2.1. Motivation
The examined sacral building consists of several functional parts (
Figure 1), erected at different periods and using different techniques, combined with a wealth of changes that have occurred over several centuries of the object’s existence. As a result, discontinuities are visible at the junction of parts created at different times and using different methods. Also noticeable are uneven vibrations in various parts of the object during its operation.
Figure 2 shows the drawings of the body of the analyzed building. The walls of the presbytery are a log structure made from cut oak logs, with a cross-section of about 13 × 50 cm, reinforced with a wooden skeleton.
The inner surfaces are partly covered with medieval iconography and ornaments [
16]. Due to their age, the ends of the logs at the junction of the walls show clear signs of degradation (
Figure 3a).The walls of the nave are a simply supported frame structure with boarding, which was rebuilt over time, weakening it—the inclination of the beams is visible (
Figure 3b), as well as the disassembly of a part of the structure at the junction of the nave and the aisle (
Figure 3c) [
18].
In buildings consisting of parts created at different times, using different methods, weak spots usually occur at the junction of different types of construction, at the point where they are connected [
20]. It is expected that the most sensitive element of the building will be the intersection of the two oldest parts, the nave and the presbytery and, in particular, the connection between the presbytery and the main nave and the chapel and the aisle in the southeastern corner of the nave.
2.2. Mathematical Model for Measuring the Movements of the Wall of a Wooden Building
Old building structures are generally stable, but this does not preclude slow tilting, subsidence, or a combination of both. In the case of impulsive external forces (e.g., gusts of wind), it had been noticed that the structure reacts with forced vibrations. Thanks to the internal damping of the structure, after the pressure is released, the vibrations disappear and, finally, the structure returns to its initial state [
18]. It can, therefore, be assumed that, as a result, there are two types of phenomena: static, consisting of slow changes in the foundation and inclination; and dynamic, expressed in the form of vibrations in response to external pressure. While static displacements are best measured using geodetic instruments, this equipment cannot keep up with the measurement of the fast-changing phenomena characterizing the dynamics of the object. In the latter case, it is reasonable to use tilt sensors, e.g., inclinometers, tilt meters, or electronic levels. The use of both measurement methods requires the determination of the relationship between the inclination and the displacement of individual elements (points) of the tested object. Secondly, it is necessary to bring the measurement results to a common frame of reference.
A mathematical model of the deformation of a column, wall, or any near-vertical element in an existing building requires certain initial assumptions. First of all, it is necessary to determine whether the lower node of the element can rotate freely (hinge) or is blocked from rotation (fixed). The second assumption concerns the flexibility of the analyzed bar. Therefore, the mathematical model of the test element enforces the number of necessary measurement points and the minimum number of inclinometers used per element.
2.2.1. One Inclinometer
In the simplest case, it can be assumed that the bottom support is a hinge, and the whole element rotates around the lower node as a rigid body (
Figure 4a). The origin of the coordinate system is taken at the bottom node, the
z-axis is directed upward along the element, and the horizontal displacements
are determined. In this case (model H1P), the displacements
are described by a linear function:
where
is the directional coefficient of the linear function,
is the angle of inclination of the element, and
is the height or the level for which the displacement is determined (
Figure 4a). A single inclinometer measurement is sufficient to determine the constant
. Therefore, in order to combine absolute and inertial measurements, it is possible to embed the tilt sensor in any place in the beam, and the tachymetric measuring target at the required height above the floor (i.e.,
).
Assuming that the beam is rigid, and the displacement vector
at height
is known, the displacement
at any level
can be calculated from the relation:
In particular, at the top of the rigid beam (), the displacement will have a maximum value .
For a rigidly fixed beam at the lower node, deformation is only possible if the element behaves elastically (
Figure 4b), meaning that the deflection line has a curvature determined by a higher-degree function. In this case (model F1P), the deflection has to be modeled by a curve described by a polynomial of at least the second degree:
where
and
are coefficients whose values should be determined. As the function representing the section rotation angle
is the derivative of the displacement function
, then:
It is worth noting that, in the case of small angles,
. From the boundary condition, one obtains:
which means that a measurement of the rotation
performed with a single inclinometer located at a given height
is enough to determine the value of the coefficient of the quadratic function:
In turn, the displacement
of any point located at height
can be obtained using the formula:
2.2.2. Two Inclinometers
In the case where the lower node is a hinge and the element deforms elastically, only the first boundary condition from set (5) is valid. To describe the deflection line with quadratic Function (3), we need two inclinometers (
Figure 4c) and two measurements of the inclination
and
in radians, at points located at heights
and
, respectively. In practice, it is necessary to precisely correlate the locations of at least two pairs of targets and inclinometers. Then, the coefficients of quadratic Function (3) have the form (8):
The displacement
of any point located at height
of any hinged-supported element (model H2P) can be calculated from the formula:
where the coefficient
and
are defined by Equation (8).
Assuming that the lower node is fixed (model F2P), using only two measurement points, the deflection line can be described by a third-degree function (see
Figure 4d):
and the inclination, or angle of rotation of the cross-section can be represented by a second-degree polynomial as the derivative of Function (10):
Using the two boundary conditions at the bottom node, the two coefficients of the above polynomials can be determined:
and with two inclination measurements
and
, the other two coefficients can be calculated as:
Finally, the displacement
of any point located at the height
of the element rigidly supported at the bottom (
Figure 4d) can be calculated from the expression:
where the coefficient
and
are defined by Equation (13).
2.2.3. Required Number of Inclinometers
Summarizing the above considerations, it can be concluded that the horizontal deflection function of any vertical element can be described by a polynomial of any degree (e.g., n). Assuming that one node is rigidly fixed, then the number of measurement points (inclinometers) one less than the degree of the polynomial (i.e., n − 1) is needed to determine the deflection function of this element. On the other hand, when the rotation of the support node is arbitrary or unknown, the same number of points at which the inclination is measured is needed as the degree of the polynomial adopted to describe the deflection (i.e., n).
All the above models assume the existence of a single beam, which is practically not reflected in spatial constructions. In each case, different inclinations can be expected along the wall of the object, and different ones across it. The former should assume values close to zero due to the high stiffness of the wall in its plane, while the latter should illustrate the reaction on forces perpendicular to the wall plane, e.g., the wind pressure on the object. It can be assumed that the complex structure of the wall causes the mutual constraint of individual structural elements, which is irrelevant in the case of movement in the transverse direction and, to a much lesser extent, reflects the interconnections of individual structural elements. The last statement is not always true, but it is sufficient to describe the object’s reaction to external pressure. In timber frame constructions, the columns are usually placed freely on the ground. However, they are connected with screws, formwork, and the possible filling of the wall plane. This limits the movement of the column in relation to the free model, but should not significantly distort the model of transverse inclination to the wall surface, especially in the simply supported variant.
A knowledge of the relationship between inclinations and displacements in the absolute system allows the definition of the relationships between the inclinometer readings and the tachymetric measurement.
2.3. Comparison of Results Obtained Using Different Mathematical Models
The results of the inclinometer measurements were used to determine the deflection line and the maximum displacement of the top of the selected column (see
Figure 5). Depending on which mathematical model was chosen for the analysis, the shape of the deflection line was different but, more importantly, different values for the maximum horizontal displacement were also obtained. Measurements from two inclinometers attached to the selected column were used for the analysis: the first (In1), marked as number 9 in
Figure 6, located at a height of
; and the second (In2) below, at a height of
. And the maximum displacement
was determined at the top of the column, at a height of
. The tilt values measured on a windy day at these points were:
In the case of the H1P model and the F1P model, measurements from only one device are sufficient to determine the deflection line.
Figure 5a shows a comparison of the results obtained when only the first inclinometer was used for the analysis (crosses on the graph) with the results when only the second inclinometer was used (circles on the graph). After the adoption of a linear deformation model for the column (model H1P) and hinge, larger displacement values were obtained than for curvilinear deformation and stiffened support (model F1P).
In the next analysis, measurements from both inclinometers were used to determine the curvilinear deformation of the second degree, with the hinge at the bottom (H2P model). The curvilinear deformation from the H2P model was also compared with the H1P straight-line models but, in this case, larger displacement values were obtained (
Figure 5b).
Finally, an analysis was performed in which measurements from both inclinometers were used, and the deformation was described by a third degree curve, with the lower node fixed (F2P model). In this case, displacement values were obtained that were closest to straight-line deformations (
Figure 5c). In addition, it was found that a single inclinometer placed at the top of the column allows the estimation of the maximum displacements in the straight-line model (H1P) similar to the curvilinear model (F2P), where measurements from two inclinometers are needed.
The maximum displacement values determined at the top of the column for the models considered are summarized in
Table 1. Based on the analysis, it can be concluded that, in the case of monitoring a historical object, when the maximum displacement values need to be determined, the best solution is to place one inclinometer in each column, but at the top of it.
2.4. Measurements
In order to check the technical conditions of, and the extent of deformations to, individual parts of the structure of the monitored building, and their reaction to external factors, the following actions were taken [
18]:
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).
Using the much more accurate tachymetric method, it was possible to correct the inclinometer results, which are less accurate due to the occurrence of possible time-varying systematic errors.
The arrangement of the target points and sensors is shown in
Figure 6. The targets are marked with numbers from 1 to 9, and the common places of targets and inclinometers are numbers 6, 7, 8 and 9. Static measurements were made in 1-month time intervals, which enabled the tracking of construction movements, and also the periodic control of the expected drift of the inclinometer indications.
2.5. Preliminary Interpretation of the Measurement Results of a Given Object
In the case of the tested object, the corners of the nave near the presbytery, i.e., in the vicinity of points 6 and 9 (shown in
Figure 6), were considered the weakest places. Both targets and inclinometers were installed there; additionally, inclinations were measured at two corner points on the rear wall of the chancel (points 7 and 8). According to the theoretical assumptions, BWsensing WF/WM 400 sensors were used to measure the inclinations, while the static displacements were measured using the Leica TCRP 1201+ total station, and weather conditions using the Sencor SWS 12500 WiFi weather station located near the facility, on its northwestern side, at a height of about 5 m. Assessing that the tested beams are not anchored to the foundations (hinges) and are characterized by a high stiffness, inclination measurements were made with one inclinometer (variant H1P) and converted to linear values for the floor level (i.e., 6.8 m). The obtained values are recorded on weekly charts. To illustrate some of these,
Figure 7 and
Figure 8 show two graphs of two-day oscillations (with lines showing the minimum and maximum values) along the
x- and
y-axes, as well as the values for the wind strength and temperature inside and outside the facility.
Figure 7 shows the state of the building in stable weather conditions; it shows smooth changes in the inclination along the
x-axis and slight vibrations in the y-direction resulting from the operation of the building.
Figure 8 illustrates the building’s response to gusts of wind. The correspondence of the inclinations with the readings for the wind gusts is visible (see
Figure 7c and
Figure 8c), confirming the validity of the assumptions made earlier. An important problem is the shift of the graphs (inclination) in relation to the initial indications, which could suggest slow changes in the inclination of the tested beams. Due to the stability of the object confirmed via the tachymetric measurement, it was assumed that this is the effect of the drift of the sensors used, so the amount of vibration of the object was determined each time in relation to the average readings for windless weather in a given month.
Based on the observation that, of all external factors, strong gusts of wind have the greatest influence on the structure, it is necessary to correlate the indications of the anemometer with the measured inclination. The dynamic nature of weather phenomena combined with the delayed reaction of the structure does not allow for a mathematical relationship between the two factors with a high frequency, so it was decided that they would be compared in 10 min blocks, based on (instantaneous) wind readings of gust strength and direction, and readings of extreme slope components, respectively, x min, x max, and y min, y max. For this purpose, the graphs illustrate the relationship between gusts of wind (as well as the temperature inside the building and outside, in its vicinity) and the inclination of the structure in selected locations. Atmospheric phenomena monitored by a weather station located near the object are also recorded in a 10 min cycle, thanks to which it measures, over the same time, the values of the same factors that affect the object.
The general correlation coefficient between the wind gust forces and slope magnitudes for the most vulnerable point No. 9 was calculated, according to the known formula:
where:
, —the i-th value of the slope of the column and the average slope at the level of the ceiling [mm],
, —the i-th gust of wind and the average value of the gust [m/s],
—the standard deviation values for the slope and for the gust.
The closer to unity (i.e., 1.0) the value of , the greater the correlation between the considered parameters. At node 9 (point No. 9), discussed here, a correlation coefficient of about 0.7 was obtained via comparing a one-day set of wind forces and slope readings from the two sensors, measured on a windy day.
As shown in
Figure 8, during strong wind, the vibrations occur in both directions, although, according to theoretical considerations, they are greater along the
y-axis. After the wind stops, there is stabilization, but residual shifts of points (column inclination) of up to about 5 mm remain. However, this is not a permanent trend because, over time, there is a return to the initial state, or a deviation of a different nature arises, although in the same range of a few millimeters.
2.6. The Problem of Compliance of the Inclinometer Indications with the Results of the Static Measurement
In order to correct the sensor indications throughout the test period, static measurements with a total station were made at 1-month intervals on quiet mornings, when there should be no vibrations in the tested structure. In this way, the components of the inclination of the measured points in the directions of the axes of the adopted coordinate system (and the component of height) were determined. Due to the analyses discussed here, the values at points 6 and 9 were the most important; the results obtained for both points in a given time interval are shown in
Figure 9. The graphs below show that the displacements along the
x- and
z-axes do not exceed 3 mm, and that those along the
y-axis do not exceed 4 mm but, most importantly, they do not show a significant trend of change.
Taking into account the standard errors in the tachymetric measurement, it was initially assumed that the object at the level of the ceiling does not show unambiguous horizontal displacements relative to the floor. Meanwhile, the inclinometer readings were characterized by a variability of a few to several millimeters even at rest and were different for different sensors. This observation is consistent with the specifications of the devices used and confirmed by the results of similar studies using inertial sensors [
18].
The heterogeneity of changes in readings suggests that they do not depend on the influence of external factors (identical measurement conditions throughout the facility), but only on the specificity of the sensor’s operation; they are also not linearly variable.
Two actions have been taken to correct the phenomenon of senor drift:
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.
The first approach was discussed in a paper at the FIG Congress in Warsaw [
18], indicating that, thanks to the second-degree correction, it is possible to “zero” the readings in a quiet time. It was also found that, for the description of the examined phenomenon of forced vibrations, the absolute values of the indications are not important, but their variability, as a result of the impulse pressure of external forces, is [
21].
The second corrective action was performed in the last phase of the research by moving sensor no. 7 to the structural element measured with sensor no. 9, but at a lower height (about 4.8 m). In further work, the upper point 9 was marked as In1, and the lower one as In2, in accordance with theoretical analyses. The angular values of the inclinations were, each time, converted into displacements under the ceiling (at the level of 6.8 m). The indications of both sensors were compared for a calm period, which was characterized by a small influence of the wind (unfortunately, no violent weather phenomena were recorded at that time). It turned out that both sensors show slightly different inclination values, which suggests that the tested beam does not meet the assumed stiffness condition, or the sensor indications are not sufficiently stable. Measurements taken with two inclinometers placed on a single column were further analyzed using mathematical models that describe the curvature of the element during its deformation.
2.7. Selected Final Results after Corrections of Inclinometers
In order to illustrate the obtained results,
Figure 10 show graphs of displacements of the same column measured by sensors In1 and In2, combining them with the indications of atmospheric conditions. Only once during the period under review—on the night of 17–18 February 2023—were significant wind gusts recorded.
The obtained results indicate a significant susceptibility of the tested structure of the nave to the impact of gusts of wind in the direction of the
y-axis. During the measurements on this very windy evening, the maximum displacement values of −12 mm were found at point 9 at the level of the ceiling (
Figure 10b). The results showing vibrations at different levels of the tested column differ slightly. This proves that the phenomenon of vibrations and, thus, horizontal displacements, is more complicated than it could initially be assumed. This, in turn, signals that, when designing a method of strengthening the discussed connection of walls, the object should be controlled at its various levels, while care is also taken of the ongoing control and correction of inclinometer readings.