Weighted average as a basis for determining the reliable tilt of a multi-segment building undergoing rectification

The tilt of a building from vertical is a common phenomenon observed in every region of the world. It affects various types of structures, including residential buildings (Kijanka & Kowalska, 2017), industrial structures (Dudek et al., 2021), historic masonry towers (Macchi, 2005), timber towers (Gromysz, 2019a) and timber/brick towers (Zhang et al., 2018), chimneys (Kaszowska et al., 2018), churches (Gromysz, 2019b), reservoirs (Gromysz, 2021), transmission towers (Liu & Liu, 2008), grain elevators (Baracous, 1957), and even high-rise buildings (Maffei & Gonçalves, 2016), bridge supports (Shi et al., 2013), piles (Peng, 2017), as well as high-voltage poles (Yuan et al., 2012) and even oil rigs (Rapoport & Alford, 1989) that are currently under construction.

The most common cause of tilting building structures from vertical is uneven subsidence of the ground. Notable examples include the 55 m high Leaning Tower of Pisa (Geng et al., 2020; Terracina, 1962), the 56 m high church tower in Bad Frankenhausen (Jänichen et al., 2020), the mid-14th century Oude Kerk church tower in Delft (Dabrowski et al., 2023), and the 500-year-old wooden tower of St. Bavo Church in Haarlem, Netherlands (Truong-Hong et al., 2021), which stands at 45 meters and has an approximate deflection of 0.63 meters. In Poland, the most common cause of tilting is underground coal mining (Lamich et al., 2016). As a result of this exploitation, uneven lowering of the ground surface of a continuous (Strzałkowski, 2022) and discontinuous (Paszek, 2024) nature occur, resulting in the leaning of structures from the vertical (Orwat, 2020).

The tilt of a structure is most often considered as a phenomenon that changes the technical properties of buildings, including their dynamic properties (Ahmari et al., 2015; Bao et al., 2021), reducing their comfort of use (Kijanka & Kowalska, 2017) and reducing the value of the real estate (Kowal, 2014). Tilting induces structural damage in wall-bearing systems (Strzałkowski, 2019), frame structures (Ren & Yan, 2015), and results in increased structural stress. Furthermore, as demonstrated by (Al’ Malul & Gadzhuntsev, 2018), tilting significantly elevates the probability of structural damage and failures.

Instances of tilt measurements, employing traditional geodetic methods, are documented in the works of (Halicka & Zyga, 2019) and (Orwat & Gromysz, 2021). These measurements facilitate the assessment of variations in the tilts and inclinations of individual structural components and enable an evaluation of the precision of building assembly. Ground-level laser measurement techniques permit real-time monitoring of tilt variations, including those induced by thermal effects and wind loads, and allow for the determination of the dynamic characteristics of structures. Conversely, aerial laser scanning provides the capacity to assess roof slopes, thereby enabling the calculation of building tilt (Yonglin et al., 2010).

A considerable number of structures exhibiting tilt are stabilized through foundation reinforcement techniques, such as the installation of driven micro-piles (Elsawwaf et al., 2023), steel piles (Yin et al., 2011) or reinforced concrete piles, in addition to ground reinforcement strategies (Macchi, 2005). In special situations, when the tilt exceeds 15 mm/m, the tilt is removed by means of hydraulic piston jacks built into the walls of buildings (rectification) (Gromysz, 2017). Rectification involves extending the pistons in the jacks, which results in the building being raised unevenly (Fig. 1a). The elevation of each corner (u_obj) depends on the building’s tilt (Fig. 1b). However, the tilt is not uniform; the slope of the floors often differs from the tilt of the walls, and each corner may tilt differently. Therefore, before any rectification can occur, it is essential to determine the reliable tilt of the building. This assessment forms the basis for calculating the required elevation adjustments for each corner.

Fig. 1.

Object rectification: a) building tilted from vertical, b) building after rectification; 1 – hydraulic piston jack, u_obj – high of the corner elevation (own research)

Rectification through uneven lifting is based on the rotation of an object about an axis that intersects the ‘zero’ point. In the context of building projections that approximate rectangular shapes (specifically, convex configurations), this ‘zero’ point corresponds to the highest positioned corner of the structure. The rotation of the building is induced by the differential elevation of its corners. The specific values required for corner elevation (u_obj) are derived from the assessed reliable tilt of the structure (Fig. 2).

Fig. 2.

Principle for determining corner elevation in a convex plan building: 1 – axis of rotation of the building, 2 – “zero” point of the segment, u_obj – high of the corner elevation, 3 – rectified building (own research)

For single-segment buildings with a load-bearing wall system, the method for determining the reliable tilt has already been developed (Gromysz, 2023). In single-storey buildings in particular, the wall tilt is the reliable value. In single-storey buildings with a usable attic, the slope of the attic floor is the reliable value. When planning the non-uniform elevating height of multi-storey buildings, the tilt from the vertical of the lift shafts should be taken as a reliable value. If there are no lifts in a multi-storey building, the tilt of an undeformed structural element can be used as a reliable reference. For buildings with rigid floor structures, the change in floor slope equals the change in wall tilt. In multi-segment buildings, there is no single method for determining the reliable deflection. This paper presents the case of a multi-segment building where the individual segments are functionally connected. The objective function for determining the reliable tilt of the building aimed to maintain its functionality after rectification.

The subject of this analysis was a school building comprising of five segments labeled “A”, “B”, “C”, “D”, and “E” (Fig. 3). The building exhibited a T-shaped plan with a total area of 1.281 m², which can be inscribed within a rectangle measuring 79.8 m by 37.7 m. The individual areas of the segments varied, with segment “B” encompassing 63 m² and segment “A” covering 354 m² “A” (Table 1). Segments “A”, “B”, and “C” contained classrooms that were accessible via corridors. Notably, these corridors traversed the expansion joints located between segments “A” and “B”, “B” and “C”, as well as “A” and “C”. The floors in these segments were on one level, which was taken as ± 0.00. Segment “D” accommodated gymnasium facilities and technical rooms, while segment “E” functioned as the gymnasium itself. The floor levels of segments “D” and “E” were situated at −1.22 m. Access from segment “C” to segment “D” was facilitated by seven steps (Fig. 3).

Fig. 3.

Plan of the five-segment school building, which features functionally interconnected segments, with the specified rectification parameters and lifting heights (u_obj) of the corners; 1 – axis of rotation of the segment, 2 – “zero” point of the segment (own research)

Segments “A”, “B”, and “C” were constructed with a full basement and two above-ground stories. The basement walls of these segments were composed of reinforced concrete, monolithic in design, and were capped with a reinforced concrete ring beam that extended beyond the outer edges of the walls. The sandwich walls of the above-ground stories rested upon this extended ring beam. The columns within these segments were supported by the reinforced concrete basement walls. The roof structure was fabricated from steel rolled profiles, with portions of the segments covered by layers of bituminous paper and others by sheet metal.

Segments “D” and “E” were designed as single-storey structures, featuring an enclosed space beneath. Both segments were supported by reinforced concrete strip foundations, which were connected monolithically to the reinforced concrete foundation walls. Additionally, within these segments, the foundation walls were capped with an extended reinforced concrete ring beam, wherein, specifically for the gymnasium segment (segment “E”), the columns and cores were effectively fixed.

The building exhibited a general tilt in a north-western direction; however, this tilt differed among the segments due to varying degrees of mining-induced deformation. This happens particularly in buildings with large projections. The tilt of each segment was evaluated through geometric levelling of the ground floor. Segment “B” demonstrated the greatest tilt towards its longer side at 49 ‰, while segment “E” exhibited the least tilt at 19.8 ‰. In terms of the shorter edge, segment “C” displayed the most significant tilt at 16.0 ‰, whereas segment “B” presented the least tilt at 11.0 ‰. The tilt measurements for each segment are depicted in Figure 3 and summarized in Table 1.

As mentioned in the introduction, for rigid buildings characterized by a load-bearing wall system, the mean tilt can be regarded as the reliable tilt. For such structures, this tilt can be ascertained by calculating the height differential between the highest and lowest points within a given storey. In the case of the school building under examination, this entailed selecting a single zero point at the southeast corner of segment “E” and evaluating the elevations for the corners of all segments. Implementing this methodology would have resulted in the segment exhibiting the greatest tilt remaining further displaced from the vertical, while the segment with the least tilt would have been oriented in the opposite direction. Conversely, establishing individual zero points for each segment and rectifying each segment independently was inadvisable, as it would have eliminated the tilt of all segments but would have introduced steps within the corridors.

Consequently, the following assumptions were established to determine the reliable tilt. It was decided that the rectification of the building must not result in the emergence of steps between segments. Specifically, the presence of steps in the corridors of the educational segments – namely, between segments “A” and “B”, “A” and “C”, and “B” and “C” – was prohibited. Thus, a uniform reliable tilt was calculated for segments “A”, “B”, and “C”. Conversely, the rectification of segments “D” and “E” could proceed independently, provided that no steps were introduced in the passage between these segments. Variations in the number of steps between segments “D” and “E” was permissible.

To determine a reliable tilt for segments “A”, “B”, and “C”, we first calculated the average slope of these segments. However, this approach proved to be wrong. The significant tilt of segment “B” with its relatively small projection (with a tilt of 49 ‰ and a projection area of 63 m²), would have disproportionately influenced the corner elevations of segment “A”, which, although exhibiting a smaller tilt, possessed the largest projection area (with a tilt of 32.5 ‰ and a projection area of 354 m²). It was therefore proposed that the tilt of segments “A”, “B”, and “C” be represented as a weighted average of the floor tilts, using the projected area of each segment as the weighting factor. The values corresponding to these weights are detailed in Table 1. Moreover, it would have been inappropriate to consider any alternative tilt, such as that derived from the edges of the walls, since the objective was to achieve level floors within the segments.

For the single-storey segments “D” and “E”, the tilt of the ground floor was similarly regarded as the reliable tilt. This determination is predicated on the presence of a void beneath the floors, which facilitated the rectification of the building, including the floor itself. Furthermore, achieving alignment of the gymnasium with the ground floor level was prioritized over minimizing the tilt of the columns.

The adoption of independent “zero” points for segments “D” and “E” resulted in a level discrepancy of 230 mm in the passage between these segments. Consequently, it was planned to uniformly elevate segment “E” by this level difference.

Based on the aforementioned considerations, the elevation heights of the corners of the individual segments were calculated. The heights (u_obj) of the characteristic points are illustrated in Figure 3. Segment “A” exhibits the highest value of u_obj at 1589 mm, while segment “E” has the lowest maximum value of u_obj at 685 mm. The maximum and minimum values of u_obj corresponding to each segment are summarized in Table 1.

The analyses conducted provided a foundation for the effective rectification of the school building. Notably, the mutual displacements among the segments resulted in a reduction of the height difference between the ground floor levels of segments “C” and “D” from 1220 mm (7 steps) to 174 mm (an imperceptible ramp). This is depicted in Figure 4a and 4b. The rectification undertaken (Fig. 5a) successfully restored the full functionality of the school building (Fig. 5b).

Fig. 4.

Mutual positioning of segments “C” and “D” of the school building: a) prior to rectification, b) subsequent to rectification (own research)

Fig. 5.

School building: a) view from the east, b) object during rectification – jacks built into the walls (own research)

The foundation of planning the rectification of tilted buildings is a reliable tilt value. For single-segment structures, this tilt is represented by the average tilt of the walls, the average tilt of the floor of each storey, or the tilt of the lift shafts, contingent upon the building’s height and number of floors.

In the context of multi-segment buildings with extensive floor plans, the tilt of individual segments may differ significantly. Research indicates that in such cases, the average tilt of the segments cannot be deemed a reliable indication of the overall tilt of the building. Consequently, the rectification of each segment should be conducted independently. However, this procedure must not compromise the building’s functionality; specifically, it is imperative that no additional steps are introduced in the expansion joints. Therefore, under certain circumstances, it may be necessary to establish a singular reliable tilt that encompasses multiple segments. Evidence suggests that this tilt may be represented as a weighted average. The weights can be derived from factors such as the projected area of the segments, their volume, or their operational significance within the facility.

In the analysis of a five-segment school building, three reliable tilts were identified based on the positioning of the expansion joints and functional considerations. A single reliable tilt, calculated as a weighted average, was determined for three of the segments, while the rectification of the remaining two segments was executed independently, based on the reliable tilt established for those specific segments.