Publikační činnost Katedry stavební mechaniky / Publications of Department of Structural Mechanics (228)
Permanent URI for this collectionhttp://hdl.handle.net/10084/64742
Kolekce obsahuje bibliografické záznamy publikační činnosti (článků) akademických pracovníků Katedry stavební mechaniky (228) v časopisech registrovaných ve Web of Science od roku 2003 po současnost.
Do kolekce jsou zařazeny:
a) publikace, u nichž je v originálních dokumentech jako působiště autora (adresa) uvedena Vysoká škola báňská-Technická univerzita Ostrava (VŠB-TUO),
b) publikace, u nichž v originálních dokumentech není v adrese VŠB-TUO uvedena, ale autoři prokazatelně v době jejich zpracování a uveřejnění působili na VŠB-TUO.
Bibliografické záznamy byly původně vytvořeny v kolekci
Publikační činnost akademických pracovníků VŠB-TUO, která sleduje publikování akademických pracovníků od roku 1990.
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Item type: Item , Solution of yielding steel arch supports used in mining(Techno-Press, 2024) Koubová, LenkaSteel arch supports are used in mines and underground structures to provide stability. Most of the supports are made up of overlapping arches. They can behave either yieldingly or unyieldingly. If the normal force at any point of overlapping equals the slip resistance, the slide occurs. This paper presents a solution procedure for determining the load -carrying capacity of steel arch supports in the yielding implementation. This solution considers the effects of several significant elements, including differing materials and the number of clamps in yielding friction joints. The direct stiffness method is applied. The solution contains geometric, physical, and structural nonlinearity. The results obtained from numerical modeling using the provided procedure are compared to laboratory tests conducted at GIG Katowice in 2012. They show a good correlation with previously collected data from equivalent laboratory conditions.Item type: Item , Gas in scattering media absorption spectroscopy for time-resolved characterization of gas diffusion processes in porous materials(Elsevier, 2024) Dostál, Michal; Suchánek, Jan; Bitala, Petr; Klečka, Vít; Nevrlý, Václav; Klímková, Lenka; Konečný, Petr; Vořechovská, Dita; Kubát, Pavel; Zelinger, ZdeněkSignal denoising is a serious problem for in-situ laser diagnostics of gases dispersed in porous materials. An optical sensor system based on absorption spectroscopy of gases in a scattering environment was built using a 3D printed cell with reference samples of polystyrene foam. Selected A-band spectral lines of molecular oxygen were investigated using wavelength modulated spectroscopy with second harmonic detection. Quantitative information on the concentration of analyte dispersed in the porous medium was obtained at extremely low signal-to-noise ratio (SNR < 10). A spectral line shape fitting procedure based on the Gabor transform followed by a filtered inverse fast Fourier transform allowed to achieve a relatively high SNR with good linearity over a range of reduced oxygen concentrations in air. Finally, the applicability of the optical sensor system to monitor the diffusion of carbon dioxide into air dispersed in a Styrofoam sample and vice versa was successfully demonstrated.Item type: Item , A conceptual open pit mine architecture for the Moon environment(Sciendo, 2024) Seweryn, Karol; Kolusz, Adam; Świca, Izabela; Tkacz, Arkadiusz; Gallina, Alberto; Katzer, Jacek; Kobaka, Janusz; Konečný, Petr; Młynarczyk, PrzemysławIn situ resource utilization (ISRU) activities are receiving increasing attention, both from space agencies and among the international science and industrial community. Prominent examples of ongoing ISRU space programs are the NASA Artemis program and the Terrae Novae program run by the European Space Agency. In technical sciences, there are at least three groups of activities related to ISRU: prospecting bodies in the context of space missions, technological investigations related to surface infrastructure and operations, and conceptual analyses of future mining activities. The present paper belongs to the third group and brings new insights into a potential open pit mine operating on the Moon. There are several novel contributions: the definition of the objectives of the mine, based on economic indicators; a conceptual description of a pit architecture dedicated to excavating ilmenite-rich feedstock; and a qualitative and quantitative description of the chosen processes and the mine's topology. In the paper, there are also added links to other papers connected with ISRU activities.Item type: Item , The influence of corrosion processes on the degradation of concrete cover(MDPI, 2024) Szweda, Zofia; Skórkowski, Artur; Konečný, PetrIn this work, two methods were used to accelerate the corrosion of concrete. In the first method, chloride ions were injected into the concrete using the migration method. The moment of the initiation of the corrosion process was monitored using an electrochemical method of measuring polarization resistance. In the next step, the corrosion process was accelerated by the electrolysis process. Changes on the sample surface were also monitored using a camera. In the second method, the corrosion process of the reinforcing bar was initiated by the use of the electrolysis process only. Here, changes occurring on the surfaces of the tested sample were recorded using two web cameras placed on planes perpendicular to each other. Continuous measurement of the current flowing through the system was carried out in both cases. It was assumed that in conditions of natural corrosion, a crack would occur when the sum of the mass loss of the reinforcing bar due to corrosion reached the same value in tcr(real) (real time) as it reached in the tcr (time of cracking) during the accelerated corrosion test. The real time value was estimated for C1 concrete with cement CEM I. The estimated value was tcr(real) = 1.1 years and for C2 concrete with cement CEM III, tcr(real) = 11.2 years. However, the main difference that was observed during the tests was the nature of the concrete cracks. In the case of the C1 concrete sample, these occurred along the reinforcing bar, while in the C2 concrete, the failures occurred on a perpendicular plane transverse to the direction of the reinforcing bar.Item type: Item , The effect of scrap tires and reclaimed asphalt pavement on the behavior of stone columns(MDPI, 2024) Bikdeli, Hoora; Sharahi, Morteza Jiryaei; Badarloo, Baitollah; Lehner, PetrThe objective of this investigation is to understand how to use waste tires to surround stone pillars and mix gravel with recycled asphalt pavement (RAP) and stone pillars to provide an environmentally friendly and cost-effective weak layer improvement method. To study the behavior of such stone columns, experiments were conducted in units consisting of a single stone column with recycled asphalt pavement as filling material and a single stone column covered with old tires. To test the effect of different mixing ratios, rapeseed content was selected from 0% to 100%. Elasticity tests were conducted on cladded and nonclad stone column samples. Furthermore, direct shear tests were conducted on samples with different ratios of gravel and rapeseed mixtures. The results of the load-bearing capacity test show that the cover of the stone columns with old tires can significantly increase the load-bearing capacity. Replacing 25% of natural stone column aggregates with RAP increases the load capacity. But as the percentage of RAP in the mixture increases from 25% to 100%, the loading capacity decreases. Another advantage is the reinforced stone column. From the point of view of ecology, an advantage is the use of recyclable materials.Item type: Item , Mechanical properties of wooden elements with 3D printed reinforcement from polymers and carbon(MDPI, 2024) Dedek, Jan; Juračka, David; Bujdoš, David; Lehner, PetrThe research presented in this article aimed to investigate the differences in mechanical properties between solid structural timber and the same reinforced element in three different ways. A three-point bending test was performed on wood elements reinforced with carbon-fiber-reinforced polymer (CFRP), 3D printed polycarbonate (3DPC) lamellas, and 3D printed polycarbonate with carbon fiber (3DPCCF) lamellas. In this comparison, the bending strength was large for CFRP samples, which have 8% higher performance than samples with 3DPCCF and 19% higher performance than samples with 3DPC. Conversely, when factoring in theoretical manufacturing costs, the performance of 3DPCCF is almost three times that of CFRP and 3DPC. In addition, 3D materials can be used for more complicated reinforcement shapes than those discussed in the paper.Item type: Item , Properties of fine graded perlite-based lightweight cement mortars subjected to elevated temperatures(MDPI, 2023) Pizoń, Jan; Konečný, Petr; Mynarz, Miroslav; Bílek, VlastimilThe present research compared the behaviours of lightweight mortars based on ordinary Portland cement (OPC), calcium sulphoaluminate cement (CSAC), and calcium aluminate cement (CAC) containing expanded perlite and subjected to elevated temperatures. The perlite substituted natural sand in amounts of 25, 50, 75, and 100% by volume. The mortars were subjected to heating at up to 300 ◦C, 650 ◦C, and 1000 ◦C at a rate of 20 ◦C/min. The consistency and density of fresh mortars, compressive strength and density of hardened mortars after heating and cooling, and absorbability were assessed. Such a holistic testing approach is the main novelty of this research, which is related to the aforementioned mixtures and elevated temperatures. The main contribution of this article is a comparison of various cement types coupled with variations in the level of sand replacement with expanded perlite. In previous studies, comparisons were made in pairs of OPC-CSAC and OPC-CAC for normal-weight concrete. There is a gap in our knowledge of triple comparisons and lightweight cement composites which is filled by the current study. The use of OPC at up to 650 ◦C is recommended because it is the most common solution, its performance is similar to that of CAC, and it is cheaper than other solutions. Above 650 ◦C and up to 1000 ◦C, CAC is the only solution because it performs better than other cements. CSAC is not suitable for use at elevated temperatures because of its poor strength performance, even if it is the best solution from an environmental point of view. Sand replacement with perlite does not increase the strength performance under elevated temperatures, but its efficiency is different for various types of cement.Item type: Item , Practical aspects of correlation analysis of compressive strength from destructive and non-destructive methods in different directions(MDPI, 2023) Badarloo, Baitollah; Lehner, PetrThe research presented here demonstrates the practical aspects of the numerical correlation of the results of the compressive strength test. The destructive test (DT) in a hydraulic press and the non-destructive test (NDT) using a Schmidt hammer in several process variations were evaluated. The aim was to evaluate the real differences between the tool supplier’s curve and testing. Therefore, 150 concrete cube specimens with an edge length of 150 mm were produced using a mixture of three types of concrete classes: C30, C35, and C40. The test was carried out 7 and 28 days of age of the concrete. The Schmidt hammer test was carried out in horizontal (θ = 0) and vertical (θ = 90) directions and using a series of 10 measurements. Furthermore, the tests were performed in two sets: first, the sample was placed on the ground, and second, under a hydraulic jack with a load of 50% of the maximum bearing capacity of specific concrete. Then, regression analysis was performed on the data sets to establish linear mathematical relationships between compressive strength and number of bounces. The results showed that the correlation between the DT and NDT tests has a high value for each group, but the correlation equations are different and must be taken into account.Item type: Item , Development of intelligent technologies in SHM on the innovative diagnosis in civil engineering - A comprehensive review(MDPI, 2023) Vijayan, Dhanasingh Sivalinga; Sivasuriyan, Arvindan; Devarajan, Parthiban; Krejsa, Martin; Chalecki, Marek; Żółtowski, Mariusz; Kozarzewska, Alicja; Koda, EugeniuszThis comprehensive review focuses on the integration of intelligent technologies, such as the Internet of Things (IoT), Artificial intelligence (AI), and Nondestructive Testing (NDT), in the Structural Health Monitoring (SHM) field of civil engineering. The article discusses intelligent technologies in SHM for residential, commercial, industrial, historical, and special buildings, such as nuclear power plants (NPPs). With the incorporation of intelligent technologies, there have been remarkable advancements in SHM, a crucial aspect of infrastructure safety, reliability, and durability. The combination of SHM and intelligent technologies provides a cost-effective and efficient building monitoring approach, significantly contributing to energy and resource conservation. This article explores using electronic instruments, such as sensors, microcontrollers, and embedded systems, to measure displacement, force, strain, and temperature, which are crucial for detecting structural damage. Implementing intelligent technologies in SHM reduces the reliance on manual and hazardous inspection practices, simplifying and reducing the cost of building monitoring. The article highlights the social, economic, and environmental benefits of adopting intelligent technologies in SHM by presenting key findings from existing research. This review aims to increase the reader’s understanding of the significance of these technologies in enhancing the efficiency of SHM in civil engineering by illuminating their advancements and applications.Item type: Item , Effect of imperial smelting process slag addition in self compacting concrete concrete on the efficiency of electrochemical chloride extraction(MDPI, 2023) Szweda, Zofia; Mazurkiewicz, Janusz; Konečný, Petr; Ponikiewski, TomaszThis paper presents the analysis of how ISP slag addition affects the effectiveness of chloride extraction from self-compacting concrete. Corrosion processes were initiated by chloride ions added to concrete by the method accelerated with an electric field. Corrosion of reinforcement was monitored using the method of linear polarization resistance (LPR). Polarization measurements of steel reinforcement and chloride profiles were analysed to evaluate the effectiveness of electrochemical extraction. Microstructural analysis was conducted on a specimen of concrete after migration and extraction of chlorides. The presence of chloride ions and the application of an electric field during migration were tested with respect to the changed microstructure of concrete evaluated on the basis of image analysis using a scanning electron microscope (SEM). The research contributes to a better understanding of the corrosion processes caused by the presence of chloride ions in concretes in which ISP slag was used as a substitute for sand in various amounts. Thanks to the treatments of concrete with already corroding reinforcement bars, it can be concluded that the moderate replacement of sand with ISP slag limited to 25% allows for the effective inhibition of corrosion processes taking place in these concretes. However, it is not possible to completely withdraw already started corrosion processes in steel. The observations of the microstructure of concrete in which sand was completely replaced with ISP slag indicate that after prolonged use of the chloride extraction process, we can expect a change in the microstructure and the formation of ettringite, which may cause the concrete structure to burst. The obtained information will contribute to the development of modelling methods for chloride ion extraction from a wide range of currently used concretes.Item type: Item , Experimental study of high thin-walled cold-rolled Z cross-sections purlins(Elsevier, 2023) Pařenica, Přemysl; Mynarčík, Petr; Lehner, PetrThin-walled cross-sections (TWCS) are used in the construction industry in many applications. The advantage of TWCS is its low weight, which is advantageous both in handling structural elements and the overall value of the structure. The presented paper is focused on the behaviour of Z high thin-walled purlins (300 and 350 mm high) at the point of connection to the supporting structure and the evaluation of whether the existing standard approach can be used to design this type of detail. A total of 60 large-scale experiments were conducted in various settings and categories. The experiments aimed to determine the bearing capacity of the connection of the roof purlins to the supporting structure. Another goal was to determine the effect of the additional reinforcement clip on the overall load-bearing capacity of the connection. Therefore, individual load assemblies were tested with and without a reinforced clip. The results obtained were compared with the analytical procedure according to the standard. Then conclusions were drawn on the applicability of existing standard approaches for the studied purlin joints. The use of analytical calculations has been found to not always be appropriate.Item type: Item , A study of lunar soil simulants from construction and building materials perspective(Elsevier, 2023) Kobaka, Janusz; Katzer, Jacek; Seweryn, Karol; Srokosz, Piotr; Bujko, Marcin; Konečný, PetrTen lunar soil simulants originated from North America and Europe were analysed in this paper. Properties important from civil engineering perspective (especially for future large-scale field experimental studies associated with lunar aggregate and lunar concrete-like composite) were of special interest. The tested properties included: bulk density in loose and vibrated state, density measured using the pycnometer method, particle size distribution, median particle size, coefficient of uniformity, coefficient of curvature, cohesion, inner friction angle and angle of repose. A microscopic analysis was also conducted. The properties of soil simulants were compared to average properties of lunar soil brought by Apollo and Luna missions. Finally the Principal Component Analysis was performed in order to select simulants most similar to the lunar soil.Item type: Item , Investigating the effects of concrete mix design on the environmental impacts of reinforced concrete structures(MDPI, 2023) Mostafaei, Hasan; Badarloo, Baitollah; Chamasemani, Niyousha Fallah; Rostampour, Muhammad Ali; Lehner, PetrThis study examines the impact of mix design parameters on the environmental effects of producing concrete and reinforced concrete buildings by conducting a life cycle assessment (LCA) and carbon footprint analysis (CFA). The study is limited to the cradle-to-gate phase, including the extraction and production of raw materials for concrete production, as well as concrete and rebar production, material transportation, and delivery to the construction site for reinforced concrete structures. Three concrete mix designs based on the American Concrete Institute (ACI) 211-09 standard, with compressive strengths of 20, 30, and 40 MPa, were analyzed. The results indicate that cement was the primary contributor to environmental impacts, accounting for approximately 90% of the carbon footprint. Sand, gravel, and admixtures followed cement in their impact on LCA results. Water usage in concrete production had a negligible effect on LCA indicators. Moreover, to determine how mix design parameters impact the carbon footprint of reinforced concrete buildings, three four-story structures were designed. The results show that in reinforced concrete buildings, concrete was a significant contributor to environmental impacts, accounting for over 50% of all indicators in the IMPACT 2002+ and CML baseline 2000 methods, except for resources and acidification. The study underscores the importance of considering mix design parameters in reducing the carbon footprint of reinforced concrete buildings and provides valuable insights into their environmental impacts. The findings indicate that cement is the main driver of environmental impacts in both assessment methods, accounting for around 90% of the carbon footprint. Additionally, concrete plays a substantial role in environmental effects, contributing to over 50% of all indicators measured in the methods used for evaluating environmental impacts.Item type: Item , Numerical simulation of a planar model of a ball absorber in a spherical dish(Budapesti Muszaki es Gazdasagtudomanyi Egyetem, 2023) Kawulok, Marek; Pospíšil, Stanislav; Freiherrová, Nela; Juračka, DavidThis paper aims to investigate the behavior of a spherical absorber composed of two parts, an inner sphere and a supporting convex spherical dish in which the ball is placed. Considering only the planar behavior of the system, a set of governing nonlinear differential equations was derived and solved numerically. Firstly, the system is exposed to the harmonic excitation of the supporting bowl and its time response is analyzed for all time dependent variables. By gradually changing the angular frequency of the excitation, a resonance curve is obtained, which is examined in detail with respect to the changing amplitudes of the excitation force and the nonlinear behavior. The effect of internal damping and different settings of the absorber characteristics are also investigated. The effect of initial conditions without the presence of an external excitation force is also numerically analyzed by means of phase portraits for selected pairs of initial conditions.Item type: Item , Evaluation of degradation and mechanical parameters and sustainability indicators of zeolite concretes(Elsevier, 2023) Lehner, Petr; Hrabová, KristýnaThe continued need of reducing greenhouse gas emissions is the main motivator for the research related to the design of new cement-based composite materials. A sustainability indicator used in this study is a suitable tool to compare a group of concrete mixes based on their mechanical and degradation properties, together with information on the production and environmental value of each raw material. The study shows the results of the analysis of two groups of concretes with the replacement of the aggregate by zeolite. Experimental data were taken from a published study and numerical evaluations were made. The two groups differ in the value of the water-cement ratio, 0.44 and 0.40 respectively. The other additional cementitious materials (SCMs) used were volcanic pumice, ground granulated blast furnace slag grades 100 and 120, silica fume and metakaolin. The results of six and sixteen concrete mixes incorporating reference concrete made of Portland cement are compared. Compressive strength and diffusion coefficient results were obtained at 28, 56, and 91 days after concreting. The values of the sustainability indicator are determined in these times and ultimately allow the sorting of all the analyzed mixes. The results show that the use of zeolite alone in concrete is not always advantageous but rather depends on the specific w/c. Among many other interesting results, one of the most important is the sustainability index value (1.99) for concrete combining 15% of zeolite and 35% ground granulated blast furnace slag of grade 120, so almost twice as good as the reference concrete. Among concretes with only zeolite as SCM, 35% of the replacement is the most suitable choice. Furthermore, the combination with other SCMs can produce a significant positive effect in terms of sustainability.Item type: Item , Hyperbolic paraboloid tensile structure - Numerical CFD simulation of wind flow in RWIND software(MDPI, 2023) Kawulok, Marek; Freiherrová, Nela; Horňáková, Marie; Juračka, David; Krejsa, MartinTensile membrane structures combine a prestressed roofing envelope material and supporting elements. To design these structures, there is a set of recommendations in the European Design Guide for Tensile Surface Structures and some other national standards. However, currently, there is no official standard related to the design process of tensile structures in the European Union. The structure studied in this project is considered as permanent roofing of an external testing device in the shape of a simple hyperbolic paraboloid without enclosing walls. Snow and wind loads were analyzed as the most critical types of loading in the location. Determining the value of the snow load is relatively simple according to the European standard. However, in the case of the wind load, this shape is not considered in the European standard and needs to be solved experimentally or by numerical simulation in a wind tunnel. The present contribution focuses on numerical analysis of the wind flow in RFEM software and simulation of the wind tunnel in RWIND software.Item type: Item , Concept of a 3D-printed Voronoi egg-shaped habitat for permanent lunar outpost(MDPI, 2023) Juračka, David; Katzer, Jacek; Kobaka, Janusz; Świca, Izabela; Seweryn, KarolThe article presents a concept of a lunar base that would take advantage of the natural shape of an egg. Several versions of egg-shaped habitat structures characterized by different sizes are presented. Possible locations of habitats both on lunar surface and in craters were discussed. Advantages and disadvantages of particular locations were also pointed out. The proposed in the paper concept of an egg-shaped structure is characterised by a spatial character based on Voronoi diagram and would be implemented using a 3D-printed method. The presented 3D-printed structure was designed to be light and suitable to be covered by lunar soil. As a necessity in the developed concept, in situ resources utilization was addressed in order to generate products using local sources to reduce the number of materials that would be required to be transported from Earth. At the end of the paper, future areas of research and tests are highlighted.Item type: Item , Comparison of standardized methods for determining the diffusion coefficient of chloride in concrete with thermodynamic model of migration(MDPI, 2023) Szweda, Zofia; Gołaszewski, Jacek; Ghosh, Pratanu; Lehner, Petr; Konečný, PetrThis research paper is the result of observations made during tests according to various standards carried out on behalf of industry. The article presents diffusion coefficient values calculated according to the thermodynamic migration model for twenty different concrete mixes and some selected mixes of the codified approaches known as ASTM 1202, NT BUILD 443, NT BUILD 492, ASTM 1556. The method used here, according to the thermodynamic model of migration, allows determination of the value of the diffusion coefficient after short studies of the migration of chloride ions into concrete and was described in earlier works by one of the authors. Unfortunately, when using standard methods, the values of diffusion coefficients differ significantly from each other. In each concrete, diffusion tests were carried out in the conditions of long-term natural diffusion to verify the values determined by standard methods and according to the thermodynamic model of migration. The analysis conducted for this research paper reveals that the chloride permeability test method according to the standard ASTM C1202-97 has an almost 2.8-fold greater dispersion of the obtained results compared to the thermodynamic model of migration. It was observed that the standard NT BUILD 492 has a 3.8-fold dispersion of results compared to the method with the thermodynamic model of migration. The most time-consuming method is the standard method NT BUILD 443. The largest 3.5-fold dispersion of values concerning the reference value are observed in that method. Moreover, a method based on a thermodynamic migration model seems to be the best option of all analyzed methods. It is a quite quick, but laborious, method that should be tested for a larger number of concrete mixes. A great advantage of this method is that it is promising for a wide range of concrete mixtures, both plain concrete and concrete with various additives and admixtures, as well as high-performance concrete.Item type: Item , Correlation study of physical and mechanical properties of concretes with crushed LCD glass(Elsevier, 2022) Badarloo, Baitollah; Lehner, Petr; Koubová, Lenka; Pirizadeh, MahboobehThe need to increase the sustainability of the construction industry is leading to the search for alternative sources of raw materials. Liquid crystal display (LCD) screens have been used for many years and are also rapidly ageing as a result of fast development. Therefore, the use of waste from LCD screens is an interesting avenue for the construction industry. This paper presents a very practical study of the use of old LCD screen of ground material to replace various fine aggregates. Fifteen concrete mixes of different strength levels and with different amounts of LCD powder were prepared. These are C20, C25, and C35 grade concretes, all of which contain LCD powder in quantities ranging from 15% to 60%. Results of density, compressive strength, and tensile strength at 7 and 28 days after concreting. The results are correlated with the number of replacement components and are also correlated with each other. The data presented to improve our understanding of the practical application of LCD waste aggregate in concrete. The results show that the amount of LCD powder has a direct linear effect on the compressive strength and also a direct lognormal effect on the tensile strength of concrete.Item type: Item , Determination of the load carrying capacity of closed steel supports used in underground construction and mining(Techno-Press, 2022) Koubová, Lenka; Janas, Petr; Janas, Karel; Krejsa, MartinClosed steel supports of different shapes are used in mining and underground constructions. The supports are prefabricated from rolled, usually robust, steel profiles. The load carrying capacity of a support is considerably influenced by the active loading and passive forces. The passive forces are induced by interactions between the support and the surrounding rock mass. The analysis herein comprises three parts: The first part consists of structural geometry processing. The second part involves finding the numerical solution of a statically indeterminate structure for a specified load. The third part is calculation of the load carrying capacity and the components of internal forces and deformations. For this, the force method and numerical integration are used. The Winkler model is applied when the support interacts with the surrounding environment. The load carrying capacity is limited by the slip resistance of the connected parts and it is limited by reaching the ultimate state of the profile. This paper serves as a comprehensive reference for the determination of the load carrying capacity of closed steel supports and includes stepwise derivations of the governing formulas.