# Two Complementary Approaches toward Geodetic Monitoring of a Historic Wooden Church to Inspect Its Static and Dynamic Behavior

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## Abstract

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## 1. Introduction

## 2. Test Object and Methodology of Measurement

#### 2.1. Motivation

#### 2.2. Mathematical Model for Measuring the Movements of the Wall of a Wooden Building

#### 2.2.1. One Inclinometer

#### 2.2.2. Two Inclinometers

#### 2.2.3. Required Number of Inclinometers

#### 2.3. Comparison of Results Obtained Using Different Mathematical Models

#### 2.4. Measurements

- an orthogonal coordinate system (grid) was defined, with the x-axis directed along the church, from the center of the entrance door to the middle of the rear wall of the presbytery, and the y-axis directed to the right, in accordance with the geodetic definition of the grid (see Figure 6),
- for static measurements (steady state of the object), the tachymetric method was selected with the use of the automated Total Station Leica TCRP 1201+, with the 1″ precision of angle measurement and 2 mm precision of measured distances, and targets made from reflective foil (for distances of about 6 m, this precision gives the standard error RMSE for determining a horizontal displacement not greater than 0.05 mm),
- for dynamic measurements (vibrations), the electronic inclinometer BWsensing WF/WM series 400 was used, with an inclination measurement accuracy of 0.005 ± 0.001° (for a wall about 7 m high, this corresponds to linear values of 0.60 mm ± 0.12 mm, where the second component is time-variable).

#### 2.5. Preliminary Interpretation of the Measurement Results of a Given Object

- ${w}_{i}$, ${\overline{w}}_{n}$—the i-th value of the slope of the column and the average slope at the level of the ceiling [mm],
- ${u}_{i}$, ${\overline{u}}_{n}$—the i-th gust of wind and the average value of the gust [m/s],
- ${\sigma}_{w},{\sigma}_{u}$—the standard deviation values for the slope and for the gust.

#### 2.6. The Problem of Compliance of the Inclinometer Indications with the Results of the Static Measurement

- the correction of indications using second-degree polynomial equations as a function of time (in a 1-month step of readings),
- the simultaneous measurement of the inclination of the same element with two sensors.

#### 2.7. Selected Final Results after Corrections of Inclinometers

## 3. Discussion

## 4. Conclusions and Further Work

- It provides the results of the classical long-term (static) monitoring of the horizontal and vertical displacements of the object at its critical points;
- It monitors vibrations—horizontal displacements caused by the dynamic impact of atmospheric factors, mainly gusts of wind;
- It enables the calibration of inertial sensors based on the results of static measurements;
- It associates the size of dynamic displacements with the measured values of the weather parameters that cause them;
- It illustrates the variability of displacements of individual fragments of the same structural element, which enables a more complete understanding of the nature of the response of the object at a given place to the external factors;
- It allows the determination of the curvature of the deformed element under load, which makes it possible to estimate the method of support (free rotation or elastic clamping).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Side view of the tested object—individual parts were created at different times and with different techniques, and their connections reveal the different behavior of the joined parts (source: own photograph).

**Figure 3.**Examples of damage or discontinuity in the wall structure: (

**a**) wear to the log construction joints, (

**b**) inclination of the frame structure beams in relation to the (newer) windows, (

**c**) removed or cut struts reinforcing the frame structure (source: own photographs).

**Figure 4.**Illustration of assumptions for modelling column deflection lines with different bottom supports: (

**a**) hinged, linear function, (

**b**) fixed, quadratic function, (

**c**) hinged, quadratic function, (

**d**) fixed, cubic function, (

**e**) fixed, higher-degree polynomial. The blue circle is the measuring point (inclinometer).

**Figure 5.**Deformation diagrams of the analyzed column determined for different models: (

**a**) hinge and linear functions (dash line), fixed with the quadratic function (solid line), (

**b**) hinge with linear (dash line) and quadratic functions (solid line), (

**c**) hinge and linear functions (dash line), fixed with the cubic function (solid line).

**Figure 6.**Scheme of the body of the church, with the location of the main components of the monitoring system—the robotic total station (RTS), targets (green squares), and inclinometers (blue circles).

**Figure 7.**Inclinations for calm weather on 30–31 July 2022: (

**a**) in point 6, (

**b**) in point 9, in the longitudinal direction x (at the top), in the transverse direction y (at the bottom), (

**c**) the temperature inside and outside the building (15 ÷ 25 °C) and wind gusts (0 ÷ 5.2 m/s).

**Figure 8.**Inclinations for windy weather on 3–4 February 2023: (

**a**) in point 6, (

**b**) in point 9, in the longitudinal direction x (at the top), in the transverse direction y (at the bottom), (

**c**) the temperature inside (7.4 ÷ 10.2 °C) and outside the building (−2.0 ÷ 5.3 °C), and wind gusts (0 ÷ 11.0 m/s).

**Figure 9.**Diagrams of horizontal displacements in 19 measurement series at points 6 and 9 most susceptible to vibrations: (

**a**) along the church—x-axis, (

**b**) across the church—y-axis, (

**c**) vertically—z-axis [mm].

**Figure 10.**Summary of the inclination values of the part of the structure in the most vulnerable location (point 9) at levels 4.6 (In2) and 6.05 m (In1): (

**a**) the inclination along the x-axis, (

**b**) the inclination along the y-axis, and (

**c**) the readings for the wind force (0.2 $\xf7$ 8.6 m/s) and temperature, in a cumulative 10 min cycle.

**Table 1.**The maximum displacement ${u}_{max}$ determined for the considered models at the top of the column (${h}_{max}=6.80\mathrm{m}$).

Model | Polynomial Degree | Bottom Support | Position and Number of Required Inclinometers | Maximum Displacement ${\mathit{u}}_{\mathit{m}\mathit{a}\mathit{x}}\left[\mathbf{m}\mathbf{m}\right]$ |
---|---|---|---|---|

H1P | 1 | hinge | In1 | 5.45 |

H1P | 1 | hinge | In2 | 6.75 |

F1P | 2 | fixed | In1 | 2.97 |

F1P | 2 | fixed | In2 | 4.60 |

H2P | 2 | hinge | In1 and In2 | 8.44 |

F2P | 3 | fixed | In1 and In2 | 5.20 |

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**MDPI and ACS Style**

Pawlak, Z.M.; Wyczałek, I.; Marciniak, P.
Two Complementary Approaches toward Geodetic Monitoring of a Historic Wooden Church to Inspect Its Static and Dynamic Behavior. *Sensors* **2023**, *23*, 8392.
https://doi.org/10.3390/s23208392

**AMA Style**

Pawlak ZM, Wyczałek I, Marciniak P.
Two Complementary Approaches toward Geodetic Monitoring of a Historic Wooden Church to Inspect Its Static and Dynamic Behavior. *Sensors*. 2023; 23(20):8392.
https://doi.org/10.3390/s23208392

**Chicago/Turabian Style**

Pawlak, Zdzisław Mikołaj, Ireneusz Wyczałek, and Piotr Marciniak.
2023. "Two Complementary Approaches toward Geodetic Monitoring of a Historic Wooden Church to Inspect Its Static and Dynamic Behavior" *Sensors* 23, no. 20: 8392.
https://doi.org/10.3390/s23208392