**1. Introduction**

In the authors' countries, in historical buildings from the 19th century, masonry arch stairs were commonly used to connect floors [1]. The arches were shaped between steel beams as a load-bearing span for the staircase landings and flights. At that time, the technology of reinforced concrete was not yet known, so steel and masonry construction was the most popular, especially in residential buildings. The construction of staircases was mostly in the form of segmental or cross vaults. The structure of the flights was made as a stair arch with the thickness of half a brick (12 cm) and elevation *f* = 1/12–1/14 *L* [1] (Figures 1 and 2).

A flight of arch stairs is characterized by a slender, long structure with a slope of approximately 35◦–40◦. One of the main permanent loads on the flight of stairs are the bricks along the entire flight. The steps have the least bricks in the middle of the stairs' arch and the most at the ends. The steps are not connected with a typical masonry bond. The bricks must be cut to fit the actual geometry of the arches.

The flights and landings were supported by steel beams. The construction of flights had a good fire resistance (except for the steel beams), stiffness and durability. The constructions of brick stairs are still in use despite many years of service. Additionally, they are a testament of old times and often protected by the conservator as a part of cultural heritage. Their condition often raises concern because of the damages, mainly cracks or erosion of joints. The typical damages of brick staircase construction are: degradation of mortar and

**Citation:** Nowak, R.; Kania, T.; Rutkowski, R.; Ekiert, E. Research and TLS (LiDAR) Construction Diagnostics of Clay Brick Masonry Arched Stairs. *Materials* **2022**, *15*, 552. https://doi.org/10.3390/ ma15020552

Academic Editor: Francesca Ceroni

Received: 28 November 2021 Accepted: 3 January 2022 Published: 12 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

bricks, flattening of vaults and cracking. The damages arise as a result of overloading, wrongly executed repair works and dynamic loads. Sometimes, one can find constructional errors such as improper fixing of gear plates on I-beam flanges.

**Figure 1.** Masonry stairs with segmental vault and arched staircase slab, based on [1].

**Figure 2.** Brick stairs with column supports based on [1]. Scheme (**a**) and partial photo (**b**): 1—arched masonry stair sides; 2—arched masonry bolt; 3—barrel staircase with support on sides; 4—groin vault in stair flight; 5—groin or barrel vaults in stair landing.

Information on calculation methods or principles of construction is often difficult to find. This type of construction was created mainly on the basis of experience with brick vaults. The steel beams supporting the vaults were calculated by the methods available at that time [1]. The operation of landing vaults is similar to balcony or roof vaults. On the other hand, staircase slabs are at an angle depending on their length and height. The distribution of stresses and internal forces is different from other staircase structures. A calculation scheme of an example of an arched masonry stairs with vaulted landings supported on steel beams is shown in Figure 3.

**Figure 3.** Calculation scheme of masonry stairs with segmental vault and arched staircase slab: 1—arched staircase slab; 2—landings.

One of the most prevalent issues in this type of stairs is the necessity of diagnostics of their technical condition in order to confirm their durability and functional use. The chosen methods used in the diagnostics of masonry structures in the authors' countries are shown in Figure 4.

**Figure 4.** Flowchart for clay brick masonry structure diagnostics.

The diagnosis of masonry structures primarily focuses on a visual assessment in order to detect cracks and damages [2–5]. The early diagnostics usually uses non-destructive measuring equipment, which allows the preliminary characteristics of existing materials or the degree of their degradation to be evaluated [6–10]. Those methods most often include sclerometric measurements (Schmidt hammer) or ultrasonic measurements. At that stage of diagnosis, infrared thermography is a novel and useful method for historic plaster and painted vaults. The management of the data derived from the application of infrared thermography, integrated with the information from visual inspections, the architectural survey and the historic analysis, allows a complete characterization of the historic plasters and painted masonry vaults to be obtained [11].

A more accurate determination of material properties in masonry structures involves sampling. In the case of brick alone, this process is feasible; however, sampling of mortar is significantly more difficult. Therefore, in many diagnostic operations, core samples are taken, also containing a fragment of mortar and masonry elements adjacent on both sides. The core samples, after certain correlations, are used to determine the load-bearing capacity of the whole masonry structure [12–17]. Even more accurate results may be obtained by test-loading a masonry section with actuators [18]. However, this method requires making cuts in the masonry in order to locate the beams to place the actuators. The method is mostly used as a validation is structural elements that are to undergo alteration, repair, or reconstruction.

A similar effect can be obtained if a section of the masonry is cut out and transported to the laboratory for experimental testing [14,18]. A less destructive solution is the flat-jack method [18,19]. It requires appropriate calibration procedures and precise preparation of the measurement base. Additionally, it may be imprecise in low buildings and lead to permanent damage when testing masonry with weak lime mortar [18]. The best measurement method is to perform a loading test with appropriate measurement [20–24], which does not lead to the destruction of the structure but allows the correlation between stresses and strains to be obtained, in order to assess the actual load-bearing capacity and stiffness. Final results on the load-bearing capacity of the structure can only be obtained from experimental destructive testing.

The finite element method (FEM) is a fundamental analysis for the assessment of masonry in seismic zones [25–28]. Nonlinear static calculation methods are commonly adopted for the evaluation of seismic performance [26–32]. Laurenco et al. [28] presented a research study on that subject with a yield criterion that included different strengths along each material axis. Baraldi et al. [31] presented the rigid beam model for studying the dynamic behavior of cantilever unreinforced masonry walls, considered along their thickness and subjected to out-of-plane loading. Celano et al. [32] presented research on the in-plane resistance of masonry walls with the use of two modeling approaches, a finite element model and a discrete macro-element model, with the use of non-linear analyses.

The TLS (terrestrial laser scanning) measurements using 3D scanners are starting to be used for structural diagnosis. Three-dimensional LiDAR (light detection and ranging) scanners have revolutionized the capability and accuracy of geometry measurements in construction, shipbuilding and other areas of science and technology [33–49]. TLS can be used to measure damage to the surface layers of materials, including bricks and mortar, as well as to track moisture in structures [33,41–49]. TLS in structural analysis is primarily used to measure strain and deflections, as well as deformations, of structural elements [25,42–50]. This technique can also be used for control measurements at the stage of constructing [51]. Some researchers use TLS technology to monitor the condition of construction to prevent possible damage [52,53]. Additionally, detailed geometry studies using TLS are useful in post-disaster analyses of structures, including earthquakes [53].

The assessment of the technical condition of brick stairs is not an easy task. The first step usually consists of initial observations of the geometry and of looking for visible signs of damage. Usually, the construction is protected by plaster, so some defects may be hidden. Modern TLS measuring methods can also be helpful in preliminary surveys. Only by uncovering, it is possible to fully assess the quality of brick and mortar and their degree of degradation. The evaluation of the material properties of the structure is possible by taking representative samples and performing laboratory tests [2,10,54–59]. A deeper analysis of the performance of the structure and the causes of its damage is possible using FEM-based 3D models [57–64]. On this basis, it is possible to precisely select the reinforcement methods. The analysis in the elastic range is rather easy to perform, but the analysis in the plastic range, after cracking has occurred, requires time-consuming and expensive studies [60].

Studies of masonry arches and vaults can be found in the works [61–105]. In these studies, masonry vaults are characterized by significant arch rise and placement of the masonry stairs. A small number of researchers has taken up the problem of load-bearing capacity of masonry stairs. Research has mostly focused on establishing the principles and methods for structural calculations [92–99]. There is a visible lack of research on masonry staircases. The design of staircases is similar to masonry arch work; however, the arrangement of loads is different. In addition, the influence of the steps above the vault on the operation of the system is also significant.

The article presents a novel approach, in which experimental tests for staircases were made in a 1:1 scale resembling a real-life construction. Together with laboratory tests, the TLS diagnosis of an arched staircase in a real-life historic building is presented. The history of the TLS LiDAR measurement method dates back only 20 years [41,42,106]. The authors' research study indicates that it can be used in the diagnosis of some types of building structures [41,42]. It is important to assess the applicability of the TLS method in the diagnosis of masonry stair structures, including arched staircases. The possibilities of its use in this area are presented in this article.

#### **2. Real-Life Structure**
