Different Approach for the Structure Inclination Determination

Abstract

The current engineering and building pace has reached localities where vast civil projects were not considered. The changes of the intravillan area may cause some vacant historical localities to become a boundary or even a part of occupied area. The proximity of designed civil projects to historical structures may have great impact on their stability, and it is recommended or even legislatively set to monitor the possible changes in their shape or position. In case of protected structures, it is convenient to find a non-invasive way to measure and monitor historical structures if possible. Many data acquisition methods used in civil engineering for various purposes have gone through significant technological progress and enable the new ways of data collection. It is needed to focus on these methods from an application and precision point of view.

1. Introduction

Developments in the field of construction have contributed to the emergence of progressive building materials and technological construction procedures. As a result, a significant number of new construction objects were created and are constantly being created, which are unique in their dimensions, shape and construction and many times have overcome the limits of human activity. The research task results from the requirements of engineering practice. In some rural areas, land management plans are limited by geomorphological and topological characteristics. The changes increase the inhabitant capacity, influencing the near surroundings of the historical structures by enabling civil processes there [1]. These civil processes, hand in hand with the geological, geomorphological and topographic conditions, may lead to the structures’ shape deformation, stability loss or even collapse. A construction represents a collection of structural, technical and technological parts, creating one compact unit with mutually defined connections. Their mutual interaction due to the operation of the object, instability of the subsoil and irregular mechanical loading, as well as external environmental influences, affect its stability. Emerging disturbances, displacements and transformations negatively affect its functionality and service life. Regular monitoring of the object or its parts, by geodetic and other means, enables emerging disturbances to be detected. It also creates space to realize the preventive steps for the object security in time. Depending on the inspection frequency of the construction object, occasional or permanent monitoring may be mentioned, chosen and used. Nowadays, classic geodetic methods and instruments have also been replaced by automated measuring systems. The automated measuring systems’ implementation is based on the integration of several electronic (geodetic and non-geodetic) devices and sensors into one unit with a precisely defined configuration, structure and parameters. The main benefit of these systems is that they enable permanent monitoring of the object in a specified time and scope without the need to restrict the operation and access of the system operator to the monitored construction object. The goal of such a monitoring system is to provide the builder, through complex information about the current state of the construction object, with the possibility of predicting its risky or emergency state to the building operator.

Not only in the field of modern architecture, but also in the field of historical structures, researchers focus their attention on safety hazards related to structures. Because the safety performance of different civil engineering structures is closely related to human life, property and health, if a building structure has safety risks, there will be unpredictable casualties [2]. For the construction of engineering structures such as buildings or bridges, as they are often affected by various external forces (whose effects need to be taken into considerations a priori in the designing and construction itself), including sudden natural disasters, i.e., earthquakes and possible consequent tsunamis depending on the locality or typhoons, the internal structure may and will over time undergo significant changes, and serious damage will come into being [3]. Ongoing progress within the civil engineering field in relation to materials and structure aging problems is gaining importance. The Ministry of Transport of the Slovak Republic conducted a study devoted to checking the quality conditions of bridges. According to the research, the number of bridges with unsatisfactory conditions compared to data from 2011 has increased by 3 times, which represents 36% of the first-class road bridges that are used on a daily basis. Mentioning the bridge structure conditions above was to demonstrate the course of the structures’ quality, and the bridges have been the only highly inspected category so far. This research reflects the relationship between residents and civil structures and points out the social debts of various structures.

To observe the negative impacts of civil processes, it is requested to monitor potentially influenced structures before the construction initiation by non-invasive methods if possible. The shape changes result from the comparison of time-series measurements with the reference measurement. The most effective method is to start with epoch measurement before the civil processes themselves begin. An important part of such research includes the clarification of the geological conditions in the locality in question, which are often anthropogenically changed. The issue of research, design and implementation of automated measuring systems and the implementation of modern technologies in the field of long-term measurement of construction object displacements and transformations is collectively referred to abroad as structural health monitoring.

2. Locality and Requirements Definition

According to the planned and designed civil processes in the examination locality, a demand for historical structure monitoring came to observe, predict and take measure of the structure’s stability. The subject of the research is a tower of the Church of St. Cosmas and Damian in the village of Sedliacka Dubová, which is located in the northern part of central Slovakia in the district of Dolný Kubín. The examined structure was built on the basis of the native gothic wooden church, whose date of establishment is not known, but the first written mention is dated back to the end of the 14th century, in 1397. This church’s significance was based on the fact of it being among the few (in total five) Christian centers of the Orava region. Since that time, the church has overcome many architectonical changes. The Renaissance stone tower, which is the subject of the observation, was built in the first half of the 17th century [4].

The nave of the church was wooden. The outline of the roof beams is still visible on the tower, which indicates that this wooden nave was somewhat smaller than the present stone nave. In the years 1735–1754, the church was rebuilt again, and the wooden nave was replaced by a stone baroque nave. The nave was covered with a stone vault, which was also protected from the weather by a wooden roof. As part of this reconstruction, two gates in the perimeter wall of the church leading to the two nearest villages—Sedliacka Dubová and Dlhá and Oravou—and two chapels built next to the Dlhá and Oravou gate were probably created.

Nowadays, the walls of the church and tower, the perimeter walls with the two gates, the ossuary behind the church and the remains of the two chapels are preserved. Until 1998, it was abandoned and almost inaccessible due to vegetation. The project “Dubova colonorum” dedicated to the preservation of the monument is trying to preserve it, so it is currently possible to visit it. The ossuary was reconstructed on the original foundations; the other parts are only preserved and cleaned.

The church, including the Renaissance tower (See Figure 1), is located on the upgrade above the village.

The slope between the church and nearest inhabited houses is quite steep, i.e., 27.5%. This so-far unbuilt steep meadow is planned to go through vast civil construction to expand the built-up part of the village.

Deformation Measurement

Construction object monitoring includes a complex of activities associated with determining the current state, as well as changes in the object, from the point of view of its static as well as dynamic properties. The information about the state of the object is obtained by measuring and comparing basic physical quantities (e.g., angle, length, acceleration, temperature, pressure and others), on the basis of which quantitative and qualitative characteristics of the object are defined using mathematical–statistical methods. International or national standards set the methods and conditions for the individual construction types’ monitoring [5,6,7,8].

The aims of the deformation measurements include the stability control of reference points, especially the positions of geodetic instruments; the spatial displacements and transformations of the supporting structure determination; the determination of transformations of exposed parts of the structure; transverse and longitudinal tilting of the supporting structure in selected places; and the dynamic transformations of the supporting structure in selected places. An equally important part of automated monitoring system design is the issue of control, testing, rectification and calibration of automated monitoring system components. The obtained information and results make it possible to respond to errors, shortcomings and software limitations of the used devices and sensors during the design and implementation of an automated monitoring system and thus build a resulting system that meets the requirements of functionality, quality and reliability.

Within deformation monitoring, the most commonly used components are universal measuring stations equipped with a system of automated aiming at the center of a reflecting prism (robotized UMS); multi-frequency GNSS equipment with the possibility of a data-recording density of 20 Hz and higher; digital leveling devices and leveling bars with a code scale; electrical sensors (tilt sensors, acceleration sensors, length sensors and sensors based on optical fibers); and meteorological stations. Other components are the signal amplifiers with A/D converters, the time server (local time server—LTS), the wi-fi antennas and network switches (switches), reflective prisms, portable computers with software equipment, central control, a registration computer (server) with software equipment, power and transmission cables and backup sources. Other components of automated monitoring systems that cannot be omitted are additional technical devices and aids for fixing and preparing sensors, reflective prisms and leveling cloths (e.g., brackets, leveling pads, spikes, lighting devices, stabilizing material and others). There is no test of the terrestrial laser scanner to be used for deformation monitoring. This method seems to be faster and cheaper, and it probably provides a significant amount of data to be post-processed. Another advantage also includes the fact of capturing the surroundings of the monitored area. These data may be used for future post-processing and better understanding of structure and its environment behaviors.

As historical structures belong under the management of competent authorities for monument and landmark protection, the choice of the measuring and monitoring process needs to meet their approval. Long-term monitoring requires signalization of clearly identifiable points, the positions of which are to be monitored to clarify the movement/deformation or static change of the structure [9,10,11,12,13]. Apart from difficult accessibility regarding the identifiable points’ stabilization, any intervention into the structure has been denied, so a non-invasive method for the data acquisition and a different approach for the data processing and evaluation must be chosen. The task was to perform non-invasive and accurate time-series monitoring measurements.

Due to the structure’s shape and positioning and due to the above-mentioned requests, the terrestrial laser scanning (TLS) method was chosen. TLS application has been so far described in many scientific works, for example in [14,15,16,17,18,19,20,21,22]. Its theoretical background is described in detail in, for example, [23]. This study’s task is to provide different approach for the analysis of the collected data within the time-series measurements to consider the influence of the nearby planned civil construction and its impact on the historical structure. In the deformation analysis, the significance of the detected displacement or deformation of the observed object is tested using the t-fold of the standard deviation, which defines the accuracy of the measurement. For this reason, part of the output of each measurement method is the definition of the internal accuracy, which is determined by the value of the a posteriori standard deviation. In this case, the shift was interpreted as a deviation of the normal vectors of the individual walls of the tower during the epoch measurements by a value of twice the standard deviation [24]:

$$\Delta \le 2{\sigma }_{\Delta },$$

(1)

If the measured difference in the position or height of the point does not exceed the value in Formula (1), it can be concluded that the recorded change in position is within the measurement accuracy, and the shift is not significant. Because it follows from the theory that the points representing the upper edge of the church tower are exposed to the greatest spatial distortion, the resulting remoteness was investigated at these points.

3. Data Acquisition and Analysis

The subject of the research was targeted from eight station positions in each epoch measurement to capture the whole structure and create a trustworthy and accurate 3D model of the structure to perform the necessary analyses for the inclination determination of the tower walls marked A–D (Figure 2).

The individual inclinations referring to the first reference epoch were essential for the spatial analysis and results evaluation. The measurement was performed using the a Leica ScanStation C10 terrestrial laser scanner with the a priori accuracy of the modelled surface set to 2 mm. The maximum mean error of the post-processed data reached 3.7 mm (the registration of all scans). To preserve the homogeneity of the epoch measurements, the object was measured with the same area density, i.e., 1.0 × 1.0 mm/10 m.

In case of the possibility of stabilizing and signalizing the observing points (targets, etc.), the task would have been solvable in a conventional geodetic way, and Fourier series and harmonic analysis could have been used. However, according to the preset conditions, consequently, the regression analysis geometry method was applied to individual epoch measurements to determine the direction and spatial orientation of the individual external planes representing the tower walls (A–D) (Figure 3). The mathematical representation of the tower walls followed the well-known formula for general plane expression [24]:

ax_i + by_i + cz_i + d = 0

(2)

where a, b, c and d are the plane parameters, and [x_i, y_i, z_i] are the spatial coordinates of random points of the plane.

The detailed theoretical background can be found in earlier research [24]. The shape of the monitored subject allowed the use of the chosen analytic approach for the determination of the possible position change and deformation of the structure. The chosen regression analysis geometry method is based on a plane estimation that represents the position and orientation of the individual tower walls. The importance of the points’ accurate position, representing the wall, is necessary, i.e., it is crucial to eliminate all gross errors within the post-processing using a proper way of testing the set of measured data on remote measurements (Fischer’s test, test of Rénvi, etc.) or other testing methods and analyses for various data types’ analysis and interpretation; see [25,26,27,28] or [29,30,31].

The mathematical interpretation of the tower wall is based on the estimated parameters of the general plane. This estimation may be performed using different mathematical models according to the type of input data. To calculate and estimate both unknown and measured parameters, it is suitable to use the Gauss–Mark model [32,33].

The acquired data, with gross errors eliminated, are used to fill the matrices and vectors of the Gauss–Mark model, the basic equation of which is in the following form [32]:

Adx + B^Tv + u = 0

(3)

The matrix A is a design matrix, the elements of which represent the calculated partial derivations of the general plane function (Formula (2)) by the unknown parameters, i.e., parameters a, b and c. In the case of matrix B, the general plane function is partially derived by the measured parameters, i.e., all the point coordinates x, y and z located on the individual observed walls A–D. After that, the vector dx, i.e., the unknown parameters vector, is calculated and includes the estimated values of the unknown parameters a, b and c of the plane. The vector u from Formula (3) is a closures vector. To define a design matrix, it is necessary to know the predefined general plane function equation. The general plane equation is defined by Formula (2), which presents the inputs, conditions and matrix dimensions defined in the least square theorem.

The theorem leads to an unequivocal result, where the vector dx^T is solved as [a,b,c,d] = [0,0,0,0]. To avoid the determinant value leading to zero within the Gauss–Mark estimation model calculation process, it is necessary to redefine Formula (2) as follows:

$$\frac{{\mathrm{ax}}_{\mathrm{i}}}{\mathrm{d}}+\frac{{\mathrm{bx}}_{\mathrm{i}}}{\mathrm{d}}+\frac{{\mathrm{cx}}_{\mathrm{i}}}{\mathrm{d}}+\frac{\mathrm{d}}{\mathrm{d}}=0$$

(4)

which may be written after substitution as:

$$\mathrm{f}:{{\mathrm{a}}^{\prime }\mathrm{x}}_{\mathrm{i}}+{{\mathrm{b}}^{\prime }\mathrm{y}}_{\mathrm{i}}+{{\mathrm{c}}^{\prime }\mathrm{z}}_{\mathrm{i}}+1=0,$$

(5)

The form of a plane in a general position in Formula (5) represents the data format to fill the matrices A and B, i.e., the format suitable for the necessary partial derivation in order to correctly define the parameters in the matrices A and B.

The parameters a, b and c of each observed wall A–D were estimated separately from the measured points resulting from post-processing of 3D measurements. The above-described scheme of parameter estimation after processing was applied to the final point cloud.

The point cloud consists of visually registered partial scans that were separated for each wall by predefined polygonal selection. The polygonal selection was performed to assure the homogeneity of all measuring epochs. Its purpose was also to eliminate the influence the objects which are not naturally part of the plane, such as windows, doors, portals and other anomalies. However, the polygonal section did not ensure an identical number of points representing the planes (A–D). The estimation analysis was applied to the selected points. The products of the analysis were in the form of normal vectors of the 3D parameters and the covariance. The vector changes between the individual epochs represented the movement of the tower walls.

4. Analysis Interpretation

The Gauss–Mark estimation provided the normal vector to express the position and orientation of every wall in every epoch, which means three normal vectors for every single wall. Because the estimation product is always the lowest combined solution, the a, b and c parameters are the least combination of three numbers [34,35]. To compare the vectors within the epochs, they must be calculated to constant values (in this case, the parameter a representing the x-axis orientation was replaced by constant 1). The other problem to be solved is the varying coordinate system defined for every epoch. It has influence on the normal vectors presented by the rotation of the vectors (Figure 4).

Before the mutual comparison, the rotation must be calculated and unified. In this case, the reference epoch represented the initial origin of the coordinate system, and the next epochs were transformed to fit the reference coordinate system of the first measurement (the scheme displayed in Figure 5). The basic characteristics of the normal vector allow focusing only on the x, y and z rotation, omitting the x, y and z translation and the scale factor change. The normal vector allows the determination of the wall inclinations relative to the horizon in general. The original normal vectors and vectors after processing are shown in

Table 1. Original normal vectors (after estimation) transformed into final shaped normal vectors (after rotation and constant valuation).

Pre-processed plane A				Post-processed plane A
Epoch	a	b	c	Epoch	a	b	c
I.	0.0802	−0.1265	0.0015	I.	1.0000	−1.5773	0.0187
II.	0.1623	−0.2024	0.0029	II.	1.0000	−1.5736	0.0180
III.	0.2778	0.0685	0.0050	III.	1.0000	−1.5653	0.0179
Pre-processed plane B				Post-processed plane B
Epoch	a	b	c	Epoch	a	b	c
I.	0.0464	0.0303	−0.0005	I.	1.0000	0.6535	−0.0101
II.	0.0442	0.0363	−0.0005	II.	1.0000	0.6553	−0.0106
III.	−0.0136	0.0525	−0.0001	III.	1.0000	0.6547	−0.0109
Pre-processed plane C				Post-processed plane C
Epoch	a	b	c	Epoch	a	b	c
I.	0.0408	−0.0616	−0.0002	I.	1.0000	−1.5100	−0.0049
II.	0.0599	−0.0726	−0.0003	II.	1.0000	−1.5186	−0.0051
III.	0.0941	0.0246	−0.0003	III.	1.0000	−1.5188	−0.0036
Pre-processed plane D				Post-processed plane D
Epoch	a	b	c	Epoch	a	b	c
I.	0.0737	0.0486	0.0002	I.	1.0000	0.6596	0.0022
II.	0.0711	0.0585	0.0001	II.	1.0000	0.6566	0.0020
III.	−0.0213	0.0818	0.0001	III.	1.0000	0.6609	0.0034

. Another way to express the change is to determine the point movement in the predefined height; in this case, the predefined height was 18 m, which is equal to the tower height.

For a reliable interpretation, the method tracks the tilt of the tower of the Church of St. Cosmas and Damian for all walls separately. Because it is a continuous object, it is appropriate to evaluate the slopes in the east–west direction, i.e., the section passing through the observed walls A and C, and in the north–south direction, i.e., the section passing through the axes of walls B and D. The angles representing the wall tilt in relation to the horizon are shown in

Table 2. Various ways of tilt interpretation.

Date of measurement	Movement representation in the E–W direction
	Wall tilt angle (°)		Shift in elev. 18 m (mm)
	A	C	A	C
30.08.	89°53′50.8″	89°58′46.08″	0.0	0.0
14.10.	89°53′47.78″	89°58′18.69″	0.2	2.4
16.11.	89°53′35.12″	89°58′14.33″	1.3	2.8
Date of measurement	Movement representation in the N–S direction
	Wall tilt angle (°)		Shift in elev. 18 m (mm)
	B	D	B	D
30.08.	89°56′32.11″	89°59′18.51″	0.0	0.0
14.10.	89°56′21.40″	89°59′14.75″	0.9	0.3
16.11.	89°56′15.23″	89°59′12.51″	1.5	0.5

. As mentioned above, the shift can be also interpreted by the point movement on the top of the wall (optional height 18 m for every wall). The changes in point deflection are also shown in Table 2. For better understanding, the varieties of tilt interpretation are illustrated in Figure 6.

5. Discussion

This paper points out the possibility of a non-invasive deformation measurement method using TLS. It shows one possible way of data processing, but post-processing of point cloud deformation calculation may be performed in other ways. For example, the point clouds during epochs can be georeferenced in a global or local coordinate system, and this can eliminate the transformation and rotation tasks. These types of collected data post-processing may be analyzed using other statistics methods, such as the Fourier transformation or harmonic analysis.

6. Conclusions

More and more structures, whether historical or other types, are becoming a part of UNESCO or other national programs on structure protection, that may cause previously non-existent obstacles such as impossibility of structural invasion and intervention to stabilize monitoring points. These obstacles may occur also within buildings which are difficult or impossible to access. A measuring method for such sites needs to meet the new requirements without losing accuracy. Geodetical methods used basically to collect large amounts of data such as TLS may find application in deformation and movement monitoring. The work with the point cloud is highly specific, and the method used for deformation measurement must put emphasis on the post-processing. This paper shows one of many ways of post-processing to enable deformation interpretation in two various ways. It also shows the hazards of such usage.

Because the results show that the standard deviation value was not exceeded in either direction between the individual epochs, no significant shift was demonstrated on the object. However, this may be due to a small interval between measurements, and for that reason, it would be advisable to repeat the measurement continuously.

The method itself seems to be precise enough but lacks the proof of the usual supplemental method. As it was impossible for monitoring points to be installed on the object, the case study did not enable comparisons. The problem should be tested and solved by measuring the object with possibility of monitoring point installation and measurement with not only TLS, but the usual geodetic measuring as well. Consequently, the new approach can be tested, and TLS may find a place among other conventional geodetic measuring methods. This type of post-processing is simplified by making the basic seven-parameter transformation into a three-parameter transformation (elimination of three translation parameters and the scale factor). To increase the reliability of the post-processing approach, it will be suitable to measure the deformation on a long-term scale with bigger time gaps between the individual epochs, or to solve the problem from other points of view, as in [36,37].