beam under lateral vibration and axial loading and (

**b**) differential segment of beam representing the positive orientation of bending moments, shear forces, and axial and inertial forces of mass.

**Figure 6.**Cathedral of St. James in Šibenik, Croatia. (

**a**) West view of cathedral and (

**b**) iron tie rods inside the cathedral.

**Figure 10.**Characteristic record of frequency domain decomposition (FDD) for the determination of natural frequencies on reference iron tie rod 6B‒C.

**Figure 11.**Comparison of the experimental mode shapes of the reference tie rod 6B‒C and the mode shapes associated initial numerical models for the (

**a**) first and (

**b**) second mode shapes.

**Figure 12.**The values of the RMSE for the observed mode shapes and different spring stiffness values.

**Figure 13.**Change of force values depending on the ratio of numerical and experimental frequency values for two mode shapes (P

_{1}, P

_{2}).

**Figure 14.**Stress levels measured on the tie rods in the Cathedral of St. James in Šibenik at level R4.

Boundary Condition | Static System | Coefficient $\mathsf{\kappa}$ | ||
---|---|---|---|---|

1st Mode | 2nd Mode | nth Mode | ||

Hinge–hinge | 3.142 | 6.283 | $\cong \mathrm{n}\mathsf{\pi}$ | |

Clamp–clamp | 4.730 | 7.853 | $\cong \frac{\left(2\mathrm{n}+1\right)}{2}\mathsf{\pi}$ | |

Clamp–hinge | 3.927 | 7.069 | $\cong \frac{\left(4\mathrm{n}+1\right)}{4}\mathsf{\pi}$ | |

Clamp–free | 1.875 | 4.694 | $\cong \frac{\left(2\mathrm{n}-1\right)}{2}\mathsf{\pi}$ |

**Table 2.**Values of the first two frequencies of experimentally-observed tie rods in the Cathedral of St. James in Šibenik at level R4.

Tie Rod | L (m) | h (mm) | b (mm) | ${\mathrm{f}}_{\mathrm{1}}^{\mathrm{exp}}$ (Hz) | ${\mathrm{f}}_{\mathrm{2}}^{\mathrm{exp}}$ (Hz) |
---|---|---|---|---|---|

2B‒C | 6.84 | 55 | 55 | 7.25 | 17.94 |

3B‒C | 6.71 | 64 | 64 | 7.56 | 19.00 |

4B‒C | 6.81 | 60 | 60 | 7.31 | 18.69 |

5B‒C | 6.87 | 68 | 68 | 7.31 | 19.56 |

6B‒C | 6.90 | 61 | 61 | 6.94 | 17.50 |

7B‒C | 6.95 | 56 | 56 | 8.25 | 18.88 |

7‒8B | 6.97 | 56 | 56 | 8.13 | 19.38 |

7‒8C | 6.98 | 60 | 60 | 8.63 | 19.38 |

RMSE Values | ||
---|---|---|

Mode Number | Hinge–Hinge | Clamp–Clamp |

1 | 7.06 | 9.15 |

2 | 8.50 | 10.66 |

**Table 4.**Values of experimentally-measured natural frequencies and computed frequencies (${\mathrm{f}}_{\mathrm{n}}^{\mathrm{num}}$) for the updated numerical model.

Mode Number | ${\mathrm{f}}_{\mathrm{n}}^{\mathrm{num}}$ (Hz) | ${\mathrm{f}}_{\mathrm{n}}^{\mathrm{exp}}$ (Hz) | Relative Error (%) |
---|---|---|---|

1 | 4.83 | 6.94 | 30.20 |

2 | 13.95 | 17.50 | 20.29 |

**Table 5.**The force values read for the first two mode shapes and the value of coefficient κ determined by the calculation of the updated model.

Mode Number | ${\mathrm{f}}_{\mathrm{n}}^{\mathrm{num}}$ (Hz) | P (kN) | σ (MPa) | κ |
---|---|---|---|---|

1 | 6.94 | 122.8 | 33.0 | 3.534 |

2 | 17.5 | 137.2 | 36.9 | 6.777 |

**Table 6.**The value of forces P

_{n}(kN) and stress levels σ

_{n}(MPa) for the observed mode shapes in the tie rods observed in the Cathedral of St. James in Šibenik at level R4.

2B‒C | 3B‒C | 4B-C | 5B-C | ||||||
---|---|---|---|---|---|---|---|---|---|

Mode Num. | κ | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) |

1 | 3.534 | 115.8 | 38.3 | 149.6 | 36.5 | 132.1 | 36.7 | 159.4 | 34.5 |

2 | 6.777 | 144.7 | 47.8 | 158.8 | 38.8 | 167.7 | 46.6 | 207.9 | 45.0 |

Mean values | 130.3 | 43.1 | 154.2 | 37.7 | 149.9 | 41.6 | 183.6 | 39.7 | |

6B‒C | 7B‒C | 7‒8B | 7‒8C | ||||||

Mode Num. | κ | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) | P_{n}(kN) | σ_{n}(MPa) |

1 | 3.534 | 122.8 | 33.0 | 170.8 | 54.5 | 166.3 | 53.0 | 215.2 | 59.8 |

2 | 6.777 | 137.2 | 36.9 | 188.7 | 60.2 | 208.1 | 66.4 | 219.6 | 61.0 |

Mean values | 130.0 | 34.9 | 179.8 | 57.3 | 187.2 | 59.7 | 217.4 | 60.4 |

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**MDPI and ACS Style**

Duvnjak, I.; Ereiz, S.; Damjanović, D.; Bartolac, M. Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique. *Appl. Sci.* **2020**, *10*, 6036.
https://doi.org/10.3390/app10176036

**AMA Style**

Duvnjak I, Ereiz S, Damjanović D, Bartolac M. Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique. *Applied Sciences*. 2020; 10(17):6036.
https://doi.org/10.3390/app10176036

**Chicago/Turabian Style**

Duvnjak, Ivan, Suzana Ereiz, Domagoj Damjanović, and Marko Bartolac. 2020. "Determination of Axial Force in Tie Rods of Historical Buildings Using the Model-Updating Technique" *Applied Sciences* 10, no. 17: 6036.
https://doi.org/10.3390/app10176036