Detailed analysis of the gravitational effects caused by the buildings in microgravity survey

In recent decades, there has been a rapid expansion of urban areas, where the phenomena that may pose threat to urban infrastructure (and therefore the residents) have been appearing with increased frequency. Some of these phenomena are related to conditions of a near-surface rock mass and can be examined by geophysical methods. An example of such phenomenon are continuous and discontinuous deformations of ground surface. They may have anthropogenic or natural origins. The most dangerous of them, are, obviously, sinkholes. The application of geophysical methods in these areas requires taking into account the urban factor, which is usually associated with the necessity to introduce appropriate corrections.

One of the geophysical methods that can be successfully applied in such conditions is microgravity method (Butler 1984; Gołębiowski et al. 2018; Loj 2014; Styles et al. 2005). An unquestionable advantage of this method lies in its simplicity, but unfortunately it has two basic limitations. One of them is an increased measurement error, resulting from increased ground vibrations, and the other is a gravity impact of a building on the measurements. As much as in the first case it is difficult or impossible to eliminate the error in the second case, but it can be reduced. One possibility is to ensure that the observation stations are not too close to the buildings. However, it reduces the scope of the method applications. The second option is to calculate a correction to eliminate the gravity effect of buildings (Śliz 1978; Panisoval et al. 2012; Dewu 2014; Dilalos et al. 2018). Building correction can also eliminate the impact of identified underground natural or anthropogenic objects (Golebiowski et al. 2016; Porzucek 2014).

Building correction calculation is connected with the determination of building geometry and mass. As it is commonly known, buildings are geometrically very complicated and for this reason in order to calculate the correction their geometry is frequently simplified. Most frequently, a building or its elements are approximated by rectangular prism or, sometimes, by polyhedron (Wójcicki 1993). It involves determining the density of the simplified objects, calculated on the basis of the building average density. This solution was submitted by Śliz in 1978. Analysing a typical four-storey tenement house, he calculated the density for each storey and presented the distribution of the building correction only for this case. The question may therefore be posed whether the average density is indeed optimal, and this is the main subject of this article.

Urbanized areas are areas that have been inhabited, built on and extended over the centuries and are characterized by the presence of buildings of various ages. Building age determines the types of walls, their thickness as well as the material they are made of. There are two types of walls: external and internal walls ([N1]PN-EN 1996-1-1+A1:2013-05/NA:2014-03 Eurokod6). Due to their function, external walls can be divided into load-bearing walls and stiffening walls, while internal walls can be divided into load-bearing walls, stiffening walls and partition walls. Load-bearing walls are used to carry vertical loads, as well as horizontal loads, limiting the displacements of the structure, while stiffening walls are used only to carry horizontal loads and maintain the stiffness of the structure. Partition walls do not carry any loads; they are only used to divide the internal space of the building. Regardless of the age, the function of the wall determines its thickness.

The analysis of a building correction distribution was carried out for three types of buildings, most frequently found in urban areas, i.e. a single-family detached house, tenement house and large-panel system building. The geometry was based on actual building plans, taking into account the type of materials to be used in construction (PN-EN ISO 10456; PN-EN ISO 6946:2017-10; Płuska 2009). For each case, we considered the variant of the building with cellar and without cellar.

The first building was the single-family detached home, composed of two storeys (ground floor and I floor) with external wall length 10 and 12 m. This house was built from external load-bearing wall and two internal load-bearing walls that are also stiffening wall. Due to the size of the wall and their thickness, two cases of house were considered. The first was the old-type house built of brick for which the thickness of the ground floor walls was 51 cm and first floor walls 38 cm. The brick density was taken as 1.8 g/cm³. The second was the modern house built of silicate brick, for which the thickness of the ground floor and I floor was the same 24 cm and the density was 1.9 g/cm³. This floor slab for both cases had a thickness of 15 cm and density of 2.3 g/cm³. On the basis of this, the wall model of these two buildings was created (Fig. 1).

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The second building was the multifamily house of medium size—the tenement house. This building had four storeys (ground floor and three floors) with external wall length 45 and 24 m. Each floor was divided into five flats with an area of about 150 or 200 m². The building was designed in a longitudinal system, which means that the external walls and longitudinal internal wall, separating the flat from the corridor, were the load-bearing walls, while the walls dividing the area into flats were the stiffening wall. Internal area of the flat was divided by partition wall. Building with the tenement house size was chosen for one important reason, namely that objects of this size often dominate in city centres, because they were built from the earliest times. Taking into account this fact, three cases of tenement houses of the sizes described above were used for the analysis.

The first case was the building from the Middle Ages for which the thickness of ground floor load-bearing walls was 90 cm, while the stiffening walls were 75 cm thick. The floor load-bearing walls were 75 cm thick and the stiffening walls were 60 cm. The partition walls for all storeys had the same thickness of 45 cm. The walls were made of solid bricks with the density of 2.0 g/cm³. The second case was the tenement house from the middle eighteenth century for which the ground floor load-bearing walls were 75 cm thick, while the stiffening walls were 60 cm. As in the previous case, the floor walls were smaller than the ground floor walls and had 60 cm thickness for load-bearing walls and 50 cm for stiffening walls. The partition walls for all storeys had the same thickness of 30 cm. The walls were made of bricks with the density of 1.8 g/cm³. The third was the new tenement house—modern apartment building. The ground floor load-bearing walls had 50 cm thickness and for floor had 38 cm thickness. The stiffening walls for all storeys had the same thickness of 38 cm. Similarly, the thickness of partition walls for all storeys was the same and had 12 cm. For the second and third cases, despite the different sizes of bricks, their density was the same and had 1.8 g/cm³ value. The floor slab for both cases had a thickness of 25 cm and density of 2.3 g/cm³. The above-described parameters were the basis for the creation of three wall models of tenement house (Fig. 2).

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The third building was from the twentieth century, named Plattenbau or large-panel system building or LPS (Basista 2001). LPS were multi-storey buildings founded on an identical segment repeated system built of two types of walls: load-bearing walls and partition walls. One segment, known as a tower block, size 72 × 24 m, consisting of 11 storeys (ground floor and ten floors) was selected for the analysis. Every storey was divided into 15 flats with an area of 62, 50 and 30 m² and two staircases. This house was built in a mixed system. It means that the load-bearing walls were both perpendicular and parallel to the longitudinal axis of the building and together with floor slab were the stiffing elements of the building and divided the internal space into flats. The partition walls were used to divide the internal flat space.

For all storeys, thickness of the load-bearing walls, partition walls and floor slab was constant and amounted to 30 cm, 14 cm and 20 cm, respectively. LPS was a building system using prefabricated concrete slabs. The slabs were made of reinforced concrete (RC) with 2.5 g/cm³ density, and this density was adopted for partition walls and floor slab. Due to need to maintain adequate wall heat, the load-bearing walls were built in three-layer system—RC, styrofoam and concrete, and for this type of wall, the density was assumed at 2.2 g/cm³. The above-described parameters were the basis for the creation of the wall models of LPS (Fig. 3).

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As described above, three types of buildings were selected for the analysis of building correction (Figs. 1, 2 and 3), which was called wall model. For the building correction analysis, it was assumed that the rectangular prism would be used to represent the shape of buildings. Therefore, in the wall model, each wall of the buildings was approximated with a single rectangular prism. The density of each prim was the same as the density of the material from which the wall was built. To calculate the building correction for wall model, the special program was created, using the Nagy algorithm (Nagy 1966). This is the algorithm used for calculating gravity effect from a rectangular prism, and before using the algorithm in the program, it was tested with Oasis Montaj and other programs for gravity modelling.

For each building, a grid of observation points was created, both outside and inside the building. Inside the building, the observation stations were located in two variants: for the building without cellar the station was on the ground floor, but for the building with cellar on the cellar floor. The value of density was calculated separately for outside and inside the building.

W_i—wall model density value, S_i—simplified model density value, n—number of stations.

These tables do not include the density for three cases: the cellar simplified model for observation stations on the ground floor; the cellar simplified model for observation station in cellar and ground floor simplified model; and whole building for observation stations on the ground floor. For two of them, it was necessary to calculate the building correction from the wall model because observation stations were affected not only by external but also by internal walls, so that the distribution of the correction was too complicated to use the simplified model to approximate a single object. In the third case, it was illogical to take measurement on the ground floor in the building with cellar.

As it could be expected, the most stations with the Δmax value above 20 nm/s² were observed in zone closest to the building. The analysis for all buildings shows that above 6 m from the building Δmax for all stations was smaller than assumed 20 nm/s².

The analysis of calculated Δmax showed that the greatest influence on correctness of calculated correction from building had incorrect density estimation for simplified model: ground floor, jointly higher floor and whole building. This is due to the fact that for these cases the Δmax was significantly increased during the determination of the optimum density values. The observation station distance was also affected, which was confirmed by the small, absolute variation of the correction for floors above the first floor. After analysing all data, it can be assumed that deviation in density from optimal ± 0.04 g/cm³ can be accepted.

The approximation of building with simplified model, usually rectangular prism, to calculate building corrections has been used for many years. Densities for these prisms have usually been taken as average densities resulting from mass of building’s parts and its volume. However, the question may be asked whether densities calculated in this way are optimal for calculating the correction. In this paper, density analysis was performed for six different buildings. The research was carried out for observation stations outside and inside the building. Inside the building, observation stations were on the ground floor and in the cellar.

The obtained results clearly confirmed that it is impossible to use the average density for simplified models of ground floor or cellar for internal station—the differences between the correction values from the wall model and the simplified model Δmax are too big. It also turned out that the calculated optimal density from external and internal stations was different. The first one was greater than the average density and the second smaller.

The analysis of the results also shows that the better results of correction value were obtained when the building was divided into the ground floor and rest of above-ground parts than when whole building was approximated by one rectangular prism. Nevertheless, problem with density of the ground floor still remains. This density was significantly different from the average value, and the differences were greater for building with thick and massive walls.

During the research, it also turned out that in case of simplified cellar model created it should be remembered that the average density should be calculated using its volume calculated on the inside the foundations and not calculated on the outside the building.

The research results clearly show that the closer building the station was the greater calculated error for simplified model was. Incorrect estimation of simplified model density increased of course building correction error calculated in stations. It is not possible to estimate what is the acceptable deviation from the optimal density but it can be assumed that if it does not exceed ± 0.04 g/cm³ the accuracy of the calculated correction is satisfactory.

It is also worth to note that the calculated error at any station for floor above the first floor was small, so that the range of the optimal density deviation was also larger.

It can be concluded that simplified model can be successfully used to calculate building correction, even for the stations within short distance to the building. However, it is important to remember about conditions specified in the article.